ABSTRACT

Title of dissertation:

MEASURING TOPOLOGY OF BECS
IN A SYNTHETIC DIMENSIONAL LATTICE
Dina Genkina
Doctor of Philosophy, 2019

Dissertation directed by:Professor Ian Spielman
Joint Quantum Institute,
National Institute of Standards and Technology,
andDepartment of Physics, University of Maryland

We describe several experiments performed on a two species apparatus capableof producing Bose-Einstein condensates (BECs) of 87 Rb and degenerate Fermi gases(DFGs) of 40 K.
We first describe computational results for observed optical depths with absorption imaging, in a regime where imaging times are long enough that recoilinduced detuning introduces significant corrections. We report that the obseved
optical depth depends negligibly on the cloud shape. We also find that the signalto-noise(SNR) ratio for low atom numbers can be significantly improved by entering
this regime and applying the appropriate corrections. We take advantage of thisSNR improvement in our subsequent experiment colliding two clouds of 40 K for different values of background magnetic field in the vicinity of a Feshbach resonance.
We directly imaged the fraction of scattered atoms, which was low and difficult to detect. We used this method to measure the resonance location to be B0 = 202.06(15)Gauss with width ∆ = 10.(5) Gauss, in good agreement with accepted values.
Next, we describe experiments creating an elongated effectively 2D lattice for a
BEC of 87 Rb with non-trivial topological structure using the technique of synthetic
dimensions. We set up the lattice by applying a 1D optical lattice to the atoms
along one direction, and treating the internal spin states of the atoms as lattice sites
in the other direction. This synthetic direction is therefore very short, creating a
strip geometry. We then induce tunneling along the synthetic direction via Raman
coupling, adding a phase term to the tunneling coefficient. This creates an effective
magnetic flux through each lattice plaquette, in the Hofstadter regime, where theflux is of order the flux quantum h/e. We detect the resulting eigenstate structure,and observe chiral currents when atom are loaded into the central synthetic site. We
further launch analogues of edge magnetoplasmons and image the resulting skipping
orbits along each edge of the strip.
We then applied a force along the real dimension of the 2D lattice and directly
imaged the resulting motion in the transverse, synthetic, direction. We performed
these measurements with 3 and 5-site width lattices along the synthetic direction.
We used these measurements to identify the value of the Chern number, the topological invariant in 2D, by leveraging the Diophantine equation derived by Thouless,
Kohomoto, Nightingale, and den Nijs. We measure Chern numbers with typical uncertainty of 5%, and show that although band topology is only properly defined in
infinite systems, its signatures are striking even in extremely narrow systems.

Measuring topology of BECs in a synthetic dimensions latticebyDina Genkina

Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
2019

Advisory Committee:
Professor Steven Rolston, Chair
Professor Mohammad Hafezi, Dean’s representative
Professor Christopher Lobb
Professor Trey Porto
Professor Ian Spielman, Advisor

c Copyright by
Dina Genkina
2019Acknowledgments
This thesis describes work that was carried out by many people, in a way thatthe multitude of authors on each of the papers only begins to convey. However,as this is the acknowledgement section of my thesis, I will endeavor to thank all
the people that guided and supported me during my admittedly long tenure as a
graduate student.
I would first and foremost like to thank my adviser, Dr. Ian Spielman, fortaking a chance on a lab-averse recovering theorist. Ian’s mentorship style wasalways patient and supportive, but without compromising on standards. Ian would
never ask ‘why isn’t this done?’, but instead ask ‘how are things going this finemorning?’ and offer insight and advice regarding whatever I happened to be working
on. He was a role model to aspire to, both in his unique combination of intuition
and rigor in physics and his unparalleled coffee consumption.
The first ‘trial’ semester of my lab work was spent in the RbLi lab at UMD.
During that time, I am grateful to have learned from Dan Campbell, who voluntarily
took on teaching me about the BEC production process one step every day, and Ryan
Price, whose magical skills with all equipment and dark humor got me through thattime. When I first joined the NIST lab, I had the great fortune of learning froma team of outstanding postdocs—the ‘old guard’—consisting of Lindsey LeBlanc,
Ross Williams, and Matthew Beeler. I thank them for an excellent introduction
to not only laboratory techniques and the physics under study but also the art ofworking as a team.
ii

An unparalleled debt of gratitude goes to Lauren Aycock, who shared the labwith me from day one and taught me most of what I know being an experimentalist.
She endured my clumsiness and obstinance and continued teaching and supportingme throughout my time at NIST. Beyond the lab, she helped teach me how todrive and pushed me to do my first half-marathon. I cannot imagine what mygraduate career would have been like without her. I would also like to thank postdocs Ben Stuhl and Hsin-I Lu for the role they played in my physics training. Ben’s
enthusiasm for electronic design was infectious and it was an honor to learn fromhim, be it about about feedback loops or version control. Hsin-I’s unparalleledproductivity in her own work didn’t stop her from taking lots of time to help me inmine. Her wisdom for knowing what the root of a problem is and how to prioritizetasks was invaluable, and she remains a role model for me today.
I am happy to have had the opportunity to work with several visiting German students: Marcell Gall, Max Schemmer, and Martin Link. Marcell excelledin balancing hard, productive work in the lab with fun in his free time, and was
generally a joy to be around. Max jumped into the research with a running start,
and it was exciting to see what he accomplished in a short amount of time. Martin(Marcell III) forced me to give up my belief that all crossfitters are annoying. It
was a pleasure to work with him as well as to exercise and hang out with him.
My later years in the lab were spent with post-doc Mingwu Lu and fellow gradstudent Alina Escalera. Mingwu has taught me so much about physics, electronics,
western philosophy and wine. I am grateful for having the opportunity to learn from
him and get to know him. It has been a pleasure to watch Alina grow in the lab,iii

and to have her understanding support during the rougher parts of paper and thesis
writing. Her tenacity and drive are to be admired. I have had few opportunities tospend time with new lab recruits, post-doc Amilson Fritsch and graduate student
Graham Reid, but even from our limited interactions I am confident the lab will beleft in good hands.
During the writing of this thesis, I have had amazing help and support frommy roommates, fellow students and friends. I would like to thank Ana Valdes-Curiel,
Paco Salces-Carcoba, Chris Billington, Kristen Voigt and Jon Hoffman for bravely
reading through entire chapters of my thesis when I couldn’t bare to even look at
it. I am extremely grateful to have been part of the amazing living community thatis the international house of physicists (IHOP)—it has felt more like a large family
than a housing arrangement. I would like to thank both the current IHOPers, Ana,
Paco, Chris and Daniel, and notable past IHOPers, Dimitry Trypogeorgos and Rory
Speirs. A special thank you to Ana and Paco for unpromptedly feeding me during
the final thesis writing days and to them and Chris for continued moral support.
Extra special thank you to Kristen, my oldest graduate school friend, for teachingme how to drive, housing me in my times of need, providing me with thesis writing
space and support, feeding me uncountable times, and telling me its okay to take anap.
I would also like to thank Jenna Beckwith for her help at the beginning of
my graduate journey, Taylor Cole for her insightful support in the last year thathas allowed me to work on my thesis almost without panic, and Dr. Diane Stabler
for her constant presence and support throughout my time here and for offeringiv

deceptively simple advice. I would also like to thank the good people at Eli Lily andCompany without whose pioneering research my work would not have been possible.
Above all I would like to thank my family. If it wasn’t for my parents’ upbringing, graduate school would never have been a possibility for me. I am gratefulto them for teaching me all that they have, supporting me throughout my time asa graduate student, but prioritizing my happiness above graduate success. I would
like to thank them, my sister Alya, and my brother Pten, for everything.

v

Table of Contents

Acknowledgementsii

List of Figuresix

1 Introduction
1
1.1 Condensed matter context . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Band topology in materials . . . . . . . . . . . . . . . . . . . 4
1.1.3 Magnetic field in 2D and the Aharonov-Bohm phase . . . . . . 6
1.1.4 Hofstadter regime . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.5 Diophantine equation . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Ultracold atoms for quantum simulation . . . . . . . . . . . . . . . . 11
1.3 Strucute of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Atom–Light Interactions
2.1 Level structure of alkali atoms . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Fine and hyperfine sturcutre . . . . . . . . . . . . . . . . . . .
2.1.2 Interaction with static magnetic fields . . . . . . . . . . . . . .
2.2 Near-resonant atom–light interaction . . . . . . . . . . . . . . . . . .
2.2.1 Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Adiabatic rapid passage . . . . . . . . . . . . . . . . . . . . .
2.3 Far–off-resonant atom–light interaction . . . . . . . . . . . . . . . . .
2.4 Absorption imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Time-of-flight and in situ imaging . . . . . . . . . . . . . . . .
2.5 One-dimensional optical lattices . . . . . . . . . . . . . . . . . . . . .
2.5.1 Lattice Hamiltonian . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Tight-binding approximation . . . . . . . . . . . . . . . . . . .
2.5.3 Pulsing vs adiabatic loading of the lattice . . . . . . . . . . .
2.6 Raman and rf coupling . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Rf coupling Hamiltonian . . . . . . . . . . . . . . . . . . . . .
2.6.2 Raman coupling Hamiltonian . . . . . . . . . . . . . . . . . .
2.6.3 Calibration of Raman and rf dressed states . . . . . . . . . . .
vi

14
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3 Ultracold Gases and the RbK apparatus
3.1 Bose-Einstein condensation . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Phase transition of a non-interacting Bose gas . . . . . . . . .
3.1.2 Interacting Bose gas . . . . . . . . . . . . . . . . . . . . . . .
3.2 Degenerate Fermi Gas . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Fermi statistics and the onset of degeneracy . . . . . . . . . .
3.2.2 Interactions and Feshbach resonances . . . . . . . . . . . . . .
3.3 RbK apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Laser beams . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Magnetic coils . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Procedure for making a BEC . . . . . . . . . . . . . . . . . .
3.3.4 Changes to apparatus for Rubidium . . . . . . . . . . . . . . .
3.3.4.1 Master laser setup . . . . . . . . . . . . . . . . . . .
3.3.4.2 Alignment imaging path . . . . . . . . . . . . . . . .
3.3.4.3 FPGA quad servo . . . . . . . . . . . . . . . . . . .
3.3.4.4 ODT beam shaping . . . . . . . . . . . . . . . . . .
3.3.5 Procedure for making a DFG . . . . . . . . . . . . . . . . . .
3.3.6 Current status of Potassium apparatus . . . . . . . . . . . . .

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4 Absorption Imaging with Recoil Induced Detuning
97
4.1 Recoil-induced detuning . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2 Perturbative treatment . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3 Stationary atom model . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4 Traveling atom model . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.5 Calibration of saturation intensity . . . . . . . . . . . . . . . . . . . . 105
4.6 SNR optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5 Imaging of Scattering Near a Feshbach Resonance111
5.1 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6 Synthetic Magnetic Fields in Synthetic Dimensions120
6.1 Synthetic dimensions setup . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 Hamiltonian of the effective 2-D system . . . . . . . . . . . . . . . . . 124
6.2.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2.2 Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.2.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2.4 Tight binding approximation . . . . . . . . . . . . . . . . . . . 134
6.3 Eigenstates of the synthetic 2-D lattice . . . . . . . . . . . . . . . . . 136
6.4 Chiral edge currents . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.5 Observation of skipping orbits . . . . . . . . . . . . . . . . . . . . . . 141

vii

7 Measuring Chern Number in Synthetic Dimesnions
144
7.1 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.1.1 Loading procedures . . . . . . . . . . . . . . . . . . . . . . . . 147
7.1.2 Application of force and measurement . . . . . . . . . . . . . 149
7.1.3 Density reduction . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.1.4 Rf correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.2 Quantum Hall Effect interpretation . . . . . . . . . . . . . . . . . . . 155
7.2.1 Microscopic view of QHE . . . . . . . . . . . . . . . . . . . . . 155
7.2.2 Chern number from Hall conductivity . . . . . . . . . . . . . . 156
7.2.3 Inadequacy for narrow systems . . . . . . . . . . . . . . . . . 158
7.3 Measuring Chern number via Diophantine equation . . . . . . . . . . 160
7.3.1 TKNN Diophantine equation in synthetic dimensions . . . . . 161
7.3.2 Prescription for identifying Chern number . . . . . . . . . . . 162
7.3.3 Properties of the method . . . . . . . . . . . . . . . . . . . . . 164
7.4 Results and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 165
A Brownian motion of solitons in a Bose–Einsten Condensate

168

B Geometrical pumping with a Bose–Einstein condensate

175

Bibliography

181

viii

List of Figures

1
2
3
4

Topology of baked goods . . . . . . . . . . . . . . . . . . . . . . . . . 3
Hall effect setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Aharanov-Bohm experiment . . . . . . . . . . . . . . . . . . . . . . . 7
Hofstadter butterfly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1
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3
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5
6
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8
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12
13

Atomic structure of the ground and first excites states . . . . . . . .
Energy structure of hyperfine states of 87 Rb and 40 K . . . . . . . . .
Adiabatic rapid passage (ARP) . . . . . . . . . . . . . . . . . . . . .
Absorption imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lattice band structure in the extended zone scheme . . . . . . . . . .
Lattice pulsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lattice pulsing for calibration . . . . . . . . . . . . . . . . . . . . . .
Adiabatic lattice loading . . . . . . . . . . . . . . . . . . . . . . . . .
Raman and rf coupling schematic . . . . . . . . . . . . . . . . . . . .
Band structure of the rf Hamiltonian . . . . . . . . . . . . . . . . . .
Band structure of the Raman Hamiltonian . . . . . . . . . . . . . . .
Pulsing on rf coupling . . . . . . . . . . . . . . . . . . . . . . . . . .
Pulsing on Raman coupling . . . . . . . . . . . . . . . . . . . . . . .

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Occupation of energy states of a 3-D harmonic oscillator . . . . . . .
Time-of-flight images of atoms . . . . . . . . . . . . . . . . . . . . . .
In situ measurement of a fraction of bose condensed atoms . . . . . .
Occupation number as a function of energy for a Fermi gas . . . . . .
Schematic of a Feshbach resonance . . . . . . . . . . . . . . . . . . .
Photograph of RbK apparatus at NIST Gaithersburg . . . . . . . . .
Schematic of RbK apparatus . . . . . . . . . . . . . . . . . . . . . . .
Schematic of magnetic coils on the RbK apparatus . . . . . . . . . .
New master board layout . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic of the bottom half of the alignment imaging system . . . .
Schematic of the new imaging lens . . . . . . . . . . . . . . . . . . . .
Schematic of the top half of the alignment imaging system . . . . . .
Turn on and off curves of the quad coils . . . . . . . . . . . . . . . .
Output current as a function of computer command voltage . . . . .

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Schematic of beam shaping optics in the path of the 0th order ODTbeam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Beam waist as a function of propagation distance . . . . . . . . . . .
Schematic of the 2D MOT setup . . . . . . . . . . . . . . . . . . . . .
Picture of 2D MOT cell . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic of 2D MOT optics . . . . . . . . . . . . . . . . . . . . . .
Picture of current 2D MOT apparatus . . . . . . . . . . . . . . . . .

1
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Recoil induced detuning . . . . . . . . . . . . . . . . . . . . . . . . . 100
Optical depth as a function of probe intensity . . . . . . . . . . . . . 102
Checking stationary atom assumption . . . . . . . . . . . . . . . . . . 103
Phase space evolution of the atomic cloud . . . . . . . . . . . . . . . 106
Comparison of stationary and moving atom simulations . . . . . . . . 107
Signal-to-noise optimization . . . . . . . . . . . . . . . . . . . . . . . 110

1
2
3

An sample absorption image after 6.8 ms TOF . . . . . . . . . . . . . 115
Experimental setup and inverse Abel transform . . . . . . . . . . . . 116
Normalized scattered population vs bias field B . . . . . . . . . . . . 119

1
2
3
4
5
6
7

Setup of effective 2-D lattice . . . . . . . . . . . . . . . . . . . . . . . 123
Band structure of the synthetic dimensions Hamiltonian . . . . . . . 129
Calibration of synthetic dimensions lattice . . . . . . . . . . . . . . . 132
Band structure of the tight binding versus full Hamiltonian . . . . . . 135
Eigenstates of the synthetic dimensions lattice . . . . . . . . . . . . . 137
Measuring chiral currents in synthetic dimensions . . . . . . . . . . . 140
Imaging skipping orbits . . . . . . . . . . . . . . . . . . . . . . . . . . 142

1
2
3
4
5
6
7

Quantum Hall effect in Hofstadter ribbons . . . . . . . . . . . . . . . 146
Band structure of the lattice-coupled system in F = 2 . . . . . . . . . 148
Application of a constant force by displacing a Gaussian beam potential150
Band structure in a 5-site wide ribbon . . . . . . . . . . . . . . . . . 151
Correction of oscillations in rf coupled data . . . . . . . . . . . . . . . 154
Hall displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Band structure of the synthetic dimesnions lattice with flux Φ/Φ0 =−4/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Chern number from the TKNN equation . . . . . . . . . . . . . . . . 161
Calculating Chern number . . . . . . . . . . . . . . . . . . . . . . . . 163
Chern number dependence . . . . . . . . . . . . . . . . . . . . . . . . 165
Chern number measurement . . . . . . . . . . . . . . . . . . . . . . . 166

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Chapter 1: Introduction

Although quantum mechanics has been well established since the early 20th
century, there are still many quantum phenomena that are not well understood andare not easy to calculate. These include high temperature superconductivity [1, 2],
fractional quantum Hall physics [3, 4], and ground states of frustrated amorphousmaterials [5]. One of the reasons these problems are proving elusive is that calculation of properties of many-body quantum mechanical systems is computationallyintensive enough to be completely prohibitive in a large class of problems.
Quantum simulation provides an attractive alternative to direct computation.
In it, a test quantum system, here ultracold atoms, is used to simulate a more
complicated, less experimentally accessible quantum system, such as a non-trivial
material from condensed matter physics. In order to get to the point where unsolved
problems can be solved with quantum simulation, tools must be built up to create
and verify Hamiltonians in the test system that are relevant to the more complex
target system. In this thesis, we present a technique for creating topologically
non-trivial Hamiltonians for ultracold atoms and experimentally measuring their
topological properties.

1.1 Condensed matter context
Topology has been a field of mathematics since the 17th century. Its importance in physics, particularly in the study of crystalline materials in condensed
matter, was first discovered by Thouless, Kohmoto, Nightingale and den Nijs [6].
1They used topology to explain the shockingly precise quantization of resistivity inthe quantum Hall effect. Since then, topology has been central to condensed matter,from topological insulators [7] to fractional quantum hall physics [3]. There havebeen many excellent pedagogical texts written on this matter. Here, we include onlya brief overview of the physics that is relevant for motivating Chapters 6 and 7 ofthis thesis.

1.1.1 Topology
Topology is the study of how things can be continuously transformed into otherthings without tearing or gluing parts together. Things that can be continuouslytransformed into each other under those rules are called homeomorphic to eachother. Classes of objects that are all homeomorphic to each other belong to thesame topological class. These classes are characterized by a topological invariant,an integer. Surfaces in 3D can be characterized by their genus g, essentially the
number of holes in the shape. Since holes cannot be opened up or closed by acontinuous transformation, the number of holes is a topological invariant that canbe used for classification.
Figure 1.1.1 shows some examples of objects with different genus g. A loaf ofbread has no holes, and is therefore topologically equivalent to a sphere, with g = 0.
A bagel has one hole, and is topologically equivalent to a torus, or a coffe mug, or
any other number of things with a single through hole, with g = 1. A pretzel has
3 through holes, and is therefore topologically distinct from both the loaf and thebagel, with g = 3.
More formally, the Gauss-Bonnet theorem proves the discrete topology of 2D
surfaces. It uses the Guassian curvature, K, defined at every point on the surface.
The Gaussian curvature is defined as follows [8]: at any point on the surface, there
is a normal vector, perpendicular to that point’s tangent plane. The set of planes
2Figure 1: Topology of baked goods. They are classified according to genus g, thenumber of holes. Baking credit: genus 0—Ana Valdes-Curiel, genus 1&3—Whole
Foods Market Riverdale.
containing the normal vector are called normal planes. Each normal plane intersects
the surface at a curve, with an assoicated curvature given by the coefficient of thequadratic term in a Taylor expansion of said curve. There is then a normal plane
giving the maximal curvature kmax and a normal plane giving the minimal curvature
kmin (if all the curvatures are the same k, as on a sphere, then kmax = kmin = k).
The Gaussian curvature is just the product of the two, K = kmin kmax . The GaussBonnet theorem [8] then states that the integral of the Gaussian curvature K overa closed surface S is an integer multiple of 2π:
1
χ=
2π

Z
KdA,

(1.1)

S

where the integer χ is called the Euler characteristic, and is related to the genus via
χ = 2−2g. Essentially, the total curvature of a closed surface is quantized to integer
values, and any closed surface can be classified by that integer. Surfaces with equalχ can be continuously transformed into each other.

31.1.2 Band topology in materials
The same general principles can be applied to the bands within the band
structure of a crystalline material [9]. Crystalline materials are characterized by aspatially periodic Hamiltonian. The primitive unit cell, or the minimal repeating
unit of the lattice, can be parametrized by primitive unit vectors ~ai , where i indexes
from 1 to the number of dimensions d. In momentum space, the repeating structure~ i . The eignestates of are labeled byis parametrized by reciprocal lattice vectors K
a momentum k and an energy E, with energies grouped into bands. According
to Bloch’s theorem [10], the eigenstate wavefunction for some band in a periodic
potential in d dimensions can be written as~|Ψ(~k)i = eik·~r |u(~k)i ,(1.2)

where ~k is the crystal momentum, ~r is the spatial coordinate, and |u(~k)i is periodic
with the reciprocal lattice periodicity. Reciprocal lattice space is continuously welldefined for an infinite system. In a finite system, the reciprocal space becomes
discrete. In the limit where the system is large compared to the primitive unit cell,
one can still approximate a sum over k states with an integral for many applications.
However, the rigorous theorems of continuous mathematics no longer apply.
There is a phase ambiguity in the definition of the Bloch wavefunction, suchthat the physics remains invariant under the transformation [9]~|u(~k)i → eiφ(k) |u(~k)i ,(1.3)

which is reminiscent of gauge invariance in electrostatics. The corresponding gauge-4

~ and is given bydependent potential is called the Berry connection A,~ = −i hu(~k)| ∇~ |u(~k)i .
A
k

(1.4)

~ goes to A
~ + ∇~ φ(~k). The
Under the transformation eq. 1.3, the Berry connection Akyxgauge invariant field, in 2 dimensions, is given by F = ∂A
− ∂A
, where F is known∂kx∂ky

as the Berry curvature.
From this, the geometric phase, or Berry phase γc [11], can be defined as thephase acquired over a closed curve c in parameter space that is independent of therate at which the curve is traversed:
Z
γc =
c

~ · d~k =AZS~ · dS,F(1.5)

where S is a surface bounded by the curve c, and in the second equality we haveinvoked Stoke’s theorem.
The Berry curvature integrated over the entire Brillouin zone, or primitive cellin reciprocal lattice space, is quantized in units of 2π and in 2D given by:
1
C=
2π

Z
BZ

Fd~k,

(1.6)

where C is an integer [12, 13]. This bears a strong similarity to the Gauss-Bonnet
theorem, eqn. 1.1, with the Gaussian curvature replaced by the Berry curvature,and the closed surface in real space replaced by the Brillouin zone in momentum
space. Similarly, the integer C is a topological invariant and can be used to classify
the topological properties of the bands. For periodic structures in 2D, this invariant
is called the Chern number.
In finite systems, the transition to momentum space is necessarily discretized.
The integral in eqn. 1.5 is no longer continuous, and the identification of the Chern5number can only be made in the continuous limit. In short, the Chern number
corresponds to the Chern number of the system in the imaginary scenario where its
bulk was extended indefinitely. In finite systems, the edges of the system become
important. The edge of the system is an interface between a topologically non-trivial
region (the system in question) and a topologically trivial region (the surrounding
air or other material). These edges support conducting edge modes. The number ofthese conducting edge modes is equal to the total Chern number of the filled bands
in the bulk. This is known as the bulk-edge correspondence [6, 14], and has been
widely used to experimentally detect the effective Chern number.

1.1.3 Magnetic field in 2D and the Aharonov-Bohm phase
Without the constraint of a lattice, the behavior of electrons in a magnetic field~ where e is the electron charge,
is well known. They experience a force F = −e~v × B,
~ is the magnetic field. If the electrons are confined to a 2D
~v is their velocity, and B
slab, as in the classical Hall effect, and the magnetic field is perpendicular to the slab(see Figure 1.1.3), they perform orbits with frequency ωB = eB/m. These orbits arecalled cyclotron orbits, and ωB is known as the cyclotron frequency. The chiralityof the orbits, i.e. whether they are clockwise or counterclockwise, is prescribed by
the direction of the magnetic field. If the system is finite, as is the case in all reality,
the electrons at the edge of the system will not be able to complete cyclotron orbits.
However, the chirality of the orbits must till be preserved, and electrons reflected off
the edge will continue semi-orbits in a defined direction along each edge, as shownin Figure 1.1.3. These are known as skipping orbits.
In the same geometry, the presence of an underlying crystal lattice constrains
the motion of the electrons. In order to understand the effect of a magnetic field inthis case, particularly in the quantum limit, it is useful to think of the field in termsof the Aharonov-Bohm effect. The setup for the Ahoranov-Bohm effect is shown in6

Figure 2: Hall effect setup. A slab of metal very thin along one axis acts as a
2D constraint for the electrons to travel in. A magnetic field normal to the planepierces the metal. Electrons travel in cyclotron orbits in the bulk and performskipping orbits along the edges.

