TOPOLOGICAL EXCITATIONS IN AN ATOMICBOSE GAS &SEXUAL HARASSMENT REPORTED BY
UNDERGRADUATE PHYSICISTS

A Dissertation
Presented to the Faculty of the Graduate Schoolof Cornell University
in Partial Fulfillment of the Requirements for the Degree ofDoctor of PhilosophybyLauren Moı̈se Aycock
August 2016c 2016 Lauren Moı̈se Aycock
ALL RIGHTS RESERVEDTOPOLOGICAL EXCITATIONS IN AN ATOMIC BOSE GAS &SEXUAL HARASSMENT REPORTED BY UNDERGRADUATE PHYSICISTS
Lauren Moı̈se Aycock, Ph.D.
Cornell University 2016

Cold atoms are highly tunable and controllable systems that can explore the impact of quantum statistics on many-body physics and realize topological states.
This thesis presents three experiments toward these goals:
1. We implemented a measurement technique to optimize imaging of s-wave
scattering halos in degenerate Fermi gases (DFGs) and took preliminary
steps toward realizing effective p-wave interactions.
2. We launched solitons in elongated Bose-Einstein condensates (BECs) andobserved a soliton’s Brownian motion. We also showed that a dilute background of impurities dramatically effects the soliton. The added impurites
scatter off the much larger soliton, contributing to its Brownian motion
and decreasing its lifetime.
3. We created a hybrid two-dimensional (2-D) lattice with synthetic flux. Thishybrid 2-D lattice was only three sites wide in one dimension, ideal forstudying edge states. We directly imaged the localized edge and bulk
eigenstates of BECs in this quantum Hall ribbon. Further, we observed
both the skipping orbits of atoms traveling down our system’s edges anda universal dynamical Hall conductivity.
For future studies, loading a soliton into a Bose quantum Hall ribbon would
add the complexity of interactions.

In the remainder of this thesis, I turned my attention to the study of thecommunity of physics, focusing on a measurement of sexual harassment experienced and observed by undergraduate women while studying physics. We conducted an internet-based survey (n = 522) of attendees of the American Physical Society’s (APS) Conferences for Undergraduate Women in Physics (CUWiP)
to measure the extent to which attendees personally experienced or observed
sexual harassment in a context associated with physics. Forty-nine percent of
the respondents reported having experienced sexual harassment in physics andforty-eight percent of respondents reported ‘sometimes’ or ‘often’ observed harassment in physics. Sexual harassment contributes to a chilly or unwelcome
climate for both harassment targets and observers. It is unknown how prevalent sexual harassment is in the field of physics and whether or not it is a contributing factor to the field’s inability to recruit and retain female students.

BIOGRAPHICAL SKETCH
Lauren Moı̈se Aycock attended the University of Illinois at Urbana-Champaign(UIUC) from 2004 to 2008, when graduated with a Bachelor of Science inPhysics. As an undergraduate, she was introduced to research with ultracold
atoms by her academic advisor, Professor Brian DeMarco. In addition to her
academic work and research in physics, she co-founded and was president of
the Society for Women in Physics at UIUC, an organization that is still goingstrong today.
In August 2008, she began her doctoral studies at Cornell University in thefield of physics and she became a National Science Foundation’s Graduate Research Fellow in 2010. From January 2009 to October 2012, she pursued her
research in the lab of Mukund Vengalattore, where she took the lead role in theconstruction of a Bose-Einstein condensate apparatus.
In January 2013, Lauren joined the Laser Cooling and Trapping Group at the
National Institutes of Standards and Technology (NIST) in Gaithersburg, Maryland to complete her graduate studies as an in-absentia Cornell student working
at the Joint Quantum Institute (JQI). More specifically, she joined the RubidiumPotassium (Rb-K, formerly known as RbII) team and her thesis research wassupervised by Dr. Ian Spielman, a NIST and JQI Fellow.

iii

For survivors,
I am sorry this happened to you.
It’s not your fault. I believe you.
You are not alone.

iv

ACKNOWLEDGEMENTSFirst, I would like thank the Nation Science Foundation’s (NSF) Graduate Research Fellowship program for their support. The research in this thesis waspartially supported by the Army Research Office’s (ARO) Atomtronics Multidisciplinary University Research Initiative (MURI) program, by the Air Force
Office Scientific Research’s (AFOSR) Quantum Matter MURI, National Instituteof Standards and Technology (NIST), and the NSF through the Physics FrontierCenter at the Joint Quantum Institute (JQI). In addition, the research presented
in Chapter 8 was supported in part by the National Science Foundation (NSF)(PHY-1346627) and by the Department of Energy (DOE) (DE-SC0011076). Opinions, findings, or conclusions expressed in this thesis are my own and do notnecessarily reflect the views of the NSF, DOE, APS, ARO, AFOSR, JQI, NIST orCornell University.
I am delighted to thank my thesis advisor, Ian Spielman for the opportunity to join his lab, his guidance and boundless enthusiasm on research fromthe mundane to the profound, and his fabrication of motivational posters. The
Laser Cooling Group at NIST and the JQI has been a fantastic community towork in. Thanks to Bill Phillips, Paul Lett, Steve Rolston, Trey Porto, Gretchen
Campbell, and Emily Edwards for their useful discussions about science andthe doing of science. My time at NIST has been fantastic thanks to all of thewonderful people I had the benefit of hobnobbing with: Jeremy Clark, Roger
Brown, Avinash Kumar, Steve Eckel, Chris Billington, Paco Salces-Carbona, Seiji
Sugawa, Andika Putra, Elizabeth Goldschmidt, Nichole Nadermann, and Katie
Schwarz. I would like to thank Victor Galitski for his fascination with having aclassical object in a BEC which inspired our results in Chapter 6. Hilary Hurst
has been an invaluable collaborator and I enjoyed our time working on soli-vtons together as an experimentalist/theorist team. My sincere thanks to Ross
Williams, Matt Beeler, and Lindsay LeBlanc for setting a high bar and their
adept handing off of the Rb-K apparatus. Heartfelt thanks go to Abigail Perry,
who pushed me and gave me a scientific and professional attitude to aspire to.
My sincere thanks to Ben Stuhl for always knowing the precisely correct wording with which to express an idea, to Hsin-I Lu for her relentless determinationand sense of humor when brilliantly tackling a puzzle, and to Dina Genkina
for her insightful questions and ideas which always kept me on my toes. I am
grateful to our German interns Marcell Gall, Max Schemmer, and Martin Link
for their fresh jolt of enthusiasm and excitement in lab.
I would like to thank my first year cohort at Cornell University, in particular Kasturi Saha, Zach Lamberty, Dusty Madison, Julia Mundy, Gala NicolasKaufman, and Kendra Letchworth-Weaver (honorary 2008 cohort member), fortheir support, community, and shared struggles. My sincere thanks to my special committee, Erich Mueller, Dan Ralph, and Carl Frank, for enabling me tocomplete my thesis research in my chosen field, for reading this thesis, and providing helpful comments and suggestions. Heartfelt thanks to Nate Ellis forteaching me to machine, and to Collin Reynolds, Kristina Colladay, and Srivatsan Chakram for being excellent lab mates to learn from and with. I grateful tothe several amazing undergraduates I had the pleasure of working with: Shannon Harvey, Scott Flanz, Sam Roland, Eli Fox, Chandler Kemp, John Lombard,
Benjamin Nosarzewski, and Airlia Shaffer.
I am profoundly grateful to Ritchie Patterson and Julia Thom-Levy for
their leadership making the physics department a more inclusive community.
Thanks to Beth Livingston, the unofficial member of my special committee,
whose insights were invaluable. I am grateful to my collaborators: Kate Clancy,viEric Brewe, Zahra Hazari, Renee Michelle Goertzen and Ted Hodapp, for assisting me with my aspirations to include a social science result in my physics
thesis.
Lastly, the completion of this thesis would have been impossible without thelove and support of William McGehee, my sisters Amanda and Lizzy, and myparents Claudia and Rickey. Thank you.

viiTABLE OF CONTENTS
Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
1 Introduction

1

2 Manipulating particles with external fields
2.1 Atomic structure and interaction with external magnetic fields . .
2.1.1 Atomic structure at zero magnetic field . . . . . . . . . . .
2.1.2 Interacting with external magnetic fields . . . . . . . . . .
2.2 Optical dipole potentials . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Optical Lattices . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Charged particle interacting with magnetic fields . . . . . . . . . .
2.3.1 Aharonov-Bohm effect . . . . . . . . . . . . . . . . . . . . .
2.3.2 Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . .
2.4 Fooling neutral atoms to act like charged particles in magnetic
fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Artificial gauge fields . . . . . . . . . . . . . . . . . . . . .

5
5
5
6
9
11
12
14
16

3

Degenerate quantum gases
3.1 Brief review of quantum statistics . . . . . . . . . . . . . . . . . . .
3.1.1 Identical particles . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Bose-Einstein Condensates . . . . . . . . . . . . . . . . . .
3.1.3 Degenerate Fermi Gases . . . . . . . . . . . . . . . . . . . .
3.2 Scattering in ultracold atomic gases . . . . . . . . . . . . . . . . . .
3.2.1 s-wave scattering length . . . . . . . . . . . . . . . . . . . .
3.2.2 Scattering between different internal states . . . . . . . . .
3.2.3 Feshbach resonances . . . . . . . . . . . . . . . . . . . . . .
3.3 Brief Intro/Overview of the Gross-Pitaevskii Equation . . . . . .

21
21
21
24
26
29
29
31
32
33

4

Rubidium-Potasium Apparatus
4.1 Experimental sequence for creation of a BEC . . . . . . . . . . . .
4.2 Experimental sequence for creation of a DFG . . . . . . . . . . . .
4.3 Lasers! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Potassium cooling lasers . . . . . . . . . . . . . . . . . . . .
4.3.2 Optical dipole trap . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Optical lattices and Raman lasers . . . . . . . . . . . . . . .
4.3.4 Green plug beam . . . . . . . . . . . . . . . . . . . . . . . .
4.3.5 Imaging systems . . . . . . . . . . . . . . . . . . . . . . . .

37
37
41
43
43
49
50
52
54

viii

17
18

4.4
5

Magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Magnetic field stabilization with microwave servo . . . . .

55
57

s-Wave Scattering of Fermions
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 s-wave scattering experiment . . . . . . . . . . . . . . . . . . . . .
5.2.1 Preparation of 50/50 spin mixtures in DFGs . . . . . . . . .
5.2.2 Mapping out the Feshbach Resonance . . . . . . . . . . . .
5.2.3 Magnetic field calibration . . . . . . . . . . . . . . . . . . .
5.2.4 Optical lattice for 40 K . . . . . . . . . . . . . . . . . . . . . .
5.3 s-wave scattering results . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Image processing . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 s-wave scattering halos . . . . . . . . . . . . . . . . . . . .
5.3.3 Determination of Feshbach resonance magnetic field position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Outlook: synthetic partial waves with 40 K . . . . . . . . . . . . . .
5.4.1 Technique: Optical dressing then lattice pulsing . . . . . .
5.4.2 Technique: Pulse Raman, pulse lattice, optically dress . . .

61
61
62
63
63
65
67
70
70
71
75
76
77
77
81

6

Solitons in the presence of Impurities
84
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.2 Solitons in elongated condensates . . . . . . . . . . . . . . . . . . . 85
6.2.1 Soliton creation and optimization. . . . . . . . . . . . . . . 86
6.2.2 Soliton oscillations . . . . . . . . . . . . . . . . . . . . . . . 87
6.2.3 Microwave tomography . . . . . . . . . . . . . . . . . . . . 89
6.3 Injecting impurity atoms . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3.1 Injection and number calibration . . . . . . . . . . . . . . . 93
6.3.2 Temperature measurements with impurities . . . . . . . . 96
6.4 Solitons in the presence of impurities . . . . . . . . . . . . . . . . . 97
6.4.1 Impact on oscillations . . . . . . . . . . . . . . . . . . . . . 97
6.4.2 Impact on lifetime . . . . . . . . . . . . . . . . . . . . . . . 98
6.4.3 Enhanced diffusion . . . . . . . . . . . . . . . . . . . . . . . 102
6.5 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . 105

7Bose-Eistein Condesates in the quantum Hall regime
107
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.2 Synthetic Dimensions: Particle in a Box . . . . . . . . . . . . . . . 108
7.3 Synthetic Flux: Quantum Hall Ribbon . . . . . . . . . . . . . . . . 114
7.3.1 The Harper-Hofstadter Hamiltonian . . . . . . . . . . . . . 115
7.3.2 Eigenstates of the Harper-Hofstadter Hamiltonian . . . . . 116
7.4 Dynamic Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.5 Edge magnetoplasmons: skipping orbits . . . . . . . . . . . . . . . 122
7.6 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . 125ix

8Physics, we have a problem: sexual harassment
126
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8.2.1 Ethics Statement . . . . . . . . . . . . . . . . . . . . . . . . 128
8.2.2 Participant Recruitment . . . . . . . . . . . . . . . . . . . . 129
8.2.3 Participant Demographics . . . . . . . . . . . . . . . . . . . 130
8.2.4 Survey Design . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.2.5 Enumeration and Analysis . . . . . . . . . . . . . . . . . . 135
8.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.3.1 Female undergraduate physics majors experience and observe sexual harassment . . . . . . . . . . . . . . . . . . . . 136
8.3.2 Correlations with belief in serious gender issues inphysics and physics GPA . . . . . . . . . . . . . . . . . . . 139
8.3.3 Study Limitations . . . . . . . . . . . . . . . . . . . . . . . . 143
8.4 Outlook and Moving Forward . . . . . . . . . . . . . . . . . . . . . 144

A Nondestructive imaging of an ultracold lattice gas

146

B Visualizing edge states with an atomic Bose gas in the quantum Hall
regime151
C Feshbach enhanced s-wave scattering of fermions: direct observation
with optimized absorption imagining
157
D Geometrical pumping with a Bose-Einstein condensate

172

Bibliography

178

xLIST OF TABLES
4.1
4.2
4.3
4.4

Bias fields during experiment sequence . . . . . . . . . . . . . . .
Potassium repumping beams. . . . . . . . . . . . . . . . . . . . .
Potassium cooling beams . . . . . . . . . . . . . . . . . . . . . . .
Optical dipole trap beam waists . . . . . . . . . . . . . . . . . . .

8.1
8.2
8.3

Sample demographics . . . . . . . . . . . . . . . . . . . . . . . . . 132
Pre-survey Questions . . . . . . . . . . . . . . . . . . . . . . . . . 134
Post-survey Questions . . . . . . . . . . . . . . . . . . . . . . . . . 135

xi

42
46
47
49

LIST OF FIGURES
2.1
2.2
2.3
2.4
2.5
2.6
2.7
3.1
3.2

Atomic Structure for 87 Rb (40 K) . . . . . . . . . . . . . . . . . . . .
Hyperfine structure of 87 Rb and 40 K . . . . . . . . . . . . . . . . .
Band structure for 1-D optical lattice . . . . . . . . . . . . . . . . .
Particle with charge q confined to a 2D plane with perpendicularmagnetic field B . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Aharonov-Bohm effect . . . . . . . . . . . . . . . . . . . . . .
Conducting edge states . . . . . . . . . . . . . . . . . . . . . . . .
Synthetic magnetic field with artificial gauge fields. . . . . . . . .

6
8
13
13
15
17
19

3.3
3.4
3.5

Identical particle under exchange . . . . . . . . . . . . . . . . . .
The chemical potential µ versus temperature T for constant density n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The condensate fraction versus temperature . . . . . . . . . . . .
Fermi distribution of occupation number versus energy . . . . .
Results from a 1-D GPE solver . . . . . . . . . . . . . . . . . . . .

23
26
27
35

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11

Schematic of Rb-K BEC/DFG machine . . . . . . . . . . . . . . .
Schematic of custom Evanescent Optics Inc. fiber splitter. . . . .
Potassium Error Signal . . . . . . . . . . . . . . . . . . . . . . . .
Potassium repumper laser system . . . . . . . . . . . . . . . . . .
Potassium cooling laser . . . . . . . . . . . . . . . . . . . . . . . .
Schematic of our 1-D optical lattices and Raman lasers . . . . . .
Beam shaping our green plug beam . . . . . . . . . . . . . . . . .
xy imaging system . . . . . . . . . . . . . . . . . . . . . . . . . . .
Large coils: H-bridge and current feedback . . . . . . . . . . . . .
Microwave source schematic . . . . . . . . . . . . . . . . . . . . .
Magnetic field stabilization . . . . . . . . . . . . . . . . . . . . . .

38
39
44
45
48
51
53
54
56
58
60

5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11

Atom loss near Feshbach resonance. . . . . . . . . . . . . . . . . .
Feshbach molecule creation and dissociation. . . . . . . . . . . .
DFG after a π/2 Raman pulse at δ = 4ER . . . . . . . . . . . . . . .
Our experimental setup. . . . . . . . . . . . . . . . . . . . . . . . .
Examples of our corrected and uncorrected absorption images. .
Corrected images of s-wave scattering halos . . . . . . . . . . . .
Inverse Abel transform of corrected image. . . . . . . . . . . . . .
Normalized scattered population plotted versus bias field B. . . .
π-pulse with low Raman coupling . . . . . . . . . . . . . . . . . .
Motion of optically dressed 40 K atoms . . . . . . . . . . . . . . . .
Pulse Raman, then pulse lattice . . . . . . . . . . . . . . . . . . . .

65
66
68
70
72
73
74
76
79
80
82

6.1
6.2
6.3

Elongated BEC setup . . . . . . . . . . . . . . . . . . . . . . . . . .
Soliton tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Soliton oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . .

86
88
90

xii

22

6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12

Microwave tomography of in-situ width . . . . . . . . . . . . . . 91
Effect of gravitational sag . . . . . . . . . . . . . . . . . . . . . . . 92
rf-Tomography of solitons . . . . . . . . . . . . . . . . . . . . . . . 94
Impurity Characterization . . . . . . . . . . . . . . . . . . . . . . . 96
Impact of impurities . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Measured survival probability. . . . . . . . . . . . . . . . . . . . . 100
Soliton lifetime in the presence of impurities. . . . . . . . . . . . . 101
Brownian diffusion for different impurity fractions. . . . . . . . . 103
Brownian diffusion constant versus impurity fraction. . . . . . . 104

7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10

Particle in a Box with Synthetic Dimensions . . . . . . . . . . . . 109
Bloch bands for hybrid 2-D lattice . . . . . . . . . . . . . . . . . . 111
Particle in a box in e s data . . . . . . . . . . . . . . . . . . . . . . . 113
Experimental schematic for hybrid 2-D lattice with flux . . . . . . 115
Bloch bands for hybrid 2-D lattice with magnetic flux . . . . . . . 117
Eigenstates for positive and negative flux . . . . . . . . . . . . . . 119
Chiral Current in Quantum Hall Strip . . . . . . . . . . . . . . . . 120
Dynamic Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Schematic of Skipping Orbits . . . . . . . . . . . . . . . . . . . . . 123
Skipping orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

8.1
8.2

Observed and experienced sexual harassment in physics . . . . . 137
Sexual harassment in physics reported by female undergraduate
physics majors by their race and ethnicity. . . . . . . . . . . . . . 138
Experienced versus frequency of observed sexual harassment. . . 140
Correlation between belief in unresolved gender issues inphysics and either frequency of observation or personally experienced sexual harassment in physics. . . . . . . . . . . . . . . . . 142
Correlation between physics GPA and either frequency of observation or personally experienced sexual harassment in physics. . 143

8.3
8.4

8.5

xiii

CHAPTER 1
INTRODUCTION

We are interested in studying new topological phases of matter, where non-local
topological properties emerge from local degrees of freedom. Cold atom systems offer a good venue to create and study topological matter because they are
tunable and easy to probe quantum systems. Twenty years ago, laser coolingand trapping of bosonic atoms led to the observation of a new state of matter,
Bose-Einstein condensates (BEC) [7]. Shortly after, the onset of Fermi degeneracy in a cold gas of fermions was observed as well [46]. A degenerate Fermi gas(DFG) was not a new state of matter, for example, the electrons in a room temperature metal (e.g. a penny). In contrast to their solid state counterparts, many
features of quantum gases can be straightforwardly calculated and give insightto many-body physics of fermions. These atomic gas samples are created in vacuum and trapped by magnetic fields and laser light, which can be configured toany arbitrary potential and even be a knob for tuning the inter-particle interactions. Experimental dynamics occur on millisecond timescales and single-atom
resolution is now possible with quantum gas microscopes [13, 119]. For thesereasons, we believe cold atom systems are ideal systems to realize topologicallyinteresting states of matter and study many-body quantum physics. This thesis
details three experiments toward that goal.
The quintessential topological effect is the quantized conductance in 2D electron gases called the Quantum Hall Effect [65]. This effect was originally observed at the large magnetic field of ∼ 15T. We experimentally realized a large
artificial magnetic field for a 87 Rb BEC in a 2-D hybrid lattice [35]. This lattice wecreated existed in both spin and position space. Introducing complex tunneling1

allowed us to create a synthetic magnetic field equivalent to ∼ 20, 000 T in a condensed matter system1 . Our spin dimension only had 3 sites, which naturally
created a hard walled potential ideal for studying edge states. We observed
localized chiral currents and cyclotron orbits[124, 93].
After measuring these single-particle effects in the quantum Hall regime, we
wanted to build on our understanding by adding a new level of complexity:interactions. Typical measurements of the effects of interactions in a BEC are
performed through observing collective modes or solitons. Once we made long
lived solitons, we began to explore measurement options. Our theorist collaborator, Victor Galitski, suggested having a small fraction of atoms in a different
internal state interacting with a soliton would be interesting. A soliton in a BEC
is similar to a classical object in a quantum gas. We observed that these impurity
atoms dramatically decreased the soliton lifetime and enhanced its Brownian
diffusion.
An another example of a topological state is a Majorana fermion. In solidstate superconductors or in DFGs, these Majorana fermions are exotic quasiparticle excitations, which are their own anti particle and exhibit non-abelian braiding statistics [131]. Interactions charaterized by p-wave and other higher odd
partial waves in a spin polarized Fermi gas are critical for engineering DFGs tosupport Majorana fermions [138]. The first measurement presented in this thesis
explored an s-wave Feshbach resonance by directly imaging elastic scattering of
two DFGs with opposite momenta [54]. We then report on preliminary attempts
to modify the fermionic interaction by laser dressing, an extension of a technique already demonstrated to introduce d- and g-wave contributions to scattering in BECs [134]. Using this technique to create p-wave interactions would
1This is calculated for a material with a 5 Å lattice spacing.

2

be a step towards observing Majorana fermions in a DFG.
In parallel with these traditional physics experiments, I turned my attention
to studying the physics professional environment, focusing on the issue of sexual harassment. At the American Physical Society’s (APS) March Meeting 2015
during the panel discussion at the end of Session F33: Supporting the Recruitment and Retention of Women in Physics, I brought up sexual harassment andassault in professional contexts and asked how it impacted women at other universities. After my question, a member of the lesbian, gay, bisexual, and transgender (LGBT) community also shared her concerns. She credited my speaking
up with her ability to speak up as well. After the session, many women approached me and shared their stories. A majority of the stories were of assault
and rape perpetrated by other physicists. Other conference attendees continued
to seek me out and share their stories for the duration of the conference.
Recently, a survey academic field experiences (SAFE) in anthropology reported 64% of respondents experienced sexual harassment and 22% experienced sexual assault [40]. I wanted to perform a similar survey in physics In
May 2015, I contacted the lead author on the SAFE study, Professor Kathryn
Clancy. In June 2015, I met with Theodore Hodapp, Director of Education at
APS, to advocate for such a survey. Together in collaboration with Eric Brewe,
Renee Michelle Goertzen, and Zahra Hazari, we added a few questions on sexual harassment to their annual evaluation survey for the APS Conference for
Undergraduate Women in Physics (CUWiP). I hope our results from this study
assist in persuading others a broader survey of physics professional environments would be a useful endeavor. I believe this result is the beginning of quantitatively studying how prevalent sexual harassment is in the field of physics

3and whether it’s a contributing factor to the field’s inability to recruit and retain
female students.
Lastly, appendices A-D are peer-reviewed, published articles to which I contributed to during the course of my PhD.

4

CHAPTER 2
MANIPULATING PARTICLES WITH EXTERNAL FIELDS

Ultracold atoms are easily manipulated with magnetic fields, laser light andradio-frequency (rf) magnetic fields. This chapter reviews some basic atomic
structure and interactions between electromagnetic fields and atoms. Additionally, we discuss an interaction of a charged particle with a magnetic field andhow charge neutral atoms behave in an analogous fashion when manipulated
by external fields.

2.1

Atomic structure and interaction with external magnetic
fieldsAll results in this thesis were with alkali atoms, which have one valence shellelectron and in their ground state the total angular momentum L = 0. This
places them in the first column of the periodic table and they are often refered
to as hydrogen-like. For our experiments with Bose-Einstein condensates, weused 87 Rb atoms. For our experiments with degenerate Fermi gases, we used
40

K. For more details on laser cooling and trapping and physical and opticalproperties of 87 Rb and 40 K, I recommend Refs. [95], [121], and [127].

2.1.1

Atomic structure at zero magnetic field

Spin-orbit coupling of the spin S = 1/2 of the one valence electron and the spin
L associated with its angular momentum gives rise to fine structure of an alkali
5

Rubidium-87
Fine Structure

Potassium-40

Hyperfine Structure

Fine Structure

Hyperfine Structure

F’=3
2

2

P3/2

~500 MHzF’=5/22F’=2~100 MHzF’=9/2F’=0

F’=11/2
2~800 MHzF’=7/2P1/2~150 MHzF’=1
D2~384THz~780.241 nm

F’=9/2
D2

D1~391THz
~766.701 nm~377 THz~794.979 nm

D1~389 THz~770.108 nm

F=2
2

S1/2

F’=7/2

F’=1

F’=2

P1/2

P3/2

F=7/2
2

~6.8 GHz

S1/2

F=1

~1.3GHz

F=9/2

Figure 2.1: Level diagram of the fine and hyperfine structure of 87 Rb (40 K). Energy levels are not to scale. Energy splitting [121, 127] is provided to showorders of magnitude.
atoms. For the first excited state L = 1, thus the total angular momentum J
can be either 1/2 or 3/2 for the Fraunhofer D1 and D2 lines, respectively. Finestructure states are further divided into hyperfine states. The total angular momentum of the electron J couples to the nuclear spin I giving the total angularmomentum of the atom F = I + J. Figure 2.1 displays the fine and hyperfine
structure for 87 Rb (40 K).

2.1.2

Interacting with external magnetic fields

An alkali atom’s hyperfine manifold consists of (2I + 1) × (2J + 1) Zeeman sublevels. An applied static magnetic field can shift the energy of these sublevels

6because magnetic moments due to spin and orbital angular momentum interact
with such fields similar to how a loop of current would1 . In terms of the atom’s
total magnetic dipole µA and external magnetic field B, the shift in energy for a
Zeeman sublevel is
HB = −µA · B =µB(gS S + gL L + gI I) · B
~

(2.1)

where g s ,gL , and gI are the electron spin, the electron orbital and the nuclear gfactors that correspond to the different magnetic dipole moments arising from
specified angular momenta. For small bias magnetic fields where the electronic
spin has not become uncoupled from the nuclear spin and F is a good quantum
number, the energy shift can be approximated by ∆E|F mF i = µB gF mF B, where
gF is the Landé g-factor of the atomic state. This regime is dubbed the linear
Zeeman regime because the energy shifts linearly depend on the magnetic field.
For the ground state of alkali atoms, HB can be exactly solved. This solution is the Breit-Rabi fomula, which describes energy shifts due to an external
magnetic field from small (linear Zeeman) to intermediate to larger (hyperfine
Paschen-Back regime) as compared to the hyperfine energy splitting. The energy shifts of 87 Rb and 40 K ground states, given by!1/2∆Eh f s∆Eh f s4mx
2
1+
+ gI µB mB ±+x
E|J=1/2 mJ I mI i = −,2(2I + 1)
2
2I + 1(2.2)

are plotted in Fig. 2.2. ∆Eh f s is the hyperfine splitting, m = mI + m J = mI ± 1/2
assigns the sign in Eqn. 2.2, and we definex=(g J − gI )µB B.
∆Eh f s(2.3)

Equation 2.2 is not correct for the “stretched” m = ±(I + 1/2) states due to a sign1

We focus on the regime where energy shifts caused by the applied magnetic field comparedto fine structure energy splitting, thus J remains a good quantum number

710
mJ=+1/2
5

mF=+2

F=2
0

mF=-2
1000

2000

3000

4000

5000

B[G]
mF=-1
F=1

5

mF=+1
mJ=-1/2

102mJ=+1/2
1 mF=+-7/2
F=7/2
mF=+7/2
0

mF=+9/2 200

400

600

800

1000

B[G]

F=9/2
1 mF=-9/2
mJ=-1/2
2

Figure 2.2: Hyperfine energy shifts of the ground states of 87 Rb and 40 K versus
external magnetic field.
ambiguity. The energy shifts of the stretched states are
E|J=1/2 mJ I mI i = −∆Eh f s∆Eh f s(1 ± x) ,+ gI µB mB +2(2I + 1)
2(2.4)

where the ± is selected to be the sign of m.
Shifting topics from native internal coupling in the atoms giving rise to its
structure, we can change the internal state of the atom by applying external
fields. We can couple different Zeeman sublevels of our atoms by illuminatingthem an oscillating magnetic field with frequency ωr f . For 87 Rb, we worked withcondensates in the F = 1 hyperfine manifold and operated at bias magnetic field
8

where both the |F = 1, mF = +1i → |F = 1, mF = 0i and the |F = 1, mF = 0i → |F =1, mF = −1i transitions were close in frequency. Our energies E+1 , E0 , and E−1 are
the solutions of Eqn. 2.2 for the |J = 1/2 m J = −1/2 I = 3/2 mI = +3/2, 1/2, −1/2i
or |F = 1, mF = +1, 0, −1i states. We can define resonance as ωr f = ωz , where~ωZ =E−1 − E+1
.
2

(2.5)

In the rotating frame, this defines a detuning δ = ωr f − ωZ and a quadratic
Zeeman shift2 ~ǫ = |E0 − (E+1 + E−1 )/2|. For 40 K, we work at fields were the|F = 9/2, mF = −9/2i → |F = 9/2, mF = −7/2i were not close in frequency toanother transition.

2.2

Optical dipole potentials

All of our experiments were performed in optical dipole traps (ODTs) [60].
Using the semi-classical approach to an atom in a light field, the electronic
states of the atom are quantized and the light is an oscillating electric fieldE(r, t) = ez E(r) exp(−iωt) + c.c.. Let’s consider that our atom has a ground state |gi
and an excited state |ei with an optical transition frequncy ω0 defined by the energy difference between |gi and |ei. Coupling to the continuium of empty modesof the electric field gives the excited state a spontaneous decay rate
ω30
Γ0 =|he|µeg |gi|23
3πǫ0 ~c(2.6)

with the dielectric constant ǫ0 , speed of light c and the dipole matrix elementµeg between the ground and excited states. For our multilevel atoms, there is2

The quadratic Zeeman shift is usually expressed as an approximation that varies quadratically with field, hence the name. Additionally, the quadratic Zeeman shift is negative, but tokeep in line with our convention where we express the quadratic Zeeman shift as explicitly
negative, I define ~ǫ as an absolute value.

9more than one ground and one excited state. We can write down our transition
matrix element more generally for states |ii and | ji as µi j = ci j ||µ||, where ci j is areal transition coefficient and ||µ|| is a reduced matrix element.
For our Hamiltonian, which describes an atom in an electric field, the temporal variation of Eis important, but the spatial variation is not. The atomic
charge distribution is much smaller than the wavelength of light. In this dipole
approximation, our Hamiltonian is HE = −µE · E with electric dipole operatorµE = −er. We use second-order time-independent perturbation theory to write
down the AC stark shift
∆Ei =

2

X |h j|H|ii|2j,i

X ci j3πc2 Γ0=I×,
Ei − E j
2ω0
∆i j
j

(2.7)

with dressed state energies Ei and Ei such that our detuning ∆i j = (Ei − Ei )/2 =ω − ω0 . This shift is linearly proportional to laser intensity I = 2ǫ0 c2 |E|2 .
Our conservative optical potentials in this thesis are all made with linearlypolarized, λ = 1064 nm light from IPG fiber amplifiers. We show the different
ground and excited states of 87 Rb in Fig. 2.1. Our 1064 nm laser light resolves thefine structure, but not the hyperfine structure. The expressions for the dipole
potential and the scattering rate simplify to!
πc2 Γ0 2
1
Udip (r) =I(r) and+2ω30 ∆2 ∆1!
πc2 Γ20 2
1
Γ sc (r) =I(r),+2ω30 ∆22 ∆21

(2.8)
(2.9)

where the optical transition frequency ω0 = (ω1 + ω2 )/2 is an average of thetransitions frequencies ω1 , ω2 for the D1 and D2 lines, respectively. Additionally,
we have defined an average spontaneous decay rate Γ0 = 3Γ1 Γ2 /(2Γ1 + Γ2 ) andassociated detunings ∆2 = ω − ω2 and ∆1 = ω − ω1 .
10

For our experiments, we used a crossed-beam ODT, with one beam frequency shifted from the other. Both of these beams were focused at the atoms.
The intensity of a focused Gaussian beam is
!
r2
2P
,
exp −2 2
IGB (r, z) =πσ2 (z)σ (z)(2.10)

where P is the bower, σ is the 1/e2 radius, and z is the direction of propagation.
p
The radius σ(z) = σ0 1 + (z/zR )2 depends on both the beam waist or minimum

radius ω0 and Rayleigh length zR = πσ0 /λ. Because our laser wavelength λ =1064 nm corresponds to a frequency that is smaller than the optical transition
frequencies, our trap is red-detuned and atoms are attracted to the intensity
maxima.

2.2.1

Optical Lattices

Two counter-propagating laser beams along e x with the same laser frequency
ω and wavevector k = 2π/λ will create an interference pattern. With V0 representing the potential at the intensity maxima, an atom will see a periodic
potential Vlat (x) = −V0 cos2 (kx) and its wavefunction have the same periodicity. Borrowing from solid state physics, we can write the atom’s wave functionikx
φk(n)x (x) = eikx · u(n)and a function uk(n)x (x) withk x (x) as a product of a plane wave e

the same periodicity of the lattice [8]. In atomic physics, we call the crystal momentum k x , the quasimomentum3 . An atom in a periodic potential obeys the
Schrödinger equation, so we insert φkx into the Schrödinger equation
"
3

#
1
2
( p̂ + k x ) + Vlat (x) uk(n)x (x) = En (x)uk(n)x (x)2m(2.11)

Rumored to be due to some one’s faulty memory of the name in Ashcroft and Mermin [8]11to find a new equation for uk(n)x (x). We can express uk(n)x (x) as a Fourier seriesP
2ikℓx
with −∞ < ℓ < ∞. Our potential Vlat (x) can also be written as a Fourierℓ cℓ eseries Vlat (x) = −V0 (eikx + e−ikx + 2)/4, we will drop the constant term because it
solely shifts the overall energy. Equation 2.11 becomesX (2~kℓ + k x )2ℓ2m

cℓ e2ikℓx +X
V0 X  2ik(ℓ+1)xcℓ e+ e2ik(ℓ−1)x = Encℓ e2ikℓx .
4 ℓℓ(2.12)

Notice that the lattice couples together the ℓ component to the ℓ ± 1 components.
We can now write our Schrödinger equation in matrix form






(k x + 2l)2 l = l′





X
X


Hlat · cℓ = Encℓ with Hlat = ER 
V/4|l − l′ | = 1




ell
ell






otherwise,
0(2.13)

where we express energy in units of recoil energy ER = ~2 kR2 /2mRb with atomic
mass mRb [57]. Additionally, the quasimomentum k x is in units of the recoil
momentum ~kR = 2π~/λL . Figure 2.3 displays band structure for varying lattice
depths from diagonalizing this Hamiltonian. We plot the lowest three bands fora numerically evaluated spectrum with −5 < ℓ < 5.

2.3

Charged particle interacting with magnetic fields

In quantum mechanics, the Hamiltonian of a charged particle q with momentum p and mass m in a magnetic field BH=1(p − qA)2 ,2m(2.14)

where A is the vector potential defining by B = ∇ × A and p is momentum
operator. A charged particle confined in 2-D subject to a perpendicular magnetic
12

Energy [ER]V0=0.0 ER

V0=4.0 ER

V0=8.0 ER

V0=12.0 ER

8

8

8

8

4

4

4

4

0

0

0

0

-4

-4

-4

-4

-0.5 0 0.5
Quasimomentum, kx [kR]

-0.5 0 0.5
Quasimomentum, kx [kR]

-0.5 0 0.5
Quasimomentum, kx [kR]

-0.5 0 0.5
Quasimomentum, kx [kR]

Figure 2.3: Band structure for 1-D optical lattice for varying lattice depths V0 in
the first Brillouin zone. For V0 = 0, the band structure is merely the free particle
dispersion folded back on itself.

B

ez

ey
ex

q

Figure 2.4: Particle with charge q confined to a 2D plane with perpendicularmagnetic field B
field B will undergo cyclotron orbits at a constant angular frequency ωc = q|B|/m
as shown in Fig. 2.4. The radius Rc = v/ωc of the orbit depends on the charged
particle’s velocity v = (p − qA)/m. For the perpendicular field B = Bez , where
B = |B|, we can make the Landau gauge choice and define A = xBey . This gauge
choice is slightly inconvenient because it breaks the rotational symmetry of ourproblem, but it will make reducing our 2-D problem to being described by a
1-D harmonic oscillator easier. Note, e x is no longer translationally invariant.

13

Inserting our gauge choice into Eqn. 2.14, our Hamiltonian becomesH=

1 2(p + (py + qBx)2 ).
2m x(2.15)The trial wavefunction ψk (x, y) = ψk (y)ψk (x) = eiky fk (x) is separable in e x and ey
and is an eigenstate of py . Combining ψk (x, y) and Eqn. 2.15, our Hamiltonian
reduces toHψk (x, y) =1 2
1 2
(p x +(py +qB)2 )eiky fk (x) =(p +(~k +qB)2 ) fk (x) = Hk ψk (x), (2.16)
2m
2m x

where
Hk =

1 2 1
p + mω2B (x + kℓ2B )22m x 2(2.17)

is a simple 1-D harmonic oscillator Hamiltonian. The physics of our 2-D electron was reduced to a 1-D harmonic oscillator problem with the center of mass√
motion shifted by −kℓ2B , where we have defined a magnetic length ℓB = ~/mωc .
The reduction from 2-D to 1-D implies that each of the harmonic oscillator energy levels of our system are degenerate.

2.3.1

Aharonov-Bohm effectA charged particle can still be impacted by a magnetic field B even where B =0. An extremely long solenoid with current I and cross sectional area A will
create a uniform B inside and outside the field will be zero. If we restrict ourcharged particle to a plane perpendicular to the solenoid, we can define a curl
free vector potential A(r) = (Φ/2πr)φ̂, where Φ = AB is the flux through thesolenoid. As a charged particle travels down one of the two paths illustrated
in Fig. 2.5, the effect of the vector potential becomes an overall phase factor
eφAB /2 on the wavefunction ψ of our charged particle. We can calculate the phase
14

B
I

A
chargedparticles

Figure 2.5: The Aharonov-Bohm effect: charged particles acquire a phase evenwhen B=0 as they travel on either side of a long solenoid because of a nonzero
vector potential A.
difference around these two paths
φAB q
=
2
~

Z

qΦ
A · dr =
~ 2π

Z

!
qΦ
1
φ̂ · (rφ̂dφ) = ± .
r2~(2.18)The total phase difference between the paths is φAB = qΦ/~ and depends on themagnetid field B. The Anahorov-Bohm effect was first observed in shifting theinterference fringes of an electron beam by adjusting the magnetic field througha solenoid [36]. In Ch. 7, we used an engineered Anahorov-Bohm phase φAB
to define a synthetic magnetic field. This phase is order unity, which is only
possible in engineered materials [53, 71], or in atomic [74, 3, 97, 79, 4, 93] andoptical [62] settings.

