Programmable arrays of alkaline earth atoms: qubits,
clocks, and the Bose-Hubbard model
by
Aaron William Young
B.A., Wesleyan University, 2017
B.S., California Institute of Technology, 2017

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2023
Committee Members:
Adam M. Kaufman, Chair
Joshua Combes
Dietrich G. Leibfried
Cindy A. Regal
Ana Maria Rey
Jun YeYoung, Aaron William (Ph.D., Physics)
Programmable arrays of alkaline earth atoms: qubits, clocks, and the Bose-Hubbard model
Thesis directed by Prof. Adam M. Kaufman

In this thesis, I report on the development of a new experimental platform that features capabilities
drawn from optical tweezer arrays, quantum gas microscopes, and optical clocks. We demonstrate that we
can trap, cool, image, and manipulate the positions of individual strontium atoms in an optical lattice using
a colocated set of optical tweezer arrays. These capabilities allow us to assemble ensembles of hundreds ofatoms while maintaining access to controls and observables that have a resolution at the level of a single
atom, and a single lattice site.
In the direction of quantum metrology, we demonstrate that the above capabilities can be combinedwith control of a long-lived optical frequency qubit encoded in the electronic states of strontium to realize a
new kind of optical clock. This “tweezer array clock” provides access to measurements not typically available
in optical clocks while also providing fractional frequency stabilities that are close to the state-of-the-art
for synchronous comparisons. We further engineer interactions between optical frequency qubits encoded
in different atoms by exciting the atoms to high-lying Rydberg states. We use the resulting interactions
to perform entangling gates, and to prepare large entangled states that enable synchronous optical clock
measurements with a precision below the standard quantum limit.
In the direction of quantum simulation, we control the motion of atoms in a tunnel-coupled opticallattice to study single- and multi-particle quantum walks. By taking advantage of the ability to modify
the local potential in the lattice using optical tweezers, we show that these quantum walks can be used
as a resource in various algorithms, including in the first experimental demonstration of spatial search byquantum walk. By further taking advantage of the ability to prepare, evolve, and detect large ensemblesof atoms in the lattice with high fidelity, we significantly advance the state-of-the-art in Fock state boson
sampling. Applying the above techniques to interacting atoms allows us to introduce a new approach to
studying both ground states and dynamics in the Bose-Hubbard model. We present preliminary results
involving the dynamics of hard core bosons, as well as the preparation of a superfluid that is assembled a

iii
single atom at a time using optical tweezers.
The combination of high fidelity control of arrays of atomic qubits with complicated many body
dynamics, and state-of-the-art capabilities in frequency metrology, results in a fruitful blurring of the lines
between quantum computation, simulation, and metrology. As the capabilities of these systems continue
to expand, we are hopeful that ideas from quantum information science can be drawn on to perform evermore precise measurements and, conversely, that metrological tools and techniques can be used as precisionprobes of complicated and poorly understood quantum many body effects.

Dedication

For mom and dad, who always answered why.

Acknowledgements

I would like to say that the first time I met Adam, we had a deep and stimulating conversation about
the future of atomic physics. In actuality, having only communicated with him via email and not knowing his
face, I asked him where the library was in the Lyman building, thanked him brusquely for his clear directions,rushed away, and promptly got lost. This interaction served as the blueprint for our work together in theyears that followed. Adam is a brilliant, careful physicist who has weaponized his “neuroticism”1 into an
uncanny ability to see to the crux of a problem, and plan well in advance to avoid any pitfalls. It has been
an absolute pleasure to work with him these last six years, even on those occasions when, despite carefulplanning, I get lost.
I feel very lucky to have worked with an incredible group of people that Adam has assembled over
the years, including past and present members of the strontium experiment: Nathan Schine, Ben Johnston,
Felix Rönchen, Nelson Darkwah-Oppong, Alec Cao, and Theo Lukin-Yelin. I am particularly indebted totwo people involved in the early phases of the experiment: Matt Norcia, who taught me, by demonstration,much of what I know about experimental atomic physics, and is the main reason that this thesis isn’t being
written in 2027; and Will Eckner, who is a very strong physicist both technically, having built all the scariest
lasers in the lab, and theoretically. More often than I care to admit, Will has listened patiently to one of myad hoc explanations, and later discreetly sent me a detailed LATEX document explaining why I was wrong.
I was similarly lucky with the fantastic people who guided me professionally early on, and led to
my eventual decision to pursue a career in experimental physics. I am deeply grateful to all of them. To
Greg MacIsaac, who instilled a desire to explore and document the world (through film, rather than atomic
physics experiments). To Lutz Hüwel, who taught a film student to document the world with more precision
1 His words not mine [162].

vi(and in much closer proximity to high power lasers). And to Oskar Painter, whose impact was twofold: hegot a mechanical engineering student so excited about quantum science that they pretended that quantum
mechanics courses were relevant to their degree,2

and instilled the drive to “set lofty goals, and work

goddamned hard.”
For most of grad school, I have lived with the same group of close friends: Shawn Geller, who has
provided years of stimulating conversations about physics that almost make up for the fact that he’s actuallya mathematician; Will Milner, who is somehow simultaneously the most hardworking and most laid backperson I know; and of course Hannah Knaack, who has been an amazing partner, despite her insistence on
working with one less electron than God intended.
I will spare you the stories about how I wanted to be a physicist ever since I was a child — that’s not
the truth. However, I will indulge in sharing a couple of anecdotes, and thanking my earliest supporters:
My grandmother Mary Young, who sent each new issue of Popular Science to me from the other side of theworld, without fail, well before I knew I had any interest in science. Perhaps she knew all along. My dad
Russell Young, who, for years, patiently explained math to what I can only describe as a hostile audience. I
wanted things to be self-evident, and connected to specific applications. He would counter by pulling tricks
and puzzles out of nowhere that would miraculously fit together to reveal deep insights. This of course still
annoys me endlessly. And finally my mom Lynn Hua, who taught me to be generous but effective withmy time, and to cherish time with family above all else. “一寸光阴一寸金,” which is why in this work wemeasure 光阴 to the 19th decimal place.
Acknowledgements are always tough, because a complete list would fill this entire thesis and more.
The following is an incomplete and unordered list3 of people who have helped me significantly throughout
my research, thank you all: Manny Knill, Scott Brown, Andrew Childs, Scott Glancy, Hans Green, Cindy
Regal, Mark Brown, J. R. Raith, Kaizhao Wang, Eric Alvarado, Dhruv Kedar, Hope Whitelock, MichaelFang, Sean Muleady, Ana Maria Rey, Terry Brown, Jie Luo, Alec Jenkins, Alex Downham, Chris Reetz, Felix
Vietmeyer, Charlie McLemore, Aruku Senoo, Jenny Wu, Paul Dieterle, Ting-Wei Hsu, Matteo Marinelli,
2 Thanks to Guillaume Blanquart for letting this fly, and suggesting that scattering problems are “close enough to fluid
mechanics and thermal transport” to justify graduation.
3 Seriously don’t read into the ordering, it’s been through a random number generator.

vii
Jun Ye, Josh Combes, Peter Zoller, Dietrich Leibfried, Joanna Lis, Lee Liu, Phillip Buelow, Ross Hutson,
Jose D’Incao, John Robinson, and William McGrew. Finally, thanks to COVID-19, for making it easier tojustify sitting in a windowless basement for most of my twenties.

Contents

Chapter
1 Introduction

1

1.1

Historical perspective. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Summary of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3

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

7

1.4

Published content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.5Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92 Prerequisites
2.1

2.2

Strontium and the alkaline earth-like atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.1.1

Level structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.1.2

Isotopes of strontium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17Electromagnetic control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.2.1

Two level approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.2.2

Dipole forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.2.3

Magic wavelengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

3 Overview of the apparatus
3.1

10

25

Vacuum chamber and atom source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.1.1

Science chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.1.2

Cold atom source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

ix
3.1.3

Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

3.1.4

Magneto-optical traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.2

Magnetic field coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.3

Optical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.3.1

Mechanical design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

3.3.2

Microscope system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

3.3.3

Optical tweezers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.3.4

Optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.3.5

Alignment of optical potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

Standard operating procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

3.4

4 Preparing and detecting single atoms

63

4.1

Magic angle trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

4.2

Preparing single atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

4.3

Controlling atomic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

4.3.1

Sideband cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

4.3.2

Sideband thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.3.3

Cooling results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

Imaging single atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4.4.1

Image analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.4.2

Image analysis in dense arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

Atom rearrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

4.4

4.5

5 Optical frequency metrology with tweezer arrays
5.1

91

A primer on quantum sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

5.1.1

Phase estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

5.1.2

On good clocks, qubits, and decoherence . . . . . . . . . . . . . . . . . . . . . . . . . .

96

5.1.3

Additional considerations in optical clocks . . . . . . . . . . . . . . . . . . . . . . . . .

98

x
5.1.4
5.2

5.3

5.4

5.5

Fluctuations and systematic shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Optical clock qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2.1

Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2.2

Single qubit gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2.3

Rabi spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.2.4

Ramsey spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Preserving coherence and minimizing frequency shifts . . . . . . . . . . . . . . . . . . . . . . 112
5.3.1

Atom loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.3.2

Clock state lifetime and coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.3.3

Tunneling-induced phase noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.3.4

Frequency shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Clock measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.4.1

Repeated interrogation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.4.2

Measurements beyond the atom-laser coherence time . . . . . . . . . . . . . . . . . . . 125

Tweezer clocks in context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6 Rydberg interactions between clock qubits
6.1

135

Rydberg states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.1.1

Rydberg spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.1.2

Rydberg interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.2

Rydberg dressing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.3

Benchmarking entangling gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.3.1

Measuring gate fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.3.2

Correcting SPAM errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.3.3

Error budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.4

Characterizing Bell state coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.5

Large entangled states for frequency metrology . . . . . . . . . . . . . . . . . . . . . . . . . . 161

xi
7 Single-particle quantum walks
7.1

7.2

165

Single atoms in a lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.1.1

Band theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

7.1.2

Wannier functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Continuous-time quantum walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.2.1

Separability of quantum walks in square lattices . . . . . . . . . . . . . . . . . . . . . 172

7.3

Calibrating lattice parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7.4

Preparing the lattice ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.5

Search by quantum walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.5.1

An analog analogue of Grover’s search . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.5.2

Spatial search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

7.5.3

Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

7.5.4

Experimental demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

7.6

Universal control of the quantum walk unitary . . . . . . . . . . . . . . . . . . . . . . . . . . 193

7.7

A note on resource costs and analog classical simulation . . . . . . . . . . . . . . . . . . . . . 194

8 Multi-particle quantum walks
8.1

The Bose Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
8.1.1

8.2

8.3

197

Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Non-interacting quantum walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
8.2.1

Boson sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

8.2.2

A note on measurements, loss, and postselection . . . . . . . . . . . . . . . . . . . . . 206

8.2.3

Effective particle statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Benchmarking atom indistinguishability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.3.1

Multimode Hong-Ou-Mandel interference . . . . . . . . . . . . . . . . . . . . . . . . . 209

8.3.2

Bunching and clouding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

8.3.3

The effect of interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

xii
8.4

Calibrating the quantum walk unitary using Fock states . . . . . . . . . . . . . . . . . . . . . 219

8.5

Measuring interference between many particles . . . . . . . . . . . . . . . . . . . . . . . . . . 220
8.5.1

Generalized bunching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

8.5.2

Tuning particle distinguishability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

8.6

Simulating non-interacting many-particle quantum walks . . . . . . . . . . . . . . . . . . . . . 225

8.7

A note on noise and the hardness of boson sampling . . . . . . . . . . . . . . . . . . . . . . . 226

8.8

8.7.1

Non-unitary errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

8.7.2

Unitary errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

Interacting quantum walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
8.8.1

Hard core bosons and fermionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

8.8.2

Assembling a superfluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

9 Conclusion and outlook

238

9.1

Directions in metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

9.2

Directions in simulations of Hubbard models . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

9.3

Combined directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Bibliography

247

Appendix
A Notes on estimation theory

269

A.1 Classical estimation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
A.2 Quantum estimation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
B Estimating indistinguishability, bunching, and clouding in the presence of parity projection

272

C Electronics

275

C.1 Clock distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

xiii
C.2 General control electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
C.3 RF synthesis with FPGAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
C.4 High current coil control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
D Laser systems

280

D.1 Diode lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
D.1.1 Repump systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
D.1.2 Cavity locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
D.1.3 689 nm system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
D.1.4 461 nm system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
D.2 Clock laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
D.2.1 Fiber noise cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
D.3 Rydberg laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
D.4 Trapping lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

xiv

TablesTable
2.1

Isotopes of strontium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

3.1

Representative conditions for magneto-optical trapping . . . . . . . . . . . . . . . . . . . . . .

33

3.2

Magnetic field coil parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.3

Representative values of the lattice depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

5.1

Clock qubit lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.1

Quantum defects for the 3 S1 and 3 D1 Rydberg series in Sr . . . . . . . . . . . . . . . . . . . . 137

6.2

Error budget for the Mølmer-Sørensen gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

8.1

Scattering properties of the isotopes of strontium . . . . . . . . . . . . . . . . . . . . . . . . . 201

8.2

Comparison of large-scale boson sampling demonstrations . . . . . . . . . . . . . . . . . . . . 227

Figures

Figure1.1

Schematic of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.1

Level diagram for 88 Sr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.2

Polarizabilities of the 1 S0 , 3 P0 , and 3 P1 states . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

3.1

Picture of the apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3.2

Layout and detailed views of the apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3.3

Manufacturing drawing for the glass cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.4

Glass cell anti-reflection coating and deformation under vacuum . . . . . . . . . . . . . . . . .

30

3.5

Cross-section of the science chamber and surrounding components . . . . . . . . . . . . . . .

34

3.6

Cartoon of the optical potentials applied to the atoms . . . . . . . . . . . . . . . . . . . . . .

38

3.7

Layout and performance of the microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.8

Layout of the tweezer systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.9

Optical lattice geometry and polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.10 Axial lattice alignment via parametric heating . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

3.11 Typical experimental sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

4.1

Magic angle trapping conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

4.2

Electronic transitions in a harmonic trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

4.3

Simulated sideband cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.4

Sideband thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

xvi
4.5

Layout of the imaging beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4.6

Image analysis for sparse arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.7

Image analysis for dense arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

4.8

Procedure for atom rearrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

5.1

Schematic for quantum sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

5.2

Clock Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.3

Effect of motion on Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4

Clock Rabi spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.5

Clock Ramsey spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.6

Ground and clock state lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.7

Trap-induced clock frequency shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.8

Ramsey spectroscopy in large arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.9

Scheme for repeated clock interrogation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.10 Repeated Rabi spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.11 Frequency comparison between tweezer-trapped atoms and an ultrastable laser . . . . . . . . 124
5.12 Ellipse fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.13 Synchronous frequency comparison via ellipse fitting . . . . . . . . . . . . . . . . . . . . . . . 128
5.14 Correlation spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.1

Rabi spectroscopy and lifetime of the 5s40d 3 D1 Rydberg state . . . . . . . . . . . . . . . . . 138

6.2

Adiabatic gate protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.3

Adiabatic gate population dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.4

Bell state fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.5

Imperfect Rydberg pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.6

Population dynamics of entangled states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.7

Ramsey coherence time of Bell states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.8

Measuring Bell state coherence using correlations . . . . . . . . . . . . . . . . . . . . . . . . . 157

xvii
6.9

Parametric plots of the evolution of Bell pairs and single atoms . . . . . . . . . . . . . . . . . 158

6.10 Correlation dynamics of entangled states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.11 One axis twisting in large ensembles of atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.12 Characterizing spin squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.13 Synchronous frequency comparisons using spin squeezed states . . . . . . . . . . . . . . . . . 164
7.1

Band structure and Wannier functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7.2

Quantum walks on a line

7.3

Quantum walks in an optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.4

Calibration measurements of the lattice potential . . . . . . . . . . . . . . . . . . . . . . . . . 174

7.5

Adiabatic preparation of an atom in the ground state of a tunnel-coupled lattice . . . . . . . 177

7.6

Robustness of ground state preparation to misalignment . . . . . . . . . . . . . . . . . . . . . 179

7.7

Operating principle of Grover’s search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

7.8

Level structure of the spatial search Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 185

7.9

Level structure for different oracle positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.10 Spatial search by quantum walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.11 Reversed spatial search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.12 Universal control of the quantum walk unitary . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.1

Symmetrization constraints on bosonic interference . . . . . . . . . . . . . . . . . . . . . . . . 204

8.2

Atom loss during quantum walk dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

8.3

Two particle quantum walks on a 2D lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

8.4

Benchmarking two particle quantum walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

8.5

Multiparticle quantum walks in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

8.6

Three particle quantum walks in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

8.7

The role of interactions in HOM interference . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

8.8

Calibrating the single particle unitary using quantum walks . . . . . . . . . . . . . . . . . . . 218

8.9Interference of large bosonic Fock states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

xviii8.10 Properties of the single particle unitary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
8.11 Dynamics of hard core bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
8.12 Phase diagram of the Bose-Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
8.13 Preparing a superfluid ground state using optical tweezers . . . . . . . . . . . . . . . . . . . . 236
C.1 Schematic for H-bridge coil switch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
D.1 Diode laser layout, stabilization, and distribution . . . . . . . . . . . . . . . . . . . . . . . . . 281
D.2 Laser stabilization to a hollow cathode lamp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
D.3 Fiber noise cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

Chapter 1

Introduction“Such awareness as there was/and there was little
confined itself to spaces smaller
than the head of a pin
where angels once danced\\”— AI Ummon

1.1

Historical perspective
In the early days of quantum mechanics, many of the properties of the new theory, in particular

the intrinsic randomness and uncertainty at its core, were treated as a nuisance that flew in the face ofwell-behaved, deterministic classical dynamics. In the decades since there has been a shift in understanding:quantum systems, particularly ones made up of many constituents, allow for fundamentally different andoften enhanced capabilities in comparison to their classical counterparts. An early hint at these capabilities
was provided in 1964 by Bell’s theorem [24], which states that quantum states can possess especially strong
correlations that cannot be explained by any classical local hidden variable theory. Such correlations can beused in various tasks in communication [121] and sensing [83, 259] to gain explicit advantages over what canbe accomplished classically. Another hint came from the general difficulties one encounters when trying tostudy quantum many body systems, where even the simplest microscopic models can end up being intractable
to solve due to the rapid growth of Hilbert space allowed by entanglement. For example, despite intensive
research [184], it is still unknown whether Hubbard’s eponymous model from 1963 [144] possesses a d-wave
superconducting ground state.1 In contrast to the Fermi-Hubbard model, the ground (and thermal) states
1 Although it is now suspected that the basic Hubbard model on a square lattice does not host such a ground state [310].

2of the Bose-Hubbard model are typically well-understood based on mean field or quantum Monte-Carlotreatments [266, 30].2

However, dynamics in any Hubbard model are generically believed to be hard tosimulate. In recent years, our understanding of the sense in which such problems are hard has sharpened.
In general, our best approximate classical methods for finding ground state or thermal expectation valuesof observables in an interacting, fermionic system are in the complexity class NP-hard [335], and solving forthe full dynamics of even a non-interacting, bosonic system is #P-hard [2].3

Although it is not believedthat quantum systems will help us solve arbitrary NP problems or simulate arbitrary quantum behaviors
in polynomial time,4 it is reasonable to believe that they will outpace (or have outpaced) our ability tosimulate the world classically [79, 334].5

In the pursuit of realizing the above advantages, there has beena concentrated effort in recent years to build large, controllable quantum systems using a zoo of different
technologies, including trapped atoms and ions, superconducting and photonic circuits, and quantum dots.
We will focus on experiments involving optically trapped and ultracold neutral atoms, which, when we
began the work presented in this thesis, had already made impressive advances in three distinct domains: (1)in simulations of various Hubbard models, (2) in programmably controlling individual atoms using opticaltweezers, and (3) in optical frequency metrology.
In (1), ultracold atomic gasses confined within optical lattices have been used to realize a variety ofdifferent Hubbard models [130]. In particular, the presence of the lattice potential allows one to reduce thekinetic energy of the atoms (by tuning the tunnel-coupling between different lattice sites) to reach regimes
where the atoms are strongly interacting. Incorporating a high resolution microscope in such experimentsallows one to perform measurements of the resulting strongly correlated states of matter with resolution on
2 Due to the lack of a sign problem [335].
3 In fact, even drawing approximate samples from the wavefunction prepared after dynamical evolution is believed to be
hard [2].
4 For example, the general problem of finding the ground state of a k-local Hamiltonian, as applies to generic Hubbard models,is QMA-complete for k ≥ 2 [168], meaning quantum computers are not expected to solve all such problems in polynomial time.
Another seminal result that should give one some pause is the fact that, in general, determining whether the ground state ofa quantum system is gapped in the infinite limit is not just a hard problem, but an undecidable one [77, 22]. To see why thisis bad, consider the fact that one could run a quantum simulation with 1023 particles, and find that the addition of a single
particle qualitatively changes the behavior of the system. However, it is not generally expected that such arguments apply tomost physical systems of interest, since the properties of real materials are almost always stable in the thermodynamic limit.
5 In systems where the many body gap closes polynomially in system size, as is expected to be true for many Hubbard
models, quantum systems can prepare the appropriate ground and excited states in polynomial time using protocols basedon adiabatic evolution [27, 5] or the variational quantum eigensolver [257, 57, 330]. The belief that polynomial time classicalalgorithms are not possible in these situations stems, in part, from Bell’s theorem: because the states of these systems canonly be described by taking non-local correlations into account, in general, we don’t expect to be able to simulate such systems
classically without keeping track of an exponentially large state vector [265].

3
the scale of a single lattice site, and sensitivity on the scale of a single atom [14, 305]. By further using the
microscope to project optical fields onto the atoms, it is possible to isolate a subset of atoms from a larger
many-body state to prepare different states of interest [349, 151, 231, 264, 166, 359].
Along similar lines, in (2), tightly focused beams of light called optical tweezers can be used to trap andmove individual atoms [19, 99]. This approach is distinct from (1) in the sense that, instead of isolating thedesired part of a large (and often challenging to prepare) many body state, one traps and deterministicallyrearranges initially isolated atoms to rapidly prepare large ensembles of atoms. The motion of the atoms canfurther be controlled using sophisticated optical cooling techniques [164, 323]. Tweezer-trapped atoms canbe made to tunnel between tweezers in similar ways to the experiments in (1) [165, 163], but it has proved
challenging to scale such experiments beyond a handful of tweezers due to difficulties associated with disorderin the depths of the tweezers on the scale of the tunneling energy [316, 359].6

However, a separate toolkit

has been developed to engineer strong and coherent interactions between up to hundreds of tweezer-trapped
atoms by exciting the atoms to high-lying Rydberg states. Since Rydberg states are strongly polarizable, theenergy of a Rydberg state atom can be affected by the presence or absence of a nearby atom in a Rydberg
state. The above techniques have been used to engineer entangling gates [190, 191, 127] and run shallow
quantum circuits [128, 32], as well as to study a variety of spin models [47, 95, 294, 297].7
In (3), one is interested in preparing ensembles of atoms or ions, with each particle in the ensemble
serving as an oscillator that keeps track of time [203]. Alkaline earth atoms are particularly suited to such
experiments (in contrast to the alkali atoms typically used in the experiments described above), because thepresence of the second valence electron allows for ground and excited electronic states that are long-lived, andinsensitive to external perturbations (see Sec. 2.1.1.3). These states form the basis of optical frequency qubits(clock qubits) whose evolution can be used to keep track of time. In the setting of frequency metrology,
quantum many body effects are often a nuisance. Instead, one wants to control or mitigate interactions
between clock qubits to approximate pristine conditions where the atoms sit motionless in a vacuum, with
their oscillations unadulterated by external perturbations from other atoms or fields. However, in the pursuit
6 Note that the stroboscopic technique developed in [359] does hold some promise for scaling to larger tunnel-coupled tweezer
arrays.
7 Specifically, models where the spins are pinned in place, and therefore where particle statistics are irrelevant. This isdistinct from the experiments in (1), where both spin and particle statistics can play a central role.

4

(b)
(a)
MO
Ts

(d)
Lo
a

d

Im
a

ge

Re
a

rra
ng

Im
a
e

ge

La
s

Im

er
-co

ol

ag
e

(c)

Figure 1.1: Schematic of a typical experiment. (a) A cold thermal cloud of atoms is prepared using several
stages of magneto optical traps (MOTs). Atoms are stochastically loaded into a tweezer array (green cones)
and transferred into an optical lattice (grey grid). By taking an image of the atoms, we can determine
their locations, and deterministically rearrange them into the desired patterns using the optical tweezers (anadditional image is typically taken to confirm that the rearrangement was performed successfully). Finally,
the atoms are laser cooled to the motional ground state of the lattice sites that they occupy. The result
is the on-demand preparation of large ensembles of atoms in states with very low entropy, which serves as
the starting point for our experiments. These experiments fall into two broad categories. (b) The atoms(circles) can be pinned in place, and an internal optical frequency qubit degree of freedom (indicated by theblue and red circles) controlled. A universal set of single qubit gates can be performed using global rotations
driven by a laser that addresses all the atoms (red arrow), and local phase shifts applied using the opticaltweezers (green cone). Interactions and entangling gates can be engineered by driving the atoms to highly
excited Rydberg states with a separate laser (purple arrow). (c) Alternatively, the atoms can be allowed tocoherently tunnel through the lattice. The resulting quantum walks can be controlled by applying a local
chemical potential (with a resolution on the scale of a single lattice site) using the tweezers. (d) In all cases, a
final image, or sequence of images, is used to measure the resulting state of the atoms. The images appearing
in this figure are single-shot experimental data, where each bright spot indicates the presence of an atom in
its electronic ground state.

of these conditions, one often still has to understand and tame complicated quantum dynamics and many
body effects. For example, the Fermi-lattice clock at JILA uses properties of the Fermi-Hubbard model toavoid overlaps between the atoms that can lead to uncontrolled shifts in their oscillation frequency [53].
Similarly, with the appropriate engineering, one can use different kinds of interactions in lattice clocks toprecisely cancel each other out [35, 3]. Clocks that use ensembles of atoms or ions as their oscillators
have achieved amazing feats, including setting bounds on possible candidates for dark matter [169], andperforming geodesy [131] and tests of relativity [67] by measuring gravitational red shifts down to the scaleof a millimeter [35].
The main objective of the work in this thesis is to apply the flexible and programmable control

5
provided by optical tweezer arrays (2), which have already been used to study spin models and quantum
circuits, to other applications in quantum science, namely quantum simulations of different Hubbard models
using optical lattices (1), and optical frequency metrology (3). We will find that by combining features fromthe three domains described above into a single apparatus, one gains significant advantages in terms of state
preparation, readout, isolation from undesirable perturbations, and the kinds of dynamics, and control overthose dynamics, that one has access to.
The primary role of the tweezer array in our experiments is to provide programmable control over bothstate preparation, and the ensuing dynamics. In terms of state preparation, the tweezer array provides fast
and flexible preparation of initial states by means of single atom trapping, optical cooling, and, optionally,
atom rearrangement (Fig. 1.1). This state preparation minimizes dead time on the experiment and improves
the rate at which one can gather statistics. For simulating Hubbard physics, atom rearrangement is particularly useful since one is able to prepare a large class of excited (but still pure) initial states8 rapidly and withhigh fidelity, making it easier to study certain kinds of dynamics. In terms of controlling the dynamics of theatoms, the tweezers can be used to implement local single qubit gates, or to modify the chemical potential
and tunnel couplings in the lattice (Fig. 1.1). For frequency metrology, access to a (nearly) universal gate
set and high-fidelity measurements (in a complete basis) allow one to explore sophisticated measurement
schemes, including those involving entanglement, that can improve the performance of a clock. For simulating Hubbard physics, the tweezers allow one to change the lattice geometry, and to implement gate-like
operations between sites in the lattice. In both cases, the above controls open the possibility of mixed analog
and digital evolution, enabling new schemes for state preparation, and access to new observables. However,tweezer arrays are not a perfect tool, and suffer from limitations on scaling due to limited laser power anddisorder in the generated optical potentials. The optical lattice nicely complements these shortcomings,
allowing one to achieve similar or improved performance in imaging, cooling, and tunneling in comparison
to tweezer arrays, while also allowing one to scale to systems with tens to hundreds of times as many atoms,
and sites that the atoms can occupy (Fig. 1.1).
8 Namely, Fock states corresponding to (nearly) arbitrary patterns with 0 or 1 atoms on each lattice site.

6

1.2

Summary of main results
With the above big-picture goals and capabilities in mind, the main results in this thesis are as

follows:9 We develop the basic tools to control the alkaline earth atom strontium (Sr) in optical tweezers,
including trapping, cooling, and imaging [241] (in parallel with work at Caltech using Sr [73], and Princetonusing Yb [286]). Using the above tools, we control an optical qubit degree of freedom in tweezer-trapped
Sr atoms, demonstrating similar coherence properties to those achieved in optical lattice clocks [240] (inparallel with work at Caltech [209]). We further show that one can engineer tweezer potentials to preserve
atomic coherence while scaling to larger atom arrays, achieving very high quality factors10 of Q ∼ 1017 ina tweezer array containing ∼ 150 atoms. This high quality factor and relatively high atom number enables
synchronous frequency comparisons in the array with a fractional frequency instability of 5.2(3) × 10−17 at
1 s of averaging [365].
We show that one can hand atoms between tweezers and an optical lattice with high fidelity, and,using the tightly confining lattice, demonstrate single qubit gates on the optical clock qubit with a π-pulse
fidelity of ≳ 99.5%, and entangling gates between clock qubits (mediated by Rydberg interactions) with afidelity of 92.8(2.0)% (with SPAM correction).11

Using the above gates, we prepare Bell states of opticalclock qubits, and show that these states could be useful for optical frequency metrology. Similar Rydbergmediated interactions are used to prepare squeezed states that allow for synchronous frequency comparisonswith a precision beyond what is achievable at the standard quantum limit (SQL), with an enhancement of
3.52(1) dB in variance below the SQL for ensembles of 16 atoms, and 1.94(1) dB in variance for ensemblesof 70 atoms [96].
Critically, the lattice is designed not just for providing tight confinement, but also to be compatible
with Hubbard physics (namely where there can be a significant tunnel-coupling between adjacent latticesites). We show that, even in a tightly spaced lattice that can achieve relatively high tunneling energies
of J/h > 100 Hz,12

it is possible to rapidly rearrange atoms with high fidelity using optical tweezers.

9 The figures of merit provided in this section are to give a general feel of the performance of the experiment. Please referto the relevant chapters and papers for more detailed accounts of errors in different experimental conditions.
10 Associated with the oscillations of an even superposition of the two optical clock qubit states.
11 Recent results from the group have improved the two qubit gate fidelities to ≳ 98.5%.
12 Using Sr atoms, which are fairly heavy and thus lead to reduced tunneling energies.

7
Quantum walks of the rearranged atoms are studied, and we further demonstrate that the tweezers can beused to programmably modify the lattice potential on the scale of the tunneling energy. This capability
is used in the first demonstration of spatial search using quantum walks [366]. We benchmark the qualityof our state preparation by studying multiparticle quantum walks, finding that the indistinguishability of
two atoms is 99.5+0.5−1.6 % with postselection on atom survival, and that the on-demand success rate without
postselection for perfectly preparing, evolving, and detecting an atom in the lattice is ∼ 92% [367].13
Given the above performance, we are able to study Fock state boson sampling problems involving up to
180 atoms occupying ∼ 1000 sites in the lattice. These demonstrations are beyond the reach of knownclassical algorithms and current photonics experiments. In addition to the above published results, we will
also present some preliminary results studying interacting Bose-Hubbard models in our apparatus.

1.3

OutlineBecause the new platform we develop in this work bridges multiple fields, we will endeavor, where

possible, to provide pedagogical introductions to the various techniques and applications that our experiments
touch on. In Ch. 2, we introduce the basic properties of the alkaline earth atom Sr, and how it is affectedby external control fields. In Ch. 3, we describe our experimental apparatus, including design considerations
and tradeoffs relating to how the various control fields are applied (especially the various optical potentials,
which are fairly unique to this apparatus). We will also cover basic features that are common to all otherexperiments presented in this work (e.g. initial trapping and cooling of a thermal gas of atoms). In Ch. 4,
we describe results relating to preparing arrays of identical Sr atoms — namely preparing and imaging
single atoms, optical cooling, and rearrangement using optical tweezers in a lattice. In Ch. 5, we will
discuss frequency metrology in tweezer arrays, beginning with a general primer on quantum sensing. We will
point out where entanglement can be beneficial in sensing, before discussing the features of tweezer arraysof alkaline earth atoms that are attractive for frequency metrology. The measurements presented in thischapter will focus on using non-entangled states of the atoms to perform comparisons between clock qubits
13 This number is primarily limited by optimizing for high post-selected indistinguishability. When the experiment is properly
tuned up, the imaging fidelity including contributions from atom loss is ≳ 99.8%, and the rearrangement success probability is∼ 99.5% per atom.

8
and a local oscillator based on an ultrastable laser, as well as on synchronous comparisons between clock
qubits. In Ch. 6, we will discuss how the atoms can be excited to Rydberg states, and how these states
can be used to engineer an entangling gate between clock qubits. We will benchmark these gates, with anemphasis on how Bell pairs prepared using these gates can be used for frequency metrology. We will also
briefly discuss results that use related techniques to generate spin squeezing in ensembles of tens of atoms.
In Ch. 7, we take a step back from systems of atomic qubits pinned in place, and consider the situation
where the atoms are free to coherently move through the lattice. We will review the underlying physics ofatoms in lattices, before framing our discussion in the language of quantum walks. We will discuss analgorithm for performing spatial search by quantum walk, and show that we can realize this algorithm byprogramming quantum walks of individual atoms. This demonstration will involve both the adiabatic (anddynamical) preparation of resource states for the algorithm, as well as the execution of the algorithm itself.
We will further comment on the degree to which the tools used in our demonstration of spatial search allow
for universal control over the quantum walk dynamics. In Ch. 8, we will extend our discussion to multiparticle
quantum walks, where different atoms are free to exchange positions. We will find that, unlike in earlier
chapters, it is no longer sensible to talk about one qubit being entangled with another because the labels
we assign to the different qubits are no longer physically meaningful, resulting in dramatically different
physical behavior. We will comment on the mechanisms leading to interactions between the atoms, andpresent preliminary results studying strongly interacting quantum walks, and a superfluid that is preparedand probed in a programmable fashion using optical tweezers. Much of the discussion in Ch. 8 will be about
non-interacting and bosonic quantum walks, which can be framed in terms of the boson sampling problem.
We will present techniques for benchmarking both state preparation and the quantum walk dynamics using
multiparticle quantum walks, as well as quantum walks with up to 180 atoms occupying ∼ 1000 distinct
sites. Although the largest-scale experiments we perform are not expected to be efficiently simulable withany known classical algorithm, we will discuss caveats to this claim, including the errors that our system
is sensitive to, and how these compare to other experimental demonstrations of boson sampling. Finally,in Ch. 9, we conclude with a general discussion of the strengths and weaknesses of our platform. We will
discuss future directions in metrology and quantum simulation using the tools developed in this thesis, as9well as combined directions that blur the lines between quantum simulation and sensing.

1.4

Published content
While pursuing their Ph.D., the author contributed to the following works:[241] M. A. Norcia, A. W. Young, and A. M. Kaufman. Microscopic control and detection of ultracold
strontium in optical-tweezer arrays. Phys. Rev. X, 8(4):041054, Dec. 2018[240] M. A. Norcia, A. W. Young, W. J. Eckner, E. Oelker, J. Ye, and A. M. Kaufman. Seconds-scalecoherence on an optical clock transition in a tweezer array. Science, 366(6461):93–97, 2019[365] A. W. Young, W. J. Eckner, W. R. Milner, D. Kedar, M. A. Norcia, E. Oelker, N. Schine, J. Ye, andA. M. Kaufman. Half-minute-scale atomic coherence and high relative stability in a tweezer clock.
Nature, 588(7838):408–413, Dec. 2020[364] A. Young, W. G. Graham, T. J. Morgan, and L. Hüwel. Laser-induced breakdown in liquid water:
Influence of repeated laser pulses on plasma formation and emission. Journal of Applied Physics,
129(18):183303, May 2021
[97] W. J. Eckner, A. W. Young, N. Schine, and A. M. Kaufman. High-power, fiber–laser-based source
for magic-wavelength trapping in neutral-atom optical clocks. Rev. Sci. Instrum., 92(9):093001,
Sept. 2021
[290] N. Schine, A. W. Young, W. J. Eckner, M. J. Martin, and A. M. Kaufman. Long-lived Bell states
in an array of optical clock qubits. Nat. Phys., pages 1–7, Aug. 2022
[366] A. W. Young, W. J. Eckner, N. Schine, A. M. Childs, and A. M. Kaufman. Tweezer-programmable
2D quantum walks in a Hubbard-regime lattice. Science, 377(6608):885–889, Aug. 2022
[96] W. J. Eckner, N. Darkwah Oppong, A. Cao, A. W. Young, W. R. Milner, J. M. Robinson, J. Ye, andA. M. Kaufman. Realizing spin squeezing with Rydberg interactions in an optical clock. Nature,pages 1–6, Aug. 2023
[367] A. W. Young, S. Geller, W. J. Eckner, N. Schine, S. Glancy, E. Knill, and A. M. Kaufman. An
atomic boson sampler. In review, 2023

1.5

Units
Unless otherwise noted, all error bars and confidence intervals provided in this work correspond to ±1

standard deviations.

Chapter 2

Prerequisites“The answer to what?”— Deep Thought
In this chapter, we review the basic properties of the alkaline earth atom Sr, and discuss the tools
we have for controlling ensembles of these atoms - namely static electric and magnetic fields, and oscillating
electromagnetic fields of variable frequency, amplitude, polarization, and geometry.

2.1

Strontium and the alkaline earth-like atoms
Alkaline earth-like atoms (AEAs) have two valence electrons, and otherwise full inner electron shells.1

The key feature of AEAs is that the spins of the two valence electrons can either cancel each other out, or
pair up to produce ground and excited states with net angular momentum J = 0, resulting in both a variety
of useful optical properties, and a nearly-complete decoupling of the nuclear spin sector from the electronic
sector [126]. The latter property is useful for preserving information stored in the state of the nuclear spin
while performing other manipulations of the electronic state [155, 206, 18, 194, 146, 207, 237], as well as for
simulations of exotic SU (N )-symmetric systems [126, 288, 56]. In this work, we primarily take advantage ofthe former property, using convenient optical transitions to manipulate the electronic and motional states ofneutral Sr atoms.
1 Atoms falling into this classification include the alkaline earth atoms Be, Mg, Ca, Sr, Ba, and Ra, the zinc-family of atoms
Zn, Cd, and Hg, as well as the lanthanide Yb and actinide No [319].

11
2.1.1

Level structure
We will concern ourselves only with the behavior of the two valence electrons of Sr, and define si andli to be the spin and orbital angular momentum vectors of the ith electron.2

si is the single electron spinquantum number, where ℏ2 si (si + 1) are the eigenvalues of the s2i operator, and similarly for li and li2 . ni
is the principle quantum number of the ith electron. S =i si , L =Pi li , and J = S + L are respectivelyP

the total spin, total orbital angular momentum, and total angular momentum vectors of the two electrons.
Similar to the single electron case, the S 2 , L2 , and J 2 operators have associated quantum numbers S, L, and
J. We will assume that the atomic Hamiltonian H commutes (at least approximately) with S 2 , L2 , and J 2 ,such that S, L, and J remain good quantum numbers. This is true, for example, when the only spin-orbit
coupling terms in H take the form L · S (this situation is known as LS coupling).3 Note that we will often
write the orbital angular momentum quantum numbers L = 0, 1, 2, 3 . . . as L = S, P, D, F . . . [318].4 With
the above definitions in mind, we adopt Russell-Saunders notation (n1 l1 n2 l2 2S+1 LJ ) to label the different
electronic states of Sr. These states and their accompanying transitions are summarized in Fig. 2.1.
The electronic states of Sr naturally separate into S = 0 (spin singlet), and S = 1 (spin triplet)
manifolds. For LS coupling, a transition between the singlet and triplet manifolds is dipole forbidden due tothe selection rule ∆S = 0.5 However, as we will see in Sec. 2.1.1.2, the large nuclear charge of Sr results inspin-orbit coupling that can change this situation, making certain singlet-triplet transitions dipole allowed.
The ground state of Sr is the 5s2 1 S0 spin singlet state.6 For bosonic isotopes of Sr, J = 0 results ina unique ground state that is first order insensitive to external magnetic fields. Note that for all electronic
excited states relevant to this work, only a single electron is excited from the ground state, with the otherbeing in the 5s state. This is because most states of Sr with two excited electrons have energies that exceed
2 We will not concern ourselves with hyperfine structure and the nuclear spin I, since all results in this work involve isotopesof Sr with I = 0.
3 The mechanism for this coupling can be understood as a relativistic effect: in the frame of the electron, the moving nucleusof the atom generates a magnetic field which interacts with the electron spin. For light elements, terms like si · sj and li · lj
dominate over terms like si · lj , and so it is reasonable to make the approximation that the spin orbit terms take the form L · S.
However, we will point out corrections to this approximation as appropriate.
4 We will also follow the convention that the single electron configurations are provided in lower case, although we will
sometimes omit these labels for convenience.
5 Such a transition is also dipole forbidden from a different perspective: due to the requirement that the overall electronic
wavefunction is antisymmetrized, the spin singlet states all have symmetric spatial wavefunctions, and the spin triplet states
antisymmetric spatial wavefunctions. For two electrons, the operator for the electric dipole moment is d ∝ x1 + x2 , where xi
is the coordinate of the ith electron. Notice that d is unchanged upon exchange of the electron positions, and so the resulting
dipole matrix element between states with different exchange symmetries must be zero [26].
6 Including the configuration of the inner electrons: 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 5s2 1 S .
05sns 3S1

5snd 3D1

12...

Repumps
λ = 707.20 nm
Γ ≃ 2π×5/9×11 MHz

5s6s 3S1

Rydbergλ ~ 317 nm

λ = 688.02 nm
Γ ≃ 2π×3/9×11 MHz

...

...

λ = 679.29 nm
Γ ≃ 2π×1/9×11 MHz~1:20,000

5s5p 1P1

5s4d 1D2

J=2
J=1

Imaging
λ = 460.862 nm
Γ = 2π×30.2 MHz

5s5p 3PJ

Cooling
λ = 689.4489 nm
Γ = 2π×7.5 kHz

J=0

Clock
λ = 698.4456 nm
Γ ~ 0 Hz5s2 1S0
1

S

1

P

1

D

Singlet

3

S

3

P

3

D

TripletFigure 2.1: Level diagram for 88 Sr. We display the relevant low-lying energy levels, and the transitions
between these levels that are of interest to our experiments. In particular, note the weakly allowed transitions
between the singlet and triplet manifolds (the clock and cooling transitions), and the strongly allowedtransitions within a manifold (the imaging and repumping transitions). The indicated Rydberg transitions
correspond to a series of Rydberg states with different properties, as described in the text.

the ionization threshold of 5.69485 eV (or 217.7 nm, expressed as a wavelength in vacuum), often resulting
in autoionization [279].

2.1.1.1

Dipole-allowed 1 S0 ↔ 1 P1 transition

The strongest transition from the ground state is the dipole allowed 1 S0 ↔ 5s5p 1 P1 “blue” transition,
which has an associated vacuum transition wavelength of 460.862 nm. The 1 P1 state has a lifetime of
5.263(4) ns [363], and a corresponding linewidth of Γb ≃ 2π × 30.2 MHz. This state decays primarily to
1

S0 , with a weak branching ratio of ∼ 1 : 20000 into the 5s4d 1 D2 state [73], which subsequently decays tothe states 5s5p 3 P1,2 . As we will see in Secs. 2.1.1.2 and 2.1.1.4, 3 P1 decays back to the ground state, and13atoms in 3 P2 can be optically pumped to the ground state. As a result, 1 S0 ↔ 1 P1 provides a strong cyclingtransition that can be used to rapidly scatter many photons off of the atoms for initial cooling and trappingof a thermal gas (Sec. 3.1.2), as well as for imaging (Sec. 4.4).

2.1.1.2

Dipole-forbidden 1 S0 ↔ 3 P1 transition

Although singlet-triplet transitions are forbidden under LS coupling, the highly charged nucleus of
Sr results in spin-orbit coupling terms of the form li · si which violate the LS coupling assumption.7 Suchterms can mix together states that differ by a spin flip. This applies to the 5s5p 3 P1 state, which gains asmall admixture of the 5s5p 1 P1 state [39]. As a result, the 1 S0 ↔ 3 P1 “red” transition is (weakly) dipole
allowed, with an associated transition wavelength of 689.4489 nm in 88 Sr [283]. The 3 P1 state decays to 1 S0with a lifetime of 21 µs, and a corresponding linewidth of Γr = 2π × 7.5(2) kHz [283]. As a result, 1 S0 ↔ 3 P1
provides a narrow-line cycling transition that can be used for various frequency selective operations (with aresolution on the scale of Γr ), including high fidelity optical cooling (in the sense that the state of the atom
has high overlap with the motional ground state, see Sec. 4.3).
Given that the 3 P1 state has J = 1, there are three degenerate states in the 3 P1 manifold with
mj = −1, 0, +1, where ℏmj are the eigenvalues of the Jz operator. However, these states are split in thepresence of a magnetic field B, the influence of which is described by the Hamiltonian [318]:
HB := µB gj J · B,

(2.1)

where µB is the Bohr magneton, and gj is the Landé gj factor. For B applied in the z direction, the 3 P1states experience a linear Zeeman shift of:8
∆EB = µB gj mj B.

2.1.1.3(2.2)

Doubly forbidden 1 S0 ↔ 3 P0,2 transitions

The 3 P0 state, also known as the “clock” state, can be extremely long lived. This is because theonly lower-lying state is the 1 S0 ground state, to which transitions are doubly forbidden in the sense that
7 The situation where these terms dominate over LS coupling is sometimes referred to as jj coupling.
8 Note that the treatment in this section is only applicable when ∆E
B is smaller than the fine structure. ∆EB larger than
the fine structure is referred to as the Paschen-Back regime, where J is no longer a good quantum number [318]. This situation
will not apply in any of our experiments.

14
they require both ∆S ̸= 0, and that J = 0 both before and after the transition. However, as first proposedin [322], one can open up the 1 S0 ↔ 3 P0 transition, which we will refer to as the clock transition, with theapplication of a magnetic field.
Specifically, a DC magnetic field vector B can couple the |3 P0 ⟩ and |3 P1 ⟩ states with:ΩB = ⟨3 P0 | µ · B | 3 P1 ⟩/ℏ(2.3)

where µ is the magnetic dipole operator. Assuming approximate values of the Landé g factors gL ≃ 1
and gS ≃ 2, |ΩB | =p

2/3µB |B|/ℏ for bosonic isotopes of Sr [322]. For |ΩB /∆01 | ≪ 1, where ∆01 =2π × 5.6 THz [322] is the detuning between the 3 P0 and 3 P1 states, |3 P0 ⟩ acquires a small admixture of|3 P1 ⟩:
|3 P′0 ⟩ = |3 P0 ⟩ +ΩB 3| P1 ⟩,∆01(2.4)

where the prime denotes the new eigenstate in the presence of the perturbing field. As a result, one candirectly drive the 1 S0 ↔ 3 P′0 transition, just with a modified Rabi frequency of Ωc = Ωr ΩB /∆01 , where Ωris the Rabi frequency associated with the 1 S0 ↔ 3 P1 transition (we will discuss Rabi driving of electronictransitions in more detail in Sec. 2.2.1). By second order perturbation theory, the above coupling also results
in an energy shift on the 3 P′0 state of:∆B = −Ω2B /∆01 ,(2.5)

which is known as a quadratic, or second order, Zeeman shift. Similarly, the linewidth of the 1 S0 ↔ 3 P′0transition is Γc = Γr (ΩB /∆01 )2 . The vacuum transition wavelength for 1 S0 ↔ 3 P0 is 698.4456 nm [283],
but one must be careful when computing the transition wavelength associated with 1 S0 ↔ 3 P′0 due to boththe quadratic Zeeman shift, and any shifts due to the presence of additional control fields (see Ch. 5). For
reasonable values of the magnetic field, the lifetime associated with 3 P′0 can be in excess of thousands of
seconds while still maintaining the possibility of reasonably fast manipulations of the 1 S0 ↔ 3 P′0 transition(with Rabi frequencies in the kilohertz range for 550 G magnetic fields and ∼ 100 mW of optical power).
This makes the 1 S0 and 3 P′0 states attractive as a long-lived and magnetically insensitive qubit degree of
freedom, which we will discuss in more detail in Chs. 5 and 6.

15
Note that the 1 S0 ↔ 3 P2 transition is also doubly forbidden, since it requires that ∆S ̸= 0 and ∆J = 2,
resulting in long lifetimes of ∼ 500 s in 88 Sr [362]. However, 1 S0 ↔ 3 P2 is a magnetic quadrupole-allowedtransition, and thus can be driven with the appropriate laser fields [333].

2.1.1.4

Optical pumping transitions

We take advantage of the dipole allowed 3 P0,1,2 ↔ 5s6s 3 S1 transitions for optical pumping. The 3 S1state has a lifetime of 13 ns [283], and decays to the 3 PJ manifold with a total rate of Γ ≃ 2π × 11 MHz, andbranching ratios of 1/9, 3/9, and 5/9 for J = 0, 1, 2 respectively [320]. The associated vacuum transition
wavelengths are 679.29 nm for 3 P0 ↔ 3 S1 , 688.02 nm for 3 P1 ↔ 3 S1 , and 707.20 nm for 3 P2 ↔ 3 S1 [283].
One can (incoherently) transfer populations from one electronic state to another by turning on theappropriate optical pumping beams. Leaving any given beam on will deplete the population from thecorresponding state, with some population decaying to the ground state via 3 P1 , and some being trapped in
the other remaining triplet state. One can return all population in 3 P0,2 to the ground state by turning on
the 3 P0 ↔ 3 S1 and 3 P2 ↔ 3 S1 optical pumping beams. The reverse process is also possible, where driving the1

S0 ↔ 3 P1 transition while applying the 3 P1 ↔ 3 S1 optical pumping beam transfers ground state population

to the 3 P0,2 states. Additionally applying the 3 P2 ↔ 3 S1 or 3 P0 ↔ 3 S1 pumping beam allows one to pump
all the population into either 3 P0 or 3 P2 . All of the above operations can be performed on a timescale of≲ 100 µs.

2.1.1.5

Rydberg states

For highly excited electronic states with n2 ≫ n1 , where i = 1, 2 are taken to refer to the second-mostand most energetic electrons respectively, the details of the orbits of the inner electrons become comparably
unimportant. Instead, the outer electron primarily sees a point charge of +e at the origin, where e is theelementary charge, with all other charges in the nucleus being screened by the inner electrons. The resulting
states are hydrogenic, and so we will label these states by the values of n, l, j, and mj associated with theouter electron, where we have dropped the i = 2 index for convenience.9
9 Here, j refers to the quantum number associated with the operator j 2 , and m

The corresponding energy levels

j with jz , where j = s2 + l2 .

16scale approximately like n−2 , and small deviations from this scaling can be captured by a quantum defectδnlj .10 Given the above definitions, the electronic energy levels are given by the empirical formula [308]:
Enlj = EI −R′,(n − δnlj )2(2.6)

where EI is the ionization energy threshold, and R′ the Rydberg energy corrected by the reduced mass:
R′ =



me
m + mem e e48ϵ20 h2


.

(2.7)

Here, m is the mass of the atom, me is the electron mass, and ϵ0 is the permittivity of free space. Noticethat in the limit of large n the difference in energy between adjacent values of n scales like n−3 .
We will discuss these states, and interactions between different atoms in Rydberg states, in more detailin Ch. 6. For now, we point out a few general scaling laws. The average size of the electron orbital scales like⟨|x|⟩ ∝ n2 , where x is the displacement vector of the electron [308]. As a result, the dipole matrix element⟨n | d | n + 1⟩ between neighboring Rydberg levels11 , where d is the electric dipole operator, also scales like⟨n | d | n + 1⟩ ∝ n2 . The increase in orbital size also reduces the overlap with the ground state, leading to a
dipole matrix element from the ground state |g⟩ of ⟨g | d | n⟩ ∝ n−3/2 . Considering the overlap with all lower
energy states, one finds that the lifetimes of the Rydberg states increase with n. Specifically, states with small
orbital angular momentum l ≲ 2 have lifetimes that scale like τ ∝ n3 when considering only spontaneousemission. States with larger l typically have longer lifetimes.12 At finite temperature, the presence of black
body radiation (BBR) can resonantly drive transitions between nearby Rydberg states, leading to a scalingof between τ ∝ n3 and τ ∝ n2 for the lifetimes of the l ≲ 2 states. Typically, one will care about the ratio
between the “good” coherent evolution with a timescale scale set by 1/|⟨g | d | n⟩| and the “bad” incoherent
evolution with a timescale set by τ . A reasonable figure of merit is |⟨g | d | n⟩|τ ∼ n1/2 to n3/2 , which scales
favorably with increased n. However, we will see in Sec. 6.1.2 that this naive scaling can sometimes be
misleading. In this work, we will be particularly interested in driving transitions between the 3 P0 state andRydberg states with n ∼ 40, for which the vacuum transition wavelength is ∼ 317 nm.
10 Note that δ
nlj ≃ 0 for l > 3 [308], since in these cases the outer electron is concentrated even further away from thenucleus.
11 We are omitting the other quantum numbers for convenience, but standard selection rules still apply.
12 For example, the so-called “circular” states with maximal values of l = n − 1 have lifetimes of up to τ ∝ n5 .

17

Sr
Sr
86
Sr
84
Sr

88

87

m

Natural abundanceI

Particle statistics

87.905612
86.908877
85.909260
83.913425

0.8258(1)0.0700(1)0.0986(1)0.0056(1)

0
9/2
0
0

Bosonic
Fermionic
Bosonic
Bosonic

1

S0 ↔ 1 P1

1S0 ↔ 3 P1

1

S0 ↔ 3 P0−46.5−124.8−270.8−62.187−163.819−351.496−62.171−162.939−349.656

Table 2.1: Isotopes of strontium. m refers to the atomic mass, and I to the nuclear magnetic moment. The
isotope shifts for the indicated transitions are provided as f (8x Sr)−f (88 Sr), where f (·) denotes the frequencyof the appropriate transition for the indicated isotope in units of megahertz. The indicated values for 87 Sr
are for the weighted mean of transitions to different hyperfine states, and are an estimate of the isotope shift
in the absence of hyperfine interactions [225]. The values appearing in this table are from [283, 319, 225].

The combined effects of reduced energy splitting between adjacent levels and ⟨n | d | n + 1⟩ ∝ n2 canresult in enhanced sensitivity to DC electric fields at large n. Specifically, consider the scalar polarizabilityof the state |n, l, j, mj ⟩ [318]:
α(0) (n, l, j) = 2e2Xn′ l′ j ′ ̸=nlj|⟨n, l, j, mj | ẑ | n′ , l′ , j ′ , mj ⟩|2,
En′ l′ j ′ − Enlj

(2.8)

where ẑ is the position operator for the outer electron along the axis of the applied field. Note that astatic electric field cannot impart angular momentum, and so we have used the selection rule ∆mj = 0
in the summation. The corresponding DC Stark shift to the energy is ∆Enlj = −α(0) (n, l, j)E 2 /2, where
E is the strength of the applied electric field. Per the arguments above, neighboring energy levels with∆n = 1 will contribute n4 to the numerator, and n−3 to the denominator, leading to an overall scaling of∆Enlj ∝ n7 . Although this large polarizability can be useful for generating strong interactions between
atoms (see Sec. 6.1.2), it also becomes very important to control stray electric fields in order to avoid large
uncontrolled DC Stark shifts of the Rydberg energy levels (see Sec. 3.1).

2.1.2

Isotopes of strontium
There are four naturally abundant isotopes of Sr, whose properties are summarized in Tab. 2.1.

Critically, the isotope shifts for the relevant transitions described in the previous section are all below a
few hundred MHz, meaning that all four isotopes can be controlled with the same laser systems, up tosmall frequency shifts that are within the bandwidth of acousto-optic modulators. Since there are an evennumber of electrons and protons in Sr, all isotopes with even mass number (namely 84 Sr, 86 Sr, and 88 Sr) also

18
have an even number of neutrons, resulting in an even overall number of fermions in each atom, and thus
bosonic composite particle statistics. Because the nucleons arrange in spin singlets in the nuclear groundstate, the total nuclear spin of these isotopes is I = 0 [319]. The isotope with odd mass number, 87 Sr, has
fermionic particle statistics, and I = 9/2. In this work, we will focus on the bosonic isotopes, avoiding details
associated with hyperfine structure due to the large nuclear spin in 87 Sr. Note that the electronic propertiesof the bosonic isotopes are almost completely identical, making it easy to switch between different isotopes.
The key differences between these isotopes come in the form of their scattering properties, which we discussin Sec. 8.1.1.

2.2

Electromagnetic control
The basic level structure of Sr described in the previous section forms a Hilbert space that we areinterested in manipulating. We briefly described the role of DC electric and magnetic fields in Sec. 2.1.1,
but now review a more complete formalism for describing the effect of both DC and AC electric fields onthe electronic states of Sr.
Consider an electric field oscillating with angular frequency ω with an initial phase ϕ, and whose
amplitude E varies as a function of position r:E(r, t) = ε̂E(r) cos(ωt + ϕ) = E + (r, t) + E − (r, t)
E ± (r, t) := ε̂E(r) ∓i(ωt+ϕ)
e
,
2

(2.9)
(2.10)

where ε̂ is a unit vector indicating the polarization of the electric field,13 and we have defined positive- andnegative-rotating components of the field E ± . For much of the following discussion, we will suppress the
position dependence with the understanding that E varies slowly over space in comparison to the spatial
extent of the electronic wavefunctions.
Under the electric dipole approximation, the interaction between a Sr atom and E is described by the
Hamiltonian:
Hi = −d · E,
13 In general, ε̂ can be complex to handle circularly polarized components of the field.

(2.11)

19
where d = −e(x1 + x2 ) is the electric dipole operator for two electrons.

2.2.1

Two level approximation
We first consider the simplified setting where there are only two relevant electronic states |g⟩ and |e⟩.

This applies, for example, when ω is close to a resonance between two states, and so the coupling between
these two states dominates over the off-resonant coupling to all other states. In this case, the Hamiltonian
Ha for the bare atom, and the full Hamiltonian H, in the presence of the drive field are:
Ha = ℏωge |e⟩⟨e|(2.12)

H = Ha + Hi ,where ℏωge = ℏ(ωe − ωg ) is the energy difference between the |e⟩ and |g⟩ states, and we have applied anoffset by the identity such that the energy of the |g⟩ state in the absence of E is 0.
Note that in the Schrödinger picture, evolution under Ha will cause the state |e⟩ to rotate with a phasefactor of e−iωge t . Therefore, for evolution under H, terms in Hi like ⟨e | d · E − | g⟩ will rotate very rapidly
in time (with an angular frequency of ωge + ω), and thus approximately average to zero.14

Taking the“rotating wave approximation” where we remove the above terms from Hi ,15 and additionally transforming
into the frame rotating at ω (indicated by |e⟩ → |e′ ⟩), we have [318]:
Ha = −ℏ∆|e′ ⟩⟨e′ |ΩHi = ℏ σ x ,2(2.13)

where σx is the Pauli x matrix, and we have defined a detuning ∆ = ω − ωge , and Rabi frequency:EΩ = − ⟨g | ε̂ · d | e⟩.
ℏ(2.14)

When ∆ = 0, H is proportional to σx . When Ω = 0, H is proportional to σz up to an offset by the identity.
As a result, application of the appropriate laser field for the appropriate duration provides a universal set ofunitary operations on the {|g⟩, |e′ ⟩} subspace.
14 Also note that ⟨g | ε̂ · d | g⟩ = 0 due to the parity of d (and similarly for e).
15 This approximation is valid for |ω
ge − ω| ≪ ωge + ω, but notably does not apply to the case of DC fields when ω = 0. We
will see how to handle this in the next section.

20
One can diagonalize H to find its eigenstates and eigenenergies:|+⟩ = sin θ|g⟩ + cos θ|e′ ⟩(2.15)|−⟩ = cos θ|g⟩ − sin θ|e′ ⟩(2.16)−∆ ± Ω′,
2(2.17)

E± = ℏ

where we have defined the generalized Rabi frequency Ω′ :=√Ω2 + ∆2 , and the angle θ ∈ [0, π/2) suchthat tan 2θ = −Ω/∆. For ∆ < 0 (red detuning) and −∆ ≫ |Ω|, |−⟩ ≃ |g⟩ − Ω/(2∆)|e′ ⟩ with an associatedeigenenergy of E− ≃ ℏΩ2 /(4∆). Therefore, the state |−⟩ is approximately equal to |g⟩, and shifted in energyby:V (r) = ℏ

Ω(r)2
,
4∆

(2.18)

in comparison to the case where Ω = 0. Note that we have reintroduced a possible dependence on the spatial
coordinate r. V is known as an AC Stark shift, and is proportional to the intensity I(r) = cϵ0 E(r)2 /2 ofthe applied oscillating field.16 If the motion of an atom is slow enough that the value of V experienced by
the atom varies slowly (in time) with respect to ∆, then the atom will simply adiabatically follow the state|−⟩. In this case, one can think of V as a conservative, spatially varying potential which allows one to apply
a mechanical force, known as an optical dipole force, to an atom. By appropriately shaping the optical field
one can make an optical dipole trap (or optical trap for short), namely a local minimum in V that atoms
are unable to escape.17
Although the above discussion provides a qualitative understanding of optical dipole forces, when ∆is large it is often no longer reasonable to make the two level approximation, and a more careful accountingof all nearby energy levels, including degeneracies due to angular momentum, is necessary.

2.2.2

Dipole forces
To accurately determine the Stark shift on a given state |β⟩, we extend the discussion in the previous

section by relaxing the rotating wave approximation (namely by including the fast counterrotating terms),
16 Note that one can change the sign of V

by changing the sign of the detuning.
comes from an oscillating electric field, is it possible to make an extremum of the field in
free space. This is to be contrasted with the case of DC fields, where no such extremum can exist since ∇ · E = ∇ × E = 0.
17 Note that because in this case V21
performing a summation over all other states |β ′ ⟩, and more carefully handling the orientation between theatom and the electric field. This leads to the Kramers-Heisenberg polarizability tensor [183, 318]:αµν (β; ω) =X 2ωβ ′ β ⟨β | dµ | β ′ ⟩⟨β ′ | dν | β⟩β′ℏ(ωβ2 ′ β − ω 2 )(2.19),

where ℏωβ ′ β = ℏ(ωβ ′ −ωβ ) is the energy difference between states |β ′ ⟩ and |β⟩, and dµ refers to one Cartesiancomponent of d.18 The associated shift in the energy of the state |β⟩ is:∆E(β; ω) = − Re[αµν (β; ω)]Eµ+ Eν− .

(2.20)

In both the above expression and throughout the remainder of this section, we are following Einstein notationand summing over repeated indices. Note that |β⟩ refers to a state labelled by n1 l1 n2 l2 2S+1 LJ and mj .
Below, when we write J or mj we are referring to the values of these quantum numbers associated with thestate |β⟩ (and similarly for |β ′ ⟩, J ′ , and m′j ).
The summation over β ′ in Eq. 2.19 can be simplified by observing the appropriate symmetries andangular momentum structure. Specifically, d is a 3-vector, so αµν (β; ω) is a rank-2 Cartesian tensor andis thus reducible to scalar, vector, and tensor irreducible parts. Following the derivation in [183, 318], we
arrive at scalar, vector, and tensor polarizabilities of:α(0) (β; ω) =X 2 ωββ ′ |⟨β | |d| | β ′ ⟩|2
β′

α

(1)

(β; ω) =

X

3

(−1)

J+J ′ +1

(−1)

′

β′

s
α

(2)

(β; ω) =X
β′(2.21)22ℏ(ωββ′ − ω )

J+Jr

6J(2J + 1)
J +1

(

1
J

1
J

40J(2J + 1)(2J − 1)3(J + 1)(2J + 3)

(

1
J′

)

1
J

1
J

ωββ ′ |⟨β | |d| | β ′ ⟩|2
2
2
ℏ(ωββ
′ − ω )2J′)

ωββ ′ |⟨β | |d| | β ′ ⟩|2
,
2
2
ℏ(ωββ
′ − ω )

(2.22)(2.23)

where { ·· ·· ·· } is the Wigner 6-j symbol. These polarizabilities produce a combined energy shift of:∆E(β; ω) = − α(0) (β; ω)|E + |2
mj
− α(1) (β; ω)(iE + × E − )z
J


+ 2
+ 2
3|Ez | − |E |(2)
− α (β; ω)2(2.24)
3m2j − J(J + 1)
J(2J − 1)!
,

18 We will use similar notation throughout this work, where a bold symbol refers to a vector, the equivalent non-bold symbolwith a subscript denotes one component of the vector, and the equivalent non-bold symbol without a subscript denotes themagnitude of the vector.

22
where the subscript z denotes the component of the indicated vector along z (recalling that mj is thequantum number associated with Jz ).19 One can think of the first line in Eq. 2.24 as the component of thepolarizability that transforms to itself under rotations (namely the spherically symmetric part), the secondline is the component that transforms like a vector, and the third line is the irreducible rank-2 component.
Note that Eq. 2.24 also applies in the DC limit where ω = 0.20
We have reduced the problem of computing the energy shifts imposed on an atom by an electric fieldof the form in Eq. 2.9 to an expression involving only terms like ωββ ′ and ⟨β | |d| | β ′ ⟩. The former terms
can be computed directly from the unperturbed energy levels of the atom. The latter terms are known as
reduced dipole matrix elements, and are dependent only on the radial part of the electronic wavefunction.
The reduced dipole matrix elements can be related to the Einstein A coefficients [318]:|⟨β | |d| | β ′ ⟩|2 =

3πϵ0 ℏc3 2J ′ + 1
Aββ ′ ,3ωββ
2J + 1′(2.25)

where Aββ ′ is the decay rate (in population) from the state |β ′ ⟩ to |β⟩ and c is the speed of light. The valuesof Aββ ′ have been extensively characterized and documented in experiments, making it possible to compute∆E(β; ω) to great accuracy.21

2.2.3

Magic wavelengths
Using Eqs. 2.21 to 2.25, and tabulated values of the A coefficients for Sr [378], one can compute theenergy shift of different electronic states in Sr as a function of the wavelength of the applied laser field. Thisis shown for a linearly polarized laser field aligned along the quantization axis z in Fig. 2.2.22

Naturally,close to specific resonances between electronic states the energy shifts diverge. However, operating away fromsuch resonances is beneficial when trying to make a conservative trapping potential using laser fields. The
benefit of operating away from resonance can be seen by returning to the simplified discussion in Sec. 2.2.1:
19 Note that we typically apply a magnetic field in our experiments, in which case it is natural to define z to be parallel tothe applied field.
20 Also note that the vector contribution vanishes for linear polarization. Since a DC electric field must be linearly polarized,
this term vanishes for DC electric fields. Both the vector and tensor terms vanish for states with J = 0.
21 It is worth keeping in mind that the discussion in this section is only second order in perturbation theory. In practice,
the approximations made are accurate for regimes we’re interested in, where the coupling between different states is far smaller
than ωββ ′ and/or ω. For optical dipole traps, it would be hard indeed to enter a regime where the Rabi frequency is comparable
to an optical frequency.
22 We thank Ross B. Hutson for generously providing us with example code in which he performed similar calculations for
87 Sr.

23

Trap dept h (arb.)

0.0
0.5
1.0
1.5
2.0
500

600

700

800
900
1000
Wavelengt h (nm )3

P1 , m j = 0

3

P0

1

S0
1100

1200

Figure 2.2: Polarizabilities of the 1 S0 , 3 P0 , and 3 P1 (mj = 0) states. The total energy shift (including
contributions from the scalar, vector, and tensor polarizabilities) for the indicated electronic states are shown(in arbitrary units) for an AC electric field which is linearly polarized along the quantization axis (z). The
intensity of the field is fixed, and the wavelength varied. We refer to wavelengths at which the energy shift of
two levels is equal as “magic” wavelengths. The magic wavelengths near 515 nm for the 1 S0 and 3 P1 states,and 813 nm for the 1 S0 and 3 P0 states, are indicated by the dashed lines, and are used extensively in this
work.

24
when coupling to an excited state with a finite decay rate Γ, the overall rate at which photons are scatteredoff of an atom in the presence of a drive field is ∝ ΓΩ2 /∆2 , whereas the conservative potential due to thelight shift is ∝ Ω2 /∆. As a result, the ratio between the depth of the conservative potential and photon
scattering strictly improves with increased ∆.
One notices in Fig. 2.2 various wavelengths at which the energy shift for two electronic states areequal, and so the applied laser field does not perturb the transition energy between the two states. Thesewavelengths are referred to as “magic” wavelengths. Of particular note to us are the magic wavelengths
for the 1 S0 and 3 P0 states at 813.42757(62) nm [4], and for the 1 S0 and 3 P1 states near 515 nm [73]. In
Sec. 4.1 we will see how it is possible to achieve a stronger magic condition where the 1 S0 , 3 P0 , and 3 P1states simultaneously experience the same energy shifts. We will return to the discussion of optical trappingin more detail as appropriate throughout this work, especially with respect to high fidelity ground statecooling of atoms (Ch. 4), and manipulations of the clock transition (Ch. 5).

Chapter 3

Overview of the apparatus“[It’s not that the glass is half full or half empty.] We’re engineers. It means the glass has
been manufactured to the wrong specifications.”— Host Bernard
Portions of this chapter have appeared in:
M. A. Norcia, A. W. Young, and A. M. Kaufman. Microscopic control and detection of ultracold
strontium in optical-tweezer arrays. Phys. Rev. X, 8(4):041054, Dec. 2018
A. W. Young, W. J. Eckner, N. Schine, A. M. Childs, and A. M. Kaufman. Tweezer-programmable
2D quantum walks in a Hubbard-regime lattice. Science, 377(6608):885–889, Aug. 2022
A. W. Young, S. Geller, W. J. Eckner, N. Schine, S. Glancy, E. Knill, and A. M. Kaufman. An
atomic boson sampler. In review, 2023

We saw in Ch. 2 that our tools for controlling Sr atoms amount to the targeted delivery of DC and AC
electromagnetic fields to the atoms. The design of the optical trapping potentials are particularly important:
we are interested in trapping large ensembles of atoms at small length scales, which enhances the energy
scales associated with experiments involving high fidelity optical cooling (Ch. 4, and all subsequent experiments), Rydberg interactions (Ch. 6), and tunneling (Chs. 7 and 8). Additional considerations associatedwith avoiding undesirable perturbations to the atoms lead to the use of several different optical potentials,
including optical tweezer arrays and optical lattices at two different wavelengths. Achieving the above properties while also maintaining single-particle-resolved control using the tweezers leads to tradeoffs between thedesign of the optical lattices, and the high resolution microscope used to generate the tweezers and image
the atoms. All the optical potentials must stay referenced to each other at the ≲ 50 nm level, leading tovarious techniques for maintaining passive stability of the alignment between the different potentials, as wellas techniques for active feedback.

26
The vacuum chamber in which our experiments take place is designed to deliver the aforementioned
optical potentials (and other control fields) to the atoms without degradation, while otherwise isolating theatoms from any perturbations. In particular, one is interested in delivering thermal atoms to the control
region while maintaining ultra-high vacuum (UHV) conditions (to avoid collisions between stray particles
and the atoms), and in keeping surfaces of the vacuum chamber far from the atoms (to avoid unwanted
interactions between the atoms and electromagnetic fields emanating from these surfaces).
For various operations, we are also interested in applying strong DC magnetic fields (of up to 550 G)
and gradients (of up to 50 G/cm). This necessitates the design of high current magnetic field coils that do
not prohibitively restrict optical access to the control region. One could also be interested in controlling
the DC electric field at the atoms using control electrodes and Faraday shielding [353]. However, for theexperiments described in this work, we find it sufficient to simply keep dielectric surfaces relatively far awayfrom the atoms.
In this chapter, we will summarize our solutions to the above challenges in our apparatus (Fig. 3.1),
including the design of the vacuum chamber and atom source in Sec. 3.1, the magnetic field coils in Sec. 3.2,
and the optical systems in Sec. 3.3. Finally, in Sec. 3.4, we will briefly discuss the typical sequence of
operations that are common to any given experiment in this work — namely how an ensemble of thermal
atoms are prepared and measured (this will be discussed in more detail as necessary in subsequent chapters).

3.1

Vacuum chamber and atom source
The core components of our apparatus are shown in Fig. 3.2, including the vacuum chamber, andsurrounding magnetic field coils and optical components. Here, we summarize key features relating to thedesign of the vacuum chamber.

3.1.1

Science chamber
We will refer to the main control region where our experiments take place as the “science chamber”

or “cell.” The role of the science chamber is to provide UHV conditions,1 while also allowing various optical
1 In fact, our vacuum borders on extremely high vacuum (XHV) conditions, with pressures below 10−11 Torr.

27

Figure 3.1: Picture of the apparatus circa 2021. To see a timelapse of the first few months of construction,
visit doi:10.5281/zenodo.10019752.

Figure 3.2: Layout and detailed views of the apparatus. The atom source is highlighted in red, and thevacuum pumping chamber in blue. The magnetic field coils are shown in gold. The callout shows detailed
views of components that are obscured by the magnetic field coils, including the optical reference plate (andmicroscope objective and lattice optics, top) and glass science chamber (bottom).

28
control fields to be delivered to atoms in the chamber without degradation — specifically, without unwanted
stray reflections or optical aberrations. To provide UHV conditions, the chamber must be constructed outof materials with low outgassing, and that are compatible with a high temperature bake. To provide low
optical aberrations, the chamber must have good optical properties, and must not undergo uncontrolled
deformations that change these properties when it is evacuated. For the above reasons, we opt to use acuvette style vacuum cell, manufactured by Japan Cell, that is constructed entirely of fused silica glass (seeFig. 3.3).
The use of fused silica is helpful due to its optical properties, being highly transparent for the fullrange of wavelengths that we work with (including wavelengths as low as 317 nm), and its low coefficient ofthermal expansion (CTE) of 0.48 × 10−6 /K, which ensures that small fluctuations in the temperature of theexperiment do not lead to significant changes in optical performance. A non-magnetic glass-to-metal seal isused to connect the fused silica cell to the metallic components of the vacuum chamber, ensuring that theentire chamber can be baked to high temperatures without cracking, and that magnetically active materials
are kept away from the region of the experiment that experiences high magnetic fields. The latter feature
helps to minimize uncontrolled fluctuations in the magnetic field that are detrimental to our experiments(see Sec. 3.2).
The cell is anti-reflection (AR) coated on the outside surfaces with a broadband coating that spansfrom < 450 nm to ∼ 950 nm for angles of incidence between 0 and 45 degrees from normal (Fig. 3.4a).2 The
inner surfaces are not AR coated in order to allow the cell walls to be optically contacted together, insteadof glued together with epoxy. This all-glass construction allows the cell to be baked at higher temperatures
than a glued cell (up to 200 ◦ C), leading to higher quality vacuum.3
The lack of AR coatings on the inner surfaces of the cell requires one to ensure that unwanted
reflections do not intersect with the laser-trapped atoms. By using a design where most beams enter the cellat a large angle of incidence, we can ensure that reflections up to the first several bounces do not intersect
2 We purchased a separate cell explicitly for the experiments involving 317 nm light in Ch. 6, since these coatings are notexpected to work at 317 nm. However, miraculously, we never had to install the second cell due to an error in the manufacturingof the original cell: during the AR coating run, it was mounted in such a way that the properties of the coating on the end
facet are substantially different from on the other surfaces, yielding ∼ 95% transmission of 317 nm light.
3 Note that another option which has become viable since the original design of our apparatus is to use a nano-textured
coating (e.g. from TelAztec) that can survive higher temperatures than typical coatings, and to assemble the cell via a fritting
process. Precision Glassblowing has now constructed such cells for a number of groups.

29with the atoms.4 An additional consideration for the geometry of the cell is that it should be large enoughthat surfaces are relatively far away from the atoms, both to improve atom capture into the magneto-opticaltraps (MOTs, see Sec. 3.1.4), and to ensure that the atoms are not strongly perturbed by stray charges
and dipoles residing on the glass surface (see Sec. 2.1.1.5). Finally, the cell should not be so large thata microscope objective placed outside the vacuum chamber must be prohibitively large to achieve a given
numerical aperture (NA).
In order to achieve diffraction-limited resolution of the microscope (Sec. 3.3.2), the windows of thecell must obey certain additional requirements with respect to their flatness, thickness, and parallelism. For
our apparatus, the target specifications for the cell are as follows: a flatness of ≤ λ/10 over 20 mm (where
λ = 461 nm is the wavelength of the imaging transition); a window thickness within ∼ 100 µm of thespecified value;5 and a parallelism corresponding to a variation of ≤ 10 µm for two sides of the same pieceof glass, and ≤ 100 µm for opposing pieces of glass,6 over the 80 mm length of the cell. Note that theabove tolerances must be maintained when the cell is under vacuum. Based on finite element analysis (inCOMSOL) we opt to use 5 mm-thick cell windows, for which the deformation of the cell under vacuum is
acceptably small (Fig. 3.4b).
Taken collectively, the above considerations lead to the cell geometry shown in Fig. 3.3. Upon receiving
the cell, we confirmed that the manufacturing was within tolerances via measurements using a homemade
white light interferometer, and some measurements on a Zygo GPI Fizeau Interferometer.

3.1.2

Cold atom source
To deliver atoms to the science chamber, we use a Sr Beam II source from AOSense (see Fig. 3.2).

This atom source consists of an oven containing several grams of strontium, a capillary array to collimate
the thermal beam of atoms released from the oven (yielding a half-angle of 17 mrad), a Zeeman slower, andtwo stages of 2D magneto-optical traps (MOTs). The thermal beam released by the oven terminates on a
4 This is primarily a concern with the lattice beams, where small stray reflections can lead to large amounts of disorder incomparison to the tunneling energy (see Ch. 7). The large (transverse) magneto-optical trapping beams (Sec. 3.1.4) also enterwith an angle of incidence of > 30 degrees, and miss the atoms on the first several bounces.
5 Upon receiving the cell, we measured this to be ∼ 20 µm using a white light interferometer.
6 Note that for a single microscope objective, the specifications for opposing pieces of glass are irrelevant to the performanceof the experiment.

30

Figure 3.3: Manufacturing drawing for the glass cell.

(a)

(b)

Figure 3.4: Glass cell anti-reflection coating and deformation under vacuum. (a) Measurements of theperformance of the anti-reflection coating applied to the outside of the glass cell, including the performanceat a 45 degree angle of incidence (AOI). (b) Simulated deformation of the glass cell when evacuated, andunder atmospheric pressure (12.1 psi because we’re in Boulder), showing maximum deviations of ≲ 25 µm
across the entire cell.

31
heated window (operated at 350 ◦ C to avoid coating the window and reducing its transmission). The twostages of 2D MOTs are tilted relative to the thermal beam, such that only atoms that are appropriately
cooled transversely are directed towards the “cold” output aperture of the atom source, and subsequently
to the science chamber. When the laser beams forming the 2D MOTs are off very few atoms in the thermal
beam make it to the cold output aperture, allowing us to rapidly turn the atom source on and off with nomoving parts.
We typically operate the oven at comparably low temperatures of 380 ◦ C, where the atom flux directly
out of the oven is 3.6 × 1011 atoms/s. The atoms exiting the cold output of the source have a transverse
temperature of < 3 mK and an axial speed ∼ 40 m/s,7 and are thus easily captured directly in a 3D MOT
acting on the broad 1 S0 ↔ 1 P1 transition in Sr (Sec. 3.1.4). Despite the presence of the oven, the pressure
attained in the atom source is fairly low, with a value of 1.6 × 10−9 Torr (as measured directly at the oven).8
The atom source includes an ion pump and two getter pumps, which help to achieve this low pressure.
Additionally, because Sr is an effective getter material, much of the chamber acts as a getter pump once theoven has been activated for a while.