Figure 3: Aharanov-Bohm experiment. An electron is sent therough two possible
paths around an infinite solenoid with a magnetic field B and corresponding flux
Φ going through the solenoid, and no field outside. An interference pattern isobserved on the other side, corresponding to a phase difference between the twopaths proportional to the enclosed flux.
7Figure 1.1.3. There is an infinite solenoid with a current running through it, resulting~ = Bz e z inside the solenoid. Outside the solenoid, thein a uniform magnetic field B~ which defines a magnetic field throughfield is zero. However, the vector potential A,
~ =∇
~ ×A
~ can be non-zero, as long as its curl remains zero. In the experiment [15],B
two electrons were sent around the solenoid, never experiencing a magnetic field.
Classically, the electrons should pass unaffected. In reality, though the electron’s
trajectory was unchanged, the quantum mechanical phase was effected. Two electrons that started out in phase acquired a phase differenceφAB = 2πΦ/Φ0 , whereΦ = A × Bz is the flux through the solenoid given by the field Bz times the area
inside the solenoid A, and Φ0 = h/e is the flux quantum.
For electrons on a 2D lattice, this provides a way to treat interpret a magnetic
field quantum mechanically. The smallest unit of the lattice, called a plaquette,
will have some magnetic flux through it Φ = A × Bz . As far the as the electron isconcerned, which is only constrained to move around the plaquette, it is as if there
is an infinite solenoid with magnetic flux Φ through it is piercing the center of theplaquette. Therefore, an electron hopping in a closed loop around the plaquette will
acquire a phase φAB = 2πΦ/Φ0 . We will use this treatment in the next section.

1.1.4 Hofstadter regime
The Hofstadter regime [16] for 2D electrons in a magnetic field occurs when themagnetic flux per individual lattice plaquette is a non-negligible fraction of a flux
quantum. This regime is hard to reach experimentally, since the typical plaquette
size in crystalline material is of order a square Angstrom, and the magnetic field
necessary to thread a magnetic flux of Φ0 through such a narrow area is of order≈ 104 Tesla, not accessible with current technology. Several platforms have howeverreached the Hofstadter regime by engineering systems with large effective plaquette
size in engineered materials [17, 18].
8

For a square lattice with sites along ex labeled by index j and sites along eylabeled by k, in the tight binding limit, the Harper-Hofstadter Hamiltonian can be
written in the Landau gauge as

H=−

X
j,k

tx eiφAB k |j + 1, ki hj, k| + ty |j, k + 1ihj, k| + h.c.,(1.7)

where tx and ty are the tunneling amplitudes along ex and ey , and h.c. is the
Hermitian conjugate. Here, we have labeled the states by site indexes along both
directions |j, ki and included only nearest neighbor tunneling. The choice of Landau
gauge is here represented by the phase factor eiφAB k being only on the tunneling termalong ex .
This Hamiltonian can be solved to find its eigenenergies for a range of phases
φAB , and therefore different fluxes per lattice plaquette Φ/Φ0 . Note that the Hamiltonian is invariant to changes of phase in integer units of 2π, and therefore thephysics is invariant under changes of magnetic flux per plaquette of Φ0 .
The spectrum of the Hamiltonian for a range of flux values is shown in Figure
1.1.4. This is known as the Hofstadter butterfly, and is remarkable for its fractal
structure. In the limit of flux Φ/Φ0 → 0, the fractal bands come together toform equally spaced composite bands, Landau levels underlying the quantum Hall
effect [19–21]. The topology of each energy level for each flux value, as defined by
the Chern number, can be found through eqn. 1.6. In the quantum Hall effect limit,
the Chern number of each Landau level is C = 1.

1.1.5 Diophantine equation
In their seminal paper explaining the topological nature of the quantum Hall
effect [6], Thouless, Kohmoto, Nightingale and den Nijs also defined an alternative
way to compute the Chern number, without resorting to eqn. 1.6. This equation

9

Figure 4: Hofstadter butterfly calculated in 2D momentum space for isotropic tunneling tx = ty .

10states that for rational flux Φ/Φ0 = P/Q (for relatively prime integers P and Q)
the integer solutions s and C to the Diophantine equation

1 = Qs − P C,

(1.8)

under the constraint |C| ≤ |Q|/2 [6, 22], uniquely determine the Chern number ofthe lowest band. We refer to this equation as the TKNN Diophantine equation, andwill make use of it in Chapter 7.

1.2 Ultracold atoms for quantum simulation
The phenomenon of Bose–Einstein condensation (BEC) was first predicted
in 1924 [23]. But it wasn’t until 1995 that techniques for cooling atoms down
to ultracold temperatures allowed for an experimental realization of this phase ofmatter. The first BEC was created and observed by two labs, one at JILA [24] andone at MIT [25]. The realization of a BEC won Eric Cornell, Carl Wieman and
Wolfgang Kettrle the Nobel prize in 2001. Since then, similar cooling techniques
have been applied to Fermionic atomic species. Though no phase transition occurs
for these atoms, at ultracold temperatures they start to differ significantly from an
ideal gas. Atoms in this regime are known as a degenerate Fermi gas (DFG), andthey were first experimentally realized in 1999 in the group of Deborah Jin [26, 27].
Since their realization, BECs and DFGs have become widely studied, bothfor their fundamental properties and as a platform for quantum simulation. A key
tool that makes ultracold atoms amenable to simulating condensed matter systems is
optical lattices. These lattices serve as analogues of crystal structure in a solid, withatoms serving as analogues of electrons. Optical lattices are created by laser lightand can be used to create almost any geometry, from square [28] to triangular [29,30],
to hexagonal [31] to Kagome [32]. This allows for simulation of almost any crystal
11structure, and even the creation of periodic structures not yet found in nature.
Moreover, these optical lattices can be tuned in situ, giving the experimenter
dynamical control of Hamiltonian parameters far beyond what is possible in condensed matter settings. This allowed for the realization of the previously predicted [33] quantum phase transition from the superfluid to the Mott insulatingphase in a BEC [34, 35].
Additionally, there are tools available to control and tune interactions between
atoms. Most notable of these are Feshbach resonances [36–40]. These have been
respnosible for significant advances, from creating BEC of attractively interacting
atoms [41], to forming molecular BECs [42–44], to allowing for realization of BECBCS crossover physics [44–46]. The control of inter-atomic scattering length allowed
by Feshbach resonance makes them an important tool and subject of study. In
Chapter 5, we describe our experiment directly imaging the scattering in the vicinityof such a resonance.
One drawback of ultracold atomic systems is that they are made with neutral
atoms, limiting their interactions with magnetic fields to those arising from theirmagnetic dipole moment. It might seem that the interesting physics of 2D systems inmagnetic fields, as described in the previous section, would not lend itself to quantum
simulation in these settings. However, many techniques have emerged for creating
artificial magnetic fields, or terms in the atomic Hamiltonian that are identicalto the charged particle–magnetic field interaction. One proposal was rotating theatoms such that the Coreolis force takes the role of the Lorentz force [47]. Anotheris engineering laser coupling in a precise geometry to induce effective magnetic
fields [48], which was successfully realized [49] and extended to creating a spin-orbit
coupled Bose gas [50].
On a lattice, the cold atomic approach of imprinting Ahoranov-Bohm phases,
rather than using large external fields, has opened the way for simulation of large

12magnetic fluxes [51–53]. Several experiments have used this approach to reach the
Hofstadter regime [54–59], and further applications are promised [60]. Furthermore,
the approach of synthetic dimensions [61] has enabled reaching the Hofstadter regime
without laser modulation. The experimental realization and detection with thistechnique as described in [62] is detailed in Chapter 6 of this thesis. It has also
been successfully used by other groups [58, 63]. In Chapter 7 of this thesis, we
further detail our experiment detecting the underlying topology of these synthetic
dimensional lattices.

1.3 Strucute of the thesis
In the second chapter, I review some background atomic physics necessary tounderstanding the experiments that follow. In the third chapter, I review basics ofBose-Einstein condensates and degenerate Fermi gases and present an overview ofthe experimental apparatus, with extra detail on a few recent upgrades. In the fourth
chapter, I describe numerical simulations of absorption imaging in the presence ofstrong recoil-induced detuning. In the fifth chapter, I describe our experiment withcolliding 40 K atom clouds in the vicinity of a Feshbach resonance and imaging theresulting s-wave scattering halos. In the sixth chapter, I describe our experimentcreating an effectively 2D synthetic dimensions lattice in the Hofstadter regime for87

Rb, and imaging skipping orbits on its edges. In the final chapter, I describe

our experiment measuring the underlying topological invariant of the same lattice.
In the appendices, I include published papers to which I contributed but did notinclude in the body of the thesis.

13

Chapter 2: Atom–Light Interactions

In this chapter, we review the physics of interactions between atoms and lightin so far as they are relevant to our experiments. We describe the level structureof alkali atoms in section 2.1, including the fine and hyperfine interactions and theeffects of external magnetic fields. We review the interaction of atoms with nearresonant light (section 2.2), pertinent to cooling and absorption imaging as well as
Raman and rf coupling. We then detail interactions of atoms with far-detuned light(section 2.3), pertinent to optical trapping and lattices.
In the following sections, we use these basic interactions to explain the tools
used in our experiments. We describe absorption imaging (section 2.4), our detection
scheme. We explain the basic physics of atoms in a 1D optical lattice potential(section 2.5), and the physics of Raman and rf coupling of hyperfine atomic levels(section 2.6).

2.1 Level structure of alkali atoms
Alkali atoms (those in the first column of the periodic table) are the most commonly used for laser cooling. Their level structure can be understood as primarily
the energy state of the single electron in the outer shell, interacting with the rest of
the atom—the nucleus and all the other electrons—as a whole. The quantum numbers that describe the energy levels of the atom are the principal quantum number
N , that electron’s spin S, the orbital angular momentum L, and finally the nuclear
spin I. The work described in this thesis was done with alkali species 87 Rb and 40 K.
14

2.1.1 Fine and hyperfine sturcutre
The fine structure of alkali atom comes from two corrections to the bare Hamiltonian. The bare Hamiltonian H0 encompasses the kinetic enegy and the Coulombinteraction between the outermost electron and the rest of the atom. The first ofthese corrections is due to special relativity—it’s the correction to the classical kinetic term. The energy shift due to this term, to leading order in the small parameterof velocity over the speed of light v/c, is given by [64]E2∆Erel = − N 2
2mc




4N
−3 ,
L − 1/2(2.1)

where EN is the unperturbed energy and m is the mass of the electron.
The second correction comes from the spin–orbit interaction. In the frame ofthe outermost electron, the nucleus is orbiting it, creating a magnetic field. The~ˆmagnetic dipole moment of the electron—given by µ~S = −gS µB S/h̄,
where gS is~ is the spin angular momenthe spin g-factor, µB is the Bohr magneton, and S
tum operator—interacts with this magnetic field. The Hamiltonian describing thisinteraction is given by [64, 65]~ˆ · S~ˆ
Ze2 gs L,
HSO =4π0 4me2 c2 r3(2.2)

where Z is a factor expressing the effective charge seen by the electron, e is theelectron charge, 0 is the vacuum permittivity and r is the radial coordinate. This~ˆ or L,~ˆ and therefore the projection
Hamiltonian does not commute with either S
quantum numbers mS and mL are no longer good quantum numbers. However,ˆ ~ˆ ~ˆ
the total angular momentum J~ = L+ S does commute with the Hamiltonian, andtherefore mJ is still a good quantum number. We can rewrite the coupling term~ˆ · S~ˆ = J~ˆ2 − L~ˆ 2 − S~ˆ2 . The leading order shift inin the spin–orbit Hamiltonian as L15

perturbation theory due to this interaction is given by
E2∆ESO = N2
mc



N [J(J + 1) − L(L + 1) − S(S + 1)]
L(L + 1/2)(L + 1)


.

(2.3)

Note that the relativistic and spin–orbit correction terms are of the same order.
We therefore combine them into a total energy shift, called the fine structure shift.
Using the fact that S = 1/2 and J = L ± 1/2, this term can be written as
E2∆Efs = N2
mc



4N
3−
J + 1/2


,

(2.4)

which splits the atomic energies according to the quantum number J.
The ground state of 87 Rb, in term notation N 2S+1 LJ is 52 s1/2 , where s is orbital
notation indicating L = 0. Since L = 0, the ground state only has one possible
value of J = 1/2, and there is no ground state fine structure splitting. The firstexcited state 52 pJ has orbital angular momentum L = 1 (as indicated by p in orbital
notation). Therefore, J can take on two different values: 1/2 and 3/2, producing
a hyperfine splitting between the 52 p1/2 and 52 p3/2 . The spectral feature associatedwith the ground 52 s1/2 and lower excited 52 p1/2 energy difference is conventionallycalled the D1 line, and the feature corresponding to the splitting between 52 s1/2 and52 p3/2 is the D2 line. For other alkalies, including 40 K, the ground state values of L
and S are identical and only the N value is different. Therefore, even though theirenergies vary, D1 and D2 lines feature in all alkalies.
There is a yet smaller correction to the bare Hamiltonian, causing what’s
known as the hyperfine splitting. This arises from the interaction of the nuclear
spin magnetic dipole moment with the magnetic field created by the electron. The
Hamiltonian for this interaction is~
Hhfs = −µ~ˆI · B,
16

(2.5)

(a)

(b)

Rb

87

K

40

F=3
266.65 MHz

2P3/2
156.95 MHz
D2
790.24 nm
384.23 THz

72.22 MHz

24.2 MHz
F=2
F=1
F=0
F=2

2P1/2

F=5/2

816.66 MHz

2P3/2

D2
766.71 nm
391.02 THz
2P1/2

F=7/2
33.3 MHz44.1 MHz

155.3 MHzF=9/2

D1
770.11 nm
389.29 THz

F=9/2

F=2
2S1/2

F=11/2
F=7/2

F=1

D1
794.98 nm
377.11 THz

F=9/2

6.8347 GHz

F=7/2
2S1/2

F=1

1.2858 GHz
F=9/2

Figure 1: Atomic structure of the ground and first excites states, with fine andhyperfine splittings. (a)87 Rb. Values from [66]. (b). 40 K. Values from [67].

17

ˆ
ˆ
where µ~ˆI = gI µB I~ is the nuclear spin magnetic dipole moment, I~ is the nuclear spin
~ is the magnetic field seen by the nucleus. The nuclear spin g-factor
operator and B
gI encompasses the entire complex structure of the nucleons, but is generally smaller
than the electron spin g-factor gS by a factor of me /mp , where me is the electronmass and mp is the proton mass, making this term very small. This magnetic field,
to leading order, is given by [66, 68]~ = 2 µ0 µB
B
4π

"
!
#
~ˆ
~ˆ · ~rˆ
L
1 ~ˆ
S
2
ˆ
~ ,
−
S − 3 2 ~rˆ + δ(~r)r̂2 S
r̂3 r̂3
r̂
3

(2.6)

where the first term arises from the field due to the orbital angular momentum of theelectron, the second term is the field created by the electron spin magnetic dipole
moment, and the final term is the contact interaction, which is only non-zero for
ˆ ~ˆ
ˆ ~ˆ
L = 0 states. The hyperfine Hamiltonian eqn. 2.5 contains both I~ · L
and I~ · S~ˆ S~ˆ and I~ˆ as well as J~ˆ no longer commute with the Hamiltonian
terms. Therefore, L,and projection quantum numbers mL , mS , mJ and mI are no longer good quantum
ˆ ~ˆ ~ˆ ~ˆ
numbers. However, the total spin including nuclear spin F~ = L+ S + I commuteswith the Hamiltonian, and total spin F and its projection mF are now good quantum
numbers.
The energy shift, to lowest order in perturbation theory, due to the hyperfine
interaction is given by [66, 68]
3
K(K + 1) − 2I(I + 1)J(J + 1)
1
2,
∆Ehfs = Ahfs K + Bhfs
2
2I(2I − 1)2J(2J − 1)

(2.7)

where K = F (F + 1) − I(I + 1) − J(J + 1), Ahfs is the magnetic dipole constant of
the electron, whose values for 87 Rb and 40 K can be found in [66, 67].
For 87 Rb, I = 3/2 and for 40 K, I = 4. The interaction with the nuclear spin
splits the ground state of 87 Rb into two manifolds, F = 1 and F = 2. Similarly
for 40 K, it splits the ground state into F = 9/2 and F = 7/2 manifolds. The
18

structure of the ground and first excited states of 87 Rb and 40 K, including both thefine and hyperfine splittings, is diagrammed in Figure 1. Note that the D1 andD2 transitions are in the optical regime, making them amenable to laser cooling.
The fine structure splitting of the excited states is in the far infrared, whereas thehyperfine splitting of the ground states is in the microwave regime.

2.1.2 Interaction with static magnetic fields~ the atomic angular momentum interIn a static background magnetic field B,
acts with the field via the Hamiltonian

HB =

µB
~ˆ + gL L~ˆ + gI I)~ˆ · B,
~
(gS S
h̄

(2.8)

where µB is the Bohr magneton, gS , gL and gI are the spin, orbital and nuclear
Landé g-factors correcting their respective magnetic dipole moments. Without loss
~ = Bz ez ,of generality, we can define the magnetic field to be in the ez direction, B
to obtain
HB =µB(gS Ŝz + gL L̂z + gI Iˆz )Bz .
h̄

(2.9)

At very low magnetic field strengths, where the energy shift due to HB is small
compared to the hyperfine splitting, the total angular momentum F remains a goodquantum number, and the Hamiltonian in eqn. 2.10 can be re-written as

HB =

µB
(gF F̂z )Bz ,h̄(2.10)

where the effective Lande g-factor is dependent on the angular momentum quantum

19

numbers:

gF = gJ

F (F + 1) − I(I + 1) + J(J + 1)
F (F + 1) + I(I + 1) − J(J + 1)
+ gI.
2F (F + 1)
2F (F + 1)
(2.11)In this regime, the levels split linearly according to the F̂z projection quantum
number, mF . For the ground state of 87 Rb, it splits into three hyperfine states in
the F = 1 manifold (mF = 0, ±1) and five hyperfine states in the F = 2 manifold(mF = 0, ±1, ±2). This regime is at fields B ≤≈ 1 Gauss for 87 Rb, as seen in Figure
2.
At fields producing energy shifts small compared to the fine structure splitting, but large compared to the hyperfine splitting, F is no longer a good quantum
number, and the relevant Hamiltonian becomes

HB =

µB
(gJ Jˆz + gI Iˆz )Bz .
h̄(2.12)

Here, since gI  gJ , the energy dependence on Bz is dominated by a linear dependence on the Jˆz projection quantum number, mJ , as seen in the higher field limit
in Figure 2.
In the intermediate regime, there is in general no analytic solution for the
eigenenergies and one must resort to numerics. However, for the specific case of
J = 1/2 applicable to alkali ground states, there is an analytic solution given by the
Breit-Rabi formula [69]:
∆Ehfs
∆Ehfs
E|J=1/2mJ ImI i = −+ gI µB mB ±2(2I + 1)
2


1/2
4mx
2
1+
+x
,
2I + 1(2.13)

where ∆Ehfs is the zero field hyperfine splitting, m = mI ± mJ , andx=(gJ − gI )µB B
.
∆Ehfs
20

(2.14)

(a)(b)

mJ = +1/2

mJ = +1/2

1

2

F =2

Energy [∆Ehfs ]Energy [∆Ehfs ]

2

0
−1

F =1

F = 7/2

10−1

F = 9/2

−2

−2
0.0

0.2

mJ = −1/2

0.4
0.6
B [Tesla]

0.8

1.0

0.00

0.05

0.10
B [Tesla]mJ = −1/2
0.15

0.20

Figure 2: Energy structure of hyperfine states of the ground state of (a) 87 Rb and(b)40 K as a function of external magnetic field strength in Gauss. Note that thezero field splitting ∆Ehfs by which the vertical axis is scaled is different for the two
atomic species.
From this we can get a better picture of the shifts at low fields, where µB B  Ehfs .
Expanding eqn. 2.13 to second order in small parameter 4mx/(2I + 1) + x2 , andneglecting the field-independent terms, we obtain∆Ehfs∆E|J=1/2mJ ImI i ≈
2

 



1 4mx
1 16m2 x2
4mx3
2
4
+x −
+
+x
.
2 2I + 1
8 (2I + 1)2 2I + 1(2.15)We recognize the term linear in mx, ∆Ehfs mx/(2I + 1) = (gJ − gI )µB Bm/(2I + 1).
In addition, there is a term quadratic in mx:∆Ehfs(gJ − gI )2 µ2B B 2 m2
m2 x2=
= (B)m2 ,(2I + 1)2
∆Ehfs (2I + 1)2(2.16)

where in the last term we have defined (B), the magnitude of this ‘quadratic’
Zeeman energy shift. For the magnetic field strengths used in experiments describedin this thesis, the linear term plus quadratic correction are sufficient for describing
the energy levels.
21

The form of the approximate Hamiltonian in this regime for any value of F isgiven by
H0 = HKE + h̄ωz F̂z + h̄F̂z2 ,(2.17)

where h̄ωz = µB gF Bz /h̄, and the kinetic energy Hamiltonian HKE = h̄2~k 2 /2mÎ, andÎ is the identity matrix.

2.2 Near-resonant atom–light interaction
In this section, we will assume the atom can be treated as a two-level system:
one with a ground state of some energy h̄ωg and excited state with energy h̄ωe ,with an energy difference h̄ω0 . The interaction with a light source results from theelectric dipole moment of the atom d~ = −0~r interacting with the electric field of~ ~r), giving [70]the beam E(t,~ ~r).
HL (t) = −d~ · E(t,(2.18)

When very weak or no light is being shined on the atom, it still experiences
the interaction with vacuum light modes. Through this mechanism, an atom inthe excited state may still decay to its ground state (emitting a photon). This iscalled spontaneous emission, and the time scale on which the decay will happen
is determined by the dipole matrix element of the transition, and is known as thenatural transition linewidth of the atomic transition Γ [70].

2.2.1 Rabi oscillations
When strong laser light is present, and on timescales short compared to 1/Γ,
spontaneous emission can be ignored [71]. Let us consider the Hamiltonian

H = H0 + HL (t),22

(2.19)

where H0 is the bare atomic Hamiltonian and HL is the interaction with the laser
beam eqn. 2.18. We can write the wavefunction Ψ as a linear combination of thetwo eigenstates (for a two-level atom) of the bare Hamiltonian H0 as
Ψ = cg (t)φg (r)e−iωg t + ce (t)φe (r)e−iωe t ,(2.20)

where cg (t) and ce (t) are the time-dependent coefficients multiplying the eigenstate
wavefuntions φg and φe of the ground and excited state respectively, and r is thespatial coordinate. Absorbing any diagonal elements of HL into H0 , multiplying
both sides of the Schroëdinger equation from the left by ψj and integrating over r,
we can write down the Schroëdinger equation as two coupled equations:ih̄dcg (t)= ce (t)HLge (t)e−iω0 t
dt(2.21)dce (t)= cg (t)HLeg (t)eiω0 t ,dt(2.22)

andih̄

where ω0 = ωe − ωg is the transition frequency, HLge (t) is the off-diagonal elementof the laser coupling Hamiltonian that couples the excited to the ground state and
HLge (t) = HLeg∗ (t). For simplicity, let’s consider light propagating along ez , linearlypolarized light along ex , E~ = (E0 , 0, 0)cos(ωt). The coupling Hamiltonian can thenbe written as [70]HLge (t) = −µge E0 cos(ωt),
where the electric dipole matrix element µge =R

φ∗g xφe . This can be written as

HLge (t) = h̄Ωcos(kz − ωL t)

23(2.23)(2.24)

withΩ=

E0 |µge |h̄(2.25)

the Rabi frequency, characterizing the coupling strength between the laser field andthe atom. Here, e is the charge of the electron.
To solve this Schroëdinger equation, we make the traditional transformation
to the rotating frame:c0g (t) = cg (t)(2.26)

c0e (t) = ce (t)e−iδt ,(2.27)

where δ = ω0 − ωL is the detuning of laser light resonance. In this frame, we from0

 cg 
can write the atom–light Hamiltonian in the   basis as:c0e−δ/2 Ω/2
H = h̄ .
Ω/2 δ/2(2.28)In the limit of no coupling, Ω = 0, in the rotating frame the eigenenegies are
E± = ±h̄δ/2. For non-zero coupling, finding the eigenvalues of H gives√
E± = ±h̄ δ 2 + Ω2 /2.