15

2.3.2

Quantum Hall Effect

The signatures of the integer quantum Hall effect (IQHE) are quantized Hall
conductance plateaus and their associated vanishing longitudinal resistivity [129]. Conceptually, the IQHE can be explained by the existence of protected edge states, often visualized as skipping orbits [31, 43, 65, 99]. Placing
our charged 2-D particle in a strong perpendicular magnetic field B closer to an
infinitely hard edge than its magnetic length ℓB , the charged particle will reflect
from the edge before completing its orbit and acquire a drift velocity along theedge. Examples of these trajectories are shown in Fig. 2.6A.
A quantum Hall bar (Fig. 2.6B) is a good conceptual tool. A voltage can beapplied across contacts 1 and 4. Contacts 2, 3, 5, and 6 are voltage probes and do
not supply or draw current. We will now consider our charged particle to be anelectron q = −e. We have N available edge states. There are edge states on bothedges, which are spatially separated and travel opposite directions. Applying a
voltage V1 at contact 1 injects an electron with energy −eV1 into an available edge
state. Because contacts 2 and 3 do not draw current, V1 = V2 = V3 . Similarly, allof the bottom contacts have the same voltage V4 = V5 = V6 . Using our total
current is I = −N(e2 /h)V1 , we can calculate the longitudinal and Hall resistance.
Our longitudinal resistance is (V3 − V2 )/I = 0. The resistance across the Hall
bar is the signature (V3 − V5 )/I = h/(Ne2 ) quantized resistance. There are many
limitations on this conceptual model, however, it provides the framework forthe appearance of analogous localized edge states in Chapter 7.

16

A

B
2

3

N

1

4

B
6

5

Figure 2.6: Conducting edge states. (A) Semi-classical skipping orbits offer avisualization of the conducting edge states. (B) A quantum Hall bar showing
conducting edge states which can be visualized as the skipping orbits above.

2.4

Fooling neutral atoms to act like charged particles in magnetic fields

One of the first ways to make neutral atoms behave like charged particles in amagnetic field was to use the similarity between the Lorentz force FL = q(v × B)
and the Coriolis force FC = −2m(Ω × v). This led to some awesome vortex lattices [2, 115]. However, because the magnetic field is directly proportional to
the rotation rate, reaching the quantum Hall regime through rotation is diffi-

17

cult. The added energy heats the gas and reaching the quantum Hall regime
requires the ratio of the number of vortices to the number of atoms to be of order unity. Another method is to create an artificial gauge field explained in thenext section. There are many methods to accomplish this, such as shaking optical lattices to introduce a vector potential [123, 79] or laser-assisted tunnelingin a titled optical lattice [3, 97]. Here, I will describe how a two-photon Raman transition and a magnetic field gradient can create an effective magnetic
field [88, 120]. For this section we will focus on 87 Rb.

2.4.1

Artificial gauge fields

We use a two-photon Raman transition to couple together different Zeeman
sublevels in the F = 1 hyperfine manifold with different relative momentum±2kR , where kR is the single-photon recoil momentum ~kR = 2π~/λ. This momentum defines our recoil energy ER = ~2 kR2 /2mRb , where mRb is the atomic
mass and λ ≈ 790 nm is the wavelength of our Raman lasers. Populations in
the different spin and momentum states will coherently oscillate with Rabi frequency ΩR set by the laser intensity. Our counter-propagating Raman lasers
have a frequency difference ∆ωR which is selected to be near resonance forωz (defined in Eqn. 2.5) at our bias field B = B0 ez . This defines a detuningδ = ΩL − ωz . Our Hamiltonian in the reference frame rotating at ∆ωL and in
the basis | + 1, k x + 2kR i, |0, k x i, | − 1, −2kR i basis is




2
+
2)
−
δ
~Ω
/2
0
(k
R
 x




2
H = ER  ~ΩR /2
k
−
ǫ
~Ω
/2
R

x



0
~ΩR /2 (k x − 2)2 + δ18

(2.19)

A

Raman laser

ey

87

ZR

ZR - 'ZR

ez
ex

Raman laser

Rb BEC(B0-b’z)

BC2

mF = -1
mF = 0

Vector potential,1İ
mF = +1

0
-1
-2

-4

-2
0
2
4
Detuning, /ER

D
Energy [ER]

1
8

8

8

4

4

4

0

0

0

-4

-4

-4

0.5
0

-2
0
2
Quasimomentum, kx [kR]-202
Quasimomentum, kx [kR]

-0.5
-2
0
2
Quasimomentum, kx [kR]

-1

Figure 2.7: Synthetic magnetic field with artificial gauge fields. (A) Experimental setup: two counter-propagating laser beams with orthogonal linear polarization and frequency difference ∆ωR . (B) Internal states of an F = 1 hyperfine manifold being addressed with a two-photon Raman transition. (C) Our
simulation’s vector potential qA x /~kr versus detuning δ. Due to the magnetic
field gradient in ez , δ is a function of z. (D) Calculated energy bands (colorful
curves) of our Raman coupling Hamiltonian for coupling strength ~ΩR = 8.0ER ,quadratic Zeeman shift ~ǫ = 0.6ER , and detunings ~δ = −5.0 ER , 0, 5.0 ER fromleft to right. The gray curves indicate the bare states and the colorbar indicatesspin.
To engineer a spatially varying vector potential to generate a synthetic magnetic
field, we create a spatial varying detuning δ.
Figure 2.7A shows a schematic of a laser geometry with an applied gradientb′ z ez to vary the detuning δ(z) ≈ gF µB ∆B(z) across a condensate. This change inδ shifts the minimum kmin of the dispersion. Our Hamiltonian for motion along19

e x becomesH x∗ ≈~2 (k x − kmin )2 ~2 (k x − qA∗x /~)2
=
,
2mRb
2mRb

(2.20)

where we have defined an engineered vector potential qA∗x /~. Our vector potential A∗ = A∗x e x is a Landau gauge choice for synthetic magnetic field B = B∗ ey ,where B∗ = ∂A∗x /∂z = (∂A∗x /∂δ)(∂δ/∂z). For our simulated qA∗x /~ in Fig. 2.7, we
assumed a realistic laboratory gradient of 5.5 G/cm and a cloud extent ∼ 50 µm
leading to a 10ER span in detuning δ across the cloud. This method for creating synthetic magnetic fields is limited by the linear extent of the BEC because
the synthetic field is proportional to the difference in detuning from one end of
the cloud to the other. In Ch. 7, we extend on this method by using an opticallattice instead of a magnetic field gradient to create spatial differences in phase.
There, the limit of accumulated phase difference is set by the ratio of the mismatched wavelengths between the Raman and optical lattice lasers as opposedto the linear extent of the cloud.

20

CHAPTER 3
DEGENERATE QUANTUM GASESIn this chapter, I will review some basic quantum statistics to define relevant energy scales for our experiments such as the chemical potential, the condensation
temperature T c for BEC, the Fermi energy ǫF and corresponding temperature T F .
Then we will discuss scattering in cold atoms, characterize atomic interactions
with the s-wave scattering length and introduce magnetic Feshbach resonances.
Finally, we will look at a derivation of the Gross-Pitaenskii equation (GPE) using
an effective interaction potential characterized by s-wave scattering. For moredetails on Bose-Einstein condensates and degenerate Fermi gases, I recommend
Refs. [107] and [44].

3.1

Brief review of quantum statistics

A feature of laser cooling and trapping atoms is that all atoms are the same. An
87

Rb atom in Ithaca, NY is identical to an 87 Rb in Gaithersburg, MD. The “quan-tum” in quantum degenerate gas is due to this indistinguishably. Degenerate
implies the gas is in a regime where this indistinguishably is relevant. For thissection, I found [80] and [69] to be useful for reviewing quantum statistics.

3.1.1

Identical particles

Imagine we have two identical particles as in Fig. 3.1. If we switch or exchange
the two particles, any description we have written down of that two particle

21

Figure 3.1: For indistinguishable particles the probability distribution has to be
the same before and after exchange because they are identical.
state must be the same as before the exchange. |Ψ(r1 , r2 )|2 is the probability offinding particle 1 at position r1 and finding particle 2 at position r2 . Similarly, if
the particles in the opposite positions, the probability for that state is |Ψ(r2 , r1 )|2 .
These probabilities must be equal. If we only consider single-valued solutions,
there are two allowed wavefunctions:|Ψ(r1 , r2 )i = +|Ψ(r2 , r1 )i(3.1)

and |Ψ(r1 , r2 )i = −|Ψ(r2 , r1 )i.
Consequently, there are two types of indistinguishable particles:

bosonsand fermions. Bosons are symmetric(+) under exchange and fermions are
antisymmetric(-) under exchange. A degenerate quantum gas is where theinter-particle spacing is approximately the size of the characteristic length of thequantum wavefunction that describes that particle. For our ultracold atomic ensembles with density n = N/V, we can describe the characteristic length of thepwavefunction as the thermal de Broglie wavelength λdB = 2π~2 /mkB T . When
nλ3dB ∼ 1, then quantum statistics becomes important and we must treat our

atoms as bosons or fermions. The chemical potential µ can be described as theamount of energy required to add an additional particle and behaves differently at low temperatures depending on the type of particle in the ensemble. In

22

chemical potential, P

HF

~TF
0
TC

temperature, T
Figure 3.2: the chemical potential µ for fermions (blue), classical particles (red)and bosons (green) at constant density n versus temperature T .
Fig. 3.2, we plot the chemical potential µ for an ideal Fermi gas, an ideal classical gas, and an ideal Bose gas at constant density n. The chemical potentialµ very clearly illustrates the impact of differences of quantum statistics in thedegenerate regime.
There are more than bosons, fermions, and classical particles. Revisiting the
wavefunction solutions for exchanging identical particles, we can write eqn. 3.1
in a more general fashion|Ψ(r1 , r2 )i = eiθ |Ψ(r2 , r1 )i,(3.2)

where θ = 0 for bosons and θ = π for fermions. Anyons are particles for which θ
is any other value besides 0 or π. Anyons with non-trivial phases can have inter23

esting properties when braided [102]. Majorana fermions are a specific type offermion, which are their own anti-particle. In this thesis, I focus on fermions andbosons. However, the quest for Majona modes in p-wave super conductors isthe motivation for the proposed experiment in Chapter 5. Majorana modes arecomposite quasiparticles where the simple physical exchange operations we describe in this section alter the quantum wavefunctions in non-trivial ways [131].

3.1.2

Bose-Einstein Condensates

A Bose-Einstein condensate (BEC) is a macroscopic occupation of the groundstate by a gas of bosons. This macroscopic occupation occurs when for a given
volume V, there are no more higher momentum states available when we add
the Nth+1 boson. Derived from the Bose distribution,n~k =1
z−1 eβǫk − 1

,

(3.3)

describes the occupation in a momentum state ~k at a temperature T with β =1/kB T , ǫk = ~2 k2 /2m, and k = |~k|. z is the fugacity, defining a chemical potential
through z = eβµ . To calculate the available ~k states for a uniform density n = N/V,
we sum over k and then take the thermodynamic limit to arrive at the integral
n=

Z

d3 k
1
1
=
(2π)3 z−1 eβǫk − 1 2π2

Z ∞
0

k2

dk −1 βǫk.
z e −1(3.4)

We will define V to be a cube with sides of length λdB . We can simplify Eqn. 3.4
pwith a substitution βǫk = x2 and k = x 2m/β~2 . Combining our substitution andspecified volume, we get

nλ3dB =

4
√
π

Z ∞
0

x2
dx −1 x2
= Li3/2 (z),z e −124(3.5)

where Li3/2 (z) is a polylogarithm. Notably, the derivative of this polylogarithm
is itself a polylogarithm
z

∞
X
zℓ
d
Li3/2 (z) = Li1/2 (z) =
.
1/2
dz
ℓ
ℓ=1

(3.6)

At z = 1, this geometric series diverges. This is unphysical. Our integral isessentially counting N atoms in a box λ3dB . The integrand can’t diverge, so z ≤ 1.
However when we took the thermodynamic limit, we assigned no weight to thek = 0 atoms because 4πk2 dk vanishes at k = 0. Once the density is high enoughor the temperature is low enough, any additional atoms will occupy the groundk = 0 state no energy cost. For non-interacting bosons, we can fit an arbitrary
number N0 atoms in the ground state. Our total N = N0 + Nex atoms is the sumof the number Nex = nλ3 atoms in states k > 0 and N0 .
The distribution of atoms in the excited states defines the temperature T of acondensate. The critical temperature T c of a condensate is when nλ3dB = Li3/2 (z =1) = ζ(3/2) ≈ 2.612 and no more atoms are allowed in the excited states. For agiven, uniform density n, the critical temperature is
n
2π~2
Tc =
mkB ζ(3/2)

!3/2

.

(3.7)

For a fixed density n, we plot the condensate fractionNexζ(3/2)
N0
T
=1−
=1−
=1−
3
N
N
Tc
nλdB

!3/2

(3.8)

as a function of temperature in units of T c in Fig. 3.3. A T = 0, all of the atoms
are in the condensate. A T increases, N0 /N decreases. However, below T c , we
see that a significant fraction of the atoms occupy the ground state. This state isa BEC.

25

1.0

N0 / N
0.8

0.6

0.4

0.2

Nex / N
0.0
0.0

0.4

0.2

0.6

0.8

1.0

T/Tc

Figure 3.3: The condensate fraction N0 /N (solid) and fraction in the excited states
Nex /N (dashed) versus temperature T/T c .

3.1.3

Degenerate Fermi Gases

A degenerate Fermi gas is a gas of N fermions below the Fermi temperature T F
derived from the energy of the N atoms at T = 0. The occupation of a fermion
in a state ~k isn~k =1
z−1 eβǫk + 1

,

(3.9)

valid for T , 0. At T = 0, n~k is one for ǫk < ǫF and zero for ǫk > ǫF , where ǫF isthe Fermi energy. Our Fermi momentum kF is the momentum k corresponding
to ǫk = ǫF . At T = 0,
N=

X

k<kF

1=

V 4π 3k(2π)3 3 F

26

(3.10)

T=0
T=0.02TF
T=0.10TF
T=0.25TF

1.0

Figure 3.4: The Fermi distribution of occupation number n(ǫ) versus energy ǫ.
The occupation at absolute zero is plotted for comparison.
atoms occupy all of the momentum states below kF . Thus, for a given atomic
density n = N/V, we can define a Fermi energy ǫk , a Fermi momentum kF and a
Fermi temperature T :kF = (6π2 n)1/3
ǫF =~2(6π2 n)2/3
2m
ǫF
TF = .
kB(3.11)In Fig.3.4, the occupation hnǫ i is plotted versus energy for different frations of
the Fermi temperature at a constant density n. As the temperature becomeslarger, there is less occupation in the lower energy states and the distribution
begins to resemble the distribution for classical atoms. For temperatures thatare small fractions of the Fermi temperature T F , the occupation deviates from27

the zero temperature distribution around ǫF .
To approximate the chemical potential for constant density, we again consider a volume with sides the length of the λdB and make the same substitution:
nλ3dB =

4
√
π

Z ∞
0

x2
dx −1 x2.
z e +1(3.12)

Because z is small for fermions at a low temperature, we cannot do the series
expansion like in the boson case. Instead, we will look at the T → 0 limit anduse that case to approximate the chemical potential at low temperatures aroundT F . As T → 0, the chemical potential µ → ǫF . At T = 0, there is a sharp change
in occupation number at ǫF . Our density as defined by the Fermi distribution at
low temperatures is
nλ3dB =

4
√
π

Z ∞
0

"
#
4
x2
π2 1
2/3
+ ···
≈ √ (ln z) +dx −1 x2√
8 ln zz e +1 3 π(3.13)

If we only keep the first term and look at the zero temperature limit and defining
n by the Fermi energy,

!2/3
√
3 π 3
TF
nλdB
,
ln z ≈=
4T(3.14)

we correctly find µ = kB T ln z = kB T F = ǫF . Including the next order, we solve for
√
!2/3
π2 1
3 (π) 3ln z =.
nλdB −√
48 ln z(3.15)

Substituting in ln z = T F /T on the right-side of the equation and using a series
expansion at T = 0, we find the chemical potential

!2 

π T  .
µ ≈ ǫF 1 −12 T F

This approximation is what is plotted in Fig. 3.2.

28

(3.16)

3.2

Scattering in ultracold atomic gases

In ultracold gases, most scattering events are s-wave and can be characterizedwith a single parameter, the s-wave scattering length. First, I will discuss calculating the s-wave scattering cross section for 2 distinguishable atoms, then for
2 identical atoms. Adding complexity, I will then discuss scattering between
atoms with different internal states. Finally, I will introduce magneticcally tunable Feshbach resonances, like the one studied in Ch. 5.

3.2.1

s-wave scattering length

For distinguishable atoms with mass m1 and m2 , the reduced mass is M =m1 m2 /(m1 +m2 ). In the center-of-mass(COM) reference frame, we can write down
their wavefunction at far distances as the sum of the incoming plane wave eik·r
and the scattered wave′√
eik ·rVψ = eik·r + ψ sc (r) = eik·r + f (k, k′ ),r

(3.17)

where |k′ | = |k| is defined by the energy of the state E = ~k2 /2M, and f (k′ ) is
the scattering amplitude. Because we work with cold gases, we assume theatoms are in the ground state and, currently, do not take into consideration any
internal degrees of freedom. Thus, the interaction is spherically symmetric andthe scattering amplitude f (k′ ) = f (θ) is solely dependent on the scattering angle
θ. Only considering s-wave scattering, f (θ) = a is fully characterized by thes-wave scattering length a.
To calculate if incoming particles will scatter, we define a differential crosssection dσ/dΩ–the number of particles scattered into the solid angle dΩ per unit
29

time divided by the incident flux of particles in a unit of cross sectional area perunit time. To define a scattering cross section σ, we integrate dσ/dΩ over all
solid angles
σ=

Z

dσ
dΩ =
dΩ

Z 1

cos θ| f (θ)|2 = 4πa2 .

(3.18)−1

In polarized Fermi gases and BECs, the particles are not distinguishable.
This has a direct effect on the scattering amplitude and σ. For indistinguishable particles, we need to symmeterize the wavefunction, which corresponds tochanging the sign of the relative position vector from r → −r and θ → π − θ. The
wave function of the COM system is now√Vψ = e

ik·r

ik·r

±e

′

 eik ·r
+ f (θ) ± f (π − θ),r

(3.19)

where the + is for bosons and the − is for bosons. The scattering amplitude
vanishes for s-wave scattering of identical fermions and doubles for identicalbosons leading. Therefore the scattering cross section




2



8πa bosons
σ=





fermions
0(3.20)

is different. For spin polarized fermions, this freezes out evaporative cooling.
For bosons, this introduces a factor of two in approximating the interaction potential for a BEC in only one internal state.
We will use the Born approximation to the first order to define an effective
interaction. A typical 87 Rb BEC with density n ∼ 1015 /cm3 has an inter-particle
spacing of ∼ 100 nm, which is much greater than the s-wave scattering lengtha ∼ 100a0 ∼ 5 nm. So, it is valid to ignore the atomic potential at short lengthscales. Defining a spherically symmetric effective potential Ueff (r), we calculate30

an interaction potentialZ
2π~2 14π~2 af (θ) ≈
Ueff (r)d3 r ≈= U0 .
M
mRb(3.21)

We will assume an effective contact interaction between atoms Ueff (r, r′ ) =U0 δ(r, r′ ) for identical bosons.

3.2.2

Scattering between different internal states

As discussed in Sec. 2.1, our alkali atoms have different internal states whoseenergy varies with an externally applied field. Considering them increases thecomplexity of our scattering problem because it introduces multiple exit channels which we have to sum over. Labeling the internal states |αβi of our incoming atoms, our non-interacting Hamiltonian isp̂2
H0 =+ HB (α) + HB (β),
2M(3.22)

where HB |αi = ǫα |αi. The total energy of our incoming atoms is
Eαβ (kαβ ) =2~2 kαβ2M+ ǫα + ǫβ .

(3.23)

Labeling our exit channel states as the states as |α′ β′ i, we can generalize oursingle channel wave function in Eqn. 3.17 to include multiple exit channels:
′

ik
·r
X
p
e α′ β′ ′ ′f (kαβ , kα′ ′ β′ )|α β i.
(V)ψ = eikαβ ·r |αβi +
r
′
′
αβ

(3.24)

When the exit channels are different than the entrance channels, the final relative momentum kα′ β′ can be different than kαβ as long as the total energy E isconserved, E = Eαβ (kαβ ) = Eα′ β′ (kα′ β′ ). kα′ β′ must satisfy~2 kα2 ′ β′2M=2
~2 kαβ2M+ ǫα + ǫβ − ǫα′ − ǫβ′ .

When kα2 ′ β′ ≤ 0, the channel is a closed.
31(3.25)

3.2.3

Feshbach resonances

A Feshbach resonance occurs when a diatomic molecular state energetically approaches the two-atom continuum [39, 128]. Diatomic molecular bound states
also have magnetic moments and the energy of those states can be tuned with astatic magnetic field. Consequently, the Feshbach resonance can be accessed bychanging the bias magnetic field. Tuning interactions in a DFG with a Feshbachresonance has led the creation of molecular Bose-Einstein Condensates (BECs)[58, 140, 78] as well as observation of the phase transition from the BardeenCooper-Schrieffer (BCS) superconducting regime to the BEC regime at sufficiently low temperatures [18, 25, 139, 111].
When the energy Eαβ of the open channel of two atoms approaches the energy of the molecule in the closed channel Em’cule , the coupling of those twostates has a dramatic impact on the scattering length a of the two atoms. We
can define the scattering length a as a function of magnetic field B and near amagnetic Feshbach resonance B0 as∆B
a = abg 1 −B − B0!

(3.26)

where abg is the s-wave scattering length of the open channel states far from theresonance B0 and ∆B is the width of the resonance related to the coupling between the closed and open channels and the wavefunction overlap of the bareatoms with the molecular states and their magnetic moments. Many experimental techniques have been used to characterize Feshbach resonances: atom
loss due to three-body inelastic scattering, re-thermalization timescales fromone axis to another, and anisotropic expansion of a cloud upon release froma confining potential [98, 73, 42, 104, 112]. In Ch. 5, we measured the location of
the Feshbach resonance by imaging the elastic s-wave scattering halos.
32

3.3

Brief Intro/Overview of the Gross-Pitaevskii Equation

When fully condensed, all the bosons in a BEC will have the same single particle
wavefunction. The Hartree approximation allows us write down the manybody wave function
Ψ(r1 , r2 , · · · , rN ) =NYφ(ri )(3.27)

i=1

for our N-particle system as a product of single particle wave functions φ(r). The
R
single particle wave function is normalized such that dr |φ(r)|2 = 1. Including

the effective interaction potential from Sec. 3.2.1, the effective Hamiltonian for
the N-particle system in a potential V(r) is
H=

N " 2
X
p

#

N

N

U0 X X+ V(ri ) +δ(ri − r j ).
2m
2 i=1 j,ii

i=1(3.28)

We find our energy functional by taking the expectation value of the Hamiltonian
E = hΨ∗ (r1 , r2 , · · · , rN |H|Ψ(r1 , r2 , · · · , rN i
 N
 N "
 N

#
Z
N
N X
X
Y
Y
X p2



U

0
i
+ V(ri ) +=dr φ∗ (ri ) δ(ri − r j )
φ(ri )
2m
2 i=1 j,i
i=1
i=1
i=1
#
"
Z
2
(N − 1)~= N dr|∇φ(r)|2 + V(r)|φ(r)|2 +U0 |φ(r)|4 ,2m
2(3.29)and simplifying by evaluating the sums and products. Our typical 87 Rb BECs
have ∼ 106 atoms, thus N >> 1 and (N − 1) ≈ N. We will assume a condensate√
wavefunction ψ(r) = Nφ(r), where the density of the particles is n(r) = |ψ(r)|2 .
The energy for our system becomes
E=

Z

#
~2
U0
2
2
4
|∇ψ(r)| + V(r)|ψ(r)| +|ψ(r)| .
dr2m
2"(3.30)

We will now assume that our number N is constant and minimize E − µN for
fixed µ. Combining this with Eqn. 3.30, we get the time-independent Gross33

Pitaevskii equation (GPE)−~2 2∇ ψ(r) + V(r)ψ(r) + U0 |ψ(r)|2 ψ(r) = µψ(r).
2m(3.31)

From this equation, we can define a length scale over which the density of ourcondensate can change with minimal kinetic energy cost. The length scale is thephealing length ξ = ~2 /2mnU0 of the condensate.
To study dynamics in a condensate, we adapted the time-independent GPE
to be time-dependent. Propagating the GPE in time is difficult due to its nonlinear term. Implementing a time-splitting spectral method (TSSP), we canstudy dynamics of our condensate [16]. I implemented a 1-D GPE solver for aharmonic trap V(x) = mω2x x2 plotted in Fig. 3.5A. For simulations, we will makeour Hamiltonian dimensionless. A natural length for a harmonic trap will be theharmonic oscillator length x s and we’ll scale the time by our oscillator frequency
ω x . Our dimensionless variables aret˜ = ω x t, x̃ =√x
, and ψ̃( x˜s , t˜) = x s ψ(x, t).
xs(3.32)The dimensionless Hamiltonian we numerically solve is
i

dψ
β d2
x2
=−
ψ(x, t) + γ|ψ(x, t)|2 ψ(x, t),
ψ(x,
t)
+
α
dt
2 dx2
2

(3.33)

where we have dropped the tildes. For the simulation, we set β = α = 1. To
determine a good value for γ, I used input parameters from our experiment.
We wanted to simulate an elongated condensate in an approximately harmonic
trap with trap frequency ω x ≈ 5.4 Hz and Thomas-Fermi radius ≈ 100 µm. The
trap frequencies in the other directions were ∼ 4 × ω x . Neither the dynamics
in these dimensions nor the changing density profile in these directions along
x are included in our modeling. I ran the simulation for different γ and selected γ = 960, where the simulated ground state’s probability density matched
34

A

B

0.10
Probability distribution

Potential energy

1200

800

400

0

-40

-20

C

0
20
position, x

40

2

4

0.08
0.06
0.04
0.02
0.00

-20

-10
0
10
position, x20

position, x

4
2
0
-2
-4
0

6
time, t8

10

12

Figure 3.5: Results from a 1-D GPE solver using a time-splitting spectral
method. All values are dimensionless, we dropped the tildes for simplicity.
(A) The simulated 1-D harmonic potential versus position. (B) The ground state(green curve) found by seeding the solver with white noise and using imaginary
time to add dissipation into the simulation. The Thomas-Fermi probability distribution (blue curve) is completely covered except at the edges. This deviation
is not visible at this scale. (C) Oscillations in the harmonic trap after displacing
the numerically found ground state to x = 5.
a Thomas Fermi probability distribution with radius ≈ 100 µm or 10.0833 in thedimensionless units.
To numerically find the ground state of the system, I set time to to be complex. Including an imaginary term in the time introduced dissipation and increasing the proportion of imaginary time in a given time step would find theground state more rapidly. When introducing imaginary time, the wavefunction was normalized after every time step. To determine if the state was theground state, I calculated the overlap integral with the state from the previous time step. If the difference of this overlap integral and 1 was < 1e − 9, the35

numerically found wavefunction was considered to be the ground state. Bothof the Thomas-Fermi and the numerically simulated ground state probability
distributions are plotted in Fig. 3.5B. The Thomas-Fermi distribution is almost
completely covered by the probability distribution of the ground state. Because
the Thomas-Fermi approximation sets the kinetic energy term to zero, there isno cost to rapidly changing the wavefunction, and the calculated probability
density is an inverted parabola with sharp cusps where the probability density
goes to zero. This is where the simulated probability deviates from the ThomasFermi approximation because the nonzero kinetic energy forces the probability
density to change over the healing length ξ. If we displace the numericallyfound ground state from the center of the harmonic trap, the probability density will oscillate with periodicity T = 2π/ω x . The average position versus timeis shown in Fig. 3.5C and we see the expected sinusoidal oscillations.

36

CHAPTER 4
RUBIDIUM-POTASIUM APPARATUS

Our experimental set-up at NIST in Gaithersburg, MD has been well documented initially in [87] and then with further detail [77]. My goal for this chapter is to document modifications and additions to the Rb-K (formerly known as
RbII) apparatus that enabled the experiments discussed in this thesis. Figure
4.1 displays the basic geometry of our apparatus with important cooling andtrapping laser beams for the creation of BECs and DFGs drawn.

4.1

Experimental sequence for creation of a BEC

We used a Zeeman slower to cool the 87 Rb atoms coming out of a combined
Rb-K oven. The atoms are cooled on the |F = 2, mF = 2i → |F ′ = 3, mF = 3itransition with a laser beam propagating down the slower in the opposite direction of the collimated atomic beam. A 2 → 2 fiber splitter combined theslower light with ≈ few mW of slower repumper light in resonance with the|F = 1i → |F ′ = 2i transition. The magnetic field provided by the Zeeman
slower shifts the energy of the atoms, so they stay in resonance with the coolingand repumper lasers as their velocity changes. The atoms are then captured ina magneto-optical trap (MOT). We typically load the 87 Rb MOT for 2.5 s. For
the MOT, we run a 25 A current through our large coils in anti-Helmholtz configuration providing B′z ≈ 13 G/ cm. The light for our MOT is sent into a custom fiber splitter (Fig. 4.2) to create 3 sets of counter propagating beams with
∼ 9 mW in each beam. The MOT cooling light was ≈ −20 MHz detuned from the|F = 2i → |F ′ = 3i transition. Our Rb MOT repump light was ≈ 1 MHz detuned
37

Slower/ Slower
RepumpMOT/
Rb Repump-1st order
dipole beam
MOT/
K Repump

rf coils
Green plug
beam

large
coils

MOT/
Rb Repump
0th order
dipole beam

MOT/
K Repump

ZeemanslowerAtomic beam

ey

ex
ez

Rb-K
oven

Figure 4.1: Schematic of Rb-K BEC/DFG machine. Atoms are laser cooled along
a Zeeman slower and captured by a MOT. The MOT beams are all directed
into the chamber with flipper mirrors (not shown). Two additional MOT beams
propagate along the ez axis. A pair of large coils are recessed into the chamber’s
bucket windows. Our PCB rf coils rest on the bucket window. A green plug
beam is directed into the chamber with a long pass dichroic. This figure is not
to scale.
from the |F = 1i → |F ′ = 2i transition. For 87 Rb, only a few milliwatts of repump are required. After we have loaded our MOT, we then turned off all thecurrents in the Zeeman slower and anti-Helmholtz coils and shut the oven shutter. Additionally, we quickly jumped the MOT cooling light detuning 33 MHzfrom resonance and reduced the MOT repump intensity to essentially leakage
light. The cooling light in the Zeeman slower was turned off as well. While the38

Rb MOT
repumpRb MOT
cooling

99.5/0.5

99.5/0.5

Monitor out

Monitor out

50/50
33/67
50/50

K MOT
cooling

Outputs
33/67

99.5/0.5

12
3

Monitor out

50/50

K MOT
repump954P Coupler

99.5/0.5

4
5
6

Monitor out

Polarizer

Figure 4.2: Schematic of custom Evanescent Optics Inc. fiber splitter. We used
this device to combine the MOT cooling and repump light for our dual species
apparatus.
atoms experienced the optical molasses created by the MOT cooling beams, we
ramped the cooling light detuning 33 MHz to 104 MHz away from resonance tofurther cool the atoms. The lack of repump caused the atoms to be optically
pumped into the F = 1 ground state. We optically pumped the atoms into the|F = 1, mF = −1i for 1 ms state by illuminating them with the slower repumping
light. We then turn on the xz imaging probe to blow away any remaining F = 2
atoms.
We then rapidly turned on a quadrupole magnetic trap with a current of
130 A in our large coils in an anti-Helmholtz configuration. After a 20 ms hold,we linearly ramped the current from 130 A to 250 A in 0.2 s to compress theatoms and increase their collision rate. After compression, we turn on our optical dipole trap. For all experiments in this thesis, the beam propagating alonge x has the smallest beam waist (See Tab. 4.4 for specific beam waist information) in ez , the direction of gravity. We turned on the dipole trap with most of
the power in this beam and at the highest power possible. We turned on our

39rf magnetic field and linearly ramped the frequency from 22 MHz to 4 MHz in
4 s. This flipped the spin of the highest energy atoms and then allowed theremaining atoms to thermalize at a lower temperature. We decompressed thequadrupole trap by ramping the current in the coils from 250 A to 60 A in an
exponential ramp with time constant τ = 1.5 s over 3 s. While decompressing,
we lowered the atoms into the dipole trap by ramping a smaller bias coil in
Helmholtz configuration providing field in ez from 10 A to 8 A. This lowers thequadrupole zero toward the dipole trap.
We completed the evaporation to BEC in the optical dipole trap.

The

quadrupole is still on, just barely compensating for gravity. We evaporated in
the optical dipole trap in either 3 s for typical BECs or 5 s for an elongated
BEC. For the overall power in the dipole trap, we either used a linear ramp
or sometimes we used an “O’Hara” ramp [103] to ramp from the initial powercommand to the final power command. The O’Hara ramp ramps the dipole potential more rapidly at first, then slows down. Also during this evaporation welinearly ramped the amplitude command to acousto-optic modulator (AOM)
controlling the power balance in the two beams of our dipole trap. This further
lowered the power in the beam along e x while increasing the power in the ey
beam. After evaporation in the dipole trap, we ramped the quad current to 0 A
with an exponential ramp with time constant τ = 0.8 s over 5 s. Ramping off thequadrupole continued the evaporation by decreasing the optical trap depth due
gravity. This sequence creates a |F = 1, mF = −1i 87 Rb BEC in an optical dipole
trap.

40

4.2

Experimental sequence for creation of a DFG

Our Zeeman slower slowed both 87 Rb and 40 K atomic beams. Because 40 K islighter than 87 Rb, the atoms will “bloom” or acquire a transverse velocity at theoutput of the slower. To increase the number of 40 K atoms ultimately captured
in the MOT, we cooled the atoms in an orthogonal direction to the slower immediately after the oven, but before the slower. We loaded the 40 K MOT first by
turning on the slower cooling, slower repump, transverse cooling, MOT cooling, and MOT repump light for 40 K, but not 87 Rb. After 7 s of 40 K MOT loading,we also loaded 87 Rb for a 1.5 s of MOT loading. The current in the large, antiHelmholtz was 25 A, similar to BEC creation.
We cooled both species in optical molasses for 2 ms. For 87 Rb, we did anabbreviated ramp of the MOT cooling light detuning from 33 MHz to 53 MHz
away from resonance. For 40 K, we lowered the intensity of the MOT coolinglight without modifying its detuning. We optically pumped both species into
their maximally stretched magnetically trappable states, |F = 9/2, mF = 9/2i for40

K and |F = 2, mF = 2i for 87 Rb. 40 K had a dedicated optical pumping beam. We

used the slower rempump and MOT repump light, as well. For 87 Rb, we pulsedon the slower light and both the slower and MOT repump light.
Both species were then loaded into a quadrupole magnetic trap with a current of 130 A in our large coils in anti-Helmholtz configuration. The currents inour 3 pairs of Helmholtz bias coils are listed in Tab. 4.2 for both the creation of a
DFG and a BEC. Similar to the BEC sequence, we compressed the magnetic trapby linearly ramping the current in in the large coils from 130 A to 160 A. After compression, we cooled our atoms evaporatively via forced rf evaporation,

41

CoilsNull

Rb mag cap

Rb opt pump

ex
ey
ez

−1.10
0.12
3.59

−6
0
10

−0.3
1
4.8

RbK mag cap RbK opt pump
−15
−4
10

1
2.35
4

Table 4.1: Bias fields during the magnetic quadrupole trap capture (mag cap)and optical pumping (opt pump) experiment sequence. The coils are labeled by
the direction they provide a bias field. All the currents are in amps. Rb(RbK)
is labeling the bias fields used in the experiment sequence for creation of a
BEC(DFG).
sweeping the rf frequency from 18 MHz to 2 MHz in 10 s. During evaporation,
the magnetic trap was plugged by a λ = 532 nm beam, tightly focused to ≈ 30 µm
and ≈ 5W in power, providing a repulsive potential around the quadrupole
zero to prevent Majorana losses. Since the 40 K atoms were spin polarized andtherefore only interacted by increasingly suppressed p-wave interactions, they
re-thermalized largely due to sympathetic cooling with 87 Rb atoms.
After the 10 s evaporation in the plugged quadrupole trap, we turned on
our crossed beam optical dipole trap at full power (∼ 6W) predominantly inthe more tightly focused beam. Both the green plug beam and the optical
dipole trap remain on as we decompressed the quadrupole trap by exponentially ramping the current from 160 A to 25.5 A with time constant τ = 1.5 sover 3 s. Next, we evaporated in the optical dipole trap by ramping the relative
power balance in the crossed beams to their final balance in 3 s. This reduced
the overall dipole trap depth because more power was in a beam with a larger
beam waist. We also ramped off the power in the green plug beam during this3 s as well. We simultaneously evaporated from the dipole trap by lowering
the overall optical power with an O’Hara ramp and using an exponential ramp
to turn off the quadrupole field with time constant τ = 1.5 s over 2 s. At this42

juncture, we had |F = 2, mF = 2i 87 Rb atoms and |F = 9/2, mF = 9/2i 40 K atoms in
our crossed beam optical dipole trap.
We then used adiabatic rapid passage (ARP) to transfer the 87 Rb atoms fromthe |F = 2, mF = 2i state to the |F = 1, mF = +1i ground state via 6.8556 GHz microwave coupling (20.02 MHz from the zero field resonance) followed by ramping the current in the bias ez coils 1 A in 50 ms. This state was chosen to minimize
spin changing collisions with 40 K atoms during further evaporation [22]. After
the ARP, we briefly applied an on-resonant probe laser, ejecting any remaining
87

Rb atoms in the F = 2 manifold from the trap. We evaporated further in the

optical trap again using an O’Hara ramp for 1 s. The optical potential from thedipole trap no longer holds the 87 Rb against gravity. We used a second ARP totransfer the 40 K atoms into the |F = 9/2, mF = −9/2i state by illuminating theatoms with a 3.3 MHz rf field and sweeping the current in the bias ez coils 5 A in
150 ms. At the end of this sequence, we had a DFG of about 75, 000 40 K atoms in
the |F = 9/2, mF = −9/2i state with temperature ≈ 0.4T F .

4.3

Lasers!

The real reason for this PhD: an excuse to play with lasers.

4.3.1 Potassium cooling lasers
Our potassium laser system at the writing of this thesis has been dismantled tobe reconfigured to provide cooling light for a 2-D to 3-D MOT system. I will
document the laser system as it was for our results discussed in Chapter 5. Our
43

Figure 4.3: A screen shot of our absorption (yellow) and error signal (pink) forlocking the potassium laser with Digilock software connected to a Toptica
Photonics Feedback Controlyzer connected to our repump laser system.
potassium laser cooling light was generated by two Toptica Photonics TA
Pro systems. To stabilize our laser’s frequency, we lock to a 39 K transition with a
93.26% natural abundance in contrast to the 0.0117% natural abundance of 40 K.
The upper hyperfine structure of 39 K is unresolved, so saturation absorption
spectroscopy of 39 K on the D2-transition has 3 features: two absorption peaks ofthe two hyperfine ground states and an enhanced depletion at their crossover
due to optical pumping effects. These 3 features are visible in Fig. 4.3, and welocked on the error signal derived from F = 2 → D2 transition, the left mostfeature. To increase our absorption signal, we heat our potassium cell to 54 ◦ C
controlled by a Thorlabs Temperature Controller TC200.
Figure 4.4 shows our repump laser system. To the left of the Toptica
Photonics TA Pro is our Doppler-free saturated absorption spectroscopy.