3.1.3

Vacuum system
In order to attain UHV conditions, one must balance outgassing rates (set by the materials and total

surface area in the chamber, as well as by the oven in the atom source) with the pumping rate, which isset by the conductances through the chamber as well as the properties of the pump used. Our experiments
all take place in the molecular flow regime,9 where the conductance is only dependent on the geometry of
the chamber and the temperature. At room temperature and in the molecular flow regime, the conductancethrough a pipe is C ≃ 12.1d3 /l, where d is diameter of the pipe and l its length (both in cm), and C is inunits of l/s (liters per second).
The atom source is attached to the science chamber via a pumping region to attain even higher
quality vacuum in the science chamber (see Fig. 3.2). Between the atom source and the pumping region is
7 Note that the atoms exiting the cold output make up only a small fraction (∼ 0.5%) of the flux exiting the oven.
8 We find that low oven temperatures are very important to the lifetime of an optically trapped atom in the vacuum chamber.

Operation at 380 ◦ C yields a lifetime of ∼ 400 s, whereas operation at 420 ◦ C yields a reduced lifetime of ∼ 160 s.
9 Meaning the mean free path of a gas particle in the vacuum chamber is significantly longer than the dimensions of thechamber.

32
a differential pumping tube with a diameter of 6.1 mm and a length of 95 mm, leading to C ∼ 0.3 l/s. The
geometry of the pumping region and the science cell is such that C ≳ 5 l/s through the whole system.10
The pumping region is constructed out of a large diameter vacuum cross, tee, and elbow. The tee houses
a titanium sublimation pump (TSP, Gamma Vacuum 3 filament), which we fired only once after the initialbake of the chamber.11 At the end of the elbow is a large ion pump (TiTan DI 75S), which has a pumping
speed of 75 l/s.
After assembling the vacuum chamber, we baked the metal components at ∼ 175 ◦ C, and the sciencechamber at ∼ 125 ◦ C, all for 15 days.12

We degassed the TSP during the bake, and then fired it after

allowing the chamber to slowly return to room temperature over the course of about two days. Althougha residual gas analyzer (SRS RGA100/12) was used during the bake, it is not permanently installed on thechamber, and we opt not to include an additional vacuum gauge. Instead, we use the current of the ion pump
as a readout, which tells us that the vacuum pressure is ≲ 1 × 10−11 Torr, and sufficient for our experiments.

3.1.4

Magneto-optical traps
The cold atom beam is trapped directly in the science chamber using a “blue” 3D MOT operating on

the broad 1 S0 ↔ 1 P1 transition, which cools the atoms to T ∼ 1.5 mK (slightly higher than the dopplerlimited temperature of T ∼ 720 µK) We then transfer the atoms into a sawtooth wave adiabatic passage(SWAP) MOT [238, 21, 229] operating on the narrow 1 S0 ↔ 3 P1 transition to rapidly reduce their temperature to T ≃ 10 µK, before performing an additional round of cooling using a conventional “red” MOT also
operating on 1 S0 ↔ 3 P1 . The red MOT compresses the atom cloud to an atom density of ∼ 1011 cm−3 , andcools it further to T ≃ 5 µK. Under typical operating conditions, we trap ∼ 107 atoms in the blue MOT in∼ 20 ms, and recapture ∼ 105 atoms from the blue MOT in the red MOT in ∼ 100 ms (including ∼ 50 ms ofoperating the SWAP MOT, and ∼ 50 ms of operating the red MOT). The general mechanisms and schemes
for magneto-optical trapping of Sr are described in detail elsewhere, including in [319] for the red and blue
10 This is primarily limited by the science chamber. The conductance through the rest of the system is significantly higher,
with C ∼ 50 l/s.
11 It is not entirely clear that this was necessary for achieving high quality vacuum, but it was included out of an abundanceof caution.
12 Most of the chamber was baked using heater tape and a makeshift oven constructed out of fiberglass insulation and vacuum
foil, but we also used a steel canister to protect the science chamber.

33
Transverse beams
Power (mW) Waist (mm)Blue MOT
SWAP MOT
Red MOT

10
0.5
0.1

10
3
3

Vertical beams
Power (mW) Waist (mm)
3
0.25
0.05

Magnetic gradient(G/cm)

2
2
2

50
4.2
4.2

Table 3.1: Representative conditions for magneto-optical trapping. Note that these are approximate values,and can vary depending on the specific experiment performed.

MOTs, and in [236] for the SWAP MOT. The final temperatures and atom densities attained using a similarcooling scheme to the one outlined above are carefully optimized in [311].
The main additional challenge in our apparatus is the fact that the high NA microscope objective
results in limited optical access on one side of the cell. Although it is possible to operate MOTs with skewed
beams that do not intersect at right angles, we instead opt to project the downwards-pointing MOT beams
through the objective itself, by focusing these beams on the back focal plane of the objective. Due to spaceconstraints on the apparatus, we are limited to using a 2′′ diameter lens with a 250 mm focal length, yielding
a beam waist13 of ∼ 2 mm when collimated by the objective. In the transverse directions, the blue MOT
beams have a waist of ∼ 10 mm,14 and the red MOT beams a waist of ∼ 3 mm. Although the parametersof the MOTs change throughout our works, representative values of the various beam powers and magnetic
gradients appear in Tab. 3.1.
Because the blue MOT beams are at the same wavelength as the imaging light, they cannot be
separated from the imaging path exiting the microscope objective using a dichroic mirror. Instead, we use
a custom 90:10 wedged non-polarizing beam splitter (Perkins Precision Developments) that results in anegligible loss of photon collection efficiency in the imaging path, at the cost of requiring extra (but readily
available) laser power in the downwards facing MOT beams (meaning the beams projected through theobjective, see Fig. 3.7). Note that although the downward facing beam is required for a stable blue MOT,
this beam is not strictly required for the SWAP MOT and red MOT, since gravity provides a restoring force
in the downward direction.15
13 Namely, the radius at which the intensity of the beam drops to 1/e2 of its maximal value.
14 But are clipped in the vertical direction to the 10 mm height of the interior of the science chamber.
15 This works in the case of the narrow line MOTs, since the temperatures achieved are such that the atoms do not undergo

excursions that are large compared to the size of the science chamber [319].

34
Microscope objective
x bias coils

Reference plate
y bias coils

2D lattice optics

Axial lattice lensGlass cellHigh field coils
z bias coils

Figure 3.5: Cross-section of the science chamber and surrounding components. Note that the lattice optics(except the axial lattice lens, which is still on a temporary mount visible as a grey bar on the left of thisfigure) and microscope objective are mounted directly to the reference plate, which is mechanically isolated
from the coils.

The position of the red MOT can be controlled on length scales smaller than the MOT beams by
simply applying bias magnetic fields which shift the position of the magnetic quadrupole that defines the
position of the MOT. We make use of this to align the MOT to our various optical trapping potentials, andto load atoms from the MOT into the optical traps, as described in more detail in Sec. 4.2.

3.2

Magnetic field coils
In order to apply the appropriate magnetic fields and gradients, the chamber is surrounded by a setof magnetic field coils. In particular, we are interested in applying magnetic gradients of up to 50 G/cm,
magnetic fields in the vertical (z) direction of up to 550 G, and moderate bias fields of ∼ 30 G that can point
in an arbitrary direction. The bias fields are applied using pairs of bias coils in the x, y, and z directions,where z is the optical axis of the microscope objective, and x and y are transverse to that axis (see Fig. 3.2).
The large fields and gradients are applied using an additional pair of high current coils oriented along the z
axis.

35
In order to ensure homogeneous magnetic fields across the atom array (except when explicitly applying
magnetic field gradients), all pairs of coils are arranged close to a Helmholtz configuration, except for the z
bias coils, which are separated by 1.7 times their radius due to geometric constraints.16 Although the z bias
coils are fairly far from the Helmholtz condition, we are less sensitive to gradients imposed by these coils,
since the resulting gradients are primarily normal to the atom array. The suboptimal design of the z bias
coils is because they were originally intended to be used only for applying our MOT gradients. However,out of a concern of heating the microscope objective,17 we instead use these coils only at low currents. To
provide the requisite magnetic gradients, we use an H-bridge circuit to switch the high current coils between
a Helmholtz and anti-Helmholtz configuration as needed (see below and Appendix C.4).
The coils and their relative orientation with respect to the vacuum chamber can be seen in Fig. 3.2,
and in more detail in Fig. 3.5. The relevant parameters associated with the coils are summarized in Tab. 3.2.
The x and y bias coils are designed to be large to avoid unwanted gradients, and use many turns of wire bothto allow for the use of low current power supplies, and to avoid heating. The z bias coils are water cooled
via an aluminum backplate, and the high current coils are cooled by running water through the core of thehollow wire used in their construction. Each high current coil is composed of two windings with 25 turns
each that are connected electrically in series using jumper bars, but in parallel for the water cooling [236].
This approach improves the water flow rate, and is critical for attaining sufficient cooling power. All coils areconstructed by wrapping wire around a support structure, and potting them in place with epoxy (Araldite2011). The support structure is 3D printed nylon plastic for all coils except the inner z coils, which arepotted directly to the aluminum cooling plate.
The bias coils are all powered by a JILA designed current controller [52]. The high current coils are
powered by a Delta electronica SM30-200 DC power supply, which is capable of outputting 200 A at 30 V.
To switch the high current coils between the Helmholtz and anti-Helmholtz configurations, we use a custom
H-bridge circuit that switches the polarity of one of the two coils (see Appendix C.4). Note that although
the coils are operated at low voltages, there is a significant amount of energy stored in the magnetic field
when they are activated. If they are disconnected suddenly, the stored energy can result in dangerous spikes
16 Namely having to fit in between the high current coils and microscope objective.
17 A concern that arose only after I melted our first microscope objective.

36

DiameterWire
Turns per coil
Resistance (Ω)
Helmholtz field (G A−1 )x biasy biasz biasHigh field

9′′
20 AWG
200
4.72
17.35

10.5′′
20 AWG
200
5.88
15.18

2.6′′
20 AWG
160
1.22
27.07

6′′
3/16′′ hollow square
50
0.1
6.15

Table 3.2: Magnetic field coil parameters. The listed Helmholtz fields are for when the pairs of coils are
wired in series (keep in mind that the z bias coils are relatively far from the Helmholtz condition). Other
parameters are listed for each individual coil.

in voltage that can destroy electrical and human components [236]. Because of this, the H-bridge circuit
includes an interlock that ensures that the polarity of the coil cannot be switched unless the current is below
a safe value. A 1 Ω flyback resistor is also included in order to dissipate the accumulated current, allowing
the coils to be switched on and off relatively quickly (on a timescale of ∼ 25 ms).
The above electrical sources provide fairly stable currents, with fractional root mean squared (RMS)
fluctuations at or below the 10−4 level, translating to similar stability in the applied magnetic field. However,magnetization of components near the coils can serve as another source of fluctuating magnetic fields. To
mitigate this effect, most components near the science chamber are non-magnetic (glass, plastic, ceramic,
aluminum, titanium, etc.). Although this limits the effect of magnetization, we still encounter issues withdrifting fields when the fields are left at a high value (> 250 G) for a long time (minutes). We suspectthat this is due to magnetization of the upper optics breadboard in the experiment (see Fig. 3.2), which isconstructed using stainless steel. To monitor the drifting fields, we use a 3 axis magnetometer (AlphaLab
VGM) installed near the science chamber. We find our magnetic field stability to be sufficient for mostapplications, including those involving precision spectroscopy (Ch. 5). However, further reducing the use ofmagnetic materials near the science chamber, and providing additional magnetic shielding, would be usefulfor experiments involving high fields and long coherence times.18
18 For example, experiments in which we wish to increase the clock Rabi frequency to perform fast single and two qubit gates,
and then use the resulting entangled states for stable frequency comparisons with a laser (see Chs. 5 and 6).

37

3.3

Optical systems
The optical design of the apparatus is centered around a custom manufactured microscope objective(from Special Optics), which both allows us to efficiently collect photons scattered from the atoms in a spatially well-resolved manner, and to project tightly confining and reconfigurable “optical tweezer” potentials
on the atoms. However, arrays of optical tweezers often suffer from disorder in the intensity and shapes of
the tweezers due to high sensitivity to optical aberrations in the entire tweezer system. Additionally, opticaltweezers are not particularly efficient with respect to optical power, since each tweezer is an independenttightly focused beam of light. This can be contrasted with optical lattices, which are more efficient withrespect to laser power,19 and can have very low disorder.20
As a result, we integrate both kinds of optical potentials in our experiment, and realize motional statepreserving, low-loss transfer between these different potentials.21 We will loosely classify these potentials as“preparation” and “science” potentials. The preparation potential is a tweezer array operating at 515 nm,
where a magic trapping condition can be achieved for the 1 S0 ↔ 3 P1 cooling transition at 689 nm via tuningof a magnetic field (see Sec. 4.1). 515 nm is a convenient wavelength for generating large arrays of tightly
confining tweezers because of the short wavelength (which allows one to generate tweezers with finer spatial
features), the availability of large amounts of laser power (≳ 10 W), and the high polarizability of groundstate Sr near this wavelength (of ∼ 900a30 in atomic units, where a0 is the Bohr radius).
The science potential is either a tweezer array or a 3D optical lattice operating at 813 nm, which is a
magic wavelength for the clock transition in Sr (see Sec. 2.2.3). The 3D optical lattice is composed of a 2D
lattice which is in the x-y plane (the same as the tweezer arrays), and an “axial” lattice that is normal tothis plane (oriented along the z axis). Note that 813 nm is less appropriate than 515 nm for initial trappingand cooling due to lower available laser power (∼ 5 − 8 W, although this may change in the future [97]),
longer wavelength, and lower polarizability of the ground state (282a30 in atomic units [39]). As a result, thescience tweezers are typically only used during phases of the experiment where shallow traps are preferable,
19 Since the same light is reused to make many tightly confining optical traps, and because a retroreflected lattice generates
features with a resolution that is akin to having a numerical aperture of 1.
20 Because they are generated using an interference effect, without using complicated high NA optics.
21 As had been accomplished in tweezer arrays [198], but not between tweezer arrays and lattices prior to our works.

38

Science tweezer

2D lattice

e

ttic

l la

ia
Ax

n
atio
par

r

eze

twe

Pre

Figure 3.6: Cartoon of the optical potentials applied to the atoms. These potentials include a 3D opticallattice (of which we use only a single 2D layer) composed of axial and 2D lattices, the 515 nm “preparation”
tweezers (green) used primarily for initial trapping and atom rearrangement, and the 813 nm “science”
tweezers used primarily for clock interrogation. Both sets of tweezers are also used to programmably modify
the lattice potential as needed during experiments involving quantum walks in the lattice. For illustration
purposes, a single atom (blue sphere) is shown being placed by a preparation tweezer into a minimum of the
3D lattice.

and therefore where these constraints do not impose a limitation on atom number or state preparation. In
contrast, the enhanced efficiency of the lattice potential more than makes up for these shortcomings, allowing
us to generate thousands of traps that are significantly more tightly confining than the preparation tweezers.
Although we use a variety of different combinations of tweezers and lattices in our experiments, a cartoonof the typical combination of optical potentials and how they interface with each other is shown in Fig. 3.6.

3.3.1

Mechanical design
In order to use the various optical potentials described above together, it is imperative that theposition of the lattice potential be well-referenced to the positions of the potentials projected through themicroscope objective. For example, a variation in the alignment between these potentials at the a/4 level,√
where a ≃ 813/ 2 nm is the lattice constant of the 2D optical lattice, would cause the experiments appearing
in Ch. 7 to fail. The optical clock laser used in the experiments in Chs. 5 and 6 must also be well-referenced
to the positions of the atoms at a similar level of precision. Our approach to addressing this challenge is39
twofold: we can actively feed back on the position of the tweezers or the phase of the optical clock laser
to correct for these drifts, but we additionally want the experiment to be as passively stable as possible tominimize the requirements on this feedback. To improve the stability of the experiment, we provide a single
rigid reference surface for all optical beams whose positions (and possibly orientations) at the atoms must
be well-controlled. This “reference plate” (see Figs. 3.2 and 3.5) is constructed using Macor glass ceramic
due to the rigidity (with a Young’s modulus of 66.9 GPa at room temperature), low CTE (9.3 × 10−6 C−1
at room temperature), and insensitivity to magnetic field22 of this material.
To first order, the position of potentials projected through the objective are determined by the positionof the objective itself,23 and so the objective is attached directly to the reference plate. Similarly, the lenses(and folding mirrors) that define the position of the 2D optical lattice are attached to the reference plate.24
The optical clock laser does not have fine spatial structure, and so we are insensitive to transverse fluctuationsin the position of this beam at the location of the atoms. However, it is critical that the spatial phase ofthis beam at the location of a given atom be consistent in a given experimental trial. To address this, we
perform fiber noise cancellation (See Appendix D.2.1), using a mirror that is attached to the reference plate
as a proxy for the position of the atoms.
Although the above design addresses most fluctuations in alignment as a result of vibrations of thevarious mechanical components in the experiment, additional drifts can occur as a result of air currents,
which affect the optical path experienced by the various laser beams, and thermal fluctuations, which cause
the various mechanical and optical components in the experiment to change their properties. As a result,great care is taken to ensure that the temperature of the experimental apparatus is stable. Due to thelarge heat load in the lab from various electronic components, the room air temperature is stabilized bypumping cold air from the building into the lab, leading to peak to peak fluctuations of ∼ 0.6 K undernormal operating conditions.25 The optical tables are completely enclosed and have less heat load than therest of the room. Therefore, their temperature can be controlled by electrically reheating the air that is sent
22 Important because the reference plate sits in the region of the apparatus that experiences high magnetic fields.
23 Although changes in the angle with which the beams enter the objective can result in differential motion between theobjective and the optical potentials, such drifts are strongly suppressed.
24 Although the initial design intention was to glue the lens defining the position of the axial lattice in place after aligning itwith a 5 axis kinematic mount (Newport 9081), we found that the stability of the axial lattice was acceptable without attaching
this lens to the reference plate. As a result, we have so far opted to avoid performing this irreversible procedure.
25 Normal meaning not during one of the many floods and power outages that have occurred in JILA during my PhD.

40
to the tables via several high efficiency particulate air (HEPA) filters, leading to peak to peak fluctuations in
the table temperature of ∼ 0.05 K assuming the heat load on the tables doesn’t change.26

To ensure thatthe stabilized air to the tables doesn’t lead to large air currents, the HEPAs are sized for a linear flow rateof up to 0.11 m/s over the tables (with 0.05 m/s being typical for day to day operation). This corresponds
to a flow rate of up to 700 cfm for the “science” table which houses the main experimental apparatus, and
350 cfm for the “laser” table which houses many of the lasers involved in the experiment (see Appendix D).

3.3.2

Microscope system
Our microscope objective is custom-made by Special Optics, with a specified numerical aperture (NA)of 0.67, a working distance of 12 mm, an effective focal length of 23.1 mm, an aperture of 27 mm, and adiffraction-limited field of view of 75 µm.27

The objective is designed to achieve the above specifications

when imaging through the 5 mm thick windows of the science chamber (Sec. 3.1.1). Given the diverse
range of wavelengths used in the experiment, the objective is also designed to achieve diffraction limited
performance between 461 nm and 950 nm. Over this range of wavelengths, the transmission efficienciesof the objective are above 87% (measured to be 91% at 515 nm),28 and the focal shift due to chromaticaberrations is below 2 µm. Like other components in the high magnetic field region of the experiment, theobjective is constructed of non-magnetic materials, namely glass, epoxy, and ULTEM PEI plastic.
When designing high NA objectives for cold atom experiments,29 it is also important to consider
realistic limitations on the manufacturing of the vacuum windows, and the degree to which alignment canbe achieved and maintained. Our objective can maintain diffraction limited performance given errors in thewindow thickness of ±20 µm by adjusting only the working distance, and errors of ±100 µm if compensatingwith both the working distance and the collimation of the input (or output) beam.30

Additionally, theobjective is insensitive to angular misalignment below 0.02 degrees with respect to the cell window. In order
26 Larger fluctuations can occur when opening the table doors, or installing new components. For the most sensitive experi-ments we have performed, we find it helpful to avoid opening the doors leading into the lab, since a perturbation to the room
temperature and air currents can also lead to measurable changes in the temperature of the optical tables.
27 The objective acts like an f -tan(θ) lens, and has negligible distortion and field curvature over the entire field of view.
28 Note that although reflections off of the various surfaces in the objective are not significant in terms of efficiencies, they
do make it challenging to project optical lattices through the objective, since even small stray reflections can interfere with thelattice beams and result in unacceptable amounts of disorder for studying Hubbard physics [13].
29 Specifically, objectives that sit outside the vacuum chamber, although other approaches which put parts of, or all of, theobjective inside the chamber come with their own strengths and weaknesses [312].
30 These considerations lead to the specifications for the science chamber in Sec. 3.1.1

41
to ease the alignment process, the front surface of the objective is flat, allowing us to use a reference beam
projected from below the science cell, and normal to the cell surface, to align the objective normal to thecell with the requisite precision (by measuring the reflection off of the objective).31

Note that this is the

only alignment step required for installing the objective in the experiment, since all other beams and opticalpotentials will be aligned to the objective itself.
Upon receiving the microscope objective, we performed a series of tests to characterize its performance,
including interferometric measurements with a dielectric bead and an optical flat, and direct measurementsof the point spread function (PSF) of the objective using a pinhole. We found the latter measurements tobe more effective, since it was challenging to ensure that other optical components in the interferometricmeasurements were of similar or better quality than the objective itself. Specifically, in a test setup that
includes both the objective and a sample of the science chamber window material, we align a 350 nm diameter
pinhole to the focus of the objective, illuminate the pinhole from the rear, and image the pinhole throughthe objective. Deconvolving the measured image with the shape of the 350 nm pinhole yields a direct
measurement of the PSF of the objective. Scanning the position of the pinhole also allows one to measure
the field of view of the objective in a straightforward manner.
Note that the ideal PSF of the microscope system is an Airy pattern, which corresponds to when theaperture of the microscope is uniformly illuminated.32 When imaging a point source, the ideal PSF yields
an intensity distribution of:I(ρ) ∝2R
J1 (kRρ/f )
ρ

(3.1)

where J1 is the first order Bessel function of the first kind, ρ is the coordinate in the image plane, k = 2π/λ
is the wavenumber of the light (with wavelength λ), R is the radius of the aperture, and f is the focal lengthof the lens.
We Fourier transform the measured PSF to attain an estimate of the field in a Fourier plane (where
the aperture of the system is defined), and fit the resulting distribution with the Zernike polynomials, which
are a complete orthonormal set of basis functions on the unit disk. The lowest order Zernike polynomials
31 The objective is mounted on a Newport HVM-2u positioner, which provides the appropriate controls for alignment.
32 In other words, it’s the Fourier transform of a disk. See [125] for a very nice introduction to Fourier optics.

42
correspond to well known optical aberrations (defocus, astigmatism, coma, etc.). Understanding these aberrations allowed us to identify and correct a manufacturing error in the original objective, and is generically
helpful once the objective is installed in the experiment to correct misalignments in the system. The final
two objectives we received were both within specification based on the above tests, with a Strehl ratio of≳ 0.8 across the entire field of view.33 The associated PSF has an effective beam waist of 440(20) nm when
imaging the atoms on the 1 S0 ↔ 1 P1 transition at 461 nm (Fig. 3.7b).
To align the objective in the experimental apparatus, we first install the tracer beam described abovethat is normal to the cell. With the objective removed from the apparatus, a tweezer array can be aligned
to the tracer beam. The objective can then be installed, and its angle aligned using the tracer beam. Analignment mirror (with markings that denote the size of the entrance pupil) is then placed at the rear of
the objective (on the side facing away from the cell, and reflecting the tweezer beam), and shimmed in placesuch that it retroreflects the tweezer beam. This mirror now serves as a reference for the optical axis of
the objective (as well the position of its entrance pupil) that can be removed and reinstalled at will. Boththe tweezer beams and imaging system can be coarsely aligned using this reference.34

Final alignment isperformed using the trapped atoms themselves as a probe, by maximizing trap frequencies and minimizing
the PSF measured in images of the atoms. The imaging and tweezer paths are separated using a custom
wedged35 dichroic mirror from Laseroptik, which reflects 461 nm and transmits 515 nm, 680-705 nm, and813 nm, when oriented at a 30 degree angle. The layout of the imaging and tweezer systems are shown
schematically in Fig. 3.7a.
Note that a relatively large magnification is required to match the PSF of the imaging system to thesize of the pixels on our camera (see Sec. 4.4). Due to challenges associated with aligning a multi-element
imaging system without degrading the performance limits imposed by the objective, we instead opt to use asingle off-the-shelf achromatic lens with a 1.9 m focal length (Edmund Optics 70-163) to achieve the desired
33 The Strehl ratio is a useful coarse-grained metric for evaluating the performance of an optical system, and is defined to be

the ratio between the peak value of the PSF of the system, and the peak value of the ideal PSF in Eq. 3.1. For small aberrations(namely due to wavefront errors of less than a wavelength), the Strehl ratio is directly related to the wavefront error of thesystem [153]. A Strehl ratio of > 0.8 is typically accepted as an operational cutoff for being “diffraction limited,” with 1 being
the maximum possible value.
34 In the latter case, we use the same trick of once again installing a tracer beam that traverses the optical path in reverse.
35 The wedge is important to avoid etaloning in the tweezer and imaging systems.

43
BS

oi

:1

0

hr
D

90

ic

hr
ic
D

AOD(y)

(x)

1.0

MOTs
AOD(y)

AOD

(x)

EMCCD
461 nmFluorescence(arb.)

m

n
813

AOD

c

(b)
oi
c

(a)

0.68 NA

0.5

9

0.0
0.0

12 mm

0.5
1.0
Radius ( m)

5 mm

515

nm
MOTs

Figure 3.7: Simplified layout of the microscope system and its performance. (a) The 515 nm and 813 nm
tweezer systems are generated using acousto-optic deflectors (AODs) (see Fig. 3.8 for more detail), combinedon a dichroic mirror, and separated from the 461 nm fluorescence from the atoms using another dichroic
mirror. The downwards-facing MOT beams come to a focus at the back focal plane of the objective, and are
combined onto the optical path using a 90 : 10 beam splitter (BS). The objective focuses (or collimates) thevarious beams, and has a numerical aperture (NA) of 0.68, a working distance of 12 mm, and is corrected toachieve diffraction-limited performance when focusing through a 5 mm thick piece of glass. (b) An averaged
image (at 461 nm) of atoms trapped in a 4 × 4 array of optical tweezers is displayed, illustrating the PSF of
the microscope system. In the background is a radial average of the PSF, yielding a waist of 440(20) nm.

magnification.36

3.3.3

Optical tweezers
The optical tweezers are simply tightly focused beams of light projected through the microscope

objective. Typically, we do not uniformly illuminate the aperture of the objective, and instead illuminate itwith a Gaussian beam, which results in an intensity profile of:I(ρ, z) = I0
w(z) :=



p

w0
w(z)

2

2

2

e−2ρ /w(z) ,1 + (z/zR )2 .

(3.2)
(3.3)

Here, z is the coordinate along the optical axis, and ρ is the coordinate transverse to z. I0 is the maximum
intensity of the beam, zR = πw02 /λ is the Rayleigh range, w(z) is the beam waist as a function of position
along the optical axis, and w0 is the minimum beam waist. The actual intensity profile of the tweezers
deviates only slightly from Eq. 3.2 due to the finite size of the microscope aperture. Per the discussion in
36 This lens has a (perhaps unnecessarily) large diameter of 128 mm, as it was originally designed for use in refractive
telescopes. This makes alignment of the imaging system quite straightforward.

AOD(x)OlympusAperture

AOD(y)

44

Path saver
Intensity
servo
Bare fiber

Polarizing beam splitter

Lens

Fiber coupler

λ/2 waveplate

Iris/beam dump

PhotodiodeFigure 3.8: Layout of the tweezer systems. Note that similar designs are used for both tweezer systems,
and that some optical elements are omitted for simplicity. The first relay from the left is actually a 3 : 4telescope in order to match the beam size to the optimal size for the AODs, and the waveplates are used toalign the beam polarization to the appropriate axis of the AODs. The right-most waveplate is used to set anappropriate pickoff fraction for the intensity servo. The path saving fiber allows the bare input fiber to be
replaced or repaired without spoiling the alignment of the rest of the tweezer system (since a newly installedfiber can just be aligned to the path saver). A typical tweezer array used in our experiments, imaged before
the objective, is shown on the right.

Sec. 2.2.1, for the appropriate choice of laser frequency the atoms experience a conservative potential thatis proportional to Eq. 3.2.37
We are interested in generating many such tweezers that can be rapidly reconfigured in a (mostly, see
Sec 3.3.3.2) independently controllable fashion, while also maintaining relatively efficient use of laser power.
Throughout the entire system used to generate the tweezers, great care must be taken to minimize aberrations
that degrade the PSF of the tweezer system (resulting in features that are larger than the diffractionlimited performance of the microscope objective).38 The system used to generate arrays of tweezers is shown
schematically in Fig. 3.8. Laser light originates from a fiber, and is collimated using a high NA lens withlow aberrations (Olympus RMS10X for the 515 nm system, and RMS4X-PF for the 813 nm system39 ). In
order to increase the optical power available, we opt to use a custom photonic crystal fiber (PCF) with amode expander (from Tratech Fiberoptics).40 The collimated beam is passed to an acousto-optic deflector(AOD), then to a second AOD oriented at 90 degrees to the first, and finally to a telescope that expands
the beam(s) to fill the aperture of the objective.41 Each element described above is separated from the next
37 For atoms that are well-cooled, the resulting potential is effectively harmonic, since the center of a Gaussian function iswell approximated by a quadratic function.
38 This amounts to controlling the wavefront of the laser through an optical system that spans several meters at a level ofsignificantly below a wavelength, which is a testament to the impressive performance of modern optical components.
39 Note that, due to poor transmission at 813 nm, the RMS4X-PF was later replaced with a Thorlabs C280TMD-B.
40 We do not find the additional structure in the output mode of the PCF to degrade the performance of the tweezer system
in comparison to a normal fiber.
41 All optical elements that are in the beam path after the beam is expanded (where we are most sensitive to imperfect optics

45
by a 4f relay,42 which images the optical field at the output of the previous element onto the input (orimage plane) of the next element. The integrated intensity of the tweezer array is stabilized using a pickoff43
located after the second AOD, and actuated with an acousto-optic modulator (AOM) located upstream of
the fiber.
AODs (and AOMs) are effectively programmable diffraction gratings, where an acoustic wave launched
into a crystal results in a spatially varying strain, and corresponding variation in the index of refraction. The
frequency fa of the modulation sets the wavelength of the acoustic wave λa = fa /v (where v is the speed ofsound in the crystal), and thus the pitch of the diffraction grating d = λa /2. For diffraction to occur, theincident angle θ of the laser with respect to the grating (in this case the k-vector of the acoustic wave) must
be close to the Bragg condition:nλ = 2d sin(θ),(3.4)

where n is the diffraction order (typically 1). Deviations from this phase matching condition cause thediffraction efficiency to drop, which sets the range of modulation frequencies over which an AOD can operate.
The linearity of the above process44 means that one can drive an AOD with multiple radio frequency(RF) tones to generate multiple superimposed diffraction gratings, each of which generates a separate spot
whose deflection angle can be controlled via the frequency of the drive tone. This allows one to turn a single
laser beam into many, whose positions can be controlled independently on a line. However, because thegrating in an AOD is moving, the diffracted light is shifted in frequency by the frequency of the acoustic
wave, resulting in tweezers that are at different frequencies.45

Although we want to make many tweezers,

we also want to ensure that beating between nearby tweezers that overlap slightly does not cause significant
parametric heating (see Sec. 5.3.1). For the trap frequencies we typically work with (see Sec. 4.3.3), we wantadjacent tweezers to be separated in frequency by more than 1 MHz, and so the number of tweezers we canintroducing aberrations) were tested carefully on a Zygo interferometer (the same one used in Sec. 3.1.1). This yielded several
surprises. For example, securing an otherwise flat 2′′ diameter mirror in a standard kinematic mount using a set screw resulted
in deviations as large as λ/2, compromising the tweezers. We found it necessary to instead glue the optics in place using slow
curing epoxy.
42 Two lenses, each of which performs a Fourier transformation, resulting in the optical field at the input of the relay being
replicated at its output.
43 Composed of two polarizing beam splitters separated by a λ/2 waveplate (see Fig. 3.8).
44 Although, in real devices the radio frequency components and acousto-optic response of the crystal are not always nicely
linear (see Sec. 3.3.3.2).
45 For a full discussion of the photoelastic effect and coupled mode theory, see [361].

46
make is primarily limited by the RF bandwidth of the AODs.
To increase the RF bandwidth of an AOD, one can use a phased array of acoustic transducers rather
than a single transducer. The relative phase of the transducers controls the angle at which the acoustic wave
is launched into the crystal, and so appropriately tailoring this phase as a function of frequency allows one
to match the condition in Eq. 3.4 for a large range of different frequencies (without changing the alignmentof the laser relative to the AOD). We use custom designed AODs from IntraAction which use this designprinciple to attain an RF bandwidth of 90 MHz about a center frequency of 180 MHz.46

To provide

reasonable grating pitches for these operating frequencies, the AODs excite longitudinal rather than shear
waves in tellurium dioxide (TeO2) crystals.47 At 515 nm (813 nm), the diffraction efficiency is 85% (70%)
at the optimal RF drive power of 1.6 W (2.8 W), the center value of the deflection is 21.8 mrad (34.4 mrad),and the deflection range is 10.9 mrad (17.1 mrad).48
The tweezer arrays at 515 nm and 813 nm are combined on a custom dichroic mirror from Laseroptik
that reflects 461 nm and 500-600 nm, and transmits 650-850 nm, when oriented at 45 degrees from theincident beams. Based on both direct measurements of the tweezer system, and measurements of the trap
frequency experienced by the atoms (see Sec. 4.3.3), we infer that the tweezers generated at 515 nm (813 nm)
have a waist of w0 = 480(20) nm (w0 = 740(40) nm), which is close to (but slightly larger than) the limit
set by the performance of the objective. Note that the number of tweezers we can currently make at 515 nm
is limited by thermal lensing in the tweezer system, which precludes optical powers in excess of 2 W out of
the fiber. We suspect this is due to lensing in the first AOD, and that using a different material like flint
glass would improve the power handling of the system.49
46 The number of resolvable spots given the ∼ 4 mm beam diameter used at the AODs ends up being ∼ 80 (this is independentof operating wavelength, and applies to both tweezer systems). Ignoring limitations on laser power, this would allow for thegeneration of several thousand independent tweezers.
47 The longitudinal waves have a speed of sound of 4260 m/s, in comparison to 617 m/s for shear waves. The higher speedof sound results in longer wavelengths for a given modulation frequency, and thus for larger frequency separations between
gratings with a given difference in their pitch.
48 Note that the higher drive power required by the 813 nm AODs necessitates the use of a heat sink. Without this, thermalchanges in the optical properties of the crystal make the alignment of the tweezer system unstable (see next paragraph).
49 Often, a high photoelastic constant comes hand in hand with sensitivity to thermal lensing, since the latter effect can be
related to a change in the index of refraction as a function of strain (although other mechanisms such as the thermo-optic effectsometimes also play a role). In places where we value optical power handling over other properties of the AOD or AOM (forexample in the AOM upstream of the fiber used to stabilize the tweezer intensity), we will often use flint glass rather than
TeO2.

47
3.3.3.1

Tweezer electronic control

When generating multiple tweezers by means of driving the AODs with multiple RF tones, one wants
the entire system to be as linear as possible. Although the AO response is inevitably non-linear when
optimizing for diffraction efficiency, and operating in the regime where the AOD is close to saturation,
the high power RF amplifiers used to drive the AODs can also be a significant source of non-linearity. Our
amplifiers (Delta RF LA10-1-525-40-A1) are chosen to have a saturation power (10 W 1 dB gain compression
point, 20 W saturation power) that will not immediately destroy the AODs, but also to have as linear a
response as possible at the desired RF drive powers. In particular, these amplifiers are optimized to have a
low third-order intermodulation distortion (IMD3) point of -30 dBc (measured for two tones with a spacingof 10 kHz and a peak envelope power of 10 W), which avoids extra drive tones appearing near the operating
range of the AODs when operating at high power and with many tones.
One of the main sources of intermodulation when driving an amplifier with many tones is simply the
maximum voltage that the amplifier is capable of outputting. If many tones phase up such that there is alarge excursion in the signal in the time domain, this spike in voltage can get cut off. As a result, one canreduce or mitigate this source of intermodulation by judiciously choosing the relative phases of the drive
tones such that these large excursions in the time domain are minimized. This is sometimes accomplishedthrough brute force optimization by computing the signal in the time domain, but such approaches arecostly with respect to computing resources, and can either slow down the experiment, or require one tochoose from a limited selection of pre-computed signals [95]. Instead, we find that applying the following
heuristic solution [295]50 provides signals with acceptably low intermodulation:
ϕi = ϕ1 − 2πi−1X(i − j)pj ,

(3.5)

j=1

where ϕi is the initial phase of the ith tone, and pi the fraction of the total signal power contained in thistone.51 This simple approach to choosing the phases allows us to synthesize our tweezer control signals inreal time without precomputing a time series (for more detail, see Appendix C.3).
50 Other heuristic approaches with low computational overhead could improve the performance further [340].
51 The intuition for this formula comes from the observation that the lowest peak factor signal corresponds to the case where
the signal is generated using only frequency modulation. By approximating a target signal with a frequency modulated signalthat simply jumps (in time) between the different frequency components of the target signal, and enforcing that the frequency
modulated signal is continuous, one arrives at Eq. 3.5.

48
For the experiments in Chs. 7 and 8, we are interested in stabilizing the total power in the tweezer
arrays (and lattices) using an additional AOM to modulate the laser power (see Fig. D.1). This stabilizationmust be done over a large dynamic range of ∼ 104 .52 To achieve this dynamic range in the presence of the
non-linear response of the tweezer system, we simply tolerate lower gain for deep traps that are closer tosaturation (the electronics used in the experiment are described in brief in Appendix C). However, upgrades
are underway to use a digital loop filter that provides greater flexibility than a typical analog loop filter,allowing one to calibrate out the non-linear response of the tweezer system electronically.53

3.3.3.2

Balancing large tweezer arrays

When making a square 2D array of tweezers, tweezers along the anti-diagonal of the array will be at
the same frequency, since the frequency shifts imposed by the two AODs relative to the mean frequency of
the array are equal in magnitude but opposite in sign.54 This results in static interference patterns leading
to uncorrectable modulations of the tweezer depths in large 2D arrays of tweezers. If the array is slightly
rectangular, tweezers along the anti-diagonal can be at difference frequencies that are relatively low and cancause significant heating, even if the frequency difference between directly adjacent tweezers is large.55 As
a result, we typically space the two axes of our array differently, with a spacing of 2 lattice sites in one
direction, and 3 in the other (where the lattice spacing is a ≃ 575 nm).56
In order to balance the depths of individual tweezers, we split off a small fraction of the light before
the objective, and measure the integrated intensity per tweezer using a CMOS camera. By adjusting therelative power in the different RF tones applied to the AODs, it is possible to balance the total optical powerin each spot to within peak to peak fluctuations of 5% for the rectangular pattern described above. The main
limitation on this balancing is a lack of fully independent control over each spot. For an Nx × Ny array, eachof the Nx + Ny RF tones has independent amplitude control.57 As a result, we only have Nx + Ny degrees
52 Note that when we are interested in applying a shallow tweezer, for example when applying the oracle in Sec. 7.5.4, we
can mitigate this difficulty by making additional deep tweezers that are far away from the region of interest, and stabilizing theintensity of the entire tweezer array [48].
53 Another option is to use an analog circuit that uses a logarithmic amplifier, as in [255].
54 Assuming the two AODs and surrounding relays are identical.
55 Avoiding this condition is important in the rearrangement procedure in Sec. 4.5.
56 Other spacings and patterns are also used extensively throughout this work. See Sec. 4.5 for a discussion of what patterns
are compatible with our current rearrangement procedure.
57 Although the phases can also be controlled, they are constrained by Eq. 3.5.

49
of freedom to balance a tweezer array with Nx × Ny spots. In comparison, a 1D array can be balanced tomuch better precision, with a peak to peak variation of ∼ 1%. While it is possible to balance the tweezer
depths using the atoms,58 we find the measurements using a camera to be sufficiently accurate (since thevariations in tweezer depth are currently dominated by a lack of independent control).

3.3.4

Optical lattices
The role of the optical lattice is to provide a static potential with many traps that are compatible

with high fidelity imaging, cooling, and clock spectroscopy. Additionally, the traps should be sufficiently
homogeneous to allow for experiments involving tunneling in large systems (Chs. 7 and 8). To fulfill theabove roles, the lattice operates at 813 nm, which is a magic wavelength for the clock transition, and wherea magic angle condition (Sec. 4.1) can be met for the narrow line cooling transition. However, to satisfy the
magic angle condition, the polarization of the lattice must be uniform and linear. Such a condition is notpossible using a retroreflected 3D lattice, since the polarization of a beam must be normal to its propagation
direction, and so not all three sets of orthogonal beams can share the same polarization. Instead, we use a
2D bowtie lattice in the plane of the tweezer arrays (the x-y plane), and a 1D crossed-beam lattice along
the remaining “axial” or z axis (Fig. 3.9).

3.3.4.1

2D lattice

The requirements on the 2D lattice are twofold: given fixed laser power, we want to generate manylattice sites that provide sufficiently strong confinement for high quality imaging and optical cooling. Additionally, we want our imaging system (with a waist of 440(20) nm) to be able to resolve individual sites in
the lattice.59

To balance the above considerations, we use a bowtie design (Fig. 3.9a) which increases thelattice spacing by a factor of√2, and the lattice depth by a factor of 8 in comparison to a retroreflectedlattice (using the same total amount of laser power). The resulting potential is given by the interference
58 E.g. via light shifts (Sec. 7.3) or measurements of the trap frequency (Sec. 4.3.3).
59 Note that our imaging resolution is potentially good enough to use a retroreflected lattice design at 813 nm, especially ifusing more sophisticated image analysis techniques like those in [149].

50

(a)

(b)

Figure 3.9: Optical lattice geometry and polarization. (a) Schematic of the 2D bowtie lattice, showing
folding mirrors (black lines) and 4f -relays (double arrows). ϕ1,2 denote the phase accrued when traversing
the indicated optical path, and the red contours denote an equipotential surface of the lattice, which ispolarized along the z axis. (b) Schematic of the crossed-beam axial lattice, showing the upper and lower
lattice beams, with polarizations ε̂u and ε̂l . Coloring denotes the polarization of the resulting standing
wave, which lies in the y-z plane at an angle θ from the z-axis. The two polarization components of thelattice beams result in two standing waves with orthogonal polarizations. For the shallow crossing angle of
36 degrees between the two beams used in our experiment, the depth Vz of the z-polarized standing wave
dominates. As a result atoms (blue disk) sit at anti-nodes of this lattice, and nodes of the y-polarized lattice.

51
between four plane waves:
V (x′ , y ′ ) = −2′′′
V0 ikx′e+ ei(ky +ϕ1 ) + ei(−ky +ϕ1 +ϕ2 ) + ei(−ky +2ϕ1 +ϕ2 )
16

(3.6)

where V0 is the maximum depth of the lattice, and k = 2π/813 nm. As shown in Fig. 3.9a, x′ and y ′ arecoordinates that are aligned with the propagation direction of the lattice beams, and rotated with respect
to x and y, and ϕ1, 2 are the phase shifts accrued in the folding arms of the lattice.60 When considering thetrapping frequencies that can be achieved in the lattice, the enhancement in depth more than compensates
for the reduced lattice constant, allowing for improved optical cooling and imaging (Ch. 4). Note that theshared paths of the four beams generating this lattice ensures that the intensity pattern is stable [296], but
this pattern can shift in position if ϕ1 or ϕ2 changes. However, the spatial phase of the lattice in the y ′direction (normal to the retroreflecting mirror) is stable with respect to the position of the retroreflecting
mirror (although the phase in the x′ direction is sensitive to the position of all the mirrors).61
The bowtie lattice is composed of a pair of 4f relays that are folded to cross at 90 degrees at their foci,where the beam waists are 55 µm (see Fig. 3.9a). To ensure that all arms of this lattice fully interfere, thepolarizations of these beams are oriented out of the plane of the lattice (in the z direction). With 540 mW
of input power, this yields average lattice depths of 416ER (where ER = h × 3.43 kHz is the recoil energy ofan 813 nm photon), and corresponding average radial trap frequencies of 98(1) kHz over a 26 × 26 µm region(corresponding to 45 × 45 sites).62

Representative values of the depth of the 2D lattice during different

phases of the experiment are summarized in Tab. 3.3.

3.3.4.2

Axial lattice

The axial lattice is composed of two parallel elliptical beams which are focused by an aspheric lens
(Edmund Optics 48-537)63 such that they cross at their foci, where the beam waists are 25 × 10 µm, withthe 25 µm axis oriented in the plane of the 2D lattice (along the y axis, see Fig. 3.9). These beams cross
60 Note that, in practice, we must also take the imperfect transmission of the lattice optics and cell into account, which
results in unbalanced optical powers in the different arms of the lattice, and slight deviations from Eq. 3.6.
61 The retroreflecting mirror could serve as a better reference for the optical clock laser than the current reference off of the
optical reference plate (Sec. 3.3.1) if the clock laser entered from this direction.
62 In more recent experiments, we use ∼ 1.5 W of optical power to achieve radial trap frequencies of ∼ 160 kHz.
63 The use of an aspheric lens is important in this application if one is interested in dynamically varying the spacing of theaxial lattice.

52

1ER
1EAx
R
1E2D
R
J0
Axial full
Axial tunneling
2D full
2D pre-tunneling
2D tunneling

ER

EAx
R

E2D
R

J0

kHz (E/h)

0.095
0.5
423
2.48
416
11.9
2.5

10.47
5.24
4430
26
-

2
0.19
0.095
846
4.97
832
23.7
5.0

21.05
2.01
10.53
52.3
249
52.6

3.43
0.328
1.72
0.163
1450
8.52
1430
40.7
8.58

Table 3.3: Representative values of the lattice depth during different phases of the experiment. Theseenergies E are provided as a frequency E/h, where h is Planck’s constant, as well as in units of the tunnelingenergy J0 (see Ch. 7), and the recoil energies of an 813 nm photon, and the axial and 2D lattices (ER ,2DEAx
R , and ER , respectively). Note that the provided values are representative values (specifically for theexperiments in [366]), but that the lattice depth, and value of the tunneling energy J0 , can differ for different
experiments performed in this work.

53
with an angle of 36 degrees, yielding a lattice constant of 1.32 µm. For these beams to interfere fully theirpolarization should be oriented in the x direction, normal to the plane in which they cross. However, thiswould result in a polarization that is orthogonal to that of the 2D lattice. Although it is possible to find an
approximate magic-angle magnetic field that is shared by both these lattice polarizations, in this situation
the atoms experience very large polarization gradients. For example, when an atom moves from a node toan antinode in either the 2D or axial lattices they experience a 90 degree rotation in the polarization of thelight field. This results in substantial broadening of the 1 S0 ↔ 3 P1 cooling transition, and thus reduced
cooling and imaging performance.
Instead, we orient the polarization of the axial lattice beams along the z axis before the final focusinglens. This results in two components of the lattice with orthogonal polarizations, and spatial phases thatdiffer by π (Fig. 3.9b). Because the crossing angle of our beams is well under 90 degrees, the axial latticeis predominantly z-polarized. As a result, atoms sit at the anti-nodes of the z-polarized lattice, and at
the nodes of the y-polarized lattice. As long as the atoms remain cold, their experience of the y-polarizedcomponent of the axial lattice is suppressed to first order in their position. In this regime the entire 3D
optical lattice, composed of the 2D bowtie lattice and the 1D crossed-beam lattice, is effectively z-polarized,
and a robust magic-angle magnetic field condition can be found. This enables high-fidelity resolved sideband
cooling and imaging (Sec. 4.3.3), with the primary tradeoff being a slightly reduced effective lattice depthof 90% in comparison to an x-polarized axial lattice. Nonetheless, with 660 mW of total optical power split
between the two arms of the axial lattice we achieve average lattice depths of 423ER , and average axial trap
frequencies of 41(1) kHz over a 26 × 26 µm region.64 The axial trap frequencies have a standard deviationof 7 kHz over this region due to the more tightly focused axial lattice beams in comparison to the 2D latticebeam.
One challenge with the crossed-beam configuration used in the axial lattice is that the relative phaseof the two beams must be carefully controlled to preserve the spatial phase of the lattice. To hand atoms
from the tweezers to the optical lattice, it is critical to ensure that the anti-node of the axial lattice lines up
with the foci of the optical tweezer arrays, as described in more detail in section 3.3.5. To ease this process
64 More recent experiments operate with ∼ 1.6 W of total optical power, and axial trap frequencies in the range of 65 kHz.

54
we ensure that the optical path length of two arms of the axial lattice are matched to within ∼ 100 µm ofeach other, which keeps the spatial phase of the axial lattice passively stable on the several-hour timescale.65
This is accomplished by sending both beams through the same optical elements, with the only exception
being an optical flat whose angle is controlled with a piezoelectric actuator in order to tune the relative phaseof the beams, and thus the spatial phase of the axial lattice.66 To additionally improve the passive stabilityof this system, optical components that are not shared by both beams are attached to a single monolithic
mount (which is visible as a silver rectangle on the left side of Fig. 3.1).

3.3.5

Alignment of optical potentials
Aligning the many optical potentials in the experiment is not a simple task. There are two tweezer

arrays, four passes of the 2D lattice beams, and two arms of axial lattice that all need to be overlappedin a volume that is ∼ 20 µm on a side, and intersect with this volume at the appropriate angles. On topof this, we want to minimize the number of degrees of freedom in the optomechanical components of theapparatus in order to improve the passive stability of optical alignment under normal operation. As a result,introducing the requisite degrees of freedom judiciously, and also ensuring access to fast, robust signals foralignment, is critical for the success of the experiment.
There are a few tricks that we use extensively when aligning almost any sensitive optical system.
When we want to save the alignment of a beam, for example when replacing an optical component, a single
coupled fiber serves as an excellent reference for both the position and angle of a beam. As described inSec. 3.3.2, we often use a tracer beam in combination with a retroreflecting mirror to establish a reference
for the optical axis of a given system. We also often use machined components and dowel pins to register
optical components, particularly components that are installed only temporarily during alignment (e.g.
retroreflecting mirrors). In systems where there are beams that do not follow the optical axis, like in theaxial lattice, it is often useful to have a permanently installed tracer beam that is on the optical axis toease alignment to the atoms. For tracer beams that are used to align to the atoms, one typically wants
65 We suspect that the remaining drift is due to drifts in the position of the final lens in the axial lattice as a result of thetemporary mount currently used (see Sec. 3.3.1).
66 Even in this case, it is important that similar optical flats are installed in both beams in order to match the optical path
lengths.

55
to choose a wavelength that is sufficiently close to the operating wavelength of the system being aligned to
avoid issues with chromatic aberrations. However, the chosen tracer wavelength must also have strong andobvious interactions with the atoms to provide a signal for alignment. Good choices for us are 461 nm and
688 nm, which can be used to deplete atoms from the MOTs, or to cause atoms to be lost (or appear to belost) between two images.67

3.3.5.1

Tweezer alignment

As mentioned in Sec. 3.3.2, atoms in the preparation tweezers serve as the reference point to which
all other optical potentials in our experiment are aligned. Aligning the 813 nm tweezers to the preparation
tweezers is straightforward because they share a common reference, namely the microscope objective used
to project them. This shared reference makes the alignment particularly robust, with negligible drifts on
the 0.5 µm scale over the course of multiple days. To perform this alignment, one can simply install
the retroreflecting mirror described in Sec. 3.3.2, and align the 813 nm tweezers to this reference. Fineradjustments (in particular of the collimation of the 813 nm tweezer system) can be performed by imaging
atoms trapped in the 813 nm tweezers.68

Final alignment of the two tweezer systems is performed byadjusting the RF control tones used to generate the 813 nm tweezers, optimizing for low atom loss andtemperature after passing atoms between the two tweezer arrays (see Sec. 3.4).

3.3.5.2

Lattices to tweezers

The lattices are aligned by first aligning a tracer beam to the atoms, and then aligning the relevant
lattice beam to the tracer.69 Final alignment of the lattice beams is performed by maximizing the light shift
induced by the lattice beams on the 1 S0 ↔ 3 P1 transition in 515 nm tweezer-trapped atoms (in magnetic
fields that are not magic for the lattice polarization).70

With the above optimization complete, one can

67 461 nm can be used to blow away atoms, and 688 nm can be used in combination with driving the 1 S ↔ 3 P transition0
1
to pump atoms into dark states (see Sec. 2.1.1.4).
68 This imaging can be destructive and low fidelity, since one is just interested in measuring the shape of the average
fluorescence signal.
69 In the case of the 2D lattice, we first align the first and second pass of the lattice beams, and then align the retroreflection
by maximizing the back-coupling into the fiber that supplies light to the lattice. If the alignment is very far off, one can alignthe back reflection before the first two passes are aligned, and look for a fluorescence signal from atoms trapped in the two
resulting 1D lattices.
70 If working with the 2D lattice it is useful to fully extinguish the tweezers during this measurement, before transferring
atoms back into the tweezers for readout, or performing readout directly in the lattice. This approach ensures that the atoms

56
verify the alignment by maximizing the observed motional trap frequencies provided by the lattices, howeverthese two procedures typically yield similar results.
One important consideration in our experiment is the possibility that the position of the lattice drifts
relative to the tweezer array, and to the masks used in image analysis (Sec. 4.4.1), which can occur on atimescale of ∼ 30 minutes. To correct this, we use the images taken before and after rearrangement to
identify the positions of the lattice-trapped atoms relative to the imaging system, and thus also relative tothe optical tweezers. This information can be used to correct any drifts by adjusting the tweezer and mask
positions on subsequent runs of the experiment. Because the repetition rate of the experiment is ∼ 1 Hz,
corrections can be made much faster than drifts can occur. With the above feedforward procedure engaged,
the RMS fluctuations in the lattice position relative to the tweezers is estimated to be ≲ a/10. The tweezer
positions can drift relative to the imaging system on a timescale of several hours, but this is readily correctedby intermittently taking images of atoms trapped in the tweezers and adjusting their positions to match thelattice.

3.3.5.3

Additional considerations for aligning the axial lattice

Alignment of the axial lattice is particularly challenging, since the foci of two separate lattice beams
need to be overlapped with the tweezer array in a 25 × 32 × 10 µm volume. While this can be accomplished
via the above procedure, the fringes of the resulting lattice must also be flattened relative to the plane of thetweezers while maintaining this overlap. Performing this flattening without misaligning the beam positions
is challenging without the appropriate controls. For this reason, we introduce an additional 4f relay to theaxial lattice before the final aspheric focusing lens. The relay allows us to install servo-actuated mirrors inboth a Fourier plane and an image plane of the tweezer array. The Fourier plane mirror can be used tocontrol a pure displacement of the lattice at the tweezer array, making it possible to compensate for drifts
in beam pointing without spoiling the flattening of the lattice. This mirror is back-side polished, such thatthe transmission can be imaged onto a camera for in-situ monitoring of the axial lattice. The image plane
mirror controls a pure tilt (rotation about x) of the lattice fringes, enabling flattening of the lattice in that
sit at the anti-nodes of the lattice, yielding consistent results, instead of being pulled away from the center of sites in the lattice
due to imperfect alignment of the tweezers.

57

(a)

(b)

(c)

Cts (arb.)

10

Flat

Skew
0

Rel. lattice phase (rad.)50
0.5 0.0

0.5

Lattice offset(periods)

Figure 3.10: Axial lattice alignment via parametric heating. (a) The upper panel shows a side view of amisaligned axial lattice (red fringes) overlapped with the tweezer array (black dots). The lower plots show
the tweezer potential (dashed curve) and the combined lattice and tweezer potential (solid red curve) at two
locations. When the lattice fringe is tilted relative to the plane of the tweezers, it shifts the centers of thetraps (as shown in the bottom left plot) in a position-dependent way. As a result, modulating the depthof the tweezers will cause different parametric heating rates at different sites in the array. (b) When thetweezers and lattice are well aligned, modulating the tweezers at the trap frequency does not cause substantial
heating. (c) Measurements of the spatial phase of the axial lattice (via position-dependent heating rates) ateach tweezer with an intentional tilt (skew) and properly aligned (flat) show that it is possible flatten thelattice relative to the entire tweezer array to within a tenth of a lattice period. In the left plot, each pixel
represents the phase measured at the location of a given tweezer. A histogram of these phases is provided
in the right plot.

58
direction.71

To control the roll (rotation about y) of the axial lattice, each lattice arm passes throughan independently controlled optical flat (the same ones used to control the relative phase of the beams in
Sec. 3.3.4.2). By controlling the angle of these optical flats, the lattice beams can be independently displacedin the Fourier plane.
To flatten the lattice fringes relative to the tweezers we parametrically heat atoms trapped in theaxial lattice with the tweezers. Specifically, in the case that a tweezer is well-centered on a lattice fringe,
modulating the power in the lattice changes the overall trap depth experienced by an atom, but not theposition of the trap center (Fig. 3.10b). This results in parametric heating with a resonance at ωm = 2ω0 ,where ω0 is the trap frequency in the absence of modulation [287, 154]. In this case, if the modulation
frequency is equal to the trap frequency, the atom is not strongly heated. If, on the other hand, the lattice
and tweezer are misaligned, modulating the power results in a shaking of the trap center, which can result inheating at ωm = ω0 (Fig. 3.10a) [287, 154]. For an appropriately chosen modulation amplitude and duration
at ωm = ω0 , we can use the observed probability of loss in a given tweezer as an indication of the relativealignment between that tweezer and the nearest lattice fringe. By scanning the phase of the lattice andfitting the phase of the resultant heating signal at each tweezer, we extract the spatial phase of the lattice atthe location of the tweezers (Fig. 3.10c). The above signal allows for alignment of the lattice to the tweezers
at the aax /10 level, where aax = 1.32 µm is the axial lattice spacing.
Once optimized, the flattening of the axial lattice has not been observed to change over multiple
months. However, as mentioned in Sec. 3.3.4.2, the spatial phase of the axial lattice drifts on the severalhour timescale. In practice the loading, cooling, imaging, and tunneling presented in this work are relatively
insensitive to drift of the axial lattice phase. As a result, this phase only needs to be adjusted when a nodeis close to the focal plane of the tweezers, where the transfer of atoms between the tweezers and the lattice
is bistable between two layers of the axial lattice, and can result in inconsistent loading and cooling.

59

Drop MOT
LACs
SB cool
Handoff
Rearrange

Blue MOT
SWAP Red MOT

ImageHigh fidelity SB cool
Adiabatic handoff
Image

...

Science

Preparation tweezers...

Handoff
Image
Lattice

V
50 ms
10 ms

50 ms

20 - 500 ms
20 - 100 ms
5 ms
30 ms*
5 ms
20 ms
10 ms

10 ms
25 - 60 ms

20 - 500 ms
5 ms

Figure 3.11: Typical experimental sequence. The steps marked in red are optional, depending on theexperiment being performed. See the text for a description of the listed operations. Note that certain steps(particularly imaging) can vary dramatically in duration depending on the specific experiment performed. In
the lower row, we show the relative depths (V , not to scale) of the lattice and preparation tweezer potentials
during a typical experiment involving clock measurements or tunneling in the lattice. However, many othervariations exist and are not depicted, including variations involving other tweezer potentials. This diagram
is intended only to provide a general feeling for the steps required in a typical experiment, and does not serve
as a source of accurate timing information for any given experiment. ∗ The rearrangement step currently
includes an additional technical delay of 110 ms, which we intend to address in future upgrades.

60

3.4

Standard operating procedures
Given the above tools, a typical experimental sequence involves the following steps (see Fig. 3.11):

First, atoms are captured from the atom beam (Sec. 3.1.2) into a blue MOT, and then transferred into a
SWAP MOT, followed by a narrow line red MOT (Sec. 3.1.4). The red MOT is overlapped with the preparation tweezers such that some of the atoms are trapped in the tweezers, and then the MOT is extinguished.72
We typically wait for the MOT to dissipate (“dropping” the MOT) before proceeding. Finally, light assisted
collisions (LACs) are used to ensure that each preparation tweezer contains exactly 0 or 1 atoms (Sec. 4.2).
Depending on the parameters of the light assisted collisions step, between 45% and 75% of the preparation
tweezers contain an atom at this stage.
The tweezer-trapped atoms are then sideband cooled. However, this cooling does not yet have to beparticularly high fidelity, since the aim is simply to shrink the atomic wavepacket to be substantially smaller
than a single site in the lattice.73

After cooling, the tweezer-trapped atoms are transfered into the lattice

via a sequence of ramps in trap depth, each 10 ms long.74

The axial lattice is ramped up first, followedby the 2D lattice, and finally the preparation tweezers are ramped down. This handoff can be performed
with 0.0+3−0 % atom loss. Once the tweezers are ramped low we fully shutter the tweezer beam, and thenramp the tweezers back up to full power. This allows us to avoid thermal lensing effects that can cause
pointing fluctuations in the tweezers, while also avoiding unwanted light shifts introduced by the tweezers.
Once the atoms are in the optical lattice, we switch from a magic magnetic field condition for the 515 nm
tweezers [241] to a magic field condition for the 813 nm optical lattices [240] and perform resolved sideband
cooling. The above preparation sequence typically takes about 200 ms.
At this stage, we take an initial image of the atoms in the lattice (see Sec. 4.4). In this image, theatoms can only occupy the subset of lattice sites which were overlapped with the preparation tweezers. Given
71 Note that one must avoid scatterers in an image plane of the atoms, since this could lead to disorder in the potential
projected onto the atoms. For this reason, we are very careful to protect the image plane mirror from dust.
72 The tweezers provide a convenient means of isolating a single 2D layer of the 3D lattice. We have, however, performed
proof of principle experiments loading directly into lattice, which enables higher atom numbers. However, some upgrades are
required to isolate a single layer of the 3D lattice when loading the lattice directly from the MOTs.
73 The procedure when working with the 813 nm tweezers as opposed to the lattice is similar, although we will comment onsome slight differences below.
74 The duration of these ramps is likely longer than necessary, especially given how quickly this transfer can happen during
rearrangement (Sec. 4.5). However, we opt to be conservative because the handoff step doesn’t result in a significant increasein the duration of the experiment, and because the atoms are not necessarily well-cooled at this stage.

61
that the preparation tweezers are spaced by more than a lattice spacing, good imaging performance can beachieved with a relatively short imaging duration of ∼ 75 ms. If rearrangement is being used, we computethe required moves based on this image, and program the RF control system in ∼ 5 ms,75 execute the movesin ∼ 30 ms (Sec. 4.5), and take an additional image to verify that the rearrangement has been performed
correctly. If the target pattern of atoms is relatively sparse, the same ∼ 75 ms image can be used. If thetarget pattern includes atoms at nearest-neighbor spacing (meaning atoms occupying adjacent lattice sites),then image durations of between 200 − 500 ms are used. In all experiments, we then perform high fidelity
resolved sideband cooling (typically using a pulsed cooling scheme as described in Sec. 4.3.1) to bring theatoms to their 3D motional ground states in the lattice (Sec. 4.3.3). The result of the above sequence is the
preparation of a (nearly, see Ch. 8) pure state corresponding to a known pattern of atoms, with each atom
being in the electronic ground state, and in the motional ground state of the lattice site that they occupy.
This is the starting point of almost all experiments described in this work.
The atoms then undergo evolution of some sort, which can involve manipulations of their electronic
state in the space spanned by the ground and clock states (Ch. 5), as well as a Rydberg state (Ch. 6), or
coherent manipulations of the positions of the atoms in the lattice (Chs. 7 and 8). After this evolution,
the positions and states of the atoms are read out, with an optional additional round of sideband coolingdepending on if any heating was expected to occur between the previous round of cooling and readout.
Depending on the experiment, readout can either be a single image or a pair of images, and possibly involves
additional control pulses such that the measurement is sensitive to the electronic state of the atoms in
addition to their position in the lattice (Sec. 5.2). As before, the duration of these final images varies
between 75 − 500 ms depending on the expected atom density. Ignoring the fact that the evolution step canhave durations between microseconds and hundreds of seconds, the preparation and readout phases of theexperiment typically vary between 250 − 1300 ms.
For experiments that use only 813 nm tweezer arrays and not the lattice the experimental sequenceis similar, with the main difference being that the images and final sideband cooling are instead performedin the 515 nm tweezers, resulting in worse performance (Secs. 4.3.3 and 4.4). However, we find the handoff
75 There is an additional 30 ms delay when reading the image off of the camera. There is also currently an additional 50 ms

delay before the moves are executed due to a computer issue, but we expect this problem to be readily solvable.

62
performance between tweezers to be similar to the performance between tweezers and the lattice.

Chapter 4

Preparing and detecting single atoms“I’ll get everything else working while you complete this first simple test, which involves
deadly lasers, and how test subjects react when locked in a room with deadly lasers.”— GLaDOS
Portions of this chapter have appeared in:
M. A. Norcia, A. W. Young, and A. M. Kaufman. Microscopic control and detection of ultracold
strontium in optical-tweezer arrays. Phys. Rev. X, 8(4):041054, Dec. 2018
A. W. Young, W. J. Eckner, W. R. Milner, D. Kedar, M. A. Norcia, E. Oelker, N. Schine, J. Ye, andA. M. Kaufman. Half-minute-scale atomic coherence and high relative stability in a tweezer clock.
Nature, 588(7838):408–413, Dec. 2020
A. W. Young, S. Geller, W. J. Eckner, N. Schine, S. Glancy, E. Knill, and A. M. Kaufman. An
atomic boson sampler. In review, 2023

This chapter is about state preparation (and by extension, detection). Given the tools described in
Chs. 2 and 3, how do we control all degrees of freedom of an atom plucked from a thermal gas? For bosonic Sr
atoms, the internal degrees of freedom are well-controlled since the electronic and nuclear ground states areboth unique and separated in energy from any excited states by far more than the available thermal energy
at room temperature (see Sec. 2.1.1). However, the motional states of the atoms, namely their positions andmomenta, are not necessarily controlled. To tame the motion of the atoms, we place them in arrays of tightly
confining optical traps (either tweezer arrays or optical lattices, as described in Sec. 3.3). This discretizes
the available motional states into states labelled by the index of the specific trap an atom occupies, and theatom’s motional state within that trap.1 In Sec. 4.1, we will describe how our traps are engineered to affectthe motion of the atoms without affecting relevant features of their internal degrees of freedom. In Sec. 4.2,
we will describe how single atoms are loaded into these traps. In Sec. 4.3, we will describe how the motional
1 We will discuss more sophisticated manipulations of the atomic motion in Chs. 7 and 8.

64
Magic Polarization
Angle (degrees)

(b)

1

(c)

1.00
30
0.75
20
0.50
20
0.25
0.00
30
513
0.0

S0 to |0B Frequency
Shift (MHz)(a)

0.2 5140.4
0.6515 0.8
Trap Wavelength (nm)

516
1.0

0.25
0.50
0.75
Tweezer Depth (mK)

1.00

1
0
1
0.00

Figure 4.1: Magic angle trapping conditions. (a) By setting the angle θ between an applied magnetic field
B = B ẑ and the polarization of the trapping light E (in this case a tweezer), we can achieve equal trap depths
between the ground state (1 S0 ) and an optically excited state (3 P1 ). In the absence of a magnetic field, thetweezer light causes energy shifts to the different spin projections of 3 P1 (where q̂ is the quantization axis
along which we have defined our states). These shifts are of opposite sign relative to the shift of the groundstate (dashed lines denote the shifted ground state offset by the unperturbed transition frequency). At the“magic” polarization angle θ, the shifts cancel for an eigenstate |0B ⟩, leading to state-insensitive trapping.
(b) Magic polarization angle as a function of tweezer wavelength, showing that a magic condition exists overa range of wavelengths. (c) Shifts of the 1 S0 to |0B ⟩ transition as a function of tweezer depth for a large(22 G) field oriented at θ = 0 (gray squares), a large (24 G) field oriented at the magic angle (black points),
and a small (7 G) field oriented at the magic angle (empty circles). All data in (c) is at a tweezer wavelengthof 515.13 nm. Grey and black lines are linear fits.

state of an atom within a given trap can be controlled optically. In Sec. 4.4, we will describe how one cancollect spontaneously emitted photons from optically cooled atoms to image the atoms with high fidelity
and low loss, and thus identify which trap they occupy. Finally, in Sec. 4.5, we will describe how theseimages can be combined with rearrangement using mobile optical tweezers to control the specific locationsof the prepared atoms. Taken collectively, the above tools allow us to control all degrees of freedom of theatoms. Later, in Sec. 8.3, we will see what this control means in a deeper sense, and conduct experimentsthat unambiguously demonstrate that, with high probability, the universe is unable to distinguish between
different atoms prepared using the prescription in this chapter.

65

4.1

Magic angle trapping
In Sec. 2.2.3, we discussed how the appropriate choice of laser wavelength can allow one to engineer

optical traps that decouple a subspace spanned by two electronic states of an atom from the motion of theatom within the trap. However, one can additionally tune the properties of the trap using the polarization(and shape [353]) of the trap. This additional tunability can be used to achieve magic-like conditions at
wavelengths where convenient laser sources exist, or to decouple a larger electronic subspace from the atomic
motion.
In our work, we use linearly polarized optical traps, and control the angle of the polarization relative
to a quantization axis set by a magnetic field [357, 241]. This approach is particularly robust because one
can take advantage of high quality polarizing optics, and, with the appropriate alignment of the polarization
relative to the optical system used to project the traps, can often avoid uncontrolled modifications to thelaser polarization due to birefringence. Similar tunability can be achieved using elliptically polarized opticaltraps [73], but this approach requires more careful consideration of how the various optics in the trapping
system affect the polarization of the trapping field.
Consider the situation where we are interested in attaining a magic condition where the polarizabilityof the 1 S0 and 3 P1 states are equal. We will apply a magnetic field B in the z direction,2 and a linearlypolarized oscillating electric field with field strength E and with a polarization vector (see Eq. 2.9) of:ε̂ = cos(θ)ẑ + sin(θ)x̂,(4.1)

where x̂ and ẑ are unit vectors along the x and z axes respectively, and θ is the angle of the electric fieldfrom the z axis. The resulting Hamiltonian for the atom is:
H = Ha + Hi + HB ,(4.2)

where Ha is the Hamiltonian for the bare atom (e.g. Eq. 2.12 for a two level atom), Hi describes the effectof the electric field on the atom (Eq. 2.11), and HB the effect of the magnetic field on the atom (Eq. 2.1).
When the coupling terms in Hi and HB are similar in magnitude, we perform exact diagonalization of Eq. 4.2
2 With z as shown in Fig. 4.1, and not along the optical axis of the tweezers as in Ch. 3.

66
numerically [241]. However, in our experiment, B is typically large enough (∼ 20 G, compared to trap depthsof ∼ 0.1 mK) that the eigenstates in the 3 P1 manifold remain approximate eigenstates of the Jz operator
with mj = −1, 0, 1. Therefore, we can proceed by computing the AC Stark shifts on these states.
Note that ε̂ is real, and so the contribution from the vector polarizability vanishes (see Eq. 2.24). The
1

S0 state has angular momentum J = 0, and so the tensor polarizability also vanishes, resulting in an AC

Stark shift of:∆E(1 S0 ) = −α(0) (1 S0 )

E2
,
4

(4.3)

where the scalar polarizability α(0) is as defined in Eq. 2.21.
The tensor contribution to the AC Stark shift on the 3 P1 states does not vanish. Plugging Eq. 4.1
into Eq. 2.24 results in a total shift of:




 E2
3 cos2 (θ) − 1,
3m2j − 2∆E(3 P1 , mj ) = − α(0) (3 P1 ) + α(2) (3 P1 )
2
4

(4.4)

where the tensor polarizability α(2) is as defined in Eq. 2.23. We find that the effective polarizabilities of
the 3 P1 states can be tuned with the value of θ. For example, for the magnetically insensitive mj = 0
state, one can tune the polarizability between α(0) (3 P1 ) + α(2) (3 P1 ) and α(0) (3 P1 ) − 2α(2) (3 P1 ). As long asα(0) (1 S0 ) falls within the above range for a given trapping wavelength, one can tune to a magic condition
were ∆E(3 P1 , mj = 0) = ∆E(1 S0 ), independent of the value of E.
In Fig. 4.1b, we perform spectroscopy of the 1 S0 ↔ 3 P1 transition, and, for different wavelengths ofthe tweezer potential, identify the angle at which this transition can be driven at its free-space transition
frequency. Notice that a magic condition can be reached for a range of wavelengths near 515 nm. In Fig. 4.1c,
we show how operation at the magic angle (for a trapping wavelength of 515.13 nm) leads to insensitivity ofthe transition frequency to trap depth.
We use the above approach to achieve magic trapping conditions for the 1 S0 ↔ 3 P1 transition bothfor a trapping wavelength of 813 nm, and for wavelengths near 515 nm. The former case corresponds to themagic wavelength for the 1 S0 ↔ 3 P0 clock transition, resulting in simultaneous magic conditions for the 1 S0 ,3P0 , and (one of the) 3 P1 states.

67
In the 3D optical lattice (see Sec. 3.3), an additional complication is that the polarization of thedifferent lattice potentials must be aligned. We achieve this by setting the fields to the magic angle withrespect to the 2D lattice in a similar fashion to the above measurements in tweezers, and then vary thepolarization of the axial lattice to minimize the shift of the 1 S0 ↔ 3 P1 transition as a function of axial lattice
depth.

4.2

Preparing single atoms
To load individual atoms into the optical traps described in the previous section, we overlap the traps

with the narrow-line MOT described in Sec. 3.1.4. The traps are typically in the range of 100 − 1000 µK
deep, far exceeding the thermal energy of the atoms in the MOT of ∼ 5 µK. Although the MOT conditions
do not correspond to the magic conditions in Sec. 4.1, residual Doppler or sideband cooling of the atoms
in the presence of the AC Stark shifts imposed by the traps is typically still sufficient to reduce the atom
temperature to well below the trap depth. The result is a Poisson distribution of atoms in each trap, witha mean of N ∼ 1 − 2 atoms per trap for typical experimental parameters.
To prepare exactly one atom in each trap, we take advantage of light assisted collisions [292]. Schematically, the mechanism for these collisions is the presence of a deep molecular potential between electronically
excited atoms.3

When one drives one or more atoms to an electronically excited state with a laser, pairsof atoms roll down this molecular potential, gaining a significant amount of kinetic energy. Eventually, theatoms decay back to the electronic ground state, effectively turning off the molecular potential. The kinetic
energy gained by the atoms during this process typically far exceeds the depth of the traps,4 and so atoms
exit the trap in a pairwise fashion [292]. The result is that traps initially filled with an even number of atoms
are left with no atoms, whereas traps initially filled with an odd number of atoms are left with a single
trapped atom. Given the above behavior, the use of light assisted collisions to map even and odd atom
numbers to 0 and 1 atoms is often referred to as “parity projection.” For large initial numbers of atoms, 50%of the traps are occupied with exactly one atom after parity projection, and the rest with zero. For the atom
3 For example, a dipole-dipole interaction between atoms in superpositions of S and P states, with a characteristic scalingof r−3 , where r is the interatomic separation.
4 Often, these collisions are excited by a red-detuned optical beam, as one might use for optical cooling. This excites theatoms to attractive molecular states with a binding energy that is almost always far in excess of reasonable trap depths.

68
numbers we typically load into each trap, the fraction of sites containing one atom is between 45% and 50%.
Note that by carefully controlling the collision channel one excites, it is possible to arrange for collisions
that are comparable in energy to the trap depth, and thus typically only remove a single atom at a time [134,189, 111, 48].5

Using these techniques, single atom loading efficiencies as high as 96(1.4)% have beenachieved in a single tweezer (and 92.78(8)% in an array of 100 tweezers) [155]. In Sr, we have used similar
techniques to achieve loading efficiencies as high as 75% in an array of 384 tweezers [367], but we leave adetailed discussion of these results for future works.

4.3

Controlling atomic motion
We are interested in controlling the motion of a trapped atom with a laser field. So far, we haveconsidered how far off-resonant laser light can generate conservative trapping potentials with different properties, but in Eq. 2.11 we neglected the fact that E can vary over the spatial extent of the atom. Including
the spatially varying phase of the electric field in Eq. 2.13 leads to [356, 185]:6Ω
Hi = ℏ (σ+ ei(k·r−∆t) + σ− e−i(k·r−∆t) ),
2

(4.5)

where r is the position operator for the atom, σ+ := |e⟩⟨g| and σ− := |g⟩⟨e| are the spin raising and lowering
operators, t is the time, ∆ is the detuning defined below Eq. 2.13, and Ω is the Rabi frequency as definedin Eq. 2.14. One can interpret Eq. 4.5 as describing the process where absorbing or emitting a photon in agiven direction results in a corresponding momentum kick in that direction.
Assuming that the trapping potential is harmonic, the Hamiltonian describing the motion of the atom
is:7
1
Hm = ℏωm (a† a + ),
2

(4.6)

where ωm is the angular frequency associated with the harmonic oscillator, and a† and a are the harmonic
5 The central trick in these “enhanced loading” techniques is to use a blue-detuned beam which excites the atoms to repulsive
molecular states. In this case, one can precisely control how far up the potential the atoms are excited to relative to the freespace energy, and thus the amount of kinetic energy gained in the collision process.
6 Keep in mind that this is in the rotating frame of the laser, and after taking the rotating wave approximation.
7 For now, we will consider the simplified scenario where the atomic motion is constrained to the propagation axis of thecontrol field, x, but this discussion is readily extended to the three-dimensional case [185]. In our experiments, we typicallydrive three orthogonal axes of a 3D trapping potential with three separate control beams, and so one can just duplicate theequations in this section for each axis.

69

Figure 4.2: Electronic transitions in a harmonic trap. The states of a harmonically trapped atom can belabelled by the internal state of the atom (e.g. the states 1 S0 and 3 P1 in this example), and the quantum
number n describing the harmonic motion of the atom. In the Lamb-Dicke regime, spontaneous emission of aphoton is unlikely to change the motional state of the atom (grey squiggly lines). However, if motional-statechanging processes (red and blue lines) are well-resolved from processes which do not change the motional
state (grey line), then one can resonantly drive these processes with a laser. Driving the red process will leadto an overall reduction of the motional state of the atom, until the atom is in the state |1 S0 , n = 0⟩. Thereis no state that differs in energy from |1 S0 , n = 0⟩ by the energy of a photon in the field of the cooling laser(red X), and so at this point the atom goes dark to the cooling light.