(2.29)The eigenenergies are shifted in the presence of the light.
Assuming the atom starts in the ground state cg (t = 0) = 1, we can solve thetime-dependent Shroedinger equation (TDSE) with the above Hamiltonian
 
 
0
0
d  cg  cg 
ih̄   = H  
dt c0
c0e
e
24

(2.30)

we obtain the oscillating excited state amplitudeΩsin2Ω + δ2c0e (t) = −i √!
√
Ω2 + δ 2 t,
2(2.31)

known as Rabi oscillations. The frequency of these oscillations is the generalized√
Rabi frequency Ω0 = Ω2 + δ 2 . The amplitude of the oscillation is at its maximum
when the laser is on-resonance, δ = 0. As the detuning increases, the contrast
in excited and ground populations decreases, while the frequency of the oscillation
increases.

2.2.2 Scattering
In the regime where spontaneous emission cannot be ignored, Rabi oscillationsof each individual atom are intermittently interrupted by decay to the ground state.
Averaging over an atomic ensemble, on the time scale of a single Rabi oscillation, theoverall excited state population reaches a steady state, and the rate of spontaneousemission becomes constant. Since during spontaneous emission the ejected photon
can go into any vacuum mode, this process can be thought of as the scattering ofphotons by the atoms. This scattering rate is given by [71]γsc =I/IsatΓ,
2 1 + 4(δ/Γ)2 + I/Isat(2.32)

where Isat is the saturation intensity. This is the intensity at which the timescale of
spontaneous emission matches the Rabi oscillation rate, reducing the capacity for
absorption of extra light.

25

2.2.3 Adiabatic rapid passage
Suppose there is a two-level atom in its ground state that an experimenter
wants to transfer into the excited state. To transfer it with perfect fidelity using Rabi
oscillations, one would need a perfectly on-resonant beam and very precise timing toshut off the coupling field at the maximum of the excited state population. This ischallenging and not very stable to small perturbations in the level splitting between
the ground and excited state (caused by small field fluctuations for the case of two
Zeeman sublevels). A more robust technique is known as adiabatic rapid passage.
Suppose the two energy levels of the atom are sensitive to some external parameter. Commonly (as in the case of hyperfine sublevels), they are sensitive to an
external magnetic field B, and in the small field limit are linear in B. Therefore,
the detuning that goes into the Hamiltonian in eqn. 2.28 becomes h̄δ = AB − h̄ωL ,where A is the coupling constant in the hyperfine (or other) Hamiltonian. In thelimit of zero coupling strength, diagonalizing the Hamiltonian amounts to calculating the eigenenergies in the rotating frame as a function of the control parameter
B. These are represented in black in Figure 3. Note that the two levels cross eachother when their energy difference matches h̄ωL , on resonance.
Once the coupling field is turned on (in the case of the hyperfine interaction,and rf-field), the two levels split near the resonance and an avoided crossing appears,
as seen in blue in Figure 3. Crucially, away from resonance towards the left, thebottom coupled state overlaps closely with the uncoupled (black state). Far fromresonance, the ground state is largely unaffected by the presence of light. On theother hand, far from the resonance on the right, the lower coupled state overlaps
almost perfectly with the excited uncoupled state. Adiabatic rapid passage (ARP)
takes advantage of this change.
The ARP protocol is as follows. Start at a bias field (or other control parame-

26

Energy [arb. u.]

4

|ei

2
0
−2
−4

|gi
0.0

Bstart

Bres
0.1

0.2

Bstop
0.3

0.4

B [Gauss]

Figure 3: Adiabatic rapid passage (ARP). Black lines represent energies of theground and excited state, in the rotating frame, with zero coupling. Blue lines
represent these bands with some coupling h̄Ω turned on. The dashed lines represent
the starting point of an ARP (Bstart ), the resonant field value (Bres ) and the end
point of an ARP (Bstop ).
ter) where turning on the coupling field does not significantly perturb the eigenstate(Bstart in the figure). Turn on the coupling field adiabatically (slow with respect tothe time scale associated with the level splitting energy at the selected detuningτ = h/(Ee − Eg )), such that the atom remains entirely in the ground state |gi.
Then, sweep the control parameter across the resonance, again adiabatically withrespect to the coupled level splitting (blue in the figure). The sweep rate can be
optimized to be faster away from resonance, where Ee − Eg is large and slower
closer to the avoided crossing. Then, at a final value of the control parameter where
the lowest coupled state overlaps almost perfectly with the bare excited state |ei,adiabatically turn off the coupling field, leaving the atoms in the excited state.
The ‘rapid’ part of adiabatic rapid passage refers to the procedure having to
be fast with respect to the spontaneous emission timescale, since in the coupledbasis there is some population in the excited state and spontaneous decay would
disrupt the process. This procedure is relatively insensitive to field fluctuations (as27long as Bres is roughly in the middle of the relatively long sweep, the procedure will
succeed). It can also be applied to multi-level situations, for example in the caseof the hyperfine states of a single manifold F , where the atoms traverse multiple
avoided crossings in the same sweep and can be efficiently transferred from one
stretch state mF = F to the other mF = −F , or vice versa.

2.3 Far–off-resonant atom–light interaction
We can infer the behavior of atoms in a far-detuned laser field by taking thesolutions from eqns. 2.29 and 2.31 in the limit δ  Ω. First, looking at the excitedstate population in eqn. 2.31, the amplitude of the excited state population oscillation approaches zero. Therefore, as expected, no absorption of the light actuallytakes place and the atom remains in the ground state. However, the light still affects the atom by shifting the eigenenergies via eqn. 2.29. Recalling that the bareeigenenergies in the rotating frame are given by E± = ±h̄δ/2, the energy shift frombare is given by√∆E± = ±h̄ δ 2 + Ω2 /2 − ±h̄δ/2.

(2.33)

Expanding the energy shift in the small parameter Ω/δ, we obtain the shifted en√ergies E± = ±h̄ δ 2 + Ω2 /2 ≈ ±(δ/2 + Ω2 /4δ). The shift from bare energy levels isthus∆E± ≈ ±h̄(δ/2 + Ω2 /4δ) − ±h̄δ/2 = ±h̄Ω2 /4δ.

(2.34)

This laser-intensity–dependent energy shift is called the AC Stark shift, and is thebasis of most laser created potentials for cold atoms.
For the ground state, and a red detuned laser beam (where the laser frequency
is lower than the resonant frequency), this creates energy minima in locations ofmaximal laser intensity. For the lattice described in this chapter, as well as for thetrapping of our atoms in the final stages of cooling, we use high power (up to 10 W)
28

lasers with wavelength λL = 1064 nm.

2.4 Absorption imaging
Absorption imaging takes advantage of the on-resonant interaction describedin the previous section. An on or near-resonant laser beam (δ/Γ  1) is shinedat the atoms, and the absorbed light acts to create a shadow in the shape of theatoms in the laser beam. This beam with the shadow is then imaged on a camera,in our case a CCD, as depicted in Figure 4a (top). This is called the atom image,and the intensity distribution over the camera is denoted by If (x, y), where thesubscript f stands for final - the intensity after the light has encountered the atoms.
To quantify the ‘shadowed out’ intensity, after the atoms have left the trap, thesame laser intensity is shined directly at the camera, as in Figure 4a (bottom). Thisis called the probe image, and the intensity distribution over the camera is denotedby I0 (x, y), where the subscript 0 indicated initial—the intensity before the light
had encountered the atoms.
To recover the atom number distribution encountered by the light, consider anatomic cloud with 3D density ρ(x, y, z). Since we can only obtain 2D informationfrom the camera, we can only hope to recover a 2D atomic column density n(x, y) =Rρ(x, y, z)dz. Focusing in on a single pixel of the camera, we can consider a single
value of I0 and If to recover a local n. As the laser light propagates throughthe atomic cloud, the intensity of the light will diminish due to absorption. Thisabsorption as a function of propagation direction z can be expressed using thescattering rate equation eqn. 2.32 as the number of photons scattered by the atoms(proportional to the atomic density times the scattering rate) times the photon
energy h̄ωL :d I(z)
I(z)/Isat= −h̄ωL ρ(z)γsc (z) = −ρ(z)σ0,2dz Isat
1 + 4δ /Γ2 + I(z)/Isat29(2.35)

n [atoms/m2 ]

3.4

×107

1.7

0.0

I0 /Isat

1.2
0.8
0.5
0

250

500

750

1000

z [µm]

(a)

(b)

(c)

Figure 4: Absorption imaging. (a) Near-resonant probe light illuminates the atoms,
and the transmitted light (containing a shadow of the atoms) is imaged on thecamera. A second image taken with no atoms provides a reference. (b) The probe
beam is partially absorbed as it traverses the cloud, and the intensity seen by atoms
further along the imaging direction ez is lowered. (c) An atomic cloud illuminatedby 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.
where the resonant scattering cross section is σ0 = 3λ20 /2π, and λ0 is the wavelength
associated with atomic resonance.
Integrating both sides of eqn. 2.35, we obtain
σ0 n = 1 + 4δ 2 /Γ2 ln(I0 /If ) + (I0 − If ) /Isat .

(2.36)The quantity OD = ln(I0 /If ) is called the optical depth of the cloud. When theprobe intensity I0 is much smaller than the saturation intensity, the second term ineqn. 2.36 becomes negligible. Assuming further that the probe light is on resonance,δ = 0, the atomic column density becomes simply σ0 n = OD. Figure 4b shows aGaussian atomic density distribution (top) and the resulting probe intensity as afunction of position in the cloud (bottom). The intensity drops from its initial to
final value gradually as it traverses the cloud.

30

However, there is an important effect that the above equations do not accountfor. Namely, as the atoms absorb light from the probe beam, they also get a momentum kick equal to the momentum of a photon during each collision h̄kr = h/λL in the
direction of propagation. It is true that the absorbed photon will then be re-emitted
by the atom, inducing a loss of momentum, but since this happens through the process of spontaneous emission into a random vacuum mode, the average momentum
kick acquired this way over many re-emissions will average to zero. On average,
each photon absorbed will induce a change in velocity of the atom of vr = h̄kr /m,
where m is the atomic mass, as depicted in Fig. 4c. As the velocity of the atomchanges, due to the Doppler effect, the apparent laser frequency will change as well.
Therefore, even if the laser light is exactly on-resonant for a stationary atom, it
will become off-resonant for longer imaging times, and eqn. 2.35 will acquire a timedependence. For most experiments, this effect is small and can be neglected. However, if the imaging time becomes of order a recoil time tr , a time after which therecoil-induced detuning δ becomes of order Γ, this effect becomes significant. We
explore this effect in Chapter 4.

2.4.1 Time-of-flight and in situ imaging
There are two commonly used protocols for measuring cold atomic clouds, insitu and time-of-flight measurements. Generally, the atomic cloud is trapped (inour case by an optical dipole trap) during the experiment. In situ is Latin for in its
original place. As suggested by the name, in situ measurements are taken while thecloud is still in its original trap, or immediately after the trap is turned off before
any dynamics have had time to occur. These measurements measure the real spatial
distribution of the atoms at the end of the given experiment. There is a difficulty
associated with making in situ measurements of BECs, however. Namely, BECs intheir original trap tend to be relatively dense, with optical depths often in excess of31

OD ≈ 20, requiring unrealistic probe light intensities to resolve. One way to bypass
this difficulty is to selectively image only a small fraction of the condensed atoms,
as was done with microwave imaging for our magnetic field stabilization feedforward protocol. Another option is to instead perform a time-of-flight measurement,
which reduce the density of the cloud but don’t give access to the original density
distribution.
In time-of-flight measurements, the trapping potential is abruptly snapped off
after the experiment, and the atoms are allowed to free fall and expand for sometime t. For our experiments, t was on the order of tens of milliseconds. In theregime where time t is long enough that the atoms travel much further than theinitial extent of the cloud in the directions transverse to the imaging axis, the final
position of the atoms is determined almost exclusively by their in situ momentum,
not their in situ position. Therefore, time-of-flight imaging in this regime measures
the atomic distribution as a function of momentum, not position.

2.5 One-dimensional optical lattices
2.5.1 Lattice HamiltonianOur 1-D optical lattice is created by retro-reflecting the λL = 1064 nm laser,
creating a standing wave of light. Via the AC Stark shift, this creates a periodic
potential for the atoms of the form
V = −V0 sin2 (kL x),(2.37)

where kL = 2π/λL is the wavenumber associated with the lattice recoil momentum.
The time-independent Hamiltonian, for some eigenenergy En , will be given by−h̄2 d2Ψn (x) − V0 sin2 (kL x)Ψn (x) = En Ψn (x).
2m dx232(2.38)

Since the potential is spatially periodic, we can invoke Bloch’s theorem [10]:Ψn,q = eiqx un,q (x),(2.39)

where q is the crystal momentum restricted to ±h̄kL , and un,q (x) is the spatiallyvarying part of the wavefunction obeying the spatial periodicity of the lattice. Plugging this in to the Hamiltonian, we obtain
h̄2
−
2m



d
d2
2
−q + 2iq+
un,q (x) − V0 sin2 (kL x)un,q (x) = En un,q (x).
dx dx2(2.40)

Expanding un,q (x) in Fourier components commensurate with the lattice period ofP
i2kL jx, we obtain
2kL as un,q (x) = ∞j=−∞ aj e
X  h̄2
j

2m

2



2

(q + 2kL ) aj − V0 sin (kL x)aj ei2kL jx = EnX

aj ei2kL jx .

(2.41)

j0

Re-writing sin2 (kL x) = (e−2ikL x + e2ikL x − 2)/4, multiplying both sides by ei2kL j x andP0
invoking cj eik(j−j ) = δjj 0 , where δjj 0 is the Kroniker delta and cj are appropriately
normalized coefficients, we get for any value of the index j
V0h̄2(q + 2kL j)2 aj − (aj+1 + aj−1 ) = En aj .
2m
4

33

(2.42)

This can be expressed in matrix form










HL = 









..



.
h̄2
(q + 4kL )2
2m

− V40

0

0

0

− V40

h̄2
(q + 2kL )2
2m

− V40

0

0

0

− V40

h̄2 2
q
2m

− V40

0

0

0

− V40

h̄2
(q − 2kL )2
2m

− V40

0

0

− V40

h̄2
(q − 4kL )2
2m

..

.

(2.43)
in the basis of momentum orders |ki = eikx given by:


..
.







|q + 4kL i






|q + 2kL i |qi  .
|q − 2kL i|q − 4k i

L 


..
.

(2.44)

This matrix can be diagonalized for every value of the crystal momentum q,with the resulting band structure shown in Figure 5. It is convenient to define thelattice recoil energy EL = h̄2 kL2 /2m. Then, we can re-write the Hamiltonian with

34










,









Energy E/EL

8

6

4

2

0
−3

−2

−1

0

1

2

3

q/kL

Figure 5: Lattice band structure in the extended zone scheme. The dashed lines
represent the limit of zero lattice depth, with the regular parabolic dispersion relation of a free particle repeating with the reciprocal lattice period. The solid lines
are the dispersion relation at V0 = 4.0EL , showing the opening of gaps at crossingsof the zero lattice depth bands. The black lines demarcate the first Brillouin zone.
V0 in units of EL and momenta q in units of kL as










HL /EL = 









..



.
(q + 4)2
− V40

− V40

0

0

0

(q + 2)2 − V40

0

0

− V40

0

0

− V40

q2

0

0

− V40 (q − 2)2

0

0

− V40

− V40
(q − 4)2

..

.










.









(2.45)In any numerical simulation, the number of momentum orders that can be
included is finite. We determine the value of the parameter n = max(|j|) as the35lowest n at which the eigenvalues stop changing to machine precision from n − 1.

2.5.2 Tight-binding approximation
In the limit of large lattice depths, V0 >≈ 5EL , the lattice Hamiltonian is wellapproximated by the tight-binding model with only nearest neighbor hopping. In
the tight-binding model, the basis is assumed to be a set of orthogonal functions,called Wannier functions, localized to each lattice site |ji. The approximation lies inassuming only nearest neighbor tunnelings between the sites, forming the nearestneighbor tight-binding Hamiltonian

Htb = −t |ji hj + 1| + H.c.,(2.46)

where t is the tunneling amplitude between nearest neighbor sites and H.c. standsfor Hermitian conjugate. We have neglected the diagonal energy term, as it will beequal for every Wannier function |ji and thus represents a constant energy offset.
All the information about the lattice depth is therefore reflected in the tunneling
amplitude t.
This Hamiltonian can also be expressed in the momentum basis by Fouriertransforming the basis functions:
1 X −ikj je|kj i ,
|ji = √
N kj

(2.47)X
1 X X −ijk1 ik2 (j+1)
tee|k1 i hk2 | + H.c = −2tcos(k) |ki hk| .
N k kk(2.48)

giving the Hamiltonian

Htb = −

1

2

From this we can directly read off the band structure of the tight binding Hamiltonian. First, we notice that we only obtain one band—to approximate higher bands
36

with the tight-binding approximation we would need to construct a different set of
Wannier functions and a different tunneling strength. Second, we see that the lowest band is simply a cosine–therefore we have solved for the band structure withouteven defining what the basis Wannier functions are! Third, the amplitude of thecosine function is given by the tunneling strength t. This gives us a good clue as to
how to determine the appropriate tunneling given a lattice depth V0 —simply find at that matches the amplitude of the lowest band, which becomes cosinusoidal in the
deep lattice limit.
The precise form of the Wannier functions depends on both the depth of thelattice and the band being reproduced. It is not necessary for us to find their full
expression, as the band structure can be calculated without them. The definition,
however, is
|ji =

Z
BZ

eiφ(q)−iqja |qi dq,(2.49)

where the integral is over the Brillouin zone, from −kL to kL , a is the lattice spacing
λL /2, and |qi is the Bloch wavefunction at crystal momentum q, and φ(q) is the phaseassociated with each Bloch wavefunction. The Bloch wavefunctions individually
have arbitrary phase. The phase plays an important role in combining the Bloch
wavefunctions into a Wannier function, finding the proper phase relationship tomake the wavefunction maximally localized at each site [72].

2.5.3 Pulsing vs adiabatic loading of the lattice
The lattice depth parameter V0 /4, for a range of values, can be well calibrated
experimentally by pulsing on the lattice. Here, the word pulsing indicates that thelattice is turned on fully non-adiabatically, if not instantaneously, such that theoriginal bare momentum state is projected onto the lattice eigenvector basis, as
shown in Figure 8a. If the atoms start out stationary in the trap, the bare state in37the momentum basis is simply
 
..
.
 
0
 
 
 
0
 
 
|Ψ0 i = 1 ,
 
 
0
 
 
0
 
 
..
.

(2.50)

as depicted in Figure 8b.
Since the lattice eigenvector basis is distinct from the bare one, instantaneously
turning on the lattice will necessarily excite the atoms into a superposition of lattice
eigenstates, each evolving with a different phase according to the eigenenergy while
the lattice is on, as shown in Figure 8c. Then, when the lattice is snapped backoff, the wavefunction is projected back into the bare basis, and the varying phaseaccumulation results in a beating of the different momentum orders, see Figure 8d.
This can be calculated simply by using the time evolution operator|Ψ(t)i = e−iHL t/h̄ |Ψ0 i .

(2.51)

By pulsing on the lattice for variable amounts of time t, we can obtain fractional
populations in the different momentum states. Time-of-flight imaging captures themomentum distribution of the cloud, and the different entries of Ψ(t) in the momentum basis will thus appear as different clouds on the absorption image, as shown in
Figure 7a. The fractional population in these clouds corresponds to a measurementof |aj |2 . Typically for our values of the lattice depth V0 < 10EL , it is sufficient tosimply count three central momentum orders, k = q, q ± 2kL . Then, we can fit eqn.
38

V0 [EL ]

(a)
8
0

time

(b)

(c)

(d)

Energy E/EL

8

6

4

2

0
−1

0

q/kL

1

−1

0

q/kL

1

−1

0

1

q/kL

Figure 6: Lattice pulsing. (a) Lattice depth as a function of time during a pulsing
experiment. The lattice is turned on instantaneously at t = 0 and held on fora variable amount of time until being turned off instantaneously at a final timet = tf . (b) Atomic population before t = 0. The dispersion relation is that of afree particle, and all of the atoms start out at q = 0 in the lowest energy level.
Here, the area of the dots is proportional to the fractional population in the energy
state. (c) Atomic population after the lattice is turned on for a lattice depth of
V0 = 8.0EL . The energy spectrum now shows the lattice band structure, and someatomic population is projected onto the excited bands. (d) Atomic population after
the lattice is snapped off at tf = 150 µs. The wavefunction is projected back onto
the bare states, with some fraction (blue circle) in the lowest band at k = 0 andsome fraction in the excited band, with equal population being projected onto thek = 2kL (green) and k = −2kL (red).

39

(b)
1.00
0.75
0.50
0.25
0.00
0

50

100

0

−2kL

Fractional population

kx

2kL

Fractional population

(a)

150

200

250

(c)
1.00

kx = +2kR
kx = 0kx = −2kR

0.75
0.50
0.25
0.00
0

20

40

60

80

100

120

140

Lattice pulse time [µs]Figure 7: Lattice pulsing for calibration. (a) An example time-of-flight image froma pulsing experiment. The three different clouds are different momentum orders.
(b) Fractional populations in the different momentum orders as a function of pulsing time at a low lattice power. Data is indicated by dots and best-fit theory isrepresented by lines. The lattice depth from fit is V0 = 5.57 ± 0.07EL . (c)Fractional
populations in the different momentum orders as a function of pulsing time at ahigher lattice power. Data is indicated by dots and best fit theory is represented bylines. The lattice depth from fit is V0 = 12.69 ± 0.07EL .
2.51 to the data with fitting parameter V0 , thus deducing the lattice depth. Some
examples of these pulsing experiments are presented in Figure 7b,c.
In contrast to pulsing, adiabatic loading turns the lattice on slowly, such thatthe atomic wavefunction starting in the bare ground state can continuously adjust
to remain in the ground state of the current Hamiltonian, without projecting onto
any of the higher bands. This process is illustrated in Figure 8. The adiabatictimescale depends on the spacing between the ground and next excited band (or if
starting in a different eigenstate, the nearest eigenstate). If the energy difference
between the ground and first excited state is ∆E, the timescale on which the lattice40

is turned on must fulfill t  h/∆E.

2.6 Raman and rf coupling
In this section, we will introduce Raman and rf coupling between the hyperfine
sublevels of the ground state of 87 Rb. While we will focus on the F = 1 and F = 2
manifolds of this ground state due to their relevance to the experiments describedin Chapters 6 and 7, the discussion can be easily extended to any value of F .

2.6.1 Rf coupling Hamiltonian
For the F = 1 manifold, there are three available spin states mF = 0, ±1.
There are many ways of introducing coupling terms between the different hyperfine
states. Here, we will explain two methods: rf coupling and Raman coupling. Rf
coupling is a radio-frequency oscillating magnetic field, in our case produced by apair of circular coils in series side by side above the atoms (see [73]). Assuming therf oscillating field is polarized along the ex , with the bias field along ez , the couplingˆ ~Hamiltonian is given by Hrf = µB gF F~ · B= µB gF F̂x Bx cos(ωt), where 2πω is the rf
frequency. The schematic of this setup is shown in Figure 9.
The eigenstates of the bare Hamiltonian H0 are the constituent mF states.
The eigenstates of the coupled Hamiltonian H0 + Hrf (t) can be expressed as a linearP
superposition of the bare eigenstates Ψ(~x, t) =x)e−iωmF t . The
mF cmF (t)φmF (~Hamiltonian in this basis can then be written as [71]−i(ωz −)t0Ωcos(ωt)e
−i(ωz −)t
Hrf = HKE + h̄ 0Ωcos(ωt)e0Ωcos(ωt)ei(ωz +)t
0
Ωcos(ωt)e−i(ωz +)t ,
0(2.52)

where ωz = (ω1 − ω−1 )/2,  is the quadratic Zeeman shift, and Ω is the Rabi fre41

V0 /EL

(a)
8
0

time

(b)

(c)

(d)

Energy E/EL

8

6

4

2

0
−1

0

q/kL

1

−1

0

q/kL

1

−1

0

1

q/kL

Figure 8: Adiabatic lattice loading. (a) Lattice depth as a function of time during
adiabatic turn-on. The lattice is ramped on starting at t = 0, slowly increasing
to a final lattice depth and turned off instantaneously at a final time t = tf . (b)Atomic population before t = 0. All atoms are at k = 0 in the lowest bare band.
(c) Atomic population after the lattice is turned on adiabatically to a lattice depthof V0 = 8.0EL . All atoms remain in the lowest band, but the band is no longer
bare. (d) Atomic population after the lattice is snapped off. The wavefunction is
projected back onto the bare states, with some fraction (blue circle) in the lowest
band at k = 0 and some fraction in the excited band, with equal population being
projected onto the k = 2kL (green) and k = −2kL (red). Since the lowest lattice
band is a superposition of bare bands, some atoms are excited to the higher baremomentum states.