44

Figure 4.4: Potassium repumper laser system

45

Beam

Path color

MOT
SlowerBeat note
Sat abs probe
Sat abs pump

Red
OrangePink
Red
OrangeAOM frequency [MHz] Shift from res.[MHz]
+80
−80
none
−80 × 2−350 × 2

0
−160
−80
−240
−940

Table 4.2: Potassium repumping beams. Resonance (res.) is the 40 K|F = 7/2i →|F′ = 9/2i transition. We used an error signal generated from saturation absorption spectroscopy (sat abs) to stabilize the laser frequency.
We wanted the frequency of our repumper laser to be 80 MHz below the 40 K|F =7/2i → |F′ = 9/2i transition. The 39 K transition we stabilized to is −511.9 MHz
from our bare laser frequency set-point. We double passed our bare laser lightthrough a 160 MHz AOM before sending the light to the spectroscopy set-up.
We then shifted our pump laser frequency by 700 MHz (double pass a 350 MHz
AOM) with respect to our probe. The shift from resonance atoms in the heatcell see is half of the frequency difference between the pump and probe beam,thus we have upshifted our bare laser light by 160 MHz + 350 MHz = 511 MHz.
We send the bare seed output to the beat-note lock for our cooling laser. The
TA output generated our repump light for our potassium MOT and Zeeman
slower. Table 4.2 lists all the beams derived from both TA output and the seed
output.
Our potassium cooling laser’s frequency was stabilized by a beat-note lockto ∼ −100 MHz from the 40 K|F = 9/2i → |F′ = 11/2i transition. The frequencyof our laser could be adjusted during the sequence because we compared thebeat-note to a signal from a Novatech 409A programmable frequency synthesizer multiplied by 16 by our electronics. During the MOT stage of our experimental sequence, we typically locked the cooling laser 1265 MHz away fromthe repumper laser. The cooling laser provided light for laser cooling, optical

46

Beam

Path color

MOT
Slower
Opt. pump
Beat note
Transverse
xz probe
xy probe

Red
OrangePurplePink
Blue
Yellow
Green

AOM frequency [MHz] Shift from res.[MHz]+80−80+80 × 2
none
+100
+100*
+100*

−24
−160
+48.7
−80
0
0
0

Table 4.3: Potassium cooling beams. Resonance (res.) is the 40 K|F = 9/2i → |F′ =11/2i transition. Our optical pumping (opt. pump) used the same fiber launchand shutter as the Zeeman slower cooling beam. *These two are the same AOM
pumping, and absorption imaging. Figure 4.5 illustrates how we derive all thelight we need from the TA output of our cooling laser system. Because so many
beams are derived from the cooling laser, several of the 0th order beams fromdifferent AOMs are used. The unshifted 0th order light is the same color as thelight heading into the AOM. The seed output was only used for diagnostics andthe beat-note lock. We used two New Focus Servo Controllers to feedback on both the current and piezo of the seed laser our TA Pro. Table 4.3 lists
the beams of our potassium cooling laser.
All of the laser light was piped from our laser systems table to our main apparatus illustrated in Fig. 4.1 with optical fibers, though how the rubidium andpotassium cooling and imaging light was combined for our dual species apparatus varied. The MOT cooling and repump light for potassium were combinedwith the rubidium MOT light as shown in Fig. 4.2. The potassium slower andrepump light were sent through a 2 → 2 fiber. The slower light was mergedwith the rubidium slower light on the apparatus table with a polarizing beam
splitter and thus has the opposite helicity compared to the rubidium light. The
transverse cooling was piped over on a optical fiber. There was no rubidiumtransverse cooling. Both the xy and xz imaging probes had there own 2 → 2
47

Figure 4.5: Potassium cooling laser

48

Experiment
Chapter 5
Chapter 7
Chapter 6

Propagation direction Order
ey
ex
ey
ex
ey
ex

0th
+1st
−1st
0th
−1st
0th

Horizontal[µm] Vertical[µm]
150
118
96
98
83
46

98
34
104
38
66
38

Table 4.4: Optical dipole trap beam waists for the different experiments presented in this thesis. The orders are of the “Split” AOM. For work with solitons,
we switched out cylindrical optics in the 0th order’s path for spherically symmetric ones.
fiber splitter to combine the rubidium and potassium probe light. The second
output port was useful for alignment and power monitoring.

4.3.2

Optical dipole trap

A IPG YDL-10-LP 10W fiber laser provided 1064 nm light for both beams ofour crossed optical dipole trap during our experiments presented in Ch. 5 andour preliminary work for Ch. 7. After our 10W fiber laser died in October
2014, we installed a IPG YDL-30-LP 30W replacement laser. This catastrophe proved useful because we modified our alignment to enable deflection of
the beam propagating along ey . The output of either laser was sent throughtwo AOMs: the “Power” AOM and the “Split” AOM. These AOMs were driven
by 80 MHz IntraAction deflector drivers with both an amplitude and a frequency modulation input. The Power AOM was used to stabilize the intensityof our optical dipole trap. A beam sampler picked off light from the Power
AOM’s 1st order and directed this light to a photodetector connected to our intensity stabilization feedback loop. The voltage to the amplitude input for thePower AOM was adjusted to maintain an intensity, linear with computer com49

mand, at a set-point. The Split AOM controlled the relative power in the 0th
and ±1st order. During the replacement process, we switched from using the0th order and the +1st order to using the −1st order and the 0th order. Withthe 0th order providing most of the confinement against gravity, we createdtime-averaged-potentials with a few hundred kilohertz sawtooth from a function generator. Additionally, deflection of our −1st order beam provides a momentum kick for our currently being implemented experiment of Bloch oscillations for optical lattices along e x .
Dipole trap alignment was slightly different depending on whether wewanted to make a DFG or a BEC. For BECs, we aligned our dipole trap beams
to the quadrupole field zero with the nulling fields listed in Tab. 4.2. Then, weapplied a bias field in the ez direction during rf evaporation in the quadrupole
trap to have the field zero above the dipole trap. This configuration led to very
efficient loading from our magnetic trap to our optical dipole trap. For DFGs,
our green plug beam created a repulsive potential at the quadrupole field zeroto prevent Majorana losses. We aligned our dipole trap about 50 µm below ourquadrupole trap zero so the repulsive potential due to the green beam was notfighting the attractive potential of our dipole trap.

4.3.3

Optical lattices and Raman lasers

We have installed two co-propagating 1-D optical lattices. Our 766.704 nm
potassium lattice light was from the monitor output for the MOT repump lightof our custom fiber splitter (Fig. 4.2). This lattice was not intensity stabilized.
We used it for rapid pulses in our scattering experiments. Our other 1-D optical50

mirrorRaman A

lensperiscope to1064 nmfiber amplifier

DMLP 900

f=400 mmfiber launch withbeam sampler

photodetector

DMSP 1000

ODT beam

f=250 mmODT beam

f=-100 mm
f=150 mm

O/4

767nm
latticeexto phase lockez

Raman Cey

Figure 4.6: Schematic of our dual species apparatus highlighting 1-D opticallattices and Raman lasers. This figure is not to scale.
lattice was generated by a IPG YAR-10K-1064-LP-SF fiber amplifier seededby 5 mW picked off from a shared NP Photonics fiber laser source. Figure 4.6
is a schematic of our optics alignment for merging these two beams.
This schematic also shows the configuration of our Raman lasers beams. The
source for these beams was a Coherent MBR-110 Ti:Sapphire (Ti:Sapph) tunable laser pumped by a Coherent Verdi V-10 at 10 W on a separate optical
table from our experiment. On our Ti:Sapph laser table was where we controlled
the relative frequency of our Raman lasers by shifting their bare laser frequency

51

with an AOM. By controlling the amplitude of the rf tone on the AOMs, we
could also control the intensity of our Raman beams. Even with the rf tone
coming from the same source, phase noise was added to our Raman beams by
traveling through the optical fibers. This additional phase noise was particularly apparent for experiments sensitive to optical phase, like our experiments
presented in Chapter 7. So using a beam sampler on the Raman C fiber launch,
we combined a small amount of light picked off from both Raman A and C andactively stabilized the phase. This stabilized the Raman frequency and optical phase at the atoms against the phase noise added by the optical fibers thatconveyed the light from the laser table to the experiment. Additionally, the deliberate choice to install the phase stabilization adjacent to the 1064 nm opticallattice retro reflection mirror passively stabilized the phase of the Raman lightto the 1-D optical lattice. Without the phase lock, we observed rapid dephasing
and heating of a BEC loaded into our 2-D hybrid lattice with synthetic flux.

4.3.4

Green plug beam

The original plan for our green plug beam to plug our quad zero with 532 nm
light generated by a Coherent Verdi V-5 operating at 5 W. This light was
to be controlled with an AOM and coupled to a NKT Photonics LMA-PM-10
high-power polarization-maintaining fiber to improve pointing stability. We
estimated we have ∼ 3 W at the atoms. For a potential barrier of 1 mK for our 40 K
atoms, we wanted a beam waist of about 10 µm. We never could get more than2 W out of the fiber, even though we did not implement this with a 5 W Verdi
but with a 18 W Verdi operating at 12.5 W. Increasing the operating power of
our Verdi did not increase the power out of the fiber. Additionaly, our green52

f2 = 300 mm
measured
beam waist~580 Pm

f3 = 250 mm
calculated
beam waist~30 Pmf1 = -75 mm

d4 = 260 mm

d2 = 210 mm

d1 = 400 mm

d3 = 490 mm

Figure 4.7: Beam shaping our green plug beam. The measured beam waist wasafter our green plug beam’s control AOM. Pictured is the telescope we used to
expand the beam before focusing it onto the quadrupole zero at the atoms. This
figure is not to scale.
beam’s position needed to be adjusted throughout the day. We monitored its
position by in-situ imaging 87 Rb atoms in the quadrupole trap with the green
plug on and adjusted its location with a mirror driven by a picomotor in thegreen beam’s path.
Often times a drop in 40 K atom number corresponded to the green beam
moving. We decided to free space couple the green plug to acheive more powerat the atoms and have a larger beam waist. We removed the fiber and profiled
the beam directly after the AOM. We designed a telescope, illustrated in Fig. 4.7
to expand the beam before the f3 = 250 mm lens focused it down to ∼ 30 µm at
the atoms. Our green plug beam was combined with our xz imaging probe beamusing a DMLP567L dichroic mirror. We used another DMLP567L dichroic mirror
to deflect the green beam after the experiment chamber. We protected our xz
imaging camera with a filter. We minimized the green plug’s beam waist at theatoms by imaging the leakage light with amplitude to the AOM commanded53

mirror

PIXIS

lens

Flea3
f=125 mmfiber launch withbeam sampler

Flea3
f=75 mm

photodetector
O/2expolarizing beam
splitter cube

f=500 mmez
ey

Figure 4.8: xy imaging system configured for phase imprinting. Our phase imprinting beam is sent backwards through our imaging system after backlighting
a razor edge mounted to a translation stage. We roughly focused the edge onto
the atoms with our relay imaging system. This figure is not to scale.
zero. Even with chromatic aberrations, this method was effective. Running the
Verdi at 12.5 W, we had ∼ 5 W at the atoms. This did reduce our potential
barrier, but the larger beam’s position did not have to constantly be adjusted.

4.3.5

Imaging systems

We had two imaging planes in our experiment. Our xz imaging plane was defined by gravity in one direction and long axis of our aparatus in the other. Historically, our slower defined the e x direction, hence the xz imaging axis name.
Our xz imaging system has a magnification of 2. Our xy imaging plane is definedby our crossed dipole trap. The xy imaging system consisted of a compound lenswith an effective focal length of 83 mm recessed in our chamber’s bucket window and a 500 mm lense shown in Fig. 4.8. These lenses gave a magnification54

of 6 and a resolution of about 2 µm. Due to its superior resolution, we used thisimaging axis to image an edge onto our condensate and launch solitons. Figure 4.8 shows our imaging system in its configuration for our experiments presented in Ch. 6. We used a polarizing beam splitter (PBS) to merge our soliton
beam into our imaging optics. A kinematic base with a gold mirror directed ourphase beam into the imaging system. When this base was removed, our Flea3
camera was focused in-situ for the microwave magnetic field lock used in Ch. 7.
We installed a relay imaging system to roughly focus our edge at the location of
the atoms. First, we focused the atoms onto our relay imaging camera. Then,
we placed our razor edge in the focal plane of our relay imaging. The positionof the razor edge for phase imprinting needed to be positioned within ≈ 5 µm.
Our relay imaging system did not have that resolution. The final focusing of
our edge was completed by looking for the existence of long lived solitons as afunction of position on the translation stage where the edge was mounted.

4.4

Magnetic fields

We have 3 pairs of bias coils in Helmholtz configuration for bias fields in the e x ,ey , and ez directions, as shown in orange in Fig. 4.1. These bias coils were used
to null fields or set bias magnetic fields < 10 G. Their currents were stabilized byproportional-intergral control circuits and could be ramped or changed during
the experimental sequence. An additional set of 4 coils were used to cancelout gradients in Bz along e x . Two coils are pictured in Fig. 4.1 and the othertwo are directly beneath them. Canceling out the gradient in e x was importantfor our studies with evenly distrubuted spin impurities in Ch. 6. We canceled
the gradient by making 50/50 mixtures of elongated 87 Rb in |F = 1, mF = −1i
55

Hall
probes1

Current
command2

Bottom coil

Servo board

4

MOSFETbank

Source

Drain

Gate

3

Main

Agilent

Top coil+15

Figure 4.9: This schematic displays our H-bridge and current feedback for ourlarge coils. When IGBT’s 2 and 3 are gated on, our coils are in anti-Helmholtz
configuration. When IGBT’s 1 and 4 are gated on, our coils are in Helmholtz
configuration.
and |F = 1, mF = 0i states because these states are miscible at our experiment’s
operating fields. During time-of-flight (TOF), we applied a Stern-Gerlach pulse
to separated the spin states. As we changed the current in the coils, the spinstates went from being on either half of the condensate, to spatially overlapped,to the opposite location. We set the current to the middle of the range where thespin states were mixed.
An H-bridge changed our large coils from anti-Helmholtz configuration for56

our MOT and quadrupole magnetic trap to Helmholtz to achieve ≈ 200 G fields
for our studies of the 40 K |F = 9/2, mF = −7/2i and |F = 9/2, mF = −9/2i Feshbach resonance in the same sequence. Figure 4.9 shows our H-bridge with 4
insulated-gate bipolar transistors (IGBTs) controlled by 2 TTL signals from ourcontrol computer to switch configurations. In either configuration, our coils’
current was supplied by an Agilent 6690A. We stabilized our current by comparing a voltage from a Hall probe to our computer command. Our servo board
sent a voltage to the gate of our MOSFET bank–an array of 19 MOSFETs inparallel–to feedback on the current.

4.4.1

Magnetic field stabilization with microwave servo

We used microwaves to address transitions between the two ground hyperfine states of 87 Rb. As shown in Fig. 2.1, these states are ∼ 6.8 GHz apart.
Our microwave source was a Stanford Research Systems SG384. We
duplicated the design of our colleagues at NIST in Trey Porto’s rubidium lab(Fig. 4.10). We manually set the output of the source to be 100 MHz awayfrom the zero field splitting. A mixer combined our microwave source withan ≈ 100 MHz frequency generated by our Novatech 409A to generate a frequency which addressed allowed transitions at a given bias magnetic field B.

We usually performed experiments with 87 Rb BECs in the F = 1 hyperfine
manifold. By pulsing on-resonant microwaves and taking an absorption image without repump light addressing the |F = 1i → |F ′ = 2i transition, we
could take images of our BEC in-situ and continue an experimental sequence57

rf Switch
ZYSWA-2-50DR

ZX60-4016E-S+

Figure 4.10: Microwave source schematic for all the components from source toantenna. Blue arrows denote the microwave frequency path, green arrows are
inputs, and red arrow indicate outputs for monitoring, debugging or tuning.
Components requiring power are marked. The high power microwave amplifier is water cooled. Modified image originally created by Lindsay LeBlanc forthe laboratory’s manual.

58

with a slightly depleted BEC. We used this technique to set up a microwave
field lock that fed back on our bias coils in the ez direction. This stabilized ourbias magnetic field against long-term drifts. During our sequence we applied a
1 ms pulse of our microwaves detuned by 500 Hz from our resonance with ourbias field set point. Resonance was determined by which transition we wereaddressing. After each pulse, we absorption imaged the transfered atoms witha probe on resonance with the |F = 2i → |F ′ = 3i transition. Figure 4.11A
shows an example of our counting the number (n1 , n2 ) of transferred atoms after each pulse versus scanning the bias field. To calibrate the sensitivity of our
microwave field lock we repeated this scan several times over a narrower field
range. The normalized asymmetry (n1 − n2 )/(n1 + n2 ) produced a signal approximately linear in a ∼ 0.5 mG range. The different markers in Fig. 4.11B represent
different scans. The long term drift of our bias field is apparent. Convertingfrom current to magnetic field, we found a slope of 0.35 ± 0.01 mG. Our fieldsensitivity was the root mean square of our residuals or 0.0027 mG. After every
sequence, we sent a voltage to our bias coil command proportional to the normalized asymmetry. Our integrator constant for our bias field lock was ≈ 60 s,
because that was the length of our sequence. We designed the rest of our sequence in phase with 60 Hz.
For our results sensitive to detuning, like the ones presented in Ch. 7, we
also used these images to provide an unbiased veto against shots with substantial field errors [85]. For our results in Ch. 6, this gave us a very reliable state
preparation which was sensitive to magnetic field. We have implemented themicrowave field lock with both xy and xz imaging1 .

1These are historical names. In the coordinate system I defined for the experiment they are
the e x ey and the (e x + ey )ez imaging systems

59

6K

4K

2K

0
12.33 12.34 12.35 12.36 12.37
Current in bias coils [A]B
Current in bias coils [A]

12.35012.34812.346

Bias magnetic field [G]Residuals [10-6 G]C
Number atoms transferred n

A

50
0
-50
1.164

1.163

12.3441.162-1.0

12.342
0.0
0.5
1.0
-1.0
-0.5
Normalized difference (n2-n1)/(n2+n1)

-0.5

0.0

0.5

1.0

Normalized difference (n2-n1)/(n2+n1)

Figure 4.11: Magnetic field stabilization. (A) The red(blue) markers represent
the number n1 (n2 ) of transfered atoms after the first(second) pulse versus current in bias coils. (B) Current in bias coils versus the normalized difference intransferred atoms between the two microwave pulses. Colors are for different
data runs. Purple was taken first, pink last. This shows drift in a short time(∼ 30 minutes). (C) Calibration for field sensitivity was calculated by finding
the RMS average of the residuals from a linear fit of the field to the normalized
difference of transferred atoms.

60

CHAPTER 5
S-WAVE SCATTERING OF FERMIONS

5.1

Introduction

We are interested in engineering topologically interesting systems that can support Majorana modes, which have non-abelian statistics[131]. One system thatcan support these modes are p x + ipy -wave superconductors that have Cooper
pairs with orbital angular momentum ℓ = 1 [131]. Section 3.2.1 discussed how
we can use one parameter, the s-wave scattering length, to characterize interaction in our cold atom samples. However, to support Majorana modes incold atom systems, interactions in the p-wave channel are required [138, 75].
For BECs, optically dressing the atoms with two-photon Raman transitions has
been shown to introduce d-wave and g-wave contributions into typical s-wave
scattering events [134]. This technique should be extend-able to fermions for
simulating odd higher-order partial waves.
Our first step in this experiment was to characterize s-wave scattering halos
for binary spin mixtures in a degenerate Fermi gases. Scattering halos appear
as circles in 2-D images of the density distribution of 3-D atoms after colliding
atomic clouds with different momenta. In BECs scattering halos are readily imaged with typical imaging techniques [55, 38, 125, 28], we found that the lower
density of our DFGs compared to typical BEC densities to be a barrier to fully
characterizing these low energy s-wave collisions. To enhance the number ofcollisions and increase our signal, we performed the measurement in the vicinity of a magnetically tunable s-wave Feshbach resonance (see Sec. 3.2.3). Even
with this enhancement, absorption imaging still had detection uncertainty that61

impacted the regions of low atomic density the most. We then used high intensity imaging to reduce our signal-to-noise. As the atoms absorb light fromthe probe beam, each scattered photon imparts momentum to the atoms. As
the atoms accelerate due to the radiation pressure of the probe beam, the atoms
are Doppler shifted out of resonance. This leads to under counting during theimaging process [82] and so we developed an image processing technique that
accounted for this [54]. We were able to characterize a s-wave Feshbach resonance in 40 K directly through measuring the fraction of atoms scattered during the collision at different bias magnetic fields and deduced the location andwidth of the resonance without relying on proxy effects. This work is publishedin New Journal of Physics 18 (2016) 013001.
We never completed the experiment to induce higher order partial waves in40

K. In the last section of this chapter, I report on a few different techniques we

explored before deciding to postpone this experiment with 40 K due to the daily
struggle with maintaining a constant, acceptably large atom number.

5.2

s-wave scattering experiment

In this section, I will discuss the procedures and ingredients for our s-wave scattering experiment that occur after creating a spin polarized |F = 9/2, mF = −9/2i
DFG of ≈ 4 × 105 atoms of 40 K at a temperature T ≈ 0.4 T F in a crossed optical dipole trap with frequencies (ω x , ωy , ωz )/2π = (39, 42, 124) Hz as described in
Chapter 4.

62

5.2.1

Preparation of 50/50 spin mixtures in DFGs

To map out the wide s-wave Feshbach resonance at 202.1 G for |F = 9/2, mF =−9/2i and |F = 9/2, mF = −7/2i 40 K atoms, we devised a standard procedure forcreating a spin mixtures above and below the resonance. We ramped the bias
magnetic field to 190.5 G (217.1 G) and turned on a 42.42 MHz (47.11 MHz) rf
field resonant with the Zeeman splitting between the two states when preparing the mixture below (above) the Feshbach resonance. We then sinusoidally
modulated the bias field at 125 Hz for 0.5 s, with a 0.14 G amplitude, producing
an equal mixture of the two hyperfine states. Because the rf circuit was optimized for frequencies of a few MHz, the rf intensity at ∼ 40 MHz was too low
for us to calibrate the coupling strength. Depolarizing a DFG decreased the T F
because the number of available states increased by a factor of two, decreasing
the Fermi momentum kF . Thus the temperature was larger in units of T F . This
depolarization allowed the fermions to re-thermalize [45]. We evaporated to alower optical potential, decreased their temperature, and narrowed our atoms’
momentum distribution.

5.2.2

Mapping out the Feshbach Resonance

After creating depolarized gases at fields ≈ ±15 G away from the 202.1 G Feshbach resonance, we used traditional methods to map out the resonance [109].
First, we measure inelastic collision losses in atom number versus bias magnetic field. To approach the set-point quickly and avoid unintended, additional
losses, we designed a two-step procedure using a pair of large coils and a concentric pair of small coils. Our large coils had a larger inductive time scale than63

our small coils. We implemented an H-bridge to change the configuration of
our large coils from anti-Helmholtz for the MOT and magnetic trapping stagedof our experiment to Helmholtz to reach magnetic fields of hundreds of Gauss(See Sec. 4.4). We approached resonance from both above and below the Feshbach resonance to map out the loss. Figure 5.1 displays both sets of data. After
creating a spin mixture at the desired field above (below) the resonance andevaporating further, we used the large coils to set the bias magnetic field B toa set-point 5.1 G above(below) the intended bias field. We held the atoms at
this field for 100 ms to allow the eddy currents induced by the large coils tosettle, and then used the smaller coils to quickly change the bias field the remaining 5.1 G. We held the bias field at its set-point for 200 ms. Our loss feature
was shifted to the lower field, repulsive scattering length side of the resonance
where a weakly-bound molecular state exists. This is consistent with previous
measurements [110].
Next, we explored Feshbach molecule creation and dissociation [111]. When
we ramped through the Feshbach resonance coming from the attractive, highfield side, Feshbach molecules were created. For accurate atom counting, wedissociated the molecules before taking an absorption image. The on-resonant
probe beam for the bare atom states was not on resonance for the Feshbach
molecule. In Fig. 5.2 we show our molecule creation and dissociation process.
During time of flight (TOF), we held at the set-point for the bias field for 1 ms.
After the 1 ms, we either continue to hold at the set-point to image or ramp awayfrom the resonance to dissociate the Feshbach molecules. To create Feshbach
molecules, we had to evaporate to a slightly lower temperature, we reduced
our dipole trap depth by 10% compared to the depth used in our loss experiments. Dissociation during TOF recovers most of the atoms above the Feshbach64A
K atom number [103]

60

40

20

0
200

201

202

203

40
K atom number [103]

B

30
20
10
0
198

200

202

204

206

Bias magnetic field B [Gauss]

Figure 5.1: Atom loss near Feshbach resonance. (A) Approaching the Feshbachresonance with a spin mixture from lower field. The atom number was measured after a 200 ms hold time at the bias field. (B) Approaching the Feshbachresonance with a spin mixture made above the resonance.The dashed line is thelocation of the resosnance B0 .
resonance.

5.2.3

Magnetic field calibration

The magnetic fields produced by our large bias coils around 200 G were independently calibrated by rf-spectroscopy on the |F = 9/2, mF = −9/2i to|F = 9/2, mF = −7/2i transition of 40 K. We prepared a spin polarized DFG as
65

Field Ramp A

Time-of-flight

B+5G(molecule creation)B
to imaging fields

Field Ramp B
B+5G(molecule creationand dissociation)
B

Detected K Atom Number [103]

5 ms

1 ms

to imaging fields
2 ms

80
60
40
20

Field Ramp A
Field Ramp B

0
200

201

202
203
Magnetic Field B [G]204

Figure 5.2: Feshbach molecule creation and dissociation. Schematics for the fieldramps for approaching the Feshbach resonance from above. Field Ramp A does
not dissociate molecules, field ramp B does. The detected atom number versus
magnetic field is plotted for each ramp.

66

described in Ch. 4 and ramped the bias coils to variable set-points while keeping the smaller coils at their zero field setting. We illuminated the atoms with arf field of frequency νr f and performed adiabatic rapid passage (ARP) by ramping the small bias coils 0.284 G or 2 A in 250 ms. We applied a Stern-Gerlach
pulse by switching our large coils back to anti-Helmholtz configuration and imaged the atoms to measure the fractional population in the |F = 9/2, mF = −9/2i
and |F = 9/2, mF = −7/2i states. We then fit the fractional population in the|F = 9/2, mF = −7/2i state as a function of current to a Gaussian function. Due
to our low rf coupling, we did not fit to the traditional Lorentzian model. Additionally, we often did not achieve complete transfer into the |F = 9/2, mF = −7/2istate. The center of the Gaussian corresponded to the resonant magnetic field
and the standard deviation of the fit parameters was our uncertainty. We used
the Breit-Rabi formula to determine the resonant field value at νr f . We did thisfor 5 different νr f , and the magnetic field produced by our MOT coils varied linearly with their commanded current. Our fit included the uncertainties in thecenter position of the Gaussian and the uncertainty of the slope gave our field
uncertainty. Our large bias coils produce 1.91 G/A with an uncertainty of 0.04 G.

5.2.4

Optical lattice for 40 K

For engineering higher order partial waves with 40 K as opposed to 87 Rb, modifications to the technique were necessary.

87

Rb is a BEC, therefore most of

the atoms occupy only on momentum state. Because the spread in momentum was smaller for 87 Rb, the scattering halos were more readily imaged dueto their higher densities as compared to DFG of 40 K. Loading a dressed statewith atoms in a equal superposition in the |F = 1, mF = ±1i internal states of67

Figure 5.3: Absorption image of a DFG after a π/2 Raman pulse at δ = 4ER . The
atoms close to 0 momentum are coupled to the other spin state.
87

Rb created two clouds with ±2kR , where ~kR = 2π~/λR is the recoil momen-

tum associated with the two-photon Raman transition. The momentum spreadof the BEC was negligible compares to kR . In contrast, for our DFG of 40 K, kR
was comparable to the Fermi momentum kF as pictured in Fig. 5.3. This image
was taken with a weak Raman π-pulse with a detuning δ = 4ER (see Sec. 5.4.1).
Additionally, we opted to perform the experiment with the |F = 9/2, mF = −9/2i
and |F = 9/2, mF = −7/2i internal states of 40 K to enhance their interactions witha Feshbach resonance. With only optical dressing, our fermions in the different
spin states would move away from each other with only 2~kR . We needed tohave a modified procedure to engineer collisions with optically dressed 40 K.
I installed a 1-D optical lattice co-propagating with our Raman A/C beams(see Sec. 4.3.3). Kapitza-Dirac pulses–the scattering of matter from pulses ofa standing wave light pulse–of this lattice created two spatially overlapping68

clouds with opposite momenta 4~kL , where ~kL = 2π~/λL is the lattice recoil momentum. The optical lattice had a beam waist of 160 µm. We used near-resonant
λL = 766.704 nm light by diverting a small amount of the 40 K repumping lightused during laser cooling. We only needed 0.06 mW for a lattice depth ≈ 8EL ,where EL = ~2 kL2 /2mK is the lattice recoil energy. We rapidly pulsed this latticeon and off with a double-pulse protocol [136]. The pulse sequence was optimized to transfer most of the atoms into the ±2~kL momentum states, shown in
Fig. 5.4B. The optimized pulse times were τ1 = 23 µs for the first pulse, τ2 = 13 µs
off interval, and τ3 = 12 µs for the second pulse. Since the initial Fermi gas had awide momentum spread and the lattice pulsing is a momentum dependent process [48], not all the atoms were transferred into the target momentum states.
We optimized our pulse times to minimize the atoms remaining in the untransferred state.
We absorption imaged the atoms after a 6.8 ms TOF and the ±2~kL atoms had
traveled ≈ 203 µm. Figure 5.5 includes examples of our absorption images. Dueto conservation of momentum, single scattering events are easily identified. After an elastic collision event between an atom traveling with −2~kL momentum
and an atom traveling with +2~kL momentum, the atoms will depart with equaland opposite momenta 2~kL at an arbitrary angle. Therefore, the single collisions-wave scattering halo should be a spherical shell with a radius corresponding
to the ±2~kL momenta imparted from the lattice and a thickness given by ~kǫ , asillustrated schematically in Fig. 5.4C.

69

A 1-D Optical Lattice

40

K DFG

B

B
Optical latticepotential ~8 ER

timeCafter time-of-flight

ey
ez

ex

Figure 5.4: (A,B) The 1D lattice was pulsed, imparting momentum to the atoms.
(C) After time of flight, the two clouds traveling along ±e x directions have separated and the atoms that underwent a single scattering event were evenly distributed in a scattering halo around the unscattered clouds.

5.3

5.3.1

s-wave scattering results

Image processing

We optimized our signal-to-noise ratio for low atom numbers by imaging withlong, high-intensity pulses [54]. Our imaging pulse was 40 µs with an intensity
0.6 IS AT . We used a probe beam approximately the size of the s-wave scattering halos to minimize interference fringes on the beam. These pulses imparted70

a non-negligible velocity from the radiation pressure and Doppler shifted theatoms out of resonance with respect to the probe. We simulated this recoil
induced detuning and corrected the images to count the fraction of scatteredatoms in our collision experiment. An example of images before and after processing are shown in Fig. 5.5. From our simulations, we created a look-up table
to go from the measured optical depth to a corrected column density along ez .
The processing constituted a ≈ 30 % change in the column density.

5.3.2

s-wave scattering halos

Our procedure for directly imaging s-wave scattering halos above and below
the Feshbach resonance were as identical as possible. We created a spin mixture above or below the Feshbach resonance, used the two-step procedure toapproach the bias field set-point, held at resonance for 3 ms, pulsed the 1-D optical lattice, and then turned off the optical dipole trap. The atoms expanded at
high-field for 2 ms. For data points below the resonance, the bias field remained
constant for these 2 ms before quickly going to the bias fields for imaging as
represented in Fig. 5.2 field ramp A. For data points above the resonance, we
held at the set-point for 1 ms and then ramped the bias field to larger bias fields
to dissociate the molecules as represented in Fig. 5.2 field ramp B. We collected
fifteen images at every set-point.
Figure 5.6 displays our corrected absorption images of s-wave scattering halos in the vicinity of a magnetically tunable Feshbach resonance. In each image, the two high-density clouds on the left and right are the atoms in the ±2kL
momentum orders that passed through each other unscattered. The cloud in71

A

Uncorrected, 196.8 G

Uncorrected, 200.4 G

3.0
2.7

2.0

y momentum

2.1
1.8
0

1.5
1.2
0.9

Optical Depth

2.4

0.6

-2.0

0.3
0.0

B

Corrected, 196.8 G

Corrected, 200.4 G

3.0
2.7

2.0

y momentum

2.1
1.8
0

1.5
1.2
0.9

Column density

2.4

0.6

-2.0

0.3
2.0

0
x momentum

-2.0

2.0

0
x momentum

-2.0

0.0

Figure 5.5: (A) Uncorrected optical depths, far from resonance (196.8 G) on theleft and close to resonance (200.4 G) on the right obtained from out unprocessed
absorption images. (B) Atomic column density obtained by applying corrections to raw optical depth above based on simulations [54].
the center were untransferred atoms after pulsing, and thus obtained no momentum. This cloud appears somewhat depleted closer to resonance due toscattering with atoms in the ±2kL momentum orders, this effect was not taken
into account in our analysis. The thin spread of atoms in the radius 2~kL withthickness 2~kǫ were the scattered atoms.

72

195.05 G

198.70 G

199.18 G

199.66 G

200.13 G

200.61 G

201.57 G

202.05 G

202.57 G

203.05 G

203.53 G

204.01 G

204.49 G

204.96 G

205.44 G

209.17 G

Figure 5.6: Corrected images of s-wave scattering halos

73

2.0

kF − kL
0
2.0

0

-2.0

5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0

Atomic density σ0 ρ × 100

R momentum [h̄kL ]kF + kL

X momentum [h̄kL ]Figure 5.7: The atoms within the Fermi momentum kF of each cloud center arein the unscattered region and counted towards the total unscattered number
Ntot . The atoms within the radius kL − kF ≤ r ≤ kL + kF but outside the unscatteredregion were counted as the number Nscat of single scattered atoms.
To extract the radial dependence of the 3D distribution from our 2D images,
we performed an inverse Abel transform assuming cylindrical symmetry alonge x , defined by the lattice [26]. We thus obtained the atomic distribution ρ(r, θ) as
a function of r, the radial distance from the scattering center, and θ, the angle between r and symmetry axis e x , integrated over φ. We then extracted the numberof atoms Nscat that underwent a single scattering event as a fraction of the total
atom number Ntot for each image, as shown in Fig. 5.7. Nscat was the numberof atoms outside the Fermi radius of the unscattered clouds, but inside the arc
created by rotating the Fermi momentum kF around the original center of thecloud (red arcs in Fig. 5.7). Ntot was the sum of Nscat and the counted atoms in
the two unscattered clouds. For both the scattered and unscattered quantities,
we extrapolated to include atoms that would fall outside the field of view of ourcamera. The undiffracted atoms were not counted as they did not contribute to
the scattering halo under study.

74

5.3.3

Determination of Feshbach resonance magnetic field position

Since we were well below the p-wave threshold temperature and p-wave collisions are energetically forbidden, our scattering cross-section σ = 4πa2 was fully
defined by the s-wave scattering length a [45]. After publishing our manuscripton s-wave scattering halos in 40 K, a result in a higher energy scattering regime
more readily imaged p-wave scattering halos because a larger fraction of theatoms scattered [126]. A single particle incident on a cloud of atoms with a surface density of N/A–where A is the cross-sectional area of the cloud and N is thenumber of atoms–will scatter with probability Pscat = σN/A. In our case, eachhalf Ntot /2 of the atoms was incident on the other half. Thus, the number of ex2/4A. Assuming A was constant for all
pected scattering events was Nscat = σNtot

our data, we defined a fit parameter b0 = 4πa2bg /4A, where abg is the backgroundscattering length. We thus adapted Eqn. 3.26 to obtain the fit function
Nscat∆= b0 1 −2
B − B0
Ntot

!2

+ C,

(5.1)

where B0 is the resonant field value, ∆ is the width of the resonance, and theoffset C accounts for any systematic difference in the initial and final intensity
images with no atoms present.
2
We extract Nscat /Ntotat each bias magnetic field by plotting the measured

Nscat /Ntot versus the total atom number Ntot for each of the fifteen images at that
set-point. These data were linear versus Ntot , and the slope and variance of thisfit gave the value and uncertainty of the data plotted in Fig. 5.8. The red curve
depicts a best fit of the model given in Eqn. 5.1 and the extracted parameters
were ∆ = 1.0(5) mT and B0 = 20.206(15) mT. For the fit, we excluded data points
75

0.00020

2
Nscat /Ntot

0.00015

B0 , this work
B0 , literature

0.000100.00005

0.00000
19.6

19.8

20.0

20.2

20.4

20.6

20.8

21.0

B [mT]

Figure 5.8: Normalized scattered population plotted versus bias field B.
Green(blue) dots represent data taken coming from below(above) the resonance
B0 . The red curve depicts the line of best fit. The hydrodynamic regime isshaded in gray, and data points within that regime were excluded from the fit.
very near the resonance. In this regime, our assumption that scattered atoms
only undergo a single scattering event breaks down and the atomic gas needs tobe treated hydronamically[39]. This led to a large uncertainty on our extracted
resonance width.

5.3.4

Conclusion

Our findings are in good agreement with the accepted values for the 40 K s-wave
Feshbach resonance for the |9/2, −9/2i and |9/2, −7/2i states are B0 = 20.210(7)mTand ∆ = 0.78(6) mT[111]. Although the data without the recoil induced
detuning correction were ≈ 30 % different from the corrected data, the optimal
parameters from fitting the uncorrected data were within our uncertainties from76

the values listed above. The resonance location was largely unaffected by thecorrection, as the scattering cross-section diverges there, making it insensitive
to the precise values. The width may have been affected, however the uncertainty on the fit is too large for the effect to be significant. Our imaging analysis,
optimization of SNR, and our study of s-wave scattering halos will be useful for
simulating higher-order partial waves with collisions between femions.
In our manuscipt, we noted that erbium and dysprosium DFGs [5, 90] havemuch higher densities than alkali DFGs and could more readily image low energy scattering halos. Recently, a result on studying dipolar interactions with
scattering halos was shared on arXiv [29].

5.4

Outlook: synthetic partial waves with 40 K

By the end of January 2014, a constantly decreasing atom number without explanation led us to postpone experiments with 40 K until after we develop and install a 2-D MOT for potassium. This was after a series of unfortunate events: theOctober 2013 federal government shutdown, running out of potassium, potassium oven controller malfunctions, unplanned air handling outages, etc. My
hope for this section is to document what we learned in combining Raman coupling and Kapitza-Dirac pulses for posterity.