70
raising and lowering operators. In this case, x = x0 (a† + a),8 where x0 :=pℏ/(2mωm ) is the harmonic

oscillator length (with m being the atomic mass). Given the above definitions, we can rewrite Eq. 4.5 as:†Ω
Hi = ℏ (σ+ eiη(a +a)−i∆t + H.C.)
2

(4.7)

where we have defined the Lamb-Dicke parameter η := kx0 .
The states of the atom can be written as |β, n⟩ := |β⟩ ⊗ |n⟩, where β ∈ {e, g} refers to the electronic
state of the atom, and n is the quantum number associated with the harmonic motion of the atom.9One

can compute the coupling induced by Hi between states |g, n⟩ and |e, n + s⟩ exactly [354, 185]:†Ωn,n+s := Ω|⟨g, n | eiη(a +a) | e, n + s⟩|sn< ! (|s|) 2−η 2 /2 |s|
L
(η ),
= Ωe
η
n > ! n<

(4.8)

(α)

where Ln (·) is the generalized Laguerre polynomial, n> = n + s and n< = n for s ≥ 0, and n> = n andn< = n + s for s < 0. Note that the states |g, n⟩ and |e, n + s⟩ differ in energy by ℏ(ωeg + sωm ), and so forΩ ≪ ωm typically only a single transition is driven at a time for a given detuning of the control field.
When the fluctuations in the atom position are small in comparison to the wavelength of the electromagnetic field, namely when ηp⟨(a† + a)2 ⟩ ≪ 1, one is in so-called the Lamb-Dicke regime. In this regime,

we can simplify the description of the dynamics by Taylor expanding Eq. 4.7 about η = 0, resulting in:ΩHi (t) = ℏ σ+ e−i∆t (1 + iηa† eiωm t + iηae−iωm t + . . . ) + H.C.
2

(4.9)

Here, ∆ is the angular frequency associated with the detuning from resonance with the electronic transition(as in Sec. 2.2.1), and we have kept the explicit time dependence of the spin- and motion-raising and lowering
operators in the rotating frame of the atom. It becomes clear that we can address a specific motion-changing
transition by picking the appropriate ∆, and perform a second round of the rotating wave approximation toremove all other off-resonant processes. The processes which are resonant at ∆ = ±ωm are known as thefirst blue and red sidebands respectively, and drive the transitions |g, n⟩ ↔ |e, n ± 1⟩. Similarly, one candrive the |s|th -order sidebands, corresponding to higher order terms in the expansion with respect to η, bypicking detunings of ±|s|ωm .
8 We have taken r → x, see previous footnote.
9 These states generically form a complete basis, and are precisely the eigenstates of the atoms for magic trapping conditions.

71
4.3.1

Sideband cooling
When combined with dissipation due to the finite decay rate γ of the |e⟩ state, driving the red

sideband(s) provides a powerful mechanism for cooling. Specifically, in the resolved sideband regime whereγ, Ω ≪ ωm , the dynamics associated with relaxation and the electromagnetic drive are slow in comparison
to the motional dynamics, and so one can spectroscopically resolve the separate resonances appearing in
Eq. 4.9.10 In this case, starting in the state |g, n⟩, one can coherently drive the atom to |e, n − 1⟩ and allow
it to spontaneously decay to |g, n − 1⟩. This procedure can be repeated until n = 0, at which point there are
no available states with lower n, and so the atom becomes dark to the drive light (see Fig. 4.2).
The evolution of the atom during sideband cooling is described by the master equation [226]:∂ρi= − [Ha + Hm + Hi , ρ] + L(ρ),∂tℏ(4.10)

where Ha , Hm , and Hi are as defined in Eqs. 2.13, 4.6, and 4.7 respectively, [· , ·] is the commutator, ρ is thedensity matrix describing the state of the atom, and L is a Liouvillian describing the damping of the atom.
To define L, we must consider how spontaneous emission of a photon affects the motion of the atom. Whenthe atom decays, it emits a photon into an emission pattern described by the angular distribution f (θ, ϕ).
In our works, we typically use transitions between states with mj = 0, where [318]:f (θ, ϕ) =3
sin2 (θ).
8π(4.11)

Emission of a photon in a direction given by the unit vector k̂θϕ , where θ and ϕ are the polar and azimuthal
angles of the vector, is described by the jump operator [185]:Lθϕ = σ− eikk̂θϕ ·r ,(4.12)

where k is the wavenumber associated with the |e⟩ → |g⟩ transition. The Liouvillian L is given by the
combination of all such decay processes, weighted by f [185, 208]:
L(ρ) = γZdΩf (θ, ϕ)Lθϕ ρL†θϕ −
1{σ+ σ− , ρ}2

(4.13)

10 Note that there are other cooling mechanisms that take advantage of transitions with small γ to reach very low temperatures.
One particularly useful example is a kind of Sisyphus cooling [73, 74] which we use in some of our experiments [240]. We opt
not to discuss this cooling mechanism in more detail here, since most of our experiments use sideband cooling to reach lower
temperatures at the cost of lower cooling rates. For a nice description of Sisyphus cooling in alkaline earth atoms, see [208].

72
where {· , ·} is the anticommutator, andRdΩ denotes an integral over all solid angles. Note that for theterm involving the anticommutator, we have taken advantage of the fact that the ordering of the operators
is such that the angle-dependent terms in the integral cancel. For approximate calculations, it is often
enough to reduce the integration over f to a summation over three appropriately weighted decay processes
corresponding to photon recoil along the three orthogonal axes of the trap.
With the above evolution in mind, the primary limitation to the temperature reached by sideband
cooling is the competition between cooling via driving the red sideband and heating due to off-resonant
driving of other processes.11 Given these mechanisms, one can arrive at an approximate, analytical solution
for the ultimate performance of resolved sideband cooling of ⟨n⟩ ≃ (γ/ωm )2 [185, 101]. The performanceof this cooling can be modified by a constant factor for different geometries and emission patterns. For
example, in [208] they numerically arrive at ⟨n⟩ ≃ 0.17(γ/ωm )2 .
Practically, one does not always saturate the above cooling limit due to geometric constraints, andadditional sources of technical heating. Noise on the drive laser can result in a spectral feature that is wider
than the expectation given γ, leading to an enhanced rate at which the various off-resonant processes are
driven. As described in Appendix D, we go to great pains to ensure that the linewidth of the drive laser iscomparable to the 7.5 kHz wide 1 S0 ↔ 3 P1 transition we use for sideband cooling. Scattering of the trapping
light can also result in recoil heating, but typically these rates are very low in comparison to the coolingrates. Additionally, modulations of the trapping potential (either due to intensity or pointing fluctuations)
can lead to parametric heating. Again, these processes are normally slow in comparison to the coolingrate, and are only relevant in specific situations. We will discuss these parametric heating mechanisms in
more detail in Sec. 5.3 in the context of experiments that are performed on very long timescales (of tens tohundreds of seconds). Parametric heating is also relevant in rearrangement, as described in Sec. 4.5, where
tweezer-trapped atoms traversing a lattice potential experience a significant modulation of the trap depth.
In our experiments, we typically utilized a pulsed cooling sequence, where three different axes arecooled sequentially (see Fig. 4.3). This allows us to tune the relative cooling performance along different
11 Namely, off resonant driving of the carrier followed by decay on the first red sideband, and driving of the first blue sideband

followed by decay on the carrier.

73

100
Axial
Radial 1
Radial 2
n

10 1

10 2

0

10

20

30

40

50

60

Time (ms)Figure 4.3: Simulated sideband cooling performance. To optically cool lattice-trapped atoms with highfidelity, we typically use a pulsed cooling sequence involving 0.4 ms axial cooling pulses, and 0.2 ms radialcooling pulses (timing diagram pictured in lower panel). We compute the expected average thermal occupation n̄ as a function of time in each of three nearly-orthogonal axes of a given site in the lattice via a
master equation calculation, yielding reasonable agreement with measured values in the experiment. Notethat in the depicted cooling sequence, we optimize for high-fidelity cooling of the axial direction at the costof slightly worse cooling in the radial directions.

74
trap axes12 without constraining the amount of optical power used to cool each axis. The pulsed procedure
additionally minimizes the effect of coherent processes where a photon is absorbed from one cooling beam,and undergoes stimulated emission into another beam.
A simulation using Eq. 4.10 of a typical pulsed cooling sequence, specifically the one used for coolingin a 3D optical lattice (with typical trap parameters listed in Tab. 3.3), as is used in the experiments in Ch. 8,
is shown in Fig. 4.3. Here, the cooling is composed of 120 pulses which alternate between cooling pulses
with a duration of 200 µs on two nearly-orthogonal “radial” axes in the plane of the 2D lattice separated by
400 µs cooling pulses on the “axial” out of plane axis. Each cooling pulse is separated by a delay of 200 µs
in order to allow the atoms to completely relax to the electronic ground state between pulses, leading to an
overall cooling sequence that is 60 ms in duration. The described cooling sequence is expected to result in anaxial motional ground state occupation of 99.58 %, and a combined ground state occupation of 97.37 % in
the radial directions, leading to a 3D motional ground state occupation of 96.97 %. This sequence is biased
towards high-fidelity cooling of the less strongly-confining axial direction at the cost of slightly worse radialcooling (by controlling the durations of the cooling pulses along different axes).13

The ultimate observed

cooling performance in our experiment is consistent with this simulation, as we will discuss in more detailin Secs. 4.3.3 and 8.3.

4.3.2

Sideband thermometry
There are many ways to characterize the final temperature of the atoms after cooling, including

release and recapture techniques [323, 73], and the interference techniques described in Ch. 8. However,one particularly simple and effective technique is known as sideband thermometry, where one performs
spectroscopy that can resolve the motional sidebands, and compares the heights of the first red and blue
sidebands.
Consider the evolution of a thermal state under a drive field on the red or blue sidebands. The initial
12 This capability is important, for example, when optimizing for high quality cooling only along the direction of a given

control beam as in Sec. 5.2, or when some thermal excitations can be filtered out using postselection as in Sec. 8.3.
13 In this specific case, we choose to make this trade-off because the axial cooling performance is more important than theradial for the experiments described in Ch. 8.

75state of the atom is described by the density matrix [185]:14
ρ(t = 0) =

∞
X

⟨n⟩n
|g, n⟩⟨g, n|.
(⟨n⟩ + 1)n+1
n=0(4.14)

Under the influence of a drive on the red sideband with duration t, the probability of observing the atom inthe |e⟩ state is:Pe(r) (t) :=∞X⟨e, n | ρ(t) | e, n⟩ =

n=0

∞
X

⟨n⟩n
sin2 (Ωn−1,n t),
n+1
(⟨n⟩
+
1)
n=1

(4.15)and similarly for the blue sideband:
Pe(b) (t) :=∞X⟨e, n | ρ(t) | e, n⟩ =

n=0

∞
X

⟨n⟩n
sin2 (Ωn+1,n t).
n+1
(⟨n⟩
+
1)
n=0

(4.16)

Shifting the summation index, and keeping in mind that Ωn,n+s = Ωn+s,n , we find that:
(r)

Pe

(b)
Pe
(r)

(b)

and so one can measure the ratio Pe /Pe

=

⟨n⟩
,
⟨n⟩ + 1

(4.17)

to extract the quantity ⟨n⟩. This measurement of ⟨n⟩ is particu-

larly robust, since Eq. 4.17 is independent of all other relevant parameters in the dynamics, including η, t, Ω,and ωm . However, when characterizing cooling along multiple axes, one must be careful that the dynamics
that occur during sideband thermometry do not perturb the temperature of the atoms.15

In practice, we

simply simulate the thermometry sequence via Eq. 4.10, and confirm that it does not significantly perturb
the atomic motion [241].
In our experiments, the procedure for sideband thermometry involves preparing the cooled atoms,
driving the 1 S0 ↔ 3 P1 transition with variable detuning ∆ and Ω ∼ 2π × 10 kHz for a duration of ∼ 100 µs.
This duration is chosen because it is long in comparison to the decay time of 3 P1 , allowing the populations
to stabilize,16

but short compared to the cooling duration, preventing the perturbations to the atomic

temperature described above. A ≃ 3 µs blow-away pulse of light resonant with the 1 S0 ↔ 1 P1 transition isused to rapidly heat and eject the atoms in 1 S0 from the traps before the 3 P1 atoms decay back to 1 S0 (see14 Note that this is often a somewhat crude approximation, since the distribution prepared by sideband cooling is notnecessarily thermal.
15 During the probe pulse, one can heat the probed axis on the blue sideband, and cool it on the red sideband, leading to agreater asymmetry in the resulting distribution, and thus artificially good inferred performance. This is particularly problematicwhen there are tradeoffs associated with cooling multiple axes.
16 Although Eq. 4.17 is not dependent on extraneous experimental parameters, it can be a problem if these parameters
fluctuate over multiple experimental trails such that the conditions under which the red and blue sidebands are measured differ.

76Sec. 2.1.1.2). Typically, the traps are extinguished for the final 1 µs of the blow-away pulse before being
turned back on, which aids in removing the heated 1 S0 atoms, but does not result in substantial loss of the3

P1 atoms. The remaining 3 P1 atoms are allowed to decay back to 1 S0 , before being imaged (see Sec. 4.4).

By taking images before and after performing the above sequence, the number of surviving atoms can beconverted into a probability of the atoms being in 3 P1 . To minimize sensitivity to imperfect calibration of themagic angle, and fluctuations in trap frequency, we typically perform the above procedure at many values ofthe detuning (not just at the location of the first red and blue sidebands), and perform maximum likelihood
estimation to fit the resulting line shape with three Lorentzians — one each for the carrier, and the red andblue sidebands. ⟨n⟩ is then inferred via Eq. 4.17 from the fitted heights of the sidebands. Note that one must
be careful when inferring ωm from the separation of the red and blue sidebands from the carrier. When Ωapproaches ωm , probing near a given sideband also results in a probe-induced AC Stark shift (see Sec. 2.2.1)of the entire spectroscopic feature including the carrier and sidebands. The result is that, when scanning
across the carrier, the red and blue sidebands appear to shift towards the carrier, resulting in artificially low
inferred values of ωm .

4.3.3

Cooling results
We perform resolved sideband cooling and characterize the resulting cooling performance using side-band thermometry in a few different conditions, which are summarized as appropriate in the works listed inSec. 1.4. Here, we will focus on two conditions that are particularly relevant to our later works:(1) Where we cool the atoms in tightly confining 515 nm tweezers and a 1D 515 nm lattice oriented
along the weakly confined axis of the tweezers (collectively referred to as the preparation potential),
and then hand the atoms into weakly confining 813 nm tweezers (science potential)(2) Where we cool the atoms directly in a tightly confining 3D optical lattice at 813 nm.
In both cases, the trapping potentials are tuned to a magic condition for the 1 S0 ↔ 3 P1 transition as
described in Sec. 4.1.
In condition (1), which applies to some of the experiments described in Ch. 5, we are interested in

77

(b)
P1 population

0.3
0.2

0.2
0.1

3

0.1

3

P1 population

(a)

0.0

0.0
-100

0

100

-100

0

100

-100

Detuning (kHz)

0

100

-100

0

100

-100 -50

0

50

100

Detuning (kHz)

Figure 4.4: Sideband thermometry in tweezers and optical lattices. (a) Performing sideband spectroscopy
before (black points) and after (grey points) adiabatically transferring the atoms to and back from thescience tweezers (after performing sideband cooling in a preparation potential composed of tweezers and a+0.08
1D axial lattice), we measure an average phonon occupation of n̄ = 0.07+0.14−0.07 , 0.06−0.06 , and 0.07 ± 0.06(n̄ = 0.25 ± 0.12, 0.31 ± 0.13, and 0.27 ± 0.10) before (after) the handoff in the axial, first, and second
radial directions respectively. Cartoons in the top left of each frame indicate the orientation of the probe
beam relative to the traps, showing probes in two orthogonal radial directions (left two subplots) and in
the axial direction (right two subplots). The right-most spectra show that, in a reduced 6 × 6 region at thecenter of the array (denoted by the right-most cartoon), the axial cooling performance is vastly improved,+0.10with an average phonon occupation of n̄ = 0.00+0.06−0.00 (0.06−0.06 ) before (after) the handoff. This is due tothe comparable extent of the lattice beams to the tweezer array (light-green contour in right-most cartoon
shows the region over which the lattice intensity stays within 90% of its maximal value). (b) Measurementsin the 3D lattice show similar performance over a larger region spanning more than 2000 lattice sites, with a+29thermal occupation of n̄ = 0.000+33−0 and 0.000−0 in the axial (grey) and radial (black) directions respectively(the relevant orientations of the probe beams are once again indicated by the inset cartoons).

78
confirming that the atoms remain cold when handing them from the preparation potential to the science
potential (as described in Sec. 3.4). We perform sideband thermometry in the preparation potential both
immediately after sideband cooling, and after adiabatically passing the atoms to the science potential, holding
for 25 ms, and passing them back. As shown in Fig. 4.4a, before the handoff we observe an average phonon+0.14
occupation of n̄ = 0.07−0.07, 0.06+0.08−0.06 , and 0.07 ± 0.06 in the axial, first, and second radial directions

respectively. After the handoff we observe an average phonon occupation of n̄ = 0.25 ± 0.12, 0.31 ± 0.13, and
0.27±0.10 (again in the axial, first, and second radial directions). Since we expect that heating occurs during
both steps of the handoff, the mean of these two measurements serves as an estimate of the temperature of
the atoms in the science potential.
Although the tweezers, and thus the radial trap frequencies, can be balanced across the entire array,
there is substantial inhomogeneous broadening of the axial trap frequencies. This is due to the relatively small
25 µm waists of the lattice beams, which are comparable to the extent of the tweezer array (Fig. 4.4a).17 In
a smaller 6 × 6 region at the center of the array the axial cooling and handoff performance is vastly improved,+0.10with an average phonon occupation of n̄ = 0.00+0.06−0.00 (n̄ = 0.06−0.06 ) before (after) the handoff.

In condition (2), the tight confinement provided by the optical lattice allows for significantly higher
quality cooling, as well as larger numbers of traps. Sideband spectroscopy after resolved sideband cooling+29
in these traps indicates average phonon occupations of n̄ = 0.000+33−0 and 0.000−0 in the axial and radial

directions respectively, and thus a 3D ground state fraction of 100+0−7 % over a 26×26 µm region corresponding
to more than 2000 lattice sites (Fig. 4.4b). This performance is consistent with the expectation of 96.97 %3D motional ground state occupation from Sec. 4.3, albeit with somewhat large statistical uncertainty. At
these very low temperatures, it can often be hard to achieve precise measurements, since as the red sideband
approaches zero, it disappears into the spectral feature associated with the carrier transition. In Sec. 8.3, we
will discuss a complementary measurement that provides tighter bounds on the cooling performance, and isalso in agreement with the expectation based on simulations.
17 This condition was chosen as a compromise between atom number and high quality state preparation and detection, given
the optical power we had available at the time.

79
Objective

ng
agi

ms

bea

Im

Figure 4.5: Layout of the imaging beams. For improved imaging performance, it is important that thepolarization of the imaging beams (small blue arrows) is oriented such that the dipole emission patternfrom the atoms (blue donut) is oriented appropriately relative to the microscope objective. Additionally,
it is useful to align the propagation axis of the imaging beams (large blue arrows) to be along the mosttightly confining axis of the optical traps, and to use two counterpropagating imaging beams with balanced
intensities.

4.4

Imaging single atoms
We are interested in imaging atoms in the 1 S0 ground state by collecting photons that are scatteredoff of the atoms. We would like to be able to do this with high fidelity (in the sense that the presence orabsence of an atom is correctly identified with high probability) and low loss (in the sense that atoms are
not lost during the measurement). Additionally, we would like to identify the positions of the atoms withgood spatial resolution, and we would like to acquire the images as quickly as possible.
To balance the above considerations, our imaging scheme consists of exciting the 1 S0 ↔ 1 P1 transition,and collecting the spontaneously emitted photons. These photons have a convenient wavelength of 461 nm,
which is short enough to provide good spatial resolution (see Sec. 3.3), and energetic enough to provide good
detection efficiencies using commercially available camera sensors.18

However, as photons are scatteredoff of this transition, the atoms experience recoil heating. To prevent atom loss due to this heating, we
simultaneously perform sideband cooling using the 1 S0 ↔ 3 P1 transition as described in Sec. 4.3. Due to themultiple decay paths from 1 P1 (see Sec. 2.1.1.1), and Raman scattering of the trapping light when occupying
the 3 P1 state, the metastable 3 P0,2 states can be populated during this process, and so it is importantto repump these states back to the ground state by additionally applying 707 nm and 679 nm repumping
18 It is also helpful that these photons are more energetic that the photons in the trapping beams. If the opposite is true,
uncontrolled fluorescence in the microscope objective as a result of the high power trapping beams can result in emitted photons
that are close in energy to the photons emitted from the atoms, and thus contribute to a background signal that is challenging
to filter out.

80
light (see Sec. 2.1.1.4). Although we have experimented with pulsing these various control fields during
imaging [241], we have not found this to make a significant difference in imaging performance, and therefore
typically apply all the control fields continuously and simultaneously.
The orientation of the 461 nm drive beam(s), which we will refer to as the imaging beam, is importantfor optimizing imaging performance (see Fig. 4.5). For scattering rates that are fast compared to the trap
frequency, photons absorbed from the imaging beam result in momentum kicks that add coherently, resulting
in faster heating of the atoms. As a result, it is helpful to orient the imaging beam along the most tightly
confining axis of the trap, and to use two counter-propagating imaging beams so that these momentum kicks
average to zero, even in the absence of the trap. To avoid interference between these beams leading to spatiallyvarying scattering rates, and thus varying imaging performance, we used two counter-propagating imaging
beams that are balanced in intensity, and detuned from each other by 20 MHz (resulting in a travellinginterference pattern that varies on timescales comparable to the linewidth of the imaging transition). The
two beams are additionally blue-detuned (on average) by 690 MHz from the free space resonance. This
detuning can help with imaging performance in tightly confining traps [217]. Whereas resonantly exciting
the atoms to the excited state can result in significant heating due to fluctuations in the applied dipole force(since the excited state is often anti-trapped, or at least experiences a very different trapping potential),
weakly dressing the atoms with the imaging light results in a dressed state that remains trapped while stillintermittently scattering photons.
To efficiently collect the spontaneously emitted 461 nm photons, one must consider the angular distribution of the emitted photons (see Eq. 4.11), and orient the polarization of the 461 nm drive field andthe microscope objective appropriately. The optimal orientation in most of our experiments is where thepolarization of the 461 nm beam is orthogonal to the axis of the microscope objective. For our microscopeand imaging setup (see Sec. 3.3), the ultimate collection efficiency of a spontaneously emitted photon froman atom onto the camera is ∼ 9%, including imperfect transmission through the various imaging optics, andthe quantum efficiency of the camera.
The collected photons are imaged onto an electron-multiplying charge-coupled device (EMCCD) camera (Andor iXon Ultra 897). The tradeoffs for different modes of operation of this camera are described81
in detail in [9]. Here, we briefly summarize a few particularly important tradeoffs between signal-to-noise
and image resolution that are relevant in our experiments. The main source of noise on an EMCCD camera
is due to clock induced charges (CICs).19

These extra charges are generated during the charge transfer

process that moves the collected signal to the readout register, and are thus indistinguishable from real
photoelectrons generated by the atoms. As this source of noise is a per-pixel effect, it can be advantageous
to concentrate the collected signal on a single pixel to maximize signal-to-noise. Additionally, CICs can be
minimized by increasing the vertical shift speeds used to transfer the collected signal to the readout registeron the sensor. However, both these choices result in reduced spatial resolution. The former because one islimited by the pixel size, rather than the optical properties of the imaging system, and the latter because
increased vertical shift speeds result in reduced charge transfer efficiency, and thus a smearing of the collectedsignal over multiple pixels. As a result, we opt to design our imaging system such that the waist of the PSF(see Sec. 3.3) corresponds to ∼ 2.5 pixels, spreading the collected atomic fluorescence over multiple pixels topreserve the spatial resolution of the imaging system. This choice enables the super-resolution techniques
described in Sec. 4.4.2. We also do not attempt to perform very fast readout of the images, using a moderate(but still fairly fast) vertical shift speed of 1.7 µs.20

4.4.1

Image analysis
For each image, we are interested in identifying which sites in the array contain an atom in the 1 S0state (occupied sites), and which do not (empty sites). To minimize the required signal and thus imaging
time, we take advantage of a few extra pieces of information. First, the array of optical traps results in a
discrete set of sites that the atoms can occupy, and the location of these sites can be characterized throughindependent measurements (Fig. 4.6a). For each site, we multiply the raw images by a mask corresponding
to the measured point spread function (PSF) of the imaging system at that location (Figs. 4.6b and 4.7b) and
19 In a conventional charge coupled device (CCD) camera, one typically also suffers from read noise associated with digitizing
the collected photoelectric signal, and thermally excited extra charges. The latter effect can be suppressed by cooling thesensor, and the former
effect can effectively be eliminated using electron-multiplying gain, at the cost of a constant reduction in√signal-to-noise of 2 from the optimal case. Below signals of a few thousand photons per pixel, read noise typically dominates,and so it is advantageous to use EM gain [9].
20 Since our exposure times are typically tens to hundreds of milliseconds, this does not pose a significant limitation to theexperiment.

82

(a)

(b)

(c)

Prob. density

10 1

Img. 0Img. 1

10 2

10 3

10 4

10 5
0

20
40
Photons collected

Figure 4.6: Image analysis for sparse arrays. (a) An averaged image of atoms loaded into ∼ 2500 sites in
the lattice is used to calibrate the positions of the lattice sites for subsequent analysis. Note that only asmall subset of these sites are loaded on any individual run of the experiment. (b) An averaged image of asingle implanted atom, surrounded by empty lattice sites denoted by the red circles. This serves as a direct
measurement of the PSF of our imaging system. (c) A histogram of the binned counts for two consecutive
images (img. 0 and 1) in typical imaging and loading conditions in the lattice. Specifically, in this data
a 16 × 24 array of preparation tweezers is loaded with ∼ 50% filling, and the atoms are implanted in a
sublattice that is spaced by 3 lattice sites in one direction (the direction with 16 tweezers), and two latticesites in the other (the direction with 24 tweezers). Finally, the atoms are imaged for 100 ms. The resulting
distribution of photon counts on each site (considering only sites where a preparation tweezer was present)
is clearly bimodal, with empty sites corresponding to the red peak, and occupied sites corresponding to thegreen peak. We estimate whether an atom is present or absent in a given experimental trial by evaluating
if the measured counts fall above or below an appropriately chosen threshold (grey dashed line). The loss
probability pL can be estimated by looking for an excess of empty sites in image 1 relative to image 0, andtakes a value of 0.015(15)% in the displayed data. The false negative probability p1→0 corresponds to the
portion of the atom peak that falls below threshold (dark green region), and the false positive probability
p1→0 to the portion of the empty peak that falls above threshold (dark red region). The shaded regions inthis plot correspond to a fit to a skewed Gaussian distribution for the empty peak, and a Poisson distribution
for the atom peak. Note that these fits are only to guide the eye, and that our procedure for estimating
the various error probabilities is independent of any assumptions about the functional form of the measured
distribution of photon counts.

83
sum the result to obtain the number of counts collected for a given site in a given image.21 For experiments
where the atoms are pinned in place (namely those appearing in Chs. 5 and 6), we ensure that the sites are
far enough apart that the overlap between the PSF from different sites is negligible.
The histograms of the counts are bimodal, with one peak near zero counts associated with the empty
sites, and one peak with higher counts associated with the occupied sites (see Fig. 4.6c). The former peakis approximately Gaussian, with scatter that is set by thermal noise and CICs on the EMCCD, as well asleakage light. The latter peak is approximately Poissonian, due to the discrete and probabilistic nature of
the collected signal. We set a threshold between these peaks to binarize the signal where, for a given image,sites with counts that exceed the threshold are identified as being occupied (a 1 event), and sites that are
below the threshold are identified as being empty (a 0 event).
Three types of errors can occur during imaging:(1) A false positive where an empty site is incorrectly identified as being occupied, which occurs withprobability p0→1 .
(2) A false negative where an occupied site is incorrectly identified as being empty, which occurs withprobability p1→0 .
(3) A false negative where an atom is lost during the course of the image, which occurs with probabilitypL .
We will sometimes collectively refer to the first two processes as infidelity. Optimizing the imaging conditions— including the parameters of the imaging and cooling beams, the duration of the image, and the selectionof the masks and threshold — amounts to minimizing p0→1 , p1→0 , and pL , with the secondary goal ofminimizing the duration of the image. The degree to which these three errors are problematic depends on
the specific conditions of the experiment. For example, if a larger fraction of the sites are occupied, p0→1
errors are less problematic than p1→0 errors (and vise versa when a small fraction of sites are occupied). To
avoid concerns of biasing, we typically opt to make a best effort attempt at minimizing the sum of all three
21 This procedure can be thought of as a version of image deconvolution which takes advantage of the additional fact thatbright points in the underlying image should come from only a discrete set of locations [305].

84
errors for conditions where half the sites are occupied, and do not subsequently adjust the image analysis orparameters as a function of experimental conditions.
To characterize the above errors, we take two images back to back and compare the histograms ofthe collected counts. One approach is to take the assumption that the empty distribution is Gaussian andthe occupied distribution is Poissonian. In this case, one can perform a maximum likelihood fit of themeasured histograms of the counts to extract the characteristic parameters of the different distributions.
p0→1 corresponds to the part of the empty distribution that exceeds the threshold, and p1→0 to the part ofthe occupied distribution that is under the threshold. 1 − pL corresponds to the ratio of the area contained
under the occupied peak in the second and first images.
Although the occupied distribution is Poissonian with high confidence, it is not the case that the empty
distribution is always Gaussian. In particular, leakage light, imperfect charge transfer on the EMCCD, andhigh-order aberrations in the imaging optics can lead to a slightly skewed distribution. Additionally, atoms
that are lost during imaging can result in a long tail that connects otherwise well-separated empty andoccupied distributions. However, we typically do not care to distinguish p1→0 from pL , since both false
negative errors affect the measured signal in the same way. When only considering the overall false positiveand false negative rates, we can measure the relevant errors in a model-independent way. Specifically, forcalibration data involving pairs of images as described previously, we can directly measure the probability
p01 of events x01 in which an empty site in the first image was occupied in the second, and probability p10
of events x10 , where an occupied site in the first image was empty in the second. Because p0→1 , p1→0 , andpL are small, we neglect any effects that are second order in these errors. For well-isolated traps with notunneling or thermal hopping, there is no physical process wherein an atom can be generated in the second
image, and so p0→1 = p01 . Similarly, any process that results in a missing atom leads to a false negative
measurement error, and so p10 = p1→0 + pL . Characterizing these errors allows us to separate the effect ofmeasurement errors from other aspects of the experiment, as we will see in more detail in Chs. 6 and 8.
Although the imaging performance can fluctuate depending on the specific experiment at hand, using
the above procedure we achieve a false positive probability of p01 ≃ 0.1% and a false negative probability of
p10 ≃ 0.1% when performing 100 ms images in the 813 nm optical lattice. In deeper 813 nm traps, even better

85
imaging performance is possible [74] at the cost of reduced atom number given a fixed amount of available
optical power.22

The imaging performance in the 515 nm tweezers is a fair bit worse, with p01 ≃ 1% andp10 ≃ 5% for 75 ms images. It is not clear exactly what the mechanism leading to increased imaging loss in
515 nm traps is. Based on the branching ratio from the 1 P1 state to the 1 D2 state, we do not believe theobserved loss is fully explained by anti-trapping of the 1 D2 state at 515 nm. We observe that the loss appears
to be related to the number of 689 nm photons scattered during cooling, and not directly to the number of
461 nm photons scattered (except to the extent that more scattered 461 nm photons necessitates additionalcooling). Note that although we do not observe any loss during the optical cooling step (without imaging),such loss has been observed in other experiments [73]. One hypothesis is that 515 nm (and 532 nm) photons
have an energy above the two-photon ionization threshold from the 3 P1 state, leading to loss, whereas 813 nm
is below this threshold and thus does not allow for this particular loss channel [207].

4.4.2

Image analysis in dense arrays
Image analysis is more challenging when the spacing between sites is comparable to the PSF of theimaging system, since an atom in a neighboring site can lead to increased background counts on an adjacentsite. In our apparatus, the imaging system is designed to be able to resolve individual lattice sites, andso this is not a significant concern. However, to improve the fidelity of the procedure described in theprevious section for denser arrangements of sites, we optimized the applied masks by training a single-layer
neural network on simulated data corresponding to 30 % random filling of the lattice (see Fig. 4.7). The
resulting masks are similar to the ones based on the measured PSF of our imaging system, but with aslight negative bias on adjacent lattice sites that reduces errors due to leakage light from one lattice site toanother (see Fig. 4.7d). Although a deep neural network can result in better performance [149], the resulting
performance gains are marginal in our setup due to the already high imaging resolution in comparison to the
22 Images using the Sisyphus cooling mechanism described in [73, 208] can be a fair bit faster than those using sideband

cooling, since each scattered 689 nm photon can remove more thermal energy. In 813 nm tweezers, we achieve p01 ≃ 1% andp10 ≃ 0.1% for 30 ms images using the Sisyphus cooling technique [240]. Note that although the same procedure is expected towork in 813 nm optical lattices, we have not had great success with this at the depths we work at (although we are optimisticthat increases of ∼ 2 − 3× in lattice depth could result in good performance). One additional consideration in a lattice is thattunneling can become significant in higher bands. As a result, in the relatively shallow lattices we work with, we must ensurethat the atoms stay primarily in the ground band for the full duration of the image, eliminating the benefit of removing multiple
motional quanta with a single scattered 689 nm photon.

86

(a)

(b)

(c)

(d)

(e)

Inaccuracy

10 1

Training
Test

10 2
10 3
10 4
0

1

0
Weight

1

149
Epoch

Figure 4.7: (a) Atoms separated by next-nearest-neighbor spacing (or more) in the lattice (lattice sites areindicated by where the grey lines cross) are well-resolved from each other, as is clear from the depicted single
shot image of rearranged atoms. (b) When the atoms are well-resolved, it is sufficient to use a completely
positive mask (as depicted) to identify the presence or absence of an atom. (c) Atoms at nearest-neighborspacing in the lattice have slightly overlapping point spread functions, as seen in this single shot image taken
after the atoms are allowed to tunnel (resulting in some atoms being in neighboring sites). (d) We train asingle layer neural network to generate the appropriate mask for identifying the presence or absence of an
atom when other atoms can occupy neighboring sites in the lattice. The resulting mask is similar to before,
but with slightly negative weights on the neighboring lattice sites to suppress spurious detection events due
to leakage light (from atoms on neighboring sites). Color scale is shared with (b). Note that the masksin (b, d) are representative examples, but that the masks applied in our analysis differ from lattice site tolattice site depending on the alignment of a given lattice site to the pixels on the camera. (e) Performanceof the optimized masks as a function of training epoch. The inaccuracy is defined to be the proportionof events where the presence or absence of an atom on a given lattice site is incorrectly identified. Noticethat whereas the performance continues to improve for the training data set, additional training does not
improve the performance with respect to a test dataset that is independent of the training data. The ultimate
performance for 500 ms images is similar to the performance for 100 ms images in sparse arrays.

87

(a)

Load

Pre-sort

1D rearrange

FilterCompress

Complete(b)

Figure 4.8: Procedure for atom rearrangement. (a) Rearrangement of atoms (blue circles) in an opticallattice (grey circles denote sites in the lattice) using optical tweezers (green) must balance several conflicting
requirements, leading to the multistep algorithm described in the text. (b) A selection of single-shot images
from the experiment, where each bright spot indicates the presence of an atom. Starting from a stochasticallyloaded array (left), one can prepare nearly arbitrary patterns of atoms in the lattice (right). Of particularnote is the second-from-the-right pattern, which shows pairs of atoms positioned in neighboring sites in thelattice, and the far-right pattern, where the target pattern is adapted in real time to make use of every
stochastically loaded atom in a given experimental trial.

lattice spacing. This being the case, we opt to use a single-layer neural network to ease the characterizationof errors and image fidelity.
The resulting analysis yields a false negative probability of p10 = 0.2(1)% for a calibration data set
using 500 ms-long exposure times, however, p10 can fluctuate day to day by ∼ 0.2% for the experiments
appearing in Chs. 7 and 8. The false positive probability is much lower, with a value of p01 ∼ 0.001%.
However, for experiments with very low density (for example when one atom occupies an analysis region
containing ∼ 1000 sites) the effect of false positives can become comparable to or larger than the effect offalse negatives when trying to correctly identify the presence and position of the atom.

4.5

Atom rearrangement
Once the atom positions are identified, we can use a mobile set of optical tweezers to deterministicallyrearrange the atoms. Rearrangement techniques have been used to great success in tweezer arrays [99, 19],and in programmable optical lattices [179], but to our knowledge, our experiments are the first to perform
rearrangement in an optical lattice that allows for appreciable tunnel coupling (as we will discuss in detail in
Chs. 7 and 8). Rearrangement in such a lattice must balance several conflicting requirements. In particular,88
our tweezers have a beam waist that is comparable to the lattice spacing (see Sec. 3.3). This means thattweezers directed at a specific lattice site still have substantial overlap with adjacent sites, and so atoms
trapped in those adjacent sites can experience higher loss rates, especially when the tweezers are moving.
Additionally, overlapping tweezers will beat against each other due to our use of crossed acousto-optic
deflectors to project the tweezer array [241]. Tweezers on adjacent lattice sites are separated in frequency
by 1.95 MHz, far higher than the relevant trap frequencies of 50-200 kHz, and so 1D tweezer arrays withequal spacing to the lattice don’t cause significant heating (see Sec. 3.3.3). However, because the lattice isslightly rectangular, placing a tweezer on every lattice site in 2D can result in lower beat frequencies along
the diagonals of the array that cause significant heating. Our algorithm for atom rearrangement seeks tobalance these concerns against the desire to minimize the total distance travelled by the atoms and the total
duration of the rearrangement procedure.
The basic premise is to perform most operations in parallel with 1D arrays of tweezers [95, 324], andto perform as many operations as possible on a sublattice that does not lead to undesirable beat frequencies
between overlapping tweezers. We choose to work with either nearest- or next-nearest-neighbor spacing
along one axis, and third-nearest-neighbor spacing along the other axis. Atoms are stochastically loaded
into tweezers on this sublattice with 50 − 75 % filling (depending on whether enhanced loading is used, seeSec. 4.2) and implanted into the lattice [366]. Rearrangement proceeds in four stages (Fig. 4.8a):(1) Pre-sorting: If a column is loaded with more atoms than are required in the target pattern,
any excess atoms are pushed to the neighboring column in a single step using a 1D tweezer array.
Similarly, if the column has too few atoms the missing atoms are pulled from the adjacent column.
In the rare case that this step fails, the experiment is terminated, and we simply load a new ensembleof atoms.
(2) 1D rearrangement: Each column is then rearranged in 1D, with excess atoms pushed to the edgesof the array.
(3) Filtering: A subset of atoms is transferred back into a 2D tweezer array that only addresses thecorrect sublattice, and the lattice potential is extinguished. Laser light that addresses the free-space

89
1

S0 ↔ 3 P1 transition is applied to resonantly blow away the excess atoms while optically coolingthe tweezer-trapped atoms (since the AC Stark shifts imposed by the tweezers shifts the 1 S0 ↔ 3 P1transition in the tweezer-trapped atoms to higher energies). The remaining atoms are subsequently
transferred back into the lattice.
(4) Compression: Optionally, the columns of rearranged atoms can be translated closer together one
column at a time to prepare denser target patterns.
The resulting algorithm allows for the preparation of nearly arbitrary patterns with 0 or 1 atoms on agiven site. However, we are currently unable to translate columns of atoms to less than next-nearest-neighborspacing without incurring significant loss in step (4). This means that although the prepared patterns canhave atoms at nearest-neighbor spacing in one axis, the other axis must be spaced at next-nearest-neighborspacing or greater.23√
The number of steps in the algorithm scales as O( n), where n is the number ofatoms in the target pattern. For target patterns with similar density to the loaded sublattice, the mean√
distance travelled by each atom is O(1), and the runtime of the algorithm scales like O( n). For dense√
target patterns the worst-case scaling of the mean distance travelled is O( n) leading to a runtime of O(n).
Throughout rearrangement the lattices are held at a constant depth of Uax /ℏ = 2π × 2.4 MHz forthe axial lattice, and U2D /ℏ = 2π × 1.7 MHz for the 2D lattice. Resolved sideband cooling is continuously
applied to the lattice-trapped atoms. To move a given set of atoms, tweezers are ramped on and off over
60 µs to a depth of Utw /ℏ ≃ 2π × 30 MHz and moved with a constant speed of 26 µm/ms. Rearrangement
occurs under magic conditions in the lattice for the 1 S0 ↔ 3 P1 transition (see Sec. 4.1), and so the tweezertrapped atoms are shifted out of resonance from the cooling light while they are being translated (since
the magic conditions for the lattice are different than for the tweezers). Because the appropriate moves in
our algorithm can be computed efficiently, and due to the architecture of our tweezer control system [365],
analyzing the atom images and programming the control system takes only ≲ 5 ms for target patterns
containing up to 270 atoms. However, there is currently a technical delay of 110 ms between taking theimage of the stochastically loaded atoms and initiating rearrangement due to the time it takes to extract
23 We expect that this limitation can be circumvented by improving the lattice alignment such that it is closer to beingsquare, but this has yet to be confirmed.

90
the image data from the camera. This could be addressed in the future by performing image processing on
the same FPGA hardware used for atom rearrangement [345]. For patterns containing up to 270 atoms,
the rearrangement itself is performed in ≲ 30 ms. The per-atom success rate for this rearrangement canbe as high as 99.5 %, and is primarily limited by imaging fidelity and loss. However, due to drifts on theexperiment, the typical per-atom success rate is 98 % for the experiments discussed in this work. Oncethe success of the rearrangement procedure is verified via a second image, the atoms are optically cooled tonear their 3D motional ground state as described in Sec. 4.3. The resulting prepared states have very low
entropy in all degrees of freedom of the atoms, and serve as the starting point for all subsequent experiments
described in this work.24

24 In some cases where the specific pattern of atoms is unimportant, we opt not to use rearrangement for convenience.

Chapter 5

Optical frequency metrology with tweezer arrays“Typically, the subject being copied is terminated.”— The Terminator
Portions of this chapter have appeared in:
M. A. Norcia, A. W. Young, W. J. Eckner, E. Oelker, J. Ye, and A. M. Kaufman. Seconds-scalecoherence on an optical clock transition in a tweezer array. Science, 366(6461):93–97, 2019
A. W. Young, W. J. Eckner, W. R. Milner, D. Kedar, M. A. Norcia, E. Oelker, N. Schine, J. Ye, andA. M. Kaufman. Half-minute-scale atomic coherence and high relative stability in a tweezer clock.
Nature, 588(7838):408–413, Dec. 2020

In Ch. 4, we showed that we can prepare ensembles of atoms in well-determined initial states, anddetect each atom individually. In this chapter, we will see how we can control a convenient two level (qubit)
degree of freedom encoded in the electronic states of the atoms for frequency metrology. We will begin in
Sec. 5.1 with a brief primer on quantum sensing, and identify the relevant features that make for a good sensor
for frequency metrology (a clock). In Sec. 5.2 we will argue that the above qubits are particularly suited tofrequency metrology, and mention some unique benefits and challenges associated with our approach. We
will also discuss how we prepare, control, and read out the qubits, and the limitations on the fidelity of these
operations. In Sec. 5.3, we will discuss the limitations on the coherence of the qubits, and how the carefully
designed optical potentials in our experiment allow us to tailor the environment of the qubits to enhancetheir coherence. Although the record-high quality factor of these qubits is beneficial for clock measurements,
we will discuss systematic perturbations to the qubit transition frequency that can affect the accuracy of suchmeasurements. In Sec. 5.4, we will describe some specific clock measurements we have performed, including
frequency comparisons between the atoms and a laser, as well as synchronous comparisons between the

(a)

(b)

92

Figure 5.1: Schematic for quantum sensing. (a) A quantum sensing problem can generally be formulated as
the task of estimating some parameter (or set of parameters) ϕ by preparing a state described by a density
matrix ρ, evolving the state under a CPTP map Λϕ which encodes the parameter of interest, performing ameasurement described by a POVM M, and using the results in an estimation procedure ϕ̌. (b) Often, thequantum sensor is composed of multiple similar subsystems. Phase estimation is a restricted measurement
problem where the signal Λϕ acts identically but independently (for example, with a local unitary Uϕ ) on eachsubsystem. Although one could perform a joint measurement on the entire system, the same performancecan be achieved using only local measurements M.

atoms. We will conclude in Sec. 5.5 with a brief discussion of how the so-called “tweezer clock” developedin our group, and in parallel at Caltech [209], fits into the broader field of frequency metrology.

5.1

A primer on quantum sensing
In almost any experiment, we are interested in extracting some parameter ϕ relating to the physicalworld from experimental data.1 This problem can, in general, be framed in terms of three steps (see Fig. 5.1):
preparation of some probe state described by a density matrix ρ, evolution of that state under a completely
positive trace preserving (CPTP) map Λϕ that encodes the parameter ϕ of interest (and can capture both
unitary and non-unitary evolution), and measurement of the resultant state described by a positive operatorvalued measure (POVM) M (which is a generalization of a typical projective measurement) [306]. Thissequence is then repeated M times to estimate ϕ to greater precision.2 The question is how to achieve thelowest uncertainty in the estimate of ϕ given restricted physical resources (typically some combination oflimitations on the total duration of the measurement, and physical constraints on the available probe states).
For starters, let us fix ρ, Λϕ , and M. In this case each measurement gives us an outcome x sampledfrom the random variable X which follows the probability density distribution p(X|ϕ). The result is a dataset
1 In principle, ϕ can be a set of multiple parameters, but we omit that possibility in this simplified discussion.
2 One can employ Bayesian techniques to adaptively adjust ρ, Λ , and M [227], but these techniques are outside the scope
ϕof this work.

93
x = {x1 , x2 . . . xM } of independent and identically distributed (IID) samples. Our goal is to construct anestimator ϕ̌(x) that takes the data x and outputs an estimate of ϕ.3 ϕ̌ should be “unbiased,” in the sensethat its expectation value is equal to ϕ:4⟨ϕ̌⟩ :=

Zdx p(x|ϕ)ϕ̌(x) = ϕ,

(5.1)

and “consistent,” in the sense that it converges to ϕ as M increases:5
lim ϕ̌ = ϕ.

M →∞

(5.2)

With this in mind, we are interested in minimizing the mean squared error6 of ϕ̌ with respect to x:∆2 ϕ̌ := ⟨(ϕ̌ − ϕ)2 ⟩ =Z

dx p(x|ϕ)(ϕ̌(x) − ϕ)2 .

(5.3)

If ϕ̌ is unbiased, it obeys the Cramér-Rao bound [120]:7∆2 ϕ̌ ≥

1,
M I (p(X|ϕ))(5.4)

where I is the classical Fisher information (see Appendix A.1). An estimator is known as “efficient” if itsaturates this bound. Notice that Eq. 5.4 includes the effect of the central limit theorem: M different IID
random variables can be combined into an estimate that averages down with a variance of 1/M .
In the previous case, the estimation was dictated by the classical probability distribution p(X|ϕ).
However, since we are dealing with a quantum state, we are free to choose our measurement M, resulting ina probability distribution p(X|ϕ, M) and associated Fisher information I(ϕ, M) := I(p(X|ϕ, M)). Maximizing the Fisher information over all M leads to the quantum Fisher information Q(ϕ) := maxM I(ϕ, M) [40]and an associated quantum Cramér-Rao bound (see Appendix A.2):∆2 ϕ̌ ≥1.
M Q (ϕ)(5.5)

3 As in [309], we use ∨ to denote an estimator rather than ∧ to avoid confusing an estimator with a quantum operator.
4 For discrete random variables, one can replace the integrals in this section with sums.
5 Note that there is a subtle difference between being consistent and being unbiased. An unbiased estimator can neverconverge and therefore be inconsistent, and a consistent estimator can be biased for a finite sample size.
6 This is equivalent to the variance in the case that ϕ̌ is unbiased [306].
7 In fact, ϕ̌ needs only be asymptotically locally unbiased to obey this bound [76], meaning that it is unbiased in the large
M limit, and about some specific value of ϕ.

94
The above discussion provides a general framework to evaluate different kinds of measurements, where
optimization of a given measurement amounts to optimizing over ρ, Λϕ , and M. Next, we look at a specificexample that is relevant to our work.

5.1.1

Phase estimation
Consider a measurement apparatus composed of N qubits.8

The single qubit Hilbert space H1 isspanned by the states {|0⟩, |1⟩}, and the combined Hilbert space is H = H1⊗N . We are interested in thesituation where Λϕ = Uϕ⊗N is a unitary transformation that acts independently but identically on each
qubit.9 Specifically, each qubit undergoes the unitary transformation:
Uϕ = e−iϕH ,(5.6)

where the generator H is some known Hermitian operator acting on H1 . The generator acting on the wholesystem is HH =PN

thqubit (and zeroes elsewhere). Additionally,
i=1 Hi , where Hi refers to H acting on the ilet ρ = |ψ⟩⟨ψ| be a pure state, and M a projection-valued measurement (PVM) in the {|0⟩, |1⟩} basis of each
qubit. In this case, the quantum Fisher information is [40, 41] (see Sec. A.2):
2
Q = 4(⟨ψ | HH
| ψ⟩ − |⟨ψ | HH | ψ⟩|2 ) = 4∆2 HH ,(5.7)

where we have defined ∆2 HH to be the variance in the expectation of HH given the state |ψ⟩.
Let the lowest and highest eigenvalues of H be λ− and λ+ respectively, with corresponding eigenstates|λ− ⟩ and |λ+ ⟩. If we restrict |ψ⟩ to product states of the different qubits then ∆2 HH =PN

2
i=1 ∆ Hi , where∆2 Hi is the variance of the measurement outcome on the ith qubit. As a result, the largest ∆2 HH we canachieve is with the state [119]:
|ψ⟩ =



|λ− ⟩ + |λ+ ⟩
√
2

⊗N
.

(5.8)

In this case ∆2 HH = N (δλ)2 /4, where δλ = λ+ − λ− , leading to a bound of:∆2 ϕ̌ ≥1.
M N (δλ)2(5.9)

8 The analysis in this section applies just as well to qudits (quantum systems with more than two levels).
9 This applies whenever the signal of interest acts globally on all qubits, but the qubits are otherwise decoupled from eachother. For example, when a uniform electric or magnetic field acts on isolated atoms which each encode a qubit degree of
freedom.

95
Eq. 5.9 is known as the standard quantum limit (SQL)10 , which can be interpreted as the lowest-varianceestimation of ϕ that one can achieve when using multiple unentangled quantum sensors. Notice that the
SQL simply corresponds to combining M N different IID measurements, where M characterizes the numberof separate measurements, N the size of the sensor (number of qubits), and δλ the energy cost.
Removing the restriction that |ψ⟩ be a product state, ∆2 HH is maximized by an even superpositionof the states with maximum and minimum eigenvalues of HH , namely:11|ψ⟩ =|λ− ⟩⊗N + |λ+ ⟩⊗N√,
2(5.10)

which is known as a Greenberger-Horne-Zeilinger (GHZ) state. For the above GHZ state, ∆2 HH = (N δλ)2 /4,
leading to a bound of:∆2 ϕ̌ ≥1.
M N 2 (δλ)2(5.11)

Eq. 5.11 is known as the Heisenberg limit (HL), which can be interpreted as the lowest variance estimationof ϕ that one can achieve with a quantum sensor composed of many identical constituents. Expressed as aq√standard deviation (∆ϕ̌ := ∆2 ϕ̌), the SQL scales like ∆ϕ ∝ 1/ N , whereas the HL scales like ∆ϕ ∝ 1/N ,leading to the often quoted quadratic enhancement with respect to N provided by entangled sensors. The
scaling of the SQL and HL with respect to all other parameters are equal. Additionally, note that the HL
can be saturated with only local measurements, meaning that entanglement is only necessarily useful in the√
preparation of ρ, and not required in M [119]. The 1/ N scaling of uncertainty in the SQL is often referred
to as the limit imposed by quantum projection noise (QPN) or “shot noise,” since it is noise associated withthe discrete nature of the measured signal.
The HL and SQL, at their core, simply describe a tradeoff between the coherent and incoherent
combinations of different signals (namely, adding amplitudes or probabilities), with the former performing
up to quadratically better than the latter. Note that, depending on the sensing problem, entanglement is
not always necessary to achieve the best possible performance. For example, if one takes a single qubit
and evolves it for N times longer than an N qubit measurement, the effect is an N times enhancement of
10 This terminology is somewhat unfortunate. Although I don’t know where it originated, people often lament that the SQL“is neither standard, quantum, nor a limit.”
11 Note that there are other entangled states that can achieve similar performance, like the spin squeezed states that we
discuss in Ch. 6.

96
δλ, and thus equal performance to the HL for the shorter-duration measurement with N qubits. Instead,entanglement should be thought of as a way to trade off between resources in time and space, as is often
true in quantum algorithms [268, 43, 120]. As a result, when discussing enhancements in metrology relating
to entanglement, one must be very careful to specify the specific sensing problem at hand, and what therelevant restricted resources are [145, 10, 120].12

5.1.2

On good clocks, qubits, and decoherence
Turning to the case of a typical clock measurement, the measured quantity is the elapsed time permeasurement run ϕ → t, and Uϕ simply corresponds to time evolution under the Hamiltonian H = δλσz /2,
where σz is the Pauli-z matrix and δλ is now an energy. We are interested in attaining the best possible
measurement of the fractional uncertainty ∆t/t for a given total measurement duration T = M t (assuming
no dead time, see Sec. 5.1.3.2). If one saturates the SQL, the resulting performance is:13
∆t
1
√
=
,
t
2πf N T t

(5.12)

where f := δλ/h is the frequency associated with the precession of the qubits. Note that in the above
expression we have fixed f to a known value, and associate fluctuations in the measurement with a fluctuation
in t. This situation applies, for example, when using a frequency reference to distribute a time standard.
Alternatively, one can fix the value of t, and take ϕ → f and H → tσz /2, arriving at a similar expression:
∆f
1
√
=
.
f
2πf N T t

(5.13)

This situation applies, for example, when measuring a frequency difference between two clocks. Given thescalings of the above measurement precisions, in this chapter, we focus on maximizing f , N , and t. 14 We
will describe the preparation of states which improve the scaling with respect to N , including those of theform in Eq. 5.10, in Ch. 6.
Maximizing f has led to atomic clocks operating at increasingly high frequencies, first in the microwave
domain [342], and more recently in the optical and ultraviolet domains [203]. Maximizing t by improving
12 This is generically true when discussing quantum enhancements of any kind, like in Sec. 7.7.
13 Note that the 2π comes from associating H with a Hamiltonian, rather than a generator of the form in Eq. 5.6.
14 We leave the maximization of 2π for later works.

97
the lifetime of the qubits is important in the SQL, but also has significant implications for any entanglementthat leads to performance beyond the SQL. For example, consider the case where the qubits decay from |λ+ ⟩to |λ− ⟩ with a time constant τ . For the GHZ state in Eq. 5.10, decay of a single qubit completely decoheres
the state, and so the effective decay time is τ /N . Similar arguments apply when τ characterizes a moregeneral decoherence model, for example involving both decay and dephasing [145, 299, 85]. If t is limited byτ /N , then any enhancement due to GHZ-like entanglement is cancelled by a reduction in t. In this scenario,
the only benefit coming from entanglement is a tradeoff between time and space: one can perform N parallelmeasurements of duration t ∼ τ with unentangled atoms, or N sequential measurements of duration t ∼ τ /Nwith maximally entangled atoms. The latter case can be beneficial in the sense that the measurement has
increased bandwidth (one can be sensitive to variations in the detected signal at higher frequencies), but does
not change the ultimate sensitivity of the measurement to a static signal. Taking the expected decoherence
into account and preparing the appropriate partially entangled state can lead to a constant enhancement
over the SQL of 1/e in variance [145], but not to a modification of the asymptotic scaling of the procedure
with N . However, an enhanced scaling is possible when t is limited by other effects [10], as we will see inmore detail in Ch. 6. Although outside the scope of this work, more sophisticated protocols involving error
correction can restore the HL in the presence of certain kinds of errors [171, 86, 84, 377].15
With the above discussion in mind, a useful figure of merit is the unitless product of τ and the angularfrequency ω := 2πf , leading to the definition of the quality factor Q = ωτ /2. If τ is exclusively due to decay,
Q can be interpreted as the total angular displacement (in radians) accrued by an oscillator before its energy
decays to 1/e of its original value.
Notice that the desirable features for quantum sensing are very similar to those for a good qubit
or quantum memory: we want the ability to prepare specific (possibly entangled) pure states with highfidelity, and to isolate these states from uncontrolled influences of the environment. The key additional
requirement for quantum sensing is that we want our qubit to be as sensitive to a specific signal of interestas possible, which, in the case of frequency metrology, means that we want our qubit states to be as far apartin energy as possible. By contrast, for typical computing applications there is no requirement on the energy
15 The necessary and sufficient condition for error correction restoring the HL in metrology ends up being quite intuitive: thegenerator H of the signal must not be contained in the linear span of all errors [377].

98
separation between qubit states, and reducing this energy separation can be useful for reducing sensitivity
to other sources of noise [155]. Another figure of merit is the number of gates that can be applied within
the coherence time of a qubit. To a certain extent this is characterized by Q, in the sense that the valueof Q tells you how many σz gates one can apply, but one must also consider the other gates making up a
universal gate set. Even when considering these other gates, we will see that the qubit we choose to workwith in the context of frequency metrology also ends up being a reasonably attractive qubit for quantum
information applications.

5.1.3

Additional considerations in optical clocks
In an atomic clock, the mechanism for timekeeping is a periodic oscillation in the electronic statesof the atoms, or, in the language of the previous section, the energy difference between the qubit levels.
However, in optical clocks these oscillations are much too fast to detect directly.16 Instead, we need accessto a reference frame that rotates at or near the difference frequency of the qubit. This is typically done witha laser,17 which additionally allows one to apply single qubit gates as described in Sec. 2.2.1. With theabove requirement in mind, the typical operation of an optical atomic clock involves a local oscillator (LO) in
the form of an ultrastable laser whose frequency is periodically compared to the atomic reference frequency,
and adjusted accordingly [203]. The ultimate limit on the frequency instability (defined in Sec. 5.1.3.1) of
the LO is set by the ability to measure the difference frequency between the LO and the atoms, and thus by
Eq. 5.13 for unentangled atoms.
To understand the joint performance of the atom-LO system, one must consider effects relating tonoise and uncertainty on both the LO and the atoms, as well as effects relating to how the LO frequency
is referenced to that of the atoms. Besides QPN,18 and decoherence leading to finite interrogation times,
one must also consider various sources of technical noise that result in uncertainty or fluctuations in H. We
will discuss the relevant perturbations to H as they apply to our experiments in more detail in Sec. 5.3. For
now, we introduce a standard figure of merit for optical clocks, point out one additional source of error that
16 The fastest photodiodes can attain impressive bandwidths of up to hundreds of gigahertz [195], but this is still a far cry
from the hundreds of terahertz at which optical clocks operate.
17 We will describe measurements where other atoms serve as the LO instead of a laser in Sec. 5.4.2.
18 Typically, it is the QPN of the atoms that dominates, since measurements of the LO benefit from large numbers of available
photons.

99
often prevents lattice clocks from reaching the performance limits set by QPN, and discuss some key sourcesof perturbations to H in optical lattice clocks.

5.1.3.1

Allan deviation

As introduced in the previous section, we are interested in characterizing the fractional frequency
difference y := (f − fLO )/f between the atomic reference frequency f and the LO frequency fLO .19
Although the variance of y comes to mind as a reasonable figure of merit, a typical variance can diverge
under commonly occurring sources of noise in a clock. Instead, we define the Allan variance [273, 275]:20
σy2 (T ) :=

M
−1
X
1
2
(⟨y(T )⟩i+1 − ⟨y(T )⟩i ) ,2M − 1 i=1(5.14)

where ⟨y(T )⟩i denotes the ith independent measurement of the average value of y over a measurement
duration T , and M total measurements are performed. The Allan variance differs from a standard variancein the sense that one is computing deviations between adjacent measurements, not deviations from the mean.
As a result, the Allan variance converges for most realistic sources of noise in a clock [275]. In practice,
one often uses the overlapping Allan variance, which behaves similarly to the Allan variance, but provides
tighter error bars on the value of the variance inferred from a given data set by removing the restriction thatthe different samples ⟨y(T )⟩i are fully independent [273, 275]. One often quotes the square root of the Allan
variance, or the Allan deviation σy (T ), which is also known as the “instability”.21√
Based on the discussion so far, σy (T ) scales like 1/ T due to the SQL, but different sources of noise
can lead to different behaviors. For example, drifts or random walks in frequency can lead to a levelling off
or even increase of σy (T ) with T [275]. It is common to quote the instability at one second of averaging,√with the understanding that a 1/ T scaling applies only at short averaging times when the contribution
from sources of noise that break this scaling is negligible.
19 Recall from Eqs. 5.12 and 5.13 that measuring frequency and elapsed time are similar.
20 Note that we use the symbol ∆2 to denote standard variances, and σ 2 to denote Allan variances. However, the contribution
to the Allan variance from QPN is the same as in Eq. 5.13 when replacing ∆ with σ.
21 Note that people sometimes interchangeably refer to this as the stability. However, in these cases, “high stability” refers
to low values of σy (T ).

100
5.1.3.2

Dick effect noise

Often, optical atomic clocks operate in a cyclic fashion where, on each cycle of duration tc , a newensemble of atoms is prepared, evolved, and then measured destructively.22

If the interrogation time is t,

the duty cycle of a clock is defined as ζ := t/tc . For neutral atom-based optical clocks initial trapping andcooling can take ∼ 100 ms (see Sec. 3.1.2) in comparison to evolution times of t ∼ 100 − 1000 ms, and so ζ
can be fairly low.
Cyclic operation of the atomic reference in combination with ζ < 1 results in increased sensitivity tofrequency noise in the LO through what is known as the Dick effect [90]. Specifically, frequency noise atharmonics of 1/tc will be downconverted to near DC, resulting in errors in the inferred center frequency of2
the LO and thus higher Allan variance. The contribution to the Allan variance from the Dick effect, σy,D,

can be calculated by computing the Fourier components of the sensitivity function associated with a given
measurement [90]. For Ramsey measurements (see Sec. 5.2.4), where the sensitivity function is a square2
wave that is high during t, and low for the remainder of tc ,23 σy,Dtakes the form [285]:2σy,D(T ) =

2
 
∞
1 X
m
sin(πmζ)Sf,
T f ζ 2 m=1tc
πm(5.15)

where Sf (·) is the one-sided power spectral density of the difference frequency f − fLO .
For clocks which operate continuously, like thermal beam clocks or masers, ζ = 1, and so the Dickeffect vanishes [203]. For ion clocks, QPN typically dominates over Dick effect noise due to both high duty
cycles,24 and small ion numbers (leading to high QPN). However, in neutral atom-based optical clocks Dickeffect noise can dominate over QPN due to low duty cycles and large atom numbers. In Sec. 5.4, we attempt
to mitigate the Dick effect by increasing ζ. Alternatively, one can circumvent the Dick effect by interleaving
measurements with two or more clocks such that the sensitivity function never drops to zero [291, 242].
22 Terminated.
23 Assuming instantaneous control pulses.
24 In ion clocks the same ion is typically used for multiple interrogation cycles (not terminated), with only a few milliseconds
of cooling and optical pumping being required to reinitialize the ions [203].

101
5.1.4

Fluctuations and systematic shifts
Fluctuations in H lead to decoherence and thus impose limits on t, and on the resulting stability ofa clock, whereas uncertainty in the characterization of H limits the accuracy of a clock. The distinction
between stability and accuracy is key when considering the performance of a clock as a frequency standard.
Whereas stability relates to how quickly a measurement converges, accuracy relates to the degree to which
the measurement provides an absolute estimate of the frequency of a given unperturbed atomic transition.
Although good stability and accuracy is always preferable, certain applications can be fulfilled with only
good stability and poor accuracy, or vise-versa.25
Variations in H can arise from any of the perturbations discussed in Ch. 2, including AC and DC
electromagnetic fields. The pristine sensitivity of optical clocks means that these perturbations can sometimes
come from surprising places. For example, assuming that the control and trapping fields are appropriately
designed, the largest systematic shifts in optical lattice clocks often come from Stark shifts imposed byblack body radiation (BBR).26 For Sr, room temperature BBR results in a systematic shift of ∼ 5 × 10−15
in fractional frequency, and is the dominant source of uncertainty for state-of-the-art lattice clocks withfractional frequency uncertainties at the low 10−18 level [234, 220, 34]. Because these BBR shifts scale withthe radiant energy density of the BBR field, they follow the T 4 scaling of the Stefan-Boltzmann law (where
T here refers to temperature). As a result, experiments are underway that try to minimize BBR-related
effects by operating optical clocks in a cryogenic environment [336].
At the 10−18 level in uncertainty, gravitational redshifts also start to play a significant role. Specifically,a fractional frequency uncertainty of 10−18 corresponds to an uncertainty in altitude at the Earth’s surface of∼ 1 cm [67, 220], which is comparable to or better than the best measurements of altitude using traditional
survey techniques [253]. This means that, in the pursuit of lower uncertainty, optical clocks could also soon
become our best tools for performing geodesy [203, 220]. In fact, for differential measurements that do not
25 For example, accuracy is paramount if one wants to compare measurements to ab-initio calculations, or to compare
measurements performed in different apparatuses or at very different times. If one is instead interested in a time-varying ordifferential signal between two or more clocks, stability with low accuracy is sometimes sufficient.
26 At room temperature, the peak in the black body spectrum is at ∼ 9.7 µm. This is a fair bit longer than relevant transition
wavelengths in Sr, and so the dominant effect of BBR is a DC-Stark shift (see Sec. 2.1.1.5) that is proportional to ⟨E 2 ⟩, where
E is the electric field that is present due to thermal fluctuations. However, for clocks operating with state-of-the-art accuracy,
corrections from the AC polarizability can be relevant too.

102
require a full accuracy evaluation, measurements of differences in the gravitational redshift at the Earth’s
surface on the millimeter scale have already been achieved using optical clocks [35].

5.2

Optical clock qubits
To summarize the last section, for the purposes of frequency metrology we are interested in working

with qubits that balance several features. For starters, we want qubits with a large energy separation, longcoherence times, and well-characterized evolution. These ingredients are met by using neutral Sr atoms
trapped in magic wavelength optical potentials (see Sec. 2.2.3), which have been extensively characterized
and used to great effect in state-of-the-art optical lattice clocks [203]. Additionally, we want to balance
the desire for large ensembles of qubits with fast, high fidelity operations (including both manipulations of
the qubits and readout). Optical lattice clocks tend to operate with large ensembles of thousands to tensof thousands of atoms, but do not provide access to controls and observables at the single atom level.27
By contrast, trapped ion clocks typically work with a single ion, but achieve fast and high fidelity control
and readout of that ion [42]. In our work, we attempt to balance the above considerations by scaling toensembles of intermediate size (∼ 100 atoms), while maintaining high fidelity, single-atom-resolved control
and readout.
Throughout this chapter, we will report on results in three slightly different experimental conditions,differentiated by the specific science potential28 used:(1) An array of 10 tightly confining clock-magic wavelength tweezers [240].
(2) An array of 320 shallow clock-magic wavelength tweezers [365].
(3) A clock-magic wavelength 3D optical lattice [290, 96].
In (2) and (3), atoms are prepared via implantation from a separate set of tightly confining preparation
tweezers (see Ch. 3). In all cases, the optical traps are initialized with atoms that are laser-cooled to near
their 3D motional ground states. The cooling performance in (1) and (2) are similar, and worse than that
27 Although some progress has been made in this direction [213].
28 As defined in Sec. 3.3.

103
achieved in (3) (see Sec. 4.3.3). With the above considerations in mind, we will use a qubit degree of freedomthat is encoded in the |1 S0 ⟩ ground and |3 P0 ⟩ clock electronic states (see Sec. 2.1.1) of an optically trapped
Sr atom. That being the case, we will sometimes refer to the atoms as “optical clock qubits” or just “clock
qubits”, with states |0⟩ and |1⟩ referring to the ground and clock states respectively.
The three different modes of operation are the result of balancing different conflicting requirements. To
minimize the effect of QPN, we are interested in preparing and interrogating large ensembles of clock qubits
in parallel. However, given finite optical power in a tweezer array, an increase in sample size comes at theexpense of trap depth and atomic confinement, with implications for detection fidelity, cooling performance,
qubit coherence, and atomic loading.29 Using different optical potentials optimized for different stages of theexperiment allows us to circumvent some of these challenges. By using an optical lattice potential in condition(3) [290, 365], we are able to make many traps that are at the clock-magic wavelength (Sec. 2.2.3) and tightly
confining, enabling high fidelity imaging, ground state cooling (Ch. 4), and single qubit gates (Sec. 5.2.2).
The combination of relatively long-lived atomic coherence with high fidelity single qubit gates will be criticalin Ch. 6. However, as we will see in Sec. 5.3, a conventional optical lattice is not ideal when optimizing
for atomic coherence as opposed to gate fidelity, motivating the experiments in condition (2). Working
in condition (2) results in relatively high atom loss incurred when imaging in 515 nm potentials [73, 241]
compared to the performance possible in (1) or (3) [74, 240, 290, 367] (see Sec. 4.4). However, this loss
is not a significant issue in the specific case where one is optimizing for high precision measurements with
non-entangled clock qubits, since the constant reduction in precision due to preparation and measurement
errors is more than offset by the enhancement in coherence time.

5.2.1

Measurement
In all experiments, we take an initial image to identify the positions of the atoms (see Sec. 3.4). Afterrunning a given experimental sequence, we can either measure the states of the qubits non-destructively or
destructively. In the former case, we simply image atoms in the |0⟩ state as described in Sec. 4.4. Optionally,
after this image one can perform a π-pulse to swap the |0⟩ and |1⟩ populations, and take an additional image
29 These considerations are what limits the system size to 10 tweezers in (1) [240].

(a)

(b)

0

Clock light shift (Hz)Clock detuning (kHz)

0

P1

104

(c)

-20

-40

-60
0

200

400
B (G)

600

100

200
0

300
600
900
Rabi frequency (Hz)Figure 5.2: Clock Rabi oscillations. (a) P1 refers to the (measurement error corrected) probability ofobserving an atom in the 3 P0 state, and is measured after initializing the atoms in the 1 S0 state and applying
a laser probe resonant with the 1 S0 ↔ 3 P0 clock transition for variable duration. (b) Due to the quadratic
Zeeman shift, the resonant frequency of the clock transition varies with the applied magnetic field. We
measure the strength of this shift to be β = −226(5) mHz/G2 . (c) The probe laser can also apply a differential
light shift to the clock transition, leading to an increasing energy shift for higher clock Rabi frequencies. For
normal operating conditions in our experiment, we observe a probe shift of −280(10) mHz/Hz2 .

in order to distinguish between atoms that are in the clock state and atoms that are lost. The above scheme
has the downside that |1⟩ can decay to |0⟩ during detection, leading to incorrect identification of the stateof a qubit. In condition (1), for 3900 to 6000 ER deep tweezers, where ER is the recoil energy of an 813 nm
photon, this leads to a 5 to 9% probability that |1⟩ decays to |0⟩ during a measurement [240]. In the caseof destructive measurements, one can mitigate errors due to decay by applying a “blow-away” pulse of lightresonant with the 1 S0 ↔ 1 P1 transition to remove atoms that are in the |0⟩ state. The blow-away operationis very fast in comparison to the lifetime of |1⟩ (< 100 µs in comparison to seconds), and so mapping |0⟩
to the vacuum state with no atom can be performed with negligible infidelity. The remaining atoms in |1⟩
can then be repumped to the ground state (see Sec. 2.1.1.4), which can also be performed with effectively
no errors, and imaged. As a result, the destructive measurement scheme is limited only by imaging fidelity
and loss, resulting in readout fidelities as high as 99.9% (see Sec. 4.4). These high detection fidelities mean
that, unlike typical lattice clocks [203], tweezer clocks have effectively no uncertainty due to fluctuations in
atom number or detection errors.

105
5.2.2

Single qubit gates
After loading ground-state-cooled atoms into the science potential of choice, we can interrogate theclock transition. We apply a magnetic field between 1.4 G and 550 G to mix the 3 P1 state into the 3 P0 state(see Sec. 2.1.1.3). By applying “probe” laser light that is referenced to an ultra-stable crystalline cavity [242](see Appendix D) and resonant with this transition, we can drive Rabi oscillations between |0⟩ and |1⟩ (seeSec. 2.2.1). These Rabi oscillations correspond to global rotations about the x axis of the Bloch sphere, andcan be performed with Rabi frequencies as high as Ω ≃ 2π × 3 kHz at B = 550 G and with ∼ 100 mWof optical power. A measurement of these Rabi oscillations under typical operating conditions is shown in
Fig. 5.2a. Global rotations about the z axis can be performed by simply detuning the probe laser from theatoms, changing the phase of the rotating frame.30

Local rotations about the z axis can be performed

by applying a local differential light shift between the ground and clock states for a given atom using a
non-magic optical tweezer. Similar local operations are used to apply the oracle in Ch. 7 [366], and aredemonstrated in the context of optical clock measurements in [96].31 The combination of local z rotations
and global x rotations constitutes a universal set of single qubit gates. We will discuss entangling gates andthe possibility of a complete universal gate set in Ch. 6.
The above scheme for performing single qubit gates comes with a few additional considerations.
Although both the ground and clock state have total angular momentum J = 0, and thus a vanishing linear
Zeeman shift, one does have to consider the quadratic Zeeman shift (Fig. 5.2b). Rewriting Eq. 2.5, we have:∆B = βB 2 ,(5.16)

where B is the strength of the applied magnetic field, and β = −233 mHz/G2 [247]. In combination withmagnetic field noise, the quadratic Zeeman shift leads to a tradeoff between the speed of the σx gates, which
prefers larger B, and the sensitivity of the qubits to magnetic field noise, which prefers smaller B. For
example, a B-field stability in the range of 1 mG is typical in most experimental labs. For B = 550 G, a
1 mG fluctuation leads to a shift of 0.26 Hz, which is enough to dramatically reduce the coherence times of
30 These z rotations can be performed with essentially perfect fidelity and on the scale of 100 ns, since they amount tochanging the phase of a digitally synthesized RF signal that controls the AODs used to switch the probe beam on and off.
31 We leave a more detailed discussion of local z gates to later works from the group.

(a)

(b)
1.0

0.6
0.4
0.2
0.0
0

2

4
6
8
Probe pulse time (ms)

10

12

9

0.15

0.8

★

8
7

0.6

0.10

6
5

0.4

4

0.05

- Log10 (ϵ)

0.8nPopulat ion fract ion Pn

P0 population

1.0

3

106

(c)

3

0.2

2

0.0

0

1
2
3
4
Mot ional st at e n

0.00
0.0

0.2

0.4

0.6

0.8

1

η

Figure 5.3: Effect of motion on Rabi oscillations. (a) Exaggerated example (with more heating than ispresent in a typical experiment) of how thermal motional excitations affect Rabi oscillations between the
ground and clock electronic states. (b) One can fit the measured Rabi oscillations to extract the occupied
motional states along the axis of the probe laser, in this case yielding a ground state fraction of ∼ 0.8. (c)Predicted π-pulse error ϵ as a function of Lamb-Dicke parameter η and the average motional state occupationn̄ (along the axis of the probe laser, and assuming a thermal distribution). Error bars in this figure correspondto 95% confidence intervals.

the atoms in comparison to the measured values (see Sec. 5.3). For B = 20 G, the equivalent shift is 9 mHz,and negligible compared to the measured coherence times.
It is also important to note that the presence of the probe light results in a differential AC Stark
shift between the ground and clock states (see Sec. 2.2.2), leading to a shift of the qubit energy which mustbe accounted for when applying control pulses (Fig. 5.2c). This probe shift is only present when the probe
beam is on, and so it has a negligible impact on clock measurements with a long dark time during which theatoms are allowed to evolve freely. However, it is important to control the homogeneity of the probe beam
so that the gate applied to different atoms in a given ensemble is identical.
The final consideration for the fidelity of the applied single qubit gates is the motion of the atoms.32 To
calculate the expected clock π-pulse fidelity achievable given our known cooling performance and confinement,
consider the motion-dependent Rabi frequency associated with the s = 0 or carrier transition in Eq. 4.8:22Ω(n) = Ωe−η /2 L(0)n (η ).

(5.17)

We can average the evolution resulting from Eq. 5.17 over a thermal ensemble (of different values of n) givenby the measured atom temperatures in our experiment to predict the optimal π-pulse fidelities achievable in
our apparatus. These fidelities are shown for different combinations of cooling performance and confinement
32 Note that, based on characterizations of the probe laser system (see Appendix D), we do not expect intensity or frequency

noise to contribute significantly to errors in the applied single qubit gates.

107
in Fig. 5.3c, along with an example where significant occupation of motionally excited states results indynamics predicted by Eq. 5.17.
In condition (2) [365], we use weakly confining magic wavelength tweezers with a typical depth of
15ER , leading to a radial trap frequency of ftrap = 6.8 kHz and η = 0.83. For measurements in these
conditions, the cooling and handoff procedure led to an inferred average phonon occupation of ⟨n⟩ ≃ 0.14
along the direction of the clock probe (see Sec. 4.3). Additionally, imaging and imperfect calibration of thehandoff procedure led to ∼ 9% additional losses in these measurements. Given these losses, the expected
π-pulse fidelity is 0.81, which is consistent with the measured value of 0.82(2) without correcting for state
preparation and measurement (SPAM) errors.
In condition (3) [290], we optimize for the fidelity of single qubit operations33 by using more tightly
confining potentials, and improving the quality of state preparation and readout. Specifically, we use a
420ER deep 2D lattice,34 resulting in η ≃ 0.22 and n̄ = 0.04+0.11−0.04 in the direction of the clock probe. In
this case, we expect to achieve a π-pulse fidelity of 0.9998+2−9 . However, by fitting the observed clock Rabi
oscillations with Eq. 5.17, we can extract the distribution of populated motional states during the single
qubit gates (see Fig. 5.3b), and find that some additional heating occurs during the experimental sequencewhich degrades the initial state preparation (see Ch. 6 for details). This heating leads to an expected clock
π-pulse fidelity of 0.9979, which is to be compared with the measured value of 0.9941(57) (with SPAM
correction). In more recent (preliminary) measurements in the lattice with reduced heating and η ≃ 0.18,
we have measured π-pulse fidelities as high as 0.999+1−2 with SPAM correction, and 0.995(2) without.

5.2.3

Rabi spectroscopy
One way to compare the LO frequency to the atomic frequency is to perform Rabi spectroscopy, where

one calibrates a resonant π-pulse as in the previous section, and then applies the same pulse to |0⟩ atoms asa function of LO frequency before projectively measuring the atoms in the z basis. For square pulses in the
33 At the cost of other performance metrics, as described in Sec. 5.3.
34 And a 400E deep axial lattice, but this is less relevant for the performance of the clock rotations, since the probe beamR
lies in the plane of the 2D lattice, and orthogonal to the axial lattice.

108

(a)

(b)
1.0

0.0
1.0

3

2

1

0

1

2

3

50

0.5
0.0

Number of traps

Measured 3P0 fraction

100
0.5

0
-0.1 0 0.1
30 20 10 0 10 20 30
Trap-dependent
Detuning (Hz)shift (Hz)Figure 5.4: Clock Rabi spectroscopy. (a) Array-averaged Rabi spectroscopy of the clock transition in
an array of 320 tweezers, with Fourier-limited linewidths of 10.1(2) Hz and 0.62(1) Hz (full width at half
maximum). Solid lines correspond to the expected spectral feature given the known probe durations usedin each case. Callout (top) shows the 0.6 Hz feature in detail. Note that the amplitude of the 0.6 Hzfeature is not reduced in comparison to the 10 Hz feature, suggesting that this feature is not yet significantly
affected by inhomogeneous broadening or other effects resulting in decoherence. (b) A histogram of the fitted
transition frequency for different tweezers yields a standard deviation of 0.039(2) Hz, providing an estimateof the magnitude of inhomogeneous, tweezer-dependent shifts of the clock transition.