42

Figure 9: Raman and rf coupling schematic. (a) Beam geometry of the Raman
beams and rf relative to the external field. The Raman beams have a frequency
difference ∆ω, and are linearly polarized in perpendicular directions. (b) Level
structure of both Raman and Rf coupling for hyperfine states of the F = 1 manifold.
The hyperfine splitting separates the levels by an energy h̄ωz . The quadratic Zeeman
shift  lowers the energy of the mF = 0 state, and the detuning δ of either the Ramanor the rf fields shifts the energies of the mF = ±1 states. Raman transitions are
two-photon, exciting up to a virtual state and coming back down to an adjacenthyperfine state, with an accompanying momentum transfer. Rf couples adjacenthyperfine states directly. Figure taken from ref. [74]quency, proportional to Bx . We can then transfer into the rotating frame c0mF =2

e−imF δt+i(1−mF )t cmF , where δ = ωz − ω. Then we apply the rotating wave approximation, that the fast oscillating terms average to zero over time scales of interest
e2iωt ≈ 0, and obtain δ Ω/2 0 
Hrf = HKE + h̄ Ω/2 − Ω/2 ,0 Ω/2 −δ(2.53)

or for any value of F
Hrf = HKE + h̄δFz + h̄Fz2 + ΩFx /2,(2.54)

which reduces to the above form for F = 1 with an overall energy shift of h̄I.
43

The band structure of this Hamiltonian can be seen in Figure 10, where we havediaganolized eqn. 2.53 for a range of momenta kx (we have isolated kx for comparison
with Raman coupling, as will be seen in the next section). The parabolas are simply
the free particle dispersion relations along one dimension, with three bands arisingfrom the three available spin states. It is convenient to define the magnetization of anPeigenstate m = mF mF pmF , where pmF is the fractional population in the mF state.
We have indicated the magnetization of the eigenstate by coloring the eigenenergies,
with m = −1 in red, m = 0 in green, and m = +1 in blue. In Figure 10a, both thedetuning and the coupling strength are zero. Therefore, there are simply three freeparticle dispersion curves, each exactly correlated with a particular spin state, themF = ±1 are degenerate and the mF = 0 state is slightly offset by the quadratic
shift h̄. In Figure 10c, the coupling strength is again zero, but the detuning has
been turned on, lifting the degeneracy between the mF = ±1 states. Figure 10b,dshows the same conditions as a,c, respectively, but with the coupling strength turned
on. In Figure 10b, where the detuning is zero and the quadratic shift is negligible
compared to the coupling strength, all states average to a magnetization of zero—
the mF = ±1 states are symmetrically populated. In Figure 10d, this symmetry is
broken by the presence of a detuning.

2.6.2 Raman coupling Hamiltonian
The counter-propagating Raman beams, as seen in Figure 9, couple the samestates as the rf. The do so via the vector light shift created by the pair of beams.
~ = E0 exp(ikR x−
The electric field due to the right going beam (red in Figure 9a) is Eiωt)ey , where E0 is the amplitude of the electric field and h̄kR = h/λR = h̄ω/c. The~ = E0 exp(−ikR x −electric field from the left going beam (gray in Figure 9b) is Ei(ω +∆ω)t)ez . This combines to give an effective field from the vector light shift [75]~ ×E
~ ∗ ∝ −E 2 cos(2kR x + ∆ωt)ex . Going through the same procedure as for
Beff ∝ E
0
44

(a)

(b)

Energy E/ER

1.00
0.75

2

0.50

0

0.25

−2

0.00

−4

(c)
2

Energy E/ER

1

4

(d)

0

4
2

1

0
0
−2
−1

−1.0

−0.5

0.0

kx /kR

0.5

1.0

−4

−1.0

−0.5

0.0

0.5

1.0

−1

kx /kR

Figure 10: Band structure of the rf Hamiltonian, eqn. 2.53, in momentum space.
For all plots, the quadratic Zeeman shift h̄ = 0.04ER , and the color represents
magnetization, labeled by the colorbar. (a) h̄Ω = 0, h̄δ = 0. No coupling or
detuning is present, so the only separation between the bands is due to the quadratic
shift h̄. (b) h̄Ω = 5.0ER , h̄δ = 0. (c) h̄Ω = 0, h̄δ = 1.0ER . Even though thecoupling strength is zero, the bands are separated by the detuning. (d) h̄Ω = 5.0ER ,h̄δ = 1.0ER .

45

the rf coupling case, including the transfer into the rotating frame and the rotating
wave approximation, we obtain the same Hamiltonian in the basis of bare spin states|−1i , |0i , |1i but with an extra phase factor:
−i2kR x

δ
Ω/2e


i2k x
HRaman = HKE + h̄ −Ω/2e R0Ω/2ei2kR x0Ω/2e−i2kR x ,−δ(2.55)

where δ = ωz − ∆ω.
This phase difference between the rf and Raman Hamiltonian has an intuitive
physical explanation. In order to undergo a Raman transition, an atom first absorbs
a photon from one beam, getting a momentum kick equal to the recoil momentumh̄kR . Then, to decay back down to an adjacent spin state, the atom undergoes
stimulated emission into the field of the other (counter-propagating) beam, acquiring
another recoil momentum kick in the same direction for a total of 2h̄kR ex . Therefore,
the Raman coupling Hamiltonian for F = 1, after transforming into the rotating
frame and performing the rotating wave approximation, can be written in the same
way as the rf Hamiltonian in eqn. 2.53 with the addition of a momentum kick—inreal space, an acquired phase—of ei2kR x .
We can again make a basis transformation to get rid of this phase. Let us
define |−1i0 = exp(−2ikR x) |−1i = |kx − 2kR , −1i , |0i0 = |0i = |kx , 0i , |1i0 =exp(2ikR x) |1i = |kx + 2kR , 1i, where for third definition we went into the momentum basis and labeled the states by a combination of their momentum and spinstate. Then, including the kinetic energy term along ex explicitly, we obtain the
Hamiltonian in the new basis as:h̄2 (kx −2kR )2
+ h̄δ
2m



(y,z)
HRaman = HKE + 



h̄Ω/2

h̄Ω/2

h̄2 kx2
− h̄
2m

0

h̄Ω/2
46

0
h̄Ω/2
h̄2 (kx +2kR )2
− h̄δ
2m



.



(2.56)

It is convenient to define the Raman recoil energy as ER =2h̄2 kR.
2m

The bandstructure of this Hamiltonian is shown in Figure 11, for several representative parameter values, with the magnetization labeled by the color. Figure 11a shows theband structure in the limit of zero coupling and zero detuning, but where we havealready gone into the basis |kx − 2kR , −1i , |kx , 0i , |kx + 2kR , 1i; therefore, the freeparticle parabola corresponding to the mF = 1 spin states is shifted to center on
kx = −2kR and the mF = −1 parabola is shifted to center on kx = 2kR . As thecoupling is turned on to h̄Ω = 1ER in Figure 11b, the points where the parabolas
cross become ‘avoided crossings’, separating into three bands where magnetization(and the underlying spin distribution) depends on the momentum kx . As the coupling strength is turned up even further to h̄Ω = 5ER in Figure 11c, the lowest bandgoes from having three minima, one corresponding to each original spin state, toonly one minimum. This transition happens at h̄Ω = 4ER [73]. In Figure 11d, weshow the band structure again in the limit of zero coupling, but this time with adetuning of h̄δ = 1.0ER . Note that the detuning tips the parabolas with respect toeach other. Figure 11e shows the detuned system with coupling strength turned up
to h̄Ω = 1ER , still in the three minima regime but with avoided crossings creating
three momentum and spin coupled bands. In Figure 11f, the detuned system isturned up to a coupling strength of h̄Ω = 5ER , creating a single minimum, this timeoffset from kx = 0.
We can write the general F version of the Raman coupled Hamiltonian in thebasis |kx + mF 2kR , mF i, where −F ≤ mF ≤ F , as:(y,z)

HRaman = HKE + h̄2 (kx I + 2kR Fz )2 /2m + h̄δFz + h̄Fz2 + ΩFx /2.

47(2.57)

Energy E/ER
Energy E/ER

(a)

(b)

(c)

(d)

(e)

(f)

1

10

5

0
0

10

5

0
−2.5

0.0

kx /kR

2.5

−2.5

0.0

kx /kR

2.5

−2.5

0.0

2.5

−1

kx /kR

Figure 11: Band structure of the Raman Hamiltonian, eqn. 2.56, in momentum
space. For all plots, the quadratic Zeeman shift h̄ = 0.04ER , and the color represents magnetization, labeled by the colorbar. (a) h̄Ω = 0, h̄δ = 0. (b) h̄Ω = 1.0ER ,h̄δ = 0. (c) h̄Ω = 5.0ER , h̄δ = 0.0. (d) h̄Ω = 0.0, h̄δ = 1.0ER . (e) h̄Ω = 1.0ER ,h̄δ = 1.0ER . (f) h̄Ω = 5.0ER , h̄δ = 1.0ER

48

2.6.3 Calibration of Raman and rf dressed states
To calibrate the rf and Raman coupling strengths, we take a similar approachto the 1-D lattice calibration: start in a pure spin state, for example mF = 0,
and turn the coupling on non-adiabatically to induce Rabi oscillations between the
coupled states. Then, during time-of-flight, apply a Stern-Gerlach gradient pulse
to separate the spin components and observe the fractional populations in different
spin states as a function of Rabi oscillation time.
Figure 12a,b shows example images obtained in time-of-flight when pulsing on
an rf coupling field for atoms in the F = 1 and F = 2 manifold, respectively. The
Stern–Gerlach gradient pulse separates the spin components along the horizontal
axis in the images. The fractional population in each state can then be obtainedby summing up the optical depth in each cloud and dividing by the total opticaldepth. Similarly, Figure 13a shows an example time-of-flight image obtained whenpulsing on a Raman coupling field on an F = 1 cloud initially in the mF = 0 spinstate. Here, the spin states are separated along the horizontal axis by the same
Stern–Gerlach pulse. In addition, the recoil momentum obtained when undergoinga Raman transition separates the different spin states along the vertical axis—
parallel to the Raman beams along ex . The direction of the Stern–Gerlach gradient
was chosen purposefully to be perpendicular to the Raman direction ex for easy
separation of the two effects.
These population oscillations can then be fit for coupling strength h̄Ω anddetuning h̄δ. Note that the quadratic Zeeman shift h̄ is set by the strength of
the bias field B0 and therefore often well known—it is not a fitting parameter.
The theoretical predictions are obtained by applying the time evolution operator
U = exp(−iHRaman/rf t/h̄) to an initial state Ψ in the appropriate basis. Figure 12cshows an example time series of rf pulsing in the F = 1 manifold, starting in the

49

(a)

(b)
0

mF = 1−1

mF = −2 −1

SG gradient

0

1

2

SG gradient

Fractional population
Fractional population

(c)
1.00

1.00

mF =+1
mF =0
mF =-1

0.75
0.50
0.25
0.00
0.0

0.5

1.0

(d)

1.5

2.0

2.5
mF =+2mF =+1
mF =0
mF =-1mF =-2

0.75
0.50
0.25
0.00
0.0

0.5

1.0

1.5

2.0

2.5

Rf pulse time [ms]Figure 12: Pulsing on rf coupling. (a) Example time-of-flight image during an
rf pulsing experiment in the F = 1 manifold. Spin states are separated via a
Stern–Gerlach pulse along the horizontal direction. (b) Example time-of-flight image
during an rf pusling experiment in the F = 2 manifold. Here, 5 spin components arepresent. (c) Pulsing experiment in the F = 1 manifold. Dots represent fractional
populations in different spin states measured from time-of-flight images, and lines
represent best-fit theory curves. Fitted parameters are h̄Ω = (0.863 ± 0.004)ER ,h̄δ = (−0.198 ± 0.007)ER . (d)Pulsing experiment in the F = 2 manifold. Dots
represent fractional populations in different spin states measured from time-of-flightimages, and lines represent best-fit theory curves. Fitted parameters are h̄Ω =(1.000 ± 0.002)ER , h̄δ = (−0.061 ± 0.001)ER . h̄ = 0.038ER for all panels.

50

mF = 0 state. The lines of best fit are overlayed on experimental data, extracting
fit parameters h̄Ω = (0.863 ± 0.004)ER and h̄δ = (−0.198 ± 0.007)ER . Figure 12dshows an example time series of rf pulsing in the F = 2 manifold, starting in the
mF = −2 state. Here, the extracted fit parameters were h̄Ω = (1.000 ± 0.002)ER
and h̄δ = (−0.061 ± 0.001)ER .
Figure 13b shows an example time series of Raman pulsing in the F = 1
manifold, starting in the mF = 0 state, with fitted parameters h̄Ω = (1.47 ± 0.01)ERand h̄δ = (0.004 ± 0.024)ER . Note that although the coupling strength is almost
double the rf coupling strength in Figure 12c, the contrast (peak to peak oscillationof the fractional population in, say, the mF = 0 state) is much lower in the Raman
data than in the rf. This is a direct consequence of the recoil momentum transfer,and can be understood by looking at the band structure. For rf, the coupled bands
at initial momentum kx = 0 are separated by the coupling strength, see Figure ??b.
For Raman, even at zero coupling strength, due to the shifting of the parabolas by
2kR , and kx = 0 the higher bands are h̄2 (2kR )2 /2m = 4ER separated from the lower
bands. Therefore, the energy difference is larger and the fraction in the excited band
will be lower, leading to lower contrast.

51

(a)
2kR

kx

0

mF = 1

−1

0

−2kR

SG gradient(b)
Fractional population

1.0
0.8
mF =+1
mF =0
mF =-1

0.6
0.4
0.2
0.0
0

20

40

60

80

100

120

140

Raman pulse time [µs]Figure 13: Pulsing on Raman coupling. (a) Example time-of-flight image during a
Raman pulsing experiment in the F = 1 manifold. A Stern–Gerlach pulse during
time-of-flight separates different spin components along the horizontal direction,and different momentum orders fly apart along the vertical direction. (b) Fractional
population in different spin states during a Raman pulsing experiment as a functionof time. Dots represent data and lines represent a best fit from theory. The fitted
parameters are h̄Ω = (1.47 ± 0.01)ER , h̄δ = (0.004 ± 0.024)ER . The quadratic
Zeeman shift was h̄ = 0.038ER .

52

Chapter 3: Ultracold Gases and the RbK apparatus

In this chapter we intoduce the basics of ultracold quantum gases. Whencooled to extremely low temperatures, bosonic atoms form Bose Einstein condensates, described in sec. 3.1. Fermionic atoms do not undergo a phase transition,
but gradually become degenerate, forming what’s known as a degenrate Fermi Gas,
described in sec. 3.2. We then give a basic overview of the rubidium-potassium(RbK) apparatus at NIST, on which the work described in this thesis was done, insec. 3.3. We detail some of the changes that have been made to the apparatus since
it was last documented.

3.1 Bose-Einstein condensation
In this section, we give some basic background on Bose-Einstein condensation,
relevant to 87 Rb atoms cooled in our apparatus.

3.1.1 Phase transition of a non-interacting Bose gas
Bose gases are characterized by the Bose-Einstein distribution giving the number of atoms n(Ej ) occupying each energy eigenstate Ej asn(Ej ) =1e(Ej −µ)/kB T − 1

,

(3.1)

where kB is the Boltzmann constant, T is the temperature in Kelvin, and µ is
the chemical potential. Assuming the total atom number N is fixed, the chemical
53

potential µ(T, N ) ensures that the total occupationPj n(Ej ) = N .

The Bose distribution leads to Bose-Einstein condensation, the collapse of amacroscopic fraction of the atoms into the ground state. This comes as a direct consequence of the Bose distribution’s characteristic −1 in the denominator. Consider
the occupation number n(Ej ). It must remain positive, as a negative occupation
number is unphysical. This implies the quantity e(Ej −µ)/kB T must remain greaterthan 1, or (Ej − µ)/kB T > 0 for all Ej . Therefore, µ ≤ E0 , where E0 is the groundstate energy.
Then, for a given temperature T , there is a maximum occupation number foreach excited state given by n(Ej ) =1. The only energy state whose occupaeEj /kB T −1

tion number is unbounded is the ground state, as n(E0 ) tends toward infinity as µ
tends towards 0. Therefore, as the temperature decreases, the maximum occupationof each excited state decreases until they can no longer support all N of the atoms.
The remaining atoms then have no choice but to collapse into the lowest energy
level and Bose condense.
We will show this quantitatively for the case of a 3-D harmonically trapped gasof non-interacting atoms, relavant to the experiments described in this thesis [76].
It is convenient to define the fugacity ζ = eµ/kB T , and re-write the Bose-Einstein
distribution for some eigenstate Ej asn(Ej ) =ζ
eEj /kB T − ζ

.

(3.2)

The harmonic oscillator potential can be written as
1
V (r) = m(ωx2 x2 + ωy2 y 2 + ωz2 z 2 ),
2

(3.3)

where ωx , ωy and ωz are the angular trapping frequencies along ex , ey , and ez . The

54

eigenenergies with this potential are1
1
1
E(jx , jy , jz ) = ( + jx )h̄ωx + ( + jy )h̄ωy + ( + jz )h̄ωz .
222
In order to find µ, we must find

P

(3.4)

jx ,jy ,jz n(E(jx , jy , jz )) and set it equal to

N . This task is greatly simplified by going to the continuum limit and finding thedensity of states. To do this, we neglect the zero-point energy (setting E0 = 0, theeffects of the zero-point energy are discussed in [77] section 2.5) and assume there
is on average one state per volume element h̄3 ωx ωy ωz . Then, the total number ofstates with energy less than or equal to some value  is given by the volume ofa prism made between points (jx , jy , jz ) = (0, 0, 0), (, 0, 0), (0, , 0) and (0, 0, ) in
units of the volume element:3.
6h̄3 ωx ωy ωz

(3.5)

d
2
G() = 3.
d3h̄ ωx ωy ωz(3.6)

G() =The density of states is given byg() =

Note that the occupation of the ground state is not included in this continuum
picture. We can therefore use it only to calculate the total number of atoms in allof the excites states:

Nex =

Z ∞
0

dg()n() =

Z ∞
0

d

2
ζ
(kB T )3=Li3 (ζ),3h̄ωx ωy ωz e/kB T − ζh̄3 ωx ωy ωz(3.7)

where Li3 (ζ) is the polylogarithm function1 . We define the mean trapping frequencyω̄ = (ωx ωy ωz )1/3 and the harmonic oscillator energy as h̄ω̄, with the thermal energy
1This calculation was done with Wolfram Alpha, not Russian algebra skills

55

in harmonic oscillator units τ = kB T /h̄ω̄, giving
Nex = τ 3 Li3 (ζ).

(3.8)

Finding the occupation number of the ground state from the Bose-Einstein
distribution
N0 =

ζ
,
1−ζ

(3.9)

we can then find the chemical potential, or equivalently the fugacity ζ, to satisfy

N = N0 + Nex .

(3.10)

This is a transcendental equation that can only be solved numerically. We present
an example of the solution in Figure 1. Here, we have calculated the fractional population in different harmonic oscillator energy levels for three different temperatures,using trapping frequencies ωx = ωy = ωz = 2π × 50 Hz, and atom number N = 106 .
For energies above the ground state (dots in the figure), we binned 50 energy levels
together, such that each dot represents the total fractional population in 50 adjacentlevels. This was obtained by integrating eqn. 3.7 from  − 25h̄ω̄ to  + 25h̄ω̄. The
stars represent the fractional population in just the ground state, calculated from
eqn. 3.9. Note that at temperature T = 255 nK (red), the ground state populationis consistent with a continuous extrapolation from the excited state populations andis almost zero. At lower temperatures, T = 180 nK (blue) the ground state population is in excess of any reasonable extrapolation from the excited state fractions,and at T = 80 nK (green) almost all the atoms are in the ground state.
The onset of Bose-Einstein condensation occurs at a critical temperature Tc .
This temperature is defined as the temperature at which the occupation number of
excited states is equal to the atom number, i.e. when the atoms have occupied all

56

1.0
T=255 nK
T=180 nK
T=80 nK

n()G()/N

0.8

0.6

0.4

0.2

0.0
0

200

400

600

800

1000

/h̄ω̄

Figure 1: Occupation of energy states of a 3-D harmonic oscillator. The trapping
frequencies are ωx = ωy = ωz = 2π × 50 Hz, and the atom number is N = 106 .
Dots represent the total fractional population in 50 ajacent energy levels, including
degeneracies. The stars represent the fractional population in just the ground state.
available excited states and any remaining atoms were forced to pile into the groundstate. Since the maximal occupation of the excited states will occur at µ = 0, theoccupation of the excited state is bounded from above by Nex (µ = 0), and thecritical temperature is defined by

N = Nex (µ = 0, T = Tc ) =(kB Tc )3Li3 (ζ = 1).
h̄3 ωx ωy ωz(3.11)

Using Li3 (1) ≈ 1.202, we obtain for a given atom number and trapping frequencies
Tc =1.202N 3h̄ ωx ωy ωz .
kB3

(3.12)For the parameters in Figure 1, Tc = 225 nK.
For temperatures below the critical temperature, the condensation fraction
fc —the fraction of atoms in the ground state—is directly related to the ratio of the57

Figure 2: Time-of-flight images of atoms. (a) Above the critical temperature - theatoms are thermally distirbuted. (b) Below the critical temperature - about half of
the atoms are condensed in the central peak. (c) Far below the critical temperature- almost all atoms are condensed in the central peak.
temperature to the critical temperature:N(kB T )3
fc = 1 −Li3 (ζ = 1) = 1 −
=1− 3
Nex
h̄ ωx ωy ωz



T
Tc

3
,

(3.13)

where in the last step we have plugged in the definition of the critical temperatureeqn. 3.12.
Figure 2 shows the progression towards condensation as the temperature of acloud of 87 Rb is decreased below Tc . The images are obtained via a time-of-flightmeasurement (see section 2.4.1), where the atoms are allowed to expand freely,
mapping the initial momentum to final position, imaged via absorption imaging(see section 2.4). The x and y axes represent momentum along x and y, while
the z axis represents the number of atoms per spatial bin. The z axis momentumis integrated over. Figure 2a shows a cloud above the condensation temperature- the momentum distribution is nearly gaussian, given by the Maxwell-Boltzmann
distribution. In fig. 2b, the temperature has been decreased below Tc , and about
half the atoms have collapsed into the ground state, producing a large peak in atom
number around zero momentum. In fig. 2c, the temperature has been decreased58

even further and almost all the atoms populate the central peak - the distribution
is no longer gaussian but a sharp peak around zero momentum.

3.1.2 Interacting Bose gas
In the previous section, we assumed there weres no interaction between theatoms other than those enforced by statistics. In this section, we will relax thisassumption somewhat and describe the condensed atomic state through its characteristic Gross-Pitaevskii equation.
Since condensation occurs at very low temperatures, and thus very low kinetic
energies, we can assume that any scattering processes between the atoms are s-wave
and can be described simply by a scattering length a (equivalent to assuming thatthe characteristic size of the atomic wavefunction, given by the thermal deBroigle
wavelength, is large compared to the scale of the interatomic potential). For 87 Rb,
relevant to experiments described in this thesis, the scattering length between two
atoms in the F = 2 hyperfine state is a = 95.44(7)a0 [78], where a0 = 5.29 × 10−11
m is the Bohr radius. The short-range interaction between two particles can beapproximated as a contact interaction with an effective strength U0 as (see [77]section 5.2.1):
U (r1 , r2 ) = U0 δ(r1 − r2 ) =4πh̄2 aδ(r1 − r2 ),m(3.14)

where m is the atomic mass and δ is the Dirac delta function. The full Hamiltonianof the many-body system is then

H=

X p2
i

i

2m

+ V (ri ) + U0

Xi<j

δ(ri − rj ),(3.15)

where i labels the particles, pi is the momentum, ri is the position, and V is theexternal potential.
We make the mean field approximation by assuming that no interactions be59

tween two atoms take them out of the ground state, and hence all atoms can beassumed to be in the same single particle wavefunction, making the overall wavefunction
Ψ(r1 , r2 , ...rN ) =NYφ(ri ),(3.16)iwhere φ is the single particle wavefunction. It is convenient to define the wave√
function of the condensed state, ψ(r) = N φ(r), making the normalization N =R
dr|ψ(r)|2 .
The energy of this wavefunction under the Hamiltonian above is given by
Z
E=


1
h̄2
2
2
4
|∇ψ(r)| + V (r)|ψ(r)| + U0 |ψ(r)| .
dr
2m
2


(3.17)

Given N particles, there are N (N − 1)/2 unique pairs of particles that can have apairwise interactions, approximately equal to N 2 /2 for large N . The N 2 is absorbed
into the definition of ψ, but the factor of 1/2 remains on the final interaction term.
The task of finding the condensate eigenstate reduces to minimizing this energyRunder the normalization constraint N = dr|ψ(r)|2 . This can be done by using themethod of Lagrange multipliers to minimize E − µN . Then, we can minimize thisquantity by finding the point where the derivative with respect to ψ and ψ ∗ is zero.
Taking the derivative with respect to ψ ∗ we obtain

−

h̄2 2
∇ ψ(r) + V (r)ψ(r) + U0 |ψ(r)|2 ψ(r) = µψ(r),2m(3.18)

which is the Gross-Pitaevskii equation. This is a non-linear equation that generally
needs to be solved numerically.
There is another approximation that can be made in cases where the atomic
density is high enough that the interaction energy is significantly larger than the
kinetic energy. Then, the kinetic term in the Hamiltonian can be neglected. This is60

called the Thomas-Fermi approximation. In this approximation, the wavefunction
is given simply by|ψ(r)|2 =µ − V (r).
U0

(3.19)

Here, the probability density simply takes the shape of the inverted potential inwhich the atoms are held. In the case of a harmonically trapped BEC, it is shaped
like an inverted parabola. The Thomas-Fermi radius, i.e. the extent of the particle
wavefunction, is the point where the probability density goes to zero: µ − V (r0 ) = 0.
For a harmonic trap, along any direction, this is given by r02 = 2µ/mω 2 .
Figure 3a shows an absorption image of a small fraction of atoms in a BEC
in situ (see section 2.4.1), meaning as it is in the trap - without expanding in timeof-flight. The x and y axes represent position, while color represents the atom
number. Figure 3b shows the atom number integrated over the y-axis in blue. The
red dashed lines represent the best fit line to a Thomas-Fermi distribution, here
an inverted parabola. The black dashed lines represent the best fit of a Gaussian
to the atomic distribution. The Thomas-Fermi distribution matches the atomic
distribution more closely in the center where the density is high, but the Gaussian
distribution does a better job at the tails of the distribution. This is due to thepresence of some fraction of uncondensed atoms, which are well approximated by a
Maxwell-Boltzmann distribution.