5.4.1

Technique: Optical dressing then lattice pulsing

We began our attempts at simulating higher-order partial waves by optically
dressing spin polarized DFGs and then pulsing the 1-D optical lattice. The plan77was to adiabatically turn on Raman coupling, wait a quarter of a trap period forthe atoms to come to rest, then use the optimized Kapitza-Dirac pulses to create two clouds with opposite momenta. We tried this, but found that KapitzaDirac pulses excited the atoms to both the higher bands of the optical lattice
and the excited band of our dressed atom state. This led to several, hard tointerpret momentum orders in our absorption images. Notably, in rubidium,
there was a fall off of synthetic d-wave contributions to the scattering halos for
Raman coupling strengths above 6ER [134]. We wanted to maximize our signal
for synthetic p-wave partial waves, so we wanted Raman couplings less than
6ER . Consequently, our designed Kapitza-Dirac pulses for the optical latticewith potential depth ≈ 8ER excited the atoms to the higher Raman band as well(the wavelengths of the 1-D optical lattice and the Raman lasers were effectively
the same). Our solution to this was to modify the procedure. We would adiabatically ramp up the Raman to the desired coupling strength, wait a quarter
trap period, then quickly increase the Raman coupling, apply the Kapitza-Diracpulses, then quickly ramp back down to the Raman coupling set-point for TOF
and the collisions. This method was effective at producing two optically dressedclouds with equal and opposite momenta.
First, we dress our spin polarized gas with a two-photon Raman transition that couples together the |9/2, −9/2i and |9/2, −7/2i states as discussed inSec. 2.4.1. For the following experiments, we used laser light with wavelength
λR = 768.49 nm, defining our recoil momentum ~kR = 2π~/λR and recoil energy
ER = ~2 kR2 /2mK . The Hamiltonian describing this coupling in the |9/2, −9/2i,|9/2, −7/2i basis with the |9/2, −9/2i atoms at rest isk2 − δ/2~ΩR /2 x
HR = ER , ~Ω /2 (k − 2)2 + δ/2
R
x
78

(5.2)

Fraction of population transferredA

B

energy [ER]
15

0.3

10

5

0.2

4

2

2
5

C

0.1

4

quasimomentum kx [kR]

10

0.0
-8

-4
Detuning [ER]0

Figure 5.9: π-pulse with low Raman coupling to find −4ER detuning. (A) Fraction of population transfered versus Raman detuning found by scanning thefrequency difference in the Raman beams. (B) Bare state dispersion relationsat δ = −4ER . The gray line represents the Fermi energy and the shading is themomentum distribution of the atoms. (C) Absorption image of the atoms after
a π-pulse at δ ≈ 4ER . The momentum states around k x = 0 were transfered to thedifferent spin state.
where k x is the quasi momentum, δ = ∆ωR −∆ωZ is the detuning from the Raman
resonance, and ~Ω is the coupling strength.
We wanted to optically dress the atoms with zero detuning δ for a given
bias magnetic field. We measured our detuning by going to small couplingstrengths ~ΩR < 1 then slicing out a narrow range of momenta to the |9/2, −7/2iand scanning the frequency difference ∆ωR in the Raman beams to determine
the experiment parameters for −4ER detuning [37]. We used Raman A/C withbeam waists ≈ 160 µm, power ≈ 4 mW and a 75 µs π-pulse. Figure 5.9 is an
example of such a scan. We also show an example absorption image and thebare states for δ ≈ −4ER .
To measure the sloshing in the optical trap after turning on the dressing, we
79

position along ex [Pm]

150
100
50
0
-50

-100
-150
0

10

20

30

40

time [ms]Figure 5.10: Sloshing of optically dressed 40 K atoms in the dipole tap. The
red(blue) markers indicate the atoms were demapped into the |9/2, −7/2i(|9/2, −9/2i) state. The gray markers indicate the position of atoms versus time
without optical dressing.
set the frequency difference between the Raman laser beams such that δ = 0.
We adiabatically ramped the Raman coupling from 0 to about 7ER (35 mW) in
2 ms with a linearly increasing command to the AO driver. The originally at
rest |9/2, −9/2i would be loaded into the dressed state with 1kR momentum andbegan to oscillate. To image the atoms, we demapped back onto the bare spinstates before ramping off the Raman. During TOF, we would wait 1ms, thenramp the field by about 0.75 G in 0.5 ms before ramping off the Raman intensity
in 0.5 ms. To map into the |9/2, −9/2i (|9/2, −7/2i), we ramped to higher(lower)
magnetic fields. The position versus time for this data is plotted in Fig. 5.10.
So, one proposed technique for seeing higher order partial waves with 40 K
is:

1. Adiabatically ramp on Raman coupling (~Ω < 8ER , about 30 mW)
2. Wait a 1/4 trap period80

3. Ramp to higher Raman coupling in 50 µs (~Ω > 8ER , about 65 − 70 mW)
4. Kapitza-Dirac pulses with 1-D optical lattice at ≈ 8ER
5. Ramp back to desired Raman coupling set-point for synthetic partial
waves in 50 µs
6. Allow atoms to expand and collide in TOF at Raman Coupling set-point
for 1 ms
7. Demap onto one spin states by ramping the bias field 0.75 G in 0.5 ms andthen Ramp down Raman intensity in 0.5 ms

This technique has the advantage that there are no losses due to the Feshbachresonance until the experiment starts. However, the disadvantage of beginningwith a spin polarized gas is that it is harder to cool well and has a larger distribution of initial momentum states populated. Additionally, waiting the 1/4
trap period to begin the experiment is complicated and heats the atoms, further
increasing the momentum distribution and decreasing the atom number.

5.4.2

Technique: Pulse Raman, pulse lattice, optically dress

Here, I will describe a technique we implemented partly. We successfully pulsed
the Raman, then pulsed the lattices to create 2 clouds of opposite momenta andspin. If we had this pulsing technique for the s-wave scattering experiment,
it would have increased our scattered signal by a factor of two. We made thedecision to no longer work on 40 K before attempting to ramp on the Raman
to optically dress the colliding atoms. I do not know what might arise whenthat step is implemented. We successfully implemented the pulsing sequence81

A

optical lattice
22 .1ER

Raman ~11 ER

SC

position ex [Pm]position ex [Pm]B

position ey [Pm]position ey [Pm]Figure 5.11: Pulse Raman, then pulse lattice. (A) Pulse sequence of light. First
pulse on Raman coupling for a π-pulse, then pulse Kapitza-Dirac pulses foratoms optimized to transfer atoms with momtentum 2kR to a different band. (B)Absorption image of atoms after this pulse sequence with a spin polarized gas.
(C) Absorbtion image of atoms after this pulse sequence on a spin mixture.
for both a spin polarized gas and a spin mixture. Here is our alternative pulse
sequence technique:

1. Begin with spin mixture at the desired bias magnetic field
2. Apply a π-pulse at large Raman coupling (11ER , 70 mW) and δ = 0
3. Apply Kapitza-Dirac pulses optimized to transfer atoms with k x = ±2kR
and 1kR width to momentum k x = ∓2kL
4. Ramp on Raman in ≈ 50 µs

82

We began implementing this pulse sequence with a spin polarized gas. Applying the π-pulse with the Raman gave the |9/2, −9/2i atoms gave them a 2kRkick in the +e x directions. We then used the optimized Kapitza-Dirac pulses
illustrated in Fig. 5.11A and saw the atoms get a 2kL kick in the −e x direction.
Figure 5.11B is an example absorption image of a spin polarized gas after thispulse sequence. In Fig. 5.11, we show the same pulse sequence applied to a spin
mixture. This creates two clouds of opposite spin and momenta.
The advantages of this technique is there is no 1/4 period trap wait timeand spin mixtures are cooled more easily. However, we would have to consider
losses due having a binary spin mixture in the vicinity of a Feshbach resonanceearlier in the sequence. Additionally, the Feshbach resonance moves to higher
field with Raman coupling [133], so that would be an additional complication.

83

CHAPTER 6
SOLITONS IN THE PRESENCE OF IMPURITIESSolitons are spatially-localized, propagating excitations resulting from an interplay between nonlinearity and dispersion. We launched long-lived dark solitons in highly elongated 87 Rb Bose-Einstein condensates (BECs) and showedthat a dilute background of impurity atoms in a different internal state dramatically affects the soliton. With no impurities and in one-dimension (1-D), these
solitons would have an infinite lifetime, a consequence of integrability. Understanding the decay and destabilization of solitons is an outstanding problem.
We contributed to understanding this problem by observing how adding impurities decreases soliton lifetime and contributes to its Brownian motion [50].
By measuring decreasing lifetime with increasing impurities, we studied how
random processes destabilize solitons. Additionally, we describe the soliton’s
diffusive behavior using a quasi-1-D scattering theory of impurity atoms interacting with a soliton, giving diffusion coefficients consistent with experiment.

6.1

Introduction

Our BECs can be modeled by the one-dimensional (1-D) Gross-Pitaevski equation (GPE) derived in Sec. 3.3. Solitons are exact excited state solutions of the
GPE. In atomic (BECs), density maxima can be stabilized by attractive interactions, i.e., bright solitons [122]; or as here, density depletions can be stabilized
by repulsive interactions, i.e., dark solitons [30, 47]. For a homogeneous 1-D
BEC of particles with mass m with density ρ0 , speed of sound c, and healing84

√
length ξ = ~/ 2mc the dark soliton solutions z − v s t 
√  v s ξϕ(z, t) = ρ0 i + tanh  √
c ξs
2ξ s

(6.1)

are expressed in terms of time t, axial position z, the soliton velocity v s , andpsoliton width ξ s = ξ/ 1 − (v s /c)2 . Such dark soliton solutions have a minimum

density ρ0 (v s /c)2 and a phase jump −2 cos−1 (v s /c) dependent upon the soliton ve-

locity v s . These solitons behave as classical objects with a negative inertial massm s , essentially the missing mass of the displaced atoms. This implies increasing
the velocity reduces the kinetic energy, thus the addition of dissipation accelerates rather than decelerates dark solitons [101]. This can also be seen from thesoliton equation of motion [49]m s z̈(t) = −γż(t) − ∂z V + f (t)(6.2)

where V is the axial confining potential, γż is the friction force and f (t) is astochastic Langevin force. f (t) is responsible for Brownian motion of the soliton
and is connected to the friction coefficient γ through the fluctuation-dissaptation
thoerem. Both can be controlled by the impurity atoms in the system.

6.2

Solitons in elongated condensates

Idealized solitons are infinitely long-lived due to the integrability of the 1-D
GPE. Integrability breaking is inherent in all physical systems, for example dueto the non-zero transverse extent of quasi-1-D systems. Indeed in experiments,
solitons are only long-lived in highly elongated geometries [20, 130, 66], where
integrability breaking is weak. Cold atom experiments have profoundly advanced our understanding of soliton instability by controllably lifting integra85

r
z
x
y

z

Figure 6.1: To make elongated BECs, we spatially modulated one beam of ourcrossed-beam optical dipole trap (ODT). Orthogonal to the plane defined by
the ODT, we back-lit a razor blade to imprint a phase shift (orange) on half thecondensate and launch solitons.
bility by tuning the dimensionality [6, 83]. Here, we studied the further liftingof integrability by coupling solitons to a reservoir of impurities.

6.2.1

Soliton creation and optimization.

We created BECs in the optical potential formed by a pair of crossed horizontal
laser beams of wavelength λ = 1064 nm [87]. The beam traveling orthogonal tothe elongated direction of the BEC was spatially dithered by modulating the frequency on an acoustic-optic modulator in the beam path at a few hundred kHz
as shown in Fig. 6.1. This created a nominally flat-bottomed, time-averaged potential. We loaded into this optical potential after evaporating in a magnetic
quadrupole trap. Our atoms were initially in the |F = 1, mF = −1i internal
state. We transfered them into the |F = 1, mF = 0i internal state by ramping86

a radio frequency (rf) magnetic field on ≈ 68 mG or 0.047 MHz away from resonance, linearly ramping to resonance in 5 ms, and then ramping off the rf in
5 ms. Magnetic field drifts would contribute to unwanted admixtures of atoms
in the |F = 1, mF = −1i and|F = 1, mF = +1i states, so we actively stabilized
the magnetic field before state transfer as described in Sec. 4.4.1. We preparedN = 8(2) × 105 atoms in the |F = 1, mF = 0i internal state at a temperature
T = 10(5) nK. To reach the extremely cold temperatures necessary to realize
long lived solitons, we evaporated to the lowest dipole trap depth in which ourtechnical stability allowed us to realize uniform BECs. In our system, number
fluctuations increased at this low trap depth. Our system’s ≈ 250 µm longitudinal extent was about 30 times its transverse Thomas-Fermi diameter set by theradial trap frequency ωr = 2π × 115(2) Hz and chemical potential µ ≈ h × 1 kHz.
We then applied a phase shift to half of a condensate by imaging a back-lit,
carefully-focused razor edge with light red detuned by ≈ 6.8 GHz from the D2
transition for 20 µs to launch long-lived dark solitons [30, 47].

6.2.2

Soliton oscillations

We absorption-imaged our solitons after a sufficiently long time-of-flight (TOF)
that their initial width ξ s ≈ 0.24 µm expanded beyond our ≈ 2 µm imaging resolution. Figure 6.2A is an image of our elongated BEC taken 2 ms after the soliton’s inception. This density depletion feature had a velocity v s in the positive ez
direction and quickly decayed into several excitations including the dark soliton
of interest. After a few hundred milliseconds, the additional excitations dissipated and the remaining soliton was identified. Figure 6.2B displays an imageof a BEC with a dark soliton taken 0.947 s after its inception. The soliton is the

87

100

100

100
z [Pm]Cz [Pm]Bz [Pm]

A

0

-100
-50
2
50 0
r [Pm]OD [a. u.]

0

-100

0

-100

-50
50 0
2
r [Pm]OD [a. u.]

50 0
-50
2
r [Pm]OD [a. u.]

Figure 6.2: Soliton tracking. Each absorption image was taken with a 19.3ms
TOF. (A) Image taken 2 ms after the phase imprint and a longitudinal density
distribution obtained by averaging over the remaining transverse direction. An
image and 1-D distribution taken at time t = 0.942 s (B) with a soliton that has≈ 30% contrast and (C) without a soliton.
easily identified density depletion sandwiched between two density enhancements. We quantitatively identified the soliton position as the minimum of thedensity depletion from 1-D distributions (right panels of Fig. 6.2A and B). By
backtracking the soliton trajectory, we were able to distinguish the soliton evenat short times where there were multiple density depletions. Often, at longer
times after the phase imprint, there would be no soliton present. Figure 6.2C isan example of an image taken with an identical procedure to Fig. 6.2B but there
is no soliton present.
In addition to the extremely low temperature requirements, a sharp edge
was essential for creating long lived solitons. Placement of the edge was difficult. Going to a few seconds after the phase imprint and adjusting the edge
position at constant intensity in the phase imprint beam was efficient for finding the soliton at the beginning of the day. We found the turning on the watercooling would shift the position of the dipole trap by a beam waist in approximately four hours after machine turn-on in the direction orthogonal to the phase
imprint beam. The ∼ 10 µm difference in position of the dipole trap caused the88edge to be out of focus at the atoms. Luckily, our magnetic field lock fedbackon xz images (see Ch. 4) and we could manually feedback on the dipole trap
position. Alternatively, we found it helpful, but risky to leave the water coolingon overnight during data collection. Leaving on the water cooling on overnight
allowed the system to be in nominal thermal equilibrium when we turned it on
in the morning, giving a one hour instead of four hour warm up.
Figure 6.3A shows a series of 1-D distributions taken from time t ≈ 0 s to
4 s after the phase imprint. These images show three salient features: (1) thesoliton underwent approximately sinusoidal oscillations, (2) there was significant scatter in the soliton position, and (3) the soliton was often absent at longtimes. Items (2) and (3) suggests that random processes were important to thesoliton’s behavior. These random processes may include but are not limited tophase and density fluctuations or interations with phonons. The soliton’s position z–when present–is represented by the light pink symbols in Fig. 6.3B andthe darker pink symbols mark the average position hzi for each time t. To collectdata, we repeated each measurement 8 times.

6.2.3

Microwave tomography

To verify we had a soliton and not some other excitation such as a solitonic
vortex, we used rf-tomography techniques to study the density profile of theatoms orthogonal to the imaging direction [83, 76]. This technique works byapplying a magnetic field gradient in the direction orthogonal to the imaging
plane, then scanning the frequency of a microwave (µ-wave) Blackman pulse
to be in resonance with an atomic transition at different spatial locations in89soliton position, z [Pm]

A
100

0

-100

soliton position, z [Pm]

B
100

0

-100
0

1

2
time, t [s]34Figure 6.3: Soliton oscillations: (A), A subset of the data where each vertical slice
is the density distribution obtained from a single realization of the experimentplotted verse time t. (B), The axial position zi of the soliton (light pink) versus
time t for different realizations of the experiment. Dashed lines represent theedges of the elongated condensate. The dark markers represent the averagesoliton position hzi at each time t.
the condensate. We opted to use an microwave pulse to transfer the atoms
to a different hyperfine state. Using pulse on resonance with a different Zeeman sublevel within the same hyperfine state would have transfered the atoms
to two different states instead of one because our majority atoms were in the|F = 1, mF = 0i state. We turned a 2.4 mG/ µm gradient in the direction orthogonal to the imaging plane, then pulsed on a microwave field detuned from the|F = 1, mF = 0i → |F = 2, mF = −1i transition. We scanned this detuning to get
the in-situ density profile in this direction. As shown in Fig. 6.4, the measured

90

Transfered fraction

0.15

0.10

0.05

0.00
-10

-5

0
y [Pm]

5

10

Figure 6.4: Microwave tomography of in-situ atoms. We plot the fraction oftransferred atoms versus the position in the orthogonal direction. Fitting thedensity distribution in this direction to a 1-D Thomas-Fermi profile we find aThomas-Fermi radius of 7.1 ± 0.3 µm.
Thomas-Fermi radius 7.1 ± 0.3 µm in this direction is larger then the ∼ 4-5 µm
we expected from our measurements of speed of sound and our atom number calibration. However, the larger Thomas-Fermi radius can be due to thegravitational sag in this direction. Comparing the trap potential to the chemical potential indicated that gravitational sag was the most likely reason for thelarger radius. Figure 6.5 plots an optical potential without sag (red). The purple
curve is the same optical potential with sag. When we measured trap frequencies, the gravitational sag was present. Without sag, for a given beam waist andoptical power, the measured trap frequency would be larger. The pink curve
represents an optical potential with sag adjusted to match the measured trap

91

2.0

Energy/

[kHz]

1.5

1.0

0.5

0

-20

0

-10

10

y [Pm]

Figure 6.5: Effect of gravitational sag on optical potential. The optical potential without gravitational sag (red) was calculated from the ODT’s measured
beam waist 35 µm and trap frequency ωr = 2π × 115(2) Hz. Other curves(pink,yellow,purple) include gravitational sag. The approximate chemical potential µ is included for reference.
frequency. The yellow curve was adjusted until the intersection of the opticalpotential with the chemical potential µ (gray) was ≈ 14 µm. Because we did notmeasure our trap frequency as a function of power, we did not have a model forour optical potential. However, gravitational sag does have a large impact on
in-situ Thomas-Fermi radii. We did convolve the Blackman window pulse weused with a Thomas-Fermi profile, but the in-situ radius from the convolved fit
was consistent with the fit absent the convolution.
We used our µ-wave tomography technique to verify we had a soliton fromimages of our out-coupled atoms. Our resolution was set by the 1 ms pulse duration and energy shift from the applied magnetic field gradient. Our calculated
resolution was ≈ 0.6 µm. To ensure mean field effects during TOF did not affectour tomography of the atoms, we doubled the gradient immediately following

92the pulse during TOF so that the transferred atoms expanded in a different spacial location than the majority atoms. The absorption imaged captured both thetransferred and untransferred atoms. The series of images in Fig. 6.6 displays
the tomography of the atoms at 1.045 s and 1.420 s after soliton inception. The
transferred atoms were likely out of focus from the added field gradient leading to the odd looking soliton feature.The tomography data supported that welaunched a soliton due to the density depletion not being localized in the direction orthogonal to the imaging direction. When a soliton feature was present,
even at long time, our µ-wave tomography data confirms that it is a soliton, nota decay product such as a solitonic vortex or ring.

6.3

Injecting impurity atoms

Having established a procedure for creating solitons, we turned to the impact
of coupling to a reservoir of impurity atoms in a different internal state, thus
further breaking integrability.

6.3.1

Injection and number calibration

We controllably introduced a uniform [51] gas of NI impurity atoms in thermal
equilibrium with our BECs using an rf pulse resonant with the |F = 1, mF = 0i
to |F = 1, mF = +1i transition prior to evaporation to degeneracy [105]. This gave
impurity fractions NI /N from 0 to 0.062 in our final BECs. We use a Blackman
enveloped rf pulse at 9 G magnetic field to transfer the |F = 1, mF = 0i atoms primarily to the |F = 1, mF = +1i internal state [76]. We varied the impurity fraction
93

A

Reference image
50 Time =1.045s

1000z [Pm]

-50
-100
100
50
0
-50
-100

B

Soliton Feature
-10

-5

0

5

10

100 Reference image
50

Time = 1.420s

0z [Pm]

-50
-100
100
50
0

Soliton Feature

-50
-100
-10

-5
0
5
r(orthogonal to imaging plane) [Pm]10

Figure 6.6: rf-Tomography of solitons. For each panel, the serires of images
on top are the untransfered atoms for reference. The bottom series of images
represennts a slice of the 3-D condensate at different z-positions using our rftomography techniques.

94

by tuning the rf amplitude. Even though the fraction of impurity atoms before
evaporation determined the fraction after evaporation, they were not equal due
to a preferential evaporation of the minority spin state [105].
We characterized the impurity fraction through careful calibration of our absorption imaging. First, we find the saturation intensity I sat and effective satef f
uration intensity I sat= αI sat that characterize our imaging system [113]. We

imaged an optically thin cloud of atoms, i.e. a gas above T c , as a function ofprobe power to determine I sat = 2.5 × 104 in counts per pixel on the camera.
We then use measurements of the probe’s beam waist and power to determine
the intensity at the atoms corresponding to counts on at the camera. Knowing
the intensity at the atoms defines α = 5.5 for our imaging system and we now
have a corrected method to go from the optical depth in an absorption image tonumber of atoms.
We can also calculate our atom number by using Castin-Dum methods to calculate an in-situ Thomas-Fermi radius from a Thomas-Fermi radius after timeof flight and the trap frequencies [34]. We found these methods to agree formost optical depths. The low optical depths, corresponding to low densities, iswhere the counting method and the Castin-Dum method differed. However we
did not expect Castin-Dum to work well at low densities due to the method’s
dependence on chemical potential. We then used our corrected counting anda Stern-Gerlach technique during TOF to measure the relative fraction of the
impurity atoms after evaporation. Typically, we prepared our spin composition
above T c . With this technique, we had some difficulty preparing the atoms only
in |F = 1, mF = 0i without impurities. We modified the procedure to prepare thespin state at an optical depth just below T c . If we preformed our state prepara-95

6

Impurities [%]

5

4

3

2

1

0

No impurities
05

10
rf amplitude [mVpp]

15

Figure 6.7: Impurities versus rf amplitude in laboratory units. The uncertainties
are the standard deviation of five data points. State preparation in the majority
spin state with thermal atoms led to a final state with 1.2% impurity atoms. Thiswas overcome by preparing the no impurities sample at a slightly lower opticalpotential before continuing to evaporate to the final trap depth.
tion in a gas above T c , we found that there were always 1.2% impurity atoms.
This may be due to gradients across the larger extent of the thermal atoms having some atoms be out of resonance with rf fields used for state preparation.

6.3.2

Temperature measurements with impurities

Below the majority atom’s condensation temperature T c = 350 nK, we removed
the majority atoms and fit the TOF expanded the impurity atoms to a MaxwellBoltzmann distribution [105]. We removed the majority atoms before TOF by96

pulsing on a µ-waves resonant with the |F = 1, mF = 0i → |F = 2, mF = −1i transition, then blew away the atoms with our probe resonant with the F = 2 → F ′ =3 optical transition. No repump light was on during this process to allow the
impurity atoms to expand unperturbed. Once the temperature was below T c forthe impurity atoms, fits to a Maxwell-Boltzmann distribution give temperature
estimates that can significantly underestimate the true temperature. With thismethod, we did observe a decrease in temperature at higher concentrations of
impurity atoms consistent with data from similar experiments [105]; however,the change in temperature we observed was smaller than the uncertainty in our
measurement. Fitting the small number of impurity atoms to a Bose distribution
was challenging due to the signal-to-noise and the addition of another free parameter. To limit the number of free parameters, we preformed a global fit of thetemperature on a few different impurity fractions and constrained the chemical
potential µ to be negative. This provided an estimation of the temperature withlarge uncertainties. We found for our usual operating parameters and based on
information from both temperature measurements, T = 10(5) nK.

6.4

Solitons in the presence of impurities

6.4.1 Impact on oscillations
Figure 6.8 displays the soliton position versus time for a range of impurity fractions. Adding impurities gave two dominant effects: further increasing the scatter in the soliton position z and further decreasing the soliton lifetime. These
effects manifested as a reduced fraction f s of images with a soliton present and97

an increase in the sample variance Var(z) =P(zi − hzi)2 / (M − 1) computed using

the number M of measured positions zi at each time. Additionally, the soliton
oscillation frequency was slightly shifted with impurities resulting from an unintentional change in the underlying optical potential. This change also slightly
reduced the BECs longitudinal extent. This is not an effect due to the presenceof the impurities.

6.4.2

Impact on lifetime

The addition of impurities had a dramatic impact on the soliton lifetime. While
we lack a quantitative model of the soliton’s decay mechanism, there are several reasons to expect a finite lifetime. When dissipation is present, solitons
accelerate to the speed of sound and disintegrate. Additionally, because ourtrap geometry has a finite transverse extent, quantified by the ratio µ/~ωr ≈ 9,solitons can be dynamically unstable and can decay into 3-D excitations [94].
Our soliton’s initial velocity v s ≈ 0.3 mm/s, roughly 1/5 the 1-D speed of soundc ≈ 1.4 mm/s[137], implies it is in an unstable regime, where, as observed, itshould decay [101]. Furthermore, numerical simulations show that in anharmonic traps solitons lose energy by phonon emission, accelerate, and ultimately
decay [106]. All of these decay mechanisms can contribute to the soliton lifetime
even absent impurities.
The added impurities act as scatterers impinging on the soliton, further
destabilizing it. This effect is captured in Fig. 6.9, showing the measured survival probability f s versus time for a range of impurity fractions. We fit the98

1.2% impurities

100
0
-100

3.1% impurities

100
0soliton position, z [Pm]-100
4.1% impurities

100
0
-100

4.7% impurities

100
0
-100

6.3% impurities

100
0
-100
0

1

2
time, t [s]34Figure 6.8: Impact of impurities. Here, we plot the position zi of the soliton(light pink) versus time t after phase imprint for different impurity levels. The
dark pink markers are the average positions hzi for each time t. Dashed lines
represent the extent of the condensate on either end versus t.

99

1.0

No impurities
0.5
0.0
1.0

1.2% impurities
0.5
0.0

survivial probability, fS

1.0

3.1% impurities
0.5
0.0
1.0

4.1% impurities
0.5
0.0
1.0

4.7% impurities
0.5
0.0
1.0

6.3% impurities
0.5
0.0

0

1

2
time, t [s]34Figure 6.9: Measured survival probability. Histograms of fractional count ofdata with solitons f s versus time t after phase imprint. The blue solid line is a
fit to a survival function for a lognormal distribution from which we extract the
lifetime τ. For each impurity fraction, we stopped collecting data when f s fell
below about 0.2.

100soliton lifetime W [s]

2.0

1.5

1.0

0.5

0.0

0.00

0.02
0.04
Impurity fraction NI/N

0.06

Figure 6.10: Soliton lifetime in the presence of impurities. The lifetime τ extracted from fits of the survival fraction f s versus impurity fraction NI /N in thecondensate.
survival probability"#− ln(t/τ)1
f s (t) = 1 − erfc
,
√
2
2σ

(6.3)

essentially the integrated lognormal distribution of decay times, suitable for decay due to accumulated random processes, to our data[41]. The survival probability f s (t) has a characteristic width parameterized by σ and reaches 1/2 attime τ which we allows us to associate τ with the soliton lifetime. Figure 6.10
shows the extracted lifetime τ versus impurity fraction NI /N, showing a monotonic decrease. Our maximum NI /N gives a factor of four decrease in lifetime τ.

101

6.4.3

Enhanced diffusion

The second important consequence of adding impurities was an increased scatter in soliton position z, reminiscent of Brownian motion. Indeed, as shown in
Fig. 6.11, this scatter, quantified by Var(z), increased linearly with time. We obtained the diffusion coefficients D as the slope from linear fits to these data andcalculated D using a quasi-1-D scattering theory. The energy of the infinitely
long 1-D system is given by the GPE energy functionalE ϕ, ψ =


Z

dz

~2
g
g′
~2
|∇ϕ|2 +|∇ψ|2 + |ϕ|2 |ϕ|2 + |ϕ|2 |ψ|2 ,
2m
2m
2
2

(6.4)

describing the majority gas interacting with itself along with the impurities withinteraction coefficients g and g′ , respectively. The fields ϕ and ψ denote thecondensate and impurity wavefunctions. Since the impurities are very dilute,
we do not include interactions between impurity atoms. Impurity scattering
states with momentum kz in the rest frame of the soliton are described by thereflection coefficientR(k) =

1 − cos(2πλ),
cosh(2πkz ξ) − cos(2πλ)(6.5)

where λ(λ − 1) = g′ /g. In 87 Rb, we have g ≈ g′ , giving λ ≈ 1.5. The scattering
problem is fully characterized by R(k) and the problem is reduced to that of aclassical heavy object moving through a gas of lighter particles.
We treat the soliton as a heavy classical object with a kinetic equation including a weak stochastic force due to elastic collisions with the much lighter impurity atoms. The collision integral can be expressed in Fokker-Planck form [86]with diffusion coefficientD=(kB T )2
,
B

(6.6)

giving Dt = Var(x). Momentum diffusion is described by the transport coeffi102

0.004

No impurities

0.002
0.000
0.004

1.2% impurities

0.002
0.000
0.004

3.1% impurities

Var(z) [mm2]

0.002
0.000
0.004

4.1% impurities

0.002
0.000
0.004

4.7% impurities

0.002
0.000
0.004

6.3% impurities

0.002
0.000
0

1

2
time, t [s]34Figure 6.11: Brownian diffusion for different impurity fractions. The uncertainties in Var(z) are the sample standard deviation. The slope of the linear fit is themeasured diffusion D. Dashed lines represent the slope’s uncertainty.

103

D [10-4 mm2/s]

20

5 nK
10 nK
15 nK

10

0
0.00

0.01

0.02
0.03
0.04
0.05
impurity fraction, NI/N

0.06

Figure 6.12: Brownian diffusion constant versus impurity fraction. The diffusion coefficient D versus impurity fraction NI /N. The experimental results(markers) are extracted from the slope of a linear fit of the sample variance Var(z)versus time t. The uncertainty in D is the uncertainty from that fit. See Sec.6.3.1
for explanation of uncertainty in NI /N. The theory curves (line and dashed lines)
are not fits.
cient [52]B = 2~X Z ∞ dkz
m,l

−∞ 2π

k2

∂ǫ
R(kz )n(ǫ) [1 + n(ǫ)] ,
∂kz

(6.7)

where ǫm,l (k) = ~2 k2 /2m + ~2 j2m,l /2mR2 is the impurities’ quasi-1D dispersion withquantized states in the radial direction, where jm,l is the lth zero of the mth Bessel
function. The quantization in the radial direction is due the tight confinementin the radial direction as compared to the temperature. n(ǫ) is the Bose-Einstein
distribution for impurity atoms excluding the condensed impurity atoms. We
model the impurity atoms as being in a uniform potential with cylindrical volume the size of the majority condensate. R(kz ) is the reflection coefficient definedin Eqn. 6.5. The term [1 + n(ǫ)] accounts for bosonic enhancement.
Figure 6.12 plots D measured experimentally (markers) and computed theoretically (solid and dashed lines) as a function of NI /N. The quasi-1-D scattering
theory accurately captures the soliton diffusion in the presence of impurities.
For the theory, the abrupt change in dependence on impurity fraction is due to104the condensed impurity atoms not contributing to the soliton diffusion. Any additional impurity atoms are added to the impurity condensate, not the thermal
reservoir of scatterers. In our quasi-1-D theory, the soliton is not reflectionless
to phonons as in the true 1-D problem, but we did not considered them in ourmodel. Thinking about how this may impact our results, adding more scatterers
would contribute to the transport coefficient B, and since D ∝ 1/B the diffusion
coefficient should be smaller with more scatterers. This is why our measurement of a suppressed diffusion coefficient at NI /N = 0 is intriguing. With such
scatterers as not reflectionless phonons, we expected some baseline diffusion
coefficient at NI /N = 0 to be larger than with impurities. Indeed, if the diffusion
at NI /N = 0 was due to technical noise in the velocity from the soliton launching
technique, we would again expect a suppression of the diffusion coefficient withthe addition of impurities, not enhancement. Thus, in the absence of impurities,
the precise mechanism of dissipation cannot be identified within the scope of
our theory and the observation of reduced diffusion remains an outstanding
problem.

6.5

Conclusions and outlook

Realizing solitons in spinor systems with impurity scatterers is an exciting
playground for studying integrability breaking and diffusion of quasi-classical,
negative-mass objects. Our observed reduction in soliton lifetime with increasing impurity fraction is in need of a quantitative theory. For the case of no
impurities there is a further open question for both theory and experiment of
whether friction and diffusion can be present even in the case of preserved integrability due to non-Markovian effects, as was recently discovered for bright105

solitons [49]. We began this experiment to learn about solitons. In particular, we
would like to load solitons into the synthetic dimensions experiment to achieve
the lofty experimental goal of chiral edge solitons. Understanding how to load
solitons into systems with optical lattices or Raman coupling would be the next
steps before continuing to add complexity. In this chapter, we studied how thesoliton interacts with uniformly distributed atoms in a different internal spinstate.

106

CHAPTER 7
BOSE-EISTEIN CONDESATES IN THE QUANTUM HALL REGIME

In this chapter, I discuss experiments in which we realized a large effective magnetic field for charge neutral atoms in a hybrid two dimensional (2-D) lattice.
Our hybrid 2-D consists of a 1-D optical lattice as one dimension and an internal
state of the atom as another, synthetic dimension. This specific choice of internal
states of 87 Rb enabled single site resolution in the synthetic dimension throughthe use of a Stern-Gerlach pulse during time of flight (TOF). We used a twophoton Raman transition to induce tunneling in the synthetic dimension and to
imprint an engineered phase onto an atom, an effective Aharonov-Bohm phase
φAB /2π. This phase fully defined an effective magnetic field. We directly imaged
the eigenstates of the hybrid 2-D system with and without the synthetic magnetic flux. Then we studied dynamics by exciting coherent super positions ofmagnetic subbands, analogs to Hofstadter bands, and observed bulk and edge
dynamics. The single site “spatial” resolution in the synthetic dimension enabled us to directly image bulk excitations, normally inaccessible in condensed
matter systems, and skipping orbits of edge site excitations. This work is published in Science 349 (2015) concurrent with a similar technique implemented
with fermionic ytterbium by Mancini et al. [124, 93].

7.1

Introduction

Cold atoms are highly tunable and controllable systems that can realize exotic
topological states. The quintessential topological state is the quantum Hall effect, a topological insulator formed in a 2-D electron gas with an insulating bulk
107and conducting, protected edge states [65]. Many groups see the realization ofquantum Hall states in cold atoms as a stepping stone to creating more exotic
topological states. Creating large artificial gauge fields, analogous to large magnetic fields, is one way to realize the extreme quantum limits required for quantum Hall-type effects in cold atoms. Previous, pioneering methods requiredlarge Raman laser coupling, leading to heating through spontaneous emission,and had the disadvantage of the realized effective magnetic field strength limited by the linear extent of the condensate [88]. Other experimental methods
for creating large artificial gauge fields reduce the limitations imposed by spontaneous emission, but introduce other sources of heating leading to lifetimes
well below 1 second. These methods include shaking optical lattices to introduce a vector potential [123, 79] and laser-assisted tunneling in a titled opticallattice [3, 97]. Our technique presented in this chapter minimizes the required
Raman laser coupling and has lifetimes due to spontaneous emission larger than
10 seconds.

7.2

Synthetic Dimensions: Particle in a Box

The tight-binding Hamiltonian
Htb = −

Xhj,mit x | j + 1, mih j, m| + t s | j, m + 1ih j, m| + h.c.

(7.1)

with hopping amplitudes t x and t s governs the motion of particles moving in a
2-D lattice [8] with sites labeled by j and m. We implemented a such 2-D latticeby combining a conventional optical lattice to define the long axis of our system(e x direction, with sites labeled by j, and effective tunneling strength t x ), withthree sequentially coupled internal “spin” states to define the short axis of our
108

A

B
1-D optical lattice87

Rb BECm = -1
B0

ey

Brf

m=0

ez
ex

İ
m = +1

C

D
j=0

j=1

m = +1

tx

m=0

tx

m = -1

tx

j=2

j=3

j=4
Energyj = -1

ts
ts

-2 -1

0 +1 +2m

Figure 7.1: Particle in a Box with Synthetic Dimensions. (A) Experimental setup
with 1-D optical lattice, bias magnetic field B0 and oscillating rf field Br f . (B)The rf field coupled the three internal states of 87 Rb with Rabi frequency ΩR and
detuning δ. The m = 0 internal state is shifted by the quadratic Zeeman shiftǫ. (C) Our 2-D hybrid lattice hoping t x in the spatial dimension e x and t s in thesynthetic dimension e s . (D) Potential along e s can be modeled as a square wellwith infinite hard walls at m = ±2
system (e s direction, with just three sites labeled by m ∈ {−1, 0, +1}, and effective
tunneling strength t s ) [23]. Figure 7.1 shows a schematic for the experimental
set up, the coupling of the different internal states, and the resulting hybrid 2-D
lattice. As shown in Fig. 7.1A, we used a 1-D optical lattice, formed by a pairof λL = 1064.46 nm laser beams counter-propagating along e x with beam waists≈ 160 µm × 135 µm for the forward beam and ≈ 126 µm × 150 µm for the retroreflection. This lattice defines the single-photon recoil momentum ~kL = 2π~/λL
and recoil energy E L = ~2 kL2 /2mRb , where mRb is the atomic mass. The tunneling t x in the spatial dimension was controlled by adjusting the intensity of thelaser beams to control the potential V due to the optical dipole force. The internal “spin” states are the Zeeman sublevels of the f = 1 hyperfine manifoldof 87 Rb and the tunneling between these sites is controlled by an rf magnetic
field. The amplitude of this field controls the tunneling strength t x along the109spin “direction” e s , proportional to the Rabi frequency ΩR . The frequency of therf field was set to be in resonance with gµB B0 /h ≈ 0.817 MHz Zeeman splitting,giving δ = 0.0. The corresponding quadratic Zeeman shift additionally lowered|m = 0i by ǫ = 0.05E L as shown in Fig. 7.1B. Since there are no other Zeeman sublevels, this system effectively has an infinite repulsive potential at sites m = −2
and m = +2 as shown in Fig. 7.1D. This allows for the formation of robust edge
states in the presence of an effective magnetic field, analogous to the conducting edge states discussed in Sec.2.3.2. Interestingly, the synthetic dimension e s
is an exact realization of the tight-binding model with only tunneling to nearest
neighboring sites and no tunneling to next to nearest neighboring sites. For thesites in the optical lattice, the tight-binding model is a very good approximation(see Sec. 2.2.1).
We began our experiments by directly imaging adiabatically loaded eigenstates of the three lowest bands of our hybrid 2-D lattice. The Hamiltonian forour system in momentum space in the |l, mi basis is





(k x + 2l)2 + mδ + (|m| − 1)ǫ












V/4
Hr f = E L 





~ΩR /2









0


l = l ′ , m = m′|l − l′ | = 1, m = m′

(7.2)

l = l , |m − m | = 1′′

otherwise,

where the quasimomentum k x is in units of kL and −∞ < l < ∞ is an integer. Now
we find the eigenstates and eigenvalues of our 2-D lattice and plot the 6 lowest
energy bands in Fig. 7.2. This was done numerically with lmax = 5, but there were
not any significant quantitative differences when lmax = 25. The parameters forthis simulation were (~ΩR , V, δ, ǫ) = (0.5, 6, 0.0, 0.05)E L . The band structure of our
hybrid 2-D lattice is similar to the band structure for a 1-D periodic potential,
110

1.0

4.0

0.5

Energy [EL]

3.0

2.0
0.0
1.0

0.0
-0.5

-1.0

-1.0

-0.5

0.0

0.5

1.0

-1.0

quasimomentum kx [kL]

Figure 7.2: Bloch bands for hybrid 2-D lattice as a function of quasimomentum
k x in the e x dimension
however, each band in our system now has 3 subbands. The colorbar represents
the magnetization hmi in each subband. Because δ = 0, the bands and subbands
all have the same magnetization hmi = 0. Adjusting the detuning δ changes hmi
for the lowest and highest energy subbands within a band by adjusting which
m site has the lowest or highest energy.
We now load our atoms into these different magnetic subbands. We can dothis by initializing the atoms in different m states within the f = 1 hyperfine
manifold, then adiabatically turning on our hybrid 2-D lattice. We initialize ourBECs, confined in our typical crossed-beam optical dipole trap, with trap fre111

quencies (ω x , ωy , ωz )/2π = (50, 40, 110) Hz, into one of three internal |mi stateswith a bias magnetic field B0 ez . To load into the ground, 1st and 2nd excited
subband, we initialized our BEC in the m = +1, 0, −1 Zeeman sublevel, respectively. Then, we adiabatically loaded into the lowest band of the optical latticeby linearly increasing the optical potential from 0 to 63 ∼ 6E L in 200 ms. The
lattice depth V ≈ 6E L (about 63 mW) gave a hopping strength |t x | ≈ 0.05E L alonge x . To turn on hopping in the synthetic dimension, we would linearly ramp
the magnetic field to (B0 − 0.06B0 )ez , and then linearly ramp on an rf magnetic
field. Then we would linearly ramp the magnetic field back to resonance B0 ez in
250ms, adiabatically turning on tunneling in the synthetic dimension.
After preparation, we used a measurement procedure common to all experiments in this chapter: we simultaneously removed all potentials and coupling
fields (toff < 1 µs), which returned the atoms to bare spin and momentum states.
The atomic cloud expanded for a ≈ 18 ms time-of-flight (TOF) period. During
TOF a 2 ms magnetic gradient pulse was applied, Stern-Gerlach separating thethree |mi states. The resulting 2-D column density was recorded using standard absorption imaging techniques, giving the normalized momentum distributions nm (k x ) with perfect single lattice site resolution along e s as shown in
Fig. 7.3A. The site-resolved fractional population distributions for each image
are displayed next to their corresponding absorption image in Fig. 7.3B.
First, we loaded into the ground subband by initializing the BEC in the|mi = +1 internal state which has the lowest energy in a bias magnetic fieldB0 ez . The top panel in Fig. 7.3A shows typical data for this procedure. Note, theRfractional population nm = dk x nm (k x ) resembles (Fig. 7.3B) that of a particle ina discretized box along e s , while the momentum distributions, typical for atoms

112

A

B
1.0

+2
+1

0.5

0

-2

0.0

+2

1.0

+1
0.5

0
-1
-2

0.0

+2

1.0

Fractional Population, nm

Lattice Site, m

-1

+1
0.5

0
-1
-2

0.0
-2.0

0.0
kX/kL

+2.0

-2

-1
0
+1
Lattice Site, m+2

Figure 7.3: Particle in a box along e s data. (A) Absorbtion images with momentum resolution in e x and site resolution in e s . (B) Fractional population nm for the
ground, 1st and 2nd excited subbands vs lattice site in e s . The gray dashed lineis the probability density for a particle in a box with infinitely hard potential
indicated by the solid gray arrows.