109
intensity of the probe beam of duration tp , the resulting spectral feature is:
p1 =

Ω2
sin2
2
Ω + δ2
tp p 2Ω + δ2 ,2(5.18)

where p1 refers to the probability of measuring the atoms in |1⟩, and δ := f −fLO is the detuning between theatomic reference and the LO. For lower Rabi frequencies and thus longer probe times, the spectral feature in
Eq. 5.18 narrows, giving a better discriminator for fluctuations in δ. At very low Rabi frequencies, differential
frequency noise between the LO and the atoms leads to a lineshape that is independent of Rabi frequency,
and typically Gaussian. In condition (1), and for B = 1.4 G and Ω = 2π × 125 mHz, we reach a linewidth
as low as 350(50) mHz, suggesting several seconds of coherence between the atoms and the LO (also see
Sec. 5.2.4). Similar performance is obtained in condition (2), as shown in Fig. 5.4.
In a typical clock operating using Rabi spectroscopy, one probes the Rabi feature at its steepest point,where the excitation probability is p1 ≃ 0.54. By additionally probing either side of the feature sequentially,
one is able to decouple variations in ⟨z⟩ due to fluctuations in δ from variations due to other effects, likefluctuations in atom number or laser power. Such a scheme is often employed to simplify the control sequencein a clock measurement and to minimize probe shifts [242], but does not saturate the SQL (although it does
follow the same scaling with respect to M and N ).

5.2.4

Ramsey spectroscopy
An alternative approach to inferring δ from the atomic population is to perform a Ramsey measure-ment. In this case, one performs a π/2 rotation about the y axis of the Bloch sphere to prepare the state√|+⟩ = (|0⟩ + |1⟩)/ 2, then allows the state to evolve under the Hamiltonian H = hδσz /2 (in the rotating
frame) for interrogation time t, and finally performs another π/2 rotation about the x axis before protectively
measuring the atoms in the z basis (see Fig. 5.5a). This leads to a signal of ⟨σz ⟩ = C sin(2πδt), where C
is the contrast of the Ramsey fringe (with C = 1 in the ideal case). Notice that |+⟩ is precisely the statein Eq. 5.8 which we used to derive the SQL. In fact, for δ ∼ 0 and C = 1, where the slope of the signal is
maximized, the estimator 2πδ = ⟨σz ⟩/t is locally unbiased, consistent, and efficient in the sense described inSec. 5.1. As a result, perfect Ramsey measurements saturate the SQL [120].

110

(a)

Clock

100
80
60
40
20

30
20

20
20
30
00

10
20
20
00

20
10
00

0
0

3

P0 transfer fraction (%)(b)

Ramsey evolution time (ms)Figure 5.5: Clock Ramsey spectroscopy. (a) Under ideal conditions, the Ramsey sequence described in the
text results in the atoms being on the equator of the Bloch sphere, and therefore maximum sensitivity todeviations in the phase of the clock laser. However, uncontrolled fluctuations unrelated to the laser itself,
for example residual motion leading to random displacements of the atoms along the propagation axis of
the laser, can also result in phase shifts. Averaging over these phase shifts results in a shortening of the
Bloch vector, and therefore a reduction in the contrast of the Ramsey fringe. (b) Measurements of Ramsey
oscillations as a function of free evolution time between the first and second Ramsey pulses. Note that inthis case the laser is intentionally detuned slightly from the atomic resonance, leading to oscillations in time.
At late times, the contrast of these oscillations decays due to the loss of coherence between the atoms andthe laser.

111
In our experiments, Ramsey measurements performed as a function t show that the contrast of theRamsey oscillations typically decay with a Gaussian time constant of 3.6(2) s [240, 365] (Fig. 5.5b).35 Thiscan be thought of as the amount of time over which the rotating frame defined by the LO remains coherent
with the rotating frame of the atoms. Because our single qubit gates depend on the presence of the LO, thisis also the coherence time against which one should compare the duration of the single qubit gates.36
In the following sections, we will see that the coherence time of the atoms relative to each other issignificantly higher than the atom-LO coherence time, and so most of this dephasing can be associated with
the LO (although at high magnetic fields, magnetic field noise and correlated fluctuations of the atomic
reference frequency can also play a role). Our clock probe light comes from a laser that is injection lockedwith light stabilized to a cryogenic silicon reference cavity [242]. In principle, this laser system can reachremarkable linewidths of a few millihertz [218, 167]. However, the performance of this laser is degraded in
the reference frame of the atoms, primarily due to mechanical vibrations that can lead to phase shifts in theapplied laser light (see Sec. D.2.1). The clock laser system is described in more detail in Appendix D. For
now, we will just point out one key challenge associated with tweezer array clocks: whereas in optical latticeclocks, the cavity mirror which defines the position of the optical lattice serves as a very good phase reference
for the position of the atoms [34], a tweezer array lacks a similar reference of the atom positions, and so we
instead resort to using the position of the microscope objective as a phase reference (see Sec. 3.3). Although
this eliminates the effect of atomic motion due to motion of the objective itself, it does not, for example,
eliminate motion due to air currents in the tweezer system.37 We suspect that the atom-laser coherence islimited by residual motion of the tweezer array (or lattice), and hope to address this shortcoming in future
upgrades (for example with a different lattice design, or more sophisticated active feedback of the tweezer
positions).
35 This decay time was measured with B = 22 G, but is not significantly affected by higher B.
36 In Sec. 5.4.2 we will see that useful things can still be done even in the limit where the LO is completely dephased withrespect to the atoms.
37 Due to the bowtie design of our lattice, the final folding mirror could act as a good phase reference for the atom positions,but we do not currently take advantage of this in our experiments.

1

0.06

S0

3

P0

0.05
0.04

25

0.03

Lifetime (s)Population loss rate (s 1)

112

0.02

50

0.01

100
200

0.00
0

10

20 30 40 50
Trap depth (ER)60

Figure 5.6: Ground and clock state lifetimes. To determine limits on atomic coherence, we measure thelifetime of both the ground (1 S0 , black points) and clock (3 P0 , open circles) states. For ground-state atoms
the lifetime saturates to 160(10) s in deep traps, with additional technical sources of atom loss contributing
in shallower traps (exponential fit to 1 S0 data, dark grey). For clock state atoms an optimal trap deptharises from a competition between depumping via spontaneous Raman scattering of the trap light (theory
prediction with no free parameters, light grey) and atom loss. The combination of the two loss mechanisms(dashed line) is in good agreement with the measured lifetimes, including with the optimal value of 46(5) s
at a depth of 14ER .

5.3

Preserving coherence and minimizing frequency shifts
So far, we have discussed how to manipulate and read out individual clock qubits. What remains isto engineer an environment that minimizes systematic shifts of the qubit transition frequency, and preserves
the coherence of the qubits to enable long interrogation times.

5.3.1

Atom loss
One significant limitation to the atomic coherence time is simply the fact that atoms can be lostfrom the optical traps. We expect the lifetime of an atom in an optical trap to be limited by collisions
with residual background gas. Assuming that the main collision partners are room temperature Σ state
H2 molecules interacting with the atoms via van der Waals forces, and following the procedure in [341], we
expect that such collisions are substantially more energetic than the trap depths we have access to. As aresult, we expect the vacuum lifetime to effectively be independent of trap depth [16]. For trap depths inexcess of ∼ 20ER , we see this borne out in measurements of the lifetime of 1 S0 ground state atoms (see113
Fig. 5.6), where we measure a trap-depth-independent lifetime of τ = 160(10) s (see Fig. 5.6).38

More

recent measurements where we operate with a lower oven temperature of 380 C instead of 420 C improve
the quality of our vacuum, and yield an improved collision-limited lifetime of ∼ 400 s.
Similar collision processes apply to atoms that are in the clock state, however atoms in electronically−2/5

excited states have larger C6 coefficients, and thus larger scattering cross sections σ ∝ C6

. The result isa reduced trap lifetime of τ ∝ 1/σ. With the known C6 coefficients for collisions between H2 and 88 Sr [224],
we calculate that the ratio between the ground and clock-state trap lifetime (τg and τe respectively) isτg /τe = 1.10, which agrees with results from [93].
The source of the dramatic reduction of trap lifetime in shallow traps with depths of ≲ 20ER (seeFig. 5.6) is not yet fully understood. However, based on the above analysis, we rule out the effect ofcollisions with background gas. One possible mechanism for the observed loss is parametric heating related
to fluctuations in the trapping potential. There are two main mechanisms for such heating: intensity noisein the trapping laser manifests as parametric modulation of the trap frequency which, assuming a flat power
spectrum in intensity noise, results in exponential heating (measured in phonon number) with a time constant
proportional to ft2 , where ft is the trap frequency [287, 115]. Similarly, pointing noise with a flat power
spectrum (in terms of fluctuations in the center position of the trap) results in linear heating with a rate
proportional to ft3 (measured in phonon number) [287, 115]. For comparison, the number of bound states,
N , in an optical trap scales roughly like N ∝√V ∝ ft , where V is the trap depth. This means that both ofthese sources of loss should improve with reduced trap depth assuming a flat noise spectrum. The intensity
noise of our trapping laser is suppressed below 10 kHz via a servo, and otherwise relatively flat over thefrequencies relevant for heating. As a result, we do not expect heating due to intensity noise to be the sourceof the observed loss in shallow traps. However, there could be increased pointing noise at lower frequencies
due to mechanical resonances and acoustic noise, and there is no convenient way of removing such noise witha servo. We therefore hypothesize that pointing noise contributes to our reduced lifetimes at and below trap
depths of 15ER (corresponding to 6.8 kHz trap frequencies). Other sources of trap-independent heating, likescattered background light, can also begin to dominate as the traps become very shallow and N becomes
38 Given this lifetime, we estimate that the fractional frequency shifts due to the background collisions [116, 34] are below
the 10−19 level, and thus negligible in our experiments.

114
Measured lifetimes (s)
1/Γtg
1/Γte
1/Γe

101(6)
92(5)∗
43(4)Expected rates (×10−3 s−1 )ΓBBR(3 P0 → 3 P0 )
0
BBR 3
Γ1
( P0 → 3 P1 )BBR 3
Γ2
( P0 → 3 P2 )R 3
Γ0 ( P0 → 3 P0 )33ΓR
12 ( P0 → P1,2 )[93]

3.45(22)
2.23(14)0.105(7)0.557 (V/ER )0.782 (V/ER )Table 5.1: Measured and theoretical values contributing to predicted Ramsey lifetime. All measured values
are for a trap depth of 15 ER , based on interpolating between the nearest points in Fig. 5.6. ∗ indicates thatthe inferred value of Γte is dependent on the calculations presented in Sec. 5.3.1. Note that ΓBBR
is smaller
2
than the error bars on the other processes, and so we neglect this process in our analysis. V is the trap
depth and ER is the recoil energy of an 813 nm photon.

small.

5.3.2

Clock state lifetime and coherence
Besides being lost from the traps, atoms in the 3 P0 clock state can also undergo transitions to otherelectronic states. Measurements of the 3 P0 state lifetime as a function of trap depth are shown in Fig. 5.6.
Note that due to our use of 88 Sr, and given the strength of the magnetic fields used, the effects of spontaneousemission from 3 P0 to 1 S0 are negligible. For typical experimental conditions,39

the dominant process

leading to transitions out of 3 P0 is Raman scattering of the trap light, with the main contributions beingfrom 3 P0 → 3 P1 and 3 P0 → 3 P2 . For π-polarized trap light, these two processes occur with rates of−1−1
and 2.84 × 10−4 s−1 ER
respectively [93]. Note that while the ratio of these two
4.98 × 10−4 s−1 ER−4 −1 −1
scattering processes is polarization-dependent, their sum, with a value of ΓR
s ER , is12 = 7.82 × 10

conserved. All population driven into 3 P1 can be assumed to immediately decay into the ground state,
whereas processes that return population in 3 P2 to the 3 P0 state are negligible. As a result, V ΓR
12 can betreated as the total rate at which population in 3 P0 is depleted due to Raman scattering of the trap light.
Another mechanism that depopulates the 3 P0 state is scattering of black-body radiation (BBR). The
dominant BBR-induced decay at room temperature is due to off-resonant driving from 3 P0 to the 3 D1 state,
39 Meaning magnetic fields of B ≪ 1 T, and trap depths of V

≳ 10ER .

115
which decays to the 3 PJ manifold with branching ratios of RJD = 59.65%, 38.52%, and 1.82% for the J = 0, 1
and 2 states respectively [93]. The dominant mechanism by which BBR contributes to loss from the 3 P0 stateis via population that branches from 3 D1 into 3 P1 , and subsequently decays into the 1 S0 ground state. This
process occurs at a rate of ΓBBR
= 2.23 × 10−3 s−1 [93] at room temperature. The sum of the above effects
1
with the rate at which 3 P0 state atoms are lost from tweezers with depth V , Γte (V ), is in good agreement
BBR
with the measured 3 P0 decay rate, Γe (V ) = Γte (V ) + V ΓR
(theory curve in Fig. 5.6).
12 + Γ1

Measurements of the 3 P0 state lifetime can be confounded by the presence of atoms pumped into the3

P2 state, which are not distinguished from 3 P0 -state atoms during our normal blow-away measurement (seeSec. 5.2.1), and can lead to an artificially long inferred lifetime. To avoid this systematic error, we perform
two separate measurements. In one measurement, we add a repumping step that depletes 3 P2 atoms before
the blow-away by driving the 3 P2 ↔ 3 S1 transition at 707 nm (see Sec. 2.1.1.4). In the second measurement,
we remove the repumping step to measure the total 3 P0 + 3 P2 population. Based on these two measurements
and the branching ratios of 3 S1 we infer the true population in 3 P0 , which appears in Fig. 5.6.
Given the decay processes described above, we can compute an expected Ramsey coherence time τ .40
Due to the magic-wavelength traps, Rayleigh scattering of the trap light does not cause decoherence [337,214, 93].41

As a result, trap-induced scattering only contributes to decay of Ramsey contrast through theRaman scattering processes described above that depopulate the 3 P0 state. Unlike Rayleigh scattering of thetrap light, BBR-driven processes that transfer population out of and back into the 3 P0 state (predominantly
via 3 D1 ) can serve as an extra source of decoherence that is not directly reflected in the measurement of the3

P0 state lifetime.42 Including all the above effects, the inferred Ramsey coherence time is:BBRτ = 2/(Γte + Γtg + V ΓR
(1 +12 + Γ1

R0D
)),
R1D

(5.19)

where Γtg,e are the vacuum loss rates of ground and clock state atoms respectively. All the relevant rates in
Eq. 5.19 are summarized in Tab. 5.1.
40 Defined as the interrogation time at which the contrast of the Ramsey fringe decays to 1/e of its zero-time value, seeSec. 5.2.4.
41 One can see this schematically from the fact that it is not possible to identify the electronic state of the atom based on
the properties of the scattered photons.
42 In the case of the weak drive provided by BBR, the dominant process is one in which the atom is transferred to the excitedstate, and then spontaneously emits a photon. Here, even if the atom returns to the 3 P0 state, it is dephased with respect toan atom that did not scatter a BBR photon. This is to be compared to the Raman process driven by the trapping light, where
the strong coherent drive leads to coherent absorption and stimulated emission into the trapping beam.

116
Given the above analysis and measurements of the trap lifetime in Sec. 5.3.1, the optimal trap depth
for performing a Ramsey measurement is approximately 14ER . Here, the measured ground state trap lifetime
is 96(8) s, leading to a predicted clock state lifetime of 44(6) s. The measured clock state lifetime is 46(5) s,
in good agreement with the predicted value. Under these conditions, the Ramsey contrast should decay
exponentially with a time constant of τ = 55(8) s. We will see in Sec. 5.4.2.2 that the measured atomic
coherence time is in agreement with the predicted Ramsey coherence time.

5.3.3

Tunneling-induced phase noise
The lifetimes measured in Sec. 5.3.2 are encouraging, but when operating at low trap depths andfor long evolution times one must be wary of tunneling or thermal hopping, where atoms are able to move
between different sites in the array of optical traps. Because the spatial phase of the LO can differ fromsite to site, such hopping effectively results in fluctuations in the phase of the LO as experienced by theatoms. A simple way to mitigate this source of phase noise is to eliminate tunneling through appropriate
control of the trapping potential [148]. In our system, we can achieve this without adjusting the trap depth
by controlling the spacing between tweezers to reduce the tunneling energy. Tunneling is further suppressedby the inherent ∼ 10−2 level disorder in the depths of the tweezers (see Sec. 3.3.3).43
For pairs of images in a given experimental trial, we expect tunneling and thermal hopping to manifest
as correlated x01 and x10 events (see Sec. 4.4.1) on nearest-neighbor sites, where an atom tunnels from onesite to an empty adjacent site between images, or pairs of x10 events, where an atom tunnels onto an occupied
adjacent site after the first image, and both atoms are lost due to light-assisted collisions during the second
image (see Sec. 4.2). For tweezers with a spacing of 1.2 µm and depths in excess of V = 6ER , we do
not observe an excess of such events beyond what is expected due to loss and imaging infidelity (for holdtimes of up to hundreds of seconds). The lack of tunneling at low trap depths allows us to operate at thenearly-optimal depth of V = 15ER to maximize the lifetime of the 3 P0 state without concerns of introducing
substantial tunneling and thus LO phase noise.
The ability to operate at very low trap depths for clock operation in a tweezer array is in contrast
43 See Ch. 7 for a more in depth discussion of tunneling.

117
to a typical retro-reflected lattice, where the lattice spacing is not independently tunable from the trapping
wavelength. In the 3D lattice clock at JILA, tunneling can limit the achievable interrogation times at depths
below ∼ 30ER along a single axis (and a combined depth of ∼ 100ER for all three axes) without additional
mitigation [148]. However, there are several approaches to eliminating LO phase noise due to tunneling inan optical lattice. For example, one can tune the lattice spacing using other lattice geometries to either
eliminate tunneling, or to make the array spacing commensurate with the spatial phase of the LO [148].
Appropriately orienting the lattice with respect to gravity can also eliminate tunneling by means of applying
a spatial gradient across the lattice [187, 35, 373, 3]. If the tunneling is suppressed in one direction, forexample along the gravitational axis, one can also orient the LO beam such that the spatial phase of the LO
is equal for all sites that are tunnel-coupled to each other, resulting in a fixed LO phase even in the presenceof tunneling. Finally, in hybrid systems like ours that integrate optical lattices with high NA microscopes,
one could use the microscope to project a disorder potential onto a shallow lattice to eliminate tunneling.

5.3.4

Frequency shifts
While a full accuracy evaluation is beyond the scope of this work, the prospects for using the tweezer

platform as an absolute frequency reference are fairly good: many of the dominant systematic effects intweezer clocks are shared with optical lattice clocks [240], which currently have accuracies at or below the10−18 level [31, 234, 34]. However, there are a couple extra considerations that one must be careful of in a
tweezer clock.

5.3.4.1

Variations in trapping wavelength

One consideration is the fact that our tweezer array is generated using AODs (see Sec. 3.3.3), leading
to different tweezers being detuned from each other, preventing us from attaining magic trapping conditions
across the entire array. Although these megahertz-scale detunings are small compared to the hundreds ofterahertz-detunings of the trapping beams from resonance, they are still sufficient to result in measurable
differential frequency shifts between atoms in different traps.
For deep traps ranging between V = 450 − 1800ER in condition (1), we perform Ramsey spectroscopy

118

Tweezer index

5 MHz

(b)

Frequency shift (Hz)

Relative phase (rad.)

(a)

1.5

0.0

1.5
0

200

400

Ramsey evolution time (ms)0.5
0.0
450 Er900 Er1800 Er

0.5

1

4

7

10

Tweezer index

Figure 5.7: Trap-induced clock frequency shifts. (a) Measured relative Ramsey phase shifts for tweezers
at different positions (and therefore frequencies), and at a depth of 1800ER . (b) Inferred relative frequency
shifts as a function of tweezer position at a range of tweezer depths. Solid lines are predictions based on the
known sensitivity of the magic wavelength to trap detuning.

Measured 3P0 fraction

119

1

1.0
0.8

0
0.0

0.6

0.1

3.0

3.1 30.0 30.1 50.0 50.1

0.4
0.2
0.0
0

10

20
30
40
Ramsey dark time (s)5060Figure 5.8: Ramsey spectroscopy in large arrays. We perform Ramsey measurements (black points) in anarray of 320 tweezers, each with a depth of 15ER , which is nearly optimal for preserving atomic coherence.
The measured clock-state lifetime (see Fig. 5.6) contributes to a decay of the Ramsey contrast with an
expected exponential time constant of 55(8) s (light-grey region). However, tweezer-dependent light shifts
further increase site-to-site dephasing, leading to Gaussian decay with an expected time constant of 33(1) s(at 15ER , and for the array geometry used in this measurement). The combination of the above two effects isdenoted by the medium-grey region. Note that each data point corresponds to a single shot of the experiment.
As a result, despite the fact that the atom-laser coherence decays with a Gaussian time constant of 3.6(2) s(dark-grey region, see also Fig. 5.5), the variance of the Ramsey signal decays on a timescale set by atomic
coherence. This is clarified by the insets, which share units with the main axes, and show detailed views of
the Ramsey fringe at a few different free evolution times. Here, it is possible to see the initial loss of phasecoherence with the laser followed, at later times, by total loss of coherence (and thus variance in this signal)in the system.

to measure the frequency shifts imposed by the tweezers (see Fig. 5.7). The results are in line with the
expectation of the differential polarizability of the clock states near the 813 nm magic wavelength for Sr (seeSec. 5.3.4.2).
In condition (2), we operate with shallow trap depths of 15ER and with large atom arrays with tweezer
frequencies differing by up to ∼ 60 MHz (at opposite corners of the array). In our measurements under these
conditions, the signal at each Ramsey time is a single-shot measurement such that even though the atom-laser
coherence decays over ∼ 3 s (see Sec. 5.2.4), we can infer the coherence of the atoms relative to each otherfrom the variance of the Ramsey signal, which remains high on much longer timescales (Fig. 5.8, see alsoSec. 5.4.2.2). At short times, the frequency of the Ramsey fringes is set by the differential light shift imposedby the probe beam on the 1 S0 and 3 P0 states. At longer times, the loss of atom-laser coherence manifests asa randomized phase of the second π/2 pulse in the Ramsey sequence, which obscures the Ramsey oscillations
but preserves the probability of large excursions due to the persistence of atomic coherence (see Sec. 5.4.2.1).

120
The Ramsey contrast inferred from this measurement decays with a 1/e time of 19.5(8) s (Fig. 5.8), corresponding to an effective quality factor of Q = 1.9(1) × 1016 . The measured decay time is slightly fasterthan the prediction of 33(1) s based on the measured lifetime, and on the expected dephasing due to thedifferential AC Stark shifts imposed by the tweezers. Note, however, that dephasing due to differential AC
Stark shifts is static (assuming the optical traps are relatively stable and do not fluctuate in intensity or
frequency), and thus can easily be eliminated with a spin echo. Additionally, alternative approaches tomaking optical traps, like tweezer arrays based on spatial light modulators or digital micromirror devices,or accordion lattices, can avoid this issue entirely while still maintaining additional control over the exact
shape and geometry of the optical potential.

5.3.4.2

Hyperpolarizability and trap geometry

To accurately determine the energy shift imposed on the clock transition due to the optical traps, onemust consider modifications to the low order expressions appearing in Sec. 2.2.2. This is particularly relevantin the case of a tightly confining tweezer array, where the intensity and polarization of the trapping fieldcan vary rapidly in space [164, 323], leading to modified expressions for the light shift. The complications
associated with higher order corrections to the trapping potential are discussed in detail in [248, 159, 49] inthe context of optical lattices, and applied to tweezer arrays in [209]. The typical approach is to compute theshift in energy of the ground and clock states due to the electric dipole α(E1) ,44 magnetic dipole α(M1) , andelectric quadrupole α(E2) polarizabilities, as well as the hyperpolarizability β. The resulting optical potential
is:45


V (ξ, f, r) = − α(E1) (ξ, f ) + α(M1) (ξ, f ) + α(E2) (ξ, f ) I(r) − β(ξ, f )I(r)2 ,(5.20)

where f is the frequency of the laser field, and ξ ∈ {e, g} denotes the electronic state of the atom.46
For the purposes of understanding the observed frequency shifts between tweezers, we are interested inisolating the lowest order terms in Eq. 5.20 that contribute to a frequency shift fLS = (V (e, f, r) − V (g, f, r)) /h
44 The discussion in Sec. 2.2.2 amounts to only handling the α(E1) contribution to the energy shift.
45 This form assumes a travelling wave, but in general one must be careful about a spatial phase shift between the α(E1) ,α(M1) , and α(E2) contributions both for lattice potentials, and very tightly focused tweezer potentials.
46 In comparison to Sec. 2.2.2, we have taken β → ξ to avoid confusion with the hyperpolarizability, and are working withfrequencies f rather than angular frequencies ω.

121of the clock transition, and depend on the detuning δm = f − fm of the trapping laser from the magic trapping frequency fm ≃ 369 THz. We can solve for fLS in a tweezer potential by Taylor expanding Eq. 5.20
(with I as defined in Eq. 3.2, and with peak intensity I0 ) up to fourth order in the longitudinal coordinate z,
and in the radial coordinate ρ, and treating the quartic terms as a perturbation to the harmonic (quadratic)
solutions. The harmonic solutions can be enumerated by nx,y,z , namely the harmonic oscillator state inthree orthogonal (Cartesian) directions. Keeping only terms that are up to linear in I, the above approachleads to an approximate expression for the shift fLS in the frequency of the clock transition of [290]: 2 
!
√ w0
√
hfLS
w0
1
√
2 (nx + ny + 1) +( ue − ug ) − (ue − ug )
≃
nz +
ER
zR
zR
2
"
!
#
(5.21) 2
1 √ w0w0
1 ∂∆α(E1)
1 √
≃
δm ,
2 (nx + ny + 1) +um − um(nz + )
2
zR
zR
2
∂f
α(E1)
f =fm
where w0 and zR are the minimum beam waist and Rayleigh range of the tweezer, and uξ = α(E1) (ξ, f )I0 /ER .
In the second line, we Taylor expand about the magic wavelength (δm = 0) keeping only the first orderterms in δm . Here, we have defined that um := ue = ug and α(E1) := α(E1) (e) = α(E1) (g) at δm = 0.
1 ∂∆α(E1)
∂f
α(E1)
f =fm

= −15.5(1.1) µHz/MHz [222, 307] is the normalized derivative of the difference in theelectric dipole polarizabilities of the e and g states, ∆α(E1) := α(E1) (e) − α(E1) (g), with respect to thetrapping frequency f and evaluated at the magic trapping frequency fm .
The above analysis illuminates the fact that the motional state of an atom determines how much of the
optical potential an atom samples, and thus how sensitive the clock transition in the atom is to imperfectmagic trapping conditions. Minimizing the effect of these shifts in the clock transition frequency is yet
another reason it is beneficial to cool the atoms to close to their 3D motional ground state (see Sec. 4.3).47
Although so far the tweezer potential seems to be an additional source of inaccuracy, there are situations where it could also be a benefit. For example, the value of β is not particularly well-characterized
for Sr [263]. The ability to independently vary the depth of the traps opens up the possibility of performing
measurements of β (and the various contributions to the polarizability) via synchronous comparisons thatare insensitive to LO noise (see Sec. 5.4.2). Such measurements are, in principle, only limited by QPN (and47 Note that, unlike the discussion in Sec. 5.2 where only the motional state along the direction of the probe laser was
important, the energy shift in Eq. 5.21 depends on all aspects of the atomic motion in 3D. This is, in part, why we opt toperform high fidelity optical cooling in the axial direction by means of employing an accordion lattice in addition to the tweezers
in [365].

122
Repeat x N
Imaging
Imaging
SB CoolingRamp to Clock Ramp to CoolBlue MOT
Red MOTs

100 msClock Interrogation

LAC

...
Tweezer
Bx
Bz

Up to 4 s

100 ms
20 ms 10 ms

8 ms

Repump

15 ms

15 ms

5 ms

20 ms

Figure 5.9: Scheme for repeated clock interrogation. After preparing an ensemble of individual atoms intweezers using several stages of MOTs and light assisted collisions (LACs), the atoms are repeatedly imaged,
sideband (SB) cooled, interrogated, and then recycled via repumping back to the ground state. Note thatthe use of deep (3900 − 6000ER ) tweezers allows for faster images than are typical for the other experimentsin this work. The lower traces denote the depth of the tweezers, and the magnitude of the magnetic fields
Bx and Bz applied along the x and z axes, during various phases of the experiment. In particular, note thatthe magnetic field conditions for sideband cooling and for clock interrogation are distinct.

your ability to calibrate relative intensities) and thus can reach arbitrary precision with sufficient averaging
time. Based on practical considerations the prospects are good for characterizing these effects at the 10−19level, which would further reduce any inaccuracies imposed by the tweezers.

5.4

Clock measurements
With an understanding of the limitations and capabilities of our system, we are interested in perform-

ing precision measurements of the clock transition frequency in tweezer-trapped 88 Sr atoms.

5.4.1

Repeated interrogation
In frequency comparisons between a laser and the atoms, we are sensitive to phase noise in the laser

due to the Dick effect (see Sec. 5.1.3.2). However, we can minimize Dick effect noise by reusing the sameatoms for multiple rounds of interrogation,48 increasing the duty cycle of the experiment.
For measurements taken in condition (1), our scheme for reusing the atoms takes advantage of thelossless detection described in Sec. 5.2.1, and is shown in Fig. 5.9 [240]. Each interrogation cycle involves
85 ms of dead time during which the atoms are measured and reinitialized, whereas it takes approximately
48 Similar to ion clocks [203].

123

a)

I)

Load
1s

Clock
Image

II)

III)

Cool/Repump
1sI) 22% Duty Cycle
nrep = 15, t = 30 ms

II) 94% Duty Cycle
nrep = 8, t = 2 sIII) 96% Duty Cycle
nrep = 4, t = 4 s

100
80
60

24(1) Hz0.49(4) Hz

0.35(5) Hz

40
20

1.
0

0.
5

0.
0

0.
5

1.
0

1.
0

0.
5

0.
0

0.
5

1.
0

10
0

50

0

50

0
10
0

3

P0 transfer fraction (%)b)

Detuning (Hz)Figure 5.10: Repeated Rabi spectroscopy. (a) Timing diagram for repeated interrogation of the clocktransition for various clock interrogation times (t) and number of repeated interrogations (nrep ): I) nrep = 15,t = 30 ms; II) nrep = 8, t = 2 s; and III) nrep = 4, t = 4 s. These values of nrep are chosen basedon experimental convenience — high repetition numbers have a more significant impact on duty cycle forshort interrogation times, but due to stochastic loading require lengthy data scans for long probe times
to achieve uniform statistical uncertainty. (b) Corresponding Rabi spectra for sequences I), II), and III).
Sinc fits shown in gray, and Gaussian fits in black. Shaded regions and labels correspond to the Gaussianfull-width-at-half-maximum.

yFract ional Allan deviat ion

124

10 15

10 16

10 17
10 1

10 0

Averaging t im e

10 2
(s)

Figure 5.11: Allan deviation of fractional frequency fluctuations between an ensemble of atoms and an
ultrastable laser. For Ramsey interrogation with a free evolution time of t = 501 ms, we find a short-timefrequency instability of 4.7 × 10−16 (T /s)−1/2 , where T is the averaging time (red line). This instability is
consistent with the limit imposed by QPN when including the imperfect contrast of the Ramsey fringe (blueline). The long-term stability of this measurement is consistent with the known flicker noise (black) anddrift (grey) of the laser (the combined effect of QPN, flicker noise, and drift is indicated by the dashed line).

300 ms to load a new ensemble of atoms into the tweezers. Each cycle of measurement and reinitialization
results in a probability of 0.001(1) that a given atom is lost, in addition to losses associated with the vacuum
lifetime (see Sec. 5.3.1), which can play a role for very long interrogation times. Practically, these low losses
mean that we can reuse the same ensemble of atoms enough times that the duty cycle of the experiment is set
predominantly by the dead time during each interrogation cycle, with the dead time due to initial preparation
being negligible. Performing repeated Rabi spectroscopy for interrogation times as long as t = 4 s allows
us to measure spectral features with a linewidth as narrow as 0.35(5) Hz, and with a duty cycle as high as
96% (see Fig. 5.10). Note that in these repeated interrogations, we do not find that the performance of laterinterrogation cycles is degraded with respect to earlier cycles, suggesting that the only factor limiting thereuse of the atomic ensemble is loss.49
We characterize the short-term frequency instability of the atoms by performing Ramsey measurements (see Sec. 5.2.4) with an interrogation time of t = 501 ms.50

At this evolution time, the contrastof the Ramsey fringe is C = 0.79(3), meaning that dephasing between the atoms and the LO does not yet
significantly impact the measurement. Measurements of the state of the atoms after the Ramsey sequencecorresponds to a measurement of σz , with an expectation value of ⟨σz ⟩ = C sin(2πδt) where δ is the de49 And limited memory in the experimental control system.
50 Note that in these measurements we did not opt to use repeated interrogation since, even when preparing a new ensembleof atoms on each measurement cycle, Dick effect noise was negligible in comparison to QPN.

125
tuning of the atomic transition frequency from its mean value on a given experimental trial. Using thesingle-shot estimates of δ from the experiment,51 we can compute the Allan deviation (Eq. 5.14) relative tothe free-running clock laser, yielding an instability of 4.7 × 10−16 (T /s)−1/2 at short averaging times T (seeFig. 5.11). This instability is entirely dominated by QPN associated with the average of 4.8 atoms loaded oneach experimental trial,52 with the performance at long averaging times dominated by the known drift andflicker noise floor of the laser [242]. Due to the relatively high duty cycle (and QPN) in these measurements,
Dick effect noise contributes to less than 1% of the measured instability and is therefore negligible.

5.4.2

Measurements beyond the atom-laser coherence time
In Sec. 5.1.3 we considered canonical optical clock measurements where a laser serves the role of an

LO whose frequency is compared to the atomic reference. In Sec. 5.4.1, we saw how the interrogation time
and therefore the instability in such a measurement was limited by the finite coherence time between theatoms and the laser. However, in certain situations one can compare one ensemble of atoms to anotherin a manner that is insensitive to laser noise, and thus extend the interrogation time to durations thatare set by the coherence time between atoms, rather than between the atoms and a laser.53

A number ofrelated techniques fall under this category, including synchronous clock comparisons [28, 203] and correlation
spectroscopy [69, 68, 147, 70, 173].

5.4.2.1

Ellipse fitting

One illustrative example is to extract the relative phase ϕ between two atomic ensembles a and b
when the laser phase ϕL is unknown, but equal for both ensembles. Consider a simplified model for the
probabilities p(a,b) that an atom in a given ensemble is detected in the |1⟩ state after performing a Ramsey
51 We assume that δ is small and that the conversion between ⟨σ ⟩ and δ is linear, and terminate the experiment before thisz
assumption fails due to uncorrected drifts of the LO frequency.
52 Based on Monte-Carlo simulations, we also include a correction factor of 1.07 to the QPN, which handles excess noise due
to fluctuations in the number of atoms loaded on each trial [240].
53 As we saw in Sec. 5.3.2, we expect the former to be much longer than the latter in our experiments.

126

Figure 5.12: Ellipse fitting. Fisher information I for measuring the relative phase ϕ between two 60 atom
ensembles with a Ramsey contrast of 0.8, when marginalizing over the global phase provided by a reference
laser. Note that I vanishes at ϕ = 0 and π. Insets depict the probability mass function for parametric plotsof the excitation fraction of the two ensembles. Color scale is arbitrary, and for illustration purposes only.

127
measurement:54
C(1 + cos(ϕL ))2
Cp(b) = (1 + cos(ϕ + ϕL )),2

p(a) =(5.22)(5.23)

where C is the Ramsey contrast. Notice that although one cannot gain any information about ϕ from p(a)or p(b) alone after averaging over all values of ϕL , this is not true if one has access to both measurements.
For example, the square of p(a) + p(b) averaged over ϕL is

R 2π0dϕL (p(a) + p(b) )2 /(2π) = 1 + cos(ϕ), whichclearly contains information about ϕ.
On each experimental trial, we measure values x(a) and x(b) , which correspond to the fraction of atoms
in the |1⟩ state in each ensemble, and are samples of the correlated random variables X (a) and X (b) . Similar
to Sec. 5.1, we end up with a dataset {x(a) , x(b) }, where the different pairs of values {x(a) , x(b) } are IID.
There are two similar approaches to efficiently estimate ϕ from {x(a) , x(b) }, which we find to offer equivalent performance. One approach is simply to perform maximum likelihood estimation on {x(a) , x(b) } (seeAppendix A.1). The other perhaps more intuitive approach is to plot the points {x(a) , x(b) } parametrically.
Given Eq. 5.22, when randomizing ϕL the points {x(a) , x(b) } fall along the edge of an ellipse, where the sizeof the ellipse is related to C, and the opening angle of the ellipse is related to ϕ (Figs. 5.12 and 5.13a). One
can fit an ellipse to the resulting distribution via the singular value decomposition to extract ϕ [213].
We are interested in identifying the optimal experimental conditions (namely the interrogation time t
and the mean value of ϕ) for measuring the relative frequencies of the atoms with high stability in condition(2). Note that in order to extract ϕ from {x(a) , x(b) }, the mean value of ϕ must be non-zero. This canbe seen by computing the Fisher information associated with the measurement (using PyTorch), which iszero at ϕ = 0, and reaches a peak at ϕ ≃ π/10 (see Fig. 5.12 and Appendix A.1).55

However, operating

under conditions where the average clock frequencies fa,b of the two ensembles are equal, and thus ϕ = 0,
is helpful for determining the optimal interrogation time t, since the mean value of ϕ will not change as afunction of t (see Fig. 5.13b). Such a measurement does not yield useful information about δ = fa − fb , butdoes yield useful information about other aspects of the measurement, including atom coherence and loss
54 A more general model including different types of loss and decoherence is provided in [213, 365].
55 Another way of seeing this is the fact that the measurements near ϕ = 0 are strongly biased [110, 213, 365].

128
(a)

Pb

1.0

0.5

0.0
0.0

1.0

0.0

0.5
Pr

(c)

1.0
y

0.5
Pr

10 17

10 18
10 18
10 19
0

10
20
30
Ramsey dark time (s)

102

103

Averaging time

Allan deviation,

10

17

(fr

fb )/f0

(b)

104
(s)

Figure 5.13: Synchronous frequency comparison via ellipse fitting. (a) We perform a synchronous clock
comparison by partitioning the atom array into two sub-ensembles (insets, red and blue), and creating aparametric plot of the 3 P0 excited-state fraction in the blue ensemble (Pb ) versus in the red ensemble (Pr )(in this case at a 15 s interrogation time in 15ER -deep tweezers). In the checkerboard (left) partitioning
there is no mean frequency shift between the two sub-ensembles, whereas in the diagonal (right) case we
expect a 7.0(1.3) mHz shift. The relative frequency between the sub-ensembles can be extracted via ellipse
fitting (red lines), which in the diagonal case yields 7.15(18) mHz. Note that such fits are biased near zero
phase shift, as is evident in the fit to the checkerboard ensemble, which returns an artificially large phaseshift. (b) To identify an optimal dark time, we compute the fractional frequency uncertainty between the
sub-ensembles as a function of Ramsey dark time. The black points (grey point) correspond(s) to 13 minutes(4.3 hours) of averaging, and are extracted from the checkerboard partitioning. Note that these values are notrepresentative of a true stability due to biasing. This is made clear by the dashed curves, which correspond to
the expected contribution from QPN, and the solid grey curves, which include an additional correction factor
calculated via Monte-Carlo simulations to account for the biased fits (shaded regions denote 1-σ confidence
interval). At 15 s interrogation times the diagonally separated sub-ensembles have a sufficient phase shift toremove the bias in the fits. This condition (red star) shows the fractional frequency uncertainty of the full
4.3 hour-long measurement, with a value of 4.2 × 10−19 . This value is in good agreement with the expected
QPN limit with no bias correction (red curve). (c) We can further compute an Allan deviation associatedwith this measurement (black points), which averages down with a slope of 5.2(3) × 10−17 (T /s)−1/2 (black
dashed line). This value is in good agreement with the expected value of 5.2 × 10−17 (T /s)−1/2 from QPN(red line). Red star is duplicated here as a point of comparison (note that this point is not strictly an Allan
deviation, and is extracted via jackknifing [365]).

(namely contributions to C in the simplified model presented in this section).
Because readout occurs in a site-resolved manner in our experiment, the partitioning of a and b
can be chosen arbitrarily. Specifically, we choose a “checkerboard” partitioning that yields no net tweezerinduced frequency shift between the two ensembles, and a “diagonal” partitioning that yields a near-maximal

129
frequency shift (Fig. 5.13a insets). Measurements using the checkerboard partitioning are compared against
Monte-Carlo simulations to identify an optimal interrogation time of t ≃ 15 s that takes into account both
atomic loss and decoherence [365].
Guided by the above measurements, we perform a 4.3 hour-long synchronous comparison using the
diagonal partitioning and with an interrogation time of t = 15 s. The expected frequency shift between
the two ensembles due to the considerations in Sec. 5.3.4 is δ = 7.0(1.3) mHz, leading to a phase shift of
ϕ = 0.219π, and thus nearly optimal performance (see Fig. 5.12). We measure an offset of 7.15(18) mHz overthe course of the full 4.3 hour-long measurement, which corresponds to a fractional frequency uncertainty of
4.2 × 10−19 . The Allan deviation associated with the above measurement is determined using a jackknifing
procedure [213, 365]. The resulting instability is 5.2(3) × 10−17 (T /s)−1/2 , in good agreement with the
expected value of 5.2 × 10−17 (T /s)−1/2 from QPN alone (Fig. 5.13c). Moreover, the instability does notdeviate from the expected T −1/2 scaling for the full duration of the measurement, meaning that, in thisspecific synchronous comparison, we are not limited by any sources of instability besides QPN down to the
4.2 × 10−19 level.
Note that we do not use repeated interrogation in the measurements described in this section due tothe significantly higher losses incurred when imaging in shallow 515 nm tweezers (see Sec. 4.4). However,the long interrogation times used here allow us to match the highest duty cycles achieved in condition (1) of
96%, even without performing repeated interrogation. As a result, although the synchronous measurementin this section is inherently insensitive to LO noise, Dick effect noise is not expected to significantly impact
the instability of an asynchronous comparison performed under the same conditions.

5.4.2.2

Correlation spectroscopy

In Sec. 5.4.2.1, we saw that we could extract useful information from the two correlated randomvariables X (a) and X (b) . Although those variables could take on many values, similar arguments apply forbinomial random variables, for example when X (i) describes the outcome of a measurement on a single
atom i. In fact, X (a) was simply an appropriately normalized sum of the form

P

i∈a X

(i)

(and similarly

for b). This suggests that one can more generally look at correlations between many X (i) s to extract useful

130

0

19

0

1

19

(c)
0.3

0.8
0.6

0.8
0.6

0.2
CA

C( r)/C2 × 2

t = 25 s

0.4

0.4

0.1

0.2
0.0

...
...

Ramsey contrast(b)t = 15 s

rx/ax
0
15

0.0
0

20
40
Ramsey dark time (s)

ry/ay

t=1s

1

15

2| eg |

(a)

0.2
0.0

0

20
40
Ramsey dark time (s)Figure 5.14: Correlation spectroscopy. (a) As a measure of atomic coherence we compute the spatiallyresolved atom-atom correlation function, C(∆r), as a function of free evolution time. These plots arenormalized by C2×2 to isolate the effects of dephasing from atom loss and decay (the various correlation
functions are defined in more detail in the text). The relative displacements ∆rx,y are normalized by thearray spacing in the relevant direction (ax = 1.5 µm, ay = 1.2 µm) such that the pixel spacing corresponds tothe tweezer spacing. Solid (dashed) diagonal lines indicate axes along which the tweezer wavelengths change(remain constant) showing accelerated (reduced) dephasing along the forward (reverse) diagonal of the arraydue to tweezer-induced frequency shifts (solid contour shows where the correlator passes through zero). The
fourth frame is a theoretical prediction at 25 s given our known tweezer frequencies and depths. (b) The
coherence of a single atom (red disk in cartoon) can be measured by computing the average correlations CA
between it and an ensemble of reference atoms (blue disks). In this case the reference ensemble is the entire
array, and the excess decay of CA (red points, averaged over a 4 × 4 block of atoms at the center of thearray) compared to the decay of the Ramsey contrast (black points) can be used to quantify the single-atomcoherence time. Fits to these quantities (dashed lines) with a Gaussian and exponential component yieldoverall 1/e times of 14.6(7) s and 19.5(8) s respectively. (c) Based on the above measurements, we infer asingle-atom coherence time of 48(8) s (dashed line), which is in good agreement with a model based on themeasured lifetimes and initial Ramsey contrast (solid line, error in grey). Open circles are CA with the decay
associated with the reference ensemble divided out — this serves as a direct measurement of the single-atomcoherence |ρeg |. In the absence of dephasing, an ensemble of atoms would have a Ramsey contrast of 2|ρeg |.
To extend this measurement from the central 4×4 region to the full array, we consider the average correlation
between all atom pairs in a 2 × 2 block averaged over all such blocks, C2×2 . In this case each atom in theblock acts as a reference for all other atoms in the block (see cartoon). The square root of C2×2 (blacksquares) decays with a fitted 1/e time of 33(2) s (double-dashed line), and serves as a lower bound on the
average atomic coherence across the entire array.

131
information. For example, correlation spectroscopy techniques with only two ions have been used to extend
the interrogation time in a differential measurement to well beyond the ion-laser coherence time [69, 68, 70,173].
Here, we are interested in using correlations between atoms to characterize tweezer-induced dephasing(see Sec. 5.3.4), and the coherence of the atoms. Specifically, we compute the g (2) correlator between atoms
in different tweezers i and j:(2)

gij = ⟨σz(i) ⊗ σz(j) ⟩ − ⟨σz(i) ⟩⟨σz(j) ⟩,(5.24)(i)where σz is the Pauli z operator acting on the ith atom. After averaging over ϕL and otherwise assuming
perfect control pulses in the Ramsey measurement, one can show that [365]:(2)(i)
gij = 2|ρeg||ρ(j)
eg | cos(ϕi − ϕj ),(5.25)(i)where ρ(i) is the density matrix describing the state of atom i, and ρeg is an off-diagonal entry of ρ(i) .56 ϕi isthe phase accrued during the Ramsey interrogation time by atom i. Eq. 5.25 provides information about the(i)relative phase ϕi − ϕj between two atoms, and about |ρeg |, which we will refer to as the atomic coherence.
We expect the tweezer-induced frequency shifts to be a function of the relative displacement ∆r of two(2)

atoms, and not of their absolute position. We therefore define C(∆r) to be the value of gij averaged over all iand j that are separated by a relative displacement of ∆r. Measurements of C(∆r) for several interrogation
times are shown in Fig. 5.14a. These measurements reveal that along the forward diagonal of the array,
where frequency offsets between tweezers — and thus clock frequency offsets — are maximal, the atoms
become uncorrelated, and eventually develop negative correlations. Along the anti-diagonal, where there isno frequency offset between tweezers, positive correlations persist over much longer timescales. Although thecoarse features of C(∆r) described above agree with expectations based on independent characterizationsof the tweezer potential (and the analysis in Sec. 5.3.4), we further observe the development of fringes inC(∆r) along the more tightly spaced axis of the array, which we hypothesize are the result of overlaps
between tweezers [365].
(i)56 ρ(i) refers to the state before the final readout pulse in the Ramsey sequence. As a result, |ρ | tells you the length of theegBloch vector, or equivalently the Ramsey contrast.

132
(2)

To efficiently extract the atomic coherence, we want to consider groupings of gij for which ϕi −ϕj ≃ 0,
(2)

(2)

and thus gij ≃ 2|ρeg |2 .57 One approach is to average gij over 2 × 2 sub-ensembles of the array, for which
we expect tweezer-induced dephasing to be suppressed to a timescale of several hundred seconds. We define
C2×2 to be the average over all such 2 × 2 sub-ensembles, which serves as a lower bound on the quantity2|ρeg |2 .58

Measurements of C2×2 provide a lower bound on the average 1/e atomic coherence time for all

atoms in the array of 33(2) s (Fig. 5.14c).
Based on our model for the tweezer-induced clock frequency shift (Sec. 5.3.4) and our balancing of
the tweezer intensities (Sec. 3.3.3.2), we expect that the average clock frequency of the entire array is equal
to the clock frequency of atoms at the center of the array. Therefore, another approach is to measure theaverage correlator between the atoms in the central 4 × 4 sites and the rest of the array, which we define to
be CA (Fig. 5.14b).59 By looking at the excess decay of CA relative to the decay of the Ramsey contrast in
the entire array, we can isolate the contribution to the decay of CA from only the central atoms. As a resultof the above analysis, we infer a single-atom 1/e coherence time of 48(8) s and a resulting atomic oscillator
quality factor of Q = 6.5(1.1) × 1016 (Fig. 5.14c) for atoms in the central 4 × 4 region. This is comparable tothe expected value of 55(8) s in Sec. 5.3.2, and corresponds to the useable interrogation time for synchronous
frequency comparisons (as in Fig. 5.13) if all tweezers were at the same wavelength, as might be achieved
with the use of a spatial light modulator. Note that a significant contribution to the coherence time issimply the vacuum lifetime of the atoms. Often, coherence times are quoted with respect to the surviving
atoms in the system. In this case, after removing the contribution from vacuum lifetime (Sec. 5.3.1), ourmeasurements correspond to a coherence time of 96(32) s, and a quality factor of 1.3(4) × 1017 .60
(i)

(j)

57 We have assumed here that |ρ | = |ρegeg |, since the quality of state preparation does not vary significantly site to site, andthus dropped the indices i and j.
58 This measurement provides a lower bound because any residual frequency differences between atoms can only serve tolower C2×2 due to the cos(ϕi − ϕj ) term appearing in Eq. 5.25.
59 C can equivalently be interpreted as the correlator between the central atoms and the total spin projection of the rest ofA
the array.
60 To put this number in context, if one were to slow the oscillations of the electrons down to more human timescales, likethe swinging of the pendulum on a grandfather clock (∼ 0.5 Hz), then this quality factor is akin to starting a pendulum at the
big bang, and finding that it is still swinging with ∼ 0.5% of its initial amplitude today.

133

5.5

Tweezer clocks in context
It is worth reflecting on the role of tweezer clocks in the broader context of establishing ever-improving

frequency standards. We have discussed a number of benefits to the tweezer system, including lossless andsingle-atom-resolved readout, very long coherence times, and high fidelity state preparation and control.
As a result, tweezer clocks serve as powerful tools for studying new measurement protocols (Sec. 5.4),
understanding the limitations in state-of-the-art optical clocks, and exploring new techniques to mitigate
those limitations (Sec. 5.3). However, the above benefits come at the cost of limited system size (and thus
higher QPN in comparison to lattice clocks), and additional complications associated with the trapping
potential (Sec. 5.3.4). It is also important to note that none of the advantages stated above are inherent
to the tweezer platform itself. Whereas the tweezer clock is the result of optimizing for flexible control
and readout, one could instead optimize for a specific metrological task, incorporating desired controls andmeasurement techniques as needed.
One can see this tradeoff play out, for example, in recent synchronous comparisons in different opticalclocks. The ability to mitigate tunneling at shallow depths in a tweezer array led to an instability of
5.2(3) × 10−17 (T /s)−1/2 for the synchronous comparison described in Sec. 5.4.2.1 [365]. At the time of
publication, this instability was comparable to the state-of-the-art value of 3.1 × 10−17 (T /s)−1/2 for asimilar comparison in a 3D lattice clock [52], despite using about a factor of 40 fewer atoms. However, these
instabilities were soon improved on by an order of magnitude to 4.4 × 10−18 (T /s)−1/2 in a synchronouscomparison in a 1D lattice clock [35].61 This was accomplished by using carefully engineered tunneling andinteractions [3] to extend the achievable interrogation times [148] without sacrificing on atom number.
A similar story arises when considering the accuracy of tweezer clocks, where the tradeoffs made to
introduce additional controls have the potential to also introduce additional sources of systematic effects(Sec. 5.3.4). This is in contrast to trapped ion systems, which maintain state-of-the-art accuracy [42] whilestill offering single-ion-resolved measurements and control that enable advanced interrogation schemes [68,70, 173], including those involving long range entanglement via quantum networks [233].
61 Using more atoms than our experiment (∼ 105 compared to ∼ 150), but a shorter interrogation time (∼ 3 s compared to15 s).

134
With the above considerations in mind, one of the main strengths of tweezer clocks is to serve as flexible
testbeds for different measurement protocols and techniques that can eventually be integrated into more
specialized sensors. For example, the combination of high fidelity control and relatively large systems sizes
makes tweezer clocks ideally suited to study how entanglement62 can be used to improve the performanceof optical clocks, which we will explore in the next chapter.

62 Particularly in relatively large systems not currently accessible in ion clocks.

Chapter 6

Rydberg interactions between clock qubits“The light that burns twice as bright burns half as long — and you have burned so very,
very brightly.”— Replicant Eldon
Portions of this chapter have appeared in:
N. Schine, A. W. Young, W. J. Eckner, M. J. Martin, and A. M. Kaufman. Long-lived Bell states
in an array of optical clock qubits. Nat. Phys., pages 1–7, Aug. 2022
W. J. Eckner, N. Darkwah Oppong, A. Cao, A. W. Young, W. R. Milner, J. M. Robinson, J. Ye, andA. M. Kaufman. Realizing spin squeezing with Rydberg interactions in an optical clock. Nature,pages 1–6, Aug. 2023

In Ch. 5 we demonstrated control over large ensembles of optical frequency qubits, and used this control
to perform precision measurements of the qubit frequency at the standard quantum limit (SQL). However,in Sec. 5.1 we saw that the SQL, in certain scenarios, can be beaten when one allows for entanglement.
As a result, we are interested in engineering entangling gates between clock qubits. There are a variety ofmechanisms which one can use to engineer such gates between neutral atoms, including collisional gates basedon spin exchange or superexchange [163, 360, 371], and gates based on exchanging photons between atoms
and an optical cavity [350, 246]. We opt to excite the atoms to high lying Rydberg states (see Sec. 2.1.1.5),
where the polarizabilities of the atoms can be so high that the presence of a nearby atom in a Rydberg state
can lead to significant energy shifts. This approach has been used to great success to engineer entangling
gates between tweezer-trapped atoms in the past [352, 150, 190, 127, 210, 103], but never in the context of
optical frequency qubits and metrology prior to our works. In Sec. 6.1 we will describe the basic properties
of Rydberg states, and the mechanisms leading to interactions between Rydberg state atoms. In Sec. 6.2,
we will see how the above interactions can be used to generate entanglement between clock qubits, and in

136
Sec. 6.3 we will benchmark the quality of a two qubit gate using the described mechanism. In Sec. 6.4, we
will use the two qubit gate to prepare Bell pairs of clock qubits, and evaluate the utility of these Bell pairs
for frequency metrology. Finally, in Sec. 6.5, we will briefly discuss recent results where we use the above
tools to prepare larger entangled states that unambiguously achieve performance beyond the SQL.

6.1

Rydberg states
As described in Sec. 2.1.1.5, Rydberg states are high lying electronic states where the outer electron

has an orbit that is much larger than usual.1

These states typically have an energy near the ionizationthreshold of the atom, and are thus challenging to excite directly with a single photon due to the shortwavelengths involved.2

However, because the 3 P0 excited state in Sr is long-lived, one can drive atoms to

the 3 P0 state, and then excite the atoms to high-lying Rydberg states with a reasonable (although still fairly
short) wavelength of λ ≃ 317 nm.3

6.1.1

Rydberg spectroscopy
The dipole allowed transitions from the 3 P0 clock state involve states in the 5sns 3 S1 and 5snd 3 D1

series. As a general rule, the 3 S1 states provide stronger interactions that are also isotropic, whereas the3

D1 states have weaker and anisotropic interactions, but provide larger dipole matrix elements from the 3 P0state. As described in Sec. 2.1.1.5, for large n these Rydberg states are approximately hydrogenic, with small
deviations in energy captured by a quantum defect parameter δn in Eq. 2.6. δn can be expressed in termsof a series expansion:δn = δ (0) +

δ (2)
δ (4)
+
+ ...,(n − δ (0) )2(n − δ (0) )4(6.1)

where the constant values δ (k) are typically determined by fitting to spectroscopic data [338]. For our
purposes, it is sufficient to assume that δ (k) = 0 for k > 4. The relevant values of δ (k) for k ≤ 4 are shownin Tab. 6.1. We find Eq. 6.1 to be accurate for all Rydberg states relevant to our works, but it is important
1 Typically, on the scale of ∼ 0.5 µm in our experiments, in comparison to the size of the ground state orbitals of the valence
electrons, which are only a few Bohr-radii in size (a0 = 52.9 pm).
2 It is worth noting that although laser sources in the deep and extreme ultraviolet exist (with wavelengths of λ ≲ 280 nm),they can be very challenging to work with due to absorption in air, photochemical damage, etc.
3 This is in contrast to approaches that drive a two-photon Raman transition from the ground state, where the intermediate
state is typically short-lived and thus can limit the fidelity with which one can drive the ground-Rydberg transition [190, 103].

137

S1
D1

3
3

δ (0)

δ (2)

δ (4)

3.371(2)2.658(6)0.5(2)
3(2)-10(20)-8800(700)Table 6.1: Quantum defects for the 3 S1 and 3 D1 Rydberg series in Sr. Values are from [23, 338].

to note that this simple scaling can be broken in rare cases where a Rydberg state from a different series ofstates is close in energy to the state of interest [113].
Recall from Sec. 2.1.1.5 that the dipole matrix element from the ground state to a Rydberg state withprinciple quantum number n scales approximately as ⟨g | d | n⟩ ∝ n−3/2 . Because this argument is based on
general scaling laws for the size of the Rydberg orbital, a similar argument applies for the dipole matrix
element from the 3 P0 state. Also recall that the Rydberg state lifetime τ has a scaling between τ ∝ n3 andτ ∝ n2 . In Fig. 6.1, we perform spectroscopy of the 5s40d 3 D1 , mj = 0 state, attaining Rabi frequencies of
up to Ω = 2π × 18 MHz using ∼ 1 W of power at 317 nm (see Appendix D.3), and measure a lifetime ofτ = 10.9(4) µs.4
For the work presented in this thesis, we measure the Rydberg state population by simply relying
on the fact that the Rydberg state mostly decays to the ground state, or to states that are lost from thetweezers. Our detection involves blowing away atoms in the ground state, and repumping clock state atoms
to the ground state for detection as in Sec. 5.2.1. The above approach results in a detection fidelity of the
Rydberg state of 89%, with 11% of the Rydberg state population decaying to the 3 P0 clock state.5

The

observed branching ratio is consistent with the expected branching of 1/9 for the 3 S1 series of states to decay
to the 3 P0 clock state in comparison to the 3 PJ̸=0 states. As we will see below, this low detection fidelity is
not important for our work, since the protocols we are interested in leave no population in the Rydberg state.
However, in more recent work we have adopted a similar approach to the one demonstrated in [200, 210],
where 407 nm light is used to drive a transition of the inner electron,6 which rapidly autoionizes the outer
4 Note that this lifetime is measured two ways: by resonantly exciting Rydberg state atoms and observing them decay [290],and by dressing the clock state with the Rydberg state, and observing the dependence of the dressed state lifetime. Bothmeasurements agree to within statistical errors (for lattice depths below ∼ 50ER ), and so the value quoted here is based on the
dressed state measurement, which has lower statistical error.
5 Note that these measurements are performed with the trap extinguished. In the case where the trap is on, the strong
antitrapping of the Rydberg states causes them to be lost, rather than decaying to low-lying electronic states.
6 We find this transition to be at 407.886 nm, which corresponds to the unperturbed transition frequency of the Sr+ ion [282].

138

(a)

(b)
1.0: |1r + |r1

0.5|1 : 0.0

0.5

100

200

300
Pulse time (ns)

400

500

0.0: |11(c)
Dressed state
lifetime ( s)Population

Survival|r : 1.0

0.5

0.0

0

10

20
Time ( s)

30

105
104
103
102
Detuning (MHz)103

Figure 6.1: Rabi spectroscopy and lifetime of the 5s40d 3 D1 Rydberg state. (a) Rabi oscillations of isolated
atoms (purple) and atom pairs (blue) when driving the clock to Rydberg transition. Notice that, as expectedin the case of Rydberg blockade, the atoms pairs oscillate between a state with no Rydberg excitations (|11⟩)
and a state with only a single Rydberg excitation (|+⟩ ∝ |1r⟩ + |r1⟩), with an oscillation frequency that is√
2 higher than that of single atoms. Solid lines are sinusoidal fits to the data, from which we extract a Rabi
frequency of Ω = 2π × 13 MHz. The displayed data is representative of the state of the experiment in 2021,
where the Rydberg state was detected via loss from the clock state. The dashed line corresponds to the
maximum expected value of the observed transfer to the Rydberg state due to the branching ratios associatedwith decay from the Rydberg state. Recent upgrades have significantly improved both the fidelity of the
Rydberg Rabi oscillations, and the fidelity with which we can detect the Rydberg state (see also [210]). We
leave a discussion of these upgrades to future works from the group. (b) The lifetime of the Rydberg state
can be measured directly by exciting atoms to the Rydberg state, holding for a variable amount of time, andde-exciting the atoms. Atoms that decay from the Rydberg state are lost during this measurement, whereas
atoms that are successfully transferred back to the clock state survive. Solid line is an exponential fit to thedata, which yields a 1/e lifetime of τ = 13(3) µs. (c) An alternative way of measuring the Rydberg state
lifetime is to prepare atoms in the clock state, and to off-resonantly dress the clock state to the Rydberg
state (in this case with a Rabi frequency of Ω = 2π × 15 MHz). The detuning is varied (negative values inblue, positive values in purple), resulting in a variation in the measured lifetime of the dressed state. The
Rydberg state lifetime extracted from this analysis is τ = 10.9(4) µs.

139electron leading to loss of the atom (now an ion). This loss can occur on timescales much shorter than the
Rydberg state lifetime, leading to high detection fidelities of the Rydberg state.

6.1.2

Rydberg interactions
The origin of the strong interactions between Rydberg state atoms is their strong polarizability (seeSec. 2.1.1.5). Schematically, the orbit of the outer electron can be so large in a Rydberg state that theelectrons in neighboring Rydberg atoms can push on each other, distorting the electron orbitals and leading
to energy shifts that are dependent on the presence or absence of a nearby atom in a Rydberg state.
Under the assumption that only a single outer electron participates in these interactions, and that theatoms are far apart relative to the size of the electron orbitals,7 the interaction Hamiltonian between two
Rydberg state atoms is simply set by Coulomb interactions between the two electrons and nuclei:
Hi ∝

1
1
1
1
−
−
+
,
|r1 − r2 − R| |r1 − R| |r2 − R| |R|(6.2)

where R is the separation between the atomic nuclei, and ri the coordinate of the outer election in atom irelative to the nucleus of the atom. Having already made the approximation that the atoms are far apart,8
one can perform a multipole expansion of Hi and neglect all but the lowest order terms. The resulting energy
shift for a pair of atoms in a given electronic state β can be expressed as [338]:
V (β, R) =X CN (β)
N

RN

,

(6.3)

where CN (β) is a constant.
In the absence of a perturbing field, the lowest order non-zero terms in Hi correspond to N = 5 and 6.9
The N = 5 contribution is associated with a resonant quadrupole-quadrupole interaction, which vanishes
for S states, and is typically relevant only at very long distances. The N = 6 contribution is associatedwith a second order dipole-dipole interaction, which is also known as a van der Waals interaction. In our
experiments, N = 6 dominates over all other effects, and so the energy shift experienced by two nearby
7 Such that one can neglect the exchange interaction [338].
8 Namely, that ⟨|R|⟩ ≫ ⟨|r |⟩.
i
9 The N = 1, 2 interactions correspond to charge-charge and charge-dipole interactions, which vanish for neutral atoms.

However, one can induce a permanent dipole moment in the state β with an applied DC or AC electromagnetic field, in whichcase the leading order term is N = 3, corresponding to resonant dipole-dipole interactions. These techniques are outside thescope of this work, but are demonstrated in [8, 269, 20, 82].

140
atoms in comparison to their free space energy is simply:
V (R) =

C6
.
R6

(6.4)

We will refer to V (R) as the interaction potential. Note that although the above discussion is helpful for
isolating the underlying physics, in practice, we often solve for the eigenstates of Eq. 6.210 using wellestablished numerical tools [308].
One can infer the rough scaling of C6 from the fact that the van der Waals interaction is fourth order inthe dipole operator, which scales like n2 (Sec. 2.1.1.5). Additionally, the energy difference between adjacent
Rydberg levels scales like n−3 (Sec. 2.1.1.5), leading to an overall scaling of C6 ∝ n11 . In combination with
the increased Rydberg state lifetime with n, this suggests that operating at higher n is strictly beneficial.
However, this is not always the case. Specifically, consider the scenario where one wants to stay in thewell-behaved large R limit where the interactions are dominated by the C6 contribution to the interaction
potential.11 The minimum spacing R0 where the above condition breaks down is when the dipole coupling
between adjacent states is comparable to their energy difference. As in Sec. 2.1.1.5, the dipole matrix element
between neighboring states scales like Ωn := ⟨n | d | n + 1⟩ ∝ n2 , and the energy separation like ∆En ∝ n−3 .
We therefore have that:Ω2n∼ ∆En → R0 ∼ n7/3 .
R03

(6.5)

Ω4n
∼ n−3 .
∆En R06(6.6)

Given R0 , the maximum value of V is:
V (R0 ) ∼This is to be compared to the Rydberg state lifetime with a scaling between n3 (in the absence of BBR)
and n2 (for the worst-case BBR spectrum). As a result, in this setting it is beneficial to operate at lower n,assuming one can arbitrarily reduce the interatomic separation. An additional benefit of operating at lower
n is the larger dipole matrix element between the ground (or 1 S0 ) state and the Rydberg state, meaning
10 Additionally including the contribution to the full Hamiltonian from the bare atomic Hamiltonians.
11 One does not necessarily have to make this restriction. For example, with better electric field control one could consider
beating many of the scaling arguments made here by going to shorter spacings, or making use of Förster resonances to boost
the interaction strength [269]. However, such a restriction can be helpful when one wants to engineer well-behaved interactions
between many atoms with variable spacing from each other, as in Sec. 6.5.

141higher Rabi frequency for fixed laser power, as well as reduced sensitivity to stray electric fields. Basedon the above analysis, we find it useful to design the experiment to minimize the spacing between atoms(see Sec. 3.3). Given fixed atom spacing there is an optimal value of n at which to operate, which for our
experiment is n ∼ 40 (given the interatomic spacing of ∼ 0.5 µm in our experiment).12
Under the presence of a (global) drive laser on the |1⟩ ↔ |r⟩ transition with Rabi frequency Ω anddetuning ∆,13 the Hamiltonian for an ensemble of atoms is:
H=

X Ω
i

2



σx(i) − ∆|1i ⟩⟨1i |+X

V (Rij )|ri rj ⟩⟨ri rj |,

(6.7)

i<j

(i)

where σx is the Pauli x matrix acting on atom i, |1i ⟩⟨1i | is the projector onto the |1⟩ state of atom i, Rij isthe separation between atoms i and j, and |ri rj ⟩⟨ri rj | is the projector onto the subspace where atoms i andj are both in the |r⟩ state. Notice that the first term in Eq. 6.7 simply comes from the familiar single-atomHamiltonian in Eq. 2.13, whereas the second term shifts the energy of states with two or more Rydberg
excitations.
When Vij ≫ Ω, ∆, one is in the so-called “blockade limit,” where states with more than a single
Rydberg excitation are shifted completely out of resonance, and so the system factorizes into subspaces with
one or fewer Rydberg excitations, and more than one Rydberg excitation. One can define a length scale
where V = Ω, which is known as the “blockade radius”:
RB =



C6
Ω

1/6
.

(6.8)

For systems where all atoms are within RB of each other, one can solve for the dynamics starting with all
atoms in the |g⟩ state, up to errors in the populations on the scale of (Ω/V )2 , by simply eliminating stateswith more than two Rydberg excitations. For example, the relevant two-atom Hilbert space in the blockade√√
limit is spanned by the states |11⟩, |+⟩ := (|1r⟩ + |r1⟩)/ 2, and |−⟩ := (|1r⟩ − |r1⟩)/ 2. Starting in the |11⟩
state, we find that for ∆ = 0, ⟨11 | H | +⟩ =√2Ω, and ⟨11 | H | −⟩ = 0.14

Therefore, the dynamics for two

blockaded atoms resulting from a drive with Rabi frequency Ω is Rabi oscillations between |11⟩ and |+⟩ with
12 Note that with fixed laser power, errors due to finite values of Ω/V are not necessarily limiting. In this case it can once
again be beneficial to go to higher values of n in order to improve the ratio of the Rydberg state lifetime to the Rabi frequency.
13 Where |1⟩ refers to the 3 P state, and |r⟩ to the Rydberg state of choice.
0
14 One way to see this intuitively is that for ⟨11 | H | +⟩ there is constructive interference from driving to the |1r⟩ and |r1⟩
components of the state, whereas for ⟨11 | H | −⟩ there is destructive interference. Similar to the discussion in Sec. 2.1.1, thedipole operator is not able to change the parity of a state.

142
effective Rabi frequency√

2Ω. We show an example of the measured time evolution of isolated atoms andpairs of blockaded atoms under the same drive in Fig. 6.1a. Notice that just by driving the above oscillations
one can apply an entangling gate that takes a pair of atoms from the product state |11⟩ to the Bell state|+⟩ (which is maximally entangled). This mechanism has been used to prepare Bell states with impressive
fidelities of > 0.991(4) (with SPAM correction) [210]. However, in this case the qubits are encoded in the{|1⟩, |r⟩} electronic states of the atoms, with the lifetime of the |r⟩ state being ∼ 80 µs. This short lifetime
makes it challenging to realize other operations on the qubits, like single qubit gates.15

6.2

Rydberg dressing
One approach to engineering a universal gate set is to use Rydberg interactions to engineer an entan-

gling gate between the long-lived clock qubits discussed in Ch. 5, for which we have already demonstrated
high fidelity single qubit gates. Given qubit states |0⟩ and |1⟩, the central idea is to drive closed loops on the{|1⟩, |r⟩}-Bloch sphere that are different depending on the presence or absence of nearby atoms in the |1⟩state, resulting in populations that remain in the {|0⟩, |1⟩} subspace at the end of the gate, but with phaseshifts that are dependent on the dynamics involving the (interacting) Rydberg states.
Given Eq. 6.7, consider the Hamiltonian H for two atoms in the blockade limit:
(1)

(2)

H = H0 + H0 + HI

(6.9)

Ω
(|ri ⟩⟨1i | + |1i ⟩⟨ri |) − ∆|ri ⟩⟨ri |
2
√
2Ω
HI :=
(|+⟩⟨11| + |11⟩⟨+|) − ∆(|+⟩⟨+| + |−⟩⟨−|),
2

(i)
H0 :=

(i)

where H0

(6.10)(6.11)

is the single atom Hamiltonian for an isolated atom i, and HI is the Hamiltonian describing(i)modifications to the single-atom behavior due to Rydberg blockade. Notice that H0
similar, with the main difference simply being the√and HI are quite

2 enhancement of Ω.

Note that resonant gate schemes involving evolution under Eq. 6.9 with ∆ ∼ 0 have been applied toqubits encoded in two hyperfine states of tweezer-trapped alkali atoms [191], including in recent time optimal
gate schemes [152] that achieve two qubit gate fidelities as high as 99.52(4)% (with SPAM correction) [103].
15 For example, the time it takes to translate the atoms by the blockade radius of ∼ 1 µm without significant heating is∼ 40 µs (see Sec. 4.5), which is comparable to the lifetime of the qubit.

143
For the experiments in this work, we focus on the situation where ∆ and Ω are varied slowly and continuously.
We choose to operate in this regime for two reasons: first, the highest values of Rabi frequency Ω we canreach correspond to timescales that are significantly faster than our ability to switch the drive laser on andoff,16 meaning that gates that take advantage of the full range of Ω that we can access necessarily involve
adiabatic ramps. Second, we will see in Sec. 6.5 that staying in the regime where Ω/∆ < 1 can be beneficial
for schemes involving interactions between many atoms at once.
Following the proposed gate scheme and discussion in [223], one can diagonalize Eq. 6.10 to find the
dressed eigenstates (Sec. 2.2.1) of a single atom:|1̃⟩ = cos(θ1 )|1⟩ + sin(θ1 )|r⟩,|r̃⟩ = cos(θ1 )|r⟩ − sin(θ1 )|1⟩,(6.12)

where tan(2θ1 ) = −Ω/∆. These states have eigenenergies:
E0± =


p
1
−∆ ± Ω2 + ∆2 .
2

(6.13)

Similarly, diagonalizing Eq. 6.11, we have:˜ = cos(θ2 )|11⟩ + sin(θ2 )|+⟩,|11⟩|+̃⟩ = cos(θ2 )|+⟩ − sin(θ2 )|11⟩√2Ω
tan(2θ2 ) = −
∆


p
1
E2± =
−∆ ± 2Ω2 + ∆2 .
2

(6.14)(6.15)(6.16)

Given the above definitions, we can define an “entangling energy” κ, which corresponds to the difference inenergy of two nearby (Rydberg blockaded) |1⟩ state atoms in comparison to two isolated |1⟩ atoms under
the influence of a global laser drive with parameters Ω and ∆:κ± := E2± − 2E0±
 p

p
1
∆
2
2
2
2
±
2Ω + ∆ − Ω + ∆ .
=2
2(6.17)

Note that which sign you take in Eq. 6.17 depends on the sign of ∆. In subsequent discussion, we will omit
the “±” for convenience.
Assuming one applies the laser drive Ω adiabatically, the atoms will follow the dressed eigenstates of˜
the system {|00⟩, |01̃⟩, |1̃0⟩, |11⟩}.
Specifically, for pulses where Ω starts and ends at zero (see Fig. 6.2), the16 Although recent upgrades combined with operation at higher n change this situation.

144
effective Hamiltonian in the {|0⟩, |1⟩} subspace is [223]:
H = −(E0 +κκ)Sz + Sz2 .
22(6.18)

Here, Sz := σz ⊗ 1/2 + 1 ⊗ σz /2 is the total spin operator, where 1 and σz are the identity and Pauli z
operators acting on a single qubit. Up to a global single qubit phase shift, Eq. 6.18 is simply the one axis
twisting Hamiltonian Sz2 with an energy set by κ/2. In fact, by performing two identical dressing pulses
separated by a spin echo — namely a π-rotation between |0⟩ and |1⟩ — one can remove the contribution
from the Sz term in Eq. 6.18, leading to an effective Hamiltonian (see Fig. 6.2):
Heff = κSz2 .
Applying Heff for duration t = π/(2κ) implements the Mølmer-Sørensen (MS) gate [223].17(6.19)
Including theeffect of the spin echo, one can think of each UV pulse as implementing a controlled phase shift of π/2√(equivalently, a controlled- z gate).
Note that, in practice, there are several errors that can affect the fidelity of the applied gate, which
we discuss in more detail in Sec. 6.3. One significant source of error is the finite lifetime of the |r⟩ state,
which we measure to be τR = 10.9(4) µs for the 5s40d 3 D1 , mj = 0 state (see Sec. 6.1.1). As a result of
this error, optimizing for the fidelity of the above MS gate involves balancing errors associated with decayof the Rydberg state with errors due to non-adiabaticity, where not all the atomic population returns to the{|0⟩, |1⟩} subspace.
For weak dressing where |∆| ≫ Ω, κ ≃ −Ω4 /(8∆3 ). However, in this limit the population in the
Rydberg state, namely the |r⟩ contribution to |1̃⟩, is ≃ Ω2 /(4∆2 ). Since the lifetime τ of the |1̃⟩ state isdominated by the contribution from the short-lived |r⟩ state, it has the same Ω2 /∆2 scaling. As a result,κ/τ ∝ 1/∆, suggesting that the quality of the gate should exclusively improve with lower ∆, as long as thefunctional form of the interaction remains appropriate.
For this reason, we choose to perform trapezoidal ramps of Ω and ∆ from a regime where |1̃⟩ ≃ |1⟩
17 Note that operations involving the clock require tight confinement to be within the Lamb-Dicke regime, as described inSec. 5.2. Conversely, operations involving the Rydberg state are improved with shallow traps, due to the strong anti-trapping
experienced by the Rydberg state (since the trapping light is blue-detuned from various transitions to nearby Rydberg states).
As a result, we perform linear ramps of the lattice depth with a duration of ∼ 1 ms between depths of 12ER (for Rydberg
dressing operations) and 42ER (for clock operations) for the gate sequences discussed in this chapter. The axial lattice is heldlow, at 8ER , for all gate operations.

145( π2 )( π2 )π
π

2 × 40Pr, P{0,1}

0.25

2 × 20

0

0

t0

t1

Time

t0

Angularfrequency (MHz)( π2 )( π2 )

Figure 6.2: Adiabatic gate protocol. The gate is composed of global single qubit rotations (red pulses), andoperations involving Rydberg dressing (purple pulses). The Rydberg dressing operations are shown in moredetail in the lower panel, where linear ramps of the Rabi frequency Ω and detuning ∆ of the laser field used
to drive the atoms to the Rydberg state result in a value of the entangling energy κ that varies in time. The
atomic populations during these ramps, as predicted via a master equation calculation, are also displayed,with P{0,1} denoting the population in the qubit subspace, and Pr the population in any state involving a
Rydberg excitation, or a lost atom. Notice that at the end of the ramp, almost all the population returns to
the qubit subspace. As a result, up to a single qubit phase shift, the dressing operations can be thought ofas applying a controlled-phase gate. The full pulse sequence depicted here implements the Mølmer-Sørensen
gate.

(a)

1.0

(b)

1.0

146

0.0

0.5

P00+P11

Sz

0.5

0.5

1.0
0.0

0.5

1.0
gat e t im e, t 1 ( s)

1.5

0.0

Figure 6.3: Adiabatic gate population dynamics. (a) We apply the gate shown in Fig. 6.2, but varying
the time t1 during which the Rydberg dressing field is held at near-resonant conditions between ramps. As
a result, pairs of atoms effectively evolve under the Hamiltonian H = κSx2 for variable time, resulting inoscillations in the measured spin projection Sz of the atoms along the z axis (blue points). In comparison,
the spin projection of isolated atoms (grey points, multiplied by two in vertical direction to match scales)
does not evolve significantly. Additionally, note that at early evolution times, pairs of atoms stay primarily
in the {|00⟩, |11⟩} subspace, as evidenced by the high probability P00 + P11 of observing either the |00⟩ or|11⟩ states. At longer evolution times, the spin echo fails to perfectly cancel the single atom phase shifts,
leading to uncontrolled global single qubit rotations and therefore dephasing (decay of the Rydberg statealso causes some additional decoherence). (b) An illustrative pair of images corresponding to the evolution
time marked by a star. Note that in this measurement we do not use rearrangement, and simply allow anarray of doublets to be loaded with 0, 1, or 2 atoms (top image, 2 atom cases circled). In the second image,atoms in the |1⟩ state appear as bright, and we find that most doublets that contained two atoms are in the|11⟩ state after the gate is applied, whereas most single atoms are in the |0⟩ state.

to a regime where |1̃⟩ ≃ |r⟩ and back. Specifically, Ω and ∆ are ramped from initial values of 0 MHz and
40 MHz respectively to values of 18 MHz and 2 MHz using linear ramps with a duration of 350 ns (seeFig. 6.2). The parameters are held at these values for duration t′ , and then ramped back to their initialvalues in 350 ns.18
In Fig. 6.3, we present results studying the dynamics of pairs of atoms evolving under Heff . Specifically,
we arrange our atoms into pairs such that atoms within a pair are fully blockaded, whereas interactions
between different atom pairs are negligible. We then perform a global rotation about y such that z → x.
This yields Heff → κSx2 , which couples the |00⟩ state directly to the |11⟩ state without ever passing throughthe |01⟩ or |10⟩ states. The resulting dynamics correspond to oscillations in the total spin projection of theatom pair ⟨Sz ⟩. However, the lack of population in the |01⟩ and |10⟩ states means that when ⟨Sz ⟩ = 0, the18 These parameters vary slightly depending on the specific experimental conditions, see [290] for details.