3.2 Degenerate Fermi Gas
In this section, we give some basic background on degenerate Fermi gases,
relevant to 40 K atoms cooled in our apparatus.

61y position(a)x position

Integrated atom number [arb. u.](b)

x position

Figure 3: In situ measurement of Bose condensed atoms. (a) Absorption image
taken of ≈ 1% of the cloud. The x and y axes represent x and y position, whilecolor represents the atom number. (b) The blue line represents atom number as afunction of position along the x axis, integrated over the y axis. The black dashedline represents the best fit of a Gaussian function to the atomic distribution. The
dashed red line represents the best fit of a Thomas-Fermi profile (integrated over
the other two dimensions) to the atomic distribution.

62

3.2.1 Fermi statistics and the onset of degeneracy
The occupation of different energy levels Ej by Fermions is given by the FermiDirac distribution:n(Ej ) =1e(Ej −µ)/kB T + 1.

(3.20)The difference from the Bose-Einstein distribution is simply the sign of the 1 in
the denominator. This has important implications, however. First, since ex varies
between 0 and ∞, the occupation n(Ej ) varies between 1 and 0 - a consequence of
the Pauli exclusion principle. Second, as the temperature T tends towards 0, there
become two distinct cases: Ej − µ > 0 and Ej − µ < 0. If Ej − µ > 0, e(Ej −µ)/kB T
tends towards ∞, and n(Ej ) tends towards 0. If Ej − µ < 0, e(Ej −µ)/kB T tends
towards 0, and n(Ej ) tends towards 1. Therefore, at T = 0, the energy states below
the chemical potential µ are maximally occupied (with probability 1) and the energy
states above the chemical potential are unoccupied.
We can use this to determine the chemical potential at T = 0 by constraining
the total atom number:N=Xn(Ej ) =jX

1.

(3.21)Ej <µAgain, we take the common example of the 3-D harmonic trap. Then the task
reduces to simply finding the number of energy levels at or below a certain energyµ. This is given by eqn. 3.5. From this, we find the chemical potential at zeroenergy, which is known as the Fermi energy EF , as
EF = (6N )1/3 h̄ω̄,(3.22)

where ω̄ = (ωx ωy ωz )1/3 is the geometric mean of the three trapping frequencies.

63

From the Fermi energy, we can define the associated Fermi temperature TF as

TF =(6N )1/3 h̄ω̄,kB

(3.23)

p
2mEF ,(3.24)and the Fermi momentum h̄kF ash̄kF =

where m is the mass of the Fermion.
For higher temperatures, we can solve for the chemical potential, or the fugacity ζ, by integrating the Fermi-Dirac distribution weighted by the density of states(eqn. 3.6) to obtain

N=

Z ∞
0

2
ζ
(kB T )3=−Li3 (−ζ),2h̄3 ω̄ 3 e/kB T + ζh̄3 ω̄ 3(3.25)

where Li3 is again the polylogarithm function. Again, this is a transcendental equation that can be solved numerically. However, in contrast to the BEC case, we do
not have to consider the ground state occupation separately, as it is bounded by 1
like every other state.
We show an example of the occupation distribution for different temperaturesin Figure 4. Here, we have used the same parameter values as for the BEC case:N = 106 and ωx = ωy = ωz = 2π × 50 Hz. The Fermi temperature for theseparameters is TF = 436 nK. For illustrative purposes, we plot n(), unweighted by
the density of states g(). At zero temperature (red line in the figure), only states
below the Fermi energy are occupied. At higher temperatures, the distribution issmoothed out (green and orange lines) until at the Fermi temperature there is almost
no significance to the Fermi energy.
In contrast with Bose-Einstein condensation, the transition to a Degenerate

64

1.0

T = TFT = 0.2TF
T = 0.05TF

0.8

T =0

n()

0.6
0.4
0.2
0.0
0

100

200

300

400

500

600

/h̄ω̄

Figure 4: Occupation number as a function of energy for a Fermi gas of N = 106
atoms in a 3-D harmonic oscillator with frequencies ωx = ωy = ωz = 2π × 50 Hz.
The Fermi temperature for these parameters is TF = 436 nK.
Fermi Gas (DFG) is not a phase transition, and there is no absolute measure of
the onset of degeneracy. Instead, a Fermi gas can be considered degenerate when
the occupation function n() differs significantly from that of a thermal gas. Thisoccurs when the temperature is of order 0.2TF .

3.2.2 Interactions and Feshbach resonances
Although the magnitude of the contact interaction U0 for DFGs is not intrinsically different from that of BECs, there are two key differences. First, thePauli exclusion principle forbids s-wave interactions between atoms of the same
spin. Higher partial wave interactions are ’frozen out’ at low temperatures, when
the impact parameter of the collision becomes larger than the effective cross sectionof interactions (see [79], sec. 2.1.2). Therefore, in order to observe interactions, andindeed to cool the gas to degeneracy, another species needs to be present so that
intraspecies s-wave interactions can occur. This can be a different atomic species
or a different spin state of the same atom.
65

Second, the densities of standard DFGs (≈ 1012 atoms/cm3 ) are much lower
than that of BECs (≈ 1014 atoms/cm3 ). Since the likelyhood of two-body collisions
is proportional to the atomic density ρ2 , this leads to a much smaller effect ofinteractions in DFGs.
A widely used technique for enhancing interaction effects in DFGs (and to amore limited extent, BECs) is Feshbach resonances [80–84]. A Feshbach resonanceoccurs between two species (either atomic species or spin species of the same atom)when the open channel, i.e. the two particles independently in their external potential, energetically approaches a closed channel, i.e. a bound molecular state of thetwo species, shown schematically in Figure 5a.
Generally, the atoms in an open channel are energetically sensitive to a back~ via the hyperfine interaction HB = −~µ · B,
~ where µ is
ground magnetic field B
the magnetic dipole moment. Tuning the magnetic field should therefore tune theenergy of the open channel with respect to the closed channel. The molecular boundstate may also have an overall magnetic moment, but it is generally not identical tothat of the two atoms in the open channel, and therefore varies differently with thebackground field. Figure 5b shows an example where the bound state has zero magnetic moment. Here, the energy of both the closed and open channel as a function~ is plotted in the vicinity of a Feshbach resonance.
of background magnetic field B
The resonance occurs at a field magnitude B0 where the energies of the two channels
coincide.
Assuming there is at least infinitesmal coupling between the closed and open
channels, as the energies of the two channels approach each other the perturbative
correction term to the energy grows and the interaction between the atoms is affected. This is most easily seen in the s-wave case through changes in the scattering
length a. In the case where there are no inelastic two-body channels, such as forthe 40 K resonance discussed in this thesis, the interatomic scattering length as a

66

Energy(a)(b)

open channel
closed channel

B0
Interatomic distance

Magnetic field

Figure 5: Schematic of a Feshbach resonance. (a) Pictoral representation of energy
as a function of interatomic distance for an open channel (red) and closed channel(blue). (b) Energy as a function of background magnetic field B for the closed (blue)
and open (red) channels. The energies coincide at the Feshbach resonance point B0 .
function of background field is given by [40]a(B) = abg 1 −∆B − B0,(3.26)

where abg is the background scattering length, ∆ is the width of the resonance, and
B0 is the field value at which the resonance occurs. The scattering length diverges
at the resonance.
The tunability of interactions provided by Feshbach resonances has allowed
for creation of molecular Bose-Einstein condensates from Fermi gases [42, 43, 45]
as well as observation of the phase transition from the Bardeen-Cooper-Schrieffer(BCS) superconduting regime to the BEC regime at sufficiently low temperatures[44, 46, 85, 86].

3.3 RbK apparatus
The rubidium-potassium (RbK) appartus at NIST Gaithersburg has been previously detailed in [73, 87, 88]. In this thesis, we will give a brief overview of the67

apparatus and how it is used to produce BECs of 87 Rb and DFGs of 40 K, andonly give detailed documentation for those parts of the apparatus that differ fromprevious works.
A photograph of the main experiment is shown in Figure 6. This is mounted
on an optical table, with the science chamber elevated above the surface of thetable. The atoms start at the ovens (off to the right, not in the photograph) andtravel down the Zeeman slower until they are trapped in the science chamber. The
optical dipole trap laser, as well as the 1-D optical lattice laser, are located on the
optical table and coupled into optical fibers, which are output on the main floor ofbreadboard before being sent towards the atoms. All other lasers are located onother optical table and brought over to the experiment table via optical fibers.

3.3.1 Laser beams
Figure 7 details the beam paths of the light going through the atoms. Figure
7a shows a side view of the apparatus. The up and down going MOT cooling beams
are shown in red, reaching the atoms when the flipper mirrors Mtop and Mbottom areflipped in. The down going probe beam, used for imaging along the x − y axis bothin situ and in time-of-flight, is shown in solid blue. The probe beam is split via apolarizing beam splitter cube to allow for both in situ and time-of-flight imaging of
the same cloud, shown in the inset in fig. 7b and described in greater detail in sec.
3.3.4. The dashed blue line represents the upward going probe beam introduced foralignment purposes, described in greater detail in sec. 3.3.4. The kinematic basemirror (green in the figure) is removable, and only inserted when the alignment
beam is in use.
Figure 7b show’s a bird’s eye view of the apparatus, with optics on the main
floor breadboard. The slower cooling (solid dark blue) and slower repump (dasheddark blue) are coming in from the left to slow the atoms as they are moving through68

Figure 6: Photograph of RbK apparatus at NIST Gaithersburg. The main sciencechamber is at the center, hidden behind optics and coils. The Zeeman slower connects the atomic ovens (not shown) to the chamber. There are several levels ofbreadboards on which optics are mounted, labelled here as basement (surface of
optical table), main floor, balcony and attic.

69

or
de
r

t

-1 s

Mtop

Zeeman slower

M2

Basement

M1

A

XY
in situ
Flea3

B







M4

floor

Mbottom

Main



C

M3





r
de
or

alignment
Flea3

th

Balcony

(b)
0

(a)
Attic

inset

XZ Flea3

Main floor

MOT cooling beams
MOT repump beams
Slower cooling beam
Slower repump beam
Optical dipole trap beams
1064 nm 1-D Lattice beam
Raman beams
Imaging beams
Alignment imaging beam
Green plug beam
767 nm 1-D lattice beam

in

inset

XY TOFPIXIStou

MirrorFlipper mirror
Dichroic mirror
Acousto-optic modulator
Optical fiber
CameraPBS cube
Quarter waveplate
Kinematic base mirror

Figure 7: Schematic of RbK apparatus. (a) Side view of apparatus. Only beams
propagating along the ez direction through the atoms are pictured. (b) Top view ofapparatus. Only beams propagating along the x-y plane are shown. Schematic is
not to scale and the angles are approximate.

70

the Zeeman slower. The remaining four MOT cooling beams, coming from four
opposing directions, are shown in red. They reach the atoms when their flipper
mirrrors, M 1 − 4, are flipped in. All six flipper mirrors are computer controlled
by the same digital channel, so they can be flipped in and out together. Only thebeams going in through mirrors M 1 and M 2 are accompanied by MOT repump light,
dashed red lines. The repump light for 87 Rb (both MOT and slower) comes from a
Toptica DL-100 laser. The cooling light (MOT, slower) as well as imaging beams,
come from a Toptica TA-100 tapered amplifier system. Both lasers are frequency
referenced to a master laser, a toptica DL-pro, which is frequency stabilized to a87

Rb atomic transition via saturated absorption spectroscopy (see section 3.3.4).
The optical dipole trap beams (solid green) come from the same 1064 nm laser(IPG YDL-30-LP), and are split via an acousto-optic modulator into two orders,which enter from opposite directions and intersect each other at approximately a
90 degree angle, providing confinement along all three axes. There is a 1-D opticallattice beam (dashed green), also 1064 nm (IPG YAR-10K-1064-LP-SF, seeded bya pick off from an NP Photonics seed laser), sent in past the M 2 mirror and retroreflected on the opposite end of the chamber to form a standing wave pattern. This
was also used for experiments in Chapters 6 and 7. There is also another imaging
beam, imaging the atoms along the x-z plane, going to a Flea3 camera.
There are three Raman beams (solid magenta): Raman A, entering past theflipped-out M 2 mirror, Raman B, at 90 degrees to Raman A entering past the M 1
mirror, and Raman C, counter-propagating with Raman A and entering past theM 4 mirror. The Raman beams are derived from a tunable Coherent MBR-110
Ti:Sapphire laser seeded by a Coherent Verdi V-10 laser. For experiments describedin Chapters 6 and 7, we used the Raman A and C beams.
When 40 K atoms are in use, the slower cooling, slower repump, MOT cooling,
MOT repump and imaging beams are all a combination of frequencies for both 87 Rb

71

Bias X-Y

Quad

Bias X+Y

Rf
Slower & Reverse

Gradient dBxy/dzGradient dBz/dz

Bias Z

Figure 8: Schematic of magnetic coils on the RbK apparatus. The black wire
frame represents the main experiment chamber, with the Zeeman slower off to theright. The Zeeman slower and reverse coils are wound around the Zeeman slower
in varying spatial frequency (magenta). The quad (orange), gradient cancellationdBz /dz (bright green) and bias Z (brown) are all pairs of identical coils on the top
and bottom of the apparatus. Bias X-Y coils (red) are a pair of identical coils aroundthe axes of the M 1 and M 3 mirrors, and the bias X+Y (dark green) are a pair ofidentical coils around the axes of the M 2 and M 4 mirrors. The rf coils (blue) area pair of circular coils on top of the experimental chamber, spaced enough to allow
the top MOT beam through. The gradient cancellation coils dBxy /dz (cyan) are
four square coils on top and bottom of the experiment along the X+Y axis.
and 40 K, fiber coupled before they were sent to the main experiment table. Boththe 40 K cooling and repump lasers are Toptica TA-pro systems, with the repump
laser frequency stabilized to the 40 K atomic transition. In addition, a green plug
beam (solid dark green in 7b) is used (see section 3.3.5), derived from a Coherent
Verdi V-5 laser. For 40 K experiments detailed in Chapter 5, we used a near resonant
retroreflected optical lattice beam, shown in dashed dark magenta entering past theM 4 flipper mirror, coming out past the M 2 mirror before getting retro-reflected.

3.3.2 Magnetic coils
Figure 8 is a schematic depiction of all the coils used to produce magnetic fields
on the RbK apparatus. The quad coils (orange in the figure) are a large pair of coils
used to produce a quadrupole field for the MOT. The top and bottom coils are72

connected through four IGBT switches, forming an h-bridge (see Figure 4.9 in [88]).
This allows switching between two configurations: anti-Helmholtz and Helmholtz.
In anti-Helmholtz configuration, the top and bottom coils conduct current in opposite directions, producing a quadrupole field gradient at the center. This is theconfiguration used for the MOT, rf evaporation, and producing a Stern-Gerlach gradient for spin resolved imaging. In Helmholtz configuration, the two coils conduct
current in the same direction, producing a strong bias field along the ez direction.
This was used to get close to the Feschbach resonance in the experiment detailed in
Chapter 5.
There are three pairs of bias coils, used to cancel constant background fields orprovide field offsets along the three axes. All three are in Helmholtz configuration.
The bias Z coils (brown) are on top and bottom of the experiment and provide aconstant Bz field at the center. The bias X+Y coils (dark green) are vertical on
two opposite sides of the apparatus along the ex +ey directions, and the bias X-Y(red) are on the other two opposing sides along the ex -ey directions. There are also
two sets of gradient cancellation coils available, although they are not subject tofeedback loops or computer control. The first is another pair of coils on top andbottom of the apparatus (bright green), connected in anti-Helmholtz configuration
to produce a small gradient dBz /dz. The second is four square coils mounted aboveand below each bias X+Y coil (cyan). Both vertically stacked pairs of coils arewound in Helmholtz configuration, and the two pairs are in series, providing a small
gradient dBxy /dz at the atoms.

3.3.3 Procedure for making a BEC
We begin with the atoms heated in the ovens, sent through a thin nozzle
allowing only those atoms with a velocity towards the science chamber to enter.
They are cooled via a Zeeman slower and captured in a Magneto-Optical trap (MOT)73

in the science chamber. During this step, the Zeeman slower is on, with both thecoils and the slower cooling and repump lights on. These beams (dark blue in Figure7b) are −148.5 MHz red detuned from the |F = 2i to |F 0 = 3i transition for coolingand −8.2 MHz red detuned from the |F = 1i to |F 0 = 1i transition for cooling (forthis, the cooling TA is locked to a 133 MHz beat note offset from the master laser).
At the same time, the flipper mirrors M 1 − 4,Mbottom ,Mtop are flipped in and the
MOT cooling and repump beams (red in Figure 7) are on. The quad coils are on inanti-Helmholtz configuration with 25 A of current running through them, producingz
a field gradient of dB≈ 13 Gauss/cm. This step can be set to take anywhere fromdz

≈ 0.7 s to ≈ 5 s depending on how many atoms are needed.
Next is the optical molasses step, during which sub-Doppler cooling of theatoms occurs. For this step, the Zeeman coils and slower lights are turned off.
The quad coil current is also switched off, leaving just the MOT cooling light andonly leakage MOT repump light. The MOT cooling light is set to 20.6 MHz below
the the |F = 2i to |F 0 = 3i transition (120 MHz beat-note command). It is thenlinearly ramped in 19 ms down to a red detuning of 90.2 MHz (50 MHz beat-note
command). Since the repump light is all but off in this step, the atoms are also
depumped into the F = 1 manifold. Then, the atoms are optically pumped into
the |F = 1, mF = −1i state to make them trappable by the quadrupole field. Thisis done by turning on the slower repump beam (dashed dark blue in fig. 7b) 1 ms.
Then, the XZ imaging beam (blue in fig. 7b) is briefly turned on to get rid of any
remaining F = 2 atoms.
Next, we compress the atoms and perform forced rf evaporation. To compress,
the quad coils are first turned on to 130 A. After holding for 20 ms, we sweep thecurrent linearly to 250 A in 200 ms. The forced rf evaporation is then performed byturning on the rf coupling field and sweeping the frequency from 20 MHz to 4 MHz
in 4 s to couple the highest energy atoms from |F = 1, mF = −1i to |F = 1, mF = 0i74and allow them to escape the trap. The slow ramp is designed to allow the system
to rethermalize through collisions as the hottest atoms are ejected. During rf evaporation, the crossed optical dipole trap (ODT) is on at an initial command power of
2.0 V and initial split (command to AOM controlling the power split between thetwo crossing beams, shown in fig. 7b) of 0.01 V. This allows any atoms that arecold enough to see the optical trap to be captured by it.
Then, the atoms are decompressed and loaded into the ODT. The quad current
is ramped down to 60 A exponentially with a time constant of τ = 1.5 s in 3 s. Thisis the quad current at which the atoms are only barely suspended against gravity by
the quadrupole trap. At the same time, the bias Z current is ramped down from 10
A to 8 A, lowering the center of the quadrupole trap to the ODT. Then, the atoms
are further evaporated in the ODT. This is done over the course of 5 s, ramping
down the depth of the ODT and allowing the hottest atoms to escape. During thisstep, the ODT power is ramped exponentially from the initial command of 2.0 V
to a final command of 0.4 V, while the split command is ramped up linearly from0.01 V to 0.65 V, effectively turning on the crossing −1st order beam. It is during
this evaporation step that the atoms are cooled below the critical temperature andBose condense.
Finally, the quad current is ramped exponentially to 0 A in 5 s, leaving theatoms optically trapped. Then any desired experiment can be performed on the
BEC. For daily checks of the BEC, no experiments are performed and the atoms
are released from the trap and allowed to expand in time-of-flight for 16.2 ms before
being absorption imaged in the XY plane by the PIXIS imaging camera, pictured
in fig. 7b.

75

Figure 9: New master board layout.

3.3.4 Changes to apparatus for Rubidium
In this section, we describe a few of the changes that were made to the apparatus since the writing of Lauren Aycock’s thesis [88]. This is not an exhaustive list,
but rather the most notable changes to the main setup affecting BEC production
or adding capabilities to the apparatus.

3.3.4.1 Master laser setup
In 2014, the master laser board was replaced by a new version, with a new laserthat was not dying. The laser was a Toptica DL-Pro, and it output approximately
80 mW, allowing for an extra beam arm that was used to imprint a phase shift on
half the cloud to produce a soliton in the soliton project (see Appendix A) [89].
The layout of the new master laser board is shown in Figure 9. The board
contains saturated absorption spectroscopy to lock the laser frequency relative tothe atomic resonance, an output port to send to the cooling and repump lasers for76

beat note locking and monitoring purposes, and an ouput port (partially dismantled
in the figure) for any use if necessary. There are two mirrors directly in at the laser
output, for easy re-alignment of the whole board if a diode is changed or otherinternal laser adjustments are made.
After hitting the two mirrors, the beam is used for saturated absorption spectroscopy, as described in section 8.3 of [90]. It is first split into two branches by apolarizing beam splitter cube (PBS). The power split between the branches can be
adjusted by a half wavpelate (HWP) preceding the PBS. One branch is used as theprobe beam in saturated absorption (red in the figure). This branch goes throughthe Rb vapor cell and is then sent to a photodetector. The photodetector reading
is sent to a scope for monitoring and to a lock-in amplifier, used to derive the error signal for frequency locking. The rest of the beam (white) then hits another
PBS cube (again preceded by a HWP to control the power split), splitting off thepump beam (green) for saturated absorption spectroscopy. This beam is sent to an
acousto-optic modulator (AOM). This AOM’s frequency is modulated by the lock-in
amplifier, with modulation frequency of 100.0 kHz, amplitude of 0.356 V and phaseshift (between the signal and photodetector response) of −115.44 degrees. The 0thorder out of the AOM is blocked by a razor blade. The 1st order is retro-reflected
in a cat’s eye configuration [91]. Note that after retro-reflection, the second passthrough the AOM also produces a 0th and 1st order beam. This 0th order beam
is not blocked, but continues along the 1st order (pump beam) path at a slightly
different angle - care must be taken to avoid aligning this order to counter-propagate
with the probe. The double-passed beam (1st order in both directions, frequency
shifted up twice) is then used as the pump and sent through the Rb vapor cell inthe opposite direction of the probe beam.
The rest of the laser beam (white) then goes through an optical isolator, toavoid any subsequent reflections off of fiber tips or anything else from disturbing the77saturated absorption frequency lock. Then, the beam hits another HWP followed
by a PBS, splitting off the former soliton beam (yellow). In the figure, the soliton
beam launch has been partially dismantled, but can be revived at any moment if
needed. The beam was double-passed through an AOM in a cat’s eye configuration
before being sent into a fiber launch. The rest of the laser power (white) is sent intoa fiber that is connected to a fiber splitter box, providing light for beat note lockingof the 87 Rb repump and cooling lasers as well as for monitoring the master laser ona wavemeter and Faby-Perot cavity.