113

in an optical lattice [59], have the same profile for each m site. This demonstrates
that the two directions are uncoupled. The grey dashed lines in Fig. 7.3B are theparticle in a box wave probability densities:|Ψ1 (qm )|2 = sin πq m2|Ψ2 (qm )|2 = cos (πqm )|Ψ3 (qm )|2 = sin

(7.3)

3πqm
,
2
!

which can be found in any quantum mechanics text book. qm is a non-integer
position coordinate along e s in units of lattice site m for purposes of plotting.
Then, we initialized the BEC in the |mi = 0 and −1 internal state and adiabatically loaded following the same procedure. For the three lowest subbands ofour system, we measured the momentum distribution along e x and the corresponding probability density for a particle in a box along e s .

7.3

Synthetic Flux: Quantum Hall Ribbon

Now, we will introduce complex hopping into our hybrid 2-D lattice by using
two-photon Raman transitions instead of an rf magnetic field to enable hopping in the synthetic m direction. The Raman lasers with wavelength λR copropagated with the 1-D optical lattice (Fig. 7.4A) and introduced a phase to
the wavefunction of the atom as it hopped from site to site directly proportional
to the ratio λL /λR . This acquired phase is an effective Aharonov-Bohm phase
φAB /2π and fully defines the effective magnetic field. For the wavelengths oflight we choose φAB /2π ≈ 4/3 around each plaquette corresponding to a fieldof ∼ 20, 000 T in material with typical lattice constant a ∼ 5 Å. By introducing
this complex hopping into our hybrid 2-D lattice, we engineered a system that
114

A

ey

Lattice / Raman+ lasers

87

Lattice / Raman- lasers

B0

ez
ex

B

Rb BECj = -1

j=0

j=1

j=2

j=3

j=4

m = +1m=0

m = -1Figure 7.4: Experimental schematic for hybrid 2-D lattice with flux. (A) Two
orthogonally-polarized Raman lasers co-propagate with the 1-D optical lattice
and induce tunneling t s . (B) The wavelength mismatch between the lattice andthe Raman lasers introduces a complex hopping along t x creating an effective
Aharonov-Bohm phase φAB .
is described by the Harper-Hofstader Hamiltonian for a charged particle in a
2-D lattice.

7.3.1

The Harper-Hofstadter Hamiltonian

The Harper-Hofstadter Hamiltonian
HHH = −

Xhit x | j + 1, mih j, m| + t s | j, m + 1ih j, m| + h.c.

(7.4)

j,m

with complex hopping amplitudes t x and t s governs the motion of charged particles moving in a 2-D lattice [64, 68] with sites labeled by j and m, the situation
which we engineered for our neutral atoms. Analogous to the Landau gauge
in continuum systems, we describe our experiment with real t s (no phase), andwith complex t x = |t x | exp(−iφAB m) dependent on m. As shown in Fig. 7.4B, the115sum of the tunneling phases around any individual plaquette is φAB .

7.3.2

Eigenstates of the Harper-Hofstadter Hamiltonian

To find the eigenstates of the Harper-Hofstader Hamiltonian, we again write
our Hamiltonian as a matrix in the |l, mi basis for our hybrid 2-D lattice. Similar
to controlling magnitude of the rf field, the intensity in the Raman lasers gavea laboratory tunable effective tunneling strength |t s | ∼ |t x | along the spin “direction” e s , proportional to the Rabi frequency ΩR . However, now there are 2~kR
momentum imparted by these transitions and this results in a spatially periodic
phase factor exp(i2kR x) = exp(iφAB j) [35]. The sign of φAB was controlled by the
relative detuning of the Raman lasers. Consequently, our Hamiltonian is





(k x + 2l + 2mφAB )2 + mδ + (|m| − 1)ǫ












V/4
HHH = E L 





~ΩR /2










0

l = l ′ , m = m′|l − l′ | = 1, m = m′

(7.5)

l = l , |m − m | = 1′′

otherwise.

Figure 7.5 plots the numerical simulation of the 3 lowest magnetic subbands
with the parameters (~ΩR , V, δ, ǫ) = (0.2, 6, 0.0, 0.05)E L . There are two very noticeable differences between this band structure with flux and the other without
flux. First, the magnetization is no longer the same for each subband. Insted themagnetization depends on the quasimomentum k x . Second, the lowest band
now has 3 local minima. This will be essential for imaging localized edge states.
In IQHE systems these states would govern the conductivity, but as individual
eigenstates they exhibit no time-dependance. We will now load into the groundband of our Hamiltonian. In contrast to the previous Hamiltonian, initializing
116

1.0

-0.7

0.5

Energy [EL]

-0.8

0.0
-0.9

-0.5

-1.0

-1.0

-0.5

0.0

0.5

1.0

-1.0

quasimomentum kx [kL]

Figure 7.5: Bloch bands for hybrid 2-D lattice with magnetic flux as a functionof quasimomentum k x in the e x dimension.
the states into the different Zeeman sublevels no longer adiabatically loads into
the different sublevels. Now, the different states will load into the bulk or edge
states for initializing in the m = 1 or m = ±1 states, respectively.
The two counter-propagating λR = 790.04 nm Raman lasers along e x haveorthogonal polarization, wave-vector kR = ±2π/λR and beam waist ≈ 175 µm,
as shown in Fig. 7.4A. These lasers counter-propagated at an angle θ = 0.75◦
away from e x established by the 1-D lattice, effectively reducing kR by ≈ 1 × 10−4
and introducing a quadratic Zeeman like term with strength −0.001E L . Both ofthese effects are negligible in our experiment. The Raman lasers’ relative phase117

was actively stabilized so the phase noise in the Raman beams did not translate
to the phase imparted to the atoms (see Sec. 4.3.3). To adiabatically load theground band, we once again ramped on the 1-D optical lattice from 0E L to 6E L
in 200 ms. Due to the next higher subband being 4E L away, we could ramp
the Raman lasers on resonance from 0E L to 0.2E L (about 1 mW) in 10 ms andadiabatically load the ground band of our system.
We directly imaged the three eigenstates of the ground Hofstadter band
along e s for both positive (Fig. 7.6A,B) and negative (Fig. 7.6C,D) flux. The data
in Fig. 7.6, with φAB , 0, are qualitatively different as a function of both k x andm. In k x for positive flux, the zero order of the 1-D optical lattice for the m = +1
is shifted by 2(kR − kL ) relative to the m = 0 site. In m, the occupation no longer
resembles the distribution expected from a particle in a box.
These differences can be understood in analogy with a 2-D electron in a perpendicular magnetic field, confined in one dimension with hard walls. Alongthe confined direction the wavefunction is localized to the scale of the magneticplength ℓB = ~/qB, with center position at k x ℓ2B in the bulk. Even though themagnetic length is a continuum concept, estimating it for our system is usefulfor conceptualizing the localized edge eigenstates. Since all physics is 2π peri-

odic in the acquired phase, our flux φAB /2π ≈ 4/3 is equivalent to φAB ≈ 2π/3.
√
In our system the magnetic length ℓ∗B = 3/2π ≈ 0.7 in units of lattice period
is of order unity. The analogue of the wavefunction’s localization is observedby comparing the significantly narrowed the bulk state (Fig. 7.6, middle row)with φAB , 0 to the φAB = 0 case (Fig. 7.3, top row). Continuing the analogy,
for large |k x |, the electron becomes localized near the edges. For data initialized
in the m = ±1 site, we observed the appearance of states localized at the sys-

118

A

B

C

D

+2

1.0

+1
0

0.5

-2

0.0

+2

1.0

+1
0

0.5

-1
-2

0.0

+2

1.0

Fractional Population, nm

Lattice Site, m

-1

+1
0

0.5

-1
-2

0.0
-2.0 0.0 +2.0
kX/kL

-2

-1 0 +1 +2
Lattice Site, m-2.0 0.0 +2.0
kX/kL

-2 -1 0 +1 +2
Lattice Site, mFigure 7.6: Eigenstates for positive and negative flux. (A,C) Absorption images
with momentum resolution in e x and site resolution in e s for positive and negative flux. (B,D) Fractional population nm for positive and negative flux. The
eigenstates are localized on the edges and on the bulk.
tem’s edges (Fig. 7.6 top and bottom row), which are completely absent for the
ground band when φAB = 0. These localized edge states are the analogue to thecurrent carrying edge states in fermionic IQHE systems. In contrast, localized
bulk states correspond to closed cyclotron orbits.

7.4

Dynamic Hall effect

Next, we loaded our systems into the bulk (m = 0 site) with t s = 0 with δ ≈ 0,
and then abruptly turned on t s , allowing tunneling along e s . Consequently, the

119

A

B
Fractional population, nm

1
0.8
0.6
0.4
0.2
0
0

1

2
3
Time, (ms)4

Figure 7.7: Chiral Current in Quantum Hall Strip. (A) The tunneling in e s was
rapidly turned on so the atoms began to coherently tunnel and feel a force perpendicular to their motion. This created edge currents Im=±1 . (B) The coherent Rabi flops between sites m for ≈ 2.4 mW in the Raman beams. The solid
curves are generated by evaluating our full Hamiltonian with best fit parameters (~ΩR , V, δ, ǫ) = (0.73, 6, 0.001, 0.05)E L .
resulting state consisted of coherent superpositions of magnetic subband eigenstates with crystal momentum q x /kL = −mφAB /π. When φAB , 0, atoms coherently tunneling along e s experienced an associated Lorentz force along e x . This
motion along m drove transverse, i.e. Hall, edge currents Im=±1 (τ) = nm (τ)hvm ialong e x (Fig. 7.7A), where hvm i is the mean velocity of atoms on site m along e x .
Figure 7.7 displays these dynamics and how balanced populations oscillated
in and out of the the originally empty m = ±1 sites as a function of time τ. At
time τ1 = 0, we linearly ramped ~ΩR from 0ER to a set-point in 300 µs. We varied ~ΩR from 0.2ER to 0.9ER (about 0.9 mW to 3 mW) changing the tunnelinganisotropy t s /t x from about one to three. The shortened time scale was adiabaticwith respect to the ≈ 5E L band spacing in the 1-D optical lattice, yet nearly instantaneous with respect to the ≈ 4t x magnetic band width. At τ2 the maximum
fractional population of atoms tunneled to the edges, acquired a transverse ve-

120locity and contributed maximally to our defined a chiral current I = I1 − I−1 .
The transverse velocity was controlled by two parameters: φAB set the crystal
momentum acquired while tunneling, and t x gave the natural unit of velocity
2t x /~kL . At τ3 , the atoms tunnel back to the bulk state and I = 0.
We confirmed the system’s chirality by inverting φAB and verifying that Ichanged sign. Additionally, we repeated the experiment with φAB = 0, andobserved no chiral current. Because the transverse velocity for the atoms was
symmetric (hv+1 i = - hv−1 i), I oscillated in phase with the fractional populationsin the edges h|m|i V as shown in Fig. 7.8A. We extracted a slope S from linear
fits to I dependence on h|m|i. Figure 7.8B displays an example of this linear fit
for positive, negative and zero flux. The small offset is due to a non-zero turnon time for tunneling in e s and is correctly predicted by our theory. The peakedge current Imax strongly depends on t s /t x (Fig. 7.8C), increasing from zerothen reaching saturation when all the atoms are no longer in the bulk, but on
the edges. This dependence and saturation of the chiral current on the tunneling anisotropy t s /t x is reminiscent of the optical lattice experiments in Ref. [10].
For small t s /t x , few atoms tunneled to the edges, giving a correspondingly small
Imax . As t s /t x increased Imax was bounded its value with all the atoms contributing to the current on the edges [70]. In contrast, S , essentially constant in t s /t x
as shown by the inset in Fig. 7.8C. The slight downward curvature in our theoretical model, plotted by the grey curve, evidences 5%-level corrections proportional to |t s |2 to the tight binding model, owing to mixing with excited Blochbands of the 1-D optical lattice.
The linear dependence of the chiral current on m demonstrates a new kind of
dynamic Hall effect. We can measure this bulk property due to simultaneously

121

B

C

2

2

1

1

0

-1
-2
-3
0

3

0
-1

2
Time,

3
(ms)

4

6

1

-2
-3

1

2
in units of3

in units of3

in units ofin units of

A

0

0.1

0.2

0.3 0.4

0.5

0

0

1

4
2
0

0

2
ts / tx

1

2 3
ts / tx

4

3

4

Figure 7.8: Dynamic Hall effect. Data shown in red, black, and empty circles
are taken for positive, negative, and zero φAB , respectively. Our theory (solid ordashed curves) uses parameters determined from Fig. 7.7B for φAB , 0, giving
t s = 0.14E L . (A) We observed in-phase oscillation of I and the combined m = ±1
populations h|m|i. The pink dashed line shows how we extracted Imax . (B) The
chiral current I = I1 − I−1 had the same sign as φAB and was linear in h|m|i. (C)The chiral current saturated in tunneling anisotropy t s /t x , but the slope of I wasessentially constant in t s /t x (inset).
resolving the momentum along e x and site m in our TOF images. Although thisbulk property is hard to access in condensed mater systems due to the symmetryof the produced current and the dominant edge properties, this behavior shouldstill exist.

7.5

Edge magnetoplasmons: skipping orbits

Edge magnetoplasmons are superpositions of edge eigenstates in different Landau levels [81, 9], or here magnetic bands. We launched these excitations andrecorded their full motion, including a chiral drift along the system’s edge andan underlying skipping motion, on either edge. Figure 7.9 illustrates this motion
starting from m = −1. The potential was tilted along e s , such that the initiallyoccupied site was at the potential minimum( Fig. 7.9B). We used our simulations of our full Hamiltonian to specify what detuning δ and Raman coupling122

B
Energy

A

-2

+1 +2
-1 0
m

Figure 7.9: Schematic of Skipping Orbits. (A) As atoms tunneled from the
m = −1 edge to the bulk, they began their orbit. Before finishing the orbit, theyencounter the infinitely hard edge of the system and reflect off it. This leads toa chiral drift along the edge. (B) The non-zero detuning δ tilts the potential.
strength ~Ω would create the large enough orbits to image easily. Our modelingof the full Hamiltonian, which does not include interactions, was more consistent with experiment at smaller atom numbers. Also based on our simulations,
we decided to operate a larger quadratic Zeeman shift to more easily image theskipping orbits, therefore we changed our bias field B0 . Thus, our Raman coupling was detuned from the gµB B0 /h ≈ 1.35 MHz Zeeman splitting by a nonzero
δ and tilted the lattice along e s as shown in Fig. 7.9. The corresponding quadratic
Zeeman shift additionally lowered |m = 0i by ǫ = 0.13E L . We loaded our systems on to either edge in m with t s = 0, and then abruptly turned on t s , allowing
tunneling along e s . The abrupt turn-on resulted in either edge state consisting ofcoherent superpositions of magnetic band eigenstates with crystal momentum
q x /kL = −mφAB /π, which began to coherently tunnel along e s and experienced
an associated Lorentz force along e x . These atoms then became cold-atom analogues to edge magnetoplasmons: they began cyclotron orbits, were reflectedfrom the hard wall, and skipped down one edge or the other.
In Fig. 7.10, we plot the time evolving average position hm(τ)i along e s

123

A

B

Lattice site m




0.5
0

0

-0.5


Lattice site m

C




0


Time,

0

3
4
(ms)




Time,

3
(ms)

4



0



í

í

0
Displacement

Figure 7.10: Skipping Orbits. Data shown in pink/blue solid circles are for systems initialized on m = 1 (m = −1). (A) Average position in m and (B) velocity
hv x i along e x versus time τ. (C) Displacement in m versus displacement along e xshows a skipping orbit trajectory.
and velocity hv x (τ)i =

P

m Im

along e x for excitations on both edges. hv x (τ)i

evolved periodically in time but with opposite velocities for the different edges.
The solid curves are numerical simulations using parameters (~ΩR , V, δ, ǫ) =(0.58, 5.2, ±0.087, 0.13)E L . The spatial trajectories are illustrated in Fig. 7.10 C,
where we obtained the displacement hδ j(τ)i by directly integrating hv x (τ)i/a,
where a = λL /2 is the lattice period. These data clearly show edge magnetoplasmons with their chiral longitudinal motion, and constitute the first experimental observation of their edge localization and transverse skipping motion.

124

7.6

Conclusions and Outlook

We engineered a 2-D lattice with square geometry for 87 Rb Bose-Eistein condensates in which we directly controlled the acquired phases as atoms traversed the lattice, giving a tunneling phase φAB /2π ≈ 4/3 around each plaquette.
Aharonov-Bohm phases of order unity are only possible in engineered materials [53, 71], or in atomic [74, 3, 97, 79, 4, 93] and optical [62] settings. These
phases take the place of the Aharonov-Bohm phases produced by true magnetic
fields and suffice to fully define the effective magnetic field. With our hard-wall
potential, a realization of the Laughlin charge pump [84] is straightforward: as
particles accelerate along e x , mass moves from one edge to the other in the orthogonal direction e s . My colleague Dina Genkina is currently implementing theLaughlin charge pump in the lab. We are also set up to extend our techniqueto periodic boundary conditions, coupling together the |m = ±1i states. Another colleague Ben Stuhl began implementing this experiment before moving
out west. The implementation of periodic boundary conditions should produce
systems exhibiting a fractal Hofstadter spectrum [35], even given a three-site extent along e s . Going beyond conventional condensed matter realities, the flexibility afforded by directly laser-engineering the hopping enables the creation ofMöbius strip geometries: topological systems with just one edge [24].

125CHAPTER 8
PHYSICS, WE HAVE A PROBLEM: SEXUAL HARASSMENT

In this chapter, I present the results from a pre- and post conference evaluation
survey of the American Physical Society (APS) Conferences for Undergraduate
Women in Physics (CUWiP). Forty-eight percent (252/521) of the respondents
reported ‘Sometimes’ or ‘Often’ observing sexual harassment in physics. Fortynine percent (255/522) of the respondents reported having personally experienced sexual harassment in physics. We also found strong correlations amongexperienced harassment, frequency of observed harassment and perception ofserious, unresolved gender issues in physics. Lastly, reporting more frequent
observations and personally experiencing sexual harassment were correlatedwith a lower GPA in physics courses. These results are part of an ongoing study
on participants’ career plans, attitudes, and prior experiences with physics to
improve undergraduate physics education and the CUWiP experience for future students. I joined the collaboration in June 2015 after advocating for the
addition of questions pertaining to sexual harassment in physics.

8.1

Introduction

The term “sexual harassment” was coined in response to the behavior of aphysics professor at Cornell University [12]. This is the story of Carmita Wood,an administrative assistant at Wilson Synchrotron Laboratory. Wood was in her
mid-forties and a mother of four in June of 1974 when she was forced to quit
her job. Wood developed severe physical pain from the constant stress of fending off the professor’s advances such as physically accosting her or mimicking
126masturbation in her presence. When the professor went on a one year leave ofabsence, the pain went away. In anticipation of his return and the return to dailystresses of harassment, Wood quit. She was denied unemployment benefits forvoluntarily leaving her job. For legal assistance and support, she approached
Lin Farley, a lecturer in the Women’s Section of Cornell University’s Human Affairs Program. Ultimately, Wood lost the appeal, but a movement against sexual
harassment in the workplace was born [27, 12, 96].
Recently, several high profile cases of sexual harassment in academia havebeen brought to light in the fields of anthropology [14], astronomy [135] andbiology [63]. These cases have indicated the need for academic communities
to address issues of sexual harassment in their professional spheres. Many climate studies have been underway in anthropology [40] and astronomy [114]. In
the field of physics, the APS ad-hoc committee on lesbian, gay, bisexual, transgender (LGBT) Issues recently released a report on a professional climate survey of the members of the LGBT community in physics [11]. In this report,
the female survey participants reported experiencing exclusionary behavior–defined as behavior that shuns, ignores or harasses a person–at three times therate of male participants. Additionally, a qualitative study of female graduate
students’ experiences in physics found that a majority of its participants experienced microaggressions–subtle insults or slights–and several reported experiencing more traditional hostile sexism [19]. We complement these climate andqualitative studies by measuring the extent to which female undergraduates
experience and observe sexual harassment in physics.
To explore the ways sexual harassment may impact female undergraduates,
we offer three hypotheses: (1) Sexual harassment occurs in physics contexts–

127defined as in research labs, instructional settings, department or student organization events–and the degree to which respondents directly experience andobserve sexual harassment will be correlated to each other, (2) reporting thatone believes that there are serious gender issues to be reconciled in physics will
be positively correlated with both experienced and observed harassment, and(3) reported GPA from physics courses will be negatively correlated with experienced and observed harassment.

8.2

Methods

In this section, I detail the methods for our internet-based survey (n = 522) ofattendees of the American Physical Society (APS) Conferences for Undergraduate Women in Physics (CUWiP). I will discuss our participant recruitment anddemographics. Then, I will discuss the question design and present the questions in their full text. Notably, we were able to survey a large proportion of thefemale undergraduate physics majors in the United States.

8.2.1

Ethics Statement

Participation was voluntary and informed consent was obtained from all survey participants. We obtained approval for human subjects research with minimal risk from the University of Maryland College Park Institutional Review
Board (project [505475-11]). In addition, I obtained an exemption (Protocol
ID# 1601006083) from Cornell University’s Office of Research Integrity and Assurance.

1288.2.2

Participant Recruitment

CUWiP are a continuation of a grass-roots collaborative effort of physicists fromaround the country that has provided conferences annually since 2006. The
goals of the CUWiP effort are to increase retention and improve career outcomes of undergraduate women in physics. As part of the effort to measure theimpact of the conference goals, online pre- and post-conference assessment surveys are administered annually. In 2016, there were 1088 attendees at CUWiP.
These attendees were predominantly undergraduate female physics majors. In
2014, 1341 women graduated with a Bachelor’s degree in physics [1]. Estimating the number of female undergraduates majoring in physics by multiplying
the number of Bachelor’s in a year by four, there are approximately 5,400 female
physics majors in the United States (U. S.). The demographic breakdown will
be discussed in the next section. If we solely look at the number of the attendees
compared to the number of female physics majors, we have the possibility ofsampling about 19% of the female physics majors in the US.
By recruiting our survey participants from this population, we have the potential to collect responses from a large fraction of the undergraduate womenin physics in the U. S. Additionally, by including our questions about observed
and experienced harassment on a post conference evaluation survey, the results
will have a different response bias than typical climate or sexual harassment surveys. So far, field specific surveys have used snowball methods, where participants recruited other participants and the survey was advertised at conferences
and through social media. These snowball climate surveys can have selection
bias, e.g. introduced through participants forwarding the survey to membersof the community who have disclosed negative experiences, and response bias,

129

e.g. people who have been harassed may be both more or less likely to respondto a survey on sexual harassment. There were no repeat survey participantsbecause each participant had a unique identifier.

8.2.3

Participant Demographics

For our study, we focused on the sample of the (890) self-identified female undergraduates of the (1088) total CUWiP attendees in January 2016. Out of the female undergraduate attendees, 524 took the post survey and n = 522 responded
to the questions on sexual harassment. Comparing the demographics of oursample to the population of female physics undergraduates is difficult because
the collection of the racial demographics of our sample was more flexible thanthe demographic data on undergraduate physics students, i.e. our survey allowed for more choice than usual demographic data. All demographic questions were asked on the pre-survey (Table 8.2). The participants were asked ifthey were Hispanic or Latino(a). Separately, they were asked with which racial
groups they identified and informed they could mark all that apply with options Black, White, Asian, Native American or Alaskan Native (NAAN), Native
Hawaiian or Pacific Islander (NHPI) or Other. Other had an option for the participant to self-identify. In contrast, the demographic data on students earning
their Bachelor’s degree in physics has these inquiries as one question, and if a
student is Hispanic or a Temporary resident, any additional information about
their racial identity is not reported. Furthermore, the data does not include
which racial groups multi-racial students identify. These differences in demographic data collection could lead to significant differences of the proportion of
students reported in different racial groups.

130

To assess whether our results give us insight into the female undergraduates’
experiences with sexual harassment, we first compare the largest demographic
group easily identified through both methods of collection. Sixty-eight percent(601/890) of the female, undergraduate CUWiP attendees identified as whiteand non-Hispanic/Latina. This is consistent with the 68% (911/1341) white,
non-Hispanic women who earned Bachelor’s degrees in 2014 [1]. If we exclude
the women who identified in multiple racial groups from the count, the attendees included closer to 62% (554/890) white, non-Hispanic women. We cannot know how the women who self-identified as multi-racial or white, Hispanic would have reported their race/ethnicity if they were only given one
option. The data indicates that the vast majority of our sample is white andnon-Hispanic, similar to the reported population of women earning Bachelor’s
degrees in Physics. The demographics of the self-identified female undergraduates were sufficiently heterogeneous with respect to the remaining demographic
characteristics of the women earning Bachelor’s degrees in physics that there isa good basis for the generality of our results. Table displays the demographic
data of our sample, the CUWiP attendees and women who Bachelor’s degrees inphysics at U. S. institutions. For our data, participants are counted in all groups
with which they identified. Due to difference in data collection, we cannot differentiate if we are oversampling under-represented minority (URM) female undergraduates or if the additional flexibility of our demographic data collectionis the explanation for the demographic differences between the attendees andthe degree earners.
We compared the demographics of the female undergraduate CUWiP attendees (890) to our sample(n = 522)–the participants who responded to the sexual
harassment questions on the post-survey–to examine the response bias. About

131

Sample%Attendees%

Degrees†

%

Total

522

100

890

100

1341

100

URM
Hispanic or Latina
Black
NAAN
NHPI
White
w/o Hispanic or Latina
and w/o multi-racial
Asian
Other
Multi-racial

174
131
19
12
2
419
365
338
77
16
39

19.6
14.7
3.6
2.2
0.4
80.2
69.9
64.8
14.8
3.1
7.5

97
75
36
17
9
691
601
554
149
24
64

18.6
14.4
4.0
1.9
1.0
77.6
67.5
62.2
16.7
2.7
7.2

146
86
54
4
2
–
–
911
89
54
49

10.8
6.4
4.0
0.3
0.1
–
–
67.9
6.6
4.0
3.7

Race or Ethnicity

Table 8.1: Racial/Ethnic Groups of our sample of and total female, undergraduate CUWiP attendees, and Bachelor’s degrees in physics earned by women at
U. S. institutions in 2014 [1]. †This demographic data was collected with slightly
different racial and ethnic groups, and only allowed for an individual to be identified in one.
15% (131/890, 75/522) of our selected population and sample reported identifying as Hispanic/Latina and about 1% (13/890, 6/522) did not specify. A majority
66% (342/522) of our sample specified being in their 3rd, 4th or 5th year of college and the remaining 34% (180/522) specified being in their 1st or 2nd yearof college. This is consistent with the overall selected population’s year in college demographics: 65% (582/890) and 35% (308/890), respectively. Ninety-fourpercent of both the female CUWiP attendees (834/890) and the participants inour sample (492/522) intend to complete a Bachelor’s degree in physics. There
are no large shifts in demographics from the demographic data about CUWiP
attendees to the sample in which we measured observations and experiences ofsexual harassment.

132

8.2.4

Survey Design

The results in this study derive from two questions on the pre-survey administered at the time of conference registration and two questions on the postconference survey after the conference was completed. Both surveys were administered online. The pre-survey question on belief in gender issues was asked
after inquiring about the participant’s career plans and in a section inquiring
about their physics attitudes and experiences. The question on the participant’s
physics GPA was in this section as well. Both of these questions came before inquiring about the participant’s demographic data. Both of these questions wereasked before the participants attended the conference as shown in Table 8.2. Additionally, the GPA inquiry was on a pre-survey which was administered prior
to the post survey with the questions on sexual harassment, hence our participants were not primed by the sexual harassment questions to report a lower
physics GPA.
The two questions on the post-survey asked about the frequency at which
the participant observed harassing behavior and whether the participant had
personally experienced this behavior limited to a context associated withphysics (Table 8.3). These questions were adapted from question 31 and 32
on the climate survey of academic field work in anthropology [40]. We do
not ask about harassment through direct query, but rather describe behavioral experiences and ask about frequency and whether or not the participant
was personally targeted. This is the preferred method for studying sexual harassment because people’s perceptions about sexual harassment change overtime. Additionally, a study on two private sector organizations and one university found that regardless of whether a women labeled her experiences as133

12. Other than representation (i.e. numbers participating), do you feel that
there are serious gender issues in physics? Not at all 0 1 2 3 4 Very much so
14. What year are you in college?
• First year
• Second year
• Third year
• Fourth year

• Fifth year
• Graduate Student• Faculty

15. Which of the following physics courses (or equivalent) have you taken
and completed in college? Mark all that apply.
• Intro Physics I• Intro Physics II• Modern Physics• Classical Mechanics (not intro)• Thermodynamics (Stat. Mech.)• Electromagnetism I• Electromagnetism II• Quantum Mechanics I• Quantum Mechanics II

16. What is your approximate average GPA (4=A, 3=B, 2=C, 1=D, 0=F) inthese physics courses-please enter a number(decimals allowed)? .
20. Are you Female or Male?
• Female• Male• Other:
21. Are you Hispanic or Latino(a)?
• No
• Yes
22. With which racial group(s) do you identify? (For multi-racial, mark allthat apply)• Black• White• Asian• Native American or Alaskan

Native• Native Hawaiian or Pacific Islander• Other:Table 8.2: Pre-survey Questions
13416. With what frequency do you observe or hear about individuals making
inappropriate or sexual remarks, comments about physical beauty, cognitive sex differences, or jokes in a context associated with physics (e.g. in
research labs, instructional settings, department or student organization
events)? These may come from other students, high school teachers, instructors or professors.
• Never• Rarely• Sometimes• Often
17. Have you ever personally experienced inappropriate or sexual remarks,
comments about physical beauty, cognitive sex differences, or jokes in a context associated with physics?
• Yes
• No
• I don’t know

Table 8.3: Post-survey Questions
sexual harassment, she experiences similar psychological, work and health consequences [91]. A study on both men and women in the military strongly replicated these results [100] suggesting they are quite generalizable. Furthermore
this study found that negative outcomes were more related to frequency of thesexually harassing behavior rather than labeling the behavior as “sexual harassment”.

8.2.5

Enumeration and Analysis

Identifying correlations required us to enumerate the responses to the sexual harassment questions. For post-survey question 16, we note the ordinal response135

has four levels, increasing with frequency. We set ‘Never’= 1, ‘Rarely’=2, ‘Sometimes’=3, and ‘Often’=4. Because the response is ordinal, we note we should notassume a linear relationship with other variables. We use a Spearman’s Rankcorrelation ρS test to look for correlations between the observed response andour other variables of experienced, belief in gender issues and GPA. For theexperienced question, we excluded the ‘I don’t know’ responses from the analysis, because there is not a clear placement relative to ‘Yes’ and ‘No’. We set‘Yes’=1 and ‘No’=2. The responses to the question about belief in gender issues
in physics was already enumerated as it is an anchored Likert-type scale from
‘Not at all’ (0) to ‘Very much so’ (4). GPA was already enumerated as well, however it saturates at 4.0. To test for correlations between the experienced response
and belief in gender issues or GPA, we used a Pearson’s product-moment correlation ρP due to the variables all qualifying as an interval scale.

8.3

8.3.1

Results and Discussion

Female undergraduate physics majors experience and observe sexual harassment

Forty-eight percent (252/521) of respondents report they observed harassment‘Sometimes’ or ‘Often’ in a context associated with physics (Fig. 8.1A). Fortynine percent (255/522) of the respondents reported being the target of sexual
harassment (Fig. 8.1B). These results support our hypothesis that sexual harassment occurs in contexts associated with physics.

136

A Observed frequency

B
Proportion of Respondents

Proportion of Respondents

0.4

0.3

0.2

0.1

0.0

Directly Experienced
0.5

0.5

(65)

(204)

(206)

(46)

Never

RarelySometimes

Often

0.4

0.3

0.2

0.1

0.0

Observed Frequency (post-Q16)

(255)

(212)

(60)

Yes

No

I don’t know

Experienced (post-Q17)

Figure 8.1: Observed and experienced sexual harassment in physics
We predicted prevalence because women in male-dominated occupations
report experiencing sexual harassment at higher rates than women in femaledominated ones [21]. The 49% of our respondents reporting observations ofsexual harassment is similar in magnitude to the 38% (184/481) of female participants who reported observations of sexual or inappropriate comments occurring ‘Regularly’ or ‘Often’ at scientific field sites [40], despite varying methodologies. Our data provides compelling evidence that students are experiencing
sexual harassment in physics contexts. For reference on the collegiate experience, the American Association of Universities conducted a climate survey
on Sexual Assault and Sexual Misconduct and found 61.9% of that sample offemale students (n = 55, 552) experienced sexual harassment on campus [32].
More broadly, a recent meta-analysis of incident rates of work-related sexual harassment studies with probability sampling and behavioral experiences survey
methods reported 58% of women (excluding studies on students) in academic
samples and 69% of women in military samples reported having experienced
harassing behaviors [72].

137Our survey questions solely focused on gender, not on any intersection ofgender and other marginalized identities, eg. race or ability. Additionally, weonly analyzed the data in the context of the female students due to the small
representation of the responses from other genders. It is important to remember that people with additional marginalized identities often face more harassment and discrimination. As shown in Fig. 8.2, we did explore if belonging
to a racial group increased or decreased the likelihood of reporting observedor experienced sexual harassment. An exact Fisher’s test performed on each

B Directly Experienced

A Observed frequency
Frequency
Hispanic

28

30

Experienced

Rarely

9

34

9

32

54

12

53

Women
of Color
Black

43

45

5

7

Often

14

7 1

3

White

166

170

33

Asian

33

28

7

NAAN/
NHPI

4

Multiple
0.0

0.2

0.4

17
0.6

0.8

208
32

45

166

10

35
9

1

4

20 1

4
0.0

1.0

0.2

No

11

8
16

Yes
I don’t know

Sometimes

0.4

18
0.6

0.8

1.0

Proportion of Respondents

Proportion of Respondents

Figure 8.2: Sexual harassment in physics reported by female undergraduate
physics majors by their race and ethnicity.
racial sub group and on Women of Color–female participants identifying withany non-white racial group–as compared to the rest of the female participants
had no statistically significant difference among racial groups. This is consistent with a study on n = 208 women faculty scientists, which found no different between the rates of reported sexual harassment among white women and
Women of Color [117]. However, an intersectional bias study of n = 557 female
STEM faculty found that through direct query–inquiring whether the participant had been sexual harassed on the survey–white women (37.2%) were the138most likely to report sexual harassment, as compared to Asian women (25%),
Latinas (21.9%) and Black women (12.5%) [132].
Prevalent sexual harassment in physics may be indicative of a deeper cultural problem surrounding women in physics. Male attitudes about women inscience have been studied through measured gender bias in their evaluationsof women in science. For example, elite male life sciences faculty employ fewer
women [118], male biology students have a strong male bias when evaluating
the knowledge of their peers [61], and male students rate female high school
science teachers significantly lower than their male counterparts on evaluations
in biology, chemistry, and physics [108]. Furthermore, women can also show
bias against women in science, particularly in the physical sciences. In the same
study that found a male bias in male student evaluation of high school scienceteachers, female students did not show a gender bias on evaluations in biology
and chemistry, but did in physics [108]. This gender bias may be illustrating theimpact on both male and female students of deep-seated cultural stereotypes
about women in physics. Knowing more about the attitudes about women inphysics in different educational environments and rates of sexual harassment inthose environments would be a potential future study.

8.3.2

Correlations with belief in serious gender issues inphysics and physics GPA

We found a strong relationship with those who directly experience and those
who frequently observe sexual harassment (ρS = −0.503, N = 462, p < 0.0001). In
Fig. 8.3, the respondents (N = 521) are grouped based on their responses to the139experienced question and calculated the percentages each group that responded‘Never’, ‘Rarely’, ‘Sometimes’ or ‘Often’ to the inquiry about frequency of observation. The area of the circles is proportional to the percentage of responses
in each ‘Yes’, ‘I don’t know’ or ‘No’ subgroup. Even though the ‘I don’t know’
responses were not included in the correlation test, overall their responses wereless extreme than the whole population and were fairly evenly balanced be-Experienced (post-Q17)

tween ‘Rarely’ and ‘Sometimes’. The strong correlation is visible.

Yes

29±3%

54±3%

48±6%

45±6%

16±2%

2±1%

I Don’t
Know

No

3±2%

3±2%

27±3%

50±3%

20±3%
2±1%

Never

RarelySometimes

Often

Observed Frequency (post-Q16)

Figure 8.3: Experienced versus frequency of observed sexual harassment. The
uncertainties are the standard error.