147
prepared state is of the form |ψ⟩ = (|00⟩ + e−iϕ |11⟩), namely a Bell state with maximal energy separation
between the two components of the state in the computational basis. Notice that this state matches theform of Eq. 5.10, and, in principle, saturates the HL for two qubits.

6.3

Benchmarking entangling gates

6.3.1

Measuring gate fidelity
The ideal solution to measuring the fidelity and general performance of our gates is to perform some-thing like gate set tomography [221] or randomized benchmarking [177]. However, due to the significant
experimental overhead posed by such techniques, it is common to perform simpler measurements that canmore efficiently (with respect to experimental controls and statistics) extract the quantities of interest (e.g.
the Rabi and Ramsey oscillations we use to characterize our single qubit gates) [17, 112].
Specifically, we are interested in applying our entangling gate to a well characterized and separable
initial state, and will use the fidelity with which the resulting maximally entangled state is prepared as astand in for the fidelity of the entangling gate. This approach makes the assumption that the sources of error
in the gate are well understood, and that there are no surprises when applying the gate to a different input
state. We will apply the MS gate (the pulse sequence in Fig. 6.2) to the |00⟩ state, resulting in the target√
state |ψ⟩ = (|00⟩ + eiϕ |11⟩) 2 (where ϕ is a well-defined and constant phase that depends on the details of
our gate sequence). The fidelity of the prepared state ρ with the target state |ψ⟩ is:
F := ⟨ψ | ρ | ψ⟩ =


1
1 iϕ
(ρ00,00 + ρ11,11 ) +e ρ11,00 + e−iϕ ρ00,11 .
2
2(6.20)

We can measure ρ00,00 and ρ11,11 directly, as in Fig. 6.3, and Fig. 6.4a at the evolution time that corresponds
to the MS gate.
To measure the off-diagonal coherences ρ00,11 and ρ11,00 we can perform a global π/2 rotation abouta variable axis with angle ϕα from the x axis [280], described by a single qubit rotation matrix:π

ϕαRπ/2 (ϕα ) := e−i 4 σx e−i 2 σz .
(2)

(6.21)

This being a global rotation, the action on the two qubits is Rπ/2 (ϕα ) := Rπ/2 (ϕα ) ⊗ Rπ/2 (ϕα ). The parity

148

(a)

( π2 )( π2 )

(b)
0.5

π
Z

π

Z

( π2 )( π2 )( π2 )( π2 )

1.0

Parity

P

0.5
0.25

0.0
0.5

0

P00 P01 P10 P11

1.0

0

/2

3 /2

2

Phase,

Figure 6.4: Bell state fidelity via population and parity measurements. (a) Measurements of the two
atom populations determine the diagonal entries of the two atom density matrix, and yield a value of
P00 + P11 = 0.955(28) (blue) after correcting for measurement errors. Dashed outlines indicate the valueof the uncorrected measurement. (b) The off-diagonal coherences in the two atom density matrices can be
determined from the contrast of the oscillations in parity when applying an additional π/2-pulse about avariable axis θ (see circuit diagram). The measured values of the parity are shown in blue (correcting formeasurement errors), with the solid line corresponding to a fit to a sin function (shaded region denotes 1σ
confidence interval). The fitted value of the oscillation contrast is C = 0.919(28) (including both statisticalerrors, and systematic errors on the experiment).

149
P := σz ⊗ σz of the resulting state is:


(2)
(2)
⟨P⟩ = Tr P Rπ/2 (ϕα ) ρ Rπ/2 (ϕα )†= ρ10,01 + ρ01,10 + e−2iϕα

ρ00,11 + e(6.22)2iϕα

ρ11,00 .

Therefore, one can simply measure P as a function of ϕα to isolate the values of ρ00,11 and ρ11,00 by looking
at the contrast of the observed oscillations. We show a measurement of these parity oscillations in Fig. 6.4b.
Note that obtaining an accurate estimate of ρ00,00 requires careful handling of the decay of the |r⟩state, which can lead to a systematic increase in this quantity. For example, the |11⟩ state can decay to thestate |1∅⟩, where |∅⟩ refers to the vacuum state where the atom is absent. The state |∅⟩ is outside the{0, 1} computational basis, but our measurements don’t explicitly differentiate between |0⟩ and |∅⟩.19

As

a result, after the final π/2 pulse in the sequence, |1∅⟩ can incorrectly be measured to be |00⟩.
To take the above effect into account in our estimate of the fidelity, we perform measurements after
performing the gate both with and without a blowaway pulse to remove |0⟩ atoms. The resulting measurement
provides a direct estimate of the proportion of atoms in the |∅⟩ state, where absent atoms correspond togenuine atom loss. We perform similar measurements with single atoms, or pairs of atoms, and define p(1)and p(2) to be the single and two atom loss probabilities due to Rydberg state decay, if the atoms all startin the |1⟩ state. The above quantities are measured by comparing experiments with and without the UV
dressing pulse:p(1) = 2 (P (a|no UV) − P (a|UV))p(1)p(2) = 4 P (aa|no UV) − P (aa|UV) −,2(6.23)

where P (a|·) and P (aa|·) refer to the conditional probabilities to observe that all the atoms survive in the
single and two atom cases respectively.20 We find that p(1) = 0.011(6) and p(2) = 0.075(24).
When in the gate sequence a decay event occurs affects how likely it is to result in a false positive
detection of the |00⟩ state. For example, if the decay occurs early in the sequence, the two atoms are in a
product state prior to the decay, and so the state of the remaining atom is not affected. The subsequent
19 Note that decay to the {0, 1} subspace is correctly handled, since the loss of coherence in these components of the state
will be reflected in the estimate of the off-diagonal elements of the density matrix.
20 Note that the factors of 2 and 4 are because we perform these measurements in a sequence where the atoms are on theequator of the (collective) Bloch sphere.

150
evolution simply corresponds to the evolution of one atom. If the decay occurs late in the sequence, thesecond atom is projected into a mixed state. Averaging over decays that can occur during the pulse sequence,
we find that the conditional probability of a false positive detection of |00⟩ given that decay occurs is
P (00|decay) = 0.75. We therefore have that the measured population in |00⟩ in the presence of these errors
is:


 (1)

p(1)
p(2)
p
p(2)
P (00) = P (00|no decay) 1 −−+ 0.75+.
2
4
2
4(6.24)

Our measurement of P (00) therefore overestimates the true value of ρ00,00 by 0.75(p(1) /2+p(2) /4) = 0.018(5).
Including an additional uncertainty of 0.006 due to the fact that P (00|decay) is based on simulations, butbounded to be between 0.5 and 1, leads us to an estimate of a systematic error in our measurement of ρ00,00
of 1.8(8)%. When including the effect of the above error, the measured Bell state fidelity is 87.1(1.6)%.
However, this fidelity includes contributions from a number of state preparation and measurement (SPAM)
errors that are unrelated to the applied MS gate.

6.3.2

Correcting SPAM errors
We opt to include the single qubit gates used in the circuit that implements the MS gate in our quoted

gate error, but are interested in estimating how much of the gate infidelity is due to measurement errors, aswell as due to additional loss during the various ramps in lattice depth. This is particularly relevant since wehave made significant improvements to both sources of errors since the above data was taken (see Secs. 3.4
and 4.4). We discussed the relevant errors in Sec. 4.4 for identifying the state of an atom on a given site,
including p01 false positive errors and p10 false negative errors (including atom loss). What remains is todetermine how these errors affect the estimate of the Bell state fidelity.
We perform calibration measurements in which the full experimental sequence is executed, except
for the clock, UV, and blowaway pulses. These calibrations yield measured values of p01 = 0.004(1) and
p10 = 0.031(3) for the data presented in this chapter (except for in Sec. 6.5). Given these calibrations, we
can construct a measurement process matrix M that maps the true underlying results n to the measured

151
values m:
m = M · n.

(6.25)

n := {n11 , n10 , n01 , n00 } is a column vector containing the true number of experimental trials where theatoms were detected21 in state |11⟩, |10⟩, |01⟩, and |11⟩ (and similarly for m and the observed results). We
have that:
1 − 2p10
 p

10
M =
 p10
0

p01
1 − p10 − p01
0
p10

p01
0
1 − p10 − p01
p10


0
p01 

,
p01 
1 − 2p01(6.26)

where we have neglected processes that are beyond first order in p01 and p01 , and assumed that these errors
are uniform across the array.22 We can apply the inverse of M on other measurements m to obtain estimatesof the true underlying values n, which can in turn be used to estimate the values of different observables(we will refer to these estimates as “SPAM corrected” values). The estimators for loss η and parity P are:
η̌ =

X
i

P̌ =

n10
n11 + n10

X n(l) n(r) − n(l) n(r) − n(l) n(r) + n(l) n(r)
i

11 11
11 10
10 11
10 10
,
(l) (r)(l) (r)(l) (r)(l) (r)n11 n11 + n11 n10 + n10 n11 + n10 n10(6.27)(6.28)

where the summation is understood to run over all sites in the array for η̌, and all doublets in the array forP̌. The superscripts (l) and (r) refer to the left and right site in a given doublet.
The uncertainty in the resulting estimates includes contributions from both statistical uncertainty anduncertainty in the SPAM correction procedure. The statistical uncertainty arises from finite sampling of amultinomial distribution. The uncertainty in the SPAM correction is related to the statistical uncertainty inthe elements of M . Although in principle Eq. 6.26 can be inverted analytically, and the errors propagatedin the standard fashion, we opt to approximate this inversion numerically. Specifically, given the known
statistical errors in the elements of M we generate 50, 000 random samples of M and invert each one toobtain a distribution in M −1 . We confirm that the uncertainties in the elements of M −1 remain Gaussian,23
and use the standard deviations of these elements to obtain an estimate of the error in the inversion process.
21 Keeping in mind that |0⟩ here includes the contribution from |∅⟩.
22 Both assumptions are reasonable, since higher order errors and the effect of inhomogeneity are below the statistical error
in our measurements.
23 Note that this need not always be the case. Certain process matrices could have inverse matrices containing skew or evenmultimodal distributions.

(a)Rabi frequency
Detuning

Rabi frequency
Detuning

25
Frequency (MHz)25
Frequency (MHz)

152

(b)

20
15
10
5
0

20
15
10
5
0

100

200

300
400
Time (ns)

500

100

200

300
400
Time (ns)500

Figure 6.5: Imperfect Rydberg pulses. (a) The average ramp in Rabi frequency applied during the gate (asmeasured directly in the apparatus using a photodiode), and the ideal ramp in detuning. (b) To generate amore realistic model for the ramps in the experiment, we use a model for the frequency noise on the laser
to generate Monte-Carlo samples of the detuning with realistic frequency fluctuations, and use measured
single-shot values of the Rabi frequency from the experiment.

Note that this approach neglects covariances between the elements of M −1 , which are clearly present given
Eq. 6.26. However, because it is straightforward to take calibration measurements with very low statisticalerror, the characterization of errors in M −1 can be seen as a rough confirmation that our measurements
errors are dominated by statistical errors in the measurement itself, and not by errors in SPAM correction.
As a result of the above analysis, we infer that with SPAM correction the fidelity of our entangling
gate (and of the prepared Bell states) is 92.8(2.0)%, in comparison to 87.1(1.6)% without SPAM correction.

6.3.3

Error budget
As described earlier, the dominant errors expected in our gate are associated with tradeoffs between

adiabaticity and the Rydberg state lifetime. We can compare the measured fidelities to a master equation
simulation based on the measured Rydberg state lifetime (Sec. 6.1.1), as well as the measured intensity andphase noise of the UV laser (Fig. 6.5). Specifically, we work in the two atom basis, with the state of each
atom being spanned by the states {|0⟩, |1⟩, |r⟩, |∅⟩}. Time evolution of the two atom density matrix ρ isgiven by:i∂ρ= − [H(t), ρ] + Lρ
∂t
ℏ


1 †
†
Lρ := γ JρJ − {J J, ρ} ,2

(6.29)(6.30)

153
EffectError

Non-adiabaticity
Rydberg state lifetime
Intensity noise
Phase noise
single qubit rotation errors
Rydberg decay systematic

0.97%
0.94+13−11 %0.73+17−17 %0.00+3−4 %0.57+20
−18 %
0.90%

Total

4.10+40−36 %Table 6.2: Error budget for the Mølmer-Sørensen gate.

where γ is the rate, and J := |∅⟩⟨r| the jump operator, associated with decay of the Rydberg state.24 H(t)is given by Eq. 6.7, with the appropriate time dependence of Ω and ∆. Laser intensity noise is incorporated
by measuring the intensity of the pulses applied on the experiment directly, and using those traces todetermine the values of Ω(t). Similarly, we measure the spectrum of the frequency noise of the laser using
the in loop error signal of the PDH lock (see Appendix D.3), and use this to generate Monte-Carlo-sampled
laser frequency trajectories that determine ∆(t). Errors in the single qubit rotations are characterizedindependently (see Sec. 5.2), and integrated into the above simulations via Monte-Carlo sampling.
By performing simulations with and without decay, and with and without laser noise, we can isolate
the contribution of each effect, and summarize the expected magnitude of the errors originating from thesesources in Tab. 6.2. The uncertainties in the estimated contribution of laser intensity and phase noise(including both the clock and UV lasers) are bootstrapped confidence intervals. The uncertainty in thecontribution from the Rydberg decay rate is based on the statistical uncertainty in the measurement of γ.
The resulting error budget explains much, but not all, of the observed infidelity in our entangling gate.
However, recent results from the group that improve the fidelity of the entangling gates to ≳ 98.5% suggestthat the remaining infidelity was due to uncharacterized errors in the single qubit rotations.25
24 Note that this model makes the simplifying assumption that decaying Rydberg state atoms are lost. This is not entirely
accurate, since the trap depths we operate at during the gate are not so high that anti-trapping of the Rydberg state is the
only source of decay (decay to the |0⟩ or |1⟩ states is still possible). However, the approximation is reasonable because multiple
rounds of excitation and decay are not expected to play a significant role in our experiment (due to the relatively low error
rates).
25 In particular, the measured atom laser coherence time (see Sec. 5.2.4) indicated that the only contribution to single qubit
rotation errors occurred during the rotations themselves, due to residual motion of the atoms and high frequency phase andintensity noise on the laser. However, in more recent measurements we find that there are two timescales associated with the
decay of atom laser coherence, leading to larger than expected errors when two single qubit rotations are separated by morethan a few milliseconds.

154
(a)

(b)

0.5

0

1

S0

3

P0

3

P0 + 3 P2Population

Survival

1

1

4
Hold time (s)

8

P10

P01

P11

0.5

0
0

P00

0

4
Hold time (s)8Figure 6.6: Population dynamics of entangled states. (a) Measurements of the 1 S0 ground and 3 P0 clockstate lifetime of single atoms. Note that in order to diagnose certain decay channels, we measure both the
population in the clock state, and the population in both the clock state and the 3 P2 state. The curves
are exponential fits, from which we extract the single atom decay rates. (b) Similar measurements of the
population dynamics for Bell pairs, where 0 and 1 refer to the ground and clock states, and the depicted
populations are for all two atom states. We primarily observe that P11 decays to {P01 , P10 }, which cansubsequently decay to P00 . A master equation model (colored curves), which uses only the measured single
atom lifetimes and initial density matrix describing the Bell pairs, is in reasonable agreement with the data.
Shaded regions are ±1σ confidence intervals in the model.

6.4

Characterizing Bell state coherenceWith Bell states prepared with relatively high fidelity, we are interested in characterizing their lifetimeand coherence, and thus their utility for performing entanglement enhanced measurements in the sense
described in Sec. 5.1.1. Similar to Ch. 5, we characterize the coherence of the prepared Bell states bymeasuring the populations of the atomic states and the contrast of their Ramsey oscillations26 as a functionof time.
Under the conditions of the above experiments, we measure a ground state lifetime of 60.5(5.2) s, anda clock state lifetime27 of 20.2(8) s (see Fig. 6.6a). Including only these single atom decay processes in amaster equation model28 yields reasonable agreement with the observed two atom populations (Fig. 6.6b),with a relatively long lifetime of 12.3(1.0) s for the |11⟩ state. However, the contrast of the Bell state parity
oscillations decays with a Gaussian 1/e time constant of τB = 407(13) ms, in comparison to exponential decayof the single atom Ramsey contrast of τ = 1.6(2) s (Fig. 6.7). The above measurements are performed withan additional spin echo pulse in the middle of the Ramsey dark time to remove effects related to variations in
26 In the case of Bell states, the parity oscillations used to characterize gate fidelity are analogous to Ramsey oscillations.
27 Including both decay from the clock state and atom loss.
28 Similar to Eq. 6.29, except where H corresponds to free evolution of the atoms in the absence of the drive, and withseparate jump operators and rates for atom loss, and decay from the clock state to the ground state.

155

(a)

( π2 )( π2 )( π2 )( π2 )
(b)

Z

π

Z

π
Z

1.0

oscillation contrast( π2 )( π2 )ππZt( π2 )( π2 )( π2 )( π2 )

π
2

t

π

t

π
2

( π2 )( π2 )2tπ2( π2 )( π2 )

1.0

0.5
0.5

0.0

0

0

0

4
Hold time (s)

0.5

1

8

Figure 6.7: Ramsey coherence time of Bell states. (a) We prepare Bell pairs via the procedure describedin the text, and inject these Bell pairs into a Ramsey measurement sequence, as shown in the upper panel(this circuit applies to the purple data appearing in this figure). In order to perform parallel measurementswith both Bell pairs and isolated atoms, we can introduce an additional rotation about the ϕ axis, which
results in a Ramsey measurement for both Bell pairs and isolated atoms, as shown in the lower panel (thiscircuit applies to the blue and grey data appearing in this figure). (b) As before, the coherence of isolated
atoms relative to the laser is characterized by measuring the contrast of the Ramsey oscillations (oscillations
in σz ) as a function of time (grey points). Similarly, the coherence of Bell pairs relative to the laser canbe characterized by measuring the contrast of parity (P) oscillations as a function of time (blue and purple
points). The solid contours are exponential (Gaussian) fits, allowing for an offset, from which we extract thecoherence time of the isolated atoms (Bell pairs), and the shaded regions are ±1σ confidence intervals.

156
the lattice intensity in combination with a slight detuning of the lattice from the magic wavelength. Despite
removing this systematic, the coherence time between the Bell states and the laser is reduced by more thana factor of two in comparison to between single atoms and the laser.29
The above measurements suggest that there are uncharacterized sources of dephasing of the Bellpairs relative to the rotating frame defined by the clock laser, but for certain measurements we are instead
interested only in the coherence of the atoms relative to each other. To study these coherences independentof effects relating to the atom-laser coherence, we turn to correlation spectroscopy techniques similar to
those in Sec. 5.4.2.2. Specifically, we prepare an ensemble containing both Bell pairs and isolated atoms in
the |0⟩ state via the gate sequence described in Sec. 6.2, and perform an additional π/2 rotation about thex axis, which leaves the parity of the Bell states unperturbed, but rotates the single atoms into the state√(|0⟩ − i|1⟩)/ 2. This initializes a Ramsey measurement involving both single atoms and Bell pairs in a single
experimental trial, allowing us to measure correlation functions between single atoms, between Bell pairs,
and between single atoms and Bell pairs. In particular, we will be interested in the single atom σz σz (or(2)

gij ) correlation function as in Eq. 5.24, the parity-parity correlation function between Bell pairs:
(PP)

Cij

:= ⟨P (i) ⊗ P (j) ⟩ − ⟨P (i) ⟩⟨P (j) ⟩,(6.31)and the parity-σz2 correlation function between one Bell pair and one single atom:(Pσz2 )Cij:= ⟨P (i) ⊗ (σz(j) )2 ⟩ − ⟨P (i) ⟩⟨(σz(j) )2 ⟩.
(PP)

In a Ramsey measurement, the parity-parity correlation function Cij(6.32)
plays an analogous role for(2)

the Bell pairs as gij plays for single atoms (see Sec. 5.4.2.2), and by a similar argument one can show thatwhen averaging over the laser phase [290]:30
(PP)

Cij
(i)

(i)

(j)

(i)

(j)

= 2|ρ00,11 ||ρ00,11 | cos(ϕ00,11 − ϕ00,11 ),(6.33)(i)where ϕ00,11 := arg(ρ00,11 ) is the phase associated with the ρ00,11 component of the density matrix for the(PP)

ith Bell pair. Therefore, by the same arguments as in Sec. 5.4.2.2, one can average Cijover all choices of

29 This is in part explained by the different functional forms used in the fits — although ideally both the single atom andBell state data would be fit with the same functional form, we found the p-value of a Gaussian fit to the decay of the parity
contrast to be lower than that of an exponential fit.
30 Under the assumption that ρ
01,10 ≃ 0, as confirmed via our characterizations in Sec. 6.3.

157(b) 0.2
0.4
correlat ion

correlat ion

(a)

0.2

0

0

4
Hold time, t (s)

8

0.1

0

0

4
Hold time, t (s)8Figure 6.8: Measuring Bell state coherence using correlations. (a) We measure the single atom coherencetime by measuring the g (2) (or σz σz ) correlator between isolated atoms (grey), and the Bell state coherencetime by measuring the parity-parity (PP) correlator between Bell pairs (blue and purple). Note that, as inFig. 6.7, the blue data is taken in parallel with the single atom data, and the purple data is taken without
the additional rotation about the ϕ axis. Although we do not expect this change in state preparation toaffect the coherence time of the resulting state, we separate the data for the two experimental conditions forclarity. (b) To compare Bell pairs to isolated atoms, we compute the Pσz2 correlator between Bell pairs andatoms. The decay of this correlator is consistent with the expected lifetimes of the Bell pairs and isolated
atoms, suggesting that there isn’t a significant additional source of decoherence between isolated atoms andBell pairs. Throughout this figure, the solid lines are exponential fits (allowing for an offset), and the shaded
regions are the ±1σ confidence intervals of the fits.

{i, j} to obtain a lower bound for 2|ρ00,11 |2 .31

We find that the above bound decays with an exponential

time constant of 5.0(3) s (Fig. 6.8a), suggesting a Bell state coherence time of τB′ = 10.0(6) s (where the′denotes atom-atom coherence, as opposed to atom-laser coherence).32

This is to be compared to the(2)single atom gij (also averaged over {i, j}), which decays with an exponential time constant of 12.2(5) s(Fig. 6.8a), suggesting a Bell state coherence time of the same value, and a single atom coherence time ofτ ′ = 24.4(1.0) s. We therefore find that the measured Bell state coherence time is close to the expectation
based on single atom effects, with possible additional sources of decoherence contributing with a combined
time constant of 55(21) s.
(Pσz2 )The parity-σz2 correlation function Cij

is chosen for a slightly more subtle reason: we are interested

in comparing the Bell pairs to a well characterized phase reference (namely single atoms), but under free
31 Note that the measurements of the various correlation functions are performed with a spin echo, meaning that we expect(i)(j)ϕ00,11 − ϕ00,11 ≃ 0. Therefore, the decay of this quantity further serves as an estimate of the decay of 2|ρ00,11 |2
32 Note that, based on physical grounds, we expect the various correlation functions to decay to zero at long Ramseyinterrogation times. However, the hypothesis that the parity-parity correlation data is better fit without an offset is weakly
disfavored with a p-value of 0.18. Therefore, in [290], we quote conservative values of a parity-parity decay time of 2.3(4) s, and′ = 4.6(7) s, based on fit functions that allow for an offset.
τB

158

(a)

(b)

(c)

(d)

1

0

-1

-1

0

( π2 )( π2 )

1

π
Z

π

Z

-1

( π2 )( π2 )

0( π2 )( π2 )

1( π2 )( π2 )-1

0( π2 )( π2 )

1

π
Z

π

Z

-1

( π2 )( π2 )

0( π2 )( π2 )

Z1( π2 )( π2 )

Figure 6.9: Parametric plots of the evolution of Bell pairs and single atoms. (a) Because the phase of
the Bell pairs precesses twice as quickly as the phase of a coherent state, a parametric plot of the Bell pairparity (⟨P⟩) against single atom projection along z (⟨σz ⟩) traces out a 2 : 1 Lissajous figure. (b) By squaring
the signal from the single atoms (measuring ⟨σz2 ⟩), we double the precession√ frequency, leading to a 1 : 1
Lissajous figure. (c, d) Similar measurements with an additional controlled- z gate (the relevant circuits forstate preparation and measurement are shown in the lower panels) applied to the atoms results in a relative
phase between the Bell pairs and single atoms, leading to a larger opening angle in the Lissajous patterns.
We attempt to apply a phase shift of π/2, and find the applied phase to be 0.94(7)π/2 via ellipse fitting.
Note that the mean has been subtracted from all the above plots, such that their center of mass lies at theorigin.

159
(i)

(i)

evolution the phase ϕ00,11 accrues at twice the rate of the phase ϕ0,1 accrued by a single atom.33

This

effect is illustrated in Fig. 6.9ac, where single-shot measurements of P for Bell pairs and σz for single atoms
are plotted parametrically. The resulting parametric plots are 1 : 2 Lissajous figures, whose opening angle
is controlled by the initial relative phase between the single atoms and the Bell pairs.34

By squaring thesingle atom signal, we can generate a signal that precesses at twice the single atom frequency, and is therefore
appropriate to compare directly to the Bell state signal. This is reflected in Fig. 6.9bd, where parametricplots of P and σz2 yield 1 : 1 Lissajous figures. Similar to above, averaging over the laser phase yields [290]:(Pσz2 )Cij
(Pσz2 )

The average of Cij

(i)

(j)

(i)

(j)

= 2 |ρ0,1 |2 |ρ00,11 | cos(2ϕ0,1 − ϕ00,11 ).

(6.34)(i)(j)

over {i, j} serves as a lower bound on the quantity 2 |ρ0,1 |2 |ρ00,11 |, and the decay of
(i)

(j)

this quantity serves as an estimate of the decay rate since 2ϕ0,1 −ϕ00,11 ≃ 0.35 We measure the above boundto decay with an exponential time constant of 5.3(4) s (Fig. 6.8b). When compared to the g (2) lifetime of
12.2(5) s, we infer that τB′ = 9.5(1.4) s from this measurement, which is consistent with the value inferredfrom the parity-parity measurement.36
Note that although our measurements match the expected behavior well at late times, deviations
from the expected behavior at early times indicate that two timescales are present in the decay of atomic
coherence. This can be seen, for example, when comparing measurements of the various correlations to
expectations based on the master equation analysis described above (see Fig. 6.10). In [290], we handle
this conservatively by allowing for an offset in the fit function used to extract the decay time, resulting inlower inferred coherence times. However, we suspect that the early time decay is the result of technicalnoise related to switching the lattice and the magnetic fields to the conditions that are maintained during
the Ramsey dark time [290]. This mechanism would result in an initial fixed reduction in coherence, withthe subsequent scaling of coherence with respect to Ramsey dark time following the less conservative values
listed throughout this section.
(i)

33 ϕ
0,1 ≡ ϕi in Eq. 5.25. The form here is chosen to make explicit the similarities between the phases in the single- andtwo-atom states.
34 We can control this relative phase using the Rydberg gates. For example, we apply an additional controlled-√z pulse, andmeasure the resulting relative phase to be 0.94(7)π/2, close to the expected value of π/2.
35 In the absence of an additional control pulse.
36 When fitting the parity-σ 2 correlator, we do not find the inclusion of an offset to be statistically significant (with a p-valuezof 0.49). However, we include an offset to provide a conservative estimate in [290] yielding an exponential time constant of′ = 3.5(1.1) s.
2.7(6) s, and a resulting inferred value of τB

160

Correlation

0.4

0.2

0

0

4
Hold time (s)8Figure 6.10: Correlation dynamics of entangled states. Similar to the population dynamics in Fig. 6.6b, we
can compare the measured correlations to a master equation prediction based only on the measured state
prepared after the applied MS gate, and on measured single-atom decay processes. These predictions (solidlines) are compared to the three measured correlation functions in this section. The shaded regions denote±1σ confidence intervals in the master equation simulation.

161
The measurements presented in this section suggest that the coherence of the Bell states (relative
to other atoms) is not significantly degraded with respect to the expectation based only on single atom
decoherence processes, and therefore that such states could be useful for precision measurements. However, as
discussed in Sec. 5.1.2, it is important to note that the benefit associated with using such states for frequency
metrology depends on the specific sensing problem at hand. For example, in synchronous measurements
where the interrogation time is limited by the coherence of the atoms themselves, such states do not confera significant benefit. One situation where entangled states can potentially be useful is when one is limitedby LO noise, and uses the atomic signal to correct the frequency of the LO [10]. It is important to note thatin the experiments in this chapter, the atom-laser coherence is almost certainly limited by common-mode
fluctuations of the atomic frequency (due primarily to magnetic field noise), and not by noise in the LO.
As a result, in order to realize an advantage from correcting the LO frequency using an entangled state asa reference, more work is required to improve the coherence of the atoms with respect to an ideal phasereference.37

6.5

Large entangled states for frequency metrology
Although entangled states involving two particles can already be useful, for example to reject certain

kinds of noise in a sensor [233], the central promise of entanglement enhanced metrology is improved scalingof the precision of a sensor with respect to physical resources, as discussed in Sec. 5.1. As a result, we areinterested in preparing useful states for frequency metrology involving many entangled particles.
We will comment briefly on experiments we have performed in this direction [96], but defer a detailed
discussion to the future thesis of William Eckner. The central idea is to observe that Eq. 6.19 is the wellknown one axis twisting Hamiltonian, which generates spin squeezed states [174, 188]. In fact, by staying in
the weakly dressed limit where Ω ≪ ∆, one will arrive at Eq. 6.19 from Eq. 6.7 even in the case where many
(more than two) atoms are present [37, 117].38

Therefore, our protocol for applying one axis twisting tolarge ensembles of atoms (Fig. 6.11) is similar to the gate protocol described in Sec. 6.2 for two atoms.
37 This is distinct from our demonstrations so far of long-lived coherence between the atoms, since such measurements are,by design, insensitive to common-mode fluctuations of the atomic transition frequency.
38 When also including the spin echo pulse used in the gate sequence to arrive at Eq. 6.19. Also note that S now refers toz
the projection of the collective spin of all the atoms onto the z axis.

162

Figure 6.11: One axis twisting in large ensembles of atoms. Coherent states of large ensembles of atoms areprepared on the equator of the Bloch sphere, and then an operation similar to the MS gate applies one axis
twisting to the entire ensemble, resulting in a squeezed state with lower variance in its spin projection along
a given axis than in the original coherent state. The low variance axis is appropriately oriented, and theninjected into a Ramsey sequence as in the measurements in Ch. 5. The resulting measurement can have lower
QPN than the equivalent measurement using a coherent state. Note that the pulse diagram and generalized
Bloch spheres appearing in this figure are not to scale. The bottom left inset is a single shot image of an
atom ensemble after rearrangement, and the bottom right inset corresponds to a projective measurementof this ensemble. The atoms appearing in this image were in the clock state, whereas the missing atoms(highlighted in red circles) were in the ground state.

163
The states that are generated by evolving under Eq. 6.19 starting from a coherent state on the equatorof the Bloch sphere are (ideally) pure states with increased variance in energy in comparison to the original
coherent state,39 and therefore by Eq. 5.7 can achieve enhanced performance in comparison to the SQL. The
ultimate limit set by one axis twisting is a reduction in measurement uncertainty by ∝ N −1/3 in comparison
to the SQL, where N is the number of atoms [174]. However, slightly modified protocols involving one axis
twisting and global rotations [160, 161] can achieve Heisenberg-limited scaling corresponding to a reductionby ∝ N −1/2 .
The enhancement provided by spin squeezed states is particularly easy to understand, because themechanism for the reduction in measurement uncertainty is simply a reduction in the variance of the spin projection measured along the axis in which displacements are generated by the signal (without a corresponding
reduction in the displacements generated by the signal). This can be characterized by the Wineland squeezing
parameter [355]:40
ξ 2 :=N (∆J⊥,min )2,
J2(6.35)

where J is the total spin vector (with length J), and (∆J⊥,min )2 is the minimum variance achieved whenmeasuring the projection of J onto an axis perpendicular to the direction of J . In Fig. 6.12 we showthat we do indeed observe such a reduction in variance without significantly reducing J when applying theappropriate evolution, leading to ξ 2 = −3.6(8) dB (in this case for ensembles of 16 atoms). In Fig. 6.13
we use similar states involving ensembles of 16 and 70 atoms in a differential frequency comparison,41 andachieve stabilities that are respectively 3.52(1) dB and 1.94(1) dB below the SQL.

39 When properly oriented on the Bloch sphere via a final global rotation
40 Where any value below 1 indicates an enhancement in comparison to the SQL.
41 Similar to the one in Sec. 5.4.2.1, but feeding back on the LO to ensure that the atom-laser phase stays near the optimalvalue for the duration of the measurement, as opposed to averaging the laser phase over all possible values.

164
Phase θL (deg.)
180

6

360
+1/2
0

θL

4

CSS

SSS

Sz/ Ntot

2
Var. reduction σα2/ σQPN
(dB)

0

−1/2

2
QPN

0
−2

α

−4

π

−6
0

45

90

π

π

2

π

2

2

135

180

Phase α (deg.)

Figure 6.12: Characterizing spin squeezing. Measuring the variance of the spin projection of the atom
ensemble σa2 , relative to the expectation for a coherent spin state (CSS), as a function of the measurement
axis α. Note that there is a measurement axis about which the spin squeezed state (SSS) has significantly
2
lower variance than the limit set by QPN (σQPN). Critically, this reduced variance is not the result of asignificantly diminished value of the total spin, which is measured in the inset by measuring the total spin
projection Sz /Ntot (where Ntot is the total atom number) as a function of the laser phase θL . Fits to thesemeasurements (solid lines) yield a Ramsey contrast of C = 0.97(1) for the CSS, and C = 0.83(2) for the
SSS. The reduction in C in the SSS is not sufficient to counteract the reduction in the variance of the SSS,
and so this state can be used in measurements that surpass the SQL.

(b)

−15

0.0
1.0

SSS

0.5

Allan deviation σy(τ)

PB

B

10

1.0
A

10−15
0.5

PB

Allan deviation σy(τ)

A

0.0
1.0
10−16
αopt

10−16
CSS
10

0

10

π

SSS
1

2

10

2

Averaging time τ (s)

10

3

0.0

0.5

PA

0.0
1.0

0.5

B

PB

1.0

CSS

10

0

π

π

2

10

1

π
2

Tdark

10

0.5

αopt

PB

(a)

π
2

2

Averaging time τ (s)

103

0.0

0.5

0.0
1.0

PA

Figure 6.13: Synchronous frequency comparisons using spin squeezed states. We measure the overlapping
Allan deviation for differential frequency comparisons between ensembles A and B, using a coherent spinstate (CSS) and a spin squeezed state (SSS). In (a), the ensembles each contain 32 atoms, and a Ramsey
dark time of Tdark = 26 ms is used. The resulting stability of the measurement with the SSS is 2.829(4) ×
10−15 (T /s)−1/2 , which is 3.69(2) dB lower than that of the CSS, and 3.52(1) dB lower than the SQL. In (b),the ensembles each contain 70 atoms, and a Ramsey dark time of Tdark = 54.5 ms is used. The resulting
stability of the measurement with the SSS is 1.087(1) × 10−15 (T /s)−1/2 , which is 2.3(1) dB lower than thatof the CSS, and 1.94(1) dB lower than the SQL. The right panels show the excitation probabilities PA and
PB of atoms in ensembles A and B, with each dark point corresponding to a single experimental trial (only
200 trials are pictured for clarity). The reduction in variance in the SSS is clearly visible. The light points
correspond to a similar measurement, in which the laser phase is not locked to the atomic reference signal,
resulting in a uniform sampling of the Ramsey fringe. This latter measurement provides an estimate of thecontrast of the Ramsey fringe for each measurement setting.

Chapter 7

Single-particle quantum walks“Nowhere. Everywhere. I’m the sum total of the works, the whole show.”— Neuromancer/Wintermute
Portions of this chapter have appeared in:
A. W. Young, W. J. Eckner, N. Schine, A. M. Childs, and A. M. Kaufman. Tweezer-programmable
2D quantum walks in a Hubbard-regime lattice. Science, 377(6608):885–889, Aug. 2022
A. W. Young, S. Geller, W. J. Eckner, N. Schine, S. Glancy, E. Knill, and A. M. Kaufman. An
atomic boson sampler. In review, 2023

To this point, we have considered the situation where many atomic qubits are pinned in place, allowing
us to manipulate their internal degrees of freedom to perform various computational and metrological tasks.
In this chapter we will focus on control over the external degrees of freedom of a single atom — namely its
position in a tunnel-coupled optical lattice — to enable the study of different kinds of dynamics. We firstpresent a brief review of band theory in Sec. 7.1, and frame our discussion in the language of continuous timequantum walks in Sec. 7.2. In Sec. 7.2, we further demonstrate that such quantum walks can be performedby implanting atoms prepared with optical tweezers into an optical lattice. In Sec. 7.3, we discuss how wecalibrate the parameters that determine the behavior of the quantum walks, and in Sec. 7.4 we manipulatethese parameters to prepare specific ground states where a single atom is delocalized across the lattice. Next,
in Sec. 7.5, we discuss (and demonstrate) how the prepared states and quantum walk dynamics can be used
as a resource in various quantum algorithms, with a particular emphasis on spatial search. We conclude inSec. 7.6 by discussing how the tools developed in this chapter could be adapted to provide universal control
over the quantum walk dynamics, and in Sec. 7.7 by commenting on how to interpret the benefits provided
by quantum algorithms (based on quantum walks or otherwise).

166

7.1

Single atoms in a lattice

7.1.1

Band theory
The Hamiltonian for a single atom in a spatially varying potential V (r) is:
H=−

ℏ2 2
∇ + V (r),
2m

(7.1)

where m is the atomic mass. If V (r) is a periodic potential, such that V (r) = V (r + R) for all Bravais
lattice vectors R (and corresponding reciprocal lattice vectors K), then the eigenstates ψ and eigenenergies
E of Eq. 7.1 can be solved by performing a discrete Fourier transformation:
V (r) =X

VK eiK·r(7.2)

ck eik·r ,

(7.3)

K

ψ(r) =

Xkwhere k is a wavevector that runs over all values allowed by the chosen boundary conditions.1

Plugging

the above equations into the time-independent Schrödinger equation (Hψ = Eψ), and using the fact thatthe plane waves form an orthonormal basis, we obtain [11]:



X
ℏ2
2
|k| ck +VK ck−K = Eck .
2m

(7.4)

K

The left-most term in Eq. 7.4 is simply the free space energy, whereas the middle term corresponds toscattering off of the Fourier component of V with wavevector K, which couples plane waves with wavevector
k to plane waves with wavevector k − K. Note that we can restrict k to the first Brillouin zone,2 since we
can always pick some reciprocal lattice vector K ′ such that k − K ′ lies in the first Brillouin zone. The key
insight in the above derivation is that the periodicity in V causes Eq. 7.1 to factorize into separate eigenvalue
problems for each value of k in the first Brillouin zone,3 since V only couples plane waves which differ by areciprocal lattice vector, and thus map to the same value of k in the first Brillouin zone. For each such valueof k there are an infinite series of solutions to Eq. 7.4, which we label by the band index n. For an infinite
lattice, the resulting energy spectrum is given by a set of continuous functions En (k), which is known asP
1

For example, for a lattice with L unit cells in each direction and cyclic boundary conditions, k =n b /L, where bi isi i ia primitive reciprocal lattice vector, and ni is an arbitrary integer.
2 Namely, the region in the reciprocal lattice that is closer to the origin than to any other reciprocal lattice site.
3 i.e. the Hamiltonian is block diagonal, with each block corresponding to one value of k.

(b)

6

M

(c)

167
1

10 1

0.3

2

10 2

y/a

J (E2D
R )

E/E2D
R

4

3

0

10

2

10 4

M

0.2

0

NN
NNN
Diag.

|w|

(a)

0.1

1
5

10

V (E2D
R )

1

0
x/a

1

Figure 7.1: Band structure and Wannier functions. (a) Band structure for a bowtie lattice with a depth
2Dof V = 5ER, plotted along a triangular path between critical points at the center, edge center, and cornerof the first Brillouin zone (Γ, X, and M respectively, see inset), where k is a wavevector and E the energy.
2DER
is the recoil energy of the 2D lattice. The free space energy is indicated by the dashed black line, andthe depth of the axial lattice by the grey dashed line. The inset shows the location of the critical points
relative to the first Brillouin zone (shaded region). (b) Under the tight binding approximation, only tunnelingprocesses between nearest-neighbor (NN) sites are significant. However, at the lattice depths we work at(vertical grey lines denote our two typical operating conditions), other processes can become significant.
After NN tunneling, the next two dominant processes involve tunneling from one site to its next-nearest
neighbors (NNN), or to its neighbors in a diagonal direction (diag.). (c) The amplitude of the Wannier
2D
function w in the lattice at a lattice depth of 6.3ER. Notice that at this depth, the Wannier function is still
well localized relative to the lattice spacing a, and approximately Gaussian.

the band structure of the lattice. The associated eigenstates are the Bloch states ψn,k (r), which, due to thediscrete translation symmetry of the problem, are delocalized across the lattice and have the property that
ψn,k (r + R) = eik·R ψn,k (r) [11].
For most optical potentials, there are only a handful of different relevant values of K. For example,
for the bowtie lattice potential in Eq. 3.6 the values of VK are:
V0

4 , if m1 = m2 = 0 V0 , if |m | + |m | = 1
1
2
8
VK =
V0



16 , if |m1 | = 1, |m2 | = 10,otherwise,(7.5)

where V0 is the lattice depth, m1 , m2 ∈ Z, and we have introduced the parameterization K = m1 b1 + m2 b2 ,where b1,2 are the primitive reciprocal vectors of the square 2D lattice. We show the numerically obtained
band structure of this lattice for parameters that are relevant to our experiments in Fig. 7.1a.
Note that although our experiments occur in a 3D optical lattice, the axial potential (in the out-ofplane or z direction) is separable from the bowtie lattice. Additionally, the axial lattice is aligned along

168
gravity and thus the atoms experience a linear gradient of V (z) = mgz, where g is the local gravitational
field strength. This gradient localizes the atoms to a single site in the axial lattice, and thus a single 2D
layer of the 3D lattice, for all parameter regimes of relevance to this thesis.

7.1.2

Wannier functions
Whereas the Bloch states are delocalized across the lattice, the states that we can readily prepare via

tweezer implantation involve atoms that are localized on a given site in the lattice. The maximally localizedstates that can be constructed out of the Bloch states are described by the Wannier functions, which havethe form [11]:wn,i (r) = N −1/2

X

e−ik·Ri ψn,k (r),

(7.6)

k

where N is a normalization factor. wn,i (r) can be interpreted as the wavefunction of an atom which primarily
occupies a specific lattice site i with lattice vector Ri (see Fig. 7.1c). In the limit of a very deep lattice,where each lattice site behaves like an independent harmonic trap, wn,i (r) is simply the harmonic oscillator
state with motional quantum number n (centered at position Ri ).
Although the Wannier states form a complete orthonormal basis, they are not eigenstates of H (asdefined Eq. 7.1). Considering how H couples the different Wannier functions we define:Jn,ij = −Z

d3 r wn,i (r)Hwn,j (r).

(7.7)

Jn,ij is known as a tunneling matrix element, and is associated with the rate at which an atom occupying agiven site i in the lattice is tunnel-coupled to another site j.4
In practice, our lattice doesn’t extend to infinity in all directions. Instead, the finite waists of thelattice beams result in a slowly varying potential (sometimes referred to as a chemical potential for historicalreasons) which is approximately harmonic, and confines the atoms to a fixed region in space. Including thischemical potential, and restricting ourselves to only the ground band where n = 0, we can rewrite H as:
H=−

X
{i,j}

Jij (|i⟩⟨j| + |j⟩⟨i|) −X

Vi |i⟩⟨i|,

(7.8)

i

4 Note that, given the definition of the Wannier functions in Eq. 7.6, Jn,ij is also directly related to the Fourier transformof the band structure.

169
where Jij ≡ J0,ij , the state |i⟩ refers to the wavefunction w0,i (r), Vi is the shift in energy of the state |i⟩ due
to the chemical potential, and the first sum runs over all pairs of sites {i, j} with appreciable tunnel coupling.
Note that although Vi breaks the initially assumed translation symmetry, the harmonic contribution to Vi
varies slowly in comparison to the lattice spacing, and so the above analysis remains accurate.5
When the Wannier functions are well-localized on a given lattice site, as applies to lattice depths ofV0 ≳ 5ER (where ER is the recoil energy of the lattice, as defined in Sec. 3.3), one typically only needs toconsider {i, j} that correspond to neighboring sites in the lattice.6

In this case, and for a region in thelattice where Vi is approximately constant, the Hamiltonian simplifies to:
H = −J

X(|i⟩⟨j| + |j⟩⟨i|) .

(7.9)

{i,j}

The key result of this analysis is that, when working in the ground band, we can abstract away the exact
form of the Wannier functions, and understand the relevant dynamics in terms of a discrete set of locations
that a point-like particle can occupy.

7.2

Continuous-time quantum walks
The dynamics of a point-like particle that can hop between a discrete set of locations is conveniently

framed in terms of a continuous-time quantum walk (CTQW). A CTQW is defined by a graph G = {V, E},
which is characterized by a set of vertices i ∈ V and edges {i, j} ∈ E [272]. G can be represented by its
adjacency matrix A, where:1, if {i, j} ∈ E
Aij =0, otherwise.

(7.10)We then define that the Hamiltonian for a CTQW is simply:
H = −A.

(7.11)

When considering the square lattice graph where vertices correspond to sites in the lattice, and edges
5 As a point of reference, under typical operating conditions the difference between V on adjacent lattice sites is significantlyismaller than the tunnel coupling J between those sites. However, Vi can change by as much as ∼ 10J when comparing a siteat the center of the lattice to one that is 25 sites away.
6 Although in our works it is sometimes important to include contributions from tunneling matrix elements that couple
diagonally spaced or next-nearest-neighbor sites in the lattice (see Fig. 7.1b).

170
to the tunnel-coupled, nearest-neighbor sites, Eqs. 7.9 and 7.11 are equivalent up to the proportionality
constant J.
To understand the expected behavior of these CTQWs, consider a CTQW on a cycle graph CN with N
total sites (a line with cyclic boundary conditions, see Fig. 7.2). The action of the corresponding Hamiltonian
H1D on a specific site |i⟩, with i ∈ [0, N − 1], is:
H1D |i⟩ = −|(i − 1)%N ⟩ − |(i + 1)%N ⟩,(7.12)

where % denotes modular division. Based on the same logic as in Sec. 7.1.1, the eigenvectors of H1D areplane waves |ψk ⟩ with eigenenergies Ek :N −1

1 X ij2πk/N|ψk ⟩ = √
e
|j⟩
N j=0
Ek = −2 cos(

2πk
),
N

(7.13)(7.14)

where k is an integer in the interval [0, N − 1].
Consider how time evolution under H1D for time t transfers an atom from site l to site m [272]:⟨m|e−iH1D t |l⟩ = ⟨m|e−iH1D t

N−1X|ψk ⟩⟨ψk ||l⟩

k=0

=

 


N −1
1 X
2πk
2πk
exp i (m − l)
+ 2t cosNNN(7.15)

k=0≃ im−l Jm−l (2t),where in the last step we have used the fact that in the limit of large N the summation can be approximatedwith an integral to yield a result in terms of the Bessel function of the first kind of order (m−l), Jm−l .7 The
result is left- and right-moving wavefronts that expand ballistically in time (with a speed of 2 sites/unit time)from the input site l. There is an exponentially suppressed tail outside the wavefronts, and a sinusoidally
modulated tail in between the wavefronts (see Fig. 7.2). This is in contrast with the behavior of classicalrandom walks, which exhibit diffusive expansion of a Gaussian probability density distribution.
7 Note that the large N limit also means that the atom never wraps around the graph. The use of a cycle graph rather thana line graph in this example is just out of convenience, as it yields a simpler expression for the eigenstates of the system.

171

0.035
0.030
0.025

p

0.020
0.015
0.010
0.005
0.000
0

25

50

75

100
Posit ion

125

150

175

200

Figure 7.2: A quantum walk on the cycle graph CN (see inset), with N = 201. The probability density
p that a quantum walker initialized on a single site (site 100, denoted by the grey vertical line) occupies
another site is shown after a unitless evolution time of t = 35 (see Eq. 7.15). Note the general features of asinusoidally modulated inner tail, and an exponentially suppressed outer tail.

172
7.2.1

Separability of quantum walks in square lattices
When considering quantum walks on a 2D square lattice graph with cyclic boundary conditions, it isuseful to work in the basis |x, y⟩ := |x⟩ ⊗ |y⟩, where x and y denote the coordinates of the atom in the two
orthogonal directions in the square lattice. In this case, the Hamiltonian for the 2D square lattice graph
H2D can be written as:
H2D = H1D ⊗ 1y + 1x ⊗ H1D ,(7.16)

where 1x is the identity operator on the x subspace (and similarly for y). The resulting unitary evolution
factorizes into operators that act only on the x and y subspaces:
U = e−iH2D t= e−iH1D ⊗1y t e−i1x ⊗H1D t(7.17)= (Ux ⊗ 1y )(1x ⊗ Uy ),where Ux = e−iH1D t is a unitary operator that acts only on the x subspace (and similarly for y). Given
the separability of the above dynamics, the behavior of a CTQW on a 2D square lattice is precisely that ofa CTQW in 1D in each axis of the lattice (with a similar argument applying to higher dimensional squarelattices).8

Along the same lines, if one bins the probability distribution resulting from the 2D CTQW

dynamics along one axis, one is simply left with a 1D CTQW along the remaining axis.
We see the above behavior borne out in experiments with a single atom placed in the lattice [366](Fig. 7.3). For the latest depicted evolution time of t = 5.0 ms, the probability density continues to exhibit
clear interference fringes, suggesting that the atom has maintained phase coherence while exploring a region
spanning approximately 200 lattice sites.

7.3

Calibrating lattice parameters
Our characterizations of the lattice potential primarily rely on either fitting CTQWs of individual

atoms, or using the atoms as spectroscopic probes of the intensity of the light field at the position of the
8 Note that the above arguments do not hold if there are terms that directly couple x to y in the Hamiltonian, for example
if there are diagonal tunneling matrix elements, or significant local energy shifts that vary site to site in the lattice and thus
depend on a specific set of x, y coordinates.

173

0.0

(a)

p

Data

0.9

0.00

0.03

0.00

0.02

0.00

0.000

0.01

0.008

Theory

(b)

1

0

p

position

9

0.1
Data

9
1

Theory
2

3

4

5

6 1
Tunneling time (ms)

2

3

4

5

6

Figure 7.3: Quantum walks in an optical lattice. (a) Atoms implanted in the lattice undergo continuous-timequantum walks in 2D, such that the probability density p corresponding to their measured position (left)
exhibits ballistic expansion, and wave-like interference. The behavior of these walks is in good agreementwith theory using fitted values of the tunneling energy (right) up to an evolution time of 5 ms, where theatoms have coherently explored a region spanning ∼ 200 lattice sites. Each pixel in these plots represents
a single lattice site. (b) Tracing out one dimension (in this case the x axis) of a 2D quantum walk on asquare lattice, yields a 1D quantum walk along the remaining axis (in this case y, left), which is also in good
agreement with theory (right) and more clearly illustrates the ballistic expansion of the wavefunction. Data
to the left of the red line is gathered in parallel by preparing atoms in nine different regions in the lattice,since inhomogeneity between regions does not play a significant role for short evolution times. Data to theright of the red line is gathered using a single atom that is placed at the center of the lattice.

174

(a)

(b)

(c)
5

47

4

y

2π t/

3
2
1
0

0
0

47

x
12

14
V (kHz)16

24Time (ms)

0.00

p

0.01

Figure 7.4: Calibration measurements of the lattice potential. (a) Spectroscopic measurements of thelattice-induced AC Stark shift of the 1 S0 ↔ 3 P1 transition are used to characterize the lattice depth V as afunction of position in the lattice (x and y are in units of the lattice spacing), here pictured at the ratio of 2D
and axial lattice depths that is typically used during tunneling. White contours correspond to a Gaussian fit
to this potential, which is in good agreement with the expected lattice beam waists. (b) Further calibrations
are performed by measuring single particle CTQWs as a function of evolution time (as in Fig. 7.3). At each
evolution time, the measured atomic densities are fit to a model that allows for independent values of theunitless tunneling times tx /τ and ty /τ in the x and y directions (red and blue circles respectively). At lateevolution times, the extracted values grow linearly with evolution time, corresponding to constant values of
the tunneling energy of Jx /ℏ = 2π × 178.1(1.1) Hz and Jy /ℏ = 2π × 148.3(1.1) Hz in the two axes (dashedlines are linear fits excluding the first 4 points). At early times the fitted tunneling times deviate from thelinear expectation due to the slow settling time of the lattice intensity control. (c) At late evolution times, inthis case at 5.2τ , the atoms begin to sample the curvature of the lattice, and the probability density deviates
substantially from the prediction in a flat lattice (inset). This behavior is qualitatively consistent with the
expectation given the independently characterized lattice potential (contours denote equipotential surfacesof the lattice).

175
atoms (we describe some more sophisticated characterization techniques in Sec. 8.4)
To spectroscopically measure our lattice potential, namely the chemical potential terms Vi appearing
in Eq. 7.8, we change the angle of the magnetic field in the experiment such that the lattice is not magicfor the 1 S0 ↔ 3 P1 transition [366]. By pinning atoms in the lattice, and performing spectroscopy on the1

S0 ↔ 3 P1 transition, we can measure the local AC Stark shift imposed on the atoms and thus the effective

chemical potential on each site in the lattice (see Fig. 7.4a). We repeat the above measurements for the axial
and 2D lattices individually, as well as with both lattices on at the ratio of lattice depths that is relevant
for tunneling. Note that these measurements are typically performed at ∼ 10 − 100× higher intensities thanthose used during the tunneling experiments in order to enhance their signal to noise. We confirm via direct
measurements of the power in the optical lattice beams that the ratios between the lattice beam powers stay
fixed when operating at different power levels.9
The above spectroscopic measurements provide a relative characterization of the chemical potential
terms, but it remains to set an overall energy scale for the Hamiltonian. To directly characterize the tunnelingrate in the lattice, we perform a maximum likelihood fit of the measured probability density distributions as
a function of evolution time t to propagation under Eq. 7.9, assuming a perfectly flat lattice with constant
Vi .10 The free parameters in these fits include independent values of the unitless tunneling times tJx /h andtJy /h in the x and y axes (Fig. 7.4b), as well as an overall contrast and offset. The offset does not evolve
appreciably over the evolution times explored in this chapter, and is consistent with overall error rates due
to imperfect imaging and cooling.
The data exhibits constant values of the tunneling energy at intermediate times, reflecting the expectedballistic expansion of the atomic wavefunction. At early times, the data deviates from this linear expansion
due to the relatively slow 0.7 ms settling time of the intensity servo used to control the 2D lattice depth,
which results in artificially low inferred tunneling rates. At late times (≳ 4 ms), the assumption of uniform
Vi breaks down, leading to a reduction in the fitted contrast. For the purposes of extracting the tunnelingenergy, we simply exclude the late time data from our fit. By either fitting the early- to intermediate-time
9 We make the assumption that the transmission of the glass vacuum cell is independent of optical power. For typical opticalintensities at the location of the cell in our experiments, we do not expect effects like thermal lensing to play a role.
10 Note that including corrections to V does not significantly alter the result of these fits.
i

176values of tJx /h and tJy /h versus t with a hyperbola, or fitting a line to only intermediate-time data, we canextract the tunneling energies. For the work described in this chapter, the lattice has tunneling energies ofJx /ℏ = 2π × 178.1(1.1) Hz and Jy /ℏ = 2π × 148.3(1.1) Hz near the lattice center. This is in good agreementwith an ab initio calculation using independent calibration data, specifically: the measured lattice beam
waists and powers, the measured transmission of the vacuum cell, the inferred lattice spacing given imagesof lattice-trapped atoms, and the magnification of the imaging system. Note that the discrepancy between
Jx and Jy is because, as a result of our alignment procedure, the lattice is not perfectly square (see Sec. 3.3).
We are able to set an absolute energy scale for Vi (see Fig. 7.4a) by combining our measurementsof Jx and Jy with the calculated band structure of our lattice as a function of lattice depth (as describedin Sec. 7.1). These calculations also provide a way to convert between the measured chemical potential
Vi and the local value of the tunneling energy Jij across the entire lattice, under the assumption that thechemical potential varies smoothly across the lattice. We find this assumption to be accurate given the
spectroscopic measurements of Vi . Constructing a full model of our Hamiltonian via the above calibrations
yields predictions that are consistent with the late time single atom CTQW data, building confidence in thismodel. Additionally, we note that inserting calibrated values of Vi into the fit function does not change theinferred values of Jx or Jy in a statistically significant way. For the remainder of this chapter, we define
J0 /ℏ = 2π × 163 Hz to be the mean of Jx and Jy . This sets a convenient energy scale, and a corresponding
characteristic evolution time of τ = h/J0 = 6.12 ms, for discussing subsequent results.
For the tunneling conditions relevant to this chapter, we operate at a 2D lattice depth of 5.0E2DR
11(where E2D
Although this depth is relatively shallow, based on
R is the lattice recoil energy, see Sec. 3.3).

the calculations in Sec. 7.1 we conclude that the effect of diagonal and next-nearest-neighbor tunneling is
negligible for subsequent results in this chapter.

177

5

1

0

0.1

p

(b)
position

(a)

5
1.0

0.5

10

J

5

E

0.0

0
0

20
40
60
Ramp time (ms)

80

P

VC

VC/J0

J/J0,

E/J0

1.0

0.5

| o| |2
f

0.0
0

25
50
75
Ramp time (ms)180

Figure 7.5: Adiabatic preparation of an atom in the ground state of a tunnel-coupled lattice. (a) An atom
implanted in the lowest-energy site in a deep lattice with no tunneling (left callout), which we label as being
in state |o⟩, can be adiabatically connected to the ground state |g⟩ of a shallow lattice with tunneling (rightcallout) via an adiabatic ramp of the tunneling energy J and the depth of the confinement tweezer VC . In
the callouts, the grey dashed line denotes the potential provided by the confinement tweezer, and the solid
grey line the sum of the tweezer and lattice potentials. The preparation tweezer used to initially implant
the atom in |o⟩ is shown schematically in green. By ramping J and VC together we can maintain a roughly
fixed energy gap ∆E between the ground and first excited states of the system throughout the ramp. Thisrelaxes the requirements on adiabaticity, and substantially increases the fidelity with which we are able toprepare the lattice ground state. (b) As an atom evolves under the above ramp, its amplitude spreads overmany sites in 2D. Here, the x coordinate has been traced out for illustration purposes, showing the spread in
the atom’s y position (top). During this ramp, the population on the initial site (grey points) decreases, andthe overlap f between the classical probability distributions corresponding to the prepared state |ψ⟩ and the
expected lattice ground state |g⟩ (black points) increases, eventually reaching 76.9(3.3)%. This value is inreasonable agreement with theory (solid lines) given the independently measured parameters in our ramp,
and an overall scale factor to account for filtering and loss due to imperfect preparation of the atoms intheir 3D motional ground states. This serves as an upper bound on the fidelity with which we can prepare
the lattice ground state, but does not certify any phase coherence between components occupying different
sites in the lattice. Nonetheless, this state is not observed to significantly evolve even after more than 100
tunneling times (right side of the broken axis), further suggesting that this is indeed the lattice ground state.

178

7.4

Preparing the lattice ground state
Instead of preparing Wannier states by implanting tweezer-trapped atoms in the lattice, one may beinterested in preparing the ground state of Eq. 7.8. In order to do this, we take advantage of the fact that wecan programmably vary the local chemical potential Vi appearing in Eq. 7.8 by shining optical tweezers on
the lattice. Specifically, we use a large 813 nm tweezer (we will refer to this as the “confinement tweezer”)
with tunable overall depth VC , and a nearly-Gaussian profile with a beam waist of 5.8a (where a is the 2D
lattice spacing). At fixed J, the lattice ground state |g⟩ is close to the uniform superposition of the atom
occupying all sites in the lattice, except with an approximately Gaussian envelope with a width determinedby the value of VC (see Fig. 7.6).12
To prepare the state |g⟩ we implant an atom in the site o, which is the deepest site of the combined
potential generated by the lattice and the confinement tweezer (Vo > Vi ∀ i ̸= o). The prepared state |o⟩ isthe ground state of the system when the lattice is deep, and J ≪ VC . |o⟩ can be adiabatically connected
to |g⟩ in a shallow lattice via a ramp of the tunneling energy (Fig. 7.5a). In practice, we perform a linear
ramp of the lattice depth VL as a function of time t, resulting in a nonlinear ramp in the tunneling energy.
We also ramp the depth VC of the confinement tweezer to maintain an approximately constant value of theenergy gap ∆E between the ground and first excited states during the ramp, which substantially relaxes
the requirements on adiabaticity and improves the fidelity with which we can prepare |g⟩. The observed
evolution during this ramp is in reasonable agreement with theory (Fig. 7.5b), where the atoms start out
primarily in |o⟩, and smoothly delocalize over many sites over the course of an 80 ms-long adiabatic ramp
of VC and J.
The prepared state |ψ⟩ can be compared to |g⟩ by computing the overlap between their populations,P √or the classical fidelity f = ( i pψ,i pg,i )2 , where pψ,i and pg,i denote the populations on site i in states|ψ⟩ and |g⟩ respectively. This constitutes an upper bound on the fidelity with which we have prepared |g⟩of 76.9(3.3)%, but does not certify any phase coherence between the amplitudes occupying different sites.
11 At these shallow depths there is only a single bound band in the 2D lattice, and so atoms occupying higher bands are
typically interpreted as lost, either because they actually leave the lattice, or leave the region of the lattice that we analyze in
our images.
12 We will sometimes refer to |g⟩ as the “resource state,” for reasons which will become clear in Sec. 7.5.

179
(a)

(b)

(c)

p

0.1

0.0

0.6
0.4
0.2
0.0
100

101

Recovered fraction

(d)

102

Ramp duration (ms)Figure 7.6:
Robustness of ground state preparation to misalignment. (a) Cartoons showing how the
combination of misalignments between the confinement tweezer (grey curves) and lattice center (dashedgrey curves), and changes in their relative depth, can lead to a shift in the minimum (vertical grey lines) of
the overall potential experienced by the atoms (black curves), and a corresponding change in the size andposition of the atomic ground state |g⟩ (blue shaded regions). These diagrams are for visualization purposes
only, and are not to scale. Furthermore, they should be understood as an average envelope spanning manylattice sites, where modulations on the scale of the lattice period have been omitted for clarity. (b) With
poor centering between the confinement tweezer and lattice center, measurements of the atomic probability
density after performing an adiabatic ramp to the ground state reveal states whose position (and size) varywith the final confinement tweezer depth VC . From top to bottom, these depths are: 2.2 J0 , the typical
depth used for resource state preparation of 1.0 J0 , and 0.0 J0 . (c) Typical atomic probability density in the
resource state, with the confinement tweezer well-aligned to the lattice center. Color scale is shared between(b) and (c). (d) As an additional test of the resource state preparation, we perform a forward ramp to theresource state, and then a reverse ramp with variable duration. For longer ramps we recover 57(5)% of theatoms, whereas for shorter ramps the evolution is diabatic, and very little of the atomic probability density
returns to the central site, beyond what was already present on that site in the resource state.

However, the prepared state is not observed to substantially evolve over t ≫ τ , and the adiabatic ramp canbe reversed to recover 57(5)% of the atoms in |o⟩ (excluding loss due to filtering of hot atoms, see Sec. 7.3).
This suggests that |g⟩ has been prepared with a fidelity of 76(3)%, in agreement with the bound set by theclassical fidelity.
If the center of the confinement tweezer and the lattice are in different locations, the adiabatic ramp
also results in a shift in the minimum of the combined potential experienced by the atoms (since the relative
depths of the different optical potentials change). For large misalignments, this shift causes the state prepa-

180
ration to fail. However, for misalignments of ≲ 5a, the center position shifts adiabatically during the ramp,
leading to the successful loading of a ground state that is displaced from the initial site that the atom wasloaded in (Fig. 7.6ab).
This robustness against misalignment is a useful tool. Because the alignment between the two tweezer
arrays is exceptionally stable (see section 3.3.5), the confinement tweezer can be used as an intermediary
between the preparation tweezer and the lattice, allowing one to load the lattice ground state even when thetweezers are significantly misaligned from the lattice center (Fig. 7.6b). Specifically, an appropriate ramp in
the depths of the confinement tweezer and 2D lattice results in a self-calibrating ramp of the center of the
combined optical potential formed by the tweezers and the lattice (Fig. 7.6a). This works even when theconfinement tweezer is fully extinguished at the end of the ramp, in which case the center position starts at
the location of the implanted atom, and ends at the location of the lattice center.13

The resulting state(Fig. 7.6b, bottom) behaves similarly to the case when the confinement tweezer is not fully extinguished: it
does not significantly evolve over time, and 50(14)% of the atom population can be recovered in the initialsite when reversing the ramp. Critically, we measure no atoms in the initial site after a similar set of ramps
without the confinement tweezer, meaning that the state preparation fails completely in this scenario.
While we do not do full state tomography to characterize |ψ⟩, there are several additional pieces ofevidence indicating that |ψ⟩ is close to |g⟩. First, the size and position of the prepared state is consistent
with expectations based on independent characterizations of the various optical potentials in the experiment.
For example, the data in Fig. 7.6b upper and middle are taken back to back, with final confinement tweezer
depths of 2.2J0 and 1.0J0 respectively. In this case we expect a ratio in the 1/e2 radius of the preparedstates of 0.78, and observe 0.83(11). Secondly, we expect forwards and reverse ramps to recover atoms in
the central site only when the evolution is adiabatic, and genuinely prepares the lattice ground state. Whenperforming these ramps, the shorter the duration of the reverse ramp, the fewer atoms return to the central
site (Fig. 7.6d), as expected. The same is also true when both the forward and reverse ramp times are
13 A similar shift in the combined center of the axial and 2D lattices can occur.

We have attempted to mitigate this byramping the axial and 2D lattice depths together, maintaining a constant ratio, but this places more stringent requirements on
the adiabaticity of these ramps, and results in poorer overall performance. Instead, because these potentials have significantly
less curvature than the confinement tweezer, we find it sufficient to align the tweezers to the combined center of the lattices at
the final ratio used during tunneling.

181
varied. Moreover, when the prepared state is perturbed by a quench in Vo (namely a sudden modificationof the chemical potential on the central site in the lattice by a few J0 ), we expect, and observe, that thereverse ramp also fails despite the fact that immediately after the quench there is more population on thecentral site than is present after only the forward ramp (see Sec. 7.5.4). This is because the quench resultsin evolution to a state that is not adiabatically connected to the ground state of the lattice when tunnelingis turned off, where the atom sits on the central site.
We have shown so far that we can implement CTQWs of single atoms in an optical lattice, and provide
evidence that we can prepare the ground state of a given CTQW Hamiltonian. Next, we discuss how thesetools can be combined to perform certain useful computational tasks.

7.5

Search by quantum walk
CTQWs provide a convenient framework for understanding a variety of quantum algorithms. Remark-ably, even when restricting oneself to undirected and unweighted graphs, CTQWs are capable of universal
quantum computation [59, 64]. This simple but intuitive framework has inspired new quantum algorithms,
including those for spatial search [62], graph traversal [60], element distinctness [6], and formula evaluation [104]. In this section, we focus on a spatial search algorithm proposed by Childs and Goldstone [63],and show how this algorithm can be performed with quantum walks of a single atom.

7.5.1

An analog analogue of Grover’s search
Spatial search by quantum walk is closely related to Grover’s unstructured search algorithm [132].

The problem statement for unstructured search is as follows: Given an unstructured database with itemsi ∈ [0, N − 1], and access to an oracle function f which has the property1, if i = w
f (i) =
0, otherwise,(7.18)

how can one find the marked item w with the fewest number of queries to f ?
In the classical case, one can do no better than O(N ) queries, since on average N/2 items must be
checked before the marked item is found. However, it is possible to do better if one is allowed to query f in

182

Figure 7.7: Operating principle of Grover’s search. Starting from the resource state |s⟩, one iteratively
applies the oracle f and the Grover diffusion operator D. Note that D is generated by a CTQW on thecomplete graph KN , and so this operation can be implemented using the appropriate CTQW dynamics. The
operations f and D leave us in the subspace spanned by√|s⟩ and the marked state |w⟩, and their combined
action is to rotate the prepared state by an angle θ ∼ 1/ N (where N is√the number of states in the search
space) closer to the marked state |w⟩. The algorithm terminates after O( N ) such rotations, at which point
the prepared state is approximately |w⟩, whose value can be read out in a single projective measurement.

superposition. In this case, we redefine f as a unitary operator:−|w⟩, if i = w
f |i⟩ =
|i⟩,otherwise,(7.19)

where items i are now states |i⟩ in some Hilbert space H. Grover’s search (see Fig. 7.7) proceeds by preparing
the uniform superposition state over all items:
1 X|s⟩ = √|i⟩,
N i

(7.20)and sequentially applying f , and then the diffusion operator D = 1 − |s⟩⟨s|. The result is a coherent build√
up of amplitude in the state |w⟩ which is of order unity after O( N ) applications of f and D, at which
point one can measure the system projectively to identify the state |w⟩. This algorithm yields a quadraticenhancement in comparison to the number of queries to f required in the classical case.
One can translate Grover’s search to the setting of continuous, Hamiltonian evolution [105]. In thiscase, we have an oracle Hamiltonian Hw , a diffusion Hamiltonian HD , and a combined Hamiltonian H:
Hw = −|w⟩⟨w|(7.21)

HD = −N |s⟩⟨s|(7.22)

H = Hw + γHD ,(7.23)

where we have introduced a parameter γ in order to tune the relative strength of Hw and HD . The resulting
time evolution is reminiscent of standard Grover’s search, where Hw results in an inversion of the |w⟩

183
component of the input state, and HD an inversion about the uniform superposition state |s⟩. The differenceis that in the case of analog Grover’s search these operations are applied continuously and simultaneously.
To generalize the query complexity to the continuous case, we want some sense in which the applicationof Hw is the resource whose use we want to minimize. A natural choice is to minimize ||Hw ||t [272], wheret is the required evolution time of the algorithm, and || · || refers to the spectral norm of a matrix. We will
typically enforce that ||Hw || = 1, and refer to t as the runtime of the algorithm.
Analog Grover’s search proceeds by preparing the state |s⟩, and evolving the system under H for theappropriate choice of γ. Notice that H does not take us out of the subspace spanned by |s⟩ and |w⟩. For√
γ = 1/N the ground and first excited states of H are |±⟩ = (|w⟩ ± |s⟩)/ 2 up to corrections of O(N −1/2 ),√and separated by an energy gap of ∆E = 2/ N [63] (see Fig. 7.8b). The resulting dynamics correspond√to Rabi oscillations between the state |s⟩ and |w⟩. As a result, one can simply wait for time π N /2, thenprojectively measure the system to identify |w⟩.14√
This algorithm has a runtime of O( N ) in comparisonto O(N ) for a classical continuous time algorithm subject to similar constraints [63], and thus maintains
the quadratic speedup associated with Grover’s search (see Sec. 7.7). It can be shown that this quadratic
runtime is optimal [105].
Notice that HD = −AKN , where AKN is the adjacency matrix of a complete graph KN . This being
the case, analog Grover’s search can be thought of as a CTQW on KN in combination with the applicationof some oracle Hamiltonian Hw .

7.5.2

Spatial search
One might expect that the quadratic speedup exhibited by analog Grover’s search relies on a highly

connected Hilbert space, where the CTQW on a complete graph KN allows one to quickly move from one
item in the search space to another. In a seminal result, Childs and Goldstone proved this to be false [63]:one can remove almost every edge from KN while still maintaining a quadratic speedup [58]. Specifically,
one can consider the more restrictive scenario (in comparison to unstructured search) of spatial search,where each item in the search space is connected to a constant number of other items, enforcing a sense of
14 Critically, these dynamics are completely independent of the specific choice of |w⟩, which is what allows one to use themto search for an unknown value of w.

184
locality [63]. Replacing KN with d > 4 dimensional square lattice graphs (with cyclic boundary conditions)√
results in a runtime of O( N ) and a success probability of O(1) per measurement for the appropriate choiceof γ. This saturates the speedup attained by Grover’s search despite going from a graph with O(N 2 ) edges√
to one with O(N ) edges. In d = 4 a reduced energy gap leads to a runtime of O( N log N ), and oscillations
with imperfect contrast lead to a reduced success probability of O(1/ log N ) per measurement. For 0 < d < 4
the algorithm continues to work qualitatively, but without a significant speedup in comparison to classicalsearch protocols. However, a speedup can be restored in d ≥ 2 by using a discrete-time quantum walk [7],or by incorporating a single additional spin degree of freedom [62].

7.5.3

Boundary conditions
To understand the asymptotic scaling of CTQW-based search algorithms, it is instructive to solve

for the spectrum of the search Hamiltonian as a function of the relative strength of the diffusion and oracle
terms. Similar to Eq. 7.23, just with an arbitrary graph G as opposed to KN :Hw = −|w⟩⟨w|(7.24)

HD = −A(7.25)

H = Hw + γHD ,(7.26)

where A is the adjacency matrix of G.
In Fig. 7.8 we study the spectrum of H as a function of γ for several different graphs. In each case wedefine the resource state, or equivalently the ground state of HD , to be |g⟩. In the typically studied case of
cyclic boundary conditions, the eigenstates of the lattice are plane waves, and so all eigenstates have equal
overlap with the oracle, and are affected by varying γ (Fig. 7.8a). In this setting, the search algorithm canbe interpreted as the formation of a subspace composed of the ground and first excited states of the system,|ψ0 ⟩ and |ψ1 ⟩, which is mostly supported by the states |g⟩ and |w⟩.
As in Sec. 7.5.1, for G = KN , and at the critical value of γ = 1/N (for large N ), |ψ0 ⟩ and |ψ1 ⟩have approximately equal contributions from |g⟩ and |w⟩ (Fig. 7.8b). The overlap of |ψ0 ⟩ and |ψ1 ⟩ with |g⟩and |w⟩ sets the contrast of the oscillations between |g⟩ and |w⟩ during the search procedure, whereas the

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Figure 7.8: Level structure of the spatial search Hamiltonian. (a) Exploring the level diagram of the search
Hamiltonian as a function of the relative strength γ of the diffusion and oracle terms, we identify a subspacespanned by states |ψ0 ⟩ and |ψ1 ⟩. This subspace is primarily supported by the ground state of the diffusion
Hamiltonian on the graph, |g⟩, and the marked state, |w⟩. In the case of a lattice with cyclic boundary
conditions (left), this subspace corresponds to the ground (red) and first excited (blue) states. In the caseof closed boundary conditions (right) and for a Gaussian confining potential (not depicted), the eigenstatesof the lattice have spatial structure, resulting in states (top and middle callouts) that do not overlap witha given oracle (positions marked by the red dots). In this case |ψ0 ⟩ and |ψ1 ⟩ (red and blue) need not be
the ground and first excited states, and are instead identified by their overlap with |g⟩ (dotted line andlower callout) and |w⟩. The behavior of |ψ0 ⟩ and |ψ1 ⟩ vary for different graph geometries, both in terms of
their overlap with |g⟩ and |w⟩, and their energy separation ∆E. Insets in (b-e) denote the relevant graphconnectivity and boundary conditions, but are for visualization purposes only and are not accurate in termsof graph size. These conditions are: (b) A fully connected graph with N = 121 nodes, as applies to the caseof continuous-time Grover’s search. (c) A 2D square lattice with cyclic boundary conditions and N = 11×11.
(d) A 2D square lattice with closed boundary conditions and N = 11 × 11. (e) A 2D square lattice with anadditional Gaussian confining potential with a waist of 5.8 lattice sites and a depth of 1.0J0 , as provided
by the confinement tweezer during the search procedure in Sec. 7.5.4. For (b, c), translational symmetry
dictates that the behavior of the system is independent of oracle position, whereas in (d, e), the behavior isdependent on oracle position, and we plot the special case of a centered oracle.

186
energy gap ∆E between |ψ0 ⟩ and |ψ1 ⟩ sets the frequency of the oscillations. For a 2D square lattice withcyclic boundary conditions (Fig. 7.8c), the ground state |ψ0 ⟩ undergoes a similar crossover from |w⟩ to |g⟩1
at a critical value of γ = 4π
ln N + 0.0488 [62], however, at this critical value the overlap between |ψ1 ⟩ and|w⟩, as well as the value of ∆E, are substantially reduced relative to the case of a fully connected graph.
This results in lower-contrast and lower-frequency oscillations during the search procedure, and thus a lessfavorable runtime of ≥ O(N/ polylog N ) [62].
In our experiment we must additionally consider the acyclic boundary conditions in the lattice. In
the case of closed boundary conditions, or in the case of an additional confining potential, translational
symmetry is broken, and the eigenstates of the lattice have distinct spatial structure. This can result ineigenstates that do not overlap with specific oracles, and thus are not affected by variations in γ (Fig. 7.8a).
In this case the search algorithm can still be interpreted in a similar way as in the case of cyclic boundary
conditions, with the formation of a subspace composed of |ψ0 ⟩ and |ψ1 ⟩ supported by |g⟩ and |w⟩, but |ψ0 ⟩and |ψ1 ⟩ need not be the ground and first excited states of the system. Furthermore, spatial structure in
the ground state can also qualitatively modify the behavior of the search algorithm. For example, in thecase of a finite 2D square lattice on a plane (Fig. 7.8d), |g⟩ has a sine-profile along each axis with reduced
overlap with vertices near the edges of the graph, since the edge vertices have reduced degree. This results indifferent behaviors for different oracle positions. For consistency, in this scenario we define a single criticalvalue, γc , corresponding to the value of γ that numerically minimizes ∆E for a centered oracle (w = o, whereo is the central site in the lattice as in Sec. 7.4). For an N = 11 × 11 square lattice with closed boundary
conditions, γc = 0.412, which is similar to the critical value of γ = 0.430 in the same lattice with cyclic
boundary conditions. Studying the behavior of the closed system at γ = γc as a function of the distanceof the oracle w from the lattice center o (Fig. 7.9a), we find that reduced overlap between |g⟩ and |w⟩ fororacles near the boundary results in reduced (increased) overlap between |ψ0 ⟩ and |g⟩ (|w⟩), and reduced∆E. Interestingly, the contributions from |g⟩ and |w⟩ to |ψ1 ⟩ remain relatively unchanged as a function oforacle distance.
Given the above analysis, the main consequence of modified boundary conditions is how those conditions change the ground state |g⟩. As described in Sec. 7.4, the confinement tweezer results in an ap-

187

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Figure 7.9: Level structure for different oracle positions. When translational symmetry is broken, thespectrum of the search Hamiltonian is dependent on the position of the oracle. Here, we plot the overlap
between the states |ψ0,1 ⟩, and the states |g⟩ and |w⟩ (top), as well as the energy gap ∆E between the states|ψ0 ⟩ and |ψ1 ⟩ (bottom), all as a function of the magnitude of the oracle’s distance from the lattice center, d(in units of the lattice spacing a). We find that the variation of the above quantities with d are primarily
related to the spatial structure in |g⟩, and the resulting position-dependent overlap with |w⟩ (bottom, ×5
scale). The above values are shown for closed boundary conditions, as in Fig. 7.8d, in (a), and for a Gaussianconfining potential, as in Fig. 7.8e, in (b).