3.3.4.2 Alignment imaging path
In 2016, there was a plan to carry out a project to create a 1D magnetic
lattice whose topological character flips in the middle of the lattice, predicted tosupport localized states at the boundary. For this, two Raman beams needed to
be overlapping, with one having a sharp phase change centered on the atoms. Thisrequired precise control of the beam phase as well as precise alignment of the beam
center to the atoms. For optimal resolution, the Raman beams were to be sent
upwards through the XY imaging system. To aid in alignment, it was decided thatan additional imaging path that could detect these Raman beams directly be built.
This alignment imaging path was implemented by Dr. Hsin-I Lu and is outlined inthis section. All figures in this section were made by Hsin-I Lu.
The bottom part of the setup, with optics on the basement level of the experiment optical table, is diagrammed in Figure 10. One of the Raman beams,
here called RamanC, first hit a spatial light modulator to imprint a phase jump.
RamanC is then combined with a second beam, here called RamanD, on a PBS.
Both overlapped beams are sent backwards through the XY imaging system and up
towards the atoms. The dichroic filter allowed the Raman beams (λ ≈ 790 nm) tobe reflected while the imaging light (λ ≈ 780.24 nm) passed through to the imaging
78

Figure 10: Schematic of the bottom half of the alignment imaging system, as wellas the Raman beam set-up.
cameras. The beam for alignment imaging is sent up to the atoms backwards along
the XY imaging beam path. It entered the path via a mirror on a kinematic mount,which could be removed to allow imaging through the usual camera focused in situ
in XY (see Figure 7).
To set up an alignment imaging system going upwards through the chamber,it was necessary to insert a new imaging lens above the chamber, as close to theatoms as possible to maximize the numerical aperture. A schematic of this lens isshown in Figure 11, it is a 25 mm diameter f = 50 mm aspheric lens, a Thorlabs
AL2550. This lens was placed above the printed circuit board (PCB) that contains
the top bias Z coil and rf coils. It was held in a custom made mount.
The top part of the imaging system is diagrammed in Figure 12. Here, thealignment probe beam is light blue. From the atoms, the alignment probe beam hits

79

Figure 11: Schematic of the new imaging lens placed on the bucket window on topof the experimental chamber.
the new imaging lens (labelled L1 ). Then, if the top MOT mirror is flipped out, ithits another additional flipper mirror (here M2 ) before reaching a second lens andhitting an additional Flea3 camera, on the ’balcony’ level of the experiment. Since
the imaging lens on top of the chamber cannot be taken in and out, it was necessary
to correct the down-going probe beam and down-going MOT beam, ensuring theyretain their size at the atoms. For the down-going probe beam, this was done byinstalling a telescope to expand the beam by a factor of 4 (L6 and L5 in the figure)and then add a lens (L4 in the figure) that forms a telescope with the imaing lensto reduce the beam back down by a factor of 4. The MOT beam was corrected byswitching the focusing lens directly after the MOT fiber to an f = 100 mm lens (L7 ),effectively expanding the beam by a factor of ≈ 2/3, and then adding an f = 75mm lens (L2 ) to form a telescope with the imaging lens, reducing the beam size bya factor of 2/3. The adjusted beams were aligned and the experiment functioned
properly. The alignment imaging system was also aligned with great effort.

80Figure 12: Schematic of the top half of the alignment imaging system, including
correction optics for the MOT beam and probe beams to undo the effects of the
new imaging lens.

813.3.4.3 FPGA quad servo
In 2014, the servo board that was used to stabilize the current in the quad coils
had failed, and rather than replacing it with an identical one a new FPGA-based
servo board design by Ryan Price was implemented. This design is described indetail in Appendix B of [92]. Here, we include a brief description of the design andimplementation details for the quad servo at RbK.
The basic operation of the board is as follows. There are four SMA connections for input signals. These signals go to a 16 bit, 8 channel analog to digital converter ADAS3022BCPZ, then through a digital isolator Si8662BC-B-IS1, to
the FPGA board. The outputs of the FPGA are sent through a similar digitalisolator, Si8660BC-B-IS1 and into a 16 bit, 4 channel digital to analog converter
AD5686R. Then, each of the four outputs is sent through a programmable gain
amplifier AD8250ARMZ. The gain setting signal is derived from the FPGA board ,by way of a serial shift register CD74HC4094. There are four SMA connections for
outputs of each of the four amplifiers.
All of these devices are powered from a +/-18 supply voltage by way of three
voltage regulators, LM2940C KTT 3 for 5V regulation, LM2990 KTT 3 for -15V
regulation, and LM2940CSX KTT 3 for +15V regulation. The FPGA communication is set up through USB. There is a USB input port that connects to a USB chip
FT232HL. The USB chip requires an EEPROM, in this case 93LC56BT is used.
Clock timing for both the FPGA and the USB chip is provided by CTX292-LVCT.
There is also a buffer SN2564BCT25244 available for amplifying digital FPGA outputs. The digital side of the board is also powered from a separate 5V supply by
way of a 3.3V regulator LM1085 KTT 3.
For quad coil current stabilization, the board receives a computer command,in volts, through one of it’s input ports, and a Hall probe reading, in amps, through82the other one. The Hall probe current sent to the servo is dropped across a stack of
two 51 Ohm resistors (located inside the servo box) for a total measured resistanceof 25 Ohms. The difference between the two inputs, in volts, is interpreted as theerror signal by the FPGA board. The control output of the board is then sent to
the gate input of a MOSFET bank. The power supply powering the quad coils isconnected to this MOSFET bank and then to the coils in series. Controlling thegate voltage of the MOSFETs controls the resistance the power supply sees and thus
the current it outputs (in voltage limited mode).
The optimal PID parameters, set via software and programed in the FPGA
board, have been found at a gain factor of −5 and integrator bandwidth of 400 Hz.
The resulting turn-on curve is shown in pink in Figure 13 a. For this curve, thecomputer command was hopped to 100 A and the resulting current as detected by
the Hall probe was observed. The turn-on curve using the preceding servo board isshown in gray.
The board is also equipped with a digital TTL input (on the back of theboard). When this digital input is high, the output control voltage is imediately
railed to its lower bound. To be compatible with the MOSFETs used, the upper andlower bounds of the control output are set to 5 and 3 V respectively. The turn-off
curve when this TTL switch is activated is shown in Figure 13 b. The timescale islikely limited by eddie currents in the chamber.
We calibrated the resulting current (as measured by the Hall probe) for different command voltages. This is shown in Figure 14. Both the schematic for the servo
board and the Box control program to talk to the FPGA are in the shared google
drive under ’RbK/Lab Notebook/Electonics/FPGA Quad Servo (From Ryan)’. Whenconnected, the box control program detects ’RbK Quad Servo’ in its device list. In
the past, several cycles of plugging and unplugging as well as turning on and off
have been necessary for the connection to be successfully made.

83

Figure 13: Turn on and off curves of the quad coils. (a) Turn-on, when computercommand is jumped from 0 to 100 A . The FPGA-based servo response is in pink.
The previous hardwired servo is in gray. The other curves are extraneous. (b).
Turn-off with the FPGA-based servo when the TTL switch is engaged.

84

Figure 14: Output current as a function of computer command voltage for the
FPGA-based servo implementation.

85

Figure 15: Schematic of beam shaping optics in the path of the 0th order ODTbeam, after the split AOM. The new lens, on a removable mount, is cylindrical,
shaping the beam along the horizontal axis only.

3.3.4.4 ODT beam shaping
In 2015, two projects were being carried out on the apparatus at the same time:
the soliton project (see Appendix A) [89] and the synthetic dimensions project detailed in Chapter 7. The soliton project used an elongated BEC, requiring the dipole
trap to be highly elongated along one direction, here ex + ey . For this, only the 0thorder of the ODT was used and it was made very tight both along the horizontal andvertical directions: with 42 and 55 µm waists, respectively. The synthetic dimensions projects suffered from momentum changing collisions, and therefore needed
the cloud to be as dilute as possible. For this, the 0th order ODT beam still needed
to be tight in the vertical direction to suspend against gravity, but needed to be as
wide as possible (while still retaining a detectable atom number) in the horizontal.
Therefore, an extra cylindrical lens on a removable, rotatable mount was added in
the beam path to switch between the two configurations.
The location of this new lens is detailed in Figure 15. Without this lens,
the beam was sent through a telescope (the f = −10 and f = 15 cm lenses before
reaching an f = 25 cm focusing lens, placed 25 cm away the center of the chamber tofocus the beam at the atoms. The beam waist as a function of propagation distance86

along this beam path is shown in blue in Figure 16 a. This graph was made by Dr.
Ian Spielman from a python calculation of Gaussian beam optics. According to thecalculation, the beam is focused down to a 45 µm waist at the atoms located at adisplacement of 1400 mm.
The additional lens used was an f = 40 cm cylindrical lens, rotated in its
rotating mount to focus the beam slightly in the horizontal direction. The effectof this lens on the horizontal beam waist along its path was calculated and plotted(again by Ian using his code) in Figure 16 b. This plot was made for a f = 75 cm
lens instead of f = 40 cm, but the qualitative effect is the same. As seen in the
figure, the waist of the beam is not significantly impacted by the addition of thelens, but the focus is shifted away from the atoms, resulting in a larger waist at
the atoms. The horizontal beam waist at the atoms with the f = 40 cm lens asmeasured by a beam profiler camera was 115 µm. This was the configuration usedin the experiments described in Chapter 7.

3.3.5 Procedure for making a DFG
To make a degenerate Fermi gas of 40 K, we followed a similar cooling procedure
as for making a BEC, with some key differences. First, as mentioned in sec. 3.2.2,
due to the Pauli exclusion principle, spin polarized 40 K atoms cannot undergo s-wave
collisions, and therefore below a certain temperature have no method to thermalizeon their own and cannot be evaporatively cooled. To overcome this problem, we
cooled a mixture of 87 Rb and 40 K, effectively using 87 Rb as a collisional bath to allow
the Fermions to thermalize. Second, 40 K is slightly below half the mass of 87 Rb.
This leads to a larger magnitude of transverse velocity for 40 K atoms in the Zeeman
slower, leading to a larger fraction of atoms missing the capture region of the MOT.
To mitigate this issue, we utilized transverse cooling of 40 K right before the Zeeman
slower. This consisted of two pairs of counter-propagating beams along the ez +ey
87Figure 16: Beam waist as a function of propagation distance as calculated by Ian
Spielman’s code. Graphs also made by Ian Spielman. The atoms are at displacement= 140 cm. (a) Without additional lens. Blue line represents horizontal beam waist.
(b) With an additional f = 75 cm lens 12 cm in before the next optic.

88and ez -ey directions, performing Doppler cooling in the directions perpendicular to
propagation (not shown in fig. 7b). The lower mass of 40 K, as well as the larger
number of available spin states, also leads to a larger number of Majorana losses near
the center of the MOT: spin flips that take the atoms out of the trappable states
because they are moving too fast to adiabatically follow the changing magnetic field
direction [93–95]. To mitigate this issue, for cooling 40 K the center of the quadrupole
trap was plugged by a tightly focused green (repulsive) laser beam (dark green in
fig. 7b).
First, 40 K atoms starting at the oven were cooled via a Zeeman slower andtransverse cooling and captured in a MOT for 7 s. Then, both 40 K and 87 Rb atoms
were slowed and MOT loaded for 1.5 s. The subsequent optical molasses step wasonly 2 ms long, with the 87 Rb MOT cooling light ramped linearly from 20.6 MHz
below the the |F = 2, mF = 2i to |F = 3, mF = 3i transition (120 MHz beat-note
command) to 40.6 MHz below the resonance (100 MHz beat-note command). In thistime, the 40 K cooling light was turned down in intensity but the detuning remained
unaltered.
Next, 87 Rb was optically pumped into the |F = 2, mF = 2i state using theslower cooling beam, while 40 K was optically pumped into the |F = 9/2, mF = 9/2i
state using a dedicated optical pumping beam in 250µs. These are magneticallytrappable states, and we subsequently turned on the quad coil current to 130 A
to capture the atoms in the magnetic trap, along with the green plug beam at thecenter to prevent Majorana losses. Both species were compressed by a linear rampof the quad current up to 160 A in 0.5 s. Then, forced rf evaporation was performedfor 10 s, sweeping the rf frequency linearly from 18 MHz down to 2 MHz.
Then, the atoms were decompressed and loaded into the ODT, similarly to
the BEC procedure. The ODT was turned on to an initial power of 2 V and an
initial split command of 0.01 V. The quad current was ramped down to 25.5 A

89

exponentially with a time constant of τ = 1.5 s in 3 s. The evaporation in the ODT
was split into two steps. During the first 3 s step, the split was ramped linearly to its
final command power of 0.65 V, putting more power into the (less tightly focused)crossing beam. The green plug beam was ramped off during this step. During thesecond, 4 s step, the overall power of the ODT was exponentially ramped down to
1.2 V, while the quad coil current was ramped exponentially to 0 A.
We then used adiabatic rapid passage (ARP, see sec. 2.2.3) to transfer the87

Rb atoms from |F = 2, mF = 2i to |F = 1, mF = +1i using a microwave coupling

field and a 50 ms ramp in bias Z coil current. Then, we pulsed on the XZ imaging
beam to eject any remaining F = 1 atoms. Then we performed one last evaporation
step in the ODT, ramping the final power down to 0.7 V. The 87 Rb atoms wereno longer suspended against gravity and fell out of the trap. We then were free toperform experiments with the degenerate 40 K cloud.

3.3.6 Current status of Potassium apparatus
At the time of writing, the 40 K part of the apparatus as described is no longer
functional. The number of 40 K atoms collected in the MOT started decaying significantly in January 2014, and by March was almost completely gone and could not
be resurrected. The specific failure point of the setup was not clear. However, othergroups have found that atomic sources are much more stable, and a higher fractionof the (expensive) 40 K sources can be utilized when the atoms were initially cooledwith a 2-D MOT rather than with a Zeeman slower [96–98]. Therefore, rather thancontinuing to attempt to revive the existing set-up, the decision was made to build
a 2-D MOT for both 40 K and 87 Rb.
The design of our 2D MOT is closely based on the design in Thomas Uehlinger’s
diplome thesis [97]. The design was developed by Dalia Ornelas, and initially implemented by Marcell Gall before it was taken over by the rest of the RbK team.
90

The schemtaic of the planned vacuum system (with attached optics) is pictured in
Figure 17. On the left side of the schematic, there are optics directing the pushing
beam into a miniconflat viewport. The viewport is part of a cross, with the 40 K and87

Rb ovens attached to the two ends of the cross, with gate valves allowing one toclose off one or both sources from the rest of the vacuum system. From there, thecross attaches to the main 2D MOT cell via a mini-conflat flange.
The cell is a custom machined stainless steel frame with rectangular antireflection (AR) coated windows on four sides and mini-conflat conectors on two
ends, pictured in more detail in Figure 18. Two aluminum mounting crosses attach
two either end of the cell. Four aluminum bars are connected between the crosses,
and the main 2D MOT optics are mounted on those four bars. The opposite end of
the cell (right in fig. 17) sandwiches a differential pumping tube in the mini-conflat
connection and connects to another cross. The top of the cross connects to a small
ion pump. The bottom connects to a rotatable feed-through mechanism with a ’flag’,a square of metal, attached inside. The rotator rotates the flag in and out of theatomic beam path, providing a means of losing off the main chamber from the atomic
beam and push beam light. The fourth end of the cross connects to another gate
valve, separating the 2D MOT vacuum system from the main experiment chamber.
The other end of this gave valve connects to a flange that is meant to connect
directly to the main experiment chamber.
A picture of the stainless cell is shown in Figure 18. Attaching the glass
windows to the stainless steel frame in a vacuum tight way proved to be quitedifficult, and this picture was taken during one of the attempts to do so using
epoxy. The clamps around the cell served to keep the windows in place as the epoxy
was curing. In the final design, the seal was made with indium, with gaskets custom
machined to press the windows onto the cell. We roughly followed the indium
sealing method presented in [99], with gaskets above and below the windows. We

91

Figure 17: Schematic of the 2D MOT setup. The mini-conflat on the right is to be
attached to the existing experiment chamber, directing the atomic beam into the
3D MOT.

92

Figure 18: Picture of 2D MOT cell during an attempt to epoxy AR coated windows
onto the stainless steel frame. The clamps serve to hold the windows in place while
epoxy cures.
also employed pre-squashing, where a metal piece in the shape of the window was
first pressed onto the indium wire to flatten it and minimize the amount of pressurethat needed to be appied to the glass window.
The main 2D MOT optics direct the cooling and repump beams into the cellfrom two directions, and retro-reflect them on the other end, producing coolingalong those two directions (hence the name 2D MOT). The optical set-up along oneof those directions is presented schematically in Figure 19. The cell is elongatedalong the atomic beam direction, to maximize the time the atoms are cooled while
travelling to the 3D MOT. Because of this, instead of a highly eliptical cooling beam,four MOT beams are launched in a row, almost overlapping, from each of the two
directions. This is accomplished by splitting one beam into four with four sequential
beam-splitting cubes (BSs).
The 2D MOT cooling and repump light is first periscoped over from a fiberand lens assembly (designed to shape the beam to be roughly 1 inch in diameter).
93

It then goes through a half-wave plate (HWP) before entering the first 70/30 beam
splitter. 30% of the light is sent through a quarter-wave plate (QWP) tuned toprovide circularly polarized light into the cell. On the other side of the cell, thebeam hits another QWP before being retro-reflected back into the cell. Since thebeam hits the QWP on the other side of the cell twice, the circular polarization ispreserved. The remaining 70% of the light goes into the next 70/30 BS, sending
21% of the total beam power into the second arm going into the cell, to be retroreflected in the same way. The remaining 49% hits a 50/50 BS, sending 25.5% of thetotal beam power into the third retro-reflected arm. The final cube is a polarizing
beam-splitter (PBS), and the HWP before the cubes is tuned to ensure all of thelight is sent into the cell on this last, fourth, arm.
The push beam enters from the oven direction and serves to provide somevelocity to the atoms along the long direction of the cell to ensure they continue
to travel to the 3D MOT, while still providing some cooling along the longitudinal
direction. In order to provide this cooling, the push beam is also retro-reflected,
with the help of a custom machined differential pumping tube (on the right in fig.
19). The differential pumping tube is machined to have a 45◦ angled polished end,
reflecting all of the light except for the central part towards a retro-reflecting mirror
outside the cell. Along the other 2D-MOT cooling direction (up and down in fig.
19), this whole set of optics is replicated, with the exception of the push beam
retro-reflection.
The current 2D-MOT setup is pictured in Figure 20. The vacuum system has
been assembled and successfully pumped down, with a octagonal test chamber inplace of the main experimental chamber. This test chamber is intended to be used
to send probe light through and detect fluorescence to characterize the atomic beam
coming out of the 2D MOT. The optics have been assembled and the quarter-wave
plates lightly epoxied onto the BS cubes and mirrors, although as can be seen in the94mirrors
QWPs

push
beam

differential
pumping tube

to 3D
MOT

PBS

50/50 BS

70/30 BSpm erisirr coor pe

MOT
cooling + repump

QWPs

70/30 BS HWP

Figure 19: Schematic of 2D MOT optics along one direction. the main cooling andrepump beam is split into four parallel arms by four beam splitter cubes. The pushbeam enters from the oven direction and is retro-reflected via a custom machined andpolished differential pumping tube. These optics are mounted on crosses attachedto the 2D MOT cell.
picture some have regrettably fallen off. Quadropole coils have been wound aroundeach of the four windows, onto 3D printed coil winding forms. Bias Z coils have also
been wound to cancel out stray gradients along the atomic beam direction.
The two Toptica TA-pro systems that were used to provide all 40 K light in
the past are still operational, and need to be re-purposed to provide both 2D MOT
and 3D MOT cooling and repump light. A new Toptica TA-pro was also purchased,with the intent to implement gray molasses cooling on the 40 K D1 line as describedin [100].

95

Figure 20: Picture of current 2D MOT apparatus. The vacuum system is in place,
optics are (mostly) mounted and coils to generate the quadrupole trap and cancel
gradients along the atomic beam direction have been wound.

96Chapter 4: Absorption Imaging with Recoil Induced Detuning

In this Chapter, we describe the simulations we performed in order to interpretabsorption images in a non-standard regime: at long imaging times, where the recoil
induced detuning needed to be taken into account. This simulation was necessary tointerpret data collected for our s-wave scattering experiment, described in Chapter
5. In this Chapter, we first describe the recoil-induced detuning effect and derive
the equations to be solved. Then, we solve these equations perturbatively, andshow that this treatment is insufficient in the regime of interest. We then perform
two versions of numerical simulations: one where the atoms are assumed to remain
stationary relative to each other during imaging, and one where they are free tomove. We show that although the atoms do move significantly during the imaging
time, this does not have a strong effect on the final observed intensity. Finally, we
use our simulated results to calibrate the saturation intensity in our camera units,
and find the parameters for optimal signal-to-noise (SNR) ratio imaging. This work
was previously reported in [101].

4.1 Recoil-induced detuning
After absorbing a number of photons N all traveling in the same direction, an
atom will obtain an average recoil velocity of N vr . Via the Doppler effect, this will
result in a detuning δ = N kr vr . This detuning will increase as more photons areabsorbed, and therefore depend on time. Since detuning affects the absorption crosssection, the absorbed intensity will also become time dependent. We can generalize97Eq. 2.35 to include a time dependence on the detuning term and therefore also theintensity:d I(z, t)I(z, t)/Isat= −ρ(z)σ0.
dz Isat
1 + 4δ(z, t)2 /Γ2 + I(z, t)/Isat(4.1)

The number of photons absorbed per atom will depend on the intensity lost, up
until the current time, at that location. The detuning will therefore be proportional
to the total number of photons lost up until time t at that location, given by theabsorbed intensity divided by the single photon energy h̄ωL , divided by the numberof atoms that participated in the absorption ρ(z) times the detuning kr vr :kr vr
δ(t, z) =h̄ωL ρ(z)Z tdI(z, τ )
dτ.
dz
0

(4.2)

These equations are interdependent, and cannot be in general solved analytically.
Figure 1a shows the velocity and detuning as a function of position in space forthree different imaging times, calculated numerically. All calculations in this chapter
were done for a cloud of 40 K atoms, as that is relevant to our experiment describedin the next chapter. The resonant wavelength is λL = 770.11 nm, the naturallinewidth of the transition is Γ = 6.035 MHz, the resulting saturation intensity andrecoil velocity are Isat = 17.5 W/m2 and vr = 0.01297 m/s.

4.2 Perturbative treatment
We can treat these equations perturbatively in time, assuming the light isinitially on resonance δ(z, 0) = 0. To first order, we can set the detuning in Eq. 4.1
to δ = 0, assume I(z) is time independent, and plug that into Eq. 4.2 to obtain
Z t
kr vr
I(z)
−ρ(z)σ0dτδ(t, z) =h̄ωL ρ(z) 0
1 + I(z)/Isatkr vr σ0 I(z)=t.
h̄ωL 1 + I/Isat
98

(4.3)
(4.4)

This can then be recursively plugged into Eq. 4.1 to obtaind I(z, t)= −ρ(z)σ0
dz IsatI(z, t)/Isat
.
2
I(z)
r vr σ02 + I(z, t)/I
1 + 4 kh̄ωtsat
L Γ 1+I/Isat

(4.5)



Integrating both sides of the above equation, we obtain a perturbative equation tosecond order in time [102]:
I0 − If (kr vr t)2
σ0 n = ln(I0 /If )+
+
Isat
3



Isat
Isat
−
+ ln
If + Isat I0 + Isat

If + Isat
I0 + Isat. (4.6)In Fig. 1b, we examine for what imaging times the above perturbative equation, as
well as the model that completely ignores recoil induced detuning, is valid. We do
this by performing numerical simulations to extract a value for the final intensity If
and using Eq. 2.36 and Eq. 4.6 to extract values σ0 n that would be deduced fromexperiment. We find that within the recoil time, both analytic expressions start todiffer from the true atomic column density by over 5%, and the perturbative modelof Eq. 4.6 quickly diverges thereafter.
In the following sections, we describe two versions of numerical simulations
that we have performed in order to appropriately extract atomic column densities
from experimental data.

4.3 Stationary atom model
In order to numerically simulate the imaging process, we assumed a Gaussian√2
2
distribution of atoms along the propagation direction, ρ(z) = n/ 2πwe−z /2w .
The dependence of the result on the choice of cloud width w is discussed in the next
section. We divided the cloud into small spatial bins of atoms along the imaging
direction z. For the initial version of the simulation, the atoms were assumed to staywithin their original bins for the entire duration of the imaging pulse, i.e. the cloud
99

3.5

1.6

3.0

t=25 µs

2.5

1.4
1.2

2.0
1.5

t=10 µs

1.0
0.0

0.2

1.0

0.4

.6

3.0

.4

1.8

Eq

3.5

2δ/Γ

Velocity [m/s]

4.0

2.0

recoil time

t=50 µs

Column density

4.5

2.5
2.0

σ0 n
1.5

Eq. 2.2

1.0

7

0

20

40

60

z [µm]

Imaging time [us]

(a)

(b)

80

100

Figure 1: (a) Dependence of velocity and detuning on position simulated for 40 K at
three different imaging times and a probe intensity I0 = 0.8Isat . (b) Column densities
deduced from optical depths obtained from recoil detuning corrected simulation ofimaging 40 K atoms at probe intensity I0 = 0.8Isat . The blue line is the true column
density σ0 n = 1.6. The green line is the high probe intensity corrected column
density given by Eq. (2.36). The red line is the column density as expanded tosecond order in time, Eq. (4.6).