One explanation for correlation between observed and experienced harassment can be because higher levels of sexual harassment occur where such behavior is perceived as being tolerated [100]. Another explanation could be thatcertain women galvanize more sexual harassment than others. To be viewed
as competent physicists, some women distance themselves from their genderand gender-roles [56]. An experimental psychology paper exploring the cause
of sexual harassment found that women who violate feminine ideals and workin male-dominated fields are the most likely to be sexually harassed to punish them for being gender-role deviant [21]. So a woman distancing herself
140

from her gender-roles to be viewed as a competent physicists, then becomes
more likely to be sexually harassed for being gender-role deviant. This tradeoff between these two ideas of being feminine or competent is referred to as
the tightrope [132]. Understanding the causes of this correlation is beyond thescope of the current study.
This correlation is unlikely to be due to some women being “more sensitive” than others. The idea that the prevalence of sexual harassment is overestimated due to women “whining” has been widely refuted [72, 91, 100]. A
meta-analysis study compared the reporting of sexual harassment at work to
the reporting of experiencing harassing behaviors at work, the labeling of sexual harassment was half the reports of harassing behavior [72]. A study in the
military found that negative occupational, psychological and health outcomescorrelate more with frequency of experiences with unwanted sexual behavior
than labeling such behavior as sexual harassment [100]. These studies and others provide evidence against the so-called “whiner” hypothesis.
We found a medium correlation (ρS = 0.350, N = 517, p < 0.0001) between frequency of observed sexual harassment and belief in gender issues inphysics (Fig. 8.4A). We found a medium correlation (ρP = −0.293, N = 457, p <0.0001, 95%CI = [−0.375, −0.207]) test between experienced sexual harassment
and belief in gender issues in physics(Fig. 8.4B). Figure 8.4 is a box-and-whisker
plot. Dividing the data into four equal parts (quartiles), the box represents the
middle half of the data. The whiskers are the minimum and the maximum of thedata, where outliers are represented by points. The thick vertical line is the median of the data. The box represents Our hypothesized correlation between belief in gender issues in physics and observed frequency and experienced sexual141

harassment was supported. Notably, in the military study about the impact oflabeling behavior as sexual harassment, the variable most related to whether or
not participants labeled the behavior as sexually harassing was whether or not
the participant viewed sexual harassment as a serious societal problem [100].

A Correlation with observed

B Correlation with experienced
150

Experienced (post-Q17)

Belief in gender issues (pre-Q12)

4

3

100

2

1

Experienced
Yes
No
50

0

0
Never

RarelySometimes

Often

Not at all

Observed Frequency (post-Q16)

Very much so

Belief in Gender Issues (pre-Q12)

Figure 8.4: Correlation between belief in unresolved gender issues in physics
and either frequency of observation or personally experienced sexual harassment in physics. (A) Belief in gender issues in physics versus frequency of observations of sexual harassment. (B) Number of participants who responded
explicitly to whether they had experienced sexual harassment versus belief in
gender issues in physics.

We found a significant correlation (ρS = −0.153, N = 500, p < 0.001) between
reported physics GPA and observations of sexual harassment (Fig. 8.5A). We
also found a significant correlation (ρP = 0.141, N = 441, p < 0.003, 95%CI=[0.048, 0.231]) between physics GPA and directly experienced harassment(Fig. 8.5B). These correlations support our hypothesis about correlations between a lower physics GPA and experienced and more frequently observed
sexual harassment. This can be due to interactions with sexist men engaging
stereotype threat for female students [89]. People typically hold different stereotypes about women than about successful scientists [33], thus observing and ex-142periencing sexual harassment could increase stereotype threat and impact thestudent’s performance in physics. The difference in the median physics GPA of
the students responding ‘Often’ compared to the students responding ‘Never’is significant for students applying to physics PhD programs due to the exponential dependence of acceptance on GPA [92].

B Physics GPA vs Experienced

4.0

Physics GPA (pre-Q16)

Physics GPA (pre-Q16)

A Physics GPA vs Observed

3.5

3.0

2.5

4.0

3.5

3.0

2.5

2.0
2.0
Never

RarelySometimes

Often

Yes

Observed Frequency (post-Q16)

No

Experienced (post-Q17)

Figure 8.5: Correlation between physics GPA and either frequency of observation or personally experienced sexual harassment in physics. (A) Physics GPA
in completed physics courses versus frequency of observations of sexual harassment. (B) GPA in completed physics courses versus personally experienced
sexual harassment.

8.3.3

Study Limitations

For this study, there was not an equal probability of selecting any undergraduate female physics majors. By selecting from CUWiP, we may only be sampling
women with strong support to continue in their physics careers. Our questions
are cumulative over the participant’s entire experience associated with physics,
so we cannot differentiate between high school experiences and collegiate experiences and found an almost linear increase in observed and experienced ha143

rassment versus year in college and number of physics courses completed. Currently, we only reported on correlations in our data. For example, participants
who believe gender issues are a serious problem in physics may also be more
likely to report observed and experienced harassment based on gender.

8.4

Outlook and Moving Forward

Our results indicate experienced and observed sexual harassment are correlated
in this sample, and that those in this sample who observe or experience sexual
harassment also tend to believe that serious gender issues are relevant to thefield of physics. The correlation between sexual harassment and physics GPA,
though weaker, could also have a dramatic impact on female students’ professional role confidence and their decisions to continue to pursue a career inphysics. Recently, a longitudinal study on women’s building of professional
identity while earning a Bachelor’s degree in engineering found many of the female participants experienced hostile sexism, sexual harassment, and isolationin their summer internships. These interactions during a professional training
lead them to question engineering being the right “fit” for them [116]. Experiencing and observing sexual harassment may have a similar effect on young
female physicists. Both experiencing a lack of fit and a depressed physics performance could be how sexual harassment affects underrepresentation of womenin physics.
Discussions of underrepresentation of women in science during high school
can have a positive effect on female students’ persistence in physics [67]. Bystander intervention programs to prevent sexual violence towards women typ-144

ically involve students discussing sexism and socialized gender-roles and how
these concepts support a rape-prone culture and how to use social pressure topositively intervene in potentially harmful situations [17]. In recent psychology
studies, these programs effectively increase intervention tendencies and reduce
attitudes that support rape culture for both male and female students [15]. Furthermore, a survey of female science faculty found that effective chair leadership could mitigate the negative job outcomes due experiences of sexual harassment and gender discrimination [117]. Moving forward, we hope the evidence
on the prevalence of sexual harassment compels the physics community to raiseawareness of sexual harassment, create guidelines for respectful behavior andimplement anti-harassment and bystander intervention training for all members of the community while evaluating the effectiveness of these programs.
This will take strong leadership to steer the community towards addressing thechilly climate experienced by many undergraduate women in a quantitative,
visible manner.

145

APPENDIX ANONDESTRUCTIVE IMAGING OF AN ULTRACOLD LATTICE GAS

146

PHYSICAL REVIEW A 90, 033422 (2014)

Nondestructive imaging of an ultracold lattice gas
Y. S. Patil, S. Chakram, L. M. Aycock, and M. Vengalattore*
Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA(Received 22 April 2014; published 24 September 2014)
We demonstrate the nondestructive imaging of a lattice gas of ultracold bosons. Atomic fluorescence is inducedin the simultaneous presence of degenerate Raman sideband cooling. The combined influence of these processes
controllably cycles an atom between a dark state and a fluorescing state while eliminating heating and loss.
Through spatially resolved sideband spectroscopy following the imaging sequence, we demonstrate the efficacyof this imaging technique in various regimes of lattice depth and fluorescence acquisition rate. Our work provides
an important extension of quantum gas imaging to the nondestructive detection, control, and manipulation of
atoms in optical lattices. In addition, our technique can also be extended to atomic species that are less amenable
to molasses-based lattice imaging.
DOI: 10.1103/PhysRevA.90.033422

PACS number(s): 37.10.Jk, 03.67.−a, 03.75.Lm, 67.85.Hj

I. INTRODUCTION

The creation, control, and manipulation of ultracold atomic
gases in tailored optical potentials has spurred enormousinterest in harnessing these mesoscopic quantum systems forthe realization of ultracold analogs of correlated electronicmaterials [1,2], studies of nonequilibrium dynamics in isolated
quantum many-body systems [3]. and quantum metrology [4].
The dilute nature of these gases and the weak interactions
impose stringent restrictions on the energy, entropy, and meansof manipulating and probing these systems. This has led tothe development of techniques to cool and image these gases
at ever increasing levels of precision and resolution. In thiscontext, the in situ imaging of lattice gases at high spatial
resolution has emerged as a powerful tool for the study of
Hubbard models [5–8] and quantum information processing[9].
In this article, we demonstrate a two-photon imaging
technique for ultracold lattice gases. The scheme relies on
extracting fluorescence from the atoms while simultaneously
cooling them to the lowest band of the lattice via Raman
sideband cooling (RSC) [10–12]. Through a combination of
sideband spectroscopy and time-of-flight measurements, wedemonstrate broad regimes of fluorescence acquisition rates
and lattice depths for which the imaging scheme preserves
the spatial location and also retains the atoms in the groundvibrational state. As a result, this imaging scheme enables
the imaging of atoms in shallow lattices with high fidelity.
In addition, the two-photon scheme is less sensitive to detailsof atomic structure thereby permitting its extension to atomic
species less amenable to molasses-based imaging.
II. IMPLEMENTATION OF THE IMAGING TECHNIQUE

The principle of the imaging sequence is depicted in Fig. 1.
Raman sideband cooling is employed to cool individual atoms
within an optical lattice to the lowest vibrational band while
simultaneously pumping the atoms to the high field seeking
state. In the case of 87 Rb atoms used in our study, this stateis denoted by |gi = |F = 1,mF = 1; ν = 0i, where ν is the*

mukundv@cornell.edu

1050-2947/2014/90(3)/033422(4)vibrational state of the atom within a lattice site. Importantly,
this state is a dark state with respect to the optical fields
used for Raman cooling. As such, the atoms do not emit
any fluorescence while in this ground state. Fluorescenceis induced in these atoms by shining a circularly polarized(σ− ) beam resonant with the F = 1 → F ′ = 0 (D2) transition.
Simultaneous use of RSC mitigates the increase in temperaturecaused by this fluorescence beam by cycling the atoms backto |gi. Due to this cycling, fluorescence can be repeatedly
extracted from the atomic distribution while leaving the atom
in its original state.
For the studies described below, we use three-dimensional(3D) optical lattices that are typically detuned 2π × 160 GHz
from the F = 1 → F ′ (D2) transition of 87 Rb. The latticeprovides both the confining potential as well as the coherent
two-photon coupling required for sideband cooling [13]. In theabsence of RSC, we measure a heating rate of 11 nK/ms dueto the photon scattering from the near-resonant lattice. While
this does not pose a limitation for the studies described in thiswork, this heating can be significantly reduced by employing
separate optical fields to provide the lattice confinement andthe Raman coupling.
Atoms are loaded into this lattice and initialized in the
ground state |gi by a 10 ms period of RSC. Based on
measurements of the atomic density within the lattice, weestimate filling fractions on the order of f = 0.20–0.25. The
average vibrational occupation number is measured using
sideband spectroscopy to be hni 6 0.01 for the entire range oflattice depths studied here. Fluorescence images are acquiredby switching on the fluorescence beam at a variable intensity.
The images are acquired within exposure times of up to 30
ms following which the number of atoms and temperature of
the atomic distribution are measured using a combination oftime-of-flight absorption imaging and sideband spectroscopy.
We perform sideband spectroscopy to accurately quantify
local changes in temperature due to fluorescence imaging.
For this, we employ a pair of counterpropagating beams
detuned 2π × 7.5 GHz from the F = 1 → F ′ (D2) transitionof 87 Rb. The beams are focused to an approximate waistof 8 µm. The measured sideband asymmetry [14] allows alocal extraction of the vibrational occupation number (Fig. 2).
An oblique orientation of these beams with respect to thelattice coordinates ensures sensitivity of the sideband spectra

033422-1©2014 American Physical Society

PATIL, CHAKRAM, AYCOCK, AND VENGALATTORE

PHYSICAL REVIEW A 90, 033422 (2014)(a)III. DISCUSSION
(c)
z

µB
|mF=0〉|mF=+1〉|F’=0,mF=0〉

(b)

RSC

Imaging|mF=+1〉|mF=0〉|mF=-1〉

FIG. 1. (Color online) (a) Lattice imaging scheme: An atom
within a lattice site is cooled to the ground state |gi ≡ |F = 1,mF =+1; ν = 0i via RSC. An auxiliary imaging beam promotes the atom
out of this state to a fluorescing state, which is subsequently cooled
back to |gi. Repeated cycles of this process extract fluorescencefrom the atom while continually restoring the atom to |gi; (b) The
near-resonance optical fields used in the imaging sequence. A coolingbeam (RSC) with σ+ and π components cools and optically pumps
the atom into the dark state |gi. A σ− beam induces fluorescence bybringing the atom out of the dark state. (c) Raman fluorescence imageof a gas of 1.5 × 106 atoms obtained within 15 ms. The field of viewis 250 µm × 250 µm.

Transition probability [arb. units]

to atomic motion in all three dimensions. The two-photon
pulses are typically 500 µs in duration with typical pump(probe) powers of 10 µW (20 nW). We have verified that thevibrational occupation number extracted from the sideband
spectra is consistent with temperatures measured by time-offlight imaging following a rapid (< 1 µs) extinction of thelattice [15]. Also, the observed width of the sidebands is
consistent with our estimate of the coherent Raman coupling
induced by the near-resonant lattice.

0.9
0.8
0.7
0.6
0.5
0.4
0.3
-100

0
100
Two-photon detuning [kHz]200

FIG. 2. Spatially resolved sideband spectroscopy of the lattice+0.03. Inset:gas following the imaging sequence yielding hni = 0.01−0.01
A time-of-flight absorption image of the ultracold gas following an
intense interrogation pulse at the two-photon resonance. The divot
near the center of the atomic distribution shows the location andrelative size of the beams used for sideband spectroscopy. The fieldof view is 600 µm × 600 µm.

Our imaging scheme, as constructed, relies on the competition between two processes: atomic fluorescence at a rate Ŵfthat yields spatial information about the atomic distribution,
and RSC at a cooling rate ŴRSC that serves to cool the atoms
back to the ground state within each lattice site. While theformer depends solely on the intensity of the fluorescence2+ 2Ä2R )beam, the latter is given by ŴRSC ∼ Ŵopt × Ä2R /(Ŵoptwhere ÄR is the coherent Raman coupling between the states|mF ,νi and |mF − 1,ν − 1i, and Ŵopt is the rate of opticalpumping to the |mF = +1i state [16]. As the fluorescence
rate is increased significantly beyond the cooling rate, atoms
can be promoted to higher bands within the lattice and cantunnel to neighboring sites. In addition to modifying the atomic
distribution, such tunneling can also lead to multiply occupied
lattice sites and subsequent loss due to light-induced collisions.
To identify the regimes of imaging where the scheme
is nondestructive, we use light-induced collisional loss as adiagnostic tool to monitor atomic tunneling across lattice sites.
Further, in order to clearly demarcate atomic dynamics dueto the fluorescence pulse from that due to RSC, we employ
a pulsed imaging sequence wherein the fluorescence pulse
and RSC are employed in rapid succession with a variable
duty cycle. As expected, the average vibrational occupation
number measured at the end of the fluorescence pulse grows
with increasing fluorescence rate [Fig. 3(a)]. However, RSC
is very efficient at cooling the atoms back to the ground stateat the end of each cycle. At the end of each RSC cycle, wemeasure average vibrational occupation numbers (∼ 0.01) thatare, within our measurement uncertainty, indistinguishable
from those measured in the absence of the fluorescence pulse(Fig. 2). The typical measured RSC cooling rates of 13 µK/ms
are also consistent with that estimated based on the intensitiesof the lattice and optical pumping beams.
While simultaneous cooling during fluorescence acquisition leaves the final vibrational occupation unaltered, thetransient increase in temperature during the fluorescencepulse can cause tunneling of atoms to neighboring latticesites. This tunneling rate depends sensitively on both theaverage vibrational occupation number as well as the lattice
depth U0 , typically parametrized in units of the recoil energy
Er = ~2 k 2 /2m. At low rates of fluorescence acquisition [17],we observe that the total number of atoms is left unchanged
subsequent to the imaging sequence indicating a negligible
level of tunneling across sites. Beyond a certain fluorescence
rate Ŵf, max , we observe two-body loss indicating the onsetof tunneling of atoms [Fig. 3(b)]. As indicated by the rapid
decrease of atoms for fluorescence rates past this maximal
value, two-body loss is a very sensitive measure of thetunneling rates induced by the imaging sequence (see also
Refs. [18,19]). The temporal evolution of the atom number
following a brief, intense fluorescence pulse indicates that RSC
cools and binds the atoms to the ground state of a lattice site
within 100 µs [see inset of Fig. 3(b)], again consistent withmeasurements of the cooling rate.
We have performed Monte Carlo simulations of the imaging
process that accurately capture the sensitivity of two-bodyloss to tunneling events and the threshold behavior arisingfrom the competition of imaging, RSC, and tunneling. For the033422-2

PHYSICAL REVIEW A 90, 033422 (2014)

NONDESTRUCTIVE IMAGING OF AN ULTRACOLD . . .
2.0

(a)

Fluor.
RSC

1.5

1.0

Fluor.
RSC

0.8

0.5

0.6

0.4
0.0
0

20

40

60

80

100

120 x10 3-1

Fluorescence Rate, Γf [s ]max Γf [s-1]

Nf /NikBT 1.0
ωL

5

10

104

0.2

0

1.0

3

N [arb. units]

Nf / N i

0.4

s

(b)

0.8

0.6

20 40 60 80 100

0.2

10

1.4
1.3
1.2
1.1
1.0
0.9
0.8

5

10

6

10

Fluorescence Rate, Γf [s-1]FIG. 4. The fraction of atoms remaining after the imaging
sequence (Nf /Ni ) vs fluorescence rates for lattice depths of U0 /Er =14.6,24.9,36.7 and 76.2 (left to right). Inset: An estimate of the
maximum fluorescence acquisition rate Ŵf, max per atom vs s =U0 /Er .
0

2

4

6

8

10

t [ms]
104

4

10

105

106

Fluorescence Rate, Γf [s-1]FIG. 3. (Color online) Regimes of fluorescence acquisition rates
for non-destructive imaging. (a) Measured temperature of the latticegas in a pulsed imaging sequence, in units of the vibrational frequency
ωL . The temperature during the fluorescence pulse grows (red) withincreasing fluorescence rates while RSC rapidly cools the atoms backto the ground state (blue). (b) Measured atom number following theimaging sequence. At low fluorescence rates, the atom number isconserved indicating negligible levels of tunneling across sites. As
the fluorescence rate is increased, the increasing temperature during
the fluorescence pulse causes tunneling followed by light-induced
loss. The shaded area represents the critical fluorescence rate forthe onset of tunneling as identified by our measurements of lightinduced loss. Inset: Evolution of atom number immediately following
an intense fluorescence pulse (Ŵf = 6 × 105 s−1 ) shows that RSC
quickly (within 100 µs) binds the atoms to the ground state of alattice site thereby drastically suppressing tunneling.

filling fractions used in this work (0.20–0.25), the measured
critical fluorescence rate, Ŵf, max , as identified by the onset oflight-induced loss, is within 20% of the critical fluorescence
rate for the onset of tunneling. We further find that thefilling fraction needs to be reduced by more than an order ofmagnitude before there is a significant probability of tunneling
events that do not lead to measurable loss. These findings
justify the correspondence between the onset of tunneling in
the lattice gas and our measured onset of two-body loss.
Similar considerations apply to the imaging of atoms inshallow optical lattices (Fig. 4). In this case, the rates oftunneling grow exponentially with decreasing lattice depth[20,21]. This leads to a reduction of the maximal fluorescence
rates that can be used while constraining atomic motion. As
expected, an estimate of this maximum allowable fluorescence
rate shows an exponential decrease with lower lattice depths(see inset of Fig. 4). Importantly, we see that fluorescenceacquisition rates greater than 104 photons/s per atom arepossible even for lattice depths around 15 recoil energies. This
makes possible the use of this imaging technique to study
lattice gases in regimes where coherent tunneling of atoms
within the lowest band occurs on experimentally relevant timescales. In addition, it augurs the intriguing possibility of using
this imaging scheme to influence or exert spatial control oversuch coherent tunneling processes. Nondestructive imaging ofatoms in even lower lattice depths could be made possible
by increasing the Raman cooling rates and operating at lower
fluorescence acquisition rates.
At the lowest lattice depths, possible limitations to ourimaging scheme include the reduced fidelity of RSC due
to a lower Lamb-Dicke parameter and off-resonant Raman
coupling to higher vibrational bands, an increased susceptibility to photon reabsorption heating [22], and faster rates oftunneling to neighboring lattice sites. As we show in Fig. 4,
these limitations can be overcome by a suitable choice of fluorescence acquisition rate and Raman cooling rates. Already,the lowest lattice depth (s ∼ 15) for which we demonstratenondestructive imaging is more than two orders of magnitude
below that required for molasses-based lattice imaging.
IV. CONCLUSIONSIn summary, we demonstrate a nondestructive imaging
technique for ultracold atoms confined in an optical lattice.

033422-3

PHYSICAL REVIEW A 90, 033422 (2014)

PATIL, CHAKRAM, AYCOCK, AND VENGALATTORE

The imaging technique is based on extracting fluorescence
while simultaneously cooling the atoms to the ground stateof the lattice via Raman sideband cooling. Using a combination of sideband spectroscopy and time-of-flight imaging,
we demonstrate a large operational regime of fluorescenceacquisition rates and lattice depths for which the imaging
scheme preserves the spatial location of the atoms whileleaving them in the ground vibrational state. At the largest
rates of fluorescence acquisition (∼ 106 photons/s/atom) andthe lowest lattice depths, the main loss mechanism occurs due
to tunneling of atoms to occupied lattice sites followed by
rapid light-induced loss. By using the light-induced loss as adiagnostic measure of tunneling, we show that this limitation
can be alleviated by a suitable choice of fluorescence andRaman cooling rates. That said, we note that the imaging
scheme demonstrated here does lead to light-induced loss inlattice sites occupied by multiple atoms. In this regard, it issimilar to molasses-based imaging in its sensitivity to the parityof lattice occupancy.
Our imaging technique represents a powerful extension oflattice imaging to the nondestructive control and measurementof lattice gases. As such, it is an enabling technique to extend
concepts of single-particle quantum control to the context
of strongly correlated many-body systems. This scheme also
permits the continuous monitoring of the out-of-equilibrium

dynamics of ultracold lattice gases. We also note that whileused primarily as a diagnostic tool here, spatially resolved
coherent two-photon processes such as the setup used forsideband spectroscopy in our work, can also be used for
subdiffraction limited quantum control of the lattice gas[23,24]. Lastly, our imaging scheme is also extendable toatomic species that are less amenable to molasses-based lattice
imaging as well as to lattice geometries [25] where molassesbased imaging can be stymied by local polarization gradients.

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M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, Nature(London) 506, 71 (2014).
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and S. Kuhr, Nature (London) 467, 68 (2010).
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and P. S. Jessen, Phys. Rev. Lett. 80, 4149 (1998).
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D. S. Weiss, Phys. Rev. Lett. 85, 724 (2000).
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84, 439 (2000).
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Itano, D. J. Wineland, and P. Gould, Phys. Rev. Lett. 75, 4011
(1995).
[15] For very large atom numbers (∼108 ) where the atomic cloudsize is commensurate with that of the lattice waist, the time-of-flight measurements indicate slightly elevated temperaturesin comparison to that extracted from the local sideband spectra.
This is presumably due to the inhomogeneous variation of latticefrequencies across the atomic distribution and a reduced efficacy
of RSC for atoms on the periphery. In such cases, we rely on the
sideband spectra for a more reliable estimate of the temperature.
[16] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom
Photon Interactions (Wiley-Interscience, New York, 1998).
[17] The fluorescence rate is calibrated by the measured rate of
Raman fluorescence acquired by the camera, and a geometricalestimate of the numerical aperture of our imaging system. Thiscalibration is very close to the rate estimated from the measured
intensity and detuning of the fluorescence beam.
[18] N. Syassen, D. M. Bauer, M. Lettner, T. Volz, D. Dietze, J. J.
Garcia-Ripoll, J. I. Cirac, G. Rempe, and S. Dürr, Science 320,1329 (2008).
[19] B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. A. Hazzard,
A. M. Rey, D. S. Jin, and J. Ye, Nature (London) 501, 521(2013).
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Phys. Rev. Lett. 81, 3108 (1998).
[21] W. Zwerger, J. Opt. B 5, S9 (2003).
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(2000).
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A. P. Chu, R. Younkin, and M. Prentiss, Science 280, 1583
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ACKNOWLEDGMENTS

This work was supported by the ARO MURI on
Non-equilibrium Many-body Dynamics (63834-PH-MUR),
the DARPA QuASAR program through a grant from the
ARO and the Cornell Center for Materials Research withfunding from the NSF MRSEC program (DMR-1120296).
L.M.A. acknowledges support from a NSF graduate research
fellowship. M.V. acknowledges support from the Alfred P.
Sloan Foundation. M.V. conceptualized the imaging scheme;
M.V., S.C., and L.M.A. contributed to the development of theapparatus; Y.S.P. and M.V. performed the experiments, thedata analysis and the simulations; M. V. wrote the paper withhelp from Y.S.P. and S.C.

033422-4

APPENDIX B
VISUALIZING EDGE STATES WITH AN ATOMIC BOSE GAS IN THE
QUANTUM HALL REGIME151QUANTUM SIMULATION

Visualizing edge states with an
atomic Bose gas in the quantum
Hall regime
B. K. Stuhl,1* H.-I. Lu,1* L. M. Aycock,1,2 D. Genkina,1 I. B. Spielman1†
Bringing ultracold atomic gases into the quantum Hall regime is challenging. We
engineered an effective magnetic field in a two-dimensional lattice with an elongated-strip
geometry, consisting of the sites of an optical lattice in the long direction and of threeinternal atomic spin states in the short direction. We imaged the localized states ofatomic Bose-Einstein condensates in this strip; via excitation dynamics, we further
observed both the skipping orbits of excited atoms traveling down the system’s edges,
analogous to edge magnetoplasmons in two-dimensional electron systems, and a
dynamical Hall effect for bulk excitations. Our technique involves minimal heating, which
will be important for spectroscopic measurements of the Hofstadter butterfly andrealizations of Laughlin’s charge pump.

n solids, the quantum Hall effects represent an extreme quantum limit, at which asystem’s behavior defies description by classical physics. The integer quantum Hall effect(IQHE) for two-dimensional (2D) electronicsystems in magnetic fields (1) was the first topo-

I
1

Joint Quantum Institute (JQI), National Institute ofStandards and Technology (NIST) and University of
Maryland, Gaithersburg, MD 20899, USA. 2Departmentof Physics, Cornell University, Ithaca, NY 14850, USA.
*These authors contributed equally to this work. †Corresponding
author. E-mail: ian.spielman@nist.gov

logical insulator (2): a bulk insulator with conducting edge states, always present in finite-sized
topological systems, which produce the IQHE’s
signature quantized Hall resistance (3).
In classical systems, the magnetic field acts entirely through the Lorentz force, whereas in quantum systems, a particle with charge q in a uniform
field B additionally acquires an Aharonov-Bohm
phase fAB/2p = AB/F0, after its path encircles anarea A normal to B. (Here, F0 = 2pħ/q is the flux
quantum, and 2pħ is Planck’s constant.) We generated an effective magnetic field in a system of cold

j = -1
Lattice / Raman+ lasers

Lattice / Raman- lasers
87

Rb BECeyez

j=0

atoms, following (4), which yielded an elongated 2D
square lattice formed from the sites of an opticallattice in the long direction and three internal
atomic spin states in the short direction (a synthetic“spatial” dimension). We directly controlled thephases acquired as atoms traversed the lattice,creating an aggregate tunneling phase fAB/2p ≈4/3 around each lattice plaquette. These phases takethe place of the Aharonov-Bohm phases producedby true magnetic fields and suffice to fully define theeffective magnetic field. Aharonov-Bohm phases oforder unity in the Harper-Hofstadter Hamiltonian,currently realized in engineered materials (5, 6) andin atomic (7–12) and optical (13) settings, fragment the low-field Landau levels into the fractalenergy bands of the Hofstadter butterfly (14). Such
Hofstadter bands are generally associated with anonzero topological index, the Chern number (3).
Topologically nontrivial bulk properties are reflected by the presence of edge channels, composed of edge states, with quantized conductance.
In fermionic systems, the number of edge channels
is fixed by the aggregate topological index of thefilled bands (2, 3, 15); this ultimately gives rise tophenomena such as the IQHE for electrons. Conceptually, the constituent edge states can be viewed
as skipping orbits (2, 16, 17): In the presence of astrong magnetic field, nascent cyclotron orbits near
the boundary reflect from the hard wall before completing a revolution, leading to skipping trajectories
that follow the system’s boundary. In contrast, localized bulk states correspond to closed cyclotron orbits.
By applying large effective fields to atomic
Bose-Einstein condensates (BECs), we directly
imaged individual, deterministically prepared bulk
and edge eigenstates. In IQHE systems, these states

j=1

j=2

j=3

j=4

m = +1
m=0

B0

ex

Brf
m = -1

Energy−0.7

m = -11

0.5

m = +1

-2

-1

0

m

+1

+2

0

m

m=0

Energy (EL)

−0.8

−0.9
−0.5

Fig. 1. Hybrid 2D lattice. (A) 87Rb BECs (containing 105 atoms), confined in
a 1D optical lattice with a bias magnetic field B0ez, were illuminated either by−1
a pair of counterpropagating Raman lasers with crossed linear polarization
−1
−1
−0.5
0
0.5
1
or by an rf magnetic field Brf. (B) The Raman or rf field linking the three internal |mi states with Rabi frequency WR was detuned by d from the Zeeman
splitting (gmBB0/h ≈ 0.817 MHz or 1.35 MHz; g, the Landé g-factor; h, Planck’s
constant). The corresponding quadratic Zeeman shift lowered |m = 0i by e = 0.05EL (for the data in Fig. 3) or 0.13EL (for the data in Fig. 4). The Raman lasers’
relative phases were actively stabilized at a beam combiner adjacent to the optical-lattice retroreflection mirror (24). (C) The lattice along es can be considered
as a square well with hard walls at m = T2, for which d 6¼ 0 tilts the potential. (D) The 2D hybrid lattice, where the nonspatial dimension is built from the internal
states |mi, with an effective magnetic flux per plaquette of F/F0 = fAB/2p. (E) The three lowest magnetic bands (rainbow colors), computed for our full latticewithout making the tight-binding approximation, with parameters (ħWR, V, d, e) = (0.141, 6, 0, 0.05)EL. The pale curves represent computations for ħWR = 0.

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R ES E A RC H | R E PO R TS

would govern the conductivity, but as individual eigenstates, they exhibit no time dependence. The
corresponding dynamical entities are edge magnetoplasmons, consisting of superpositions of edge
eigenstates in different Landau levels (18, 19) or, in
this case, magnetic bands. We launched these excitations and recorded their full motion, observing
both a chiral drift along the system’s edge and theunderlying skipping motion.
The Harper-Hofstadter Hamiltonian
H¼−

X½tx jj þ 1; mihj; mj þj;mts jj; m þ 1ih j; mj þ h:c:(

ð1Þ

governs the motion of charged particles moving
in a 2D lattice (14, 20)—i.e., the situation that
we engineered for our neutral atoms—with complex hopping amplitudes tx (long direction) and ts(short direction) and sites labeled by j and m (h.c.,Hermitian conjugate). Analogous to the Landau
gauge in continuum systems, we describe our ex-

Lattice site, m

2

Fractional population, nm

1
0
-1
-2

2
1
0
-1
-2

Fractional population, nm

2Lattice site, m

Fig. 2. Adiabaticallyloaded eigenstates.
(Left panels) Siteresolved normalized
momentum distributions nm(kx) obtainedby absorption imaging.
(Right panels) Fractional population nm.
(A and B) Atoms
loaded into the single
ground state present
for fAB = 0 show
the expected separable behavior along ex
and es. (C to H) Atoms
loaded into the upperedge, bulk, and loweredge states (top,middle, and bottom
panels, respectively)
present for fAB/2p ≈4/3 demonstrate
the coupling between
ex and es and the
localization along es.

periment with real ts (no phase) and with complex
tx = |tx|exp(–ifAB m), dependent on m. The sum of
the tunneling phases around any individual plaquette is fAB (Fig. 1D).
We created a 2D lattice by combining a conventional optical lattice (21) to define the long axis of
our system (ex direction, sites labeled by j) withthree sequentially coupled internal spin states
to define the short axis (es direction, three sites
labeled by m ∈ {–1,0,+1}); in parallel to the work
described here, an analogous scheme has been realized for fermionic ytterbium (12). This system effectively has an infinite repulsive potential for |m| ≥2 (Fig. 1C), allowing for the formation of robustedge states. In each band of our engineered lattice(Fig. 1E), the momentum along ex specifies the position in es (denoted by color on the curves), as for
2D electrons in Landau levels.
We used 87Rb BECs in the f = 1 ground-state
hyperfine manifold (22), confined in an optical
dipole potential from two 1064-nm laser beams
aligned along ex and ey, with trap frequencies(wx , wy, wz)/2p = (50, 40, 110) Hz (Fig. 1, A and B).
We adiabatically (23) loaded these BECs onto the

1
0
-1
-2
2
1
0
-1
-2
-2

0

2

-2

-1

0

1

Lattice site, m

SCIENCE sciencemag.org

2

sites of the 1D optical lattice, formed by a pair oflaser beams (wavelength, lL = 1064.46 nm) counterpropagating along e x. The motion of atoms
along ey and ez within each 2D layer formed by
the 1D optical lattice was largely unaltered by thepresence of the lattice, which only affected motion
along ex. The lattice lasers’ wavelength defines the
single-photon recoil momentum ħkL = 2pħ/lL andrecoil energy EL= ħ 2k L2/2mRb, where ħ is the reduced Planck’s constant, and mRb is the atomic mass.
The lattice depth V ≈ 6EL gave a hopping strength|tx| ≈ 0.05EL along e x.
We then coupled the three |mi sublevels, eitherwith two-photon Raman transitions or with a radiofrequency (rf) magnetic field. The Raman lasers (lR =790.04 nm) also counterpropagated approximately
along ex (24), with wavenumber kR = ±2p/lR; the rf
coupling effectively had a kR of 0. Either field produced a laboratory-tunable effective tunnelingstrength |ts| ~ |tx| along e s, proportional to the Rabi
frequency WR. The 2ħkR momentum imparted
from these transitions resulted in a spatially periodic phase factor exp(i2kR x) = exp(ifAB j) accompanying the change in m, where fAB/2p = kR/kL ≈±4/3 for Raman coupling, and fAB/2p = 0 for rf
coupling (25, 4). The sign of fAB was controlled byreversing the relative detuning of the Raman lasers(Fig. 1, A and B), effectively swapping the laser
directions.
Figure 1E shows the band structure for oursystem, featuring the three Hofstadter bands expected for fAB/2p ~ 4/3, with our boundary conditions. The system is most easily understood after
making the local gauge transformation | j,mi→exp(–ifAB jm) | j,mi, which transfers the Peierls
phase from ts into t x. This fully maps our system
to Eq. 1 and results in an effective flux F/F0 =fAB/2p per plaquette (Fig. 1D).
We began our experiments by directly imaging adiabatically loaded eigenstates of the ground
Hofstadter band with either fAB/2p = 0 or fAB/
2p ≈ 4/3, in an isotropic lattice with |tx| ≈ | ts|. After
preparation, we used a measurement procedurecommon to all experiments: We simultaneously
removed all potentials and coupling fields (witha turnoff time toff < 1 ms), which returned theatoms to bare spin and momentum states. The
atomic cloud expanded for an ~18-ms time-offlight (TOF) period (21). During TOF, a 2-ms magnetic gradient pulse was applied, and Stern-Gerlach
spin separation was used to separate the three|mi states. The resulting 2D column density wasrecorded using standard absorption imaging techniques, giving the normalized momentum distributions nm(kx) with perfect single-latticesite resolution along es.
Figure 2A shows typical data for fAB = 0, underwhich conditions we adiabatically loaded the
BEC into the ground state and observed nm(kx).
The fractional population nm = ∫dkxnm(kx) (Fig.
2B) resembles that of a particle in a discretized box
along es, whereas the momentum distributions,typical for atoms in an optical lattice (26), have thesame profile for each m site. This demonstrates
that the two directions are uncoupled at fAB = 0.
The data in Fig. 2, C to E, for which fAB ≠ 0, arequalitatively different as a function of both kx
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RE S EAR CH | R E P O R T S

R ES E A RC H | R E PO R TS

1

3
2

0.8

1
0.6

in units of

0.4

-1
-2

0.2

0
0

0

-3
1

2

Time,

3

4

0

1

(ms)

2

3

Time,

4

(ms)

3
3
2

0
-1

2

6
in units ofin units ofin units of

1

1
-2

4
2
0
0

1

2

3

4

-3
0

and m. These differences can be understood byanalogy with a 2D electron system in a perpendicular magnetic field, confined in one dimension
with hard walls. Along the confined direction, thewavefunction is localized
pffiffiffiffiffiffiffiffiffito
ffi the scale of the magnetic length ℓB ¼ ℏ=qB, with the center position at kxℓB2 in the bulk state, and where ħkx isthe electron’s canonical momentum. At large |kx|,
the electron becomes localized near the edges,
lifting the degeneracy of the otherwise macroscopically degenerate Landau levels. Each ofthese points has an analog in our observations(Fig. 1E and Fig. 2, Fpto
ffiffiffiffiffiH).
ffiffiffiffiffi In our system, the+magnetic length ℓB e 3=2p ≈ 0:7, in units of lattice period, was of order unity (23); this considerably narrowed the bulk state (Fig. 2G), as
compared with conditions in which fAB = 0 (Fig.
2B). In addition, we observed the appearance ofstates that were localized at the system’s edges(Fig. 2, F and H), which were completely absent
when fAB = 0. These localized edge states are theanalog to the current-carrying edge states infermionic IQHE systems (27).
Having described the static properties of thissystem, we now turn to dynamics. We loaded oursystem onto different m sites with ts = 0 and then1516

25 SEPTEMBER 2015 • VOL 349 ISSUE 6255

0.1

0.2

0.3

0.4

0.5

abruptly (24) turned on ts , allowing tunnelingalong es. The resulting initial states all consisted ofcoherent superpositions of magnetic-band eigenstates with crystal momentum qx /kL = –mfAB /p,which began to coherently tunnel along es and tomove along ex as a result of the associated Lorentz
force. Atoms initialized on the bulk m = 0 site exhibited a dynamic Hall effect. Those starting on theedge sites became cold-atom analogs to edge magnetoplasmons: They began cyclotron orbits, werereflected from the hard wall, and skipped down
one edge or the other.
The dynamics of atoms initialized in the bulk
state (on the m = 0 site) are presented in Fig. 3.
As schematically illustrated in Fig. 3A and plottedin Fig. 3B, a balanced population oscillated in andout of the originally empty m = ±1 sites, as a function of time t. When fAB ≠ 0, this motion drove
transverse (i.e., Hall) edge currents Im=±1(t) =nm(t) hVmi along ex (Fig. 3A), where hVmi is themean velocity of atoms on site m along ex .
A chiral current I = I1 – I–1 developed, with
its overall sign following that of fAB (Fig. 3C).
As atoms tunneled to the edges, they acquireda transverse velocity controlled by two parameters: fAB determined the crystal momentum

0
0

1

2

3

4

acquired while tunneling, and tx set the natural unit of velocity 2tx/ħkL. This led to theobserved in-phase oscillation of I and the combined m = ±1 populations h|m|i.
This synchronous oscillation implies a linear
dependence of I on h|m|i, the slope S of which isplotted in red and black symbols in Fig. 3D. We
confirmed the system’s chirality by inverting fAB
and verifying that S changed in sign. For comparison, we repeated the experiment with fAB = 0 andobserved no chiral current (open symbols). These
data are in good agreement with our theory (solid
curves in Fig. 3D), which uses parameters obtained
from fits to Fig. 3B (24).
The dependence of the chiral current on thetunneling anisotropy ts/tx (Fig. 3E) is reminiscentof the optical lattice experiments in (28). The chiralcurrent remained linear in h|m|i, with an essentially constant slope S ≈ 2t x l Lsin(fAB)/ħ; this demonstrates a dynamic Hall effect (29) governed by
the magnetic flux and the optical lattice strengthand period (24). In contrast, the peak edge current
Imax (pink dashed line in Fig. 3C) strongly depends on ts/tx (Fig. 3E), increasing from zero andthen reaching saturation. For small t s /t x, few
atoms tunneled, giving a correspondingly small
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Fractional population, nm

Fig. 3. Chiral edge currents. (A) Schematic ofsystem dynamics: A system prepared on site m = 0
at time t1 = 0 obtains a chiral edge current at t2,
which returns to zero at t3. (B) Measured fractional population versus time for atoms initialized on m = 0 at time t = 0. The dynamics are thesame for positive and negative values of fAB.
Colors in (A) and (B) correspond to m sites, as inFigs. 1 and 2. (C) Chiral edge current I versus
time. Data shown in red, black, and open circles
correspond to positive, negative, and zero valuesof fAB, respectively. (D) I plotted against h|m|i.
The solid curves (theory) use parameters (ħWR,V, d, e) = (0.52, 6, 0.001, 0.05)EL, determinedfrom the data in (B) for fAB 6¼ 0 and giving ts =0.14EL, and parameters (ħWR, V, d, e) = (0.33, 6,–0.01, 0.05)EL for fAB = 0. (E) Maximum edge
current versus asymmetry (ts/tx). (Inset) Slope
S [taken from data as in (D)] is nearly independent of ts/tx.