188
proximately Gaussian ground state, which is similar to the sine-shaped ground state resulting from closed
boundary conditions. Consequently, the behavior of the system under these conditions (Figs. 7.8e, 7.9b) issimilar to those in the closed-boundary case, leading to a numerically computed critical value of γc = 0.404,
and thus an optimal oracle depth of Vw = J0 /γc = 2.48J0 . Note that in this case as we move away fromthe center of |g⟩, the optimal depth of the applied oracle is reduced, which can be interpreted as the atoms
effectively taking more time to reach the marked site. As a result, at intermediate oracle distances, and at
the fixed value of γ = γc , the application of a deeper-than-optimal oracle leads to atoms being more tightly
localized on the marked site in the state |ψ0 ⟩. This leads to the peak observed in Fig. 7.9b. At even further
distances the oracle has very little overlap with |g⟩, and effectively leaves the system unperturbed such that|ψ0 ⟩ ≃ |g⟩.
While the position-dependent behavior of the system with respect to different oracles with fixedparameters seems antithetical to search, in practice, these changes are small enough that the above search
procedure still works for fixed system sizes. However, more theoretical analysis is required to understand how
different boundary conditions can change the asymptotic scaling of a given search algorithm (see Sec. 7.7).

7.5.4

Experimental demonstration
We perform a proof of principle demonstration of spatial search by CTQW on 2D square lattice.

To accomplish this, we prepare the resource state |g⟩ by preparing the ground state of HD as describedin Sec. 7.4, and then quench to H (Eq. 7.26) by turning on the Hw term for some arbitrary choice of w.
After performing the quench, we observe coherent oscillations in the population on the marked site |w⟩
for a selection of different values of w (Fig. 7.10b). At the peak of these oscillations, the marked site w isreadily identified as the highest amplitude site in the lattice. The amplitudes of these oscillations are in good
agreement with theory, and limited in magnitude not by technical noise, but by the expected performanceof this search procedure in a 2D square lattice (see Sec. 7.5.3).
Hw is controlled by applying a tweezer at the appropriate location w, and with tunable depth Vw .
Although the analysis in Sec. 7.5.3 suggests that Vw = 2.48J0 is optimal, in practice, we operate at aconstant, deeper-than-optimal oracle depth of Vw = 12.55(65)J0 . At these depths, even when the applied

189

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5

10

0
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Distance from origin (a)Figure 7.10: Spatial search by quantum walk. (a) We perform the described spatial search algorithm
by applying the oracle Hamiltonian Hw using a tweezer (green) that selects for a specific (and arbitrary)marked site |w⟩, in addition to quantum walk dynamics governed by HD . The spectrum of the system
includes ground and first excited states that are approximately the even and odd superpositions of |w⟩
and the lattice ground state, or resource state, |g⟩. As a result, quenching on the contribution from Hw
starting from the state |g⟩ results in coherent oscillations between |g⟩ and |w⟩. (b) We observe corresponding
oscillations in the population on the marked site (black points) for a selection of oracles. These oscillations
are in good agreement with theory, up to an overall offset, which uses independently characterized values ofthe state preparation fidelity, tunneling rate, and confinement potential (black lines). For comparison, we
also plot the population in |o⟩ at the center of the lattice (grey points and theory curves). Insets show themeasured populations on different sites at the peak of these oscillations, where the location of the relevant
oracle is marked by a red point, and the location of the origin marked by a cross. (c) A given marked
site can be found by measuring the state of the system after a half-period of the above oscillations. The
population on the marked site after a 2.46 ms quench is plotted for a selection of different oracle positions(each oracle is marked by a red point, with the corresponding population shown in the surrounding circle,
and the origin is marked by a cross), showing a noticeable increase in amplitude relative to the amplitudes
present in the prepared resource state (background image, with each pixel corresponding to a lattice site.
See also Fig. 7.6c). (d) To quantify the range of the search procedure we plot the above amplitudes as afunction of the distance of the oracle from the center of the lattice in units of the lattice spacing a (black
points). The result is in good agreement with a theory prediction with no free parameters (black curve).
At short distances, corresponding to a region spanning ∼ 13 lattice sites, the marked site can be found bylooking for the highest amplitude site after the quench. At longer distances the amplitude at the origin is not
significantly modified by the quench, and remains near its value in the resource state |g⟩ (light grey shading),
which can exceed the amplitude on the marked site. Nevertheless, in a region spanning ∼ 45 lattice sites
there is a several-fold increase in the amplitude on the marked site relative to both the theoretical (dark grey
shading) and measured (grey points) amplitude that was present before the quench. Color scale is sharedacross all parts of this figure.

190
oracle fluctuates low due to the tweezer-lattice alignment concerns described in section 3.3.5, we expect theresultant dynamics to closely follow a sinusoidal oscillation of fixed amplitude and frequency. This meansthat a clear signal can be observed even when averaging over different oracle depths. By contrast, if operatingcloser to the optimal oracle depth, fluctuations to lower depths can lead to more complicated dynamics thatare not strictly sinusoidal, and so averaging over these dynamics can lead to misleading signals. We took
some data at the closer-to-optimal oracle depth of Vw = 3.1J0 , which resulted in higher transfer into themarked site [366]. However, at the time of taking this data, it was challenging to maintain the alignment
between the tweezers and the lattice to properly characterize the coherent dynamics contributing to spatial
search in this regime.15
To calibrate the depth of the oracle, we operate at a fixed evolution time after the quench that is
optimal based on calculations of the desired oracle depth, and perform the search procedure as a functionof the applied oracle depth at a fixed location. At the depth that maximizes population on this marked
site, we characterize the behavior of the quench as a function of both oracle position and time to confirm
that the desired coherent behavior is occurring (see Fig. 7.10b). The resultant calibration of the oracle
depth is in reasonable agreement with independently characterized values of the tweezer shapes and depths,
and time evolution under these conditions matches up well with experimental observations (see Fig. 7.10b).
Interestingly, the amplitude of both the calculated and observed oscillations are higher than what is suggestedby the overlap between |ψ1 ⟩ and |w⟩ alone, indicating that higher-energy eigenstates also play a role in theevolution.
As described in Sec. 7.5.3, in the presence of the confinement tweezer the oscillations between |g⟩ and|w⟩ can be dependent on the specific position of |w⟩. This position-dependent behavior sets the effective
size of the search space, where at greater range reduced overlap between |w⟩ and |g⟩ yields oscillations
with reduced amplitude. In our experiment, for an oracle a distance of√

2a away from the center of theconfinement tweezer (site o), the optimal evolution time after the quench is 2.46 ms. Performing this quench
and evolution for a variety of different oracles (Fig. 7.10c), we find that the amplitude on the marked site
after the evolution decreases as a function of distance from the origin, in agreement with theory (Fig. 7.10d).
15 Note that for the experiments presented in this chapter, we had not yet implemented the feedforward protocol describedin Sec. 3.3.5 for controlling the tweezer-lattice alignment.

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Within 2a, corresponding to a region spanning ∼ 13 lattice sites, we can blind ourselves to the positionof the oracle tweezer, and identify the marked site by looking for the most probable location of the walker
after the quench. At longer range the amplitude that remains at the origin after the quench and subsequentevolution can exceed that of the marked site. However, within√

13a, corresponding to a region spanning∼ 45 lattice sites, there is still a several-fold increase in the amplitude on the marked site relative to what
was present in the resource state. In principle, the effective size of the search space could be increased withconstant overhead by measuring the atomic probability density as a function of evolution time after thequench, removing effects relating to variable oscillation frequencies for different oracles, as well as the large
amount of amplitude that remains near the origin for distant marked sites.
In the specific context of spatial search, the runtime of the demonstrated algorithm does not exhibit
a quadratic speedup in comparison to classical search due to our use of a 2D square lattice [62]. However,√as mentioned earlier, a runtime of O( N log N ) is achievable with a single particle in such a lattice if anadditional spin-1/2 degree of freedom is used to implement a Dirac Hamiltonian [63], or a discrete-timequantum walk [7].16

7.5.4.1

Lattice ground state preparation by reverse search

One might be concerned that adiabatic preparation of the resource state as described in Sec. 7.4
overwhelms any potential speedups associated with spatial search by CTQW. However, since the timescaleof both the search dynamics and the required adiabatic ramp are set by ∆E, the runtime of both thesearch algorithm and the adiabatic procedure for state preparation have similar asymptotic scaling. An
alternative route to preparing |g⟩ is to run the search procedure backwards. By starting with the atom ata known location w, and quenching with an oracle at that location, the atom should oscillate from |w⟩ to|g⟩ (Fig. 7.11). This “reverse search” procedure for state preparation has exactly the same runtime as thesearch algorithm.
We perform a proof of principle demonstration of reverse search at an oracle depth of 6.2J0 , leading
16 One could consider implementing this using a modified optical lattice containing an array of doublets [61], or the opticalclock qubit in strontium, especially since it is possible to engineer state-dependent optical potentials for this qubit [138] torealize a broad class of discrete-time quantum walks [158, 175].

192

p
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Theory0.01 0.1 1

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Quench duration (ms)Figure 7.11: Reversed spatial search. By implanting an atom in the central lattice site |o⟩, and quenchingwith an oracle at that site with a depth of 6.2J0 , we observe coherent oscillations in the atomic probability
density (black points) on this site due to the same dynamics as in the search algorithm. Callout: characterizing the atomic probability density as a function of position at the first minimum in this oscillation (data,left), we observe the expected behavior where much of the atom population remains on the central site, butsome has oscillated into a superposition that is qualitatively similar to the desired resource state (theory
with no free parameters, right). For comparison, similar quenches are performed with an oracle depth of
4.1J0 (dark grey points), and without the oracle (light grey points). In the absence of an oracle, the atom
simply leaves the central site, and does not return on the timescales explored in this work. Theory curves
with no free parameters match the deeper quench well, and are in reasonable agreement with the shallower
quench, whose behavior is more sensitive to fluctuations in the depth of the applied oracle.

(a)

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193

~

Variable phase shifter

Variable beam splitter

Figure 7.12: Universal control of the quantum walk unitary. An arbitrary unitary transformation between
input and output modes of a system (a) can be efficiently decomposed into the appropriate application ofvariable beam splitters and phase shifters (b) [270]. An arrangement that is more efficient in terms of circuit
depth (and still universal) is shown in (c) [71].

to slower and slightly larger amplitude oscillations than those in Sec. 7.5.4. When reducing the oracle depth
even further, to 4.1J0 , we observe less clearly sinusoidal behavior, and the resulting dynamics are moresensitive to fluctuations in the applied oracle depth. The behavior of the reverse searches are in reasonableagreement with theory with no free parameters.17

However, due to the limitations of spatial search in a

2D square lattice, the state preparation fidelity achieved by reverse search is significantly worse than the
adiabatic protocol.18 Nevertheless, reverse search serves as an additional test of the dynamics contributing
to our demonstration of spatial search, and, by extension, that the adiabatic protocol is indeed preparing
the appropriate resource state.

7.6

Universal control of the quantum walk unitary
In the previous sections, we have shown that through the appropriate application of optical tweezer

potentials to a tunnel-coupled 2D optical lattice, we can program quantum walk dynamics to perform usefultasks. In principle, the demonstrated programmability allows for universal control, in the sense that any
unitary transformation between some set of input and output sites in the lattice can be achieved. Notethat in this case we are referring to a single particle unitary, namely a unitary transformation of the single
particle state space (and not, for example, on the state space of many particles occupying the lattice, which
we will discuss in Ch. 8).
17 Note that the theory predictions for reverse search only use information about the independently characterized opticalpotentials, since the adiabatic state preparation step is omitted.
18 However, in a more connected lattice, this protocol could be a viable way of preparing the resource state with identicalscaling to the search procedure, removing any additional overhead associated with adiabatic state preparation.

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Such control can be realized in a few ways. It has been shown that CTQWs on sparse and unweighted
graphs are universal for quantum computation. Specifically, one can construct a graph that prepares a plane
wave input from a walker that is initially localized on a single site, and connect that to a graph that realizes
an arbitrary scattering process for an incident plane wave [59]. Using only static tweezer potentials, one could
implement such a graph.19 However, this approach suffers from the fact that performing an arbitrary single
particle unitary transformation in general requires a polynomial increase in the total number of vertices in
the graph in comparison to the number of input and output vertices.
Alternatively, by applying time-varying tweezer potentials, one can apply an arbitrary single particle
unitary transformation between all sites in the lattice [230]. Specifically, if one isolates a single site iand applies a tweezer with depth Vi for time t, an atom occupying that site experiences a phase shift
ϕi = −Vi t/ℏ relative to the case where no tweezer is applied. Isolating two adjacent sites i and j andallowing an atom to tunnel between these sites for a variable evolution time implements a tunable beam
splitter. For example, evolution for a quarter of a tunneling time, t = τ /4, results in a balanced beamsplitter√U = (|i⟩⟨i| − i|i⟩⟨j| − i|j⟩⟨i| + |j⟩⟨j|)/ 2. As is well known in the context of linear optics, a network of(local) beam splitters and phase shifters is sufficient to implement any unitary transformation between the
inputs and outputs of the network [270, 54, 71] (see Fig. 7.12). Therefore, the appropriate application oftime varying tweezer potentials to an optical lattice can similarly implement any single particle unitarytransformation between the sites in the lattice.20

7.7

A note on resource costs and analog classical simulation
Through the course of working on the results presented in this chapter, I became concerned about

what it means to actually have an asymptotic “quantum speedup”. For example, a CTQW of a single particle
on any graph with N vertices is fully captured by a classical wave equation involving N coupled oscillators,√
including the dynamics resulting in the O( N ) runtime for searching a sufficiently connected graph [133].21
19 In the construction in [59], all required graph elements can be implemented by removing vertices and edges from a 2D
square lattice with diagonal tunneling by shining tweezers on and in between vertices.
20 One does not have to restrict oneself to the described construction of sequential beam splitters and phase shifters. Often,
performing these operations in parallel can result in more efficient realizations of arbitrary unitaries [230].

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In fact, dynamics in any discrete Hilbert space can be captured in such a manner. That being the case, isthis whole endeavor bogus?
The answer, of course, is no. However, when discussing algorithmic speedups, one must be very careful
to define precisely what the relevant resource costs are. In the standard oracular model of spatial search [1],an algorithm must alternate between queries that determine whether the current location is marked, andoperations that can only perform local interactions on the graph. Schematically, one can think of this as thecase where a single local “robot” is allowed to traverse a memory that is spread out in physical space, andreport the result of the computation back to the experimenter.22 In this model, any classical algorithm must
make O(N ) queries to find the marked item, even if the graph is complete. The classical wave dynamics
that reproduce the behavior of a single-particle quantum walk can be classically simulated in this model
only with significant overhead. As a result, in this setting there can be an asymptotic quantum speedup
for search by quantum walk even when working with a single particle (or robot) traversing a graph whose
vertices correspond to physical locations in space.
A distinct formulation of the search problem is to find one of N = 2n items using a single registerof n bits. In the classical case, one can check only one item at a time, leading to a runtime of O(N ). The√
O( N ) runtime of Grover’s search on N items using a register of n qubits leads to the oft-quoted quadratic
speedup. However, unlike in the case of the quantum robot, in this setting the speedup is dependent on
the exponential scaling of Hilbert space with additional qubits. In our demonstration each state in Hilbert
space corresponds to a physical location in a real lattice, and so one could argue that the physical resource
cost scales like O(N ) instead of O(n). Additionally, we read out the position of the walker by performinga parallel measurement on all N sites in the lattice. In the classical case, this would correspond to having
access to a register with O(N ) bits that can be read out in parallel. In this setting, one can simply assign a
bit to each item in the search space, flip the bit corresponding to the marked item, and read out the wholeregister to find the marked item in a runtime of O(1). This illustrates that, in some settings, the speedup
21 If you are interested in seeing a second machine that I built for performing spatial search which uses wood and string

rather than ultra-high vacuum and lasers, visit doi:10.5281/zenodo.10019752.
22 While this may seem like an absurd restriction at first glance, there are fundamental (albeit somewhat extreme) situations
where such a model could apply. For example, if information is distributed in a space that imposes a fundamental speed limit,
like the speed of light, and one wants to send out a robot (or collection of robots) to search that space and return with some
information [1].

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associated with quantum search algorithms requires both reduced runtime, and a state space that scales
rapidly with physical resources. We consider such a state space in the next chapter.

Chapter 8

Multi-particle quantum walks“That’s the best thing about being me... there are so many ‘me’s.”— Agent Smith
Portions of this chapter have appeared in:
A. W. Young, S. Geller, W. J. Eckner, N. Schine, S. Glancy, E. Knill, and A. M. Kaufman. An
atomic boson sampler. In review, 2023

In this chapter, we will extend the single atom dynamics described in Ch. 7 to the multiparticle
case. In Sec. 8.1, we will review the mechanisms leading to interactions between ground state atoms in a
lattice, and show how for certain experiments these interactions are negligible. In Sec. 8.2 we will focus on
non-interacting multi-particle CTQWs. One might naively expect that in the absence of interactions, themany-particle dynamics arise trivially from that of a single particle. However, we will find that particle
statistics play a crucial role. In specific cases, including for what is known as boson sampling [2], noninteracting dynamics can still result in behavior that is not believed to be possible to simulate classically inan efficient manner.
Interestingly, non-interacting bosonic dynamics lie in a convenient middle ground where, on the one
hand, their behavior is in general hard to simulate. On the other hand, the lack of interactions introduce
additional structure to the dynamics that enable powerful techniques for benchmarking. In Sec. 8.3, we will
use the properties of non-interacting dynamics to develop and apply various techniques to benchmark thequality of our state preparation. In Sec. 8.4, we will briefly discuss related measurements that also allow us tocharacterize the applied unitary evolution directly. In Sec. 8.5, we will extend the above measurements andbenchmarking techniques to large systems involving many atoms, including the largest-scale demonstrations

198of boson sampling to date, which are beyond our ability to simulate using current classical algorithms. We
will comment on the classical simulations we perform in Sec. 8.6. In Sec. 8.7, we will conclude our discussionof non-interacting quantum walks by describing how noise affects the sample complexity of boson sampling,
and comparing our results to previous results using photons.
The capabilities demonstrated in this chapter, namely the ability to rapidly prepare nearly arbitrary
patterns of identical atoms in a tunnel coupled optical lattice using optical tweezers, provide a new approachto studying Hubbard models using ultracold atoms. We will conclude the chapter in Sec. 8.8 by presenting
preliminary results where interactions between atoms are reintroduced in our experiment to study stronglyinteracting quantum walks, as well as the preparation of a superfluid ground state.

8.1

The Bose Hubbard model
In order to extend the discussion in Sec. 7.1 to multiple non-interacting bosonic atoms, we definebosonic creation operators a†i which create an atom with the localized wavefunction w0,i (as defined in
Eq. 7.6). Similarly, we can define bosonic annihilation operators ai and number operators ni = a†i ai . ai anda†i encode the symmetrization required by bosonic particle statistics (namely that the wavefunction remains
the same upon exchanging particles), and have the standard bosonic operator algebra:[ai , aj ] = [a†i , a†j ] = 0,a†i |ni ⟩ =√ni + 1|ni + 1⟩,

ai |ni ⟩ =[ai , a†j ] = δij√

ni |ni − 1⟩,

ai |0i ⟩ = 0(8.1)

ni |ni ⟩ = a†i ai |ni ⟩ = ni |ni ⟩,where |ni ⟩ is a Fock state containing ni atoms on site i (namely atoms with the wavefunction w0,i ). Given
the above definitions, we can write the many-boson equivalent of Eq. 7.8 as:
H=−

X
{i,j}


 X
Jij a†i aj + a†j ai −V i ni ,(8.2)

iwhere, as before, the first sum runs over all pairs of sites {i, j} with appreciable tunnel coupling. Assuming
the Vi ’s are uniform, the evolution of a Fock state of atoms occupying different sites under this Hamiltonian
corresponds to a multi-particle CTQW.

199
8.1.1

Interactions
It is not always the case that lattice-trapped atoms are non-interacting. When two atoms come close

together with relative displacement r, they experience an interaction potential U (r) (due to van der Waals
forces, Coulomb forces, etc.). The details of this potential can, in general, be quite complicated. However,for most atomic gasses U (r) can be assumed to be spherically symmetric, and U (r) → 0 for |r| ≫ r0 , wherer0 is some characteristic length scale of the potential (typically on the scale of angstroms) [65]. Given theabove assumptions, we can consider the scattering problem where an atom initially in a plane wave statewith momentum k oriented in the z direction scatters off of another atom located at the origin. The steadystate atomic wavefunction in the far field (|r| ≫ r0 ) is [281]:
ψ(r) = eikz + fk (θ)

eikr
,
r

(8.3)

where θ is the angle r makes with respect to the z-axis, and z is the projection of r onto that axis. fk is ascattering amplitude that, due to the assumed spherical symmetry of U (r), depends only on the angle θ.
One can expand fk in the basis of spherical harmonics to obtain [281]:∞

fk (θ) =1 X(2l + 1)Pl (cos θ)(e2iδl − 1),
2ik

(8.4)

l=0

where Pl are the Legendre polynomials, and δl is the phase shift acquired by the lth partial wave due toscattering. For k ≪ 1/r0 , and when there isn’t a nearby molecular resonance, δl ∼ k 2l+1 [281]. As a result,for bosonic atoms with very small k (as applies to our experiments), only the l = 0 (also known as s-wave)
component contributes to scattering. In this low energy case, one can show that [281]:
lim k cot δ0 = −

k→0

1
,
as

(8.5)

where as is a single constant quantity known as the scattering length which determines the entirety of thelow energy scattering behavior. It is challenging to perform an ab initio calculation of as for atoms that are
significantly more complicated than hydrogen. However, as can be measured directly from experiments, theresults of which are summarized in Tab. 8.1 for the naturally abundant isotopes of Sr.
Because the low energy scattering is found to be independent of the exact functional form of U (r),we can choose a pseudopotential that is easier to work with, but nevertheless results in the same behavior.

200
For example, we can choose the delta, or contact, potential:
4πℏ2
δ(r),
m

U (r) = as

(8.6)

where m is the atomic mass. Using this pseudopotential, we can calculate the energy shift due to interactions
between two atoms occupying the same site in the lattice [293]:4πℏ2 asUc =m

Z

d3 r|w0,i (r)|4 ,(8.7)

where w0,i is as defined in Eq. 7.6.
As long as Uc is small compared to the band gap, and the atom density is relatively low (meaning
there aren’t ever significantly more than two atoms on a given lattice site), we can assume that s-wave
interactions between the atoms do not dramatically modify the functional form of w0,i . In this case we canwrite the full, interacting Bose-Hubbard Hamiltonian as:
H=−

X
i,j

Jij â†i âj −Xi

Vi n̂i +Uc X
n̂i (n̂i − 1),
2 i

(8.8)

where the last term accounts for all pairwise interactions that can occur on a given site.
Often, one is interested in studying different regimes of Eq. 8.8 by controlling the value of Uc /J. Thiscan be achieved by varying the lattice depth, which alters J much more rapidly than Uc . However, lower
J leads to increased sensitivity to noise (e.g. atom loss, or disorder in the lattice potential), resulting ina practical limit to how much one can reduce J. In experiments with alkali atoms, Uc /J can additionallybe tuned by adjusting the value of as using a magnetic Feshbach resonance [65]. Unfortunately, the J = 0
ground state of Sr (where J here refers to the total angular momentum), makes it insensitive to magnetic fieldtuning.1 Luckily, the values of as for the naturally abundant bosonic isotopes of Sr span a convenient rangeof values (see Tab. 8.1). Experiments with 88 Sr, where Uc ≪ J, are well captured by the non-interacting
Hamiltonian in Eq. 8.2.2

Experiments with 86 Sr, where Uc ≫ J, are well captured by Eq. 8.83 andapproximate hard core (infinite strength) interactions. Experiments with 84 Sr allow us to access the regime
where Uc ∼ J.
1 It is possible to achieve similar tuning in alkaline earth atoms using optical Feshbach resonances, where one uses laser lightto couple the scattering atoms to a specific molecular state [100, 358]. However, to date, such techniques have also resulted insignificant amounts of accompanying inelastic scattering [235].
2 We will comment on the accuracy of this approximation in Sec. 8.7
3 Although these large values of U begin to approach the limit where wc0,i is modified due to the strength of Uc [51].

88

Sr

87

Sr

86

Sr

84

Sr

Sr
Sr
86
Sr
84
Sr

-1.4(6)-55.0(2)92.6(1)-97.4(1)162.5(5)823(24)-1790(130)-56(1)
31.9(3)122.7(3)Uc /h (Hz)-1.7(7)109.4(1)970(30)144.9(4)

88
87

201

Table 8.1: Scattering properties of the naturally abundant isotopes of strontium. Upper rows list the s-wave
scattering length between isotopes, provided in units of the Bohr radius a0 . These values are obtained
from [216]. Lower row lists the values of Uc /h for interactions between atoms of the same isotope in units ofhertz and for the typical lattice parameters used in this chapter, where J/h = 119 Hz.

8.1.1.1

Inelastic interactions

Due to conservation of energy and momentum, the dominant inelastic interactions in our experiment
involve three body processes, where two atoms form a molecular bound state and a third atom carries away
the binding energy. Such a process typically results in the loss of all three atoms, since the binding energy ismuch larger than the depth of the lattice potential. A loose upper bound for the three body loss coefficientin 88 Sr is βggg < 10−27 cm6 s−1 [314], with an expected value closer to (and likely lower than)4 the measured
value of βggg = 2.0(2) × 10−30 cm6 s−1 in 87 Sr [122]. The resulting rate equation for the total atom number
N is:dN= −βggg
dt
where ρ(r) =Z

d3 rρ(r)3 ,

(8.9)

2
i ⟨ni ⟩|w0,i (r)| is the atom density. At early times, we can define an effective single particle

P

loss rate associated with three body processes:
−1
τggg
=

1 dN
.
N dt t=0(8.10)The highest densities we operate at using 88 Sr are ⟨ni ⟩ = 1/4, which for typical lattice parameters and theabove loss coefficients yield τggg ≳ 6 × 102 s and τggg ≳ 3 × 105 s respectively. Note that the effect of bosonicinterference leads to a worst case correction of 1/3! to the above lifetimes [157, 50]. As a result, we conclude
that the effect of inelastic collisions are negligible for all experiments using 88 Sr in this work.
4 Typically, three body collision rates follow the scaling β

2 4
ggg ∝ n as [106], which would indicate a vanishingly small three88
body loss rate for Sr due to the small value of as in comparison to the other isotopes of Sr. However, this approximation is

only accurate for |as | larger than the van der Waals radius of the atom.

202
We have performed some preliminary experiments involving quantum walks of atoms in both theground and clock electronic states. In this case, new channels involving electronic-state changing collisions
open up, leading to strong two-body loss.5 Specifically, the two body loss coefficient for a collision between
one ground (g) and one clock (e) state 88 Sr atom is βge = 5.3(1.9) × 10−13 cm3 s−1 , whereas the coefficient
for a collision between two clock state 88 Sr atoms is βee = 4.0(2.5) × 10−12 cm3 s−1 [196]. The resulting rateequation for the total atom number N is:dN= −βgedt

Z

d3 rρg (r)ρe (r) − βeeZd3 rρe (r)2 ,(8.11)

where ρx denotes the density of atoms in the electronic state x. For typical lattice parameters, and for two
clock state atoms occupying the same site, this results in an effective single atom lifetime of τee ∼ 10 ms.
Similarly, for one clock and one ground state atom occupying the same site, this results in an effective single
atom lifetime of τge ∼ 200 ms.

8.2

Non-interacting quantum walks
As we saw in the previous section, ground state 88 Sr atoms behave like non-interacting bosons in our

experiments, and thus the evolution of these atoms are well-captured by Eq. 8.2. Consider the simplified
scenario of a CTQW on a cycle graph, as in Sec. 7.2.6

As before, the single atom eigenstates are planewaves,7 and so we can define creation (and annihilation) operators in the eigenbasis of plane waves [230]:
1 X ij2πq/m †eaj ,b†q = √
m j

(8.12)

where q is the wave number, and m the number of lattice sites (vertices in the graph). The many body time
evolution operator is then:
Ub (t) =X(8.13)

eit2J cos(2πq/m) nq ,qwhere J is the tunneling energy, and nq := b†q bq is the number operator in the plane wave basis. Althoughtime evolution in this situation is trivial, predicting the outcome of measurements in the position basis is not.
5 In these cases, the emission of a photon can preserve conservation of energy and momentum.
6 This corresponds to Eq. 8.2 for a 1D lattice with cyclic boundary conditions, V
7 Eq. 7.13, with energies given by Eq. 7.14.

i = 0, and Jij = J.

203
This is because when making predictions regarding such measurements, one must keep track of the interplay
between the phases accumulated during time evolution, and the symmetrization constraints encoded in thebosonic operators. In fact, even in this simple scenario of non-interacting bosonic CTQWs on a cycle graph,
it is conjectured that it is not possible to efficiently sample from the output distribution in the position basis
classically [230].
More generally, we are interested in the probability of observing a specific pattern of atoms |k′ ⟩after arbitrary (non-interacting) time evolution starting from a given initial pattern |j ′ ⟩. The state |j ′ ⟩(and similarly |k′ ⟩) is conveniently described in the site occupation basis. Given a site occupation list′), where ji′ counts the number of atoms occupying a given site i ∈ {1, . . . , m} (and m is thej ′ := (j1′ , . . . , jm

total number of sites in the lattice), we define that: ji′
ma†iYp ′ |0⟩.
|j ′ ⟩ :=ji !
i=1(8.14)The effect of the non-interacting Hamiltonian dynamics is to apply a transformation to the bosonic operators [325]:a†j → a†j (t) :=X

U (t)ji a†i ,(8.15)iwhere U (t) = e−iHt/ℏ is the single atom time evolution operator (t is the evolution time and, for our
experiments, H is as defined in Eq. 7.8). Therefore, under time evolution, the many atom state undergoes
the transformation:|j ′ ⟩ → |j ′ , t⟩ :=

m
Y
i=1



ji′
a†i (t)p ′ |0⟩.
ji !

(8.16)

After time evolution, the probability of observing a specific output state |k′ ⟩ is:
P B (k′ |j ′ , U ) := |⟨k′ |j ′ , t⟩|2
′

2

′
m
m
j
Y
(ap )kp Y (a†q (t)) qp ′p ′ | 0⟩ .
= ⟨0 |jq !
kp ! q=1
p=1(8.17)

To simplify Eq. 8.17, we introduce the site-assignment basis |j⟩, which is equivalent to the site occupation basis (|j⟩ ≡ |j ′ ⟩). Here, n atoms occupy the sites j = (j1 , . . . , jn ), with j1 ≤ · · · ≤ jn , where

204

Figure 8.1: Symmetrization constraints on bosonic interference. In this cartoon, we assume that the bosonicparticles (blue disks) occupy distinct positions in an optical lattice (black contours), and that the evolution
is non-interacting, and thus described by a single particle unitary transformation U . We are interested incomputing the probability that one observes an arrangement of particles |k⟩ after preparing the arrangement|j⟩ and applying U . Due to the indistinguishability of the particles, in order to perform this calculation one
must coherently sum over all trajectories (blue lines) that can take one from |j⟩ to |k⟩, leading to complicatedinterference effects.

jl ∈ {1, . . . , m}. Plugging Eq. 8.15 into Eq. 8.17, and performing some combinatorial manipulations, one
can arrive at the expression [289, 2, 325]:
P B (k|j, U ) ≡ P B (k′ |j ′ , U ) =1|Perm(Ukj )|2 ,k!j!

(8.18)

where Perm(·) is the matrix permanent, and Ukj is the n × n submatrix of U that contains only the rows
corresponding to sites k, and columns corresponding to j. The normalization constant is k! :=Qm′
i=1 ki ! (andsimilarly for j!), where we have used the representation of the state in the site occupation basis.8
A first quantized interpretation of Eq. 8.18 is often adopted in the literature on boson sampling [2],where the interference between multiple particles can be understood as a summation over all permutationsof atom labels on the input going to atom labels on the output (see Fig. 8.1). The equivalent calculation for
distinguishable atoms yields:
P D (k|j, U ) =1
Perm(|Ukj |2 ),k!j!

(8.19)

which can be interpreted as simply combining the probability distributions resulting from the single particle
CTQWs of each atom.

8.2.1

Boson sampling
Computing the permanent of an arbitrary matrix is in the complexity class #P-hard [2, 339], and soeven sampling from the probability distribution resulting from the evolution described in Sec. 8.2 is believed
8 Note that these normalization terms are equal to 1 unless a collision occurs (in the sense of multiple atoms occupying thesame site). In our experiments, where the lattice is initialized with at most one atom per site, j! = 1.

205

1.00

Survival

0.95
0.90
0.85
2 particle NN
1 particle

0.80

0.9

1.0

1.1
1.2
Tunnel time (ms)6.45

Figure 8.2: Atom loss during quantum walk dynamics. We measure the atom loss in experiments with one
atom, and with two atoms initially positioned in nearest-neighbor (NN) sites in the lattice, as a function ofevolution time. The pictured evolution times are near the Hong-Ou-Mandel dip (see Sec. 8.3.1), and at thelatest evolution time explored in this work of t = 6.45 ms (horizontal line corresponds to the mean single
particle loss). Data points marked by a star are taken at t = tHOM := 0.96 ms. Notice that single particle
loss is not observed to vary over the evolution times explored in this work. The excess loss in the two particle
data is associated with collisions between atoms during readout. As the atoms spread out and the atom
density goes down, the gap between the measured one and two particle loss closes.

to be intractable for more than ∼ 50 atoms using classical techniques [72]. By contrast, approximating thepermanent of a non-negative real matrix can be performed in polynomial time, and so the corresponding
sampling task for distinguishable atoms can efficiently be accomplished classically [326]. Different degrees ofatom distinguishability result in behaviors that lie in between these two scenarios [326, 94].
Because most of the discussion surrounding boson sampling is framed in terms of implementations
using photonics experiments, it is useful to point out a few differences in terminology: for us, the modes thatthe bosonic particles (atoms) can occupy are sites in the lattice, and the single particle unitary transformation
U is the result of Hamiltonian evolution on the lattice and the associated CTQW dynamics. In photonics,
the modes that the bosonic particles (photons) can occupy are optical modes, and U is the transformation
applied between the input and output modes of a linear optical network. We will use the above terminology
interchangeably as appropriate.

206
8.2.2

A note on measurements, loss, and postselection
Implementing boson sampling in our apparatus amounts to preparing specific patterns of identicalatoms using the prescription in Ch. 4, performing a CTQW by evolving under Eq. 8.2, and measuring theatoms in the position basis. However, due to the parity projection that occurs during imaging (see Sec. 4.2),
our measurements are not number resolving, and instead detect atom number parity on each site. For
most experiments in this chapter, we will operate in a regime where the probability that more than one
atom occupies a given site is low, minimizing the effect of parity projection.9

Additionally, except forexperiments that involve large atom numbers (namely those in Sec. 8.5), we postselect our measurements on
perfect rearrangement, and on observing no lost or extra atoms after the quantum walk dynamics.
Postselection is further useful for suppressing the effect of imperfect state preparation. Because atoms
occupying higher bands in the lattice experience significantly higher tunneling energies (Sec. 7.1.1), for our
experimental parameters these atoms are either lost, or exit the analysis region and are interpreted as lost
during the quantum walk dynamics. Therefore, postselection on atom number ensures that all remaining
atoms were in the ground band in the plane of the 2D lattice. We observe a loss of 5.0(2) % (Fig. 8.2),
which is mostly explained by the expected combined ground state occupation (see Sec. 4.3.1) of 97.37 %in the radial directions (leading to 2.63% loss). The remaining loss is likely due to imperfect adiabaticity
of ramps in the lattice depth, as well as imperfect overlap between the Wannier functions of the atoms in
the lattice before and after the lattice is quenched to the conditions used for tunneling. Note that we donot see increased single particle loss as a function of evolution time, even at the longest evolution times
explored in this chapter, suggesting that the loss is indeed due to filtering of imperfectly cooled atoms. It is
important to note that avoiding the effects of loss that scales with evolution time is critical for maintaining
the computational complexity of the boson sampling problem when scaling to larger systems, since one must
apply a single particle unitary that adequately mixes the different modes in the system [114, 243]. Ultimately,
effects like parametric heating and vacuum lifetime will result in single particle loss that is dependent on
the evolution time. However, we expect these effects to become relevant only on the timescale of seconds
9 In fact, although our measurements are not in a complete basis, similar restrictions are also made in the original proposal
for boson sampling out of consideration for the use of bucket or click detectors in photonics [2], which distinguish only between
no photons, and the presence of one or more photons.

Labeled

0.2

9
y

6
0.0

3
0

(c)

1D atomdensity

9

0.0
0

3

6

9 0

3

6

Labeled

207

Data

0.8
0.4

Unlabeled

9x

Theory

6

0.2

3

p

Unlabeledx2(b)Atom density

(a)

0

0.0
0

3

6

9

x1

Figure 8.3: Two particle quantum walks on a 2D lattice. (a) The interference between atoms can bestudied by measuring correlations in the occupation of specific sites in the lattice (red). Partitioning thelattice into arbitrary subsets of sites κ (grey) can allow one to more efficiently (in terms of the number ofexperimental trials) study the effects of interference. Due to the separability of quantum walk dynamics ona square lattice, it is particularly useful to partition the lattice into columns indexed by their x coordinates(blue). (b) Specifically, by binning 2D quantum walk data over the y axis, one can study 1D quantum
walks along x, where the initial coordinate in the y axis acts as an additional “hidden” degree of freedomthat can modify the distinguishability of the atoms in the remaining “visible” x axis. For example, one canprepare two atoms with the same (unlabeled) or different (labeled) y coordinates (sites initialized with anatom are marked in red). The atom number density on each site after the evolution of these atoms is shown
here at an evolution time of t = tHOM := 0.96 ms. (c) After binning to 1D, each two particle output is
uniquely labeled by the x coordinates of the two atoms (x1 , x2 ), with x2 ≤ x1 . The measured probabilities
p of observing a given two particle output are in agreement with theory, where the unlabeled atoms (topleft) agree with the expectation for perfectly indistinguishable bosons (bottom left), and labeled atoms (topright) with distinguishable particles (bottom right). The states highlighted in red once again denote theprepared initial states before the atom evolution.

(Sec. 5.3.1) in comparison to the millisecond-scale experiments explored in this chapter.
Finally, the above postselection also results in a quadratic suppression of imaging errors. Given
the already low imaging error rates (Sec. 4.4.2), measurements involving such postselection are therefore
negligibly impacted by imaging errors. For experiments in which we do not postselect on the number ofatoms remaining after the quantum walk dynamics (namely the experiments appearing in Sec. 8.5) imaging
errors can play a role. Specifically, for the atom and mode numbers relevant to Sec. 8.5, the combined effectsof false positives and negatives lead to overall error rates of ∼ 1 % in our ability to correctly identify thepresence and location of a given atom. In experiments that are sensitive to imaging errors, we calibrate theeffect of the errors and include them in our simulations (see Sec. 8.6).

208
8.2.3

Effective particle statistics
Although our atoms are fundamentally bosonic composite particles, they may not behave bosonicallyon the lattice. This is because the single-atom state space includes degrees of freedom (DOFs) other thanjust the location of the atom in the lattice. For example, the atom can be in different electronic states, orin different motional states in the direction that is normal to the lattice.10

The single atom state space isthus a tensor product of the Hilbert spaces HV (“visible” DOFs) spanned by |i⟩ (sites), and HH (“hidden”
DOFs) spanned by |h⟩ (labels). In our system, the overall unitary dynamics is non-interacting for the atoms,
and the single atom Hamiltonian acts independently on the hidden and the visible DOFs, which meansthat it is of the form HV ⊗ 1H + 1V ⊗ HH (where the subscripts indicate the subspace that each operator
acts on, and H is a Hamiltonian acting on the indicated subspace). Due to this independence, the hidden
DOFs can affect the visible behavior of the atoms only by changing their effective particle statistics. For
example, if we prepare some number of atoms in specific visible sites, but each atom has a different hidden
label, then the visible behavior of the atoms is that of perfectly distinguishable particles. Other types ofvisible particle statistics are also possible, including fermionic statistics if the multi-particle wavefunction
is antisymmetric in the hidden DOFs [284, 219]. The tests in this chapter primarily focus on determining
limits on the deviation from bosonic behavior due to the hidden DOFs without characterizing the specificparticle statistics exhibited in the visible behavior. Note that errors in the visible DOFs are unlikely due toour procedure for state preparation11 and postselection (see Sec. 8.2.2).12
There is flexibility in how to partition between visible and hidden DOFs (Fig. 8.3a), which we takeadvantage of in tests of indistinguishability involving up to 8 atoms. In a square lattice, H takes the form
Hx ⊗ 1y + 1x ⊗ Hy where x and y denote the spatial coordinates of atoms (see Sec. 7.2.1), and so we canconsider one of the spatial coordinates as hidden. For example, if we ignore the y coordinate of the atoms
the resulting visible behavior is that of a 1D multiparticle quantum walk along x (Fig. 8.3bc). We refer tothis procedure as “binning,” since it is accomplished by summing the observed numbers of atoms along the y
10 Recall that in-plane motional excitations lead to loss (Sec. 8.2.2).
11 Namely, optical cooling and atom rearrangement as described in Ch. 4.
12 Because both cooling and imaging involve the spontaneous emission of photons from atoms occupying different latticesites, our state preparation is also expected to be perfectly dephasing. This rules out error models for boson sampling like themean field sampler that rely on coherences between different input modes [327].

209
axis in the final image. Binning is convenient for two reasons. First, it allows us to operate in a regime where
it is rare that there is more than one atom on a given lattice site at the end of the dynamics. As a result,parity projection with binning is effectively equivalent to number-resolved measurements of the visible sites.
Second, because we can control the initial y coordinates of the atoms via rearrangement, we can control oneof the hidden DOFs and thus the distinguishability of the atoms on the visible DOF (Fig. 8.3bc). We willrefer to this approach to preparing distinguishable atoms as position labeling.13
An alternative way of simulating distinguishable particle statistics is to perform a pair of experiments
where only one atom is prepared at a time, and then combine the data in subsequent analysis (we will referto this as time labeling). One can view time labeling as being akin to introducing a time delay between
otherwise identical atoms, where the time when a particle is measured serves as a hidden DOF.14 For data
sets in which we use time labeling to simulate n partially or fully distinguishable particles by combining
multiple runs with fewer than n particles, we compensate for parity projection during imaging by summing
the resulting processed images and taking the result mod 2.

8.3

Benchmarking atom indistinguishability
As stated in Sec. 8.2.1, the degree to which it is computationally hard to sample from non-interacting

quantum walk dynamics depends on particle statistics, with distinguishable (and fermionic) particles being
easy to simulate classically. We are therefore interested in characterizing the indistinguishability of the atoms
in our experiment, which tells us the degree to which they behave like identical bosons.

8.3.1

Multimode Hong-Ou-Mandel interference
The indistinguishability J of two atoms is defined to be the Hilbert-Schmidt inner product of thedensity matrices of the two one-atom wavefunctions on the hidden DOFs. It can be inferred from a Hong-OuMandel (HOM) experiment [143] by comparing the coincidence probability of the atoms to the corresponding
coincidence probability for perfectly distinguishable atoms. We measure HOM interference of atoms [165,
13 Referring to the fact that the initial positions of the atoms in y act as labels that cause the atoms to be distinguishable.
14 This is similar to standard techniques in photonics, where a time delay is used to tune the distinguishability of two or
more photons.

210
201, 276] by studying quantum walks in 1D via the binning procedure described in Sec. 8.2.3 (Figs. 8.3 and
8.4). Schematically, one can associate the evolution time with a variation in the reflectivity of the beam
splitter in a standard HOM measurement (Fig. 8.4ab).
However, whereas in a typical HOM experiment there are only two modes, in our experiment there aremany modes, and so the probability of coincidence of two atoms in any specific two sites is small. Therefore,
it is useful to bundle many sites together. Given partially distinguishable atoms that start on sites k, l withk ̸= l, the probability that they arrive at sites i, j is:15
|Uik |2 |Ujl |2 + |Uil |2 |Ujk |2 + 2J Re Uik Ujl U ∗ U ∗ , if i ̸= j
jk il
partialp(ij|kl) =(1 + J )|U |2 |U |2 ,otherwise.
ikil
One can see the role that interference plays by noting the cross terms that vanish with J → 0.16

(8.20)
The

probability we observe a coincident detection of partially distinguishable atoms in disjoint sets of sites S1and S2 is:Xppartial
S1 ,S2 =ppartial (ij|kl).

(8.21)i∈S1 ,j∈S2

The corresponding probability pdist
S1 ,S2 for distinguishable particles is obtained when setting J = 0. Analogous to a typical HOM experiment, we can consider the ratio of such coincident detections for partially
distinguishable and distinguishable particles:
QHOM
S1 ,S2 :=ppartial
S1 ,S2pdist
S1 ,S22∗∗i∈S1 Re(Ui,k Uj,l Uj,k Ui,l )j∈S2=1+J P.
2222i∈S1 (|Ui,k | |Uj,l | + |Ui,l | |Uj,k | )
j∈S2

|

P

{z

:=−τ (S1 ,S2 )(8.22)}

Note that the ratio −τ (S1 , S2 ) ≥ −1, and so we have the inequality:
QHOM
S1 ,S2 ≥ 1 − J .

(8.23)We therefore have that the distinguishability (1 − J ) of two atoms is upper-bounded by the ratio QHOM
S1 ,S2 forany choice of S1 and S2 .
When equality holds in Eq. 8.23,17 we say that S1 and S2 satisfy the balanced condition, in analogy
to the case of a balanced beamsplitter. Such a balanced condition can easily be found in our experiment
15 Assuming linear optical evolution, and a label degree of freedom that evolves independently of the visible degrees of
freedom.
16 Note that the enhancement by a factor of two for two indistinguishable atoms (with J = 1) to bunch in the same mode i
is precisely the two atom case of the more general enhancement of n! for full bunching of n atoms (see Sec. 8.3.2).
17 So that τ (S , S ) = 1 and QHOM = 1 − J .
12
S1 ,S2

(a)

211

(c)
0.00

0.02

p partial
S1,S2

47

++

0.01

++

--

(b)

10 1

1 2

p partial/p dist
S ,S
S ,S

1 2

y

100

10 2

0
0.0

0.5

1.0

1.5

0

x

47

t (ms)

Figure 8.4: Benchmarking two particle quantum walks. (a) Two atoms (blue disks) are prepared on adjacentsites in a 1D chain and undergo CTQW dynamics. At the appropriate evolution time t, the operation applied
between the two initially populated sites is precisely a beam splitter, and we can measure the resulting HongOu-Mandel (HOM) interference. (b) pS1 ,S2 refers to the probability of coincident detection of one atom
dist
on each of the two input columns (sites in 1D after binning). We plot the ratio ppartial
S1 ,S2 /pS1 ,S2 , where thesuperscripts “dist” and “partial” respectively refer to experiments with and without introducing an additional
distlabel for distinguishability. At an evolution time of t = tHOM (grey dotted line) ppartial
S1 ,S2 /pS1 ,S2 should vanishfor perfectly indistinguishable bosons, and for an ideal lattice with only nearest-neighbor tunneling (greytheory curve). Higher order tunneling terms in our lattice (diagonal and next-nearest neighbor) result inimperfect visibility even for identical bosons (black theory curve). (c) HOM measurements at different
locations in the lattice do not show a strong dependence on location over a region that spans all input sites
used in this work (colored circles show the quality of the HOM dip at the corresponding position). Note thatthe HOM data appearing in (b) is averaged over three regions centered at x = 27 and y = (16.5, 24.5, 32.5).

by considering the binning procedure described in Sec. 8.2.3, which converts a 2D quantum walk into a 1D
quantum walk (Fig. 8.3). In the 1D quantum walk, an evolution time of tHOM := 0.96 ms in our experimentapproximates a balanced beam splitter between two adjacent sites. Therefore, at tHOM , QHOM
S1 ,S2 is a tightupper bound for the distinguishability of two atoms when S1 and S2 each correspond to all the sites in oneof the two columns that initially contained an atom before the quantum walk dynamics.18
Performing measurements of QHOM
S1 ,S2 , including measurements of distinguishable atoms using both
position and time labeling,19 we infer a lower bound on the atom indistinguishability of 97.1+1.0−1.5 %. Due tothe presence of small diagonal and next-nearest-neighbor tunneling terms (see Sec. 7.1.1), we are not able
to perfectly implement the balanced condition for reasonable choices of t, S1 , and S2 . However, via the18 One must be careful to include the effect of parity projection in the measurement of QHOM , which we discuss in Appendix B,
S1 ,S2but the effect of this correction is multiplicative, and fairly small.
19 We find that both approaches agree to within statistical errors.

212
calibrations discussed in Secs. 7.3 and 8.4, we estimate the magnitude of the resulting error and apply it
20
which is consistent withas a correction. This correction yields an estimated value of J = 99.5+0.5−1.6 %,

what we expect based only on errors due to imperfect optical cooling.21 We do not expect other sources of
distinguishability to play a significant role in our experiments.
Note that the above measurements are averaged over three different regions in the lattice. To confirm
that the quality of our state preparation does not vary significantly across the lattice, we perform additional
HOM measurements as a function of position in the lattice, and do not observe significant variation in
performance across a region containing all input sites used in this chapter (Fig. 8.4c).

8.3.2

Bunching and clouding
Similar to HOM interference and the resulting bunching for two atoms, the visible particle statistics

of many atoms can be tested by measuring full bunching [315] (Fig. 8.5acd), namely the probability that
all n atoms occupy the same output site.22

In fact, the full-bunching probability is n! larger when theparticle statistics are perfectly bosonic as opposed to when they are perfectly distinguishable, independentof the unitary applied [315]. This can be seen by comparing Eq. 8.18 and Eq. 8.19: because full bunchingcorresponds to a situation where all elements of k are equal, all the rows of Ukj are equal. The permanentis then simply a sum of n! identical terms, and so squaring inside or outside the permanent results in adifference of n!.23
The full-bunching probability is uniquely maximized by bosonic particle statistics in comparison toother particle statistics, and so maximum full bunching for the visible DOFs implies that the visible particle
statistics are bosonic (for non-zero full bunching). We note that we normalize the bunching probabilityof distinguishable particles by the probability of full survival for bosonic particles to account for the effectof parity projection (see Appendix B). We see the expected enhancement in full bunching borne out in
20 We also performed a separate measurement of the indistinguishability that does not rely on either postselection or the
separability of the two dimensions of the quantum walk dynamics, and found that the two measurements were consistent [367].
21 Based on the master equation simulation in Sec. 4.3.1, we predict an axial motional ground state occupation of 99.58 %,and a corresponding indistinguishability of 99.27 %.
22 Keeping in mind that in our binned measurements a site refers to a column of the lattice.
23 This is an example of what is sometimes known as “kinematic” interference, namely interference that only depends
on particle statistics, and the structure those statistics impose on the relevant Hilbert space. This is to be contrasted with“dynamical” interference, namely the (likely more familiar) interference that arises due to specific details regarding the evolutionof a system. For an excellent discussion of different kinds of interference, see [325].

213

(a)

+
p S ,S
1

+

-

-

+

2

+-

pbunch pcloud
Data

LabeledNN

Theory

(c)

0.4

Dist.
NN

Indist. NN
Indist. NNN
Dist. NNDist. NNN

0.3

pbunchUnlabeled
NN

(d)

0.2
0.1

Indist.
NN

Indist.
NNN

5pcloudUnlabeledNNN

0.00 0.03
Atom density

0.6

pcloud

x2

10

0.4
0.2

0

10 4
10 6
100

0.0
0.8

15

Indist.
1 Dist.
Dist.

10 2

pbunch

(b)

10 1

10 2

0.0
0

5

10
x1

15

0
0.00

p

0.05

1

2
t (ms)

3

4

2

4

6

8

Atom number, nFigure 8.5: Multiparticle quantum walks in 1D. (a) Quantum walks in 1D exhibit dramatically different
behaviors for different degrees of distinguishability, and for different input states. We consider three waysof coarse-graining multi-atom distributions (after binning) to characterize the degree to which the atoms
are behaving like indistinguishable bosons. pS1 ,S2 (black output) refers to the probability of coincident
detection of one atom on each of two input sites, pbunch to a coincident detection of all atoms on thesame site (summing over red outputs), and pcloud to all atoms appearing on the same half of the array(summing over blue outputs). Example trajectories that fulfill these conditions are shown on the left. The
rightmost two panels provide an intuitive, first quantized picture for understanding how constructive (+)
and destructive (−) interference of different input states (open squares) lead to the observed probability ofoccupying a given output state (shaded squares). (b) Two particle quantum walks in 1D at an evolution timeof t = 4.23 ms. The relevant input state for each subpanel is highlighted in red, with the atoms prepared ineither neighboring sites in the lattice (nearest-neighbor, NN), or separated by one site (next-nearest-neighbor,NNN). The theoretical predictions are for perfectly distinguishable particles (Dist.) and indistinguishable
bosons (Indist.). The associated measurements either introduce (Labeled) or do not introduce (Unlabeled)a time label to distinguish between the different atoms. For the remainder of this figure we use “Dist.” or“Indist.” to refer to both data and theory for consistency, with the understanding that the experimental
measurements may not correspond to perfectly indistinguishable atoms. (c) Measurements of pbunch andpcloud for two atoms as a function of evolution time, and for both NN and NNN inputs, are in good agreementwith theory. (d) Measurements of pbunch and pcloud with up to 5 and 8 atoms respectively are also in good
agreement with theory, and indicate that there is no observable degradation of particle indistinguishability
when scaling to more atoms. For comparison, we also display a prediction for when each trial of theexperiment contains one randomly selected atom that is distinguishable from the rest (1 Dist.). The displayed
experiments are performed at an evolution time of t = (n−1)tHOM , where all n input sites are approximately
uniformly coupled to each other after the quantum walk dynamics (see inset for an example of the measured
atom density in 2D with 8 prepared atoms, red region highlights input sites). The atoms are prepared with
NN spacing in 1D, which, unlike NNN spacing, is an arrangement that can generate a significant increasein pcloud for identical bosons in comparison to distinguishable particles. The theory predictions appearing
throughout this figure are for error-free preparations of atoms with the indicated particle statistics. The
theory curves in (d) are performed using Monte-Carlo methods, and thus the shaded regions denote ±1σ
confidence intervals that include both sampling errors, and systematic errors relating to fluctuations in theapplied unitary evolution.

214

(a)

0

Data

0

0

5

x3

0

0

0
5

x3

0

5

10
x3

15

5

0
10
15
0

0.03
p

5

10
x3

5

10
x3

15

5

10
x3

15

5

10
15
0

15

0.00 0.01
p

(c)

(d)
Indist. NN
Indist. NNN
Dist. NNDist. NNN

0.10

0.5
0.4
pcloud0.08
pbunch

0
10
15

x2

0.00

0

5

5

10
15

0
5
10
15

x2

x3

10

10

5

0

0
5
10
15

x2

0

5

x2

x2

10

0
5

10

10
5
x3

15

x1

10

0

0
10
15
0

0
5
10
15

5

5

x1

10

10
5
x3

15

x1

x1

x1

5

0
10
15

x2

10

x3

5

x2

5

10

x2

0

x2

x2

10

0
5

0
5
10
15

x1

0
10

Indist. NNN

0
5
10
15

x1

5

10
5

Indist. NN

Dist. NN

0
5
10
15

x1

x1

x1

5

10
0

Theory(b)Indist. NN

Dist. NN

0

0.06

0.3
0.2

0.04

0.1

0.02
0.00

0.0
0.0

0.5

1.0

1.5

2.0
t (ms)

2.5

3.0

3.5

4.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

t (ms)

Figure 8.6: Three particle quantum walks in 1D. The output distributions resulting from three particle
quantum walks at evolution times of (a), 1.97 ms and (b), 4.23 ms are in good agreement with theory.
Similar to the two particle case, each three particle output can be uniquely labeled by the coordinates of thethree particles (x1 , x2 , x3 ), with x3 ≤ x2 ≤ x1 . The probability p of measuring an output state (x1 , x2 , x3 ) isdenoted by both the size and color of the circle at the corresponding coordinates. Note that, when referring
to data, “indistinguishable” denotes the lack of an explicit time label but does not necessarily mean thatthe atoms behave as perfectly indistinguishable bosons. For input patterns with nearest-neighbor (NN)spacing, indistinguishable bosons (Indist.) exhibit enhanced probability to lie near the leading edge of the
distribution along the main diagonal (x1 = x2 = x3 ) in comparison to distinguishable particles (Dist.). Thistendency disappears for input patterns with next-nearest-neighbor (NNN) spacing. In both the NN and NNN
cases the probability to lie directly on the main diagonal is enhanced in the indistinguishable case, since
this corresponds to bunching. (c), (d), Like in the two particle case, we can coarse-grain the three particle
distributions by measuring bunching and clouding, and find good agreement with theory as a function ofevolution time. All theory predictions in this figure correspond to error-free preparations of atoms with theappropriate particle statistics.

215
experiments with up to 5 atoms (Fig. 8.5cd). However, the full-bunching probability diminishes exponentiallyin atom number, becoming infeasible to measure for more than 5 atoms given the data rate currently attainedin our setup.
In order to test the indistinguishability of larger numbers of atoms it is beneficial to look for signalsthat are sensitive to interference, but bundle many correlation functions together into a single quantity thatcan converge more quickly than any specific correlation function. One can build an intuition for the kindsof interference that occur in multiparticle CTQWs in 1D by noting that a CTQW of two distinguishable
particles in 1D is isomorphic to a single particle CTQW in 2D — for example, one can associate the positionof particle 1 with the x coordinate and particle 2 with the y coordinate. Similar arguments apply for moreparticles and higher dimensional square lattices. Therefore, one can apply intuitions about single particle
CTQWs in square lattices to the visualizations of the full output distributions for multiparticle CTQWs in
1D appearing in Figs. 8.3 and 8.5a.
In the following discussion, we will refer to the space that these multiparticle output distributions
occupy (namely states labeled by the coordinates of multiple particles) as a virtual lattice (by analogy
to a real lattice, where the states are labeled by real coordinates). Consider the two particle input statek = (k0 , k1 ) in a first quantized picture:1|k⟩ = √ (|k0 ⟩|k1 ⟩ + |k1 ⟩|k0 ⟩),2(8.24)

where |i⟩ refers to the single atom state where an atom occupies site i in a real 1D lattice (Fig. 8.5a). One
can consider |k⟩ to be the state where one is in a uniform superposition of two states across the diagonal of a2D virtual lattice. Under CTQW dynamics, the two components of |k⟩ will interfere.24 By virtue of matrix
exponentiation, each “hop” from one site to another in the virtual (and real) lattice is associated with a phaseshift of π/2. Note that the minimum path length connecting the two components of |k⟩ to a site lying on the
diagonal of the virtual lattice is equal. Therefore, the leading edge of the probability distribution along the
diagonal of the virtual lattice always experiences constructive interference, as predicted by the full bunchingprobability.25 The interference on the leading edge along the anti-diagonal of the virtual lattice can either
24 Keep in mind that these two “components” are not physical, since the labels we have assigned the two indistinguishable
bosons here are not physical.
25 Keep in mind that the leading diagonal of the virtual lattice precisely corresponds to states where all the particles have216
be constructive or destructive. If, for example, k0 and k1 are adjacent sites in the real lattice, then the twocomponents of the wavefunction in Eq. 8.24 are separated by two hops on the virtual lattice,26 resulting in aphase shift of π and thus destructive interference (Fig. 8.5a, b middle). If k0 and k1 are separated by one site
in the real lattice, then the two components of the wavefunction are separated by four hops on the virtual
lattice, resulting in a phase shift of 2π and thus constructive interference (Fig. 8.5a, b bottom). As one
rotates from an axis oriented along the anti-diagonal to the diagonal of the virtual lattice, the relative phaseassociated with the minimum paths that connect the two components of the state to the leading edge of the
distribution along the axis will interpolate between 0 and 2π.27 Note that for distinguishable particles, one
instead starts in a statistical mixture of |k0 ⟩|k1 ⟩ and |k1 ⟩|k0 ⟩,28 and therefore all the effects of interference
described above disappear (Fig. 8.5b top).
When extending the above picture to more particles it becomes clear that, for particles initiallyarranged in adjacent sites in 1D, there will be a general tendency for interference to yield an enhanced
probability for being close to the main diagonal, and a reduced probability to be close to the antidiagonal(s)of the virtual lattice. This tendency can be quantified by clouding [55], which is defined as the probability
that all atoms end up on the same half of the 1D array (Fig. 8.5).29 We study clouding for atoms preparedin neighboring sites in the lattice as a function of the atom number n, and with an evolution time oft = (n−1)tHOM such that all input sites are approximately uniformly coupled to each other after the quantum
walk dynamics. In clouding measurements with up to 8 atoms (Fig. 8.5cd), the behavior of the atoms is inline with the prediction for ideal bosons and clearly separated from both measurements with time-labeled(and thus distinguishable) atoms, and theoretical predictions for partially distinguishable atoms. Similar tothe bunching scenario, we normalize the clouding probability of distinguishable particles by the probabilityof full survival for bosonic particles to account for the effect of parity projection (see Appendix B). Note
that, unlike full bunching, the enhancement of clouding for identical bosons in comparison to distinguishable
particles is strongly dependent on both the system evolution, and the specific input state. For example, per
the same coordinate, i.e. bunch on the same site.
26 Equivalently, it takes two hops to exchange the positions of two distinguishable particles occupying adjacent sites.
27 As k and k are separated by more and more sites, this phase will interpolate from 0 to larger multiples of π, leading to
0
1
more lobes of constructive interference, separated from each other by regions of destructive interference.
28 This state is mixed because we do not keep track of the labels. If keeping track of the labels, one would instead be in apure state corresponding to a specific site in the virtual lattice.
29 Equivalently, in a quadrant of the virtual lattice that touches the main diagonal.

(a)

(b)

1.0

10

3

1

10 10
|Uc /J|

1

HOM dip error

B
D
p11
/p11

1.5

0.5
0.0
0.0

0.1

0.2

0.3
t/

0.4

0.5

0.6

217
10

0

10 2
10 4
10 6

10 2

10 1
|Uc /J|

100

101

Figure 8.7: The role of interactions in HOM interference. (a) We simulate the evolution of two nearestneighbor atoms in a uniform 2D lattice under the influence of a variable contact interaction Uc in comparison
to the nearest-neighbor tunneling energy J. We define τ = 1/J. (b) The deviation in the visibility of the
HOM dip from unity is characterized as a function of interaction strength, with the experimentally relevant
values marked by the grey lines.

the arguments above, preparing atoms at next-nearest-neighbor spacing in the lattice can make the difference
in clouding for bosonic and distinguishable atoms vanishingly small, as confirmed by measurements with two
and three atoms (Fig. 8.5bc, Fig. 8.6). As a result, although our measurements of clouding provide strong
evidence of indistinguishable, bosonic particle statistics for up to 8 particles, interpreting these measurements
requires precise knowledge of the atom evolution, namely the single-particle unitary U .

8.3.3

The effect of interactions
To confirm that the interactions between 88 Sr atoms are indeed negligible for our experiments, we

simulate the interacting dynamics exactly for two and three particles (Fig. 8.7) via matrix exponentiationin QuSpin [348], and compare these results to the non-interacting case (see also Sec. 8.6). For all signals
involving two and three particles, we find that the resulting errors are at or below the 10−4 level. Although
we do not explicitly check if these interactions affect any of the signals at higher particle number, we do
confirm that the measured signals are not significantly affected by triple occupation or higher of a given site.
Since ni must reach a value of ∼ 10 for the interaction term in Eq. 8.8 to be similar to the hopping energy fora single atom, we do not expect these interactions to significantly affect any of the signals at higher particle
number [94].

218

(a)

(b)

(c)

50

Pt . t o Model
Pt . t o Freq.
Model t o Freq.

0.02
0.00

Prob. densit y

Atom density

40
30
20
10
0
0.00

0.02

0.04
0.06
0.08
TVD t o point est im at e

0.10

Figure 8.8: Calibrating the single particle unitary using quantum walks. (a) We directly characterize theevolution of pairs of atoms from the sites circled in red to those marked by green crosses, which is determinedby a 5 × 4 submatrix M of the single particle unitary U . (b) Depicted are density plots of the one- andtwo-particle quantum walks that are used to infer M , with sites that are populated in a given preparation
circled in dark red (unpopulated sites in light red). The magnitudes of the entries of M can be inferred using
the one-particle quantum walk data, and the phases of the entries of M relative to the first column and row
can be inferred using two-particle quantum walks [180]. (c) We perform maximum likelihood (ML) inference
using the above data, and compare the point estimate to ML estimates of bootstrap resamples of the data,
and to the spectroscopic calibration (see Sec. 7.3). To quantify this comparison, we compute the one- andtwo-particle distributions generated by the inferred parameters, and compute the total variation distances(TVDs) of these distributions, then take the maximum of the TVDs over the prepared input patterns. We
call this quantity the max TVD between two sets of distributions. The depicted histogram is the max TVD
between the point estimate and the ML estimates of the bootstrap resampled data. Shown also are the max
TVD between the frequencies of the data (Freq.) and the point estimate (Pt.), and that between the point
estimate and the spectroscopic model (Model). The bootstrap histogram gives a sense of the size of thestatistical fluctuation of the max TVD between the point estimate and the truth. The max TVD between
the spectroscopic model and the point estimate is large compared to the bulk of the histogram, which isthe expected behavior because statistical fluctuations in the model add to the statistical fluctuations in the
point estimate.

219

8.4

Calibrating the quantum walk unitary using Fock states
The calibrations of U discussed so far involve performing spectroscopic measurements that use theatoms as local probes of the lattice depth, and to combine these measurements with band structure calculations to generate a model of the lattice Hamiltonian (see Sec. 7.3). Although this approach is expected
to be accurate for quantum walks in the lattice, direct measurements of U that do not rely on band theory
are important for future studies that attempt to program U using local, possibly time-varying potentials
imposed by the optical tweezers [366] (see Sec. 7.6).
For comparison, in photonics experiments, direct characterizations of U can be performed efficiently
and with very low noise by taking advantage of bright coherent states containing a macroscopic number ofphotons [376, 211]. A challenge in our experiment is the inability to prepare equivalent states of atoms.
Instead, as a proof of principle, we show that U can be inferred directly by measuring the interference
resulting from different preparations of Fock states [180] (Fig. 8.8). Specifically, we perform a maximum
likelihood (ML) fit of the desired parameters in the unitary using data involving preparations of one and two
atoms in the lattice, which results in better precision than previous approaches [180].30

Using this fitting

procedure, we perform characterizations of the terms in U corresponding to four input sites and five output
sites at an evolution time of t = 1.46 ms using 4440 separate experimental trials (Fig. 8.8). We find that theML fit is within statistical variation of the spectroscopic characterization of U (see Fig. 8.8c). In principle,
these fits allow one to characterize all parameters in U at constant precision with a number of experimental
trials that scales polynomially in m. However, the number of parameters in U that we can directly infer
using quantum walk data is currently limited by the combination of high constant overhead in the required
number of experimental trials, and drifts in the experiment that modify U . As a point of reference, it isreasonable for us to collect ∼ 104 useful experimental trials each day, which is comparable to the numberof trials collected in the above calibration. Although U is stable on the scale of a single day (based on the
spectroscopic calibrations), it is prone to drifting over the course of a few days. Therefore, it is not currently
feasible for us to use the above procedure to perform calibrations of U at significantly later evolution times,
and for significantly more input and output sites.
30 We leave a detailed discussion of this fitting procedure to the future thesis of Shawn Geller.

220

8.5

Measuring interference between many particles
With an understanding of the quality of our state preparation, and the applied CTQW dynamics inthe lattice, we perform experiments involving large numbers of particles that are beyond the reach of current
classical simulations.

8.5.1

Generalized bunching
A natural way of measuring interference between many identical particles is to measure connected

correlation functions of increasingly high order. However, the difference in such correlations for atoms withdifferent particle statistics typically decreases exponentially in atom number. Instead, we are interested
in measuring observables that are sensitive to interference, but also converge quickly with respect to thenumber of experimental trials. One option is to appeal to generalized notions of bunching, where one
measures the probability pκ that all n atoms appear in an arbitrary subset κ of sites. Although it was
originally suspected that pκ was maximized by bosonic particle statistics in comparison to other particle
statistics for any κ where the number of sites |κ| is ≥ n or |κ| = 1 [304], recent counter-examples haveshown that this is false [298, 262].31 Nevertheless, except in certain fine-tuned cases, pκ serves as a sensitive
signal of interference (meaning that the number of required measurements scales only polynomially withatom number). Specifically, partial distinguishability errors rapidly reduce the value of pκ in comparison to
the value for indistinguishable bosons.
The above generalization of bunching is also useful in that it unifies our earlier measurements, wherebinning columns of sites and measuring coincidences, full bunching, or clouding simply correspond to specificchoices of κ (Fig. 8.3a). For appropriate selections of the size of κ, the difference in generalized bunchingprobabilities of bosonic behavior in comparison to other behaviors is expected to converge in a number ofmeasurements that is polynomial in n [304, 302]. Specifically, we choose |κ| = ⌊m − m/n⌉ (where ⌊·⌉ denotes
rounding to the nearest integer), and study generalized bunching as a function of atom number for square
√

n×

√

n input patterns with next-nearest neighbor spacing. We perform these measurements at a fixed

31 These counterexamples are interesting in their own right, in that they break some long-standing mathematical conjectures,
including various stronger versions of Lieb’s permanental dominance conjecture [303, 346, 262].

221
evolution time of t = 6.45 ms, and with m = 500, which is approximately the number of sites that are likely
to contain an atom after the quantum walk dynamics. We cannot directly measure pκ , and instead measure
the probability p′κ that all observed atoms on a given run of the experiment appear in the set of sites κ. p′κ
differs from pκ because even-numbered occupancy of a site not contained in κ contributes to a “successful”
event where all remaining atoms after parity projection appear within κ. We find in numerics that p′κ , likepκ , serves as a useful observable: it converges in a reasonable number of measurements, and distinguishes
between a family of experimentally relevant models for the hidden DOFs (defined below).

8.5.1.1

Averaging generalized bunching

To avoid concerns of biasing in our choice of κ, we average p′κ over all choices of a given size |κ|to compute the quantity p′κ . Despite the large number of such selections, this quantity can be computed
efficiently via a combinatorial argument. Let Sm = {1, . . . , m} be shorthand for the set of output sites, and
let S ⊆k Sm denote a subset of sites of size k. Let G be the random variable denoting the site occupationof the output, with g being a single sample of that random variable. Then, the average probability that alln particles arrive in a set of size k = |κ| ≥ n is
 −1 Xm
Pr(G ⊆ S)
kS⊆k Sm −1 XXm=
Pr(g)I(g ⊆ S)kg
S⊆k Sm −1 X
m − C(g)
m
Pr(g)
=
k
k − C(g)gp′κ =(8.25)(8.26)(8.27)

where Pr(f ) refers to the probability of event f occurring, and I is the indicator function that is 1 when its
argument is true, and 0 otherwise. The sum over g ranges over all possible mode occupations of the output,
and C(g) is the number of nonzero entries of g. Note that the main contribution to p′κ is bunching and theresulting loss as a result of parity projection. As a result, we do not postselect on the number of surviving
atoms after the CTQW dynamics in our measurements of p′κ .

222
(a)

(c)
1 label100

2 labelProbability density

10

0

4 label

10

53
2.
52
0.
17
0.
00

0.

0

9 label
9 labelsimulation

10

(b)

0

180 labelsimulation

10

0.0
0.2
Atom density

0
0.6

0.7

0.8

0.9

Atom survival fraction

Figure 8.9:
Interference of large bosonic Fock states.√ (a) √Measurements of the generalized bunchingprobability p′κ as a function of atom number for square n × n input patterns at next-nearest-neighborspacing and at a fixed evolution time of t = 6.45 ms with (triangles) and without (circles) introducing timelabels that make the atoms fully distinguishable. These measurements can be compared to a model forpartial distinguishability due to thermal motion normal to the lattice (colored lines), which is described by aharmonic oscillator with quantum number nax and thermal expectation value ⟨nax ⟩. The fully distinguishable
case (dashed line) corresponds to infinite temperature. Notice that simulations for low temperatures andlarge atom numbers are absent due to computational overhead. Insets in (a) and (b) show the atom density
after their evolution, and share a color bar. The input patterns relevant to each inset are highlighted inred. (b) Measurements of p′κ with input patterns containing 180 atoms as a function of distinguishability(circles). The atom distinguishability is controlled by partitioning the input state into nlabels sub-ensembles,such that only atoms within a sub-ensemble share a time label and thus can interfere. For particles thatare sufficiently distinguishable to simulate, we can compare these measurements to theory (solid line). (c)The distribution in the observed fraction of surviving atoms on each shot of the experiment is also sensitive
to the effects of interference due to parity projection. For the case of 9 labels (second from bottom panel),simulations capture the measured distribution of atom survival probabilities. Measurements with different
numbers of labels (and thus atom distinguishability) are clearly resolved both from each other, and froma simulation of the fully distinguishable case with 180 labels (bottom panel). For ease of comparison, thevertical lines denote the mean of each distribution (dotted lines are measurements, solid lines are theory).
Insets in (c) denote the relevant assignment of labels for each data set in both (b) and (c), with each color
corresponding to a unique time label for a subset of input sites. Shaded regions about all theory curves inthis figure denote ±1σ confidence intervals, including systematic errors relating to fluctuations in the applied
unitary.

223
8.5.1.2

Measuring generalized bunching

Measurements of p′κ show a clear separation between the distinguishable and bosonic visible behaviors (Fig. 8.9a). As expected for pκ [302], this separation grows with the particle density ρ = n/m. We
expect that the main source of distinguishability in our experiment is due to thermal motional excitations
normal to the lattice (see Sec. 8.3). This motional DOF is well-approximated by a harmonic oscillator withmotional quantum numbers nax , where atoms that possess different values of nax are distinguishable. We
numerically show that this model for distinguishability leads to expected measurements that monotonicallyinterpolate from the distinguishable to the indistinguishable bosonic case as the temperature is reduced.
Our measurements of p′κ are consistent with a thermal occupation of ⟨nax ⟩ = 0 corresponding to the fully
indistinguishable case, and inconsistent with significantly higher thermal occupation (⟨nax ⟩ ≳ 0.167). It is
important to note that, due to the lack of postselection on the number of surviving atoms, measurements ofp′κ (and atom survival) are significantly more sensitive to the effects of single-particle loss and certain kindsof detection errors (see Sec. 8.2.2) than the measurements in Sec. 8.3. As a result, these effects must be
calibrated and included in our simulations (Sec. 8.6). Despite being sensitive to calibration errors, the agreement between our measurements of p′κ and low temperature simulations suggest that the indistinguishability
measured via few particle calibrations in Sec. 8.3 is not noticeably degraded when scaling our experiments
up to more particles. This agreement motivates experiments with large ensembles of atoms whose behavior
we are unable to simulate.
The largest input patterns we study in this work contain 180 atoms (Fig. 8.9bc). Although thesepatterns can be prepared with no defects, in subsequent measurements we no longer enforce perfect rearrangement to avoid incurring significant overhead in the number of required experimental trials. However,
because we image the atoms after rearrangement and before their evolution, we can identify the locationsof any defects. This procedure results in a version of scattershot boson sampling [204], but with much lessvariation in input states than is typical [25]. Based on the few-particle characterizations of atom indistinguishability performed across relevant regions in the lattice (see Sec. 8.3.1), we expect the on-demand
success rate for preparing a single ground-state atom, evolving it with no loss, and detecting its position

(a)

(b)

0

500

1000

Lattice mode

102

Prob. density

10 4

0

0

500

Prob. density

|Uij|

10 2

500

224

(d)

1000

100

arg(Uij)

1000

(c)

101
100

0

0.2

0.1

0.0

0

500

1000

Lattice mode

0.00

0.01
|Uij|20.02

0arg(Uij)

Figure 8.10: Properties of the single particle unitary. The modulus (a) and argument (b) of the single
particle unitary U is depicted here for an evolution time of t = 6.45 ms, as is relevant to the measurements
in Fig. 8.9. The finite waist of the optical lattice beams and resulting harmonic confinement means that U
does not significantly couple all sites to each other, and thus is not Haar random. (c), (d) However, U isapproximately Haar random when considering only a 15 × 15-site region near the center of the lattice. In
this region, the distribution of the norm-square of the amplitudes in U (c) are well-captured by the PorterThomas distribution for 385 outputs (black line). The distribution of the phases in U (d) is well-capturedby the uniform distribution (black line).

in an arbitrary site to be ∼ 92 %. The dominant source of deviation from perfect boson sampling is from
5.0(2) % atom loss due to imperfect cooling in the in-plane directions, with additional contributions fromimaging, rearrangement, and distinguishability. Given the above performance we expect that, on an average
run of the experiment, ∼ 166 of the 180 input sites are populated with identical bosons that evolve to 1015
output sites in the lattice with no loss or detection errors.
Although U couples all input sites to all output sites, only some of these couplings are significant,primarily due to the finite size of the lattice beams and resulting harmonic confinement (see Fig. 8.10ab andSec. 7.3).32

The harmonic confinement is negligible in a 15 × 15-site region near the center of the lattice,in which the distribution of the norm-square of the elements in U is well-captured by the Porter-Thomas
distribution for 385 outputs (Fig. 8.10cd), indicating behavior that is close to Haar-random (in the sense thatthe distribution is spiky as a result of single-particle interference, and not necessarily that the distribution
is truly Haar-random). Over the full 1015-site region considered in this section,33 U couples each site toan average of 83 of the 180 input sites with an amplitude of ≥ 10−3 . Some sites are coupled by ≥ 10−3 toas many as 156 input sites. While these numbers suggest that the experiment samples from a large effective
state space, it is important to stress that this is not a certification of computational complexity, since there
32 The effect of this harmonic confinement manifests in the “eye” shape of the amplitudes appearing in Fig. 8.10a, where, at
a fixed evolution time, atoms starting further from the center of the lattice spread out less than atoms near the center.
33 This region includes sites that can contain an atom at the latest evolution times explored, but for which the amplitudes of
U are not necessarily well-captured by the Porter-Thomas distribution.

225
is no known efficient means to verify that the machine is behaving correctly in this regime.

8.5.2

Tuning particle distinguishability
One way to build additional confidence that the many particles maintain coherence during their evo-lution, leading to high sample complexity, is to tune the visible particle statistics into a regime that issimulable. Specifically, multiple experiments performed with different subsets of atoms in the full input
pattern can be combined to simulate partially distinguishable atoms (Fig. 8.9bc), where only atoms thatshare a time label are able to interfere. We observe the expected qualitative behavior where, as the distinguishability is increased by increasing the number of labels, p′κ is reduced. Once the atoms are sufficiently
distinguishable it is once again possible to simulate their evolution, yielding good agreement with theory.
Due to parity projection, the number of detected atoms after their evolution (Fig. 8.9c) serves as a similar
measurement to p′κ , and is akin to standard measurements performed in the photonics community using
click detectors [376, 375]. Measurements of the atom survival do indeed yield a clear separation between
experiments with different visible particle statistics. Additionally, a theoretical prediction for the partially
distinguishable case with 9 extra labels captures the distribution of atom survival probabilities (Fig. 8.9c).
The above measurements serve as indirect evidence that the experiment with no additional time labels is
behaving in line with expectations for bosonic particle statistics, and sampling from a very large effective
state space.

8.6

Simulating non-interacting many-particle quantum walks
The conclusions drawn in Sec. 8.5 (and some of the conclusions in Sec. 8.3) primarily rely on com-paring the performance of the experiment to simulations. For all simulations in Secs. 8.3 and 8.5, we use
the spectroscopic calibration of the lattice Hamiltonian (Sec. 7.3) to infer the single particle unitary U . For
up to 3 particles, we solve for the full output distribution exactly by evaluating the permanent in Eq. 8.18
(or in Eq. 8.19 for distinguishable particles) for all possible outputs via Glynn’s formula. For larger particle numbers, we follow the approach of Clifford and Clifford [72] to sparsely sample from the full output
distribution.