100

shape remained constant. We then used eqns. 4.1-4.2 to numerically propagate theprobe intensity and detuning as a function of both time and space. The algorithm
used is detailed by Alg. 1.
Algorithm 1 Stationary atom modelI[n = 0, t] = I0 {n is the bin index, t is the time index, I is in units of Isat }δ[n, t = 0] = 0 {light initially resonant, δ in units of Γ/2}
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 = σ0 ρ[n]dz {dz is the size of spatial step}B = vr dt/(h̄cρ[n]) {dt is the size of the time step}I[n, t] = I[n − 1, t] − AI[n − 1, t]/(1 + δ[n, t − 1]2 + I[n − 1, t]) {Eq. (4.1)}
δ[n, t] = δ[n, t − 1] + B (I[n − 1, t] − I[n, t]) {Eq. (4.2)}
end forHf = Hf + I[N, t]dt {collecting total fluence seen by the camera}end for
ODsim1 = − ln (Hf /I0 tf )We call the optical depth obtained in this way ODsim1 , to distinguish if fromthe simulated optical depth via the method described in the next section.
The validity of this model can be checked by considering limits where theequations are analytically solvable. For short imaging times, the recoil-induced
detuning should not contribute to the optical depth, and therefore Eq. 2.36 should
become exact. This is seen in Fig. 2a, where the imaging pulse is only 3 µs longand the simulated optical depth (blue dots) agrees with that given by Eq. 2.36 for
all intensity regimes.
Even at longer imaging times, the problem can be analytically solved for limitsof both high and low intensity compared to the saturation intensity. At intensities
I  Isat , even far detuned atoms will scatter light at their maximum, and we canassume δ 2 /Γ2  I/Isat , reducing back to Eq. 2.36. At extremely low intensities,
atoms will scatter very little light and the detuning δ 2 /Γ2  1, again reducing backto Eq. 2.36. As seen in Fig. 2 b,c the simulation agrees with the analytic Eq. 2.36
in the limit of both high and low intensities. But, as the imaging time increases,101t = 3 µs

t = 25 µs

t = 50 µs

1.6

Optical Depth

1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
10−2

101

I0 /Isat104 10−2

101

104 10−2

I0 /Isat

101

104

I0 /Isat

Figure 2: Optical depth as a function of probe intensity as predicted by the simulation (blue symbols) and by Eq. (2.36) (green curves), for three different imaging
times. As expected, the predictions agree in both the high and low intensity limits,and differ for probe intensities comparable to the saturation intensity and longer
imaging times.
the disagreement due to recoil induced detuning grows.
The simulation allows us to extract both the intensity and the detuning as afunction of both time and position. We can use this information to infer the velocity
and therefore the displacement of the atoms during the imaging pulse, and check
if our assumption that the atoms stay in their original bins during the image pulse
is valid. Figure 3a shows the position, deduced by integrating the recoil-induced
velocity, as a function of time of the first (closest to light source), middle, and last
(furthest from light source) spatial bin for a probe intensity slightly above saturation,
I = 1.2Isat . As seen in the figure, not only do the atoms move beyond their bins,
but also at long imaging times the first atoms (which have absorbed the most light)
overtake the last ones. Therefore, the atomic cloud does not maintain shape during
the imaging pulse, and our initial assumption is invalid.

102t = 10

12

0 µs

250
200
150
100
50
0

t = 50µs

8v[m/s]z position [µm]

10

6
4

back
middlefront
0

t = 10µs

2
0
20

40

60

80

0

2

4

6

time [µs]I0 /Isat

(a)

(b)

8

10

Figure 3: (a) Position of atoms as a function of imaging time for atoms in thefirst (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 inthis calculation was 1.2 Isat , and the column density was σ0 n = 1.6. (b) The velocity
of a single composite atom as a function of probe intensity for various imaging times.
Simulation data (dots) and numerical solutions of Eq. (4.7) (lines) are in agreement.

1034.4 Traveling atom model
To model the recoil-induced detuning effect during the imaging pulse taking
into account the potentially significant spatial displacement of the atoms, we performed a second version of our simulation. In this version, we clumped Nca ≈ 1011
atoms per unit transverse area into a single composite atom, and then tracked thedetuning, velocity and position of each composite atom as a function of imaging
time. Tacking individual atoms would be computationally inaccessible for reasonable cloud sizes. The algorithm used is given by Alg. 2.
Algorithm 2 Travelling atom modelz[n] = z0 , δ[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] = I0 { t is the time index, I is in units of Isat }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 N do {loop over superatoms}
n = O[i] {apply probe intensity to composite atoms in order of appearance}
A = σ0 Nsa dz {dz is length over which atoms were grouped into single composite atom}
B = vr dt/(h̄cNsa ) {dt is the time step}I[n, t] = I[n − 1, t] − AI[n − 1, t]/(1 + δ[n]2 + I[n − 1, t]) {Eq. (4.1)}
δ[n] += B (I[n − 1, t] − I[n, t]) {Eq. (4.2), detuning in units of Γ/2}
z[n] += dtΓδ/2k {k is the wavenumber, Γδ/2k is the velocity at δ 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 /I0 tf )To check the validity of this version of the simulation, we check the velocity
of a composite atom as a function of time in an analytically solvable limit. In thiscase, we take the limit of a single composite atom, such that the intensity seen by
the composite atom becomes time independent. This simplifies Eqs. 4.1 and 4.2 to
only carry time dependence in the detuning term, and we can then plug Eq. 4.1
104

into Eq. 4.2 and differentiate both sides with respect to time to obtaindδ(t)Γkr vrI/Isat=.
2
dt
2 1 + 4δ /Γ2 + I/Isat(4.7)

Equation (4.7) can be solved numerically, and is in agreement with our simulation,
as seen in Fig. 3(b).
We then used this version of the simulation to look at the motion of composite
atoms as a function of imaging time in phase space (i.e., velocity and position). Some
examples of this motion can be seen in Fig. 4. As seen in the figure, the atomic
cloud is significantly distorted during the imaging pulse and the atoms perform somecrazy acrobatics.
It remains to check how the atoms’ acrobatics affect the resulting optical depth,
ie the attenuation of the probe beam. To do this, we compare the optical depths
generated by our stationary atom model, ODsim1 , and by our traveling atom model,
ODsim2 . The results of this comparison are seen in Fig. 5(a). As seen from the figure,the optical depths predicted by the two versions of the simulation are negligibly
small - ODsim1 − ODsim2 /ODsim1 ≤ 0.005. We also checked the effect of having
different initial distributions of atoms in space by varying the initial function ρ(z)and keeping the total atom number constant. We found the effect of this to be
negligible as well. Therefore, to infer atomic column densities from observed opticaldepths, it is sufficient to use the stationary atom model.

4.5 Calibration of saturation intensity
Saturation intensity is an intrinsic property of the atom, so the idea of calibrating it may be confusing. However, there are several experimental parametersthat may influence exactly what value of Isat is appropriate to use in eqn. 4.1 and
4.2, such as losses in the imaging system and polarization of the probe beam. In
105

Cloud width =10.0 µm
t = 0.0 us

t = 40.0 us

t = 80.0 us

t = 120.0 us

t = 160.0 us

t = 200.0 us∆v [m/s]

0.4
0.2
0
-0.2
-0.4

∆v [m/s]

0.4
0.2
0
-0.2
-0.4
−20

−10

0

10

∆z [µm]

20−20

−10

0

10

∆z [µm]

20−20

−10

0

10

20

∆z [µm](a)

Cloud width =1.0 µm
t = 0.0 us

t = 40.0 us

t = 80.0 us

t = 120.0 us

t = 160.0 us

t = 200.0 us∆v [m/s]

0.06
0.03
0
-0.03
-0.06

∆v [m/s]

0.06
0.03
0
-0.03
-0.06
−2

−1

0

∆z [µm]

1

2 −2

−1

0

∆z [µm]

1

2 −2

−1

0

1

2

∆z [µm](b)

Figure 4: Phase space evolution of an atomic cloud exposed to probe light withintensity I˜0 = 1.2. We defined ∆v = v − hv(t)i and ∆z = z − hz(t)i, subtracting out
the center of mass position and velocity of the cloud. The column density σ0 n is 1.6,
and the initial cloud is a Gaussian with a width of 10 µm in (a) and 1 µm in (b).
The center of mass velocities hvi are (0, 3.41, 5.26, 6.52, 7.50, 8.32) m/s sequentially,
and are the same for both initial cloud widths. Color represents original position of
the composite atom along ez .

106

0

1.4
1.2

−2

OD

(2)

1.0

OD

Optical Depth
Fractional difference

1.6

OD(1)
ODsim

0.004

0.8
0.6
0.4

0.002

0.2
0.000
0

1

2

3

4

5

I0 /Isat

0.0

0.5

1.0

1.5

2.0

2.5

I0 /Isat(a)(b)

Figure 5: (a) Top. Optical depth as a function of probe intensity for an imaging
time t = 100 µs. OD(1) and OD(2) are optical depths predicted from a given column
density by Eq. (2.36) and (4.6) respectively. The two versions of simulated opticaldepth, ODsim1 (green curve) and ODsim2 (green dots) are plotted and overlapping
each other. Bottom. The fractional difference between two versions of the simulated
OD, ODsim1 − ODsim2 /ODsim1 . (b) The optical depth as a function of probe
intensity for three imaging times: t = 40 µs (blue), t = 75 µs (green), t = 100µs (red). The dots represent experimental data and the lines represent the best fit
of simulated data. The optimal fit parameters pictured are a σ0 n of 1.627(5) andsaturation intensity of 29(7) counts/µs.

107

3.0

addition, we have no direct experimental access to the total radiant fluence (time integral of intensity) seen by the camera. Instead, the light hitting the charge-coupleddevice (CCD) camera triggers some number of photoelectrons to be registered. The
proportionality between the number of photons hitting the camera and the numberof photoelectrons it triggers is called the quantum efficiency qe of the camera. The
number of these photelectrons, after some electronic gain and noise introduced during the readout process, is then read out as a number of ‘counts’ registered on eachpixel. The camera-dependent factors influencing how the number of counts depends
on the number of incoming photons can be convolved with the experimental factors
such as probe polarization and optical loss into a single calibration of the effective
saturation intensity in units of ‘counts’ output by the camera per unit time.
To calibrate this effective Isat in camera counts per unit time, we absorption
imaged our cloud of 40 K atoms for a range of probe intensities for three different
values of imaging time: 40 µs, 100 µs, and 200 µs. We select a small region in
the center of the cloud, where we can assume the atomic column density σ0 n, andthe initial probe intensity I0 to be roughly constant. We then average the valuesof I0 and If over this region and plot the final intensity If as a function of I0 . We
then used the optical depth predicted by our simulation ODsim and used that tosimultaneously fit the three curves with Isat and σ0 n as fit parameters, as shownin Fig. 5(b). As can be seen from the figure, this procedure not only allows us toread off Isat in units of camera counts per µs, but also shows that our simulation
accurately reproduces the differences in OD dependence on imaging time.

4.6 SNR optimization
We consider Poisson distributed photon shot noise, converting into shot noiseon photoelectrons triggered inside the CCD. The standard deviation will then bepproportional to qe Np , where qe is the quantum efficiency of the camera and Np is108the photon number. This uncertainty can be then propagated via the lookup table
into uncertainty on the measured atomic column density δσ0 n . The signal-to-noise
ratio (SNR) can then be expressed as σ0 n/δσ0 n .
We study the SNR as a function of imaging time and initial probe intensity
for a few different atomic column densities. Some representative data is shown in
Fig. 6. As seen in Fig. 6(a), for a wide range of atomic column densities, extending
the imaging time beyond 40 µs no longer yields significant improvements in SNR.
There is, however, a factor of 1.5 improvement between using an imaging time of 10µs, where the simple model given by Eq. 2.36 is appropriate, and 40 µs. Therefore,
there are significant gains that can be made by going to longer imaging times andmaking use of the simulated lookup table.
This simulation allowed us to interpret experimental data. For a given imaging
time, we created a look-up table of predicted optical depth as a function of probe
intensity and atomic column density. We then found the observed optical depth on
this table, with the given probe intensity, and inferred the atomic density. The uncertainty in the measured intensities can be propagated through this procedure, andwe established optimal imaging parameters to maximize the SNR of this detection
scheme. Figure 6(b) illustrates that the optimal initial probe intensity is different
for different atom numbers. For low atom numbers, σ0 n ≈ 0.1, a probe intensity ofI0 ≈ 0.6Isat is best.

109

σ0 n=10σ0 n=10

103

103

102

101

SNR

SNR

σ0 n=1.0102

σ0 n=1.0

σ0 n=0.1100101

σ0 n=0.1
0

20

40

60

80

0

1

2

3

time [µs]I0 /Isat

(a)

(b)

4

Figure 6: 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 I0 = 5.0Isatand (b) SNR as a function of probe intensity for an imaging time of 50 µs.

1105Chapter 5: Imaging of Scattering Near a Feshbach ResonanceIn this chapter, we describe our experiment directly imaging s-wave scattering
halos of 40 K atoms in the vicinity of a Fesbach resonance between the |F = 9/2, mF = −9/2i
and |F = 9/2, mF = −7/2i internal states. We used this data to extract the locationof the magnetic fields resonance of 20.206(15) mT and a width of 1.0(5) mT, similar
to the accepted values of 20.210(7) mT and 0.78(6) mT [44]. The data presented inthis chapter was previously reported in [101].
We first introduced Feshbach resonances in section 3.2.2. Although Feshbachresonances are extremely useful for studying and manipulating Fermi gases, theirresonant magnetic field values are difficult to predict analytically and are commonly
computed via numerical models based on experimental input parameters [103–105]
or determined experimentally [41, 106]. There have been a variety of experimental
techniques used to characterize Feshbach resonances, including measuring atom loss
due to three-body inelastic scattering, measurement of re-thermalization timescales,
and anisotropic expansion of the cloud upon release from a confining potential, allof which infer the elastic scattering cross section from collective behavior of thecloud [80, 107, 108].
Here, we present an alternative technique, where we directly image the enhancement in elastic scattering due to the resonance. We collided pairs of ultra-cold
Fermi gases and directly imaged the resulting s-wave scattered atoms as a functionof magnetic field strength. This allowed us to observe the enhancement in scattering without relying on proxy effects. We measured the fraction of atoms scattered111

during the collision, and from this fraction deduced the resonant magnetic field andthe width of the resonance.
In our dilute DFGs, even with the resonant enhancement of the scattering crosssection, only a small fraction of the atoms scattered as the clouds passed througheach other. This made direct detection of scattered atoms difficult due to detection
uncertainty that disproportionately 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, where the atoms acquired a velocity during imaging and the resulting Doppler-shift was non-negligible.
Simulation of the absorption imaging process was necessary for an accurate interpretation of these images, as described in Chapter 4. Using the simulation-corrected
images, we extracted the fraction of atoms scattered in our collision experiment.

5.1 Experimental procedure
We prepared our degenerate 40 K clouds as described in section 3.3.5. After
this preparation, we used adiabatic rapid passage (ARP) to transfer the degenerate
cloud of 40 K atoms in the |F = 9/2, mF = 9/2i state into the |F = 9/2, mF = −9/2istate by using a 3.3 MHz rf field and sweeping the bias magnetic field from -0.518
mT to -0.601 mT in 150 ms.
Following the state transfer, we had two versions of the protocol – one forapproaching the Feshbach resonance from higher fields and one for approaching itfrom lower fields. For approaching the resonance from lower fields, we proceededby ramping the bias magnetic field to 19.05 mT, turning on a 42.42 MHz RF field,
and then sinusoidally modulating the bias field at 125 Hz for 0.5 s, with a 0.14 mT
amplitude, decohering the 40 K state into an equal mixture of |F = 9/2, mF = −9/2i
and |F = 9/2, mF = −7/2i. For approaching the resonance from higher fields, thesame was done at a bias field of 21.71 mT and an RF frequency of 112.3 MHz. The
112

depolarization allowed the 40 K atoms to interact and re-thermalize, allowing us tofurther evaporate in the dipole trap [109]. Since 87 Rb is heavier than 40 K, we wereable to evaporate the 40 K atoms past the point where 87 Rb atoms were no longer
suspended against gravity and had been completely removed. These hyperfine statesof 40 K were then used to study their Feshbach resonance.
After evaporation, we ramped the bias field in a two-step fashion to the desired
value B near the Feshbach resonance. We approached the field using our quad coils
in Helmholtz configuration (0.19 mT/A, see sec. 3.3.2) to bring the magnetic field
to a setpoint 0.59 mT away from B, B − 0.59 mT when approaching from below andB + 0.59 mT from above. We held the atoms at this field for 100 ms to allow theeddy currents induced by the large quad coils to settle, and then flipped the currentof our lower inductance biasZ coils from −18 to 18 A (0.017 mT/A, see sec. 3.3.2)
to quickly change the field the remaining 0.59 mT. This allowed us to study theresonance from both sides without the added losses associated with going throughthe resonance [40].
Once at the intended bias field, we split the cloud into two spatially overlapping
components with opposing momenta and observed scattering as they moved througheach other and separated. These counter-propagating components were createdusing an 8EL deep near resonant (λL =766.704 nm) 1-d retro-reflected optical lattice(see sec. 3.3.1), where EL = h̄2 kL2 /2mK is the lattice recoil energy and h̄kL = 2πh̄/λis the recoil momentum. We rapidly pulsed this lattice on and off with a doublepulse protocol [110, 111]. The pulse sequence was optimized to transfer most of
the atoms into the ±2h̄kL momentum states. Since the initial Fermi gas had awide momentum spread (in contrast to a BEC, which has a very narrow momentum
spread), and the lattice pulsing is a momentum dependent process [110], not all
the atoms were transferred into the target momentum states. We experimentally
optimized our pulse times to minimize the atoms remaining in the zero momentum

113state. The optimized pulse times were 23 µs for the first pulse, 13 µs off interval,and 12 µs for the second pulse [111].
We then released the atoms from the trap and allowed 1 ms for the two opposite
momentum states within the cloud to pass through each other, scattering on theway. For the data taken coming from below the Feshbach resonance, we then simply
ramped down the field and imaged the atoms. For the data taken coming fromabove the Feshbach resonance, we ramped the field back up, retreating through theresonance if it had been crossed and thereby dissociating any molecules that werecreated, and then quickly ramped the field back down and imaged the atoms. We
used a 40 µs imaging pulse with I0 /Isat ≈ 0.6 at the center of the probe laser. The
total time-of-flight was tT OF = 6.8 ms.
The magnetic fields produced by the combination of our quad and biasZ coils in
the regime of interest were independently calibrated by rf-spectroscopy. We prepared40

K atoms in the |F = 9/2, mF = −9/2i state and illuminated them with and rf-fieldwith some frequency νrf . We then ramped our high-inductance coils to variable set
points, followed by an adiabatic 250µs ramp of 2.84 mT in the lower inductancecoils. We then used Stern-Gerlach and observed the fractional population in the|F = 9/2, mF = −9/2i and |F = 9/2, mF = −7/2i states as a function of the highinductance coil current. We fit the fractional population curve to a Gaussian, andconsidered the center of the fit to be on-resonant, with an uncertainty given by theGaussian width. We used the Breit-Rabi formula (see sec. 2.1.2) to determine theresonant field value at νrf . We did this for 5 different rf frequencies, and acquireda field calibration with an uncertainty of 0.3 mT, which was included in the listed
uncertainty on the center field of the Feshbach resonance.

114

2.7

2.7
2.0

2.1
1.8
0

1.5
1.2
0.9
0.6

-2.0

Y momentum [h̄kL ]

2.4

2.4
2.1
1.8

0

1.5
1.2
0.9
0.6

-2.0

0.3
2.0

0

-2.0

Column density σ0 nsim

3.0

Optical Depth

Y momentum [h̄kL ]

2.0

3.0

0.3

0.0

2.0

0

X momentum [h̄kL ]X momentum [h̄kL ]

(a)

(b)

-2.0

0.0

Figure 1: An example of our absorption image after 6.8 ms TOF. The 1-D lattice
imparts momentum along ex . The two large clouds on the left and right are theatoms in the ±2kL momentum orders that passed through each other unscattered.
The smaller cloud in the center is the atoms that remained in the lowest band of thelattice after pulsing, and thus obtained no momentum. The thin spread of atoms
around these clouds is the atoms that underwent scattering. This image was taken
coming from below the Feshbach resonance at 20.07 mT. (a) Raw optical depth,(b) atomic column density obtained by comparing to simulated ODs, σ0 nsim

5.2 Data analysis
We first processed each image by comparing the obsereved ODs to simulations
taking into account the recoil induced detuning as described in Chapter 4. An
example of images before and after processing are shown in Fig. 1. To improve
the signal and mitigate our shot to shot number fluctuations, we took 15 nominallyidentical images for each data point.
We counted the fraction of atoms that experienced a single scattering event foreach of the fifteen images at a given bias magnetic field. Single scattering events areeasily identified, as two atoms that scatter elastically keep the same amplitude of
momentum, but depart along an arbitrary direction. Therefore, an atom traveling

115

2.0

kF − kL

0

2.0

0

-2.0

0.050
0.045
0.040
0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000

X momentum [h̄kL ]

(a)

(b)

Figure 2: (a) Our experimental setup. After time-of-flight, the two clouds traveling along ±êx directions have separated and the atoms that underwent a single
scattering event were evenly distributed in a scattering halo around the unscatteredclouds. The 1-D lattice defined the axis of cylindrical symmetry. (b) Inverse Abeltransformed image. The atoms within the Fermi momentum kF of each unscatteredcloud center are in the unscattered region and counted towards the total unscatterednumber. The atoms outside the radius kL − kF but inside kL + kF while also beingoutside the unscattered region are counted towards the number of single scatteredatoms.
at 2h̄kL to the right that collides elastically with an atom traveling at −2h̄kL to
the left will depart with equal and opposite momenta 2h̄kL at an arbitrary angle,
and in a time-of-flight image such atoms will lie in a spherical shell, producing thescattering halo pictured in Fig. 2(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 Abel transform. The
inverse Abel transform assumes cylindrical symmetry, which was present in our case,with the axis of symmetry along ex , defined by the lattice. We neglect the initialasymmetry of the trap, as during time-of-flight the atoms travel far beyond theinitial extent of the cloud (rx , ry , rz ) ≈ (45,48,15) µm, while the cloud width afterTOF is ≈ 82 µmin each direction. 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
116

Atomic density σ0 ρR momentum [h̄kL ]kF + kLbetween r and symmetry axis ex , integrated over φ, the azimuthal angle around thex axis.
We then extracted the number of scattered atoms Nscat as a fraction of thetotal atom number Ntot for each image, as shown in Fig. 2(b). The unscatteredatom number was the number of atoms in the two unscattered clouds. The numberof atoms that underwent a single scattering event was the number of 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 the cloud (red arcs in Fig.
2(b)). For both the scattered and unscattered numbers, we accounted for atoms
that fell outside the field of view of our camera by multiplying the counted atom
number by a factor of the total area as defined by the radii divided by the visiblearea on the camera. The atoms in the center region were not counted as they wereoriginally in the zero momentum state and could not contribute to the scattering
halo under study.
We fit the fraction of scattered atoms versus the total atom number for eachof the 15 images taken at the same bias magnetic field to a line constrained to be2
at that bias
zero at zero. The slope of this fit was taken to be the value of Nscat /Ntot

magnetic field, and the variance of the fit gave the uncertainty on that data point.
This uncertainty reflected our shot to shot number fluctuations, which produced
variable atomic densities and thus influence the scattered fraction.
We then deduced the resonant field value B0 and width of the resonance ∆,the parameters in Eq. (3.26). Since we were in the low energy regime (the atomic
momentum was much smaller than the momentum set by the van der Waals lengthkL + kF  1/lvdW , and we were well below the p-wave threshold temperature [109]),
the scattering cross-section was given by σ = 4πa2 .
The scattering cross-section σ gives the probability Pscat = σN/A that a single
particle will scatter when incident on a cloud of atoms with a surface density of117

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 half of the initial cloud, with atoms number Ntot /2,
is incident on the other half. Thus, the number of expected scattering events is
2
Nscat = (Ntot /2)σ(Ntot /2) = σNtot/4A. Assuming A is constant for all our data, we

can define a fit parameter b0 = 4πa2bg /4A, where abg is the background scattering
length. We can thus adapt Eq. (3.26) to obtain the fit function

2
Nscat
∆
= b0 1 −
+ C.
2
Ntot
B − B0(5.1)

We found that our imaging noise skewed towards the positive, giving rise to a small
background offset. We accounted for this in our fit by including a constant offset
parameter C.

5.3 Results
Our final data is presented in Fig. 3. The red curve depicts a best fit of themodel given in Eq. (5.1). The fit parameters we extracted were ∆ = 1(5) mT,
B0 = 20.206(15) mT, b0 = 5(3) × 10−3 and C = 8(1) × 10−4 . To obtain the fit, weused data taken by approaching the resonance from above for points above where
we expected the resonance to be and data taken approaching the resonance frombelow for points below. We also excluded from the fit data points very near the
resonance, as there the assumption σρ  1, where ρ is the atom number per unit
area, is no longer valid and the problem must be treated hydrodynamically.
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) mT and ∆ = 0.78(6) mT [44], which is ingood agreement with our findings. Some potential sources of systematic uncertainty
that we did not account for include scattering with atoms that did not receive amomentum kick from the lattice pulsing or the impact of multiple scattering events.
118

0.0202
Nscat /Ntot

0.015

B0

0.010

δB
0.005

0.000
19.6

19.8

20.0

20.2

20.4

20.6

20.8

21.0

B [mT]

Figure 3: Normalized scattered population plotted versus bias field B. Green dots
represent data taken coming from below the resonance, and blue dots represent thedata taken coming from above the resonance. The red curve depicts the best fit,
where data coming from above the resonance was used above the resonance and data
coming from below the resonance was used below the resonance to create the fit; theunused data points are indicated by hollow dots. The regime where the scattering
length is likely large enough for the atoms to behave hydrodynamically is shaded ingray, and data points in that area were also excluded from the fit. Resonant fieldvalue B0 as found in this work and our systematic uncertainty in the bias magnetic
field δB0 are indicated.