RE S EAR CH | R E P O R T S

1EnergyLattice site, m

0.1

-1
-2

0

+1

0.5
0

0

-0.5
-0.1

+2

m

-1

0

1

2
Time,

3
(ms)

4

−1

0
Displacement,

0

1

2
Time,

3
(ms)

4

Imax ; as ts/tx increased, Imax began to saturate as
essentially all atoms tunneled (27).
We then shifted our focus from bulk excitations to edge excitations, which we studied bylaunching edge magnetoplasmons, or superpositions of edge states across magnetic bands withcrystal momentum qx/kL = ∓fAB/p. We createdthem on either edge, with the potential tilted alonges (Fig. 4, A and B), so that the initially occupied
site was at the potential minimum. The timeevolving average position hm(t)i along es and theXvelocity hnx ðtÞi ¼
Im along ex are shown inmFig. 4, C and D. Data shown in pink and blue
solid circles are for initial sites hm(t = 0)i = ±1, bothof which evolved periodically in time but with opposite velocities. The spatial trajectories are illustrated in Fig. 4E; we obtained the displacement
hdj(t)i by directly integrating hvx(t)/ai, where a =lL/2 is the lattice period. These data show edge
magnetoplasmons with their chiral longitudinalmotion and constitute an experimental observation of their edge localization and transverse
skipping motion.
This and related approaches (12) have a practical advantage over other techniques for creating
artificial gauge fields, in that minimal Ramanlaser coupling is required [typically 10 to 50 times
less than in previous experiments using Raman
coupling (30)], thereby minimizing heating fromspontaneous emission and enabling many-bodyexperiments that require negligible heating rates.
Lifetimes from spontaneous emission with thistechnique are in excess of 10 s (corresponding to aheating rate of <10–3 tx/2pħ), whereas other approaches for creating large artificial gauge fields
have lifetimes well below 1 s (8–10, 31).
The experiments described here used nearlypure BECs, either in adiabatically prepared eigenSCIENCE sciencemag.org

0

-1

−2

states or evolving after sudden changes to the
Hamiltonian. In the former case, interactions did
not affect our measurements, whereas in the latter case, collisions during the dynamical evolution
gradually populated additional states and contributed to considerable dephasing within 10 ms. Because our approach of using the internal atomic
spin states as lattice sites involves different “synthetic sites” residing in the same location in space,
the interactions between atoms are anisotropic—
short-ranged along ex but long-ranged along es. In(32), it was shown theoretically that even suchanisotropic systems can support fractional quantum Hall states.
With our hard-wall potential, a realization of
the Laughlin charge pump (33) is straightforward:
As particles accelerate along ex, mass moves fromone edge to the other in the orthogonal direction
es. Extending our technique to periodic boundary
conditions—i.e., coupling the |m = ±1i states—should
produce systems exhibiting a fractal Hofstadter
spectrum (4), even with only a three-site extent
along es. Going beyond conventional condensedmatter realities, the flexibility afforded by directly
laser-engineering the hopping enables the creation of Möbius strip geometries, or topologicalsystems with only one edge (34).
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23. Because all physics is 2p-periodic in the acquired phase, ourflux fAB/2p ≈ 4/3 is equivalent to fAB ≈ 2p/3. For the purposeof estimating the magnetic length,pwhich
ffiffiffiffiffiffiffiffiffiffi is a continuum
concept, it is suitable to use ℓ+B ≈ 3=2p.
24. Materials and methods are available as supplementarymaterials on Science Online.
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30. Y. J. Lin, R. L. Compton, K. Jiménez-García, J. V. Porto,
I. B. Spielman, Nature 462, 628–632 (2009).
31. J. Struck et al., Phys. Rev. Lett. 108, 225304(2012).
32. A. Petrescu, K. Le Hur, Phys. Rev. B 91, 054520 (2015).
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M. Lewenstein, New J. Phys. 17, 045007 (2015).
AC KNOWLED GME NTSThis work was partially supported by the Army Research Office’s
Atomtronics Multidisciplinary University Research Initiative(MURI), the Air Force Office of Scientific Research’s Quantum

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Fig. 4. Skipping orbits. (A and B) Schematic of dynamics, starting fromm = –1 (indicated by the blue ellipse) and including a nonzero detuning thattilted the lattice along es. (C) Mean displacement hmi versus time t for
excitations on both edges. Data for systems initialized on m = 1 (m = –1)
are depicted by pink (blue) circles. (D) Group velocity along ex versus
time for both edge excitations. Numerical simulations (solid curves) use
parameters (ħWR, V, d, e) = (0.41, 5.2, ± 0.087, 0.13)EL from fits to population dynamics, such as those shown in (C). In both cases, the potential
gradient along es from d was selected so that the initial site (m = T1) had
the lowest energy. (E) Edge-magnetoplasmon trajectories, where the displacement hdj(t)i was obtained by integrating hvx(t)/ai.

Lattice site, m

1

R ES E A RC H | R E PO R TS

H.-I.L., L.M.A, and I.B.S. performed numerical and analytical
calculations. All authors contributed to writing the manuscript.
I.B.S. proposed the initial experiment.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/349/6255/1514/suppl/DC1
Materials and Methods

NANOMATERIALS

Atomically thin two-dimensionalorganic-inorganic hybrid perovskites
Letian Dou,1,2* Andrew B. Wong,1,2* Yi Yu,1,2,3* Minliang Lai,1 Nikolay Kornienko,1,2Samuel W. Eaton,1 Anthony Fu,1,2 Connor G. Bischak,1 Jie Ma,2 Tina Ding,1,2
Naomi S. Ginsberg,1,2,4,5,6 Lin-Wang Wang,2 A. Paul Alivisatos,1,2,5,7 Peidong Yang1,2,5,7†
Organic-inorganic hybrid perovskites, which have proved to be promising semiconductormaterials for photovoltaic applications, have been made into atomically thin two-dimensional(2D) sheets. We report the solution-phase growth of single- and few-unit-cell-thicksingle-crystalline 2D hybrid perovskites of (C4H9NH3)2PbBr4 with well-defined squareshape and large size. In contrast to other 2D materials, the hybrid perovskite sheets
exhibit an unusual structural relaxation, and this structural change leads to a band gap
shift as compared to the bulk crystal. The high-quality 2D crystals exhibit efficientphotoluminescence, and color tuning could be achieved by changing sheet thickness aswell as composition via the synthesis of related materials.

he organic-inorganic hybrid perovskites,
especially CH3NH3PbI3, have recently beenused in solution-processable photovoltaic devices that have reached 20% power
conversion efficiency (1–4). These layeredmaterials have a general formula of (RNH3)2(CH3NH3)m-1AmX3m+1, where R is an alkyl oraromatic moiety, A is a metal cation, and X isa halide. The variable m indicates the numberof the metal cation layers between the two layers of the organic chains (5–11). In the extreme
case where m = V, the structure becomes athree-dimensionally bonded perovskite crystal
with a structure similar to BaTiO3. In the opposite extreme where m = 1, the structure becomes
an ideal quantum well with only one atomic
layer of AX42– separated by organic chains, inwhich the adjacent layers are held together by
weak van der Waals forces.
This arrangement is fundamentally different
from transition metal dichalcogenides, in which
one layer of the metal ions is sandwiched between two hexagonal layers of S or Se atoms,
affording a rigid backbone. In contrast, the lay-T

1

Department of Chemistry, University of California, Berkeley, CA
94720, USA. 2Materials Sciences Division, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA. 3School ofPhysical Science and Technology, ShanghaiTech University,
Shanghai, 201210, China. 4Department of Physics, University ofCalifornia, Berkeley, CA 94720, USA. 5Kavli Energy NanoScience
Institute, Berkeley, CA 94720, USA. 6Physical Biosciences Division,
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.
7
Department of Materials Science and Engineering, University ofCalifornia, Berkeley, CA 94720, USA.
*These authors contributed equally to this work. †Corresponding
author. E-mail: p_yang@berkeley.edu

1518

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Supplementary Text
Fig. S1
Reference (35)
Database S1

ered hybrid perovskites normally have a tetragonal or orthorhombic structure and are inherently
more flexible and deformable (5–11). By varying
the value of m, the thickness and the related
optoelectronic properties of the quantum well
can be tuned. To date, many organic amines,
metal cations (Cu2+, Mn2+, Cd2+, Ge2+, Sn2+, Pb2+,
Eu2+, etc.) and halides (Cl, Br, and I) have beenused to construct such layered materials (m =1 ~ 3), and their corresponding optoelectronic
properties have been well studied (12–15). Previous reports have claimed that the organic layers
effectively isolate the two-dimensional (2D) quantum wells in each layer from electronic coupling,if the organic chain is longer than propyl amine(16). This means that the properties of the atomically thin 2D quantum well should be the sameas those of the bulk layered material (microscopic
crystal, powder, or film). This hypothesis, as wellas the technical difficulty of separating individual
layers, has probably delayed investigation of freestanding single layers of such 2D materials. Veryrecently, attempts to obtain ultrathin 2D perovskite samples by spin coating, chemical vapor deposition, or mechanical exfoliation methods havebeen made with limited success (17–19).
Here we report the direct growth of atomicallythin 2D hybrid perovskites [(C4H9NH3)2PbBr4]
and derivatives from solution. Uniform squareshaped 2D crystals on a flat substrate with highyield and excellent reproducibility were synthesized by using a ternary co-solvent. We investigated the structure and composition ofindividual 2D crystals using transmission electron microscopy (TEM), energy-dispersive spec-

9 February 2015; accepted 28 July 201510.1126/science.aaa8515

troscopy (EDS), grazing-incidence wide-angle
x-ray scattering (GIWAXS), and Raman spectroscopy. Unlike other 2D materials, a structural relaxation (or lattice constant expansion)
occurred in the hybrid perovskite 2D sheetsthat could be responsible for emergent features.
We investigated the optical properties of the 2D
sheets using steady-state and time-resolved photoluminescence (PL) spectroscopy and cathodoluminescence microscopy. The 2D hybrid perovskite
sheets have a slightly shifted band edge emissionthat could be attributed to the structural relaxation. We further demonstrated that the as-grown
2D sheets exhibit high PL quantum efficiency aswell as wide composition and color tunability.
A structural illustration of a monolayer 2D
perovskite (Fig. 1A) shows the case with six Br
atoms surrounding each Pb atom, and the fourin-plane Br atoms are shared by two octahedrons,
forming a 2D sheet of PbBr42–. The negative
charges are compensated for by the butylammonium that caps the surfaces of the 2D sheet.
This structure is amenable to facile solution synthesis. The ionic character of such materials isstronger than the transition metal disulfides
and diselenides, and the bulk solid is soluble inpolar organic solvents such as dimethylformamide(DMF) (20). To grow 2D sheets, a very dilute precursor solution was dropped on the surface of a
Si/SiO2 substrate and dried under mild heating[see the supplementary materials (21)]. A DMF
and chlorobenzene (CB) co-solvent was initiallyinvestigated, because CB helps to reduce the solubility of (C4H9NH3)2PbBr4 in DMF and promote crystallization. Because CB has a similarboiling point and evaporation rate as DMF, thedrying of the solvents and the crystallization
process were uniform across the whole substrate.
We examined the products of this reaction by
optical microscopy and atomic force microscopy(AFM), but instead of monolayers, thick particles with random shapes formed on the substrate (fig. S1). Hybrid perovskites have limited
solubility in acetonitrile, and the solvent has
been used previously for making microscopic
hybrid perovskite single crystals (22). In thiscase, acetonitrile evaporates more quickly andhelps induce the formation of the ultrathin 2D
hybrid perovskite sheets. When acetonitrile wascombined with DMF and CB, uniform square
sheets grew on the substrate (Fig. 1B). The edge
length of the square crystals ranged from 1 to
10 mm, with an average of 4.2 mm (the size distribution statistics can be found in fig. S2). The
detailed synthetic procedure and discussion of
the role of each solvent can be found in the supplementary text (21).
sciencemag.org SCIENCE

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Matter MURI, NIST, and NSF (through the Physics Frontier Centerat the JQI). B.K.S. is a NIST–National Research Council
Postdoctoral Research Associate. L.M.A. was supported by the
NSF Graduate Research Fellowship Program. All authors except
I.B.S contributed to the data collection effort. B.K.S. and L.M.A.
configured the apparatus for this experiment. H.-I.L. led theteam on all aspects of the edge-current and skipping-orbit
measurements. H.-I.L., L.M.A., and B.K.S. analyzed data. B.K.S.,APPENDIX CFESHBACH ENHANCED S-WAVE SCATTERING OF FERMIONS: DIRECTOBSERVATION WITH OPTIMIZED ABSORPTION IMAGINING

157

New J. Phys. 18 (2016) 013001

doi:10.1088/1367-2630/18/1/013001

PAPER

OPEN ACCESS

Feshbach enhanced s-wave scattering of fermions: direct
observation with optimized absorption imaging

RECEIVED

30 September 2015

D Genkina1,4, L M Aycock1,2,4, B K Stuhl1, H-I Lu1, R A Williams3 and I B Spielman1

REVISED

1

20 November 2015

2

ACCEPTED FOR PUBLICATION

3

24 November 2015

4

Joint Quantum Institute, National Institute of Standards and Technology, and University of Maryland, Gaithersburg, MD 20899, USA
Physics Department, Cornell University, Ithaca, NY 14850, USA
National Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, UK
These authors contributed equally to the work.

PUBLISHED

17 December 2015

E-mail: ian.spielman@nist.gov
Keywords: quantum gases, atomic physics, Feshbach resonances, absorption imaging

Content from this work
may be used under theterms of the Creative
Commons Attribution 3.0
licence.

Abstract40
Any further distribution of We directly measured the normalized s-wave scattering cross-section of ultracold K atoms across athis work must maintainmagnetic-ﬁeld Feshbach resonance by colliding pairs of degenerate Fermi gases (DFGs) and imaging
attribution to thethe scattered atoms. We extracted the scattered fraction for a range of bias magnetic ﬁelds, andauthor(s) and the title of
the work, journal citation
measured the resonance location to be B0=20.206(15) mT with width Δ=1.0(5) mT. To optimize
and DOI.
the signal-to-noise ratio (SNR) of atom number in scattering images, we developed techniques tointerpret absorption images in a regime where recoil induced detuning corrections are signiﬁcant.
These imaging techniques are generally applicable to experiments with lighter alkalis that would
beneﬁt from maximizing SNR on atom number counting at the expense of spatial imaging resolution.

1. Introduction
Feshbach resonances are widely used for tuning the interaction strength in ultracold atomic gases. In degenerate
Fermi gases (DFGs), the tunability of interactions provided by Feshbach resonances has allowed for studies of
the creation of molecular Bose–Einstein condensates (BECs) [1–3] as well as observation of the phase transition
from the Bardeen–Cooper–Schrieffer (BCS) superconducting regime to the BEC regime at sufﬁciently low
temperatures [4–7]. Conversely, measuring interactions as a function of controlled parameters can be used tocharacterize a Feshbach resonance.
A Feshbach resonance occurs when a diatomic molecular state energetically approaches the two-atom
continuum [8, 9]. For a magnetic-ﬁeld Feshbach resonance, a bias magnetic ﬁeld deﬁnes the relative energy ofthe free atomic states in two hyperﬁne sublevels and the molecular state. Consequently, the Feshbach resonancecan be accessed by changing the bias ﬁeld. In cold atomic systems where only s-wave channels contribute toscattering, the interactions are entirely characterized by the scattering length a. In the simple case where there are
no inelastic two-body channels, such as for the 40K resonance discussed in this work, the effect of the resonanceon the scattering length between two free atoms is [8]⎛% ⎞a (B)  a bg ⎜ 1 ⎟,⎝
B  B0 ⎠(1)

where abg is the scattering length far from any resonance (background scattering length), Δ is the width of theresonance, and B0 is the ﬁeld value at which the resonance occurs.
The exact value of the resonant ﬁeld B0 is difﬁcult to calculate analytically and is commonly computed via
numerical models based on experimental input parameters [10–12] or determined experimentally [13, 14].
Many experimental techniques have been used to characterize Feshbach resonances. These include theobservation of atom loss due to three-body inelastic scattering, measurement of re-thermalization timescales,
and anisotropic expansion of a cloud upon release from a conﬁning potential, which infer the elastic scatteringcross section from collective behavior of the cloud [15–17]. More recently, precise measurements of the© 2016 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

New J. Phys. 18 (2016) 013001

D Genkina et al

molecular binding energy have been performed, which can be used in conjunction with theoretical modeling to
extract the scattering cross section [18, 19].
Here direct scattering was the primary probe of the location and width of a Feshbach resonance. We collided
pairs of DFGs and imaged the resulting s-wave scattered atoms as a function of bias magnetic ﬁeld. This allowed
us to observe the enhancement in scattering without relying on proxy effects. We measured the fraction of atoms
scattered during the collision at different bias magnetic ﬁelds and deduced the location and width of theresonance.
In contrast to BECs, where scattering halos are readily imaged [20–22], the density of Fermi clouds istypically ≈100 times less than that of BECs5, making it necessary to enhance the strength of inter-atomic
interactions to directly detect the scattered atoms. In our dilute DFGs, even with the resonant enhancement of
the scattering cross section, only a small fraction of the atoms was scattered. Using typical absorption imaging,direct detection of scattered atoms was difﬁcult due to detection uncertainty that particularly affected regions of
low atomic density. To optimize the signal-to-noise ratio (SNR) for low atom numbers, we absorption imaged
with fairly long, high-intensity pulses—a non-standard regime—which imparted a non-negligible velocity andtherefore Doppler shift to the atoms. Simulation of the absorption imaging process was necessary for an accurateinterpretation of these images. Using the simulation-corrected images, we extracted the fraction of atoms
scattered in our collision experiment.
This paper is divided into two parts. First, we study absorption imaging in the presence of a signiﬁcant timedependent Doppler shift and show how we use our results to interpret data. Second, we describe our s-wave
scattering experiment and extract a measure of the location and width of the Feshbach resonance in 40K.

2. Absorption imaging in the presence of strong recoil induced detuning
Absorption imaging measures the shadow cast by an atomic ensemble in an illuminating probe laser beam withangular frequency ωL. This imaging technique relies on optical transitions between ground and excited atomic
states. Such atomic transitions have an energy difference X0, and a natural transition linewidth Γ. Wheninteracting with a laser ﬁeld an atom scatters photons from the ﬁeld into the vacuum modes. In the two-level
atom approximation, the rate of scattering is [25]
Hsc 

(
21

I˜
2
4E˜

I˜

,

(2)

where I˜  I Isat is the laser intensity in units of the saturation intensity, and E˜  E ( is the detuningE  X L  X0 in units of the natural linewidth.
An absorption image is obtained by shining an on- or near-resonant probe beam (generally E˜  1) onto theatomic cloud. Some of the light is scattered by the atoms, and the shadow cast by the atoms in the probe beam,I˜f (x , y ), is imaged onto a camera, as depicted in ﬁgure 1(a) (top). The probe light is reapplied with the atoms
absent to calibrate the intensity I˜0 (x , y ) of light unaffected by the atoms (bottom).
Consider the light as it travels along the imaging axis ez through a 3D atomic density proﬁle ρ(x, y, z). We
focus on a single pixel of the camera: sensitive to a single column of atoms ρ(z), integrated in x and y over the
pixel, giving a single value of I˜0 and I˜f . Every atom scatters light according to equation (2). Therefore, the atoms
further along the imaging axis ez experience a reduced optical intensity due to attenuation of the laser ﬁeld by theother atoms (ﬁgure 1(b)). The intensity change from scattering as a function of z isdI˜(z )  X L S (z ) Hsc (z )   S (z ) T0dz
1

I˜(z )
2
4E˜

I˜

,

(3)

where σ0 is the resonant scattering cross section. Integrating this equation [26] yields a straightforward relation
between the observed intensities I˜0 and I˜f and the atomic column density n  ¨ S (z ) dz :(T0 n   1

4E˜

2

) ln ( I I )f0I˜0  I˜f .

(4)

We call the column density deduced from this relation T0 n(1). When the probe intensity is much smaller than thesaturation intensity, I˜0  1, and the probe light is on resonance, E˜  0, the right-hand side of equation (4)reduces to the optical depth, deﬁned as OD  ln (If I0) [26], giving the simple relationship T0 n(0)  OD. In
all other regimes, the optical depth is not constant and depends on the probe intensity and imaging time.
Equations (3), (4) neglect the atomic recoil momentum and the resulting Doppler shift [27]. When an atom
absorbs a photon from the laser light ﬁeld it acquires a momentum kick k r in the ez direction. The associatedrecoil velocity is vr  k r m, where m is the atomic mass and k r  h M is the recoil momentum from the laser
with wavelength λ. Each re-emitted photon imparts a similar recoil momentum pe , but over many scattering
5This is not the case for recently realized erbium and dysprosium DFGs [23, 24], where strong dipolar interactions are present.

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Figure 1. Absorption imaging. (a) Near resonant probe light illuminates the atoms, and the transmitted light (containing a shadow of
the atoms) is imaged on the camera. A second image taken with no atoms provides a reference. (b) The probe beam is partiallyabsorbed as it traverses the cloud, and the intensity seen by atoms further along the imaging direction ez is lowered. (c) An atomic
cloud illuminated by a probe light ﬁeld absorbs photons from the probe and re-emits them in all directions. This process results in a
net acceleration of the cloud in the direction of the probe light as well as diffusive spreading in the transverse directions.

Figure 2. Dependence of velocity and detuning on position simulated for 40K at three different imaging times and a probe intensity
I˜0  0.8.

events this momentum distribution averages to zero. Therefore the atom will only acquire an average velocity
per photon vr along ez . The variance of pe , however, is not zero, allowing the atoms to acquire some momentum
transverse to the laser ﬁeld. While we ignore this correction, it results in the reduction of spatial resolution in theﬁnal image and its effect on the atomic cloud is pictured in ﬁgure 1(c).
The average atomic velocity parallel to the light ﬁeld after scattering N photons is N vr and the laser frequency
as seen by the atoms is Doppler shifted E  k r Nvr from resonance. After an atom scatters Nphotons  ( 2k r vr, itgets Doppler detuned by half a linewidth. For a probe intensity of I˜0  1, the time it takes a single atom to scatteron average that many photons is given by t recoil  Nphotons Hsc  2 k r vr. For 40K atoms imaged on the D2
transition, the case relevant for our experiment, Nphotons  178 and trecoil=18.76 μs—for imaging times
longer than that the recoil induced detuning correction cannot be neglected. Furthermore, this detuning varies
both with imaging time t and with distance along the propagation direction ez (ﬁgure 2). Thus, the laserʼs
spatially varying intensity proﬁle in the atomic cloud also depends on time:dI˜(t , z )  T0 S (t , z )dz
1I˜(t , z ).
4E˜ (t , z )2 I˜(t , z )Assuming that the atoms do not move signiﬁcantly during the imaging time (we will remove this assumption
shortly), the dimensionless detuning is
3(5)

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Figure 3. Optical depth as a function of probe intensity as predicted by the simulation (dots) and by equation (4) (curves), for threedifferent imaging times, and for column densities σ0n=1.6 (blue) and 0.2 (green). As expected, the predictions agree in both the highand low intensity limits, and differ for probe intensities comparable to the saturation intensity and longer imaging times.

E˜ (t , z ) 

kr vr
2T0 S (t , z )t¨0dI˜(z , U )
dU ;
dz

(6)

the relationship between the atomic density and the observed intensities is no longer straightforward.
Peturbative treatments of these equations also prove insufﬁcient (see appendix A.1).
2.1. Simulation
To obtain a relationship between the atomic density and the observed intensities in this non-standard regime, we
numerically simulated the imaging process, including the recoil induced detuning. We used parameters relevant
to our experiment—the D2 transition of 40K, with λ = 766.701 nm, Γ/2π = 6.035 MHz, vr = 1.302 cm s−1 andIsat = 1.75 mW cm−2 [28]. The simulation obtained I˜f as a function of imaging time t, atomic density σ0n, andprobe intensity I˜0.
We performed two versions of this simulation. First, we took a simplistic approach where the spatial
distribution of atoms did not change appreciably during the imaging time: vt  I0 (X L Hsc S )—the stationary
assumption. Starting with a Gaussian density proﬁle, we numerically integrated equations (5) and (6) andobtained a simulated optical depth for a range of input probe intensities and atomic column densities. We used
the results of this simulation to check the self-consistency of the stationary atom assumption and found it to beinvalid (see appendix A.2).
To account for the changing atomic distribution during the imaging pulse, we numerically simulated theclassical kinetics of atoms subject to the recoil driven optical forces, and obtained a dynamics adjusted version of
the simulated optical depth. We compared the optical depths predicted by each of the two simulations in theparameter range 0.3 - t - 100 μs, 0.01 - I˜0 - 50 and 0.01 - T0 n - 2 and found that the predicted opticaldepths were hardly changed by including the full time evolution (see appendix A.3). Thus, for the purposes ofdeducing the atomic density from experimental optical depths, the stationary atom simulation is sufﬁcient in theexperimentally relevant parameter regime we explored. However, since there is no a priori reason why thisshould be sufﬁcient, it is possible that for some range of experimental optical depths or imaging times thiscorrection is signiﬁcant, or that it has some impact on the acquired transverse velocity of the atoms—an effectnot considered in our simulations. Furthermore, we simulated a range of initial density proﬁles ρ(z), and foundtheir impact on the simulated OD to be negligible—the only observable is the integrated atomic density n, and
3D atomic densities cannot be reconstructed.
Figure 3 illustrates the effect of the recoil induced detuning correction as obtained from our simulations. In
the limit of low probe intensity, I˜0  1, the atomic velocities are hardly changed and the recoil induced
detuning correction is negligible. In the limit of high probe intensity I˜0  E˜ , even far detuned atoms will scatter
light at their maximum rate and the overall absorption will again be unaffected by the correction. In theintermediate regime, there is a signiﬁcant deviation between the optical depth predicted by equation (4) and thesimulated optical depth, and this deviation becomes stronger with longer imaging times.
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Figure 4. SNR for three different column densities after correcting for recoil induced detuning. (a) SNR as a function of imaging time
for a probe intensity of I˜0  5.0 and (b) SNR as a function of probe intensity for an imaging time of 50 μs.

This simulation provided us with a correction procedure to interpret experimentally observed I˜f and I˜0. For
a given imaging time, the simulation predicted a ﬁnal intensity as a function of probe intensity and atomic
column density. We inverted this prediction to obtain an atomic column density given our observed intensities.
For interpreting experimental images, we used the optical depths predicted by the traveling atom simulation,
ODsim.

2.2. SNR optimization
We added shot noise to our simulation and established optimal imaging parameters to maximize the SNR of thisdetection scheme. We considered Poisson noise on the detected arriving photons (i.e., photoelectrons) in boththe initial and ﬁnal images I0 and If, with standard deviation proportional to qe Np , where qe is the quantum
efﬁciency of the camera (0.66 for our camera) and Np is the photon number. We then propagated the shot noisein the two images using standard error propagation techniques through the correction scheme described in
section 2.1 to obtain the uncertainty in a deduced column density, ET0 n. We deﬁne the SNR as T0 n ET0 n.
As seen in ﬁgure 4(a), after about 40 μs extending the imaging time no longer yields appreciableimprovement in SNR. Imaging for 40 μs as opposed to 10 μs, where the uncorrected model is appropriate,
improves the SNR by a factor of 1.5. We therefore performed the experiments described in the second section at
40 μs imaging time. Figure 4(b) shows that the optimal probe intensity varies with the atomic column density.
For low atom numbers, T0 n x 0.1, a probe intensity of I˜0 x 0.6 is best. However, in our experiment the probe
intensity had a Gaussian proﬁle and was not uniform over the whole image. The typical probe intensities used inour experiments varied over the I˜0  0.1–0.7 range.

2.3. Calibration of saturation intensity
The calibration of the observed signal in units of the saturation intensity is crucial to our measurement of thecolumn densities. Our absorption images were taken using a charge-coupled device camera. For each pixel, thecamera returned an integer number of counts proportional to the radiant ﬂuence seen by that pixel. However,the proportionality constant depended on many factors, such as the quantum efﬁciency of the camera, theelectronic gain during the readout process and losses in the imaging system. One way to determine this
proportionality constant is to experimentally calibrate the saturation intensity in counts per unit time.
To calibrate the saturation intensity in camera counts per unit time, we took absorption images of the atoms
at three different imaging times (40, 100, and 200 μs) with varying probe intensities. For each image we obtainedI˜0 and I˜f in counts per microsecond by averaging over a few pixels in a region of constant atomic column density.
We then simultaneously ﬁt our simulated optical depth ODsim to this full data set, with the atomic density σ0n
and Isat in counts per microsecond as free parameters. As seen in ﬁgure 5, the model produced a good ﬁt to theexperimental data, and provided a calibration of the saturation intensity for our experiment.

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Figure 5. The optical depth as a function of probe intensity for three imaging times: t=40 μs (black), t=75 μs (blue), t=100 μs(red). The dots represent experimental data and the curves represent the best ﬁt of simulated data. The optimal ﬁt parameters pictured
are a σ0n of 1.627(5) and saturation intensity of 29(7) counts/μs. The dashed curve represents the theoretical prediction without recoil
induced detuning corrections.

3. s-wave scattering experiment
For our measurement of the Feshbach resonance location and width, we collided two counter-propagating 40K
clouds in a spin mixture of ∣F  9 2, mF  9 2§ and ∣F  9 2, mF  7 2§ hyperﬁne states and observed
the resulting s-wave halo of scattered atoms. We measured the dependence of the scattered atomic fraction on
the bias magnetic ﬁeld in the vicinity of the Feshbach resonance. We used this data to extract the location of theresonance at 20.206(15) mT with width 1.0(5) mT, consistent with the accepted values of 20.210(7) mT and 0.78(6) mT [7].
3.1. Experimental procedure
Our experiment is a hybrid 40K and 87Rb apparatus, previously described in [29–31]. Initially, we prepared a spinpolarized ∣F  9 2, mF  9 2§ DFG of x4 q 105 atoms of 40K at a temperature of T≈0.4 TF, where TF is
the Fermi temperature, in a crossed optical dipole trap with frequencies (Xx , Xy , Xz ) 2Q  (39, 42, 124) Hz(see appendix B).
To map out the entire Feshbach resonance without the added losses associated with going through theresonance [8], we needed to create equal spin mixtures of ∣F  9 2, mF  9 2§ and ∣F  9 2, mF  7 2§
on either side of the resonance. We ramped the bias magnetic ﬁeld to 19.05 mT (21.71 mT) and turned on a
42.42MHz (47.11 MHz) rf ﬁeld6 resonant with the Zeeman splitting between the two states when preparing the
mixture below (above) the Feshbach resonance. We then sinusoidally modulated the bias ﬁeld at 125 Hz for
0.5 s, with a 0.14 mT amplitude, producing an equal mixture of the two hyperﬁne states. The depolarization
allowed the fermions to re-thermalize, allowing us to further evaporate in the dipole trap [32]. These hyperﬁnestates of 40K were then used to study their Feshbach resonance.
After evaporation, we ramped the bias ﬁeld in a two-step fashion to the desired ﬁeld value near the Feshbachresonance. The two-step procedure was designed to allow us to approach the set-point quickly and avoid
additional losses. This procedure used two sets of Helmholtz coils—large coils that provided the majority of thebias ﬁeld but had a long inductive timescale, and smaller coils only capable of generating 0.59 mT of bias, butwith a shorter inductive timescale. We approached the ﬁeld using the large coils to bring the magnetic ﬁeld to a
set-point 0.59 mT above or below the intended bias ﬁeld. We held the atoms at this ﬁeld for 100 ms to allow theeddy currents induced by the large coils to settle, and then used the smaller coils to quickly change the bias ﬁeld
the remaining 0.59 mT. For all set-points, the data was taken approaching from both above and below the
Feshbach resonance7.
Once at the intended bias ﬁeld, we split the cloud into two spatially overlapping components with opposite
momenta and observed scattering as they separated. These counterpropagating components were created using
Kapitza-Dirac pulses of a 1D retro-reﬂected near-resonant optical lattice (λL=766.704 nm) with 8EL depth,
where EL  2kL2 2mK is the lattice recoil energy and kL  2Q M is the recoil momentum. We rapidly pulsed
6
7

The rf intensity at these frequencies was too low for us to calibrate the coupling strength.
An extra data point was taken on each side far from the resonance using only one approach.

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this lattice on and off with a double-pulse protocol [33]. The pulse sequence was optimized to transfer most of
the atoms into the o2kL momentum states. Since the initial Fermi gas had a wide momentum spread (here,
2kL x 2.5kF, where kF is the Fermi momentum), and the lattice pulsing is a momentum dependent process[34], not all the atoms were transferred into the target momentum states. We optimized our pulse times tominimize the atoms remaining in the zero momentum state.
We then released the atoms from the trap and allowed 1 ms for the two opposite momentum states to passthrough each other while interacting at the magnetic ﬁeld set-point. For data taken approaching the set-pointfrom below, we then ramped down the ﬁeld and imaged the atoms. For data taken approaching the set-pointfrom above, molecules may have been created when crossing the Feshbach resonance. Therefore, we ﬁrst
ramped the ﬁeld up to a point above the resonance to dissociate any molecules that were created and thenquickly ramped the ﬁeld back down and imaged the atoms. After a total time-of-ﬂight tTOF=6.8 ms, we used a
40 μs imaging pulse with I˜0 x 0.6 at the center of the probe laser, chosen for SNR optimization as described in
section 2.2.
3.2. Magnetic ﬁeld calibration
The magnetic ﬁelds produced by our coils in the regime of interest were independently calibrated by rfspectroscopy on the ∣F  9 2, mF  9 2§ to ∣F  9 2, mF  7 2§ transition. We prepared a spinpolarized state and ramped the large coils to variable set-points. We then illuminated the atoms with a rf ﬁeld
with frequency νrf and performed adiabatic rapid passage (ARP) by ramping the smaller coils 0.0284 mT in
250 ms. We applied a Stern–Gerlach pulse and imaged the atoms to measure the fractional population in the∣F  9 2, mF  9 2§ and ∣F  9 2, mF  7 2§ states. We ﬁt the fractional population as a function ofcurrent to a Gaussian function8. The center of the Gaussian corresponded to the resonant magnetic ﬁeld, which
was produced by the high inductance coil setpoint plus half the ARP, 0.0142 mT, with an uncertainty given by
the Gaussian width. We used the Breit–Rabi formula to determine the resonant ﬁeld value at νrf. We did this forﬁve different rf frequencies, and acquired a ﬁeld calibration with an uncertainty of 0.004 mT, which was
included in the listed uncertainty on our measured value of B0.
3.3. Methods
We ﬁrst processed the s-wave scattering images by comparing the observed OD to simulations taking into
account the recoil induced detuning as described in section 2. An example of images before and after processing
are shown in ﬁgure 6. The processing constituted a ≈30% change in the column density.
We counted the fraction of atoms that experienced a single scattering event for each image. Single scattering
events are easily identiﬁed, as two atoms that scatter elastically keep the same amplitude of momentum, but
depart along an arbitrary direction. Therefore, an atom traveling at 2kL to the right that collides elastically withan atom traveling at 2kL to the left will depart with equal and opposite momenta 2kL at an arbitrary angle,
and after a time-of-ﬂight sufﬁciently long to convert initial momentum into position, as ours was, such atoms
will lie in a spherical shell, producing the scattering halo pictured in ﬁgure 7(a).
Absorption images captured the integrated column density along ez , a projected 2D atomic distribution. To
extract the radial dependence of the 3D distribution from the 2D image, we performed a standard inverse Abeltransform [35]. The inverse Abel transform assumes cylindrical symmetry, which was present in our case, withthe axis of symmetry along ex , deﬁned by the lattice. We thus obtained the atomic distribution ρ(r, θ) as afunction of r, the radial distance from the scattering center, and θ, the angle between r and symmetry axis ex ,integrated over f, the azimuthal angle around the x axis.
We then extracted the number of scattered atoms Nscat as a fraction of the total atom number Ntot for each
image, as shown in ﬁgure 7(b). The unscattered atom number was the number of atoms in the two unscatteredclouds. The number of atoms that underwent a single scattering event was the number of atoms outside theFermi radius of the unscattered clouds, but inside the arc created by rotating the Fermi momentum kF aroundthe original center of the cloud (red arcs in ﬁgure 7(b)). For both the scattered and unscattered quantities, we
extrapolated to include atoms that would fall outside the ﬁeld of view of our camera. The atoms in the centerregion were not counted as they were originally in the zero momentum state and could not contribute to thescattering halo under study. We did not account for possible s-wave scattering between the atoms found in thezero momentum state and one of the ±2kL momentum states in this treatment.
Since we were in the low energy regime (the atomic momentum was much smaller than the momentum set
by the van der Waals length kL kF  1 l vdW, and we were well below the p-wave threshold temperature [32]),
the scattering cross-section was given by σ=4π a2. The scattering cross-section σ gives the probability
Pscat=σ N/A that a single particle will scatter when incident on a cloud of atoms with a surface density of N/A,
where A is the cross-sectional area of the cloud and N is the number of atoms in the cloud. In our case, each half8

Due to our low rf coupling and high noise, we did not ﬁt to the traditional Loretzian model.

7

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Figure 6. Examples of our absorption images after 6.8 ms TOF. The 1D lattice imparts momentum along ex . In each image, the twolarge clouds on the left and right are the atoms in the ±2 kL momentum orders that passed through each other unscattered. The
smaller cloud in the center is composed of the atoms that remained in the lowest band of the lattice after pulsing, and thus obtained no
momentum. This cloud appears somewhat depleted closer to resonance due to scattering with atoms in the ±2 kL momentum orders—an effect not taken into account in our analysis. The thin spread of atoms around these clouds are the atoms that underwent
scattering. (Top) Raw optical depths, far from resonance (19.68 mT) on the left and close to resonance (20.04 mT) on the right,(Bottom) atomic column density σ0nsim obtained by applying corrections to raw optical depth above based on simulations(section 2.1).

Figure 7. (a) Our experimental setup. Top. The 1D lattice was pulsed, imparting momentum to the atoms and deﬁning the axis ofsymmetry. Bottom. After time of ﬂight, the two clouds traveling along± ex directions have separated and the atoms that underwent asingle scattering event were evenly distributed in a scattering halo around the unscattered clouds. (b) Inverse Abel transform ofcorrected image. The atoms within the Fermi momentum kF of each unscattered cloud center are in the unscattered region andcounted towards the total unscattered number. The atoms within the radius kL  kF - r - kL kF but outside the unscatteredregion are counted towards the number of single scattered atoms.