226
The simulations involving thermal occupation of the motional degree of freedom normal to the lattice in
Fig. 8.9a assume that the evolution of this hidden degree of freedom is independent of the visible evolutionof the atoms, and dephased by our state preparation. As a result, the evolution of atoms with a mixed
motional state can be simulated by assigning a specific motional state drawn from the appropriate thermal
distribution to each atom in a given simulated sample. For each subset of atoms that share a motional state
we draw a sample using the approach of Clifford and Clifford [72], and combine these samples into a single
sample of partially distinguishable atoms.
For experiments without postselection, we calibrate the false positive and false negative detection
error rates (see Sec. 4.4.1), and take advantage of the fact that the absence of an atom, whether due to loss
or detection errors, is expected to be an effect that commutes with the linear optical dynamics [44].34 Given
this observation, our simulations proceed by simulating samples of the experiment without imaging errors,
applying single atom loss (including due to detection errors), then parity projection, and finally detection
errors that lead to extra sites that are observed to contain atoms.

8.7

A note on noise and the hardness of boson sampling
Boson sampling [2] and most of its variants, including Gaussian boson sampling [137, 88], are strongly

believed to be hard to simulate classically in the absence of noise.35

In this scenario, the hardness of thesampling task is effectively characterized by the number of bosons n and number of modes m involved.36
However, the hardness of these problems can change in the presence of noise. Such noise can broadly be
categorized into non-unitary errors relating to state preparation, detection, and loss, and errors relating touncontrolled parameters that govern the unitary evolution of the particles. Here, we compare our approachto pioneering experiments that study boson sampling using photons (Tab. 8.2) through the lens of theabove errors. In summary, although our experiment achieves significantly lower non-unitary error rates thanphotonics experiments, an outstanding challenge is to characterize the applied unitary to a level that iscomparable to what can be achieved in photonics experiments using large coherent states of many photons.
34 Due, in part, to the absence of interactions (see Sec. 8.1.1).
35 The main assumption being that the polynomial hierarchy does not collapse to the third level.
36 Up to requirements that the unitary transformation applied between modes is sufficiently mixing [114, 243], and that one
doesn’t use a pathological input state (e.g. all bosons on one mode).

[344]
[376]
[211]
[87]
[367] (our work)

n

P

J

r

m

Loss

20
50∗
216∗
50∗
180

0.975
0.938
0.962

0.93-0.954(1)0.995+5−16

1.34-1.84∼ 1.1
1.2-1.6
-

60
100
216
144
∼ 101526 %∼ 45 %∼ 67 %
57 %∗∗
5.0(2)%†

Detection
60-82 %
73-92 %‡
95 %
99.8(1) %††

Click
Click
Counting
PPNRD
ParityInput

Evolution

Fixed
Fixed
TunablePhase
Pattern

Fixed
Fixed
TunableFixed
Hamiltonian

Table 8.2: Comparison of large-scale boson sampling demonstrations. For works involving Fock state boson sampling, n denotes both the particle
number and the number of input modes. Works involving Gaussian boson sampling are marked with a ∗ , in which case n corresponds only to thenumber of input modes. P = 1 − g (2) (0) is typically referred to as the “purity” in photonics experiments, and is measured via second order correlationsin Hanbury-Brown-Twiss-like experiments. † To the extent that these measurements characterize the single-particle nature of the input field [344], inour experiments P ≃ 1 and is lower-bounded by our imaging fidelity of 0.998(1). However, our state purity is primarily limited by thermal motional
excitations normal to the lattice, and can be estimated using the measured particle indistinguishability of J = 0.995+5−16 , which is an estimate of thepurity assuming that the single-particle wavefunctions in the out-of-plane motional DOF are equal. To characterize state preparation, we list J for
Fock state boson sampling results and the squeezing parameter r for Gaussian boson sampling results. m denotes the number of output modes in thelinear optical network. “Loss” denotes the fraction of particles lost during evolution, including incoupling from the source to the linear-optical network,
loss in the network, and outcoupling to the detectors. Detection is characterized by the detection efficiency, and the type of measurement performedon each output mode. “Click” refers to detecting the presence or absence of particles, “parity” to detecting particle number parity, “PPNRD” topseudo-photon-number-resolving detection, and “counting” to full particle number-resolved readout. The work marked with ‡ includes fiber couplingloss in the estimate of detection efficiency, and the work marked with ∗∗ includes detection efficiency in the quoted value for loss. †† The listed value
for our work is a detection fidelity rather than an efficiency, and includes contributions from both particle loss and infidelity. Converting the otherlisted values to detection fidelities would involve including the effects of leakage light and dark counts, resulting in slightly lower values. “Input” refers
to the class of states that can be prepared as inputs to the linear optical network, with “phase” referring to tunability of the phases of the preparedsqueezed states, and “pattern” to nearly arbitrary Fock states with occupations of 0 or 1 on each input mode (see Sec. 4.5). “Evolution” refers to thefamily of linear optical networks that can be applied in a given system, with “Hamiltonian” referring to unitary evolution for variable time under afixed Hamiltonian. For both “input” and “evolution”, “fixed” refers to a single instance, and “tunable” to flexible, but not universal, programmability.
The numbers appearing in this table are representative values for approximate comparison only, please refer to the original publications for details.

227

228
8.7.1

Non-unitary errors
In photonics experiments, it can be challenging to generate and interfere large Fock states of photons

due to transmission losses, and the probabilistic techniques that are often [256, 249, 284, 46, 317, 329, 25,
54, 250] (but not always [202, 344]) used for single photon generation. To circumvent this difficulty, recent
experiments have performed modified versions of boson sampling that take advantage of more easily accessible
non-classical states of light [204, 25], including demonstrations of Gaussian boson sampling [376, 375, 211, 87].
The above demonstrations have focused on scaling up the number of modes in the linear optical
network, as well as the number of photons detected at the output of the network. For example, in [211], they
achieve impressive results involving up to 219 photons detected in 216 modes. However, recent theoreticalresults suggest that the high transmission losses in these experiments (∼ 67% in [211]) make it possible toefficiently simulate their behavior classically. In particular, because loss occurring either during evolution
or detection is expected to commute with the applied linear optical evolution [44],37

one can treat all

realistic non-unitary errors as a state preparation error, resulting in reduced quantum correlations on top ofa classical background. State-of-the-art tensor network algorithms take advantage of this simplification tosimulate the largest photonics-based boson sampling experiments in about an hour [244]. Therefore, beyondn and m, important figures of merit for the hardness of a given boson sampling demonstration include thestate preparation fidelity, loss, and detection errors.
In our experiment, we benefit from low loss that is not observed to scale with evolution time (seeSec. 8.2.2) [114], high state preparation and detection fidelity, and many output modes (Tab. 8.2). Although
it is likely that modern tensor network approaches developed for Gaussian boson sampling [244] can be
modified to simulate our experiment, it is not expected that these approaches will provide a significant
speedup over preexisting simulations [72] (Sec. 8.6) given the loss rates and detection fidelities achieved inour experiment.
37 Given certain assumptions regarding how the linear optical network is implemented.

229
8.7.2

Unitary errors
The comparably low non-unitary error rates achieved in our experiment, and the challenges they

impose for classical simulation, are encouraging. However, another source of errors comes from the imperfectapplication or characterization of a given target unitary. In our experiment, these errors primarily come fromfluctuations in the intensity of the optical lattice, and thus a Hamiltonian whose energy scale fluctuates:
H(s) = sH0 ,(8.28)

where s is an unknown unitless real number, and H0 is the target Hamiltonian that generates the desired
unitary transformation. Assuming that s follows a Gaussian distribution with mean 1 and standard deviation
σs , we show in [367] that the fidelity F of the prepared state to the ideal state with no unitary errors obeys
the lower bound:(nσs W t)2F ≥ exp −2

.

(8.29)

Here, W ≃ 8J is the bandwidth of the single particle Hamiltonian (Eq. 7.9) and n is the atom number.38
In our experiments, W ≃ 1 kHz and t ≤ 6.45 ms. Fast shot-to-shot fluctuations in the lattice intensity resultin σs ≃ 10−3 ,39 which does not significantly affect our experiments with up to n = 180.
However, the uncertainty in our characterization of the applied unitary is much larger, due to the
combined effect of drifts and the time-consuming calibration procedures we currently use (see Secs. 7.3 and
8.4). The resulting value of σs ≳ 10−2 dramatically affects the fidelity of the prepared state.40

It is very

possible that such unitary errors affect the hardness of the sampling problem realized, although I cannot
find much discussion of this limitation in the boson sampling literature.41
Additionally, like in previous large-scale demonstrations of boson sampling [376, 375, 211, 87], we
currently can only apply a restricted family of unitaries. These unitaries possess additional structure that
could, in principle, be taken advantage of in efficient classical simulations. Haar-averaging of the unitary
38 Note that to obtain an evolution time when all input sites are overlapping (in a 2D lattice), we need that t ≳ √n, meaningthat the exponent in Eq. 8.29 scales like n3 .
39 These errors are comparable to errors due to shot-to-shot fluctuations in the optical path length achieved in photonics
experiments, e.g. [376].
40 Note that residual interactions between the atoms enter at a similar order (Sec. 8.3.3). However, these interactions do notfluctuate significantly in strength, and are not expected to affect the hardness of the sampling problem [374].
41 Note that similar errors can occur in photonics experiments given limitations on phase locking large linear optical networks [376, 211].

230
would remove the possibility of such simulations, and provide access to additional tests of boson samplingthat rely on random matrix theory [304, 302, 33]. Based on the results in Ch. 7, universal control over the
single particle unitary (see Sec. 7.6) can in principle be implemented using the same optical tweezers we
use for atom rearrangement without introducing additional loss. However, an important open problem is toimprove the efficiency of protocols for directly calibrating the applied unitary for large systems, as well as
improving the stability of our experiment.42

8.8Interacting quantum walks
The tools demonstrated in the previous sections for controlling non-interacting quantum walks are justas easily applied to interacting systems. As discussed in Sec. 8.1.1, one way to reintroduce interactions into
our quantum walks is to simply work with a different isotope of Sr. In this section, we discuss preliminary
experiments we have performed with 86 Sr, which provides strong interactions on the scale of Uc /J ≳ 8
in our experiments.43

Note that, despite the fact that non-interacting quantum walk dynamics can bechallenging to simulate (as we saw in the previous sections), the lack of interactions still results in a significant
simplification in the behavior of the system.44 However, incorporating almost any interaction into quantum
walks of identical particles makes such quantum walks universal for quantum computing [64].45

Here, we

will briefly discuss preliminary experiments we have performed involving both interacting dynamics, and the
preparation of a non-trivial and interacting ground state.
42 In particular, note that the quoted fluctuations in H apply only to the lattice potential. We expect that drifts in thetweezer potential relative to the lattice (see Sec. 3.3.5) will result in significantly larger errors in our current apparatus.
43 Upgrades are underway to work with 84 Sr, which will provide intermediate interaction strengths of U /J ∼ 1.
c
44 For example, boson sampling is not believed to be universal for quantum computing, although the addition of mid-circuit
measurement and feed forward is enough to recover universality [176].
45 This is distinct from the universality of single particle quantum walks [59] — although single particle quantum walks canbe universal, they operate on a Hilbert space that grows linearly in physical resources (see Sec. 7.7). Interacting multiparticle
quantum walks are universal on a Hilbert space that grows exponentially in the particle number, and therefore can in principle
fulfill the full promise of efficient and universal quantum computing.

231
8.8.1

Hard core bosons and fermionization
Our use of 86 Sr results in Uc ≫ J, which is often referred to as the “hard core” limit. In this limit,

the bosonic operators in Eq. 8.1 take on modified commutation relations [139]:[ai , aj ] = [a†i , a†j ] = [ai , a†j ] = 0 for i ̸= j
{ai , ai } = {a†i , a†i } = 0,(8.30){ai , a†i } = 1,

where the relations on the second line simply encode the fact that double occupancy of a given site isenergetically forbidden. Other than the modified operator algebra in Eq. 8.30, the Hamiltonian governing
hard core bosons (HCBs) is the same as in Eq. 8.2. Notice the similarity of Eq. 8.30 to the fermionicanticommutation relations:{fi , fj } = {fi† , fj† } = 0,

{fi , fj† } = δij ,(8.31)

where fi† is the fermionic creation operator on site i (and fi the corresponding annihilation operator). One
can map the behavior of HCBs to fermions by using the Jordan-Wigner transformation, which restores thefull set of anticommutation relations associated with fermions by including the appropriate phase factors in
the operators [274]:46a†j = fj†

j−1
Y

†

(8.32)

e−iπfl fl ,l=1

and similarly for aj . For nearest-neighbor tunneling in 1D, namely where:
H = −J

X(a†i ai+1 + a†i+1 ai ) +iX

Vi a†i ai ,(8.33)ithe phase factor in Eq. 8.32 cancels47 leading to a Hamiltonian describing non-interacting (or free) fermions:
H = −J

X
i†(fi† fi+1 + fi+1
fi ) +

X

Vi fi† fi .

(8.34)i

Therefore, for quantum walks on a 1D chain, the only difference between HCBs and free fermions is the phasefactor appearing in Eq. 8.32. This phase factor is irrelevant for measurements involving particle positions,46 Note that the HCB system can be mapped to a spin model by observing that each lattice site only has two allowed states(0 or 1 bosons), and therefore can be represented with a single spin 1/2 degree of freedom. From this point the Jordan-Wignertransformation can proceed in the standard way.
47 Note that in the hopping terms, the cancellation of the phase factors additionally relies on the fact that a fermion cannot
hop onto an occupied site.

232
or correlations between particle positions (namely for the sampling problems discussed in this chapter), andso all predictions for such measurements are identical for HCBs and free fermions in a 1D chain. The above
equivalence is sometimes referred to as the “fermionization” of HCBs.
Interestingly, the fermionic equivalent of boson sampling (fermion sampling) is trivial to simulateclassically,48 since the anticommutation relations result in a determinant49 rather than a permanent in
Eq. 8.18 [2, 328]:
P F (k|j, U ) = |Det(Ukj )|2 ,(8.35)

where P F is the probability of observing a particular output event for fermions. Note that the normalization
appearing in Eq. 8.18 is 1 here due to Pauli exclusion and the resulting lack of collisions. In our experiment,
we expect Eq. 8.35 to describe the behavior of 86 Sr atoms in a 1D lattice (up to small corrections due toimperfect realization of the HCB condition).50
Interactions break the separability of quantum walks on square lattices discussed in Secs. 7.2.1 and8.2.3.51 Therefore, in order to study the dynamics of HCBs in 1D, we isolate individual rows or columns of
the lattice using optical tweezers.52 This is done either by applying 515 nm tweezers to all sites in the 1D
region of interest, or by applying tweezers to sites away from the region of interest to form a pair of barriers.53
In both cases, we have experienced issues with atoms leaking out of the 1D region of interest into other sites
in the lattice on a timescale of ∼ 10 − 100 ms. Although this is not an issue for short time dynamics,
48 I have long wondered about whether there is a deep reason that there is an inversion in the computational complexity
associated with bosons and fermions with and without interactions. Whereas non-interacting fermionic dynamics are trivial (andnon-interacting bosonic dynamics complicated, as we saw in previous sections), the opposite is true when studying interacting
ground states. Here, the non-negativity of bosonic ground states makes quantum Monte-Carlo techniques very effective, whereas
properties of fermionic ground states remain challenging to compute due to the sign problem [335]. As far as I can tell, thespecial case of free fermions stands alone because the cancellations that make the determinant efficient to compute are veryfine-tuned (i.e. most matrix immanants are hard to compute [78]), but I would be very interested in any insights the reader
may have.
49 Which can be computed efficiently, for example via Gaussian elimination (although more efficient polynomial time algorithms for computing the determinant also exist).
50 The ability to change the effective particle statistics of the atoms (either by studying HCBs in 1D, or by changing to afermionic isotope) could be useful in future schemes for benchmarking boson sampling in atomic systems. Specifically, withoutchanging other aspects of the experiment, one can switch between a situation that is easy to simulate and therefore benchmark,and a situation that is hard to simulate. Such a change would be challenging to implement using photons, especially without
affecting other aspects of the experiment [284, 219].
51 Because the presence or absence of interactions depends on the exact location of a given particle in the lattice, rather thanjust the occupied row or column.
52 Note that the main utility of the mapping between 2D and 1D in the non-interacting case was to suppress the effectof parity projection (see Sec. 8.2.2). In the case of HCBs, double occupancy and thus parity projection is suppressed due tointeractions, and so mapping from 2D to 1D is no longer necessary for achieving measurements in an effectively complete basis.
53 Note that the 515 nm tweezers are attractive, meaning that the former approach leads to sites in the region of interest
being lower in energy than other sites in the lattice, whereas in the latter approach sites in the region of interest are higher inenergy.

88

86

Sr

Sr

(b)

Dat a

10.0
Evolution time (ms)

x2

20

233

(c)

Theory

10

1.00
7.5

0.75
0.50

5.0

0.25

Atom density

(a)

88

Sr

86

Sr

0.00

2.5

0

0

10
x1

20

10

0
y

10

Figure 8.11: Dynamics of hard core bosons (preliminary). (a) Two particle quantum walks in 1D (notappealing to binning 2D quantum walks) exhibit dramatically different behaviors for 88 Sr, which behaves
like a non-interacting boson, and 86 Sr, which behaves like a hard core boson (HCB). In the former case weobserve similar interference to that of previous sections, with constructive interference along the diagonal
and destructive interference along the anti-diagonal in the space of {x1 , x2 } detection events. In the latter
case, the fermionization of HCBs leads to signatures of fermionic interference — namely constructive interference along the anti-diagonal, and destructive interference along the diagonal. (b) Similar experiments withquantum walks of many hard core bosons in 1D are underway. Here we display the density distribution fora quantum walk of 7 HCBs in 1D. (c) In 2D, one can qualitatively observe a crossover between ballistic anddiffusive transport for non-interacting bosons and HCBs. Here, a regular pattern of atoms is prepared (single
shot image, top), and the atoms are allowed to tunnel for 8.5 ms. Averaged images (bottom) exhibit theapproximately rectangular distribution characteristic of non-interacting quantum walks for 88 Sr (as exploredin previous sections), but a diffusive, approximately Gaussian distribution for 86 Sr.

it does significantly limit our ability to perform adiabatic ramps of the lattice parameters. One might beconcerned that applying tweezers to every lattice site would pose challenges in terms of inhomogeneity, but
it is important to note that this approach is distinct from performing tunneling directly in the tweezers.
Here, the tweezers are providing a shift in the chemical potential that makes certain tunneling processes
off-resonant, requiring depths on the scale of ∼ 10J ≃ 1 kHz. This is to be compared with depths of≳ 5ER ≃ 10 kHz (where ER is the recoil energy associated with our lattice spacing) for realizing tunneling(in the tight binding limit) directly between the tweezers.
Two particle quantum walks in 1D performed using 88 Sr and 86 Sr (Fig. 8.11a) nicely illustrate thedifferent dynamics resulting from bosonic and fermionic interference, with the familiar bunching and cloudingfrom Sec. 8.3.2 exhibited in the former case, and anti-bunching and anti-clouding in the latter case.54 Similararguments to those in Sec. 8.3.2 apply to the case of multiparticle fermionic quantum walks in 1D (and byextension, HCBs in 1D), leading to constructive interference along the leading edge of the anti-diagonal(s)54 Namely, an enhanced probability for the particles to be far apart from each other.

234
in the virtual lattice.
Note that although the phases in Eq. 8.32 do not affect the HCB sampling problem, they are still
physical, and can result in interesting effects. For example, they are responsible for what is known as“dynamical quasicondensation” [274], which has been investigated previously in the quantum gas community [343]. These studies focus on the situation where particles initialized at nearest neighbor spacing areallowed to expand in 1D. Although the density profiles (and correlation functions involving density) of freefermions and HCBs undergoing such expansion in 1D are identical, their description in the momentum basis
is different. Free fermions initialized in Fock states are in a superposition of all momentum states, and remain
this way because they are non-interacting. In contrast, a significant fraction55 of the HCBs condense intomodes with definite wavenumbers k = ±π/(2a) (where a is the lattice spacing) [274]. This partial (quasi)
condensation is precisely due to the non-local phase factors appearing in Eq. 8.32. To investigate dynamicalquasicondensation in our system, we have extended the two particle experiments in 1D described above tomore particles (Fig. 8.11b). However, it remains to perform direct measurements of the momentum after
performing these experiments.
Although the above picture mapping HCBs to free fermions breaks in a 2D lattice, the effect ofdynamical quasicondensation is expected to persist [156]. However, simulations of such effects are challenging
without making certain mean field approximations [156]. To our knowledge, quasicondensation has not
yet been observed experimentally in more than one dimension, and performing such experiments to verify
the accuracy of approximate (classical) simulations is an ideal proving ground for quantum simulators.
Additionally, quasicondensation has only been explored in the context of quenching a Mott insulating state.56
The ability to rapidly prepare arbitrary patterns of atoms in our experiment opens up the unique possibilityof studying similar dynamics starting from other input patterns, which could dramatically alter the behaviorof the atoms in momentum space.57
55 ∼ √n, where n is the number of particles. Note that the fact that the number of condensed particles does not scale like n
is the sense in which this is only “quasi” condensation.
56 Namely, a state where an equal, integer number of atoms are prepared on each lattice site in a given contiguous region.
57 Similar to how changing the input pattern can switch clouding on and off in 1D CTQWs of (possibly interacting) bosons.

235
1.5

1.5

1

1

0.5

0.5

0

0
0

0.05

0.1

0.15

0.2

0.25

Figure 8.12: Phase diagram of the Bose-Hubbard model as a function of the chemical potential V and thetunneling energy J (normalized to the contact interaction Uc ). Z is the coordination number of the lattice(e.g. 2 for a 1D lattice, and 4 for a 2D square lattice). The Mott insulating phase is shown in the shaded
regions, where the number of atoms on each lattice site ni takes on well-defined integer values. The white
region corresponds to the superfluid phase. The dotted lines indicate contours of constant particle density
ρ. Notice that for any non-integer value of ρ, the system is superfluid. This figure is adapted from [193].

8.8.2

Assembling a superfluid
Beyond the studies of dynamics described in the preceding sections, we are also interested in exploring

the utility of optical tweezers for preparing interesting ground states (or other steady states) in different
Hubbard systems. Along similar lines to the experiments in Sec. 7.4, we can take advantage of programmable
tweezer potentials to adiabatically connect a state that is easy to prepare — namely patterns of ground state
cooled atoms in the lattice — to a different ground state of interest.
For bosons with repulsive interactions, the zero temperature ground state phase diagram is shownin Fig. 8.12. There are two possible phases, including a Mott insulating (MI) phase where there is a welldefined, integer number of atoms on each lattice site,58 and a superfluid phase, where there is long range
phase coherence59 between different sites in the lattice.60

Typically, the Bose-Hubbard phase diagram

is studied in quantum gas experiments by modifying the chemical potential, leading to the observation ofisolated MI and superfluid regions depending on the local chemical potential [15]. Our experiment allows
for operation in a unique, and perhaps somewhat counter-intuitive regime where the number of atoms in the
58 The MI phase additionally has zero compressibility, and a finite energy gap for generating particle-hole excitations.
59 In the sense that the relative phase of the wavefunction on two sites in the lattice is well-defined, in contrast to the MIphase.
60 The superfluid phase additionally has finite compressibility, and (sometimes) a vanishing energy gap.

1

Data

2
2

Theory

2

Data

236

(b)
Single atom return probability

1
Ramp

(a)

1.0
0.8
0.6
0.4
2×2
3×3
4×4

0.2
0.0

0

50

100

150

200

Ramp time (ms)Figure 8.13: Preparing a superfluid ground state using optical tweezers (preliminary). (a) Scheme forpreparing a superfluid at fixed atom density (top left) using an adiabatic ramp of optical tweezers (green)interfaced with an optical lattice (black curves). Preparation tweezers are used to implant atoms in a subset
of sites in the lattice (top right), and then the tweezers are adiabatically ramped off. When the tweezers are
extinguished, the expected ground state of the system is superfluid, with the atoms delocalized across thelattice. After the tweezers are extinguished adiabatically, we measure an atom density (bottom right) that is
in line with a theoretical prediction based on independent characterizations of our lattice potential (bottom
left). (b) To provide evidence that the prepared state is indeed a ground state (or at least an eigenstate), wereverse the adiabatic ramp, and measure the probability of returning to the original atom distribution as afunction of the duration of the ramps. The single atom return probability is the probability that a given site
in the initial pattern is populated with an atom after the forward and reverse ramps. For fast ramps thatare not sufficiently adiabatic, the return probability is negligible. For slower ramps and for input patterns
containing up to 3 × 3 atoms, we observe a significant probability of recovering the original patterns, evenafter holding for many tunneling times between the forward and reverse ramps. For 4 × 4 input patterns,
the probability of perfectly recovering the input pattern is negligible, but there is an enhanced probability
to recover a pattern that is close to the original (in comparison to a model where the tweezer-addressed sites
are randomly populated with atoms, as one might expect for a thermal state).

system is fixed. If the chemical potential is uniform, and the number of atoms in the system is not an integer
multiple of the number of lattice sites, the Bose-Hubbard ground state is always superfluid (see Fig. 8.12).61

Notice that the states that we can prepare via rearrangement and optical cooling look a lot like a MI
state, with exactly one atom on each lattice site (or a subset of sites). With this in mind, our procedure
for preparing the superfluid state is as follows (Fig. 8.13a): we prepare a pattern of atoms with the desireddensity, and place a tweezer on each site that contains an atom. The tweezers are made deep enough thattunneling is off resonant, and so the prepared state corresponds to the MI state, just on the sublattice (orother pattern of sites) that is marked by the tweezers. The tweezers can then be adiabatically ramped
61 One can think of this in the following way: HCBs obey a kind of Pauli exclusion principle in real space, leading to MI
behavior. However, unlike fermions, they do not obey such an exclusion principle in momentum space and so can condense
into plane-wave-like states. Even for large atom numbers (> 1 atom per site), one can think of the coherence in the superfluid
phase as being the result of a small number of condensed atoms on top of a MI background.

237
off to prepare the ground state in the regime where there is significant tunneling. Preliminary results
performing the above protocol using modest numbers of atoms (4 − 16) are shown in Fig. 8.13ab. Note thatthe density profile of the atoms after the forward ramp matches expectations based on simulations using
the density matrix renormalization group (DMRG) and independent calibrations of the lattice potential(Sec. 7.3). Additionally, we can reverse the adiabatic ramp and recover the original pattern prepared after
rearrangement with a fidelity of ∼ 70% for four atoms, even when holding the atoms for many tunneling times(> 50 ms) between the forward and reverse ramps. This recovery suggests that we have indeed preparedan eigenstate of the Bose-Hubbard Hamiltonian under conditions where we expect the ground state to bea superfluid. However, it remains to perform a direct measurement of the phase coherence expected in the
superfluid state, for example via well-established time of flight techniques [129].
The additional control provided by optical tweezers opens up a variety of future directions for bothstate preparation and measurement. For example, one could isolate subsets of sites in the lattice using thetweezers to efficiently (with respect to computational resources and the number of required experimental
trials) measure certain non-standard observables [332], including the phase coherences characteristic of thesuperfluid state [172].62

Although it is not yet clear that the tweezers can help prepare arbitrary groundstates with higher fidelity than conventional techniques involving evaporation, they are already helpful for
preparing specific patterns of identical atoms rapidly and with high fidelity. Critically, the schemes that
we have explored so far for preparing the superfluid ground state do not take advantage of the full set ofcontrols that our apparatus provides.63

It is possible that more sophisticated variational [57, 330], time

optimal [91], or circuit inspired [251] techniques could offer significantly improved performance. We havejust started to scratch the surface of what can be achieved with Hubbard quantum simulators that possess
additional programmable controls, and are excited by what such systems could offer in the future [79].

62 Many of these ideas come to us from Daniel Mark.
63 Including control over the initial states of atoms that we can prepare and the chemical potential and tunneling energies in
the lattice, and also over internal degrees of freedom as discussed in Ch. 5 and 6.

Chapter 9

Conclusion and outlook“Thank you for a most enjoyable game.”— HAL 9000

In this work, we have endeavored to apply the flexible control provided by optical tweezer arrays tofrequency metrology, namely by working with the alkaline earth atom (AEA) strontium, and to simulationsof Hubbard models, by incorporating features of a quantum gas microscope (QGM) in our experiment.
In the direction of frequency metrology, we demonstrate fast assembly of large ensembles of Sr atoms,
and engineer the tweezers to preserve the coherence of the atoms. High-fidelity readout in a complete basis
provides access to new observables in our experiments, enabling the use of various correlation spectroscopy
techniques. We further engineer high fidelity single qubit gates, and entangling gates using Rydberg interactions. The achieved fidelities and coherence times are close to the state-of-the-art for any neutral atomic
qubits [103], but our clock qubits have the additional benefit that the qubit states are separated by an opticalfrequency, and are thus useful for frequency metrology. The above capabilities enable synchronous frequency
comparisons with competitive fractional frequency instabilities at the 10−17 level at 1 s of averaging, andseparate measurements using entangled states that achieve instabilities below the standard quantum limit.
A key outstanding challenge is to preserve the controls provided by tweezer clocks without sacrificing
on other desirable features, like low systematic uncertainty and large atom numbers (and thus low instability).
Because the controls provided by tweezer arrays are primarily used during state preparation, and possibly
readout, and because the various control beams can be fully extinguished during clock interrogation, theprospects for maintaining accuracy in such systems is fairly promising. However, tweezer clocks currently

239
make significant sacrifices in terms of atom number in comparison to leading lattice clocks, which work withhundreds of thousands of atoms [35]. Depending on the specific sensing problem at hand, universal control
over clock qubits is not always necessary [92, 301], and so it could be beneficial to give up some control in favorof larger system sizes, for example by generating useful large-scale entangled states without resorting to thebrute force solution of a universal gate set [254, 277, 96]. With the above considerations in mind, it is likelythat the main role of tweezer clocks will be to serve as universal testbeds for developing new measurement
protocols, especially those that draw on ideas developed in the context of quantum computing [171, 86, 84,377]. Lessons learned from these experiments can then be applied to more specialized quantum sensors thatare optimized for specific measurement tasks.
In the direction of simulating Hubbard models, we demonstrate fast and high fidelity state preparation
in a QGM using tweezer rearrangement and optical cooling. We also demonstrate the ability to programmably
modify the lattice potential using additional optical potentials projected onto the lattice.1 Using the above
tools, we perform spatial search using quantum walks of single atoms, large-scale boson sampling using
quantum walks of many atoms, and proof of principle preparations of a superfluid. It is worth reflecting
on the strengths and weaknesses of our approach to state preparation in comparison to standard techniques
used in QGMs. One useful capability that is uniquely provided by the rearrangement-based approach tostate preparation is the possibility of performing powerful kinds of postselection. For example, postselection
based on measurements taken both before and after the dynamics of interest allows us to study many
particle quantum walks in the absence of atom loss and collisions. Such postselection would not be possible
in conventional QGMs, where the procedure for state preparation precludes taking an image before thedynamics.
For the specific states prepared using rearrangement and optical cooling — namely Fock states of 0 or
1 atom per site — we reach state preparation fidelities that are comparable to or exceed the performance ofstandard QGMs.2 These states serve as a strong starting point for studying dynamics, for example problems
in transport and thermalization (see Sec. 9.2), or in the demonstrations of boson sampling performed in this
1 It is worth noting that this capability is not unique to our experiment — similar techniques are already in use in other

QGMs (e.g. in [379]).
2 This is in part due to the additional overhead required to prepare such states using “cookie-cutting” techniques to isolate
a subset of atoms from a larger many body state in standard QGMs [349, 151, 231, 264, 166, 359].

240
work. It is also worth noting that we expect there is significant room to improve the fidelity of our state
preparation, since the remaining errors are well understood, and primarily due to the optical cooling step.
The performance of the optical cooling is consistent with the ratio between trap frequency and the linewidthof the cooling transition, suggesting that cooling schemes based on the clock transition could result insignificantly improved performance [372, 271].
Although the above results are encouraging, the degree to which tweezers help with preparing interesting ground states is not yet clear. For example, we still rely on slow adiabatic ramps to prepare the superfluid
ground state, as well as the resource state used for spatial search. The utility of atom rearrangement forstudying ground state physics will hinge on the existence of states that are both easy to prepare with thetweezers, and can be adiabatically connected to ground states of interest efficiently.3

There is some hopethat this could be the case, especially when, as in our experiment, one has access to single and two qubit
gates in addition to atom rearrangement and cooling (see Sec. 9.2).
Although, in principle, the tweezers provide universal control over quantum walk dynamics in thelattice (Sec. 7.6), and the ability to perform circuit-like operations between sites in the lattice (Sec. 8.8.2),
these operations currently are prone to drifts and calibration errors.4 Even the improved calibration schemes
we develop in this work remain costly to run in terms of the required experimental trials, and cannot yet
be applied to large systems due to the comparable timescales associated with the calibration procedure, anddrifts in the experiment.5

Improving the fidelity of tweezer-controlled operations in the lattice is a key

outstanding challenge that could be addressed by reducing drifts on the experiment, or by developing faster
calibration schemes.

9.1

Directions in metrology
There are a number of exciting directions in performing frequency metrology using tweezer arrays.

One can show that the states that are most sensitive to a global signal (in the sense of Sec. 5.1.1) are in the
3 Specifically, that the requirements on adiabaticity are similar to (or more relaxed than) those for adiabatic ramps startingfrom states that are easily prepared via techniques like evaporation [30].
4 Primarily uncompensated drifts in the alignment between the tweezers and the lattice.
5 Although, fortunately, these schemes scale only polynomially in the number of lattice sites, providing some hope foraccessing high fidelity operations on large systems in the future.

241
symmetric subspace, since the quantum Fisher information is upper bounded by the length of the total spin
vector [259]. However, often, as in Sec. 6.5, we only have access to interactions that explicitly take us out of
the symmetric subspace. One interesting direction is to devise improved schemes that use local interactions
to generate metrologically useful states (namely ones that reside in the symmetric subspace). This could
be achieved by introducing the appropriate transverse field to the interactions used in Sec. 6.5 [369], or via
variationally optimized sequences using the same transverse field and interactions [160, 161, 161]. Given
possible challenges associated with collective loss mechanisms in large ensembles of Rydberg atoms [370, 38,368, 135, 107], another possibility is to perform variational optimization of a simple gate sequence that only
involves pairwise interactions (which we know can be high fidelity based on measurements presented in thiswork).
More broadly, being able to move the atoms without destroying the coherence of the clock qubits
would, in combination with the results demonstrated in this thesis, constitute a universal gate set.6 Such agate set would open the door to a variety of directions, including schemes to mitigate [170] or correct [182]errors to improve the performance of a quantum sensor. The required moves can be performed with mobile
813 nm tweezers [301], although using a different magic wavelength that is more amenable to making tightly
confining traps (like the magic wavelength near 497 nm) could be helpful when attempting to perform rapid
moves in a large array.
The additional controls provided by our system could be particularly valuable when considering moregeneral formulations of the sensing problem. For example, one could take a Bayesian approach to sensing andoptimize entire sequences of measurements to more rapidly converge on a precise estimate of the parameterof interest given prior knowledge about the parameter [228]. Another interesting possibility is to study
situations where the symmetry of the sensing problem is explicitly broken by a signal or noise that does not
act uniformly on all qubits [181].7 In these situations, the enhancements provided by entangled states canbe more significant than when one is detecting a global signal.
6 Similar capabilities have been demonstrated in the alkali atoms [32], however, clock qubits suffer from the additional
challenge that the qubit frequency is significantly more sensitive to the exact characteristics of the trap.
7 For example when the signal varies in space, and acts on an extended array of atoms.

242

9.2

Directions in simulations of Hubbard models
We have just started to explore interacting Hubbard models in our apparatus, and are excited by

the near- and long-term directions enabled by tweezer-controlled QGMs. One particularly exciting nearterm direction, mentioned earlier, is to use the fast preparation of excited states in our apparatus to study
transport problems. For example, the high repetition rate of the experiment could allow for studies ofdensity- or energy-dependent modifications to the diffusion constant [118, 36], which can be challenging
to distinguish from typical diffusion without significant statistics and large system sizes. The ability torepeatedly prepare specific patterns of atoms could also allow one to study how a given input pattern seeds
the dynamics leading to universal transport behaviors [252, 36, 347]. The ability to dynamically alter thelattice potential further provides access to new observables. For example, using an appropriately designed
quench of the local chemical potential imposed on the lattice using tweezers, one could measure separations
between charge and energy transport by directly measuring the local energy density [332].
In the direction of non-interacting quantum walks, fine-tuned programming of the quantum walk unitary could enable stronger certifications of the many particle interference at the core of boson sampling [327].
Additionally incorporating a controllable label degree of freedom, encoded in the axial motion of the atoms,
or in their nuclear or electronic states, would enable studies of new and counter-intuitive kinds of interference between partially distinguishable particles [298, 262]. The ability to prepare the appropriate entangledstates on the label degree of freedom would allow for more exotic particle statistics, including the ability tosimulate fermions [284], and certain kinds of interference problems involving anyons [219].
In some applications, the clock qubit is an attractive candidate for a controllable label. This isespecially true in cases where one wants the different qubit states to have dramatically different properties.
For example, the large energy separation between the clock qubit states makes it straightforward to engineer
spin-dependent optical potentials [142], which has enabled studies of mass-imbalanced Hubbard models [81,80], and possible future studies of Kondo lattice models [109, 108]. However, the clock qubit introduces
additional complications associated with the strong inelastic (and elastic) interactions between pairs of clockstate atoms, as well as, to a lesser degree, between clock and ground state atoms (see Sec. 8.1.1).8In

8 We have performed some exploratory measurements with tunneling clock state atoms, and find reasonable performance for243applications where one does not want the label to affect the evolution of the system (other than modifying
the particle statistics), one possibility is to map the state of the clock qubit to the axial motion of the atoms(by simply driving a π-pulse on the red sideband of the clock transition).9

Another possibility is to takeadvantage of the nuclear spin of fermionic isotopes of Sr (see below).
In this work, we control the strength of the interactions between atoms by simply working with different
isotopes of Sr, which provides limited tunability. More flexible control of the interactions could be achievedby taking advantage of the Rydberg dressed interactions we demonstrated in the context of performing
entangling gates between clock qubits. Proof of principle demonstrations using itinerant, Rydberg-dressed
atoms have already yielded exciting results [135], allowing one to study extended Hubbard models withinteractions that are tunable in both strength and range. Although there are challenges associated withminimizing inhomogeneity in the lattice introduced by the dressing light,10 as well as the collective loss
mechanisms described earlier, we suspect that such challenges could be addressed using the appropriate
Trotterized control sequences. For example, to address lattice inhomogeneity, one could perform a spinecho sequence similar to the gate sequence in Sec. 6.2 that removes the contribution from the single atom
light shift. Trotterized evolution involving interactions generated using the above sequence interspersed withtunneling could introduce long-range interactions without any lattice inhomogeneity. Furthermore, it has
been shown that short dressing pulses like those that would appear in the above Trotterized evolution are
beneficial because they allow atoms in contaminant Rydberg states to decay before they seed any collective
decay processes, leading to a dramatic improvement in coherence time [141]. For the fidelities and lifetimes
achieved in our experiments, we expect that one could study dynamics involving up to hundreds of thedescribed Trotter steps.
It is worth noting that the toolset associated with tweezer-controlled QGMs is broadly applicable toother controllable particles, including dipolar atoms [66, 321] or molecules [278, 192] that natively realize longrange interactions in their ground states. Working with a fermionic AEA like 87 Sr is of particular interest.
small ensembles and short evolution times. However, for denser ensembles and long evolution times (for example for adiabatic
ramps of ∼ 100 ms in duration) we measure significant loss as a result of inelastic two body collisions.
9 As we saw in the boson sampling demonstrations in this work, the axial motion serves as a label that is otherwise decoupledfrom the effectively non-interacting dynamics of ground state 88 Sr atoms.
10 For reasonable parameters in our experiment, controlling the chemical potential introduced by the dressing light at thelevel of the tunneling energy would require homogenization of the dressing beam at the ∼ 10−4 level in intensity.

244
The interactions between AEAs in different nuclear spin states are symmetric in the electronic ground orclock states (due to these states having angular momentum J = 0),11

and therefore provide a direct

implementation of highly symmetric SU (N ) Hubbard models, with N as large as 10 in 87 Sr [126, 56, 212].
Such systems could host a variety of exotic behaviors, including chiral spin liquids that host non-Abelian
anyons [140]. The SU (2) case is particularly well-studied and significant, since it corresponds to the standard
spin-1/2 Fermi-Hubbard model that is believed to capture the physics of high Tc superconductivity in the
cuprates [184]. Therefore, applying the controls developed in this work (namely the ability to perform gateson the spin degree of freedom, and gate-like operations between sites in the lattice) to fermionic atoms
could be extremely fruitful [124].12

For state preparation, variational or circuit-inspired techniques [57,330, 91, 251] could potentially be used to prepare ground states in these models more effectively thanconventional techniques involving evaporation followed by adiabatic ramps [30, 102]. For measurement,
gates between spins could allow one to access complicated spin correlation functions, likely with relatively
high fidelity. Modifications of the lattice potential and tunneling dynamics could further provide accessto nearly arbitrary observables [332], although likely with lower fidelity that the spin gates.13

Access tonon-standard measurements of this type could be extremely valuable. For example, whereas different pairing
mechanisms leading to superconductivity can be hard to distinguish using conventional measurements, using
the above tools one could directly measure the appropriate order parameter to distinguish s-wave fromd-wave pairing in a Hubbard model that hosts high Tc superconductivity [332].

9.3

Combined directions
In addition to the directions outlined above, there are a number of directions that combine aspectsof Hubbard physics with precision measurement and control. We discussed in Sec. 9.2 how the clock qubit
can be used to alter the distinguishability of different atoms. This serves as a useful control in precision
measurements. For example, in [3], the distinguishability of the atoms is used to tune the relative strength of
11 Unlike collisions between atoms in different electronic states, these collisions also do not suffer from the lossy two bodycollisions.
12 Fermionic particle statistics can only be simulated on standard quantum architectures with significant (albeit polynomial)
overhead in either runtime or physical resource cost. As a result, implementing analog evolution of a fermionic system incombination with the ability to perform appropriate gates and circuits could be useful for a broad set of simulation problems
in quantum science beyond just Hubbard physics, ranging from high energy physics to chemistry [124].
13 Due to the challenges associated with implementing these operations with low noise, as described above.

245
p- and s-wave interactions and cancel the interaction shift, enabling clock measurements with large ensemblesof atoms and long interrogations times, and thus record-low instability. Distinguishabilty could also be
an interesting signal in and of itself. Although landmark experiments have measured the gravitational
redshift using optical clocks [67, 35], in these measurements the proper time for each ensemble is well-definedclassically (in the sense that the atomic ensembles being compared remain separate). The ability to perform
clock measurements with the atoms in a spatial superposition allows for fundamental tests of relativity andquantum mechanics,14

where one expects to observe interference effects that can only be explained byproperly incorporating general relativistic notions of proper time into quantum theory [380, 381].15

Suchmeasurements also relate to universal sources of decoherence due to time dilation [260, 261], which have notyet been observed.
Beyond measuring fundamental quantities and behaviors, frequency metrology techniques can beapplied to probe and control large systems with complicated emergent properties. Already, high resolution
spectroscopy has played a key role in studying many body quantum systems, where the clock transition canbe used as a spectrally narrow probe which allows one to directly study the energy landscape of a given
system with very fine resolution [215]. Such techniques have been used to precisely measure the strength ofspin exchange [288] and many-body interactions [122] in systems involving lattice-trapped fermionic AEAs,
including in implementations of the Fermi-Hubbard model in 3D [53]. Calibrations based on the above
techniques are already relevant in our simulations of the Bose-Hubbard model [367].
One can also use the clock transition to explicitly engineer different kinds of dynamics, for example
by driving transitions that couple the electronic (or spin) state of the atoms to their motion. Spin orbit
coupling of this kind has been used to engineer artificial gauge fields [199, 178, 45] in the synthetic latticewhere one spatial dimension corresponds to an internal degree of freedom of the atoms.16

Combining

the above techniques with state- and site-resolved readout, as well as the preparation of low entropy initialstates, could provide a route towards studying exotic many body behaviors in lattices subjected to strong
14 These tests are distinct from what people normally refer to as quantum gravity, and are not related to detecting gravitons.
15 Schematically, the idea here is that different components of the atomic wavefunction experience different proper time,leading to entanglement between the trajectories and the internal states of the atoms.
16 Similar demonstrations have been performed in pioneering experiments using Raman transitions [123]. Implementations
using single-photon transitions between long-lived clock states have the benefit of eliminating scattering off of the short-lived
intermediate states, potentially improving coherence. However, one must be careful of the additional challenge associated withdecoherence due to the inelastic collisions between clock state atoms mentioned above.

246
fields, like chiral Mott insulators [89, 258] and topological superfluidity [267].

Bibliography[1] S. Aaronson and A. Ambainis. Quantum search of spatial regions. In 44th Annual IEEE Symposium
on Foundations of Computer Science, 2003. Proceedings., pages 200–209, Oct. 2003.
[2] S. Aaronson and A. Arkhipov. The computational complexity of linear optics. In Proceedings of the
Forty-Third Annual ACM Symposium on Theory of Computing, STOC ’11, pages 333–342, New York,NY, USA, June 2011. Association for Computing Machinery.
[3] A. Aeppli, A. Chu, T. Bothwell, C. J. Kennedy, D. Kedar, P. He, A. M. Rey, and J. Ye. Hamiltonian
engineering of spin-orbit–coupled fermions in a Wannier-Stark optical lattice clock. Science Advances,8(41):eadc9242, Oct. 2022.
[4] T. Akatsuka, M. Takamoto, and H. Katori. Three-dimensional optical lattice clock with bosonic 88 Sr
atoms. Physical Review A, 81(2), Feb. 2010.
[5] T. Albash and D. A. Lidar. Adiabatic quantum computation. Rev. Mod. Phys., 90(1):015002, Jan.
2018.
[6] A. Ambainis. Quantum walk algorithm for element distinctness. In 45th Annual IEEE Symposium on
Foundations of Computer Science, pages 22–31, Oct. 2004.
[7] A. Ambainis, J. Kempe, and A. Rivosh. Coins make quantum walks faster: Sixteenth Annual ACMSIAM Symposium on Discrete Algorithms. pages 1099–1108, 2005.
[8] W. R. Anderson, J. R. Veale, and T. F. Gallagher. Resonant Dipole-Dipole Energy Transfer in a
Nearly Frozen Rydberg Gas. Phys. Rev. Lett., 80(2):249–252, Jan. 1998.
[9] Andor. iXon Ultra and Life 897, Sept. 2023.
[10] A. André, A. S. Sørensen, and M. D. Lukin. Stability of Atomic Clocks Based on Entangled Atoms.
Phys. Rev. Lett., 92(23):230801, June 2004.
[11] N. W. Ashcroft and N. D. Mermin. Solid State Physics. Holt, Rinehart and Winston, New York, 1976.
[12] X. Baillard, A. Gauguet, S. Bize, P. Lemonde, Ph. Laurent, A. Clairon, and P. Rosenbusch.
Interference-filter-stabilized external-cavity diode lasers. Optics Communications, 266(2):609–613, Oct.
2006.
[13] W. S. Bakr. Microscopic Studies of Quantum Phase Transitions in Optical Lattices. PhD thesis, 2011.
[14] W. S. Bakr, J. I. Gillen, A. Peng, S. Fölling, and M. Greiner. A quantum gas microscope for detecting
single atoms in a Hubbard-regime optical lattice. Nature, 462(7269):74–77, Nov. 2009.
[15] W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and M. Greiner.
Probing the Superfluid–to–Mott Insulator Transition at the Single-Atom Level. Science, 329(5991):547–
550, July 2010.

248[16] S. Bali, K. M. O’Hara, M. E. Gehm, S. R. Granade, and J. E. Thomas. Quantum-diffractive backgroundgas collisions in atom-trap heating and loss. Phys. Rev. A, 60(1):R29–R32, July 1999.
[17] C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas. High-Fidelity Quantum Logic
Gates Using Trapped-Ion Hyperfine Qubits. Phys. Rev. Lett., 117(6):060504, Aug. 2016.
[18] K. Barnes, P. Battaglino, B. J. Bloom, K. Cassella, R. Coxe, N. Crisosto, J. P. King, S. S. Kondov,
K. Kotru, S. C. Larsen, J. Lauigan, B. J. Lester, M. McDonald, E. Megidish, S. Narayanaswami,
C. Nishiguchi, R. Notermans, L. S. Peng, A. Ryou, T.-Y. Wu, and M. Yarwood. Assembly andcoherent control of a register of nuclear spin qubits. Nat Commun, 13(1):2779, May 2022.
[19] D. Barredo, S. de Léséleuc, V. Lienhard, T. Lahaye, and A. Browaeys. An atom-by-atom assembler ofdefect-free arbitrary 2d atomic arrays. Science, 354(6315):1021–1023, Nov. 2016.
[20] D. Barredo, H. Labuhn, S. Ravets, T. Lahaye, A. Browaeys, and C. S. Adams. Coherent excitationtransfer in a spin chain of three Rydberg atoms. Phys. Rev. Lett., 114(11):113002, Mar. 2015.
[21] J. P. Bartolotta, M. A. Norcia, J. R. K. Cline, J. K. Thompson, and M. J. Holland. Laser cooling by
sawtooth-wave adiabatic passage. Phys. Rev. A, 98(2):023404, Aug. 2018.
[22] J. Bausch, T. S. Cubitt, A. Lucia, and D. Perez-Garcia. Undecidability of the Spectral Gap in One
Dimension. Phys. Rev. X, 10(3):031038, Aug. 2020.
[23] R. Beigang, K. Lücke, D. Schmidt, A. Timmermann, and P. J. West. One-Photon Laser Spectroscopy
of Rydberg Series from Metastable Levels in Calcium and Strontium. Phys. Scr., 26(3):183, Sept. 1982.
[24] J. S. Bell. On the Einstein Podolsky Rosen paradox. Physics Physique Fizika, 1(3):195–200, Nov.
1964.
[25] M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J.
Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino. Experimental scattershot
boson sampling. Science Advances, 1(3):e1400255, 2015.
[26] H. A. Bethe and E. E. Salpeter. Quantum Mechanics of One- and Two-Electron Atoms. Springer
Berlin Heidelberg, Berlin, Heidelberg, 1957.
[27] J. D. Biamonte, V. Bergholm, J. D. Whitfield, J. Fitzsimons, and A. Aspuru-Guzik. Adiabatic quantum
simulators. AIP Advances, 1(2):022126, May 2011.
[28] S. Bize, Y. Sortais, P. Lemonde, S. Zhang, P. Laurent, G. Santarelli, C. Salomon, and A. Clairon.
Interrogation oscillator noise rejection in the comparison of atomic fountains. IEEE Transactions on
Ultrasonics, Ferroelectrics, and Frequency Control, 47(5):1253–1255, Sept. 2000.
[29] E. D. Black. An introduction to Pound–Drever–Hall laser frequency stabilization. American Journalof Physics, 69(1):79–87, Jan. 2001.
[30] I. Bloch, J. Dalibard, and W. Zwerger. Many-body physics with ultracold gases. Rev. Mod. Phys.,
80(3):885–964, July 2008.
[31] B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L.
Bromley, and J. Ye. An optical lattice clock with accuracy and stability at the 10−18 level. Nature,506(7486):71–75, Feb. 2014.
[32] D. Bluvstein, H. Levine, G. Semeghini, T. T. Wang, S. Ebadi, M. Kalinowski, A. Keesling, N. Maskara,
H. Pichler, M. Greiner, V. Vuletić, and M. D. Lukin. A quantum processor based on coherent transport
of entangled atom arrays. Nature, 604(7906):451–456, Apr. 2022.
[33] S. Boixo, S. V. Isakov, V. N. Smelyanskiy, R. Babbush, N. Ding, Z. Jiang, M. J. Bremner, J. M.
Martinis, and H. Neven. Characterizing quantum supremacy in near-term devices. Nature Phys,
14(6):595–600, June 2018.

249[34] T. Bothwell, D. Kedar, E. Oelker, J. M. Robinson, S. L. Bromley, W. L. Tew, J. Ye, and C. J. Kennedy.
JILA SrI optical lattice clock with uncertainty of 2.0 × 10−18 . Metrologia, 56(6):065004, Oct. 2019.
[35] T. Bothwell, C. J. Kennedy, A. Aeppli, D. Kedar, J. M. Robinson, E. Oelker, A. Staron, and J. Ye.
Resolving the gravitational redshift across a millimetre-scale atomic sample. Nature, 602(7897):420–
424, Feb. 2022.
[36] I. Bouchoule and J. Dubail. Generalized hydrodynamics in the one-dimensional Bose gas: Theory andexperiments. J. Stat. Mech., 2022(1):014003, Jan. 2022.
[37] I. Bouchoule and K. Mølmer. Spin squeezing of atoms by the dipole interaction in virtually excited
Rydberg states. Physical Review A, 65(4), Apr. 2002.
[38] T. Boulier, E. Magnan, C. Bracamontes, J. Maslek, E. A. Goldschmidt, J. T. Young, A. V. Gorshkov,
S. L. Rolston, and J. V. Porto. Spontaneous avalanche dephasing in large Rydberg ensembles. Phys.
Rev. A, 96(5):053409, Nov. 2017.
[39] M. M. Boyd. High Precision Spectroscopy of Strontium in an Optical Lattice: Towards a New Standard
for Frequency and Time. PhD thesis, 2007.
[40] S. L. Braunstein and C. M. Caves. Statistical distance and the geometry of quantum states. Phys.
Rev. Lett., 72(22):3439–3443, May 1994.
[41] S. L. Braunstein, C. M. Caves, and G. J. Milburn. Generalized uncertainty relations: Theory, examples,and Lorentz invariance. Annals of Physics, 247(1):135–173, Apr. 1996.
[42] S. M. Brewer, J.-S. Chen, A. M. Hankin, E. R. Clements, C. W. Chou, D. J. Wineland, D. B. Hume,and D. R. Leibrandt. 27 Al+ quantum-logic clock with a systematic uncertainty below 10−18 . Phys.
Rev. Lett., 123(3):033201, July 2019.
[43] H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest. Measurement-based
quantum computation. Nature Physics, 5(1):19–26, Jan. 2009.
[44] D. J. Brod and M. Oszmaniec. Classical simulation of linear optics subject to nonuniform losses.
Quantum, 4:267, May 2020.
[45] S. L. Bromley, S. Kolkowitz, T. Bothwell, D. Kedar, A. Safavi-Naini, M. L. Wall, C. Salomon, A. M.
Rey, and J. Ye. Dynamics of interacting fermions under spin–orbit coupling in an optical lattice clock.
Nature Phys, 14(4):399–404, Apr. 2018.
[46] M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White.
Photonic boson sampling in a tunable circuit. Science, 339(6121):794–798, Feb. 2013.
[47] A. Browaeys and T. Lahaye. Many-body physics with individually controlled Rydberg atoms. Nature
Physics, 16(2):132–142, Feb. 2020.
[48] M. O. Brown, T. Thiele, C. Kiehl, T.-W. Hsu, and C. A. Regal. Gray-molasses optical-tweezer loading:Controlling collisions for scaling atom-array assembly. Phys. Rev. X, 9(1):011057, Mar. 2019.
[49] R. C. Brown, N. B. Phillips, K. Beloy, W. F. McGrew, M. Schioppo, R. J. Fasano, G. Milani, X. Zhang,
N. Hinkley, H. Leopardi, T. H. Yoon, D. Nicolodi, T. M. Fortier, and A. D. Ludlow. Hyperpolarizability
and Operational Magic Wavelength in an Optical Lattice Clock. Phys. Rev. Lett., 119(25):253001, Dec.
2017.
[50] E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and C. E. Wieman. Coherence,
Correlations, and Collisions: What One Learns about Bose-Einstein Condensates from Their Decay.
Phys. Rev. Lett., 79(3):337–340, July 1997.

250[51] T. Busch, B.-G. Englert, K. Rzażewski, and M. Wilkens. Two Cold Atoms in a Harmonic Trap.
Foundations of Physics, 28(4):549–559, Apr. 1998.
[52] S. L. Campbell. A Fermi-degenerate Three-Dimensional Optical Lattice Clock. PhD thesis, Oct. 2017.
[53] S. L. Campbell, R. B. Hutson, G. E. Marti, A. Goban, N. Darkwah Oppong, R. L. McNally, L. Sonderhouse, J. M. Robinson, W. Zhang, B. J. Bloom, and J. Ye. A Fermi-degenerate three-dimensional
optical lattice clock. Science, 358(6359):90–94, Oct. 2017.
[54] J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt,
N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto,
J. L. O’Brien, and A. Laing. Universal linear optics. Science, 349(6249):711–716, Aug. 2015.
[55] J. Carolan, J. D. A. Meinecke, P. Shadbolt, N. J. Russell, N. Ismail, K. Wörhoff, T. Rudolph, M. G.
Thompson, J. L. O’Brien, J. C. F. Matthews, and A. Laing. On the experimental verification ofquantum complexity in linear optics. Nature Photon, 8(8):621–626, Aug. 2014.
[56] M. A. Cazalilla and A. M. Rey. Ultracold Fermi gases with emergent SU(N) symmetry. Reports on
Progress in Physics, 77(12):124401, Dec. 2014.
[57] M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai,
X. Yuan, L. Cincio, and P. J. Coles. Variational quantum algorithms. Nat Rev Phys, 3(9):625–644,
Sept. 2021.
[58] S. Chakraborty, L. Novo, A. Ambainis, and Y. Omar. Spatial Search by Quantum Walk is Optimal
for Almost all Graphs. Phys. Rev. Lett., 116(10):100501, Mar. 2016.
[59] A. M. Childs. Universal Computation by Quantum Walk. Phys. Rev. Lett., 102(18):180501, May 2009.
[60] A. M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, and D. A. Spielman. Exponential algorithmic
speedup by quantum walk. Proceedings of the thirty-fifth ACM symposium on Theory of computing- STOC ’03, page 59, 2003.
[61] A. M. Childs and Y. Ge. Spatial search by continuous-time quantum walks on crystal lattices. Phys.
Rev. A, 89(5):052337, May 2014.
[62] A. M. Childs and J. Goldstone. Spatial search and the Dirac equation. Phys. Rev. A, 70(4):042312,
Oct. 2004.
[63] A. M. Childs and J. Goldstone. Spatial search by quantum walk. Phys. Rev. A, 70(2):022314, Aug.
2004.
[64] A. M. Childs, D. Gosset, and Z. Webb. Universal Computation by Multiparticle Quantum Walk.
Science, 339(6121):791–794, Feb. 2013.
[65] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga. Feshbach resonances in ultracold gases. Rev. Mod.
Phys., 82(2):1225–1286, Apr. 2010.
[66] L. Chomaz, I. Ferrier-Barbut, F. Ferlaino, B. Laburthe-Tolra, B. L. Lev, and T. Pfau. Dipolar physics:
A review of experiments with magnetic quantum gases. Rep. Prog. Phys., 86(2):026401, Dec. 2022.
[67] C. W. Chou, D. B. Hume, T. Rosenband, and D. J. Wineland. Optical Clocks and Relativity. Science,
329(5999):1630–1633, Sept. 2010.
[68] C. W. Chou, D. B. Hume, M. J. Thorpe, D. J. Wineland, and T. Rosenband. Quantum coherence
between two atoms beyond Q = 1015 . Phys. Rev. Lett., 106(16):160801, Apr. 2011.
[69] M. Chwalla, K. Kim, T. Monz, P. Schindler, M. Riebe, C. Roos, and R. Blatt. Precision spectroscopy
with two correlated atoms. Appl. Phys. B, 89(4):483–488, Dec. 2007.

251[70] E. R. Clements, M. E. Kim, K. Cui, A. M. Hankin, S. M. Brewer, J. Valencia, J.-S. Chen, C.-W. Chou,
D. R. Leibrandt, and D. B. Hume. Lifetime-limited interrogation of two independent 27 Al+ clocks
using correlation spectroscopy. Phys. Rev. Lett., 125(24):243602, Dec. 2020.
[71] W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley. Optimal
design for universal multiport interferometers. Optica, OPTICA, 3(12):1460–1465, Dec. 2016.
[72] P. Clifford and R. Clifford. The classical complexity of boson sampling. In Proceedings of the 2018
Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Proceedings, pages 146–155. Society
for Industrial and Applied Mathematics, Jan. 2018.
[73] A. Cooper, J. P. Covey, I. S. Madjarov, S. G. Porsev, M. S. Safronova, and M. Endres. Alkaline-earth
atoms in optical tweezers. Phys. Rev. X, 8(4):041055, Dec. 2018.
[74] J. P. Covey, I. S. Madjarov, A. Cooper, and M. Endres. 2000-times repeated imaging of strontium
atoms in clock-magic tweezer arrays. Phys. Rev. Lett., 122(17):173201, May 2019.
[75] D. R. Cox and E. J. Snell. A General Definition of Residuals. Journal of the Royal Statistical Society.
Series B (Methodological), 30(2):248–275, 1968.
[76] H. Cramér. Mathematical Methods of Statistics. Mathematical Methods of Statistics. Princeton
University Press, Princeton, NJ, US, 1946.
[77] T. Cubitt, D. Perez-Garcia, and M. M. Wolf.
528(7581):207–211, Dec. 2015.

Undecidability of the Spectral Gap.

Nature,[78] R. Curticapean. A full complexity dichotomy for immanant families. In Proceedings of the 53rd
Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pages 1770–1783, New
York, NY, USA, June 2021. Association for Computing Machinery.
[79] A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller. Practical quantum
advantage in quantum simulation. Nature, 607(7920):667–676, July 2022.
[80] N. Darkwah Oppong, G. Pasqualetti, O. Bettermann, P. Zechmann, M. Knap, I. Bloch, and S. Fölling.
Probing Transport and Slow Relaxation in the Mass-Imbalanced Fermi-Hubbard Model. Phys. Rev.
X, 12(3):031026, Aug. 2022.
[81] N. Darkwah Oppong, L. Riegger, O. Bettermann, M. Höfer, J. Levinsen, M. M. Parish, I. Bloch, andS. Fölling. Observation of Coherent Multiorbital Polarons in a Two-Dimensional Fermi Gas. Phys.
Rev. Lett., 122(19):193604, May 2019.
[82] S. de Léséleuc, D. Barredo, V. Lienhard, A. Browaeys, and T. Lahaye. Optical Control of the Resonant
Dipole-Dipole Interaction between Rydberg Atoms. Phys. Rev. Lett., 119(5):053202, Aug. 2017.
[83] C. L. Degen, F. Reinhard, and P. Cappellaro. Quantum sensing. Rev. Mod. Phys., 89(3):035002, July
2017.
[84] R. Demkowicz-Dobrzański, J. Czajkowski, and P. Sekatski. Adaptive Quantum Metrology under General Markovian Noise. Phys. Rev. X, 7(4):041009, Oct. 2017.
[85] R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă. The elusive Heisenberg limit in quantumenhanced metrology. Nat Commun, 3(1):1063, Jan. 2012.
[86] R. Demkowicz-Dobrzański and L. Maccone. Using Entanglement Against Noise in Quantum Metrology.
Phys. Rev. Lett., 113(25):250801, Dec. 2014.
[87] Y.-H. Deng, Y.-C. Gu, H.-L. Liu, S.-Q. Gong, H. Su, Z.-J. Zhang, H.-Y. Tang, M.-H. Jia, J.-M.
Xu, M.-C. Chen, J. Qin, L.-C. Peng, J. Yan, Y. Hu, J. Huang, H. Li, Y. Li, Y. Chen, X. Jiang,
L. Gan, G. Yang, L. You, L. Li, H.-S. Zhong, H. Wang, N.-L. Liu, J. J. Renema, C.-Y. Lu, and J.W. Pan. Gaussian Boson Sampling with Pseudo-Photon-Number-Resolving Detectors and Quantum
Computational Advantage. Phys. Rev. Lett., 131(15):150601, Oct. 2023.

252[88] A. Deshpande, A. Mehta, T. Vincent, N. Quesada, M. Hinsche, M. Ioannou, L. Madsen, J. Lavoie,
H. Qi, J. Eisert, D. Hangleiter, B. Fefferman, and I. Dhand. Quantum computational advantage via
high-dimensional Gaussian boson sampling. Science Advances, 8(1):eabi7894, Jan. 2022.
[89] A. Dhar, M. Maji, T. Mishra, R. V. Pai, S. Mukerjee, and A. Paramekanti. Bose-Hubbard model ina strong effective magnetic field: Emergence of a chiral Mott insulator ground state. Phys. Rev. A,
85(4):041602, Apr. 2012.
[90] G. J. Dick. Local Oscillator Induced Instabilities in Trapped Ion Frequency Standards. In Proceedingsof the 19th Annual Precise Time and Time Interval Systems and Applications Meeting, pages 133–147,
Dec. 1989.
[91] P. Doria, T. Calarco, and S. Montangero. Optimal Control Technique for Many-Body Quantum
Dynamics. Phys. Rev. Lett., 106(19):190501, May 2011.
[92] S. Dörscher, A. Al-Masoudi, M. Bober, R. Schwarz, R. Hobson, U. Sterr, and C. Lisdat. Dynamical
decoupling of laser phase noise in compound atomic clocks. Commun Phys, 3(1):1–9, Oct. 2020.
[93] S. Dörscher, R. Schwarz, A. Al-Masoudi, S. Falke, U. Sterr, and C. Lisdat. Lattice-induced photon
scattering in an optical lattice clock. Phys. Rev. A, 97(6):063419, June 2018.
[94] G. Dufour, T. Brünner, A. Rodríguez, and A. Buchleitner. Many-body interference in bosonic dynamics. New J. Phys., 22(10):103006, Oct. 2020.
[95] S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar,
H. Pichler, W. W. Ho, S. Choi, S. Sachdev, M. Greiner, V. Vuletić, and M. D. Lukin. Quantum phasesof matter on a 256-atom programmable quantum simulator. Nature, 595(7866):227–232, July 2021.
[96] W. J. Eckner, N. Darkwah Oppong, A. Cao, A. W. Young, W. R. Milner, J. M. Robinson, J. Ye, andA. M. Kaufman. Realizing spin squeezing with Rydberg interactions in an optical clock. Nature, pages
1–6, Aug. 2023.
[97] W. J. Eckner, A. W. Young, N. Schine, and A. M. Kaufman. High-power, fiber–laser-based source
for magic-wavelength trapping in neutral-atom optical clocks. Rev. Sci. Instrum., 92(9):093001, Sept.
2021.
[98] B. Efron and R. J. Tibshirani. An Introduction to the Bootstrap. Chapman and Hall/CRC, New York,May 1994.
[99] M. Endres, H. Bernien, A. Keesling, H. Levine, E. R. Anschuetz, A. Krajenbrink, C. Senko, V. Vuletic,
M. Greiner, and M. D. Lukin. Atom-by-atom assembly of defect-free one-dimensional cold atom arrays.
Science, 354(6315):1024–1027, Nov. 2016.
[100] K. Enomoto, K. Kasa, M. Kitagawa, and Y. Takahashi. Optical Feshbach Resonance Using theIntercombination Transition. Phys. Rev. Lett., 101(20):203201, Nov. 2008.
[101] J. Eschner, G. Morigi, F. Schmidt-Kaler, and R. Blatt. Laser cooling of trapped ions. J. Opt. Soc.
Am. B, JOSAB, 20(5):1003–1015, May 2003.
[102] T. Esslinger. Fermi-Hubbard Physics with Atoms in an Optical Lattice. Annu. Rev. Condens. Matter
Phys., 1(1):129–152, Aug. 2010.
[103] S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T.
Wang, N. Maskara, H. Levine, G. Semeghini, M. Greiner, V. Vuletić, and M. D. Lukin. High-fidelity
parallel entangling gates on a neutral-atom quantum computer. Nature, 622(7982):268–272, Oct. 2023.
[104] E. Farhi, J. Goldstone, and S. Gutmann. A Quantum Algorithm for the Hamiltonian NAND Tree.
Theory of Computing, 4(1):169–190, Dec. 2008.

253[105] E. Farhi and S. Gutmann. Analog analogue of a digital quantum computation. Phys. Rev. A,
57(4):2403–2406, Apr. 1998.
[106] P. O. Fedichev, M. W. Reynolds, and G. V. Shlyapnikov. Three-Body Recombination of Ultracold
Atoms to a Weakly Bound s Level. Phys. Rev. Lett., 77(14):2921–2924, Sept. 1996.
[107] L. Festa, N. Lorenz, L.-M. Steinert, Z. Chen, P. Osterholz, R. Eberhard, and C. Gross. Blackbodyradiation-induced facilitated excitation of Rydberg atoms in optical tweezers. Phys. Rev. A,
105(1):013109, Jan. 2022.
[108] M. Foss-Feig, M. Hermele, V. Gurarie, and A. M. Rey. Heavy fermions in an optical lattice. Phys.
Rev. A, 82(5):053624, Nov. 2010.
[109] M. Foss-Feig, M. Hermele, and A. M. Rey. Probing the Kondo lattice model with alkaline-earth-metal
atoms. Phys. Rev. A, 81(5):051603, May 2010.
[110] G. T. Foster, J. B. Fixler, J. M. McGuirk, and M. A. Kasevich. Method of phase extraction between
coupled atom interferometers using ellipse-specific fitting. Opt. Lett., 27(11):951, June 2002.
[111] Y. H. Fung and M. F. Andersen. Efficient collisional blockade loading of a single atom into a tightmicrotrap. New J. Phys., 17(7):073011, July 2015.
[112] J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill,+
D. Leibfried, and D. J. Wineland. High-Fidelity Universal Gate Set for 9 Be Ion Qubits. Phys. Rev.
Lett., 117(6):060505, Aug. 2016.
[113] T. F. Gallagher. Rydberg Atoms. Cambridge University Press, 1 edition, Sept. 1994.
[114] R. García-Patrón, J. J. Renema, and V. Shchesnovich. Simulating boson sampling in lossy architectures. Quantum, 3:169, Aug. 2019.
[115] M. E. Gehm, K. M. O’Hara, T. A. Savard, and J. E. Thomas. Dynamics of noise-induced heating in
atom traps. Phys. Rev. A, 58(5):3914–3921, Nov. 1998.
[116] K. Gibble. Scattering of cold-atom coherences by hot atoms: Frequency shifts from background-gas
collisions. Phys. Rev. Lett., 110(18):180802, May 2013.
[117] L. I. R. Gil, R. Mukherjee, E. M. Bridge, M. P. A. Jones, and T. Pohl. Spin squeezing in a Rydberg
lattice clock. Phys. Rev. Lett., 112(10):103601, Mar. 2014.
[118] S. Gil-Gallegos, R. Klages, J. Solanpää, and E. Räsänen. Energy-dependent diffusion in a soft periodic
Lorentz gas. Eur. Phys. J. Spec. Top., 228(1):143–160, May 2019.
[119] V. Giovannetti, S. Lloyd, and L. Maccone. Quantum Metrology. Phys. Rev. Lett., 96(1):010401, Jan.
2006.
[120] V. Giovannetti, S. Lloyd, and L. Maccone. Advances in quantum metrology. Nature Photon, 5(4):222–
229, Apr. 2011.
[121] N. Gisin and R. Thew. Quantum communication. Nature Photon, 1(3):165–171, Mar. 2007.
[122] A. Goban, R. B. Hutson, G. E. Marti, S. L. Campbell, M. A. Perlin, P. S. Julienne, J. P. D’Incao,
A. M. Rey, and J. Ye. Emergence of multi-body interactions in a fermionic lattice clock. Nature,563(7731):369–373, Nov. 2018.
[123] N. Goldman, G. Juzeliūnas, P. Öhberg, and I. B. Spielman. Light-induced gauge fields for ultracold
atoms. Rep. Prog. Phys., 77(12):126401, Nov. 2014.

254[124] D. González-Cuadra, D. Bluvstein, M. Kalinowski, R. Kaubruegger, N. Maskara, P. Naldesi, T. V.
Zache, A. M. Kaufman, M. D. Lukin, H. Pichler, B. Vermersch, J. Ye, and P. Zoller. Fermionic
quantum processing with programmable neutral atom arrays, Mar. 2023.
[125] J. Goodman. Introduction to Fourier Optics. Roberts and Company Publishers, Englewood, Colo, 3rdedition edition, Dec. 2004.
[126] A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S. Julienne, J. Ye, P. Zoller, E. Demler, M. D.
Lukin, and A. M. Rey. Two-orbital SU(N) magnetism with ultracold alkaline-earth atoms. Nature
Physics, 6(4):289–295, Apr. 2010.
[127] T. M. Graham, M. Kwon, B. Grinkemeyer, Z. Marra, X. Jiang, M. T. Lichtman, Y. Sun, M. Ebert, and
M. Saffman. Rydberg-mediated entanglement in a two-dimensional neutral atom qubit array. Phys.
Rev. Lett., 123(23):230501, Dec. 2019.
[128] T. M. Graham, Y. Song, J. Scott, C. Poole, L. Phuttitarn, K. Jooya, P. Eichler, X. Jiang, A. Marra,
B. Grinkemeyer, M. Kwon, M. Ebert, J. Cherek, M. T. Lichtman, M. Gillette, J. Gilbert, D. Bowman, T. Ballance, C. Campbell, E. D. Dahl, O. Crawford, N. S. Blunt, B. Rogers, T. Noel, and
M. Saffman. Multi-qubit entanglement and algorithms on a neutral-atom quantum computer. Nature,604(7906):457–462, Apr. 2022.
[129] M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch. Quantum phase transition from asuperfluid to a Mott insulator in a gas of ultracold atoms. Nature, 415(6867):39–44, Jan. 2002.
[130] C. Gross and I. Bloch. Quantum simulations with ultracold atoms in optical lattices.
357(6355):995–1001, Sept. 2017.

Science,[131] J. Grotti, S. Koller, S. Vogt, S. Häfner, U. Sterr, C. Lisdat, H. Denker, C. Voigt, L. Timmen, A. Rolland,
F. N. Baynes, H. S. Margolis, M. Zampaolo, P. Thoumany, M. Pizzocaro, B. Rauf, F. Bregolin,
A. Tampellini, P. Barbieri, M. Zucco, G. A. Costanzo, C. Clivati, F. Levi, and D. Calonico. Geodesy
and metrology with a transportable optical clock. Nature Phys, 14(5):437–441, May 2018.
[132] L. K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of theTwenty-Eighth Annual ACM Symposium on Theory of Computing, STOC ’96, pages 212–219, New
York, NY, USA, July 1996. Association for Computing Machinery.
[133] L. K. Grover and A. M. Sengupta. From coupled pendulums to quantum search. In Mathematics ofQuantum Computation. Chapman and Hall/CRC, 2002.
[134] T. Grünzweig, A. Hilliard, M. McGovern, and M. F. Andersen. Near-deterministic preparation of asingle atom in an optical microtrap. Nature Phys, 6(12):951–954, Dec. 2010.
[135] E. Guardado-Sanchez, B. M. Spar, P. Schauss, R. Belyansky, J. T. Young, P. Bienias, A. V. Gorshkov,
T. Iadecola, and W. S. Bakr. Quench Dynamics of a Fermi Gas with Strong Nonlocal Interactions.
Phys. Rev. X, 11(2):021036, May 2021.
[136] J. L. Hall, M. S. Taubman, and J. Ye. Laser stabilization. In Handbook of Optics, volume II of
Handbook of Optics. McGraw-Hill Professional, New York, NY, 3 edition, 2009.
[137] C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex. Gaussian boson
sampling. Phys. Rev. Lett., 119(17):170501, Oct. 2017.
[138] A. Heinz, A. J. Park, N. Šantić, J. Trautmann, S. G. Porsev, M. S. Safronova, I. Bloch, and S. Blatt.
State-Dependent Optical Lattices for the Strontium Optical Qubit. Phys. Rev. Lett., 124(20):203201,
May 2020.
[139] I. Hen and M. Rigol. Superfluid to Mott insulator transition of hardcore bosons in a superlattice.
Phys. Rev. B, 80(13):134508, Oct. 2009.

255[140] M. Hermele, V. Gurarie, and A. M. Rey. Mott Insulators of Ultracold Fermionic Alkaline Earth Atoms:
Underconstrained Magnetism and Chiral Spin Liquid. Physical Review Letters, 103(13), Sept. 2009.
[141] J. A. Hines, S. V. Rajagopal, G. L. Moreau, M. D. Wahrman, N. A. Lewis, O. Marković, and
M. Schleier-Smith. Spin Squeezing by Rydberg Dressing in an Array of Atomic Ensembles. Phys.
Rev. Lett., 131(6):063401, Aug. 2023.
[142] T. O. Höhn, E. Staub, G. Brochier, N. D. Oppong, and M. Aidelsburger. State-dependent potentials
for the 1 S0 and 3 P0 clock states of neutral ytterbium atoms, May 2023.
[143] C. K. Hong, Z. Y. Ou, and L. Mandel. Measurement of subpicosecond time intervals between two
photons by interference. Phys. Rev. Lett., 59(18):2044–2046, Nov. 1987.
[144] J. Hubbard. Electron Correlations in Narrow Energy Bands. Proceedings of the Royal Society of
London. Series A, Mathematical and Physical Sciences, 276(1365):238–257, 1963.
[145] S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac. Improvementof Frequency Standards with Quantum Entanglement. Phys. Rev. Lett., 79(20):3865–3868, Nov. 1997.
[146] W. Huie, L. Li, N. Chen, X. Hu, Z. Jia, W. K. C. Sun, and J. P. Covey. Repetitive Readout andReal-Time Control of Nuclear Spin Qubits in 171 Yb Atoms. PRX Quantum, 4(3):030337, Sept. 2023.
[147] D. B. Hume and D. R. Leibrandt. Probing beyond the laser coherence time in optical clock comparisons.
Phys. Rev. A, 93(3):032138, Mar. 2016.
[148] R. B. Hutson, A. Goban, G. E. Marti, and J. Ye. Engineering quantum states of matter for atomic
clocks in shallow optical lattices. Phys. Rev. Lett., 123(12):123401, Sept. 2019.
[149] A. Impertro, J. F. Wienand, S. Häfele, H. von Raven, S. Hubele, T. Klostermann, C. R. Cabrera,
I. Bloch, and M. Aidelsburger. An unsupervised deep learning algorithm for single-site reconstructionin quantum gas microscopes. Commun Phys, 6(1):1–8, July 2023.
[150] L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and
M. Saffman. Demonstration of a Neutral Atom Controlled-NOT Quantum Gate. Phys. Rev. Lett.,
104(1):010503, Jan. 2010.
[151] R. Islam, R. Ma, P. M. Preiss, M. Eric Tai, A. Lukin, M. Rispoli, and M. Greiner. Measuring
entanglement entropy in a quantum many-body system. Nature, 528(7580):77–83, Dec. 2015.
[152] S. Jandura and G. Pupillo. Time-Optimal Two- and Three-Qubit Gates for Rydberg Atoms. Quantum,
6:712, May 2022.
[153] A. Janssen, S. Van Haver, P. Dirksen, and J. Braat. Zernike representation and Strehl ratio of opticalsystems with variable numerical aperture. Journal of Modern Optics, 55(7):1127–1157, Apr. 2008.
[154] R. Jáuregui. Nonperturbative and perturbative treatments of parametric heating in atom traps. Phys.
Rev. A, 64(5):053408, Oct. 2001.
[155] A. Jenkins, J. W. Lis, A. Senoo, W. F. McGrew, and A. M. Kaufman. Ytterbium nuclear-spin qubits
in an optical tweezer array. Phys. Rev. X, 12(2):021027, May 2022.
[156] M. Jreissaty, J. Carrasquilla, F. A. Wolf, and M. Rigol. Expansion of Bose-Hubbard Mott insulatorsin optical lattices. Phys. Rev. A, 84(4):043610, Oct. 2011.
[157] Yu. Kagan, B. V. Svistunov, and G. V. Shlyapnikov. Effect of Bose condensation on inelastic processes
in gases. Soviet Journal of Experimental and Theoretical Physics Letters, 42:209, Aug. 1985.
[158] M. Karski, L. Forster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera. Quantum Walkin Position Space with Single Optically Trapped Atoms. Science, 325(5937):174–177, July 2009.

256[159] H. Katori, V. D. Ovsiannikov, S. I. Marmo, and V. G. Palchikov. Strategies for reducing the light shift
in atomic clocks. Phys. Rev. A, 91(5):052503, May 2015.
[160] R. Kaubruegger, P. Silvi, C. Kokail, R. van Bijnen, A. M. Rey, J. Ye, A. M. Kaufman, andP. Zoller. Variational spin-squeezing algorithms on programmable quantum sensors. Phys. Rev. Lett.,
123(26):260505, Aug. 2019.
[161] R. Kaubruegger, D. V. Vasilyev, M. Schulte, K. Hammerer, and P. Zoller. Quantum Variational
Optimization of Ramsey Interferometry and Atomic Clocks. Phys. Rev. X, 11(4):041045, Dec. 2021.
[162] A. Kaufman. Laser Cooling Atoms to Indistinguishability: Atomic Hong-Ou-Mandel Interference andEntanglement through Spin Exchange. PhD thesis, 2015.
[163] A. M. Kaufman, B. J. Lester, M. Foss-Feig, M. L. Wall, A. M. Rey, and C. A. Regal. Entangling two
transportable neutral atoms via local spin exchange. Nature, 527(7577):208–211, Nov. 2015.
[164] A. M. Kaufman, B. J. Lester, and C. A. Regal. Cooling a single atom in an optical tweezer to its
quantum ground state. Phys. Rev. X, 2(4):041014, Nov. 2012.
[165] A. M. Kaufman, B. J. Lester, C. M. Reynolds, M. L. Wall, M. Foss-Feig, K. R. A. Hazzard, A. M.
Rey, and C. A. Regal. Two-particle quantum interference in tunnel-coupled optical tweezers. Science,
345(6194):306–309, July 2014.
[166] A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner. Quantum
thermalization through entanglement in an isolated many-body system. Science, 353(6301):794–800,
Aug. 2016.
[167] D. Kedar. A Fully Crystalline Cryogenic Reference Cavity. PhD thesis, 2023.
[168] J. Kempe, A. Kitaev, and O. Regev. The Complexity of the Local Hamiltonian Problem. In
K. Lodaya and M. Mahajan, editors, FSTTCS 2004: Foundations of Software Technology andTheoretical Computer Science, Lecture Notes in Computer Science, pages 372–383, Berlin, Heidelberg,
2005. Springer.
[169] C. J. Kennedy, E. Oelker, J. M. Robinson, T. Bothwell, D. Kedar, W. R. Milner, G. E. Marti, A. Derevianko, and J. Ye. Precision Metrology Meets Cosmology: Improved Constraints on Ultralight Dark
Matter from Atom-Cavity Frequency Comparisons. Phys. Rev. Lett., 125(20):201302, Nov. 2020.
[170] E. M. Kessler, P. Kómár, M. Bishof, L. Jiang, A. S. Sørensen, J. Ye, and M. D. Lukin. Heisenberglimited atom clocks based on entangled qubits. Phys. Rev. Lett., 112(19):190403, May 2014.
[171] E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin. Quantum Error Correction for Metrology. Phys. Rev. Lett., 112(15):150802, Apr. 2014.
[172] S. Keßler and F. Marquardt. Single-site-resolved measurement of the current statistics in opticallattices. Phys. Rev. A, 89(6):061601, June 2014.
[173] M. E. Kim, W. F. McGrew, N. V. Nardelli, E. R. Clements, Y. S. Hassan, X. Zhang, J. L. Valencia,
H. Leopardi, D. B. Hume, T. M. Fortier, A. D. Ludlow, and D. R. Leibrandt. Improved interspecies
optical clock comparisons through differential spectroscopy. Nat. Phys., 19(1):25–29, Jan. 2023.
[174] M. Kitagawa and M. Ueda. Squeezed spin states. Phys. Rev. A, 47(6):5138–5143, June 1993.
[175] T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler. Exploring topological phases with quantum
walks. Phys. Rev. A, 82(3):033429, Sept. 2010.
[176] E. Knill, R. Laflamme, and G. J. Milburn. A scheme for efficient quantum computation with linear
optics. Nature, 409(6816):46–52, Jan. 2001.

257[177] E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland. Randomized benchmarking of quantum gates. Phys. Rev. A, 77(1):012307,
Jan. 2008.
[178] S. Kolkowitz, S. L. Bromley, T. Bothwell, M. L. Wall, G. E. Marti, A. P. Koller, X. Zhang, A. M. Rey,and J. Ye. Spin-orbit-coupled fermions in an optical lattice clock. Nature, 542(7639):66–70, Feb. 2017.
[179] A. Kumar, T.-Y. Wu, F. Giraldo, and D. S. Weiss. Sorting ultracold atoms in a three-dimensional
optical lattice in a realization of Maxwell’s demon. Nature, 561(7721):83–87, Sept. 2018.
[180] A. Laing and J. L. O’Brien. Super-stable tomography of any linear optical device, Aug. 2012.
[181] D. Layden and P. Cappellaro. Spatial noise filtering through error correction for quantum sensing. npj
Quantum Inf, 4(1):1–6, July 2018.
[182] D. Layden, S. Zhou, P. Cappellaro, and L. Jiang. Ancilla-free quantum error correction codes forquantum metrology. Phys. Rev. Lett., 122(4):040502, Jan. 2019.
[183] F. Le Kien, P. Schneeweiss, and A. Rauschenbeutel. Dynamical polarizability of atoms in arbitrary
light fields: General theory and application to cesium. Eur. Phys. J. D, 67(5):92, May 2013.
[184] P. A. Lee, N. Nagaosa, and X.-G. Wen. Doping a Mott insulator: Physics of high-temperature superconductivity. Rev. Mod. Phys., 78(1):17–85, Jan. 2006.
[185] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland. Quantum dynamics of single trapped ions.
Reviews of Modern Physics, 75(1):281–324, Mar. 2003.
[186] D. R. Leibrandt and J. Heidecker. An open source digital servo for AMO physics experiments.
arXiv:1508.06319 [physics], Nov. 2015.
[187] P. Lemonde and P. Wolf. Optical lattice clock with atoms confined in a shallow trap. Physical Review
A, 72(3), Sept. 2005.
[188] I. D. Leroux, M. H. Schleier-Smith, and V. Vuletić. Implementation of Cavity Squeezing of a Collective
Atomic Spin. Phys. Rev. Lett., 104(7):073602, Feb. 2010.
[189] B. J. Lester, N. Luick, A. M. Kaufman, C. M. Reynolds, and C. A. Regal. Rapid production of
uniformly filled arrays of neutral atoms. Phys. Rev. Lett., 115(7):073003, Aug. 2015.
[190] H. Levine, A. Keesling, A. Omran, H. Bernien, S. Schwartz, A. S. Zibrov, M. Endres, M. Greiner,
V. Vuletić, and M. D. Lukin. High-fidelity control and entanglement of Rydberg-atom qubits. Phys.
Rev. Lett., 121(12):123603, Sept. 2018.
[191] H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T. Wang, S. Ebadi, H. Bernien, M. Greiner,
V. Vuletić, H. Pichler, and M. D. Lukin. Parallel implementation of high-fidelity multiqubit gates withneutral atoms. Phys. Rev. Lett., 123(17):170503, Oct. 2019.
[192] J.-R. Li, K. Matsuda, C. Miller, A. N. Carroll, W. G. Tobias, J. S. Higgins, and J. Ye. Tunable itinerant
spin dynamics with polar molecules. Nature, 614(7946):70–74, Feb. 2023.
[193] F. Lin, E. S. Sørensen, and D. M. Ceperley. The superfluid-insulator transition in the disordered
two-dimensional Bose-Hubbard model. Phys. Rev. B, 84(9):094507, Sept. 2011.
[194] J. W. Lis, A. Senoo, W. F. McGrew, F. Rönchen, A. Jenkins, and A. M. Kaufman. Mid-circuit
operations using the omg-architecture in neutral atom arrays, May 2023.
[195] S. Lischke, A. Peczek, J. S. Morgan, K. Sun, D. Steckler, Y. Yamamoto, F. Korndörfer, C. Mai,
S. Marschmeyer, M. Fraschke, A. Krüger, A. Beling, and L. Zimmermann. Ultra-fast germanium
photodiode with 3-dB bandwidth of 265 GHz. Nat. Photon., 15(12):925–931, Dec. 2021.

258[196] Ch. Lisdat, J. S. R. V. Winfred, T. Middelmann, F. Riehle, and U. Sterr. Collisional Losses, Decoherence, and Frequency Shifts in Optical Lattice Clocks with Bosons. Phys. Rev. Lett., 103(9):090801,
Aug. 2009.
[197] J. Liu, H. Yuan, X.-M. Lu, and X. Wang. Quantum Fisher information matrix and multiparameter
estimation. J. Phys. A: Math. Theor., 53(2):023001, Jan. 2020.
[198] L. R. Liu, J. D. Hood, Y. Yu, J. T. Zhang, K. Wang, Y.-W. Lin, T. Rosenband, and K.-K. Ni. Molecularassembly of ground-state cooled single atoms. Phys. Rev. X, 9(2):021039, May 2019.
[199] L. F. Livi, G. Cappellini, M. Diem, L. Franchi, C. Clivati, M. Frittelli, F. Levi, D. Calonico, J. Catani,
M. Inguscio, and L. Fallani. Synthetic Dimensions and Spin-Orbit Coupling with an Optical ClockTransition. Phys. Rev. Lett., 117(22):220401, Nov. 2016.
[200] G. Lochead, D. Boddy, D. P. Sadler, C. S. Adams, and M. P. A. Jones. Number-resolved imaging
of excited-state atoms using a scanning autoionization microscope. Phys. Rev. A, 87(5):053409, May
2013.
[201] R. Lopes, A. Imanaliev, A. Aspect, M. Cheneau, D. Boiron, and C. I. Westbrook.
Hong–Ou–Mandel experiment. Nature, 520(7545):66–68, Apr. 2015.

Atomic[202] J. C. Loredo, M. A. Broome, P. Hilaire, O. Gazzano, I. Sagnes, A. Lemaitre, M. P. Almeida, P. Senellart,
and A. G. White. Boson sampling with single-photon Fock states from a bright solid-state source. Phys.
Rev. Lett., 118(13):130503, Mar. 2017.
[203] A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt. Optical atomic clocks. Rev. Mod.
Phys., 87(2):637–701, June 2015.
[204] A. P. Lund, A. Laing, S. Rahimi-Keshari, T. Rudolph, J. L. O’Brien, and T. C. Ralph. Boson samplingfrom a Gaussian state. Phys. Rev. Lett., 113(10):100502, Sept. 2014.
[205] L.-S. Ma, P. Jungner, J. Ye, and J. L. Hall. Delivering the same optical frequency at two places:
Accurate cancellation of phase noise introduced by an optical fiber or other time-varying path. Opt.
Lett., 19(21):1777, Nov. 1994.
[206] S. Ma, A. P. Burgers, G. Liu, J. Wilson, B. Zhang, and J. D. Thompson. Universal Gate Operations
on Nuclear Spin Qubits in an Optical Tweezer Array of 171 Yb Atoms. Phys. Rev. X, 12(2):021028,
May 2022.
[207] S. Ma, G. Liu, P. Peng, B. Zhang, S. Jandura, J. Claes, A. P. Burgers, G. Pupillo, S. Puri, and J. D.
Thompson. High-fidelity gates with mid-circuit erasure conversion in a metastable neutral atom qubit,
May 2023.
[208] I. S. Madjarov. Entangling, Controlling, and Detecting Individual Strontium Atoms in Optical Tweezer
Arrays. PhD thesis, 2021.
[209] I. S. Madjarov, A. Cooper, A. L. Shaw, J. P. Covey, V. Schkolnik, T. H. Yoon, J. R. Williams, and
M. Endres. An atomic-array optical clock with single-atom readout. Phys. Rev. X, 9(4):041052, Dec.
2019.
[210] I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R.
Williams, and M. Endres. High-fidelity entanglement and detection of alkaline-earth Rydberg atoms.
Nature Physics, pages 1–5, May 2020.
[211] L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Rortais, T. Vincent, J. F. F. Bulmer, F. M. Miatto,
L. Neuhaus, L. G. Helt, M. J. Collins, A. E. Lita, T. Gerrits, S. W. Nam, V. D. Vaidya, M. Menotti,
I. Dhand, Z. Vernon, N. Quesada, and J. Lavoie. Quantum computational advantage with a programmable photonic processor. Nature, 606(7912):75–81, June 2022.

259[212] M. Mamaev, T. Bilitewski, B. Sundar, and A. M. Rey. Resonant Dynamics of Strongly Interacting
SU(n) Fermionic Atoms in a Synthetic Flux Ladder. PRX Quantum, 3(3):030328, Aug. 2022.
[213] G. E. Marti, R. B. Hutson, A. Goban, S. L. Campbell, N. Poli, and J. Ye. Imaging optical frequencies
with 100µHz precision and 1.1µm resolution. Phys. Rev. Lett., 120(10):103201, Mar. 2018.
[214] M. J. Martin.
Quantum Metrology and Many-Body Physics:
Optical Lattice Clock. PhD thesis, 2013.

Pushing the Frontier of the[215] M. J. Martin, M. Bishof, M. D. Swallows, X. Zhang, C. Benko, J. von-Stecher, A. V. Gorshkov, A. M.
Rey, and J. Ye. A quantum many-body spin system in an optical lattice clock. Science, 341(6146):632–
636, Aug. 2013.
[216] Y. N. Martinez de Escobar, P. G. Mickelson, P. Pellegrini, S. B. Nagel, A. Traverso, M. Yan, R. Côté,
and T. C. Killian. Two-photon photoassociative spectroscopy of ultracold 88 Sr. Phys. Rev. A,
78(6):062708, Dec. 2008.
[217] M. Martinez-Dorantes, W. Alt, J. Gallego, S. Ghosh, L. Ratschbacher, and D. Meschede. Statedependent fluorescence of neutral atoms in optical potentials. Phys. Rev. A, 97(2):023410, Feb. 2018.
[218] D. G. Matei, T. Legero, S. Häfner, C. Grebing, R. Weyrich, W. Zhang, L. Sonderhouse, J. M. Robinson,
J. Ye, F. Riehle, and U. Sterr. 1.5 µm lasers with sub 10 mHz linewidth. Physical Review Letters,
118(26), June 2017.
[219] J. C. F. Matthews, K. Poulios, J. D. A. Meinecke, A. Politi, A. Peruzzo, N. Ismail, K. Wörhoff, M. G.
Thompson, and J. L. O’Brien. Observing fermionic statistics with photons in arbitrary processes. Sci
Rep, 3(1):1539, Dec. 2013.
[220] W. F. McGrew, X. Zhang, R. J. Fasano, S. A. Schäffer, K. Beloy, D. Nicolodi, R. C. Brown, N. Hinkley,
G. Milani, M. Schioppo, T. H. Yoon, and A. D. Ludlow. Atomic clock performance enabling geodesy
below the centimetre level. Nature, 564(7734):87–90, Dec. 2018.
[221] S. T. Merkel, J. M. Gambetta, J. A. Smolin, S. Poletto, A. D. Córcoles, B. R. Johnson, C. A. Ryan,
and M. Steffen. Self-consistent quantum process tomography. Phys. Rev. A, 87(6):062119, June 2013.
[222] T. Middelmann, S. Falke, C. Lisdat, and U. Sterr. High accuracy correction of blackbody radiation
shift in an optical lattice clock. Phys. Rev. Lett., 109(26):263004, Dec. 2012.
[223] A. Mitra, M. J. Martin, G. W. Biedermann, A. M. Marino, P. M. Poggi, and I. H. Deutsch. Robust Mølmer-Sørensen gate for neutral atoms using rapid adiabatic Rydberg dressing. Phys. Rev. A,
101(3):030301, Mar. 2020.
[224] J. Mitroy and J. Y. Zhang. Dispersion and polarization interactions of the strontium atom. Molecular
Physics, 108(15):1999–2006, Aug. 2010.
[225] H. Miyake, N. C. Pisenti, P. K. Elgee, A. Sitaram, and G. K. Campbell. Isotope-shift spectroscopy of
the S 0 1 → P 1 3 and S 0 1 → P 0 3 transitions in strontium. Phys. Rev. Research, 1(3):033113, Nov.
2019.
[226] K. Mølmer, Y. Castin, and J. Dalibard. Monte Carlo wave-function method in quantum optics. J.
Opt. Soc. Am. B, JOSAB, 10(3):524–538, Mar. 1993.
[227] M. Mullan. Quantum Algorithms for Atomic Clocks. PhD thesis, 2013.
[228] M. Mullan and E. Knill. Optimizing passive quantum clocks. Phys. Rev. A, 90(4):042310, Oct. 2014.
[229] J. A. Muniz, M. A. Norcia, J. R. K. Cline, and J. K. Thompson. A Robust Narrow-Line MagnetoOptical Trap using Adiabatic Transfer. arXiv:1806.00838 [physics, physics:quant-ph], June 2018.

260[230] G. Muraleedharan, A. Miyake, and I. H. Deutsch. Quantum computational supremacy in the samplingof bosonic random walkers on a one-dimensional lattice. New J. Phys., 21(5):055003, May 2019.
[231] S. Murmann, A. Bergschneider, V. M. Klinkhamer, G. Zürn, T. Lompe, and S. Jochim. Two fermionsin a double well: Exploring a fundamental building block of the Hubbard model. Physical Review
Letters, 114(8):080402, Feb. 2015.
[232] L. Neuhaus, R. Metzdorff, S. Chua, T. Jacqmin, T. Briant, A. Heidmann, P.-F. Cohadon,and S. Deléglise. PyRPL (Python Red Pitaya Lockbox) — An open-source software package for FPGA-controlled quantum optics experiments.
In 2017 Conference on Lasers andElectro-Optics Europe European Quantum Electronics Conference (CLEO/Europe-EQEC), pages 1–
1, June 2017.
[233] B. C. Nichol, R. Srinivas, D. P. Nadlinger, P. Drmota, D. Main, G. Araneda, C. J. Ballance, and D. M.
Lucas. A quantum network of entangled optical atomic clocks. Nature, 609(7928):689–694, Sept. 2022.
[234] T. Nicholson, S. Campbell, R. Hutson, G. Marti, B. Bloom, R. McNally, W. Zhang, M. Barrett,M. Safronova, G. Strouse, W. Tew, and J. Ye. Systematic evaluation of an atomic clock at 2 × 10−18
total uncertainty. Nat Commun, 6(1):6896, Nov. 2015.
[235] T. L. Nicholson, S. Blatt, B. J. Bloom, J. R. Williams, J. W. Thomsen, J. Ye, and P. S. Julienne.
Optical Feshbach resonances: Field-dressed theory and comparison with experiments. Phys. Rev. A,
92(2):022709, Aug. 2015.
[236] M. A. Norcia. New Tools for Precision Measurement and Quantum Science with Narrow Linewidth
Optical Transitions. PhD thesis, 2017.
[237] M. A. Norcia, W. B. Cairncross, K. Barnes, P. Battaglino, A. Brown, M. O. Brown, K. Cassella, C.-A.
Chen, R. Coxe, D. Crow, J. Epstein, C. Griger, A. M. W. Jones, H. Kim, J. M. Kindem, J. King, S. S.
Kondov, K. Kotru, J. Lauigan, M. Li, M. Lu, E. Megidish, J. Marjanovic, M. McDonald, T. Mittiga,
J. A. Muniz, S. Narayanaswami, C. Nishiguchi, R. Notermans, T. Paule, K. Pawlak, L. Peng, A. Ryou,
A. Smull, D. Stack, M. Stone, A. Sucich, M. Urbanek, R. van de Veerdonk, Z. Vendeiro, T. Wilkason,
T.-Y. Wu, X. Xie, X. Zhang, and B. J. Bloom. Mid-circuit qubit measurement and rearrangement ina 171 Yb atomic array, June 2023.
[238] M. A. Norcia, J. R. K. Cline, J. P. Bartolotta, M. J. Holland, and J. K. Thompson. Narrow-line Laser
Cooling by Adiabatic Transfer. New Journal of Physics, 20(2):023021, Feb. 2018.
[239] M. A. Norcia and J. K. Thompson. Simple Laser Stabilization to the Strontium 88 Sr Transition at 707
nm. Review of Scientific Instruments, 87(2):023110, Feb. 2016.
[240] M. A. Norcia, A. W. Young, W. J. Eckner, E. Oelker, J. Ye, and A. M. Kaufman. Seconds-scalecoherence on an optical clock transition in a tweezer array. Science, 366(6461):93–97, 2019.
[241] M. A. Norcia, A. W. Young, and A. M. Kaufman. Microscopic control and detection of ultracold
strontium in optical-tweezer arrays. Phys. Rev. X, 8(4):041054, Dec. 2018.
[242] E. Oelker, R. B. Hutson, C. J. Kennedy, L. Sonderhouse, T. Bothwell, A. Goban, D. Kedar, C. Sanner,
J. M. Robinson, G. E. Marti, D. G. Matei, T. Legero, M. Giunta, R. Holzwarth, F. Riehle, U. Sterr,and J. Ye. Demonstration of 4.8 × 10−17 stability at 1 s for two independent optical clocks. Nature
Photonics, 13(10):714–719, Oct. 2019.
[243] C. Oh, Y. Lim, B. Fefferman, and L. Jiang. Classical simulation of boson sampling based on graphstructure. Phys. Rev. Lett., 128(19):190501, May 2022.
[244] C. Oh, M. Liu, Y. Alexeev, B. Fefferman, and L. Jiang. Tensor network algorithm for simulating
experimental Gaussian boson sampling, June 2023.
[245] N. D. Oppong. Towards a Degenerate Fermi Gas of Strontium in an Optical Lattice. PhD thesis, 2015.

261[246] T. Ðorđević, P. Samutpraphoot, P. L. Ocola, H. Bernien, B. Grinkemeyer, I. Dimitrova, V. Vuletić,
and M. D. Lukin. Entanglement transport and a nanophotonic interface for atoms in optical tweezers.
Science, 373(6562):1511–1514, Sept. 2021.
[247] S. Origlia, M. S. Pramod, S. Schiller, Y. Singh, K. Bongs, R. Schwarz, A. Al-Masoudi, S. Dörscher,
S. Herbers, S. Häfner, U. Sterr, and Ch. Lisdat. Towards an optical clock for space: Compact, highperformance optical lattice clock based on bosonic atoms. Phys. Rev. A, 98(5):053443, Nov. 2018.
[248] V. D. Ovsiannikov, V. G. Pal’chikov, A. V. Taichenachev, V. I. Yudin, and H. Katori. Multipole,nonlinear, and anharmonic uncertainties of clocks of Sr atoms in an optical lattice. Phys. Rev. A,
88(1):013405, July 2013.
[249] J. O. Owens, M. A. Broome, D. N. Biggerstaff, M. E. Goggin, A. Fedrizzi, T. Linjordet, M. Ams, G. D.
Marshall, J. Twamley, M. J. Withford, and A. G. White. Two-photon quantum walks in an ellipticaldirect-write waveguide array. New J. Phys., 13(7):075003, July 2011.
[250] S. Paesani, Y. Ding, R. Santagati, L. Chakhmakhchyan, C. Vigliar, K. Rottwitt, L. K. Oxenløwe,
J. Wang, M. G. Thompson, and A. Laing. Generation and sampling of quantum states of light in asilicon chip. Nat. Phys., 15(9):925–929, Sept. 2019.
[251] F. A. Palm, J. Kwan, B. Bakkali-Hassani, M. Greiner, U. Schollwöck, N. Goldman, and F. Grusdt.
Growing Extended Laughlin States in a Quantum Gas Microscope: A Patchwork Construction, Sept.
2023.
[252] P. B. Patel, Z. Yan, B. Mukherjee, R. J. Fletcher, J. Struck, and M. W. Zwierlein. Universal sounddiffusion in a strongly interacting Fermi gas. Science, 370(6521):1222–1226, Dec. 2020.
[253] N. K. Pavlis and M. A. Weiss. A re-evaluation of the relativistic redshift on frequency standards at
NIST, Boulder, Colorado, USA. Metrologia, 54(4):535–548, Aug. 2017.
[254] E. Pedrozo-Peñafiel, S. Colombo, C. Shu, A. F. Adiyatullin, Z. Li, E. Mendez, B. Braverman,
A. Kawasaki, D. Akamatsu, Y. Xiao, and V. Vuletić. Entanglement on an optical atomic-clock transition. Nature, 588(7838):414–418, Dec. 2020.
[255] A. W.-C. Peng. Quantum Gas Microscope With Optical Lattice. PhD thesis, May 2010.
[256] A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini,
N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien. Quantum
walks of correlated photons. Science, 329(5998):1500–1503, Sept. 2010.
[257] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L.
O’Brien. A variational eigenvalue solver on a photonic quantum processor. Nat Commun, 5(1):4213,
July 2014.
[258] A. Petrescu and K. L. Hur. Chiral Mott Insulators, Meissner Effect, and Laughlin States in Quantum
Ladders. Phys. Rev. B, 91(5):054520, Feb. 2015.
[259] L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein. Quantum metrology with
nonclassical states of atomic ensembles. Rev. Mod. Phys., 90(3):035005, Sept. 2018.
[260] I. Pikovski, M. Zych, F. Costa, and Č. Brukner. Universal decoherence due to gravitational time
dilation. Nature Physics, 11(8):668–672, Aug. 2015.
[261] I. Pikovski, M. Zych, F. Costa, and Č. Brukner. Time dilation in quantum systems and decoherence.
New J. Phys., 19(2):025011, Feb. 2017.
[262] L. Pioge, B. Seron, L. Novo, and N. J. Cerf. Enhanced bunching of nearly indistinguishable bosons,Aug. 2023.

262[263] S. G. Porsev, M. S. Safronova, U. I. Safronova, and M. G. Kozlov. Multipolar Polarizabilities andHyperpolarizabilities in the Sr Optical Lattice Clock. Phys. Rev. Lett., 120(6):063204, Feb. 2018.
[264] P. M. Preiss, R. Ma, M. E. Tai, A. Lukin, M. Rispoli, P. Zupancic, Y. Lahini, R. Islam, and M. Greiner.
Strongly correlated quantum walks in optical lattices. Science, 347(6227):1229–1233, Mar. 2015.
[265] J. Preskill. Quantum Information Notes, 2015.
[266] W. Purwanto and S. Zhang. Quantum Monte Carlo method for the ground state of many-boson
systems. Phys. Rev. E, 70(5):056702, Nov. 2004.
[267] X.-L. Qi and S.-C. Zhang. Topological insulators and superconductors. Rev. Mod. Phys., 83(4):1057–
1110, Oct. 2011.
[268] R. Raussendorf, D. E. Browne, and H. J. Briegel. Measurement-based quantum computation withcluster states. Physical Review A, 68(2), Aug. 2003.
[269] S. Ravets, H. Labuhn, D. Barredo, L. Béguin, T. Lahaye, and A. Browaeys. Coherent dipole–dipole
coupling between two single Rydberg atoms at an electrically-tuned Förster resonance. Nature Phys,
10(12):914–917, Dec. 2014.
[270] M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani. Experimental realization of any discrete unitary
operator. Phys. Rev. Lett., 73(1):58–61, July 1994.
[271] I. Reichenbach and I. H. Deutsch. Sideband cooling while preserving coherences in the nuclear spinstate in group-II-like atoms. Phys. Rev. Lett., 99(12):123001, Sept. 2007.
[272] D. Reitzner, D. Nagaj, and V. Bužek. Quantum Walks. Acta Phys. Slovaca., 61(6), Dec. 2011.
[273] F. Riehle. Frequency Standards: Basics and Applications. Wiley, 1 edition, Sept. 2003.
[274] M. Rigol and A. Muramatsu. Emergence of Quasicondensates of Hard-Core Bosons at Finite Momentum. Phys. Rev. Lett., 93(23):230404, Dec. 2004.
[275] W. J. Riley. Handbook of Frequency Stability Analysis. 2008.
[276] C. Robens, I. Arrazola, W. Alt, D. Meschede, L. Lamata, E. Solano, and A. Alberti. Boson samplingwith ultracold atoms, Aug. 2022.
[277] J. M. Robinson, M. Miklos, Y. M. Tso, C. J. Kennedy, D. Kedar, J. K. Thompson, and J. Ye. Directcomparison of two spin squeezed optical clocks below the quantum projection noise limit, Nov. 2022.
[278] J. S. Rosenberg, L. Christakis, E. Guardado-Sanchez, Z. Z. Yan, and W. S. Bakr. Observation of the
Hanbury Brown–Twiss effect with ultracold molecules. Nat. Phys., 18(9):1062–1066, Sept. 2022.
[279] J. R. Rubbmark and S. A. Borgström. Rydberg Series in Strontium Found in Absorption by Selectively
Laser-Excited Atoms. Phys. Scr., 18(4):196, Oct. 1978.
[280] C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette,
W. M. Itano, D. J. Wineland, and C. Monroe. Experimental entanglement of four particles. Nature,404(6775):256–259, Mar. 2000.
[281] J. J. Sakurai and J. J. Napolitano. Modern Quantum Mechanics. Pearson Education, 2nd edition,
2011.
[282] J. E. Sansonetti. Wavelengths, Transition Probabilities, and Energy Levels for the Spectra of Strontium
Ions (Sr II through Sr XXXVIII ). Journal of Physical and Chemical Reference Data, 41(1):013102–
013102–119, Mar. 2012.

263[283] J. E. Sansonetti and G. Nave. Wavelengths, Transition Probabilities, and Energy Levels for theSpectrum of Neutral Strontium (SrI). Journal of Physical and Chemical Reference Data, 39(3):033103,
Sept. 2010.
[284] L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame. Twoparticle bosonic-fermionic quantum walk via integrated photonics. Phys. Rev. Lett., 108(1):010502,
Jan. 2012.
[285] G. Santarelli, C. Audoin, A. Makdissi, P. Laurent, G. J. Dick, and A. Clairon. Frequency stability
degradation of an oscillator slaved to a periodically interrogated atomic resonator. IEEE Trans Ultrason
Ferroelectr Freq Control, 45(4):887–894, 1998.
[286] S. Saskin, J. T. Wilson, B. Grinkemeyer, and J. D. Thompson. Narrow-line cooling and imaging ofytterbium atoms in an optical tweezer array. Phys. Rev. Lett., 122(14):143002, Apr. 2019.
[287] T. A. Savard, K. M. O’Hara, and J. E. Thomas. Laser-noise-induced heating in far-off resonance opticaltraps. Phys. Rev. A, 56(2):R1095–R1098, Aug. 1997.
[288] F. Scazza, C. Hofrichter, M. Höfer, P. C. De Groot, I. Bloch, and S. Fölling. Observation of two-orbital
spin-exchange interactions with ultracold SU(N)-symmetric fermions. Nature Phys, 10(10):779–784,
Oct. 2014.
[289] S. Scheel. Permanents in linear optical networks. arXiv:quant-ph/0406127, June 2004.
[290] N. Schine, A. W. Young, W. J. Eckner, M. J. Martin, and A. M. Kaufman. Long-lived Bell states inan array of optical clock qubits. Nat. Phys., pages 1–7, Aug. 2022.
[291] M. Schioppo, R. C. Brown, W. F. McGrew, N. Hinkley, R. J. Fasano, K. Beloy, T. H. Yoon, G. Milani,
D. Nicolodi, J. A. Sherman, N. B. Phillips, C. W. Oates, and A. D. Ludlow. Ultrastable optical clockwith two cold-atom ensembles. Nature Photonics, 11(1):48–52, Jan. 2017.
[292] N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier. Sub-poissonian loading of single atoms in a
microscopic dipole trap. Nature, 411(6841):1024–1027, June 2001.
[293] P.-I. Schneider, S. Grishkevich, and A. Saenz. Ab initio determination of Bose-Hubbard parameters
for two ultracold atoms in an optical lattice using a three-well potential. Phys. Rev. A, 80(1):013404,
July 2009.
[294] P. Scholl, M. Schuler, H. J. Williams, A. A. Eberharter, D. Barredo, K.-N. Schymik, V. Lienhard,
L.-P. Henry, T. C. Lang, T. Lahaye, A. M. Läuchli, and A. Browaeys. Quantum simulation of 2D
antiferromagnets with hundreds of Rydberg atoms. Nature, 595(7866):233–238, July 2021.
[295] M. Schroeder. Synthesis of low-peak-factor signals and binary sequences with low autocorrelation(Corresp.). IEEE Transactions on Information Theory, 16(1):85–89, Jan. 1970.
[296] J. Sebby-Strabley, M. Anderlini, P. S. Jessen, and J. V. Porto. Lattice of double wells for manipulating
pairs of cold atoms. Phys. Rev. A, 73(3):033605, Mar. 2006.
[297] G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kalinowski, R. Samajdar, A. Omran, S. Sachdev, A. Vishwanath, M. Greiner, V. Vuletić,
and M. D. Lukin. Probing topological spin liquids on a programmable quantum simulator. Science,
374(6572):1242–1247, Dec. 2021.
[298] B. Seron, L. Novo, and N. J. Cerf. Boson bunching is not maximized by indistinguishable particles.
Nat. Photon., 17(8):702–709, Aug. 2023.
[299] A. Shaji and C. M. Caves. Qubit metrology and decoherence. Phys. Rev. A, 76(3):032111, Sept. 2007.

264[300] J. Shao. Mathematical Statistics. Springer Texts in Statistics. Springer, New York, 2nd ed edition,
2003.
[301] A. L. Shaw, R. Finkelstein, R. B.-S. Tsai, P. Scholl, T. H. Yoon, J. Choi, and M. Endres. Multi-ensemble
metrology by programming local rotations with atom movements, Mar. 2023.
[302] V. Shchesnovich. Distinguishing noisy boson sampling from classical simulations. Quantum, 5:423,
Mar. 2021.
[303] V. S. Shchesnovich. The permanent-on-top conjecture is false. Linear Algebra and its Applications,490:196–201, Feb. 2016.
[304] V. S. Shchesnovich. Universality of generalized bunching and efficient assessment of boson sampling.
Phys. Rev. Lett., 116(12):123601, Mar. 2016.
[305] J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr. Single-atom-resolved
fluorescence imaging of an atomic Mott insulator. Nature, 467(7311):68–72, Sept. 2010.
[306] N. Shettell. Quantum Information Techniques for Quantum Metrology. PhD thesis, Jan. 2022.
[307] C. Shi, J.-L. Robyr, U. Eismann, M. Zawada, L. Lorini, R. Le Targat, and J. Lodewyck. Polarizabilitiesof the 87 Sr clock transition. Phys. Rev. A, 92(1):012516, July 2015.
[308] N. Šibalić, J. D. Pritchard, C. S. Adams, and K. J. Weatherill. ARC: An open-source library forcalculating properties of alkali Rydberg atoms. Computer Physics Communications, 220:319–331,
Nov. 2017.
[309] J. S. Sidhu and P. Kok. Geometric perspective on quantum parameter estimation. AVS Quantum
Science, 2(1):014701, Feb. 2020.
[310] Simons Collaboration on the Many-Electron Problem, M. Qin, C.-M. Chung, H. Shi, E. Vitali, C. Hubig, U. Schollwöck, S. R. White, and S. Zhang. Absence of Superconductivity in the Pure TwoDimensional Hubbard Model. Phys. Rev. X, 10(3):031016, July 2020.
[311] S. Snigirev, A. J. Park, A. Heinz, I. Bloch, and S. Blatt. Fast and dense magneto-optical traps forStrontium. Phys. Rev. A, 99(6):063421, June 2019.
[312] M. Sohmen, M. J. Mark, M. Greiner, and F. Ferlaino. A ship-in-a-bottle quantum gas microscope
setup for magnetic mixtures. SciPost Physics, 15(5):182, Nov. 2023.
[313] L. I. Sonderhouse. Quantum Gas Engineering for Atomic Clocks. PhD thesis, 2020.
[314] F. Sorrentino, G. Ferrari, N. Poli, R. Drullinger, and G. M. Tino. Laser cooling and trapping of atomic
strontium for ultracold atom physics, high-precision spectroscopy and quantum sensors. Mod. Phys.
Lett. B, 20(21):1287–1320, Sept. 2006.
[315] N. Spagnolo, C. Vitelli, L. Sansoni, E. Maiorino, P. Mataloni, F. Sciarrino, D. J. Brod, E. F. Galvão,A. Crespi, R. Ramponi, and R. Osellame. General rules for bosonic bunching in multimode interferometers. Phys. Rev. Lett., 111(13):130503, Sept. 2013.
[316] B. M. Spar, E. Guardado-Sanchez, S. Chi, Z. Z. Yan, and W. S. Bakr. Realization of a Fermi-Hubbard
Optical Tweezer Array. Phys. Rev. Lett., 128(22):223202, June 2022.
[317] J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta,
N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A.
Walmsley. Boson sampling on a photonic chip. Science, 339(6121):798–801, Feb. 2013.
[318] D. A. Steck. Quantum and Atom Optics. 2017.
[319] S. Stellmer. Degenerate Quantum Gases of Strontium. PhD thesis, May 2014.

265[320] S. Stellmer and F. Schreck. Reservoir spectroscopy of 5s5p 3 P 2 –5snd 3 D1,2,3 transitions in strontium.
Phys. Rev. A, 90(2):022512, Aug. 2014.
[321] L. Su, A. Douglas, M. Szurek, R. Groth, S. F. Ozturk, A. Krahn, A. H. Hébert, G. A. Phelps, S. Ebadi,
S. Dickerson, F. Ferlaino, O. Marković, and M. Greiner. Dipolar quantum solids emerging in a Hubbard
quantum simulator. Nature, 622(7984):724–729, Oct. 2023.
[322] A. V. Taichenachev, V. I. Yudin, C. W. Oates, C. W. Hoyt, Z. W. Barber, and L. Hollberg. Magnetic
field-induced spectroscopy of forbidden optical transitions with application to lattice-based opticalatomic clocks. Phys. Rev. Lett., 96(8):083001, Mar. 2006.
[323] J. D. Thompson, T. G. Tiecke, A. S. Zibrov, V. Vuletić, and M. D. Lukin. Coherence and Raman
sideband cooling of a single atom in an optical tweezer. Phys. Rev. Lett., 110(13), Mar. 2013.
[324] W. Tian, W. J. Wee, A. Qu, B. J. M. Lim, P. R. Datla, V. P. W. Koh, and H. Loh. Parallel assemblyof arbitrary defect-free atom arrays with a multitweezer algorithm. Phys. Rev. Appl., 19(3):034048,
Mar. 2023.
[325] M. C. Tichy. Entanglement and Interference of Identical Particles. PhD thesis, Uni. Freiburg, 2011.
[326] M. C. Tichy. Sampling of partially distinguishable bosons and the relation to the multidimensional
permanent. Phys. Rev. A, 91(2):022316, Feb. 2015.
[327] M. C. Tichy, K. Mayer, A. Buchleitner, and K. Mølmer. Stringent and efficient assessment of BosonSampling devices. Phys. Rev. Lett., 113(2):020502, July 2014.
[328] M. C. Tichy, M. Tiersch, F. Mintert, and A. Buchleitner. Many-particle interference beyond manyboson and many-fermion statistics. New J. Phys., 14(9):093015, Sept. 2012.
[329] M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther. Experimental boson
sampling. Nature Photon, 7(7):540–544, July 2013.
[330] J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth,and J. Tennyson. The Variational Quantum Eigensolver: A review of methods and best practices.
Physics Reports, 986:1–128, Nov. 2022.
[331] G. Toth and I. Apellaniz. Quantum metrology from a quantum information science perspective. J.
Phys. A: Math. Theor., 47(42):424006, Oct. 2014.
[332] M. C. Tran, D. K. Mark, W. W. Ho, and S. Choi. Measuring Arbitrary Physical Properties in Analog
Quantum Simulation, Dec. 2022.
[333] J. Trautmann, D. Yankelev, V. Klüsener, A. J. Park, I. Bloch, and S. Blatt. The 1 S0 -3 P2 magnetic
quadrupole transition in neutral strontium, Nov. 2022.
[334] R. Trivedi, A. F. Rubio, and J. I. Cirac. Quantum advantage and stability to errors in analogue
quantum simulators, Dec. 2022.
[335] M. Troyer and U.-J. Wiese. Computational Complexity and Fundamental Limitations to Fermionic
Quantum Monte Carlo Simulations. Phys. Rev. Lett., 94(17):170201, May 2005.
[336] I. Ushijima, M. Takamoto, M. Das, T. Ohkubo, and H. Katori. Cryogenic optical lattice clocks. Nature
Photonics, 9(3):185–189, Mar. 2015.
[337] H. Uys, M. J. Biercuk, A. P. VanDevender, C. Ospelkaus, D. Meiser, R. Ozeri, and J. J. Bollinger.
Decoherence due to Elastic Rayleigh Scattering. Phys. Rev. Lett., 105(20):200401, Nov. 2010.
[338] C. L. Vaillant, M. P. A. Jones, and R. M. Potvliege. Long-range Rydberg–Rydberg interactions incalcium, strontium and ytterbium. J. Phys. B: At. Mol. Opt. Phys., 45(13):135004, July 2012.

266[339] L. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8(2):189–201,
1979.
[340] A. Van Den Bos. A new method for synthesis of low-peak-factor signals. IEEE Transactions on
Acoustics, Speech, and Signal Processing, 35(1):120–122, Jan. 1987.
[341] J. Van Dongen, C. Zhu, D. Clement, G. Dufour, J. L. Booth, and K. W. Madison. Trap-depth
determination from residual gas collisions. Phys. Rev. A, 84(2):022708, Aug. 2011.
[342] R. F. C. Vessot. A review of atomic clock technology, the performance capability of present spaceborne
and terrestrial atomic clocks, and a look toward the future. In Relativistic Gravitational Experimentsin Space, Aug. 1989.
[343] L. Vidmar, J. P. Ronzheimer, M. Schreiber, S. Braun, S. S. Hodgman, S. Langer, F. Heidrich-Meisner,
I. Bloch, and U. Schneider. Dynamical Quasicondensation of Hard-Core Bosons at Finite Momenta.
Phys. Rev. Lett., 115(17):175301, Oct. 2015.
[344] H. Wang, J. Qin, X. Ding, M.-C. Chen, S. Chen, X. You, Y.-M. He, X. Jiang, L. You, Z. Wang,
C. Schneider, J. J. Renema, S. Höfling, C.-Y. Lu, and J.-W. Pan. Boson sampling with 20 input photons
and a 60-mode interferometer in a 1014 -dimensional Hilbert space. Phys. Rev. Lett., 123(25):250503,
Dec. 2019.
[345] S. Wang, W. Zhang, T. Zhang, S. Mei, Y. Wang, J. Hu, and W. Chen. Accelerating the assembly ofdefect-free atomic arrays with maximum parallelisms. Phys. Rev. Appl., 19(5):054032, May 2023.
[346] I. M. Wanless. Lieb’s permanental dominance conjecture. In The Physics and Mathematics of
Elliott Lieb, volume 2, pages 501–506. EMS Press, Berlin, June 2022.
[347] D. Wei, A. Rubio-Abadal, B. Ye, F. Machado, J. Kemp, K. Srakaew, S. Hollerith, J. Rui, S. Gopalakrishnan, N. Y. Yao, I. Bloch, and J. Zeiher. Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion. Science, 376(6594):716–720, May 2022.
[348] P. Weinberg and M. Bukov. QuSpin: A Python Package for Dynamics and Exact Diagonalisation of
Quantum Many Body Systems. Part II: Bosons, fermions and higher spins. SciPost Phys., 7(2):020,
Aug. 2019.
[349] C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau, P. Schauß, T. Fukuhara, I. Bloch, and S. Kuhr.
Single-spin addressing in an atomic Mott insulator. Nature, 471(7338):319–324, Mar. 2011.
[350] S. Welte, B. Hacker, S. Daiss, S. Ritter, and G. Rempe. Photon-Mediated Quantum Gate between
Two Neutral Atoms in an Optical Cavity. Phys. Rev. X, 8(1):011018, Feb. 2018.
[351] C. E. Wieman and L. Hollberg. Using diode lasers for atomic physics. Rev. Sci. Instrum., 62(1):20,
1991.
[352] T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys. Entanglement of two individual neutral atoms using Rydberg blockade. Phys. Rev. Lett., 104(1):010502,
Jan. 2010.
[353] J. T. Wilson, S. Saskin, Y. Meng, S. Ma, R. Dilip, A. P. Burgers, and J. D. Thompson. Trapping
Alkaline Earth Rydberg Atoms Optical Tweezer Arrays. Phys. Rev. Lett., 128(3):033201, Jan. 2022.
[354] D. Wineland, C. Monroe, W. Itano, D. Leibfried, B. King, and D. Meekhof. Experimental issues incoherent quantum-state manipulation of trapped atomic ions. Journal of Research of the National
Institute of Standards and Technology, 103(3):259, May 1998.
[355] D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen. Squeezed atomic states and projection
noise in spectroscopy. Phys. Rev. A, 50(1):67–88, July 1994.

267[356] D. J. Wineland and W. M. Itano. Laser cooling of atoms. Physical Review A, 20(4):1521–1540, Oct.
1979.
[357] R. Yamamoto, J. Kobayashi, T. Kuno, K. Kato, and Y. Takahashi. An ytterbium quantum gas
microscope with narrow-line laser cooling. New Journal of Physics, 18(2):023016, Feb. 2016.
[358] M. Yan, B. J. DeSalvo, B. Ramachandhran, H. Pu, and T. C. Killian. Controlling Condensate Collapse
and Expansion with an Optical Feshbach Resonance. Phys. Rev. Lett., 110(12):123201, Mar. 2013.
[359] Z. Z. Yan, B. M. Spar, M. L. Prichard, S. Chi, H.-T. Wei, E. Ibarra-García-Padilla, K. R. A. Hazzard, and W. S. Bakr. Two-dimensional programmable tweezer arrays of fermions. Phys. Rev. Lett.,
129(12):123201, Sept. 2022.
[360] B. Yang, H. Sun, C.-J. Huang, H.-Y. Wang, Y. Deng, H.-N. Dai, Z.-S. Yuan, and J.-W. Pan. Coolingand entangling ultracold atoms in optical lattices. Science, June 2020.
[361] A. Yariv. Quantum Electronics. John Wiley & Sons, 3 edition, 1989.
[362] M. Yasuda and H. Katori. Lifetime measurement of the 3 P2 metastable state of strontium atoms.
Phys. Rev. Lett., 92(15):153004, Apr. 2004.
[363] M. Yasuda, T. Kishimoto, M. Takamoto, and H. Katori. Photoassociation spectroscopy of 88 Sr:
Reconstruction of the wave function near the last node. Phys. Rev. A, 73(1):011403, Jan. 2006.
[364] A. Young, W. G. Graham, T. J. Morgan, and L. Hüwel. Laser-induced breakdown in liquid water:
Influence of repeated laser pulses on plasma formation and emission. Journal of Applied Physics,
129(18):183303, May 2021.
[365] A. W. Young, W. J. Eckner, W. R. Milner, D. Kedar, M. A. Norcia, E. Oelker, N. Schine, J. Ye, andA. M. Kaufman. Half-minute-scale atomic coherence and high relative stability in a tweezer clock.
Nature, 588(7838):408–413, Dec. 2020.
[366] A. W. Young, W. J. Eckner, N. Schine, A. M. Childs, and A. M. Kaufman. Tweezer-programmable
2D quantum walks in a Hubbard-regime lattice. Science, 377(6608):885–889, Aug. 2022.
[367] A. W. Young, S. Geller, W. J. Eckner, N. Schine, S. Glancy, E. Knill, and A. M. Kaufman. An atomic
boson sampler. In review, 2023.
[368] J. T. Young, T. Boulier, E. Magnan, E. A. Goldschmidt, R. M. Wilson, S. L. Rolston, J. V. Porto, andA. V. Gorshkov. Dissipation-induced dipole blockade and antiblockade in driven Rydberg systems.
Phys. Rev. A, 97(2):023424, Feb. 2018.
[369] J. T. Young, S. R. Muleady, M. A. Perlin, A. M. Kaufman, and A. M. Rey. Enhancing spin squeezing
using soft-core interactions. Phys. Rev. Res., 5(1):L012033, Mar. 2023.
[370] J. Zeiher, R. van Bijnen, P. Schauß, S. Hild, J.-y. Choi, T. Pohl, I. Bloch, and C. Gross. Many-bodyinterferometry of a Rydberg-dressed spin lattice. Nature Physics, 12(12):1095–1099, Dec. 2016.
[371] W.-Y. Zhang, M.-G. He, H. Sun, Y.-G. Zheng, Y. Liu, A. Luo, H.-Y. Wang, Z.-H. Zhu, P.-Y. Qiu,
Y.-C. Shen, X.-K. Wang, W. Lin, S.-T. Yu, B.-C. Li, B. Xiao, M.-D. Li, Y.-M. Yang, X. Jiang, H.-N.
Dai, Y. Zhou, X. Ma, Z.-S. Yuan, and J.-W. Pan. Scalable Multipartite Entanglement Created by SpinExchange in an Optical Lattice. Phys. Rev. Lett., 131(7):073401, Aug. 2023.
[372] X. Zhang, K. Beloy, Y. S. Hassan, W. F. McGrew, C.-C. Chen, J. L. Siegel, T. Grogan, and A. D.
Ludlow. Subrecoil Clock-Transition Laser Cooling Enabling Shallow Optical Lattice Clocks. Phys.
Rev. Lett., 129(11):113202, Sept. 2022.
[373] X. Zheng, J. Dolde, V. Lochab, B. N. Merriman, H. Li, and S. Kolkowitz. Differential clock comparisonswith a multiplexed optical lattice clock. Nature, 602(7897):425–430, Feb. 2022.

268[374] Y.-G. Zheng, W.-Y. Zhang, Y.-C. Shen, A. Luo, Y. Liu, M.-G. He, H.-R. Zhang, W. Lin, H.-Y. Wang,
Z.-H. Zhu, M.-C. Chen, C.-Y. Lu, S. Thanasilp, D. G. Angelakis, Z.-S. Yuan, and J.-W. Pan. Efficiently
Extracting Multi-Point Correlations of a Floquet Thermalized System. arXiv:2210.08556, Oct. 2022.
[375] H.-S. Zhong, Y.-H. Deng, J. Qin, H. Wang, M.-C. Chen, L.-C. Peng, Y.-H. Luo, D. Wu, S.-Q. Gong,
H. Su, Y. Hu, P. Hu, X.-Y. Yang, W.-J. Zhang, H. Li, Y. Li, X. Jiang, L. Gan, G. Yang, L. You,
Z. Wang, L. Li, N.-L. Liu, J. J. Renema, C.-Y. Lu, and J.-W. Pan. Phase-programmable Gaussianboson sampling using stimulated squeezed light. Phys. Rev. Lett., 127(18):180502, Oct. 2021.
[376] H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding,
Y. Hu, P. Hu, X.-Y. Yang, W.-J. Zhang, H. Li, Y. Li, X. Jiang, L. Gan, G. Yang, L. You, Z. Wang,
L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan. Quantum computational advantage using photons. Science,
370(6523):1460–1463, Dec. 2020.
[377] S. Zhou, M. Zhang, J. Preskill, and L. Jiang. Achieving the Heisenberg limit in quantum metrology
using quantum error correction. Nat Commun, 9(1):78, Dec. 2018.
[378] X. Zhou, X. Xu, X. Chen, and J. Chen. Magic wavelengths for terahertz clock transitions. PhysicalReview A, 81(1), Jan. 2010.
[379] P. Zupancic, P. M. Preiss, R. Ma, A. Lukin, M. E. Tai, M. Rispoli, R. Islam, and M. Greiner. Ultraprecise holographic beam shaping for microscopic quantum control. Opt. Express, OE, 24(13):13881–
13893, June 2016.
[380] M. Zych, F. Costa, I. Pikovski, and Č. Brukner. Quantum interferometric visibility as a witness ofgeneral relativistic proper time. Nat Commun, 2(1):505, Oct. 2011.
[381] M. Zych, I. Pikovski, F. Costa, and Č. Brukner. General relativistic effects in quantum interference of"clocks". J. Phys.: Conf. Ser., 723:012044, June 2016.

Appendix A

Notes on estimation theory

A.1

Classical estimation theory
To determine how much information a probability distribution p(x|ϕ) contains about some parameterϕ, we are interested in something like the variance of ∂p(x|ϕ)/∂ϕ. It is useful to include a logarithm inthe above quantity to ensure that the resulting measure of information is additive when considering joint
probability distributions of independent random variables. The above considerations lead to the definitionof the “score” s:s(ϕ) :=∂ ln p(x|ϕ)
.
∂ϕ

(A.1)

Taking the variance of the score leads to the definition of the Fisher information I:1
I (p(x|ϕ)) :=Z

dxs(ϕ)2 .

(A.2)

I is non-negative and additive, meaning that when x corresponds to N independent measurements of therandom variable X, it has the property:
I (p(x|ϕ)) = I (N p(X|ϕ)) .

(A.3)

By taking the derivative of Eq. 5.1 with respect to ϕ and applying the Cauchy-Schwarz inequality,
one arrives at the Cramér-Rao bound [76]:∆2 ϕ̌ ≥1.
I (p(x|ϕ))(A.4)

1 Note that it is straightforward to generalize the Fisher information to multiple parameters [197], but we omit these details

in this simplified discussion.

270
As discussed in Sec. 5.1, a protocol for estimating ϕ is efficient if it saturates this bound.
One general-purpose scheme for parameter estimation is maximum likelihood estimation (MLE). The
likelihood function is defined as L(ϕ|x) = p(x|ϕ), and can be interpreted as a measure of how well theobserved data x is explained by the parameter ϕ. Maximizing L(ϕ|x) over ϕ provides an estimator ϕ̌ thatis consistent and efficient in the infinite-data limit. Although MLE can yield results that are biased fora finite number of samples, a number of techniques can be used to correct this bias [75, 300], typicallyresulting in leading-order contributions to the bias of only O(1/N 2 ) (where N is the number of independentmeasurements).

A.2

Quantum estimation theory
Extending the discussion in the previous section to the quantum case is, at least conceptually, relatively

straightforward. Once one has picked a measurement basis, each measurement of a quantum system provides
a sample of an IID random variable, and all the statements in the previous section apply. As described inSec. 5.1, maximizing the Fisher information I over all possible measurements leads to the definition of thequantum Fisher information Q.
More explicitly, we can generalize the score s(ϕ) (Eq. A.1) to the quantum case by considering how
quickly a density matrix ρ(ϕ) varies with respect to a parameter ϕ. Note that s(ϕ) has the property:∂p(x|ϕ)= p(x|ϕ)s(ϕ).
∂ϕ

(A.5)

We can analogously define the symmetric logarithmic derivative L [331]:2
1∂ρ(ϕ)= {L, ρ(ϕ)},
∂ϕ
2

(A.6)

where the anticommutator avoids any ambiguity about operator ordering. Recall from Sec. 5.1 that thegenerator H of the signal transforms ρ(ϕ) according to the von Neumann equation:∂ρ(ϕ)= −i[H, ρ(ϕ)].
∂ϕ
2 See [309] for a nice geometric interpretation of the score and symmetric logarithmic derivative.

(A.7)

271
The equality of Eqs. A.6 and A.7 yields [331]:
L = 2iX λk − λlkl

λk + λl|k⟩⟨l|⟨k | H | l⟩,(A.8)

where |k⟩ is an eigenstate of ρ(ϕ) with eigenvalue λk . Given the above definitions, the quantum Fisher
information Q is simply the expectation of the variance of L:
Q := Tr(ρ(ϕ)L2 ).
Note that substituting a pure state into the above expressions leads to Eq. 5.7.

(A.9)

Appendix B

Estimating indistinguishability, bunching, and clouding in the presence ofparity projection

1
Continuing the discussion in Sec. 8.3.1, we are interested in estimating the quantity QHOM
For theS1 ,S2 .

discussion below, we will need to keep track of the input sites defining S1 and S2 , and therefore make thesubstitution S1 → Ck and S2 → Cl . Ck is the column of lattice sites that includes site k, and similarly for
Cl and l.
partialHOM
dist
We will construct the estimate of QHOM
S1 ,S2 (QCk ,Cl ) from separate estimates of pCk ,Cl and pCk ,Cl , which

are respectively the joint detection probabilities of one atom in Ck and one atom in Cl for distinguishable
atoms, and nominally indistinguishable atoms.2 pdist
Ck ,Cl can be estimated from data where a single atom isprepared in either k or l on a given experimental trial:
pdist
Ck ,Cl = p(Ck |k, ¬λ)p(Cl |l, ¬λ) + p(Ck |l, ¬λ)p(Cl |k, ¬λ),(B.1)

where p(Ci |j, ¬λ) is the probability that an atom is observed in Ci , given that a loss event λ did not occur(¬λ), and that the atom was initially prepared in state j (with i, j ∈ {k, l}). Eq. B.1 is a multilinear
polynomial of the single atom probability distributions, and so the plug-in estimator3 is unbiased.
ppartial
Ck ,Cl is estimated directly from measurements with two atoms. However, the observed coincidence
partialprobability in the experiment pobs
Ck ,Cl is distinct from pCk ,Cl due to parity projection:

partialpobs
Ck ,Cl = p(ν2 |kl)pCk ,Cl ,1 The analysis in this section is primarily the work of Shawn Geller.
2 With postselection on no atom loss.
3 Namely, where one estimates pdist
Ck ,Cl by plugging the estimates of the single atom probabilities into Eq. B.1.

(B.2)

273
where ν2 refers to an event where neither of the two atoms was lost due to single-atom loss processes (i.e.
loss not due to parity projection). Assuming that loss acts independently and identically on each site, we canestimate p(ν2 |kl) by estimating the single atom loss probability pλ . Specifically, we measure the probabilityof an event β, where exactly one of the two atoms is lost:p(β|kl) = 2(1 − pλ )pλ .

(B.3)

We therefore have:
pλ =

1−

p
1 − 2p(β|kl)
,
2

(B.4)

and:
p(ν2 |kl) = (1 − pλ )2 .

(B.5)

Using Eqs. B.2 to B.5, we can construct a plug-in estimate of QHOM
Ck ,Cl . Note that the plug-in estimator is
not unbiased in this situation, and so we use the delta method [300] to obtain an estimate that is firstorder unbiased. We perform 1000 bootstrap estimates of QHOM
Ck ,Cl and apply the bias-corrected percentile
method [98] to obtain a confidence interval.
Note that small diagonal and next-nearest neighbor tunneling terms in the lattice break the balanced
condition for Ck and Cl (see Sec. 8.3). Based on our calibration of the lattice Hamiltonian (see Secs. 7.3 and
8.4), we calculate a correction factor τ (Ck , Cl ), leading to a corrected indistinguishability estimate of:J =
1
1 − QHOM
Ck ,Cl .
τ (Ck , Cl )(B.6)

Similar to before, the confidence interval in our estimate of J is computed using 1000 bootstrap estimates,
and the bias-corrected percentile method. The combination of statistical fluctuations and τ < 1 can lead tobootstrap samples that go above 1, and so we clip the upper end of our distribution to 1 in both our confidence
interval and our point estimate. We do not expect the uncertainty in τ (due to imperfect calibration of thelattice potential) to be significant in comparison to the statistical fluctuations in our estimate of J , and so
we ignore this source of uncertainty in our confidence interval.
The point estimates and confidence intervals for full bunching, clouding, and generalized bunchingappearing in Ch. 8 use a similar treatment to what is described above, where we account for the differing274
effects of parity projection on multi- and single-atom measurements, and then construct point estimates andbootstrapped confidence intervals for the desired quantities. For a full accounting of the above estimation
procedures, see [367].

Appendix C

Electronics

A significant number of different custom and off-the-shelf electronics are used in our experiment.
Summarizing the designs of all of these systems is beyond the scope of this work, but we will endeavor inthis section to give a few guiding principles of electronics design, and provide some useful references.

C.1

Clock distribution
In order to synchronize various parts of the experiment and generate the appropriate control signals,

a radio frequency (RF) clock signal must be distributed to various electrical systems in the lab. We use
a 10 MHz reference, either from a rubidium frequency standard (Stanford Research Systems FS725), or a
10 MHz maser standard distributed from NIST Boulder. The 10 MHz reference is distributed around thelab as necessary using a distribution amplifier (Stanford Research Systems FS730).
To generate digital clock signals, we use a high performance jitter attenuator (EVAL-HMC7044). This
device is locked to the 10 MHz reference signal, and is able to output up to 14 independent digital clock
signals with different frequencies, and with different clock standards (including LVDS, LVPECL, and CML).
Although analog signals with fixed frequencies are generated in a variety of different ways in the lab, signalsthat must be well-referenced to the 10 MHz clock are typically generated by converting a digital clock fromthe HMC7044 board to an analog signal using direct digital synthesis (e.g. using an EVAL-AD9959 board).

276

C.2

General control electronics
The experimental control system involves multiple devices that are programmed with a set of in-structions, and are then synchronized and triggered using TTL signals generated using a Spartan 6 fieldprogrammable gate array (FPGA). Analog control voltages are generated using a National Instruments
control system (PXI-1042Q, PXI-6733, BNC-2110). To communicate between the control computer and thevarious control electronics, we use either USB connections,1 or, in cases where data transfer rate is critical(for example when programming tweezer moves as in Sec. C.3), UDP over ethernet.2
There are a number of electronic servos in the lab, most of which have been described in detailelsewhere. To control the currents in our magnetic field coils, we use a JILA designed current controller
similar to the one in [52]. To control laser diode currents and temperatures, we typically use off the shelfcontrollers from Thorlabs (LDC2xx and TED200C), or JILA controllers of a similar design.
We use a number of different high speed loop filters, both analog and digital. Besides standard JILA
designs [136], we also use the Newport LB1005-S, Toptica FALC pro, and Vescent D2-125 controllers. For
applications that do not require very high bandwidth (≲ 1 MHz), we often opt to use digital loop filters dueto the increased convenience they offer.3 We have used digital loop filters based on Red Pitaya boards [232],
as well as other open source designs [186]. For stabilizing laser powers using AOMs, we use both the digital
loop filters described above, and specialized analog loop filters that take the significant phase delay imposed
by AOMs into account [245].

C.3

RF synthesis with FPGAs
There are a number of cases where we have to synthesize RF signals that are modulated in frequency

and/or amplitude. In general, we do this via direct digital synthesis (DDS) controlled with an FPGA,
however the precise manner in which this is done depends on the specific application.4
For the synthesis of general purpose signals with carrier frequencies of ≲ 150 MHz and relatively simple
1 For long USB cable runs, we opt to use USB to fiber optic converters.
2 For local connections with short cable runs the chances of an error are extremely low, and so UDP ends up being both

faster and easier to deal with than TCP. With UDP, it is relatively straightforward to achieve data rates in excess of 1 Gbps.
3 For example, the ability to easily include non-linear behaviors, auto re-locking features, etc.
4 Most of our FPGA-based control electronics were developed by Felix Vietmeyer in the JILA electronics shop.

277
modulation (jumps and/or slow ramps of frequency and amplitude), we control several EVAL-AD9959 boards
using a Spartan 6 FPGA.5
The tweezer control system has more stringent requirements, where up to four separate outputs (onefor each AOD in the two tweezer systems) each needs to be driven with multiple RF tones (where each tone
defines the position of one row or column of tweezers and therefore must be independently tunable). To
generate the above control tones, we use a custom frequency synthesizer composed of an FPGA (Kintex
UltraScale KCU105) driving a high speed (2.5 Gs/s) quad-channel 16-bit digital to analog converter (DAC,
Texas Instruments DAC38J84EVM).6 Specifically, the FPGA runs 512 DDS cores,7 which are interleaved
to generate 256 outputs with independently tunable frequency, phase, and amplitude. The 256 outputs are
split among the four DAC channels, which each drive one of the four AODs used in our tweezer systems.
The outputs are clocked at 750 MHz (but can be clocked in the gigahertz range if desired), corresponding
to a maximum usable frequency of ∼ 300 MHz (for our AODs, we operate in the range of 125-235 MHz).
Each analog output is amplified using two stages of linear RF amplifiers, with the final stage being a highpower (10 W) amplifier that delivers ∼ 2 W (∼ 5 W) of total RF power to each of the 515 nm (813 nm)
AODs (see Sec. 3.3.3.1). The final result is that each AOD can be controlled with up to 64 independent RF
drive tones, where each tone has 36 bits of frequency resolution, 12 bits of phase resolution, and 10 bits ofamplitude resolution.
The tweezer control system can be programmed with multiple settings, and can linearly interpolate
between these settings. Therefore, to execute a move between two patterns of tweezers, one only needs toprogram the appropriate frequencies, amplitudes, and phases of each RF drive tone at the beginning and endof the move, as well as the rates at which the frequencies and amplitudes should be varied during the move.
This simple scheme allows the tweezer control system to be programmed with hundreds of moves involving
hundreds of distinct tweezers in under 1 ms (using a gigabit Ethernet connection).8
The system used to perform arbitrary waveform generation (AWG)9 is similar to the tweezer control
5 We refer to this system as the “Minimoog,” in honor of the Moog synthesizer that revolutionized electronic music.
6 We refer to this system as the “Gigamoog,” for obvious reasons.
7 Implemented on the fabric of the FPGA itself, and not in a standalone DDS chip as in the Minimoog.
8 To specify a similar control sequence as a time series, as is typically required when using off the shelf arbitrary waveformgenerators, many orders of magnitude more data would have to be transferred from the control computer to the signal generator,
possibly resulting in significant experimental delays.
9 This is used, for example, for modulation of the UV laser to perform the entangling gates described in Ch. 6.

278
system in terms of hardware, with the only difference being the use of a different FPGA (Zynq UltraScale+
ZCU106).10 Zynq FPGAs integrate an FPGA and an ARM CPU on the same die, with the CPU providing
a number of convenient features for steps that are not sensitive to timing (like communication over Ethernet).
The main difference between the tweezer control system and the AWG system is how the desired signals
are programmed into the device and subsequently synthesized. Note that specifying the output of an AWG
as a time series can be very costly in terms of how much data must be transferred and stored in memory.
However, the signals we are interested in generating typically involve comparably slow modulation of a fast
carrier signal. Therefore, instead of specifying a time series defining the desired output voltage on eachclock cycle, we program the AWG with a series of “snapshots.” Each snapshot includes a timestamp, andthe desired phase,11 amplitude, and frequency of the output signal at that time. The AWG will linearlyinterpolate from the frequency and amplitude settings of one snapshot to the next, allowing us to specify
the desired modulated signals without sending a significant amount of data to the AWG.12 16384 snapshots
can be stored in memory, and the maximum update rate to go from one snapshot to another is 153.6 MHz.

C.4

High current coil control
Our high current coils (see Sec. 3.2) are switched between a Helmholtz and anti-Helmholtz configura-tion using a custom H-bridge circuit (see Fig. C.1). The circuit is designed with an interlock which ensuresthat the H-bridge does not switch while there is a significant amount of current in the coils. We are also
careful to optically isolate sensitive components of the circuit (and other electronics in the experiment) fromthe parts of the circuit that can experience large voltage spikes when the coils are switched.

10 We refer to this system as the “Subharmonicon,” for less obvious reasons.
11 Each snapshot also includes a setting that allows one to ignore this phase, and simply roll over continuously based on thephase of the signal at the end of the previous snapshot.
12 For example, a linear ramp between two settings of the amplitude and frequency involves only two snapshots, and thetransfer of a few bytes of data (even if the duration of the ramp is very long in comparison to the period of the carrier).

279

Figure C.1: Schematic for H-bridge coil switch. This circuit was designed by Carl Sauer in the JILA
electronics shop.

Appendix D

Laser systems

As described in Sec. 2.1.1, we are interested in addressing a number of different electronic transitionsin Sr using the appropriate lasers. Here, we briefly summarize the key laser systems in the lab.

D.1

Diode lasers
For applications involving low power (≲ 100 mW), we typically use homemade external cavity diodelasers (ECDLs). An ECDL uses a laser diode as the gain medium, and provides feedback using an external
cavity. In order to isolate a single mode of the cavity, it is helpful to have an additional frequency selective
element that is broader than the cavity linewidth, but narrower than the bandwidth of the gain medium.
An optical grating is commonly used both as the intermediate frequency selective element, and to form theoptical cavity [351]. We instead opt to use an interference filter1 as the intermediate frequency selective
element. The corresponding laser design is shown schematically in Fig. D.1a, and is sometimes referred to as
an interference filter diode laser (IFDL). The main benefit of the IFDL design is that tuning the interference
filter (via its angle) does not significantly change the optical length of the cavity, improving both the stability
and tunability of the laser, as described in detail in [12].
Note that ECDLs are sensitive to optical feedback from reflections off of optical elements downstreamof the laser.2

To minimize the effect of these reflections, we typically use two stages of optical isolators

directly after the laser output.
1 The specific parts differ for different lasers, but we have had good luck with filters from Iridian Spectral Technologies andSemrock.
2 In particular, coupling into fibers can often lead to reflections that are well-aligned to the laser diode, and form an additional
cavity mode that can compete with the external cavity in the laser.

281

(a)

Laser diode

Outcoupling mirror

Shutter AOM
Laser output

Interference filter
To lock (wavemeter/cavity)

Loop filter(b)φFrom laser

To injection lock
EOM

Or

λ/2

λ/4
(d)

(c)
AOM

λ/2

λ/2

Shutter AOM

Laser output
Seed laser
Loop filter...

Laser diode

(e)
Lens

Mirrors

Partial mirror

PBS

Photodiode

Mixer

Isolatorφ

Phase shift

Waveplates

Fiber w/ coupler

SignalFigure D.1: Diode laser layout, stabilization, and distribution. Note that the depicted diagrams do notnecessarily correspond to any specific diode laser system, and simply illustrate design elements that arecommonly used in the various laser systems in the lab. Additionally, many optical elements (including thosefor mode shaping and alignment) are omitted for clarity. Throughout this figure, the grey shaded regions
correspond to optical elements that are sometimes omitted, or replaced with a different design, dependingon the specific laser system. (a) IFDL design, corresponding to a laser diode that is stabilized using an
external cavity in a “cat’s eye” configuration, with an intracavity interference filter used for coarse tuningof the dominant (highest quality) cavity mode. One or two optical isolators are used to prevent opticalfeedback from stray reflections downstream of the laser. Part of the laser output is used for monitoring or
frequency stabilization, whereas the output to the atoms is switched using an AOM and optical shutter. (b)Some lasers are stabilized to a high finesse optical cavity using a conventional PDH locking scheme. In thecase of the 689 nm laser system, the cavity is used to filter out unwanted laser noise at frequencies higher
than the cavity linewidth. (c) When more laser power is required, we injection lock an additional laser diodeusing ∼ 0.5 − 1 mW of seed power. In some systems, the output laser power is stabilized using the depicted
scheme (in the grey region). (d) In cases where the laser light must be distributed to multiple places in theexperiment, we use an optical switchyard similar to the depicted design. Each beam can be switched using an
independent AOM, and fully extinguished using an optical shutter. (e) Legend for the optical and electronic
components appearing in this figure. PBS refers to a polarizing beam splitter. Note that the figures in thissection use graphics that are adapted from the ComponentLibrary package by Alexander Franzen.

282
D.1.1

Repump systems

Our IFDLs typically have a free-running instantaneous linewidth of ∼ 100 kHz (full width at half
maximum, FWHM).3

For repumping on the ∼ 1 MHz wide transitions at 679, 688, and 707 nm (seeSec. 2.1.1.4), no additional narrowing of the IFDLs are required. However, the center frequency of thelasers can drift, necessitating additional stabilization on long timescales. For our repump lasers, we simply
monitor the output frequency of a given laser on a wavemeter (HighFinesse WS6-600), and adjust the laser
diode current, and the cavity length (using a piezo-actuated mirror), a few times a second to maintain theappropriate output frequency.
The repumping beams are delivered to the experimental apparatus using optical fibers, and switchedusing acousto-optic modulators (AOMs) located before the fibers. Especially for experiments where theatoms are held in place for a long time, it is important to prevent even small amounts of leakage light fromreaching the atoms. For this reason, we also incorporate optical shutters that fully extinguish the laser
beams when they are not in use (see Fig. D.1a).

D.1.2

Cavity locking

For applications where the laser linewidth must be below ∼ 100 kHz, we perform additional narrowingof the laser output by locking it to a high finesse optical cavity (Stable Laser Systems SLS-6010-1-4bore). We
use a conventional Pound-Drever-Hall (PDH) locking scheme [29], where the output of the laser is frequency
modulated using an electro-optic modulator (EOM), and its reflection off of the optical cavity monitored(see Fig. D.1b).

D.1.3

689 nm system

Although PDH locking can substantially narrow the central spectral feature in the output of the laser(in our case to a few kilohertz), it can also result in added noise when the phase delay of the locking systemapproaches π (the resulting noise peaks are often referred to as “servo bumps”). In our laser system at
689 nm, which is used for optical cooling, these servo bumps are offset by ∼ 500 kHz from the carrier.
3 Whereas we typically use cavity length of ∼ 5 cm, free running linewidths as low as 10 kHz have been achieved using asimilar design with a 15 cm cavity in James Thompson’s lab.

283
The servo bumps therefore add a significant amount of noise at frequencies close to multiples of the trap
frequency (see Sec. 4.3.3), which could lead to reduced cooling performance by resonantly driving processes
that can heat the atoms. To avoid such reductions in cooling performance, we opt to filter out the servo
bumps by using light that is transmitted through the PDH locking cavity (Fig. D.1b).4
The cavity transmission is significantly attenuated relative to the full output power of the IFDL,5 andmust be amplified to achieve the optical powers required by the experiment. We perform this amplification
by injection-locking an additional laser diode, as depicted in Fig. D.1c.6

The optical power provided by

the injection lock (as measured at the output of an optical fiber) is stabilized by actuating the laser with an
AOM. The frequency- and power-stabilized laser light is then distributed to various parts of the apparatus
using the scheme shown in Fig. D.1d.

D.1.4

461 nm system

Our 461 nm laser (for driving the 1 S0 ↔ 1 P1 transition) is a commercial laser system (Toptica DLC
TA-SHG pro), which is composed of an ECDL operating at 922 nm that is amplified using a tapered amplifier,and doubled to 461 nm in a resonant doubling cavity. We use the scheme described in [239] to lock this laser
to a see-through hollow cathode lamp (Spectrolamps HC054ST), as depicted in Fig. D.2. The scheme forpower stabilization and distribution of this light is similar to what is shown in Fig. D.1cd.

D.2

Clock laser
Our clock (698 nm) laser system has undergone a number of revisions during the course of this work,

depending on the specific experiment being performed. Typically, this laser is referenced to laser lightstabilized to a cryogenic silicon reference cavity in the Ye lab at JILA [242]. This referencing has been done
either by beatnote locking an independent laser to the reference laser, or by injection locking a laser diodeusing the stabilized laser light.7 Here, we will describe a simplified setup used in many of our experiments
4 The cavity bore used for stabilizing the 689 nm laser has a linewidth of ∼ 70 kHz, which is significantly smaller than theoffset of the servo bumps from the carrier.
5 ∼ 1 mW transmitted power, in comparison to ∼ 15 mW directly out of the IFDL.
6 In some cases, we use multiple stages of injection locks of a similar design.
7 It is worth noting that injection locks are almost unreasonably effective in preserving the spectral properties of the seed
light, and can maintain linewidths on the scale of tens of millihertz [218, 242].

284

φ

λ/2

Hollow cathode lamp

Toptica
TA-SHG pro

Probe
λ/4

EOM
AOM
Pump
To wavemeter

30 kHz chop
Loop filter

Figure D.2: Laser stabilization to a hollow cathode lamp. Doppler free spectroscopy of atoms in a hollow
cathode lamp is used to stabilize the frequency of our 461 nm laser via the scheme described in [239]. Notethat the output from the left-most fiber is not the full output of the TA-SHG pro system, and contains only
∼ 5 mW of power. “30 kHz chop” refers a square wave at 30 kHz. The actual signal delivered to the AOM
is a sinusoidal signal at 220 MHz that is modulated on and off with this square wave. This figure shares thelegend presented in Fig. D.1e.

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where we forgo any additional amplification, and simply deliver stable laser light from the Ye lab to theatoms in our lab (via a 50 m optical fiber) with minimal degradation of the spectral properties of the light.

D.2.1

Fiber noise cancellation

A dominant source of noise when delivering stabilized laser light to a distant location is fluctuationsin the optical path length, leading to a modulation of the laser phase (and thus frequency). To counter this,one can measure the path length fluctuations interferometrically, and apply the appropriate correction using
an AOM [205].8

Three additional considerations inform the design of our path length stabilization: first,

we wish to minimize the number of optical and RF components present in the Ye lab; second, we wish to
avoid laser beat notes at uncontrolled frequencies in the Ye lab; and third, we must apply the appropriate
frequency shift to the clock laser light to compensate for the different isotope shifts of the clock transition in87

Sr (used in the Ye lab) and 88 Sr (used in our lab). To balance the above considerations, we operate in a

feed-forward configuration as shown in Fig. D.3. One AOM applies a fixed frequency shift to a beam that isreflected off of a surface that is referenced to the atom positions (Sec. 3.3.1), generating the beat note used
to measure fluctuations in optical path length. A second AOM is used to undo the measured fluctuations,
to apply the requisite frequency offset to address 88 Sr atoms, and to stabilize the laser power delivered tothe atoms. Note that we endeavor to keep the two laser beams (the reference beam and the beam going to
the atoms) as close together as possible to avoid differential fluctuations in the optical path length.9

D.3

Rydberg laser
For driving transitions between the 3 P0 clock state and different Rydberg states, we are interested in

preparing high power and spectrally pure ultraviolet (UV) laser light near 317 nm. For a detailed descriptionof this laser system, see [290] and the future thesis of William Eckner. Schematically, the UV laser system iscomposed of a pair of high power (∼ 15 W and ∼ 10 W respectively) fiber lasers operating at 1066 nm and1565 nm. These two lasers undergo frequency summing in a MgO:PPLN crystal, resulting in up to ∼ 9.5 W
8 See [313] for a nice description of the path length stabilization up stream of our experiment in the Ye lab.
9 Based on measurements with different lengths of uncanceled optical paths, we do not expect that differential fluctuations
between the reference beam and the atom beam pose a significant limitation to our experiment [240].

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Ye lab

Kaufman lab
Reference surface-1

Stable laser

AOM

AOM

+1

To atoms
λ/2

FNC
Loop filter

Figure D.3: Fiber noise cancellation. In order to deliver stabilized laser light to our lab (from the Ye lab)
without degradation, we measure fluctuations in the optical path length experienced by the laser along areference path, and apply the appropriate correction to the laser light delivered to the atoms. This setup
additionally stabilizes the laser power delivered to the atoms, and allows us to scan the laser frequency.
“FNC” refers to a JILA-designed electronic circuit for fiber noise cancellation, which locks a voltage-controlled
oscillator (VCO) to the measured beat note, and divides the output frequency by 2. ±1 denotes the diffraction
order of the AOM. This figure shares the legend presented in Fig. D.1e.

287(∼ 4.5 W typical) of optical power at 634 nm. 634 nm is a particularly convenient wavelength, since it can belocked to the same optical cavity described in Sec. D.1.2. Finally, the 634 nm light is doubled in a resonant
doubling cavity using a BBO crystal, resulting in ∼ 1 W of optical power at 317 nm, and delivered to theatoms using similar intensity stabilization techniques to those described above.

D.4

Trapping lasers
In addition to resonantly (or near-resonantly) addressing various electronic transitions, we are alsointerested in optically trapping the atoms using far-detuned laser light at 515 nm and 813 nm (see Sec. 2.2.2).
Our 515 nm laser source is a commercial fiber laser system (Azurlight Systems ALS-GR-514.85-10-A-SF),
which delivers up to 10 W of optical power with a linewidth below 200 kHz. We find the output frequencyof the 515 nm system to be sufficiently stable for our experiments without any additional stabilization. The
output power is stabilized using the scheme shown in Fig. D.1c (see also Fig. 3.8). Our 813 nm laser system
is a Ti:Sapphire laser (M Squared SolsTiS or Spectra Physics Matisse), which is stabilized to a wavemeter.
We have also explored fiber-based laser sources at 813 nm [97], which could offer a higher power and morestable (but less tunable) solution.