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Figure 8. Normalized scattered population plotted versus bias ﬁeld B. Green dots represent data taken coming from below theresonance, and blue dots represent the data taken coming from above the resonance. The red curve depicts the line of best ﬁt. The
regime where the scattering length is likely large enough for the atoms to behave hydrodynamically is shaded in gray, and data points
in that area were excluded from the ﬁt. Resonant ﬁeld value B0 from literature and as found in this work are indicated.

of the initial cloud, with atoms number Ntot/2, was incident on the other half. Thus, the number of expected2
scattering events was Nscat  (Ntot 2) T (Ntot 2)  TNtot
4A. Assuming A was constant for all our data, we
2deﬁned a ﬁt parameter b0  4Qabg 4A, where abg is the background scattering length. We thus adapted
equation (1) to obtain the ﬁt function
2
⎛
% ⎞
Nscat


b
1
⎜
⎟
0
2
⎝
B  B0 ⎠
Ntot

C,

(7)

where B0 is the resonant ﬁeld value and Δ is the width of the resonance, the parameters in equation (1), and theoffset C accounts for any systematic difference in the initial and ﬁnal intensity images with no atoms present.
For each value of the bias magnetic ﬁeld, we took 15 nominally identical images, allowing us to compensate
for shot-to-shot atom number ﬂuctuations. We ﬁt the fraction of scattered atoms Nscat/Ntot versus the total
atom number Ntot for each of these 15 images to a line. The slope of this ﬁt was taken to be the value of Nscat/N2tot
at that bias magnetic ﬁeld, and the variance of the ﬁt gave the uncertainty on that data point.
3.4. Results
Our data is presented in ﬁgure 8. The red curve depicts a best ﬁt of the model given in equation (7). The ﬁtparameters we extracted were Δ=1.0(5) mT and B0=20.206(15) mT. To obtain the ﬁt, we used data taken byapproaching the resonance from above for points above where we expected the resonance to be and data taken
approaching the resonance from below for points below. We also excluded from the ﬁt data points very near the
resonance, as there the assumption Tn  1 is no longer valid and the problem must be treatedhydrodynamically [8]. Due to this, we could not obtain usable data very close to the resonance, explaining thelarge uncertainty on the resonance width.
The accepted values for the 40K s-wave Feshbach resonance for the ∣9 2,  9 2§ and ∣9 2,  7 2§ states are
B0=20.210(7) mT and Δ=0.78(6) mT [7], which is in good agreement with our ﬁndings. Although the data
without the recoil induced detuning correction were ≈30% different from the corrected data, the optimal
parameters from ﬁtting the uncorrected data were within our uncertainties from the values listed above. The
resonance location was largely unaffected by the correction, as the scattering cross-section diverges there,
making it insensitive to the precise values. The width may have been affected, however the uncertainty on the ﬁt
is too large for the effect to be signiﬁcant. Performing the correction was still necessary to ensure the values were
proportional to the scattering cross section and to obtain a trustworthy result.
Some potential sources of systematic uncertainty that we did not account for include scattering with atoms
that did not receive a momentum kick from the lattice pulsing and the impact of multiple scattering events.

4. Conclusion
We studied the effects of recoil-induced detuning on absorption images and found an imaging time thatmaximized SNR to be ≈40 μs for 40K atoms. We used these results to directly image s-wave scattering halos of
the Fermi gas around the ≈20.2 mT Feshbach resonance and verify the resonance location and width. Our
imaging analysis can be used in any absorption imaging application where SNR optimization is critical.
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Acknowledgments
We thank Marcell Gall for helpful discussions. We thank William R McGehee and Elizabeth A Goldschmidt for acareful reading of the manuscript. This work was partially supported by the AROs Atomtronics MURI, by the
AFOSRs Quantum Matter MURI, NIST, and the NSF through the PFC at the JQI. BKS acknowledges support
from the National Research Council Research Associateship program. LMA acknowledges support from the
National Science Foundation Graduate Research Fellowship program.

Appendix A
A.1. Peturbative treatment
By considering equations (5), (6) perturbatively in imaging time, we can obtain corrections to the column
density due to recoil induced detuning to second order [36]:T0 n(2)  c 0

c1 t

c2 t 2, where2c 0  T0 n(1) , c1  0, c2 

( k v ) ⎡⎢ 1r r

3⎢⎣ I˜f

1



1
I˜0

1

(A.1)

⎛ I˜f
ln ⎜
⎝ I˜0

1 ⎞⎤
⎟ ⎥.
1 ⎠ ⎥⎦(A.2)

However, as shown in ﬁgure A1(a), the perturbative treatment is only accurate to times up to the recoil time
trecoil—deﬁned as the time it takes a single atom to get Doppler shifted from resonance by half a linewidth—after
which this prediction begins to diverge. To adequately correct for the recoil induced detuning of the atoms,
numerical simulation is necessary.
A.2. Stationary atom model
To solve equations (5), (6), we divided the cloud into spatial bins. In this approximation, the number of atoms ineach bin was time-independent. The algorithm used is shown in algorithm 1, in which we took a Gaussian
proﬁle for our initial density distribution. We call the optical depth simulated by this algorithm the simulated
optical depth ODsim1.
Algorithm 1. Stationary atom modelI˜[n  0, t ]  I˜0 {n is the bin index, t is the time index}
E˜ [n, t  0]  0 {light initially resonant}Hf=0 {Radiant ﬂuence seen by camera after passing through cloud}for t=0 to tf do {loop over time steps}
for n=1 to N do {loop over bins, N is total bin number}A=σ0ρ[n] dz { dz is the size of spatial step}B  vr dt (cS [n]) {dt is the size of the time step}I˜[n, t ]  I˜[n  1, t ]  AI˜[n  1, t ] (1 4E˜ [n, t  1]2 I˜[n  1, t ]) {equation (5)}E˜ [n, t ]  E˜ [n, t  1] B (I˜[n  1, t ]  I˜[n, t ]) {equation (6)}end forHf  HfI˜[N , t ] dt {collecting total ﬂuence seen by the camera}end for
ODsim1  ln (Hf I˜0 t f )We checked the validity of our simulation in the limits where the problem is analytically solvable. In the limit
where the probe intensity is much weaker than the saturation intensity, I˜0  1, the atoms’ velocities are hardly
changed, and equation (5) reduces todI˜(z )
  ST0 I˜(z ) , from which we recover the analytic form,(A.3)
dz
T0 n(0)   ln I˜0 I˜f .
(A.4)
In the limit where the probe intensity is much larger than the saturation intensity, I˜0  E˜ , even far detuned
atoms will scatter light at their maximum rate. The time dependence of the detuning can thus be neglected, andequation (5) becomesdI˜(z )
  ST0, which integrates to
(A.5)
dz
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T0 n  I˜0  I˜f .

(A.6)

We recognize the right-hand sides of equation (A.4) and equation (A.6) as the two terms in equation (4). Thus, as
shown in ﬁgure 3, ODsim1 coincides with the optical depth as predicted by equation (4) in both the small andlarge probe intensity limits.
We used the results of this simulation to check the self-consistency of the stationary atom assumption, i.e.
the distance traveled by the atoms (as deduced from integrating the acquired recoil velocity over the imagingtime) is less than the bin size. As shown in ﬁgure A1(b), not only do the atoms travel more than the bin size, butthey travel far beyond the initial extent of the cloud. Moreover, owing to the higher initial scatter rate, the back of
the cloud overtakes the front for long imaging times. Thus, the atomic distribution as a function of positionchanges dramatically during the imaging pulse, and the stationary assumption is invalid.
A.3. Traveling atom model
To account for the changing atomic distribution during the imaging pulse, we numerically simulated theclassical kinetics of atoms subject to the recoil driven optical forces. To simulate large ensembles in a reasonable
time, we modeled composite atoms, each describing the aggregate behavior of Nca atoms. The amended
algorithm is shown in algorithm 2.
Algorithm 2. Travelling atom modelz[n]=z0, E˜ [n]  0 {initialize position and detuning for each composite atom, labeled by index n}O[i]=n {make a list of composite atom indexes, ordered by position}I˜[n  0, t ]  I˜0 {t is the time index}
Hf=0 {Radiant ﬂuence seen by camera after passing through cloud}for t=0 to tf do {loop over time steps}
for i=1 to Nca do {loop over composite atoms}n=O[i] {apply probe intensity to composite atoms in order of appearance}A=σ0Nca dz {dz is length over which atoms were grouped into single composite atom}
B  vr dt ( cNsa ) {dt is the time step}I˜[n, t ]  I˜[n  1, t ]  AI˜[n  1, t ] (1 4E˜ [n]2 I˜[n  1, t ]) {equation (5)}E˜ [n]
 B (I˜[n  1, t ]  I˜[n, t ]) {equation (6)}z [n] dt (E˜ 2k r { (E˜ 2k r is the velocity at Ẽ detuning}end forO[i]=sort(n, key=z[n]) {sort composite atom indexes by current position}
Hf Hf
I˜[N , t ] dt {collecting total ﬂuence seen by the camera}end for
ODsim2  ln (Hf I˜0 t f )To validate our code, we again checked the velocity predicted in this model against known limits. One such
limit is that of a single composite atom. In this case, there is no attenuation, and the intensity seen by thecomposite atom is constant at I˜0. Only the detuning evolves in time, and equations (5) and (6) givedE˜ (t )
kv
 r r
dt
2 1

I˜
2
4E˜

I˜

.

(A.7)

Equation (A.7) can be solved numerically, and is in agreement with our simulation, as seen in ﬁgure A2(a).
We used this model to study the time evolution of the cloud shape during imaging and visualized the phasespace evolution of superatoms, shown in ﬁgure A3. The cloud shape is strongly distorted during imaging.
We compared the optical depths predicted by each of the two models, ODsim1 and ODsim2. As seen ﬁgureA2(b), the predicted optical depths were hardly changed by including the full time evolution:
ODsim1  ODsim2 ODsim1 - 0.01 for times up to 100 μs, I˜0 up to 50 and σ0n up to 2.0.

Appendix B
We used a Zeeman slower to slow both 87Rb and 40K before capturing in a magneto-optical trap (MOT). After 7
seconds of MOT loading 40K followed by 1.5 s of loading both 40K and 87Rb, we cooled both species in opticalmolasses for 2 ms. We optically pumped both species into their maximally stretched magnetically trappablestates, ∣F  9 2, mF  9 2§ for 40K and ∣F  2, mF  2§ for 87Rb. Both species were then loaded into aquadrupole magnetic trap with a gradient of ≈7.68 mT cm−1 along ez , and cooled evaporatively via forced rf
evaporation, sweeping the rf frequency from 18 to 2 MHz in 10 s. The magnetic trap was plugged by a
λ = 532 nm beam, tightly focused to ≈30 μm and ≈5 W in power, providing a repulsive potential around the
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Figure A1. (a) Column densities deduced from simulated optical depths of on-resonant imaging at probe intensity I˜0  0.8. The
input column density was σ0n=1.6. σ0n1 is the high probe intensity corrected column density given by equation (4). σ0n2 is thecolumn density as expanded to second order in time, equation (A.1). (b) Position of atoms as a function of imaging time for atoms inthe ﬁrst (solid green), middle (dashed red), and last (dotted blue) bins of the simulated density distribution for an initial cloud 50 μm
in extent. The probe intensity used in this calculation was 1.2 Isat, and the column density was σ0n=1.6.

Figure A2. (a) The velocity of a single composite atom as a function of probe intensity for various imaging times. Simulation data
(dots) and numerical solutions of equation (A.7) (lines) are in agreement. (b) Top. Optical depth as a function of probe intensity for an
imaging time t=100 μs. OD1 and OD2 are optical depths predicted from a given column density by equation (4) and (A.1)
respectively. The two versions of simulated optical depth, ODsim1 (green curve) and ODsim2 (green dots) are plotted. Bottom. The
fractional difference between two versions of the simulated OD, ODsim1  ODsim2 ODsim1.

zero ﬁeld point to prevent Majorana losses. Since the 40K atoms were spin polarized and therefore onlyinteracted by the strongly suppressed p-wave interactions, they re-thermalized largely due to sympathetic
cooling with 87Rb atoms.
We then loaded the atoms into a crossed optical dipole trap, provided by a 1064nm ﬁber laser, and continued
evaporative cooling by slowly ramping down the dipole trap to trap frequencies of (Xx , Xy , Xz ) 2Q
 (39, 42, 124) Hz (for potassium atoms) in the three spatial directions, while also turning off the quadrupoleﬁeld. We then used ARP to transfer the 87Rb atoms from the ∣F  2, mF  2§ state to the ∣F  1, mF 
1§absolute ground state via 6.8556 GHz microwave coupling (20.02 MHz from the zero ﬁeld resonance) followed
by a magnetic ﬁeld sweep from −0.469 to −0.486 mT in 50 ms. This state was chosen to minimize spin changing
collisions with 40K atoms during any further evaporation [37]. We then brieﬂy applied an on-resonant probe
laser, ejecting any remaining 87Rb atoms in the F=2 manifold from the trap. We again used ARP to transfer the40
K atoms into the ∣F  9 2, mF  9 2§ state by using a 3.3 MHz rf ﬁeld and sweeping the bias magnetic ﬁeld
from −0.518 to −0.601 mT in 150 ms.
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Figure A3. Phase space evolution of an atomic cloud exposed to probe light with intensity I˜0  1.2. We deﬁned %v  v   v (t )and %z  z   z (t ) , subtracting out the center of mass position and velocity of the cloud. The column density σ0n is 1.6, and theinitial cloud is a Gaussian with a width of 10 μm in (a) and 1 μm in (b). The center of mass velocities v are (0, 3.41, 5.26, 6.52, 7.50,
8.32) m s−1 sequentially, and are the same for both initial cloud widths.

References[1] Greiner M, Regal C A and Jin D S 2003 Emergence of a molecular Bose–Einstein condensate from a Fermi gas Nature 426 537–40[2] Zwierlein M W, Stan C A, Schunck C H, Raupach S M F, Gupta S, Hadzibabic Z and Ketterle W 2003 Observation of Bose–Einsteincondensation of molecules Phys. Rev. Lett. 91 250401[3] Jochim S, Bartenstein M, Altmeyer A, Hendl G, Riedl S, Chin C, Hecker Denschlag J and Grimm R 2003 Bose–Einstein condensation ofmolecules Science 302 2101–3[4] Bartenstein M, Altmeyer A, Riedl S, Jochim S, Chin C, Hecker Denschlag J and Grimm R 2004 Collective excitations of a degenerate gas
at the BEC-BCS crossover Phys. Rev. Lett. 92 203201[5] Bourdel T, Khaykovich L, Cubizolles J, Zhang J, Chevy F, Teichmann M, Tarruell L, Kokkelmans S J J M F and Salomon C 2004
Experimental study of the BEC-BCS crossover region in lithium 6 Phys. Rev. Lett. 93 050401[6] Zwierlein M W, Stan C A, Schunck C H, Raupach S M F, Kerman A J and Ketterle W 2004 Condensation of pairs of fermionic atoms
near a Feshbach resonance Phys. Rev. Lett. 92 120403[7] Regal C A, Greiner M and Jin D S 2004 Observation of resonance condensation of fermionic atom pairs Phys. Rev. Lett. 92[8] Chin C, Grimm R, Julienne P and Tiesinga E 2010 Feshbach resonances in ultracold gases Rev. Mod. Phys. 82 1225–86[9] Timmermans E, Tommasini P, Hussein M S and Kerman A 1999 Feshbach resonances in atomic Bose–Einstein condensates Phys.
Rep.-Rev. Section Phys. Lett. 315 199–230[10] Tiesinga E, Verhaar B J and Stoof H T C 1993 Threshold and resonance phenomena in ultracold ground-state collisions Phys. Rev. A 47
4114–22[11] Lysebo M and Veseth L 2009 Ab initio calculation of Feshbach resonances in cold atomic collisions: s- and p-wave Feshbach resonancesin 6Li2 Phys. Rev. A 79 062704[12] Gao B 2011 Analytic description of atomic interaction at ultracold temperatures: II. Scattering around a magnetic Feshbach resonancePhys. Rev. A 84 022706[13] Inouye S, Andrews M R, Stenger J, Miesner H J, Stamper-Kurn D M and Ketterle W 1998 Observation of Feshbach resonances in a
Bose–Einstein condensate Nature 392 151–4[14] Cornish S L, Claussen N R, Roberts J L, Cornell E A and Wieman C E 2000 Stable Rb-85 Bose–Einstein condensates with widely tunableinteractions Phys. Rev. Lett. 85 1795–8[15] Regal C A and Jin D S 2003 Measurement of positive and negative scattering lengths in a Fermi gas of atoms Phys. Rev. Lett. 90 230404[16] O’Hara K M, Hemmer S L, Gehm M E, Granade S R and Thomas J E 2002 Observation of a strongly interacting degenerate Fermi gas of
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414–7[18] Bartenstein M et al 2005 Precise determination of 6Li cold collision parameters by radio-frequency spectroscopy on weakly boundmolecules Phys. Rev. Lett. 94 103201[19] Zürn G, Lompe T, Wenz A N, Jochim S, Julienne P S and Hutson J M 2013 Precise characterization of 6Li Feshbach resonances using
trap-sideband-resolved RF spectroscopy of weakly bound molecules Phys. Rev. Lett. 110 135301[20] Chikkatur A P, Görlitz A, Stamper-Kurn D M, Inouye S, Gupta S and Ketterle W 2000 Suppression and enhancement of impurity
scattering in a Bose–Einstein condensate Phys. Rev. Lett. 85 483–6[21] Thomas N R, Kjærgaard N, Julienne P S and Wilson A C 2004 Imaging of s and d partial-Wave interference in quantum scattering ofidentical Bosonic atoms Phys. Rev. Lett. 93 173201[22] Williams R A, LeBlanc L J, Jiménez-García K, Beeler M C, Perry A R, Phillips W D and Spielman I B 2012 Synthetic partial waves inultracold atomic collisions Science 335 314–7

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D Genkina et al[23] Aikawa K, Frisch A, Mark M, Baier S, Grimm R and Ferlaino F 2014 Reaching Fermi degeneracy via universal dipolar scattering Phys.
Rev. Lett. 112 010404[24] Lu M, Burdick N Q and Lev B L 2012 Quantum degenerate dipolar Fermi gas Phys. Rev. Lett. 108 215301[25] Metcalf H J and van der Straten P 1999 Laser Cooling and Trapping (Graduate Texts in Contemporary Physics vol 26) (New York:
Springer)[26] Reinaudi G, Lahaye T, Wang Z and Guéry-Odelin D 2007 Strong saturation absorption imaging of dense clouds of ultracold atoms Opt.
Lett. 32 3143–5[27] Konstantinidis G O, Pappa M, Wikstroem G, Condylis P C, Sahagun D, Baker M, Morizot O and von Klitzing W 2012 Atom number
calibration in absorption imaging at very small atom numbers Cent. Eur. J. Phys. 10 1054–8[28] Tiecke T G 2009 Feshbach resonances in ultracold mixtures of the fermionic quantum gases 6Li and 40K PhD Thesis University of
Amsterdam[29] Williams R A, Beeler M C, LeBlanc L J, Jiménez-García K and Spielman I B 2013 Raman-induced interactions in a single-component
Fermi gas near an s-wave Feshbach resonance Phys. Rev. Lett. 111 095301[30] Lin Y-J, Perry A R, Compton R L, Spielman I B and Porto J V 2009 Rapid production of 87Rb Bose–Einstein condensates in a combined
magnetic and optical potential Phys. Rev. A 79 063631[31] Jimenez-Garcia K 2012 Artiﬁcial gauge ﬁelds for ultracold neutral atoms PhD Thesis Joint Quantum Institute, National Institute ofStandards and Technology, and the University of Maryland[32] DeMarco B, Bohn J L, Burke J P, Holland M and Jin D S 1999 Measurement of p-wave threshold law using evaporatively cooled
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A 71 043602[34] Edwards M, Benton B, Heward J and Clark C W 2010 Momentum-space engineering of gaseous Bose–Einstein condensates Phys. Rev.
A 82 063613[35] Bracewell R 1965 The Fourier Transform and its Applications (New York: McGraw-Hill)[36] LeBlanc L J 2011 Exploring many-body physics with ultracold atoms PhD Thesis University of Toronto[37] Best T 2010 Interacting Bose–Fermi mixtures in optical lattices PhD Thesis Johannes Gutenberg-Universitat

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APPENDIX DGEOMETRICAL PUMPING WITH A BOSE-EINSTEIN CONDENSATE

172

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PHYSICAL REVIEW LETTERS

PRL 116, 200402 (2016)

Geometrical Pumping with a Bose-Einstein Condensate
H.-I Lu,1 M. Schemmer,1,2 L. M. Aycock,1,3 D. Genkina,1 S. Sugawa,1 and I. B. Spielman1,*
1

Joint Quantum Institute, National Institute of Standards and Technology,
and University of Maryland, Gaithersburg, Maryland 20899, USA
2
École Normale Supérieure de Lyon, F-69364 Lyon, France
3
Cornell University, Ithaca, New York 14850, USA(Received 17 August 2015; revised manuscript received 21 April 2016; published 20 May 2016)
We realized a quantum geometric “charge” pump for a Bose-Einstein condensate (BEC) in the lowest
Bloch band of a novel bipartite magnetic lattice. Topological charge pumps in filled bands yield quantizedpumping set by the global—topological—properties of the bands. In contrast, our geometric charge pumpfor a BEC occupying just a single crystal momentum state exhibits nonquantized charge pumping set by
local—geometrical—properties of the band structure. Like topological charge pumps, for each pump cycle
we observed an overall displacement (here, not quantized) and a temporal modulation of the atomic wave
packet’s position in each unit cell, i.e., the polarization.
DOI: 10.1103/PhysRevLett.116.200402

Ultracold atoms in optical lattices provide a uniquesetting for experimentally studying concepts that lie at
the heart of theoretical condensed matter physics, but areout of reach of current condensed matter experiments.
Here, we focus on the connection between topology,
geometry, and adiabatic charge pumping [1–7] for BoseEinstein condensates (BECs) in cyclically driven latticepotentials.
Particles in periodic potentials form Bloch bands withenergy ϵn ðqÞ and eigenstates jΨn ðqÞi ¼ expðiqx̂Þjun ðqÞilabeled by the crystal momentum q along with the band index
n. The states jun i retain the underlying periodicity of thelattice, set by the unit cell size a. Motion in lattices isconventionally understood in terms of these bands: metals
are materials with partially filled bands, while insulators havecompletely filled bands. In this context, a topological charge
pump is a counterintuitive device, where charge motion—
conduction—accompanies the adiabatic and cyclic drive of
an insulating lattice’s parameters. Thouless showed that thisconduction is quantized, completely governed by the band
topology [8,9]. Although various charge pumps have beenrealized in condensed matter devices—such as modulated
quantum dots [10–12], one-dimensional (1D) channels
driven by surface acoustic waves [13], and superconducting
qubits [14]—Thouless pumps remain unrealized in condensed matter settings but have been demonstrated in recent
experiments with cold-atom insulators [15,16].
Here, we break from this established paradigm forinsulators and create a quantum charge pump for a BEC
in a 1D lattice [17–19] occupying a single crystal momentum state q. This charge pump gives nonquantized motion
sensitive to the Berry curvature at q integrated over thewhole pump cycle, a local geometric quantity, rather than aglobal topological quantity. Berry curvatures play animportant role in condensed matter systems. An iconic
0031-9007=16=116(20)=200402(5)example is the integer quantum Hall effect, where theelectrons acquire an anomalous transverse velocity proportional to the Berry curvature and the quantized Hall
conductance is given by the Berry curvature integrated
over the whole two-dimensional (2D) Brillouin zone (BZ)[20]; recent cold-atom experiments in 2D have measured
such curvatures integrated over part [21,22] or all [23] ofthe BZ. In an analogous way, 1D lattice systems, driven
cyclically in time t, have a generalized Berry curvaturedefined on the 2D effective BZ in q, t space. This curvatureis the source of an anomalous velocity [24], utilized to drive
an adiabatic quantum pumping process.
The Rice-Mele model [25–28] of a bipartite lattice with aunit cell consisting of A and B sites is the paradigmatic
system for understanding quantum pumps. The
Hamiltonian for this tight-binding model is
X
ĤRM ¼ −½ðt þ δtÞb̂†j âj þ ðt − δtÞâ†jþ1 b̂j þ H:c:(j

þΔXðâ†j âj − b̂†j b̂j Þ;jð1Þwhere â†j and b̂†j describe the creation of a particle in unit
cell j and sublattice site A or B, respectively. The nominal
tunneling strength t is staggered by δt, and the sublatticesites are shifted in energy by Δ.
We investigated quantum pumping in a novel 1D (alongex ) bipartite magnetic lattice (building on Refs. [29,30])
that in effect allowed independent control of t, δt, and Δ. As
shown in Figs. 1(a) and 1(b), our magnetic lattice for 87 Rb
arose from the interplay of one rf and two Raman fields that
coupled the jf ¼ 1; mF ¼ )1; 0i “spin” states comprising
the f ¼ 1 ground state hyperfine manifold, which wereZeeman split by ℏωZ. The natural units of momentum andenergy are given by the single photon recoil momentum

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PHYSICAL REVIEW LETTERS

FIG. 1. Bipartite magnetic lattice. [(a) and (b)] Dipole trapped87
Rb BECs subject to a bias magnetic field B0 ez had a
Zeeman splitting ωZ =2π ¼ 0.817 MHz and a quadratic shiftℏϵ ¼ 0.03ER . These BECs were illuminated by four Raman
beams and an rf magnetic field. Each of the two Raman couplings(strengths Ω) ) was derived from two cross-polarized Raman laser
beams with frequency components ω and ω þ δω. (c) Adiabatic
potentials colored according to hmx i computed for ℏðΩ̄; Ωrf ; δÞ ¼
ð6; 2.2; 0ÞER , δΩ=Ω̄ ¼ −0.1, and ϕ ¼ π=4. The dashed curves
plot the )ℏΩx contributions to the potential experienced by states
jmx ¼ )1i. (d) Lowest two energy band energies plotted as afunction of ϕ, otherwise with the same parameters as (c).

ℏkR ¼2πℏ=λR and its corresponding energy ER ¼ℏ2 k2R =2m,
where m is the atomic mass. In the frame rotating at the rf
frequency δω and under the rotating wave approximation,
the combined rf-Raman coupling lead [31] to the overall
HamiltonianĤ ¼ℏ2 k̂2xþ Ωðx̂Þ · F̂ þ ĤQ ;2m

ð2Þ

where F̂ is the total angular momentum vector operator. We
interpret Ωðx̂Þ p¼ffiffi½ΩrfpcosðϕÞþ Ω̄ cosð2kR x̂Þ; −Ωrf sinðϕÞ−
ffiffiδΩ sinð2kR x̂Þ; 2δ(= 2 as a spatially periodic effective
Zeeman magnetic field, in which Ωrf is the rf couplingstrength; Ω̄ ¼ Ωþ þ Ω− and δΩ ¼ Ωþ − Ω− are derived
from the individual Raman coupling strengths Ω) ; δ ¼
δω − ωZ is the detuning from Raman-rf resonance; and ϕ is
the relative phase between the rf and Raman fields.
Additionally, HQ ¼ −ϵðℏ2 Î − F̂2z Þ=ℏ describes the quadratic Zeeman shift, where Î is the identity operator.
This spatially varying effective magnetic field produces a
1D bipartite lattice [2,32] with lattice constant a ¼ λR =2
and with adiabatic (Born-Oppenheimer) potentials depicted
in Fig. 1(c). This magnetic lattice is most easily conceptualized for small δΩ: the Ω̄ cosð2kR x̂Þ term provides

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periodic potentials for the jmx ¼ )1i states spatiallydisplaced from each other by a=2 [dashed curves inFig. 1(c)]; the resulting mx ¼ )1 sites are then staggered
pffiffiin energy, giving Δ ≈ Δmax cosðϕÞ, with Δmax ¼ Ωrf = 2.
The Ωy term couples these sublattices together: the rf
term −Ωrf sinðϕÞ generates constant height barriers(largely specifying t), which become staggered by the−δΩ sinð2kR x̂Þ contribution (largely specifying δt).
Figure 1(d) plots the energies of the resulting lowest twobands as a function of ϕ (modulating Δ cosinusoidally).
Although our lattice is not in the tight-binding limit, theband structure qualitatively matches that of the Rice-Melemodel. In the remainder of this Letter, we focus on thelowest band n ¼ 0 and henceforth omit the band index.
As illustrated by the shading in Fig. 1(c), in each unit cellthe sublattice sites are labeled by their F̂x spin projection
with the jmx ¼ −1i site on the left and jmx ¼ þ1i site on
the right. To confirm this, we adiabatically loaded
jmz ¼ −1i BECs into the lattice’s ground state by simultaneously ramping the detuning from 5ER to 0 while
ramping on the coupling fields in 10 ms. Following
preparation, our measurement sequence began with a
π=2 spin rotation along ey , allowing us to measure theeigenstates of F̂x in our F̂z measurement basis. We
achieved this π=2 rotation (rot) with a 44 μs pulse froman additional rf field with phase ϕrot ¼ π=2 and strengthℏΩrf;rot ¼ 2.2ER , applied while the Raman coupling was
greatly reduced (Ω̄ ≪ Ωrf;rot ) and the lattice rf coupling wasoff (Ωrf ¼ 0). We then abruptly removed the remaining
control fields along with the confining potential andabsorption imaged the resulting spin-resolved momentum
distribution after a 20 ms time-of-flight period in thepresence of a magnetic field gradient along ey .
Figure 2 shows the measured F̂x spin composition [33]and magnetization for adiabatically loaded BECs as afunction of ϕ with δΩ ¼ 0. Because ΔðϕÞ controls therelative depth of the jmx ¼ )1i wells, we observe groundstate spin populations that follow this “tilt.” For example,
when ϕ ¼ 0 or π the double well is strongly tilted and we
observe the near perfect spin magnetization, consistent withatoms residing in the individual sublattices; in contrast,
when ϕ ¼ π=2, the double wells are balanced and we
observe equal populations in each jmx i state as expected forequal occupancy of both sublattices. Thus, the magnetization [Fig. 2(b)] measures the mean atomic position withineach unit cell, i.e., the polarization.
Having constructed a physical realization of the RiceMele model, and demonstrated the requisite control andmeasurement tools, we now turn our attention to topologicaland geometrical charge pumping. These fundamentallyquantum mechanical effects rely on the canonical commutation relation between position and momentum. Consider
a finite wave packet with a center of mass (c.m.) positionhxi ¼ hΨjx̂jΨi, subject to a lattice Hamiltonian Ĥ that is

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PHYSICAL REVIEW LETTERS

FIG. 2. Ground state spin projections. (a) Ground state spin
projections at various ϕ along with the predicted populations forℏðΩ̄; δΩ; Ωrf ; δÞ ¼ ð4.4; 0; 2.2; 0ÞER . The associated adiabatic
potentials [insets in (b)] have minima with spin projection
following the observed population’s trends. (b) Magnetization
derived from data in (a).

adiabatically modulated with period T, i.e., ĤðtÞ ¼
Ĥðt þ TÞ. After one cycle, any initial crystal momentum
state is transformed, jΨðqÞi → exp½iγðq̂Þ(jΨðqÞi, at mostacquiring a phase, where q̂ is the crystal momentum
operator; this defines the single-period evolution operator
ÛT ¼ exp½iγðq̂Þ(. The time-evolved position operator
Û†T x̂ÛT ¼ x̂ − ∂ q̂ γðq̂Þ is displaced after a single pump cycle.
The displacement is particularly simple in two limits:
when just a single crystal momentum state is occupied orwhen every crystal momentum state in the BZ,−π=a ≤ q < π=a, is occupied with equal probability. As
for our BEC, when a single jq0 i state is occupied thedisplacement is Δxðq0 Þ ¼ −∂ q γðqÞjq0 . Both the dynamical(D) phase γ D ðqÞ ¼ −ϵ̄ðqÞT=ℏ from the time-average energyϵ̄ðqÞand the geometric Berry (B) phase γ B ðqÞ ¼
R
i 0T huj∂ t uidt contribute to γðqÞ ¼ γ D ðqÞ þ γ B ðqÞ. In agreethis predicts
ment with conventional descriptions [18,24,27],
Ra mean velocity v̄ðqÞ ¼ ∂ q ϵ̄ðqÞ=ℏ − T −1 0T Fðq; tÞdt. The
first term is the usual group velocity and the second term—
the anomalous velocity—derives from the Berry curvatureFðq; tÞ ¼ iðh∂ q uj∂ t ui − h∂ t uj∂ q uiÞ. In our experiment, the
BEC occupied the minimum of ϵðq; tÞ at q ¼ 0 during thewhole pump cycle giving ∂ q ϵ̄ðqÞ ¼ 0, so only the geometricphase γ B ðqÞ contributed
to the per-cycle displaceR
ment Δxðq ¼ 0Þ ¼ − 0T Fðq ¼ 0; tÞdt.
In the contrasting case of a filled band, the average groupvelocityis also 0 and the displacement is Δx ¼
R
−a BZ ∂ q γ B ðqÞdq=2π; this is often expressed as Δx ¼
R
R
a 0T ∂ t γ Zak ðtÞdt=2π. The Zak phase γ Zak ¼ i BZ huj∂ q uidq,
a topological property of 1D bands, is the Berry’s phaseassociated with traversing the 1D BZ once, in the same waythat γ B ðqÞ is a Berry’s phase taken over a pump cycle.

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Our lattice’s Zak phase is plotted in Fig. 3(a); this Zak
phase is qualitatively indistinguishable from that of the
Rice-Mele model, with singularities at ϕ ¼ )π=2 andδΩ ¼ 0, signaling topological phase transitions acrossthese points. For filled band experiments, pumping trajectories encircling these points give quantized charge pumping [15,16]. Figure 3(b) shows the richly structured Berrycurvature Fðq ¼ 0; ϕÞ relevant to our experiment, which is
explored next.
For our charge pump experiments, we linearly ramped
the pump control parameter ϕðtÞ ¼ 2πt=T, effectively
modulating the lattice potential in two qualitatively different regimes (separated by a critical jδΩ=Ω̄j ≈ 0.63). In thefirst [Fig. 3(c), left panel] the sublattice sites rise and fall
but the local potential minima are essentially fixed in space;in the second [Fig. 3(c), right panel] each minimum is only
present for part of the pump cycle (the potential appears to“slide” by )a per cycle). As these schematics imply, theassociated pumping process gives either no displacement ora quantized per-cycle displacement )a for classical trajectories [34]. In quantum systems, however, geometrical

FIG. 3. Band geometry and topology computed forℏðΩ̄; Ωrf ; δÞ ¼ ð6; 2.2; 0ÞER . [(a) and (b)] Zak phase and q ¼ 0
Berry curvature showing the dependence on both δΩ=Ω̄ and ϕ. In(b), the arrows show experimental charge pump trajectories inFig. 4(b). (c) Adiabatic potentials (displaced vertically for clarity)
computed for a range of ϕ constituting a complete pump cycle atδΩ=Ω̄ ¼ −0.4 (left panel) and −0.8 (right panel). Filled circles
mark the local energy minima.

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pumping is controlled by the Berry curvature, giving
nonquantized per-cycle displacements that can, in principle, take on any value.
We studied adiabatic charge pumping in this lattice in
two ways: in the first we observed the F̂x magnetization,
giving the polarization within the unit cells, and in thesecond we directly measured the displacement Δx of our
BEC. In both cases we loaded into the lattice’s ground stateand linearly ramped ϕ ¼ 2πt=T, driving the Hamiltonian
with period T [34]. As shown in Fig. 4(a), the magnetization oscillated with the T ¼ 2 ms period, demonstrating
the periodic modulation of polarization per cycle. In good
agreement with our data, the solid curves in Fig. 4(a) show
the predicted behavior given our known system parameters.
This agreement persists to long times: for example, after 50
pumping cycles (for t ¼ 100 ms to 110 ms) the contrast isunchanged, confirming the adiabaticity of the process [34].
Lastly, we performed a charge pumping experiment bydirectly measuring the cloud’s position in situ for a rangeof δΩ=Ω̄. We obtained in situ density distributions
using partial-transfer absorption imaging [35] in which≈6.8 GHz microwave pulses transferred ≈5% of the atoms
from jf; mz i ¼ j1; −1i to j2; 0i where they were absorption
imaged. This technique allowed us to repeatedly measure
the in situ density distribution for each BEC. Each observed
displacement was derived from differential measurementsof the cloud position taken just before and just after thepumping process, rendering our observations insensitive to
micron-level drift in the trap position between different
realizations.
Figure 4(b) shows data taken for δΩ=Ω̄ ¼ 0.7, 0, and −0.7
along trajectories i, ii, and iii, respectively, with bothincreasing and decreasing phases. Our data display two
expected symmetry
R properties. First, since the displacement
Δxðq ¼ 0Þ ¼ − Fðq; ϕÞdϕ depends on the sign of theacquired phase, the direction of motion is reversed when theramp direction is inverted. Second, as shown in Fig. 3(b),week ending
20 MAY 2016Fðq ¼ 0; ϕÞ is an odd function of δΩ=Ω̄, so the direction of
motion is also reversed when δΩ=Ω̄ → −δΩ=Ω̄. Thus, Δx is
an odd function of both ϕ and δΩ=Ω̄, and as expected weobserve no motion when δΩ=Ω̄ ¼ 0.
The displacement was markedly nonlinear when thepumping time became comparable to our trap’s 80 ms
period, showing the influence of the confining potential[36]. We included the harmonic potential in our real-space
simulations by directly solving the time-dependent
Schrödinger equation for our system [37]. The simulated
results [Fig. 4(b), solid curves] agree with our observations.
To extract the per-cycle displacement due to geometricpumping, we fit the sinusoidal predictions of our model toeach data trace, with only the overall amplitudes and asmall vertical offset as free parameters, giving the shorttime per-cycle displacement [34]. Figure 4(c) shows theseper-cycle displacements for a range of Raman imbalances.
The in situ cloud typically had a Thomas-Fermi radius of
30 μm, corresponding to a small momentum width of
0.004kR for our BEC. We estimated the thermal fraction
to be ≈5% given by our ≈20 nK temperature (momentum
width of 0.24kR ). Moreover, the per-cycle displacement isnearly independent of q for jqj < 0.25kR [34]. These allow
us to compare the data with the expected displacement from
integrating q ¼ 0 Berry curvature [Fig. 4(c), solid line],showing an excellent agreement and confirming the geometric origin of our quantum charge pump.
Our magnetic lattice enables new experiments with 1D
topological lattices. Berry curvatures at q ≠ 0 can beprobed by performing the charge pump pairwise at )jqj
(for example, prepared via Bloch oscillations [38]). The
dynamical phases in these cases are opposite and therefore
cancel while Berry curvatures (even in q) contributeequally to the displacements [34]. Furthermore, protectededge states, a hallmark of topological systems, are present
at the interface between regions characterized by different

FIG. 4. Geometric charge pumping. (a) Magnetization measured while linearly ramping ϕ with period T ¼ 2 ms, along with theprediction for ℏΩ̄ ¼ 6.38ð2ÞER, ℏδΩ ¼ 4.50ð2ÞER , and ℏΩrf ¼ 2.20ð3ÞER . (b) Displacement plotted versus ϕ=2π (number of pumpcycles). Trajectories i–iii are taken at δΩ=Ω̄ ¼ 0.7, 0, and −0.7, respectively; in each case ℏΩ̄ ≈ 6ER and ℏΩrf ¼ 2.20ð3ÞER . Solid
curves: simulation of a charge pump in the trap. The small displacement near ϕ ¼ 0 is introduced by our loading procedure.
(c) Measured displacement Δx per pump cycle (symbols), along with the prediction obtained by integrating the Berry curvature over our
pumping trajectory (solid curve). The uncertainty bars represent the 95% confidence interval.

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topological invariants [39–41]. Since in our lattice thetopological index is set by the rf phase, a bulk topologicaljunction can be generated by replacing the rf field with anadditional copropagating pair of Raman laser beams inwhich just one beam has an abrupt π phase shift in its
center. This provides a static model of the soliton excitation
mode in polyacetylene [25,42]. Terminating our lattice with
hard-wall boundaries gives rise to similar end states—
somewhat analogous to Majorana fermions in 1D topological superconductors [40,43]—with a spin character.
We appreciate the constructive discussions with W. D.
Phillips, E. Mueller, L. J. LeBlanc, and L. Wang. This work
was partially supported by the ARO’s Atomtronics MURI,and by the AFOSR’s Quantum Matter MURI, NIST, and the
NSF through the PFC at the JQI. M. S. was supported by
Ampère Scholarships of Excellence of the École Normale
Supérieure (ENS) de Lyon. S. S. acknowledges support from
JSPS Postdoctoral Fellowship for research abroad.

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