119Chapter 6: Synthetic Magnetic Fields in Synthetic Dimensions

In condensed matter, 2-D systems in high fields have proved to be of great technological use and scientific interest. The integer quantum Hall effect (IQHE) [20],with its quantized Hall resistance, has given rise to an ultra-precise standard forresistivity. It was also one of the first examples of topology playing an importantrole in physics—the precise quantization of the Hall conductance is guaranteed by
the non-trivial topology of the system [6]. This leads to the precisely quantized‘plateaus’ in the transverse resistivity that occur as different multiples of the magnetic flux quantum Φ0 = h/e, where e is the electron charge, are threaded throughthe material.
In the IQHE system, the underlying lattice structure of the host semiconductor is effectively washed out—the magnetic flux per individual lattice plaquette is
negligible. However, new physics arises when the magnetic flux per plaquette isincreased to some non-negligible fraction of the flux quantum, giving rise to the
Hofstadter butterfly [16]. These regimes are hard to reach experimentally, since thetypical plaquette size in crystalline material is of order a square Angstrom, and themagnetic field necessary to thread a magnetic flux of Φ0 through such a narrow area
is of order ≈ 104 Tesla, not access able with current technology.
Several platforms have however reached the Hofstadter regime by engineeringsystems with large effective plaquette size, in engineered materials [17, 18], and in
atomic [51, 54–58] and optical [112] settings. Here, we use the approach of synthetic dimensions [61] in atomic BEC of 87 Rb to reach the Hofstadter regime. We

120

demonstrate the non-trivial topology of the system created, and use it to image
skipping orbits at the edge of the 2-D system—a hallmark of 2-D electron systems
in a semi-classical treatment.
We first describe the experimental setup of the effective 2D lattice with a large
effective magnetic flux through it created with synthetic dimensions in sec. 6.1. We
write down the Hamiltonian of the system, calculate the band structure and discuss
its basic features in sec. 6.2.1. We then describe the results of our experiments.
First, we describe the measured eigenstates of the system in sec. 6.3. Second,
we detail the measurement of chiral edge motion when atoms are loaded into thecentral site of the lattice along the synthetic direction. Finally, we describe ourobservation of skipping orbits along the edges of the system, when atoms are loaded
into the edge sites along the synthetic direction. This measurement represents thefirst direct observation of the phenomenon of skipping orbits. The work describedin this chapter was published in [62].

6.1 Synthetic dimensions setup
Any internal degree of freedom can be thought of as a synthetic dimension—
the different internal states can be treated as sites along this synthetic direction.
As long as one can define a metric along this direction, i.e. some of the internal
states are ‘nearest neighbors’ while others are not, it is sensible to consider it is adimension. In our case, an effective 2-D lattice is formed by sites formed by a 1-D
optical lattice along a ‘real’ direction, here ex , and the atom’s spin states forming
sites along a ‘synthetic’direction, es .
The experimental setup for this system is schematically represented in Fig.
1a. The BEC is subject to a 1-D optical lattice, formed by a retro-reflected beamof λL = 1064 nm along ex . A bias magnetic field B0 along ez energetically separates
the different spin states. The spin states can be thought of a sites along a synthetic121dimension even without any coupling field. However, only once a coupling field
is present do they acquire a sense of distance. We couple them via rf or Raman
coupling, which only couples adjacent spin states. The Raman beams illuminating
the atoms are shining in opposite directions along the same ex axis as the 1-D opticallattice. The rf field has components both along the ex and ey .
Figure 1b sketches out the effective 2-D lattice created. Here, we have labeled
the lattice sites along the ‘real’ direction ex by site index j. In the tight binding
approximation, we can describe a lattice hopping between adjacent sites with tunneling amplitude tx . Similarly, the sites along the ‘synthetic’ dimension are labeled
by site index m (identical to spin projection quantum number mF ), and the rf or
Raman coupling here plays the role of a tunneling amplitude ts . In the case of rf
coupling, there is no momentum kick associated with spin exchange, and both tx
and ts are real.
In the case of Raman coupling, however, there is a momentum kick of 2kR
associated with every spin transfer, and therefore a phase factor of exp(2ikR x) withevery spin ‘tunneling’ event. Since position x is set by the 1-D lattice, xj = jλL /2 =jπ/kL , and the space dependent phase factor is exp(2πikR /kL j). An absolute phasechange in the wavefunction is not meaningful. However, a phase acquired whengoing around a plaquette and coming back to the same place is meaningful, as one
could imagine one atom staying at the same site and the other going around aplaquette and coming back to detect the phase difference. In this setup, the phases
acquired while going around a single plaquette are, starting at some lattice site|j, mi, are: 0 (for tunneling right to |j + 1, mi), 2πikR /kL (j + 1) (for tunneling up
to |j + 1, m + 1i, 0 (for tunneling left to |j, m + 1i) and −2πikR /kL j (for tunnelingback down to |j, mi). The total phase acquired is thus φAB = 2πkR /kL , independentof the starting lattice site. Since the absolute phase does not matter and only thevalue of φAB , we can perform a phase transformation that shifts the tunneling phase1221064 nm lattice

(a)
ey

ez

Rb
BEC
87

ex

Raman ωBrf

B0 Raman ω+Δω

(b)

es

ex

Figure 1: Setup of effective 2-D lattice. (a) Beam geometry. The BEC is subject toa bias magnetic field B0 in the ez direction. The 1-D lattice beam and Raman beams
are both along the ex direction, and the rf field can be applied with projections onto
both ex and ey . (b) Schematic of the effective 2-D lattice. Sites along ex are formedby the 1-D optical lattice and labeled by site number j. Sites along the synthetic
direction es are formed by the spin states: 3 sites for atoms in the F = 1 manifold
and 5 sites for atoms in F = 2. These sites are labeled by m. Raman transitions
induce a phase shift, which can be gauge transformed into a tunneling phase along
the ex direction. This leads to a net phase when hopping around a single latticeplaquette of φAB .

123onto the spatial direction, defining tx = |tx |exp(iφAB m) and ts = |ts |, as labeled in
Figure 1b.
To see how this phase implies an effective magnetic field, we draw an analogy to
the Aharonov-Bohm effect [15, 113] from quantum mechanics. This effect considers
an infinite solenoid with an electric current running through it. The magnetic field
B in this setup exists only inside the solenoid, while the magnetic vector potential
persists outside the solenoid. However, if two electrons are sent on a trajectoryaround the solenoid, even though they never pass through any magnetic field, they
nevertheless acquire a relative phase that can be detected by interfering them witheach other. This relative phase is given by φAB = 2πΦ/Φ0 , where Φ = B ∗ A is themagnetic flux through the solenoid (A is the area inside the solenoid, pierced by themagnetic field) and Φ0 = h/e is the flux quantum, with e the electron charge. Sincein our system, the atoms acquire a phase when they perform a closed loop arounda single lattice plaquette. Therefore, they behave as though there was an infinite
solenoid piercing each plaquette with a magnetic field going through it, and the flux
per plaquette in units of the flux quantum is Φ/Φ0 = φAB /2π = kR /kL . For the caseof rf coupling, the phase acquired at every transition is 0 and the flux Φ/Φ0 = 0.
In this way, we are able to engineer large fluxes per individual plaquette simply
by adjusting the ratio kR /kL , allowing us to sidestep the need for experimentally
inaccessible field strengths and reach the Hofstadter regime.

6.2 Hamiltonian of the effective 2-D system
6.2.1 Hamiltonian
The full Hamiltonian of this system, without making the tight binding approximation, can be written down by combining the lattice Hamiltonian (eqn. 2.43) and124

the rf (eqn. 2.53) or Raman Hamiltonian (eqn. 2.56):H=h̄~k 2Ω− V0 sin2 (kL x) + h̄δFz + h̄Fz2 + Fx
2m
2

(6.1)

for rf coupling andH=h̄ky2 + h̄kz2
kx I + kR Fz )2Ω+ h̄2− V0 sin2 (kL x) + h̄δFz + h̄Fz2 + Fx
2m
2m
2

(6.2)

for Raman coupling. Here, k is the wavenumber, V0 is the lattice depth, h̄kL is thelattice recoil momentum, h̄kR is the Raman recoil momentum, Fz and Fx are theangular momentum matrices, δ is the rf or Raman detuning, Ω is the rf or Raman
Rabi frequency,  is the quadratic Zeeman shift, and I is the identity matrix. To findeigenstates of these Hamiltonions, we write them in a new basis that encompasses
both the momentum and the spin degrees of freedom. For the lattice Hamiltonian,
we used the momentum basis



..
.







|q + 4kL i






|q + 2kL i |qi  .
|q − 2kL i|q − 4k i

L 


..
.

(6.3)

For the Raman Hamiltonian in the F = 1 manifold, we used the spin and momentum
basis




|kx − 2kR , −1i.
|kx , 0i|kx + 2kR , 1i

(6.4)

In a lattice, the momentum kx becomes crystal momentum q. For every state125

in the lattice basis, we now expand to three states, one for each spin state, with theappropriate momentum shifts, giving
..
.









|q + 2kL − 2kR , −1i






|q + 2kL , 0i






 |q + 2kL + 2kR , 1i 
 |q − 2kR , −1i 





.
|q, 0i|q + 2kR , 1i






|q − 2kL − 2kR , −1i






|q
−
2k
,
0i
L






 |q − 2kL + 2kR , 1i 


..
.

(6.5)

In this basis, we combine the lattice and Raman Hamiltonians (omitting the
kinetic energy in the other two directions) in an infinite block matrix form as






H=






..



.
HR (2kL )

V0
4

0

V0
4

HR (0)

V0
4

0

V0
4

HR (−2kL )
..

126

.






,






(6.6)

where HR (x) is the Raman Hamiltonian with a momentum shift of x:h̄2 (q+n2kL −2kR )2
+ h̄δ
2m



HR (n2kL ) = 




h̄Ω/2

h̄Ω/2

h̄2 (q+n2kL )2
− h̄
2m

0

h̄Ω/2

0
h̄Ω/2
h̄2 (q+n2kL +2kR )2
− h̄δ
2m



,


(6.7)

the matrix V40 is a 3x3 diagonal matrix lattice coupling strength V40 on the diagonal,
and 0 is a 3x3 matrix of zeros. This extends in both directions with HR (2nkL ) on
the diagonal blocks and V40 as the first off-diagonal blocks and 0 everywhere else.
This Hamiltonian is easily extended to higher F values by replacing the Raman
blocks HR (x) with the corresponding Raman coupling Hamiltonian from eqn. 2.57,
and extending the diagonal matrix V40 and the zero matrix 0 to be (2F +1)x(2F +1).
For computational convenience, we convert to lattice recoil units, EL = h̄2 kL2 /2m,
kL = 2π/λL . Then the diagonal blocks become




2

h̄Ω/2
0
(q + 2n − 2φAB /2π) + h̄δ



,
2
HR (n)/EL = 
h̄Ω/2
(q
+
2n)
−
h̄
h̄Ω/2




0
h̄Ω/2
(q + 2n + 2φAB /2π)2 − h̄δ(6.8)
where h̄δ, h̄Ω and h̄ are now written in units of EL , q is written in units of kL and
we have used the fact that φAB /2π = kR /kL . The off-diagonal blocks V40 will be thesame 3x3 diagonal matrices, with V40 in units of EL .
This Hamiltonian can be written for general values of F in the presence ofRaman coupling and a 1-D optical lattice as

H=

F,∞
X

H0 + HR + HL ,m=−F,n=−∞127

where the diagonal term

H0 =h̄2 (q − 2mΦ/Φ0 − 2n)2 kL2 /2m + h̄δm + h̄m2

|q + n2kL , mi hq + n2kL , m|
includes the kinetic energy as well as the two-photon Raman detuning from resonance δ and the quadratic Zeeman shift . The second term represents the Raman
coupling with coupling strength h̄Ω, with anisotropic tunneling arising from thespin-dependent prefactor (Clebsch-Gordan coefficient):

HR =
h̄Ω

p
√
F (F + 1) − m(m + 1)/2 2 |q + n2kL , mi hq + n2kL , m + 1|
+H.c.

Here, H.c. stands for Hermitian conjugate. The third term represents lattice coupling to higher order lattice states, with lattice depth V0 :HL = V0 /4 |q + n2kL , mi hq + (n + 1)2kL , m| + H.c.

6.2.2 Band structure
The band structure of this Hamiltonian is presented in Figure 2. Here, wehave restricted ourselves to the lowest lattice band. We can do this because theenergy splitting between the lowest and second lowest lattice band is of order 4EL(see Figure 5), while the width of the lowest band, given by the amplitude of theapproximate sinusoid, is of order 0.3EL for lattice depths around 5.0EL , relevant toour experiment. As long as the Raman coupling stays small compared to the lattice

128

(a)

(b)
1
−0.4
−0.5

−0.5

0
−0.6

−0.6

hmi

Energy [EL ]

−0.4

−0.7
−2

−1

0

1

−1.0

2

(c)

−0.5

0.0

0.5

1.0

(d)
2

−0.4

Energy [EL ]

−1

−0.4

−0.5

1

−0.6

−0.6
−0.7

0
−1

−0.8
−2

−1

0

1

2

q [kL ]

−1.0

hmi

−0.7

−0.5

0.0

0.5

1.0

−2

q [kL ]

Figure 2: Band structure of the synthetic dimensions Hamiltonian, eqn. 6.6. For
all panels, the detuning h̄δ = 0 and the quadratic shift h̄ = 0.02EL . (a) F = 1,h̄Ω = 0.0. The color represents hmi, magnetization along es . (b) F = 1, h̄Ω = 0.5.
(c) F = 2, h̄Ω = 0.0. (d) F = 2, h̄Ω = 0.5.

129band spacing, the higher lattice bands are energetically separated enough that they
can be ignored.
Therefore, we can think of the Raman coupling analogously to the free space
Raman coupling (see section 2.6.2), except instead of free space parabolas each spinstate gets a lowest lattice band sinusoid. Figure 2a shows this in the limit of no
Raman coupling, Ω = 0, but with the 1D lattice on at V0 = 4.0EL . The quadratic
Zeeman shift is h̄ = 0.02EL and the detuning δ = 0. The mF = −1 sinusoid is
shifted 2kR , similarly to section 2.6.2, but since the sinusoid is periodic with 2kL , itfolds into the first Brillouin zone of the lattice, such that the nearest minimum toq = 0 is at q = 2kR − 2kL = (2φAB /2π − 2)kL . The edges of the Brillouin zone arePmarked by horizontal lines. The color indicates magnetization hmi = mF mF nmF ,where nmF is the fractional population in the mF state. In synthetic dimensions
language, hmi is the expectation value of position along es .
In Figure 2b, we have restricted ourselves to the first Brillouin zone and turned
the Raman coupling to h̄Ω = 0.5EL . This results in avoided crossings in Figure 2a,and the lowest band now has a spin dependence on crystal momentum. Figure 2c-dshows the same progression for the F = 2 manifold. Figure 2c is taken in the limitof h̄Ω = 0. All of the 5 spin states get ‘folded’ back into the first Brillouin zone dueto the lattice periodicity of the bands. The different overall energies of the sinusoids
are due to the quadratic Zeeman shift h̄ = 0.02EL . The lattice depth is again
V0 = 5.0EL and detuning h̄δ = 0. In Figure 2d we have restricted ourselves to thefirst Brillouin zone and turned on the Raman coupling to h̄Ω = 0.5EL . Note thatthe inverted hyperfine structure in Fig. 2c (meaning that the quadratic shift pushed
the mF = 0 state up rather than down in energy compared to the others), combinedwith the Raman coupling serves to make the lowest band in the F = 2 manifold
close to flat. This makes the band more similar to a quantum Hall Landau level,
and also shows promise for potential simulation of fractional quantum Hall physics,

130which require bands to be very flat.

6.2.3 Calibration
To calibrate the lattice depth V0 in the synthetic dimensions system, we cansimply calibrate the lattice depth without Raman or rf coupling as described in
Section 2.5.3. However, we are operating at very low Raman coupling strengths,h̄Ω ≈ 0.5EL . This is necessary because in the synthetic dimensional system theRaman coupling plays the role of tunneling, which has to be small, ts ≈ 0.1EL , toapproximate the tight binding limit. At these low Raman couplings, simple pulsing
as described in Section 2.6.3 is not useful for calibration, as the contrast of the Rabi
oscillations would be too low to resolve. Therefore, we calibrate the Raman couplingand detuning with the full synthetic dimensions system. The ‘folding in’ effect of
the lattice (meaning, the folding in of the sinusoids into the first Brillouin zone, orUmklapp processes) makes the higher Raman bands much closer energetically than
without the lattice, leading to larger contrast and allowing for accurate calibration.
To do this, we must first adiabatically load the lowest 1-D lattice band. To
do that, we must ramp on the lattice potential on a time scale slow compared tothe band spacing, ≈ 4EL . This gives t ≈ h/4EL = 0.12 ms. Figure 3a shows thefull ramping scheme. We ramp the lattice on in ≈ 20 ms. Then, we must pulseon the Raman coupling on a time scale fast compared to the spin sub-band levelspacing to produce Rabi oscillations, but still adiabatic with respect to the latticespacing to avoid exciting to the higher lattice band. We ramp the Raman beams
on in 300 µs. Then, the system is held on for a variable amount of time before alllight is snapped off and the atoms are allowed to expand in time-of-flight. For thecase of F = 2 atoms, the transfer to the F = 2 manifold is done in the 1-D lattice
before the Raman beams are ramped on to minimize the time spend in the F = 2
manifold.
131

Energy [EL ]

(a)
5

V0

t

h̄Ω
0
0

5

10

mF =+1
mF =0
mF =-1

0.6
0.4
0.2
0.0
0.5

1.0

1.5

2.0

2.5

momentum along ex

0.8

2kR
2kL
2kR − 2kL0

2kL − 2kR

−2kL
−2kR
+1

3.0

mF =+2mF =+1
mF =0
mF =-1mF =-2

0.6
0.4
0.2
0.0
1.0

1.5

-1

(e)

0.8

0.5

0

2.0

2.5

Raman pulse time t [ms]3.0

momentum along ex

Fractional population

(d)
1.0

0.0

25

(c)

1.0

0.0

20

Time [ms](b)
Fractional population

15

-2

-1

0

+1

+2

position along es

Figure 3: Calibration of synthetic dimensions lattice. (a) Ramping procedure. The
blue line represents the 1-D lattice depth as a function of time and the red line
represents Raman coupling as a function of time. Both are held on for a variable
amount of time t, producing Rabi oscillations. (b) Example of fractional populations
in different m states as a function of time t in the F = 1 manifold. Dots indicate
data and lines indicate the best fit to theory, with parameters h̄Ω = 0.56 ± 0.01EL
and h̄δ = 0.029 ± 0.002EL . The quadratic shift was constrained at the calibrated
value h̄ = 0.02EL . (c) Example time-of-flight image in the F = 1 manifold. A
Stern-Gerlach gradient pulse separates different m states along the horizontal axis,
while the lattice and Raman beams give momentum along the vertical axis. (d)Example of fractional populations in different m states as a function of time t in the
F = 2 manifold. Dots indicate data and lines indicate the best fit to theory, withparameters h̄Ω = 0.61±0.002EL and h̄δ = 0.002±0.001EL . The quadratic shift wasconstrained at the calibrated value h̄ = 0.02EL . (e) Example time-of-flight image
in the F = 2 manifold. A Stern-Gerlach gradient pulse separates different m statesalong the horizontal axis, while the lattice132and Raman beams give momentum along
the vertical axis.

Figure 3c,d shows sample time-of-flight images during the calibration procedure for F = 1 and F = 2 respectively. The vertical axis is ex , aligned with thelattice and Raman beams. Since the atoms have expanded in time-of-flight, this axis
corresponds to the momentum kx . The horizontal axis of the image is the axis along
which a Stern-Gerlach magnetic field gradient, separating the different spin states, isapplied. Therefore, this axis is the position m along the synthetic dimension es . In
the effectively 2-D synthetic dimensions lattice language, this is a ‘hybrid’ imaging
technique, imaging momentum along one lattice direction and position along theother.
Figure 3c labels some notable momentum orders. The central order is at
kx = 0, where the atoms start before the experiment. Two higher lattice orders,
at kx = ±2kL , are populated for the same spin m = 0. kx = ±2kR is labeled, but
not visibly populated, to indicate where the orders would appear of only Raman
coupling was present with a higher coupling strength. Due to the ’folding in’ effectof the lattice, the brightest orders of the m = ±1 states appear at kx = ±(2kL −2kR ).
The F = 2 states follow the same pattern, not labeled in Figure 3e as there are too
many orders.
For each value of the time t we sum up the total optical depth in all of theorders of each spin state to obtain fractional populations for each spin state as afunction of time. An example scan in the F = 1 manifold is shown in Figure 3 b. The
colored dots represent the data for different spin states, and the lines represent thebest fit to theory. Here, the significant detuning makes populations in the m = ±1
states unequal. An example scan in the F = 2 manifold is shown in Figure 3e. Here,the detuning is small and states with opposite spin oscillate in approximate unison.
This technique allows us to calibrate our experimental parameters.

133

6.2.4 Tight binding approximation
The synthetic dimensions Hamiltonian can be approximated in the tight binding limit as:

H=−

X
j,m

|tx |eiφAB m |j + 1, mi hj, m| + ts (m)|j, m + 1ihj, m| + Am |j, mihj, m| + h.c.,(6.9)

where j and m label sites along ex and es respectively, as shown in Figure 1b.
ts = |ts | and tx = |tx |exp(iφAB m) are the associated tunnelings. Am captures
the spin dependent diagonal elements, detuning h̄δ and quadratic shift h̄. Here, wehave implicitly restricted ourselves to the lowest 1-D lattice band, and assumed thattight binding, i.e. confinement at discrete lattice sites, is a good approximation (see2.5.2). ts is not a spin dependent quantity for F = 1 atoms, but is for F = 2, where
differences in Clebsch-Gordan coefficients create non-uniform tunneling. In the limitof zero detuning and neglecting the quadratic shift as well as the ts dependence
on spin, this becomes the traditional Harper-Hofstadter Hamiltonian 1.7 with asynthetic dimension:

H=−

X
j,m

tx eiφAB m |j + 1, mi hj, m| + ts |j, m + 1ihj, m| + h.c.

(6.10)

We can transform this Hamiltonian into momentum space along ex by plugging
the Fourier transform formula
1 X −ikj je|kj , mi,|j, mi = √
N kj

134

(6.11)

(a)(b)

Tight binding
Full Hamiltonian

−0.70

−0.90

−0.75

Energy [EL ]Tight binding
Full Hamiltonian

−0.95

−0.80
−0.85

−1.00

−0.90

−1.05

−0.95

−1.10

−1.00
−1.0

−0.5

0.0

0.5

1.0

Crystal momentum qx [kL ]

−1.0

−0.5

0.0

0.5

1.0

Crystal momentum qx [kL ]Figure 4: Band structure of the tight binding versus full Hamiltonian. V0 = 6.0EL ,giving |tx | = 0.1EL , h̄δ = 0, h̄ = 0.02EL , h̄Ω = 0.5EL . (a) F = 1, fitted value ts =0.154EL . (b) F = 2, fitted value ts = 0.284. Here, the tight-binding Hamiltonian
included anisotropic tunneling ts due to Clebsch-Gordan coefficients.
with N the number of sites j, into the above Hamiltonian to obtain

H=−

1 X
ts |kj , m+1ihkj , m|+h.c.+2|tx |cos(kj −φAB )|kj , mihkj , m|+Am |kj , mihkj , m|
N k ,mj(6.12)
To approximate the full Hamiltonian, eqn. 6.6, by the tight binding Hamiltonian, eqn. 6.9, we must find appropriate values for ts and tx . We find |tx | by treating
the 1-D lattice independently, and matching the tight binding band to the lowest
full lattice band. For most of the experiments described in the chapter, the lattice
depth was V0 = 6EL , corresponding to |tx | ≈ 0.01EL . To find the appropriate valueof ts , we fit the lowest 3 bands of the full synthetic dimensions band structure to
the tight binding band structure eqn. 6.12 with ts as fitting free parameter.
Figure 4 shows the overlayed band structure of the full Hamiltonian, eqn.
6.6, and the best fit tight binding band structure, eqn. 6.12. To fit, we minimize
the square difference between the energies in the lowest two bands, relevant to our135experiment.

6.3 Eigenstates of the synthetic 2-D lattice
After calibrating the synthetic dimensional lattice via pulsing, we can study the
eigenstates of the lowest band of the system by adiabatically loading, i.e. ramping
both the lattice and Raman or rf coupling on on a time scale slow compared to themagnetic band spacing. Along the synthetic direction, in the F = 1 manifold, there
are no m = ±2 sites. This can be thought of as hard wall boundary conditions at
the m = ±2 sites, confining the atoms in the allowed m = 0, ±1 sites. Therefore,
we can consider the position eigenstates along the synthetic direction in relation toeigenstates of a square-well potential.
Figure 5a shows a time-of-flight image of an adiabatically loaded synthetic
dimensions lattice eigenstate with rf coupling along the synthetic direction. The
vertical axis is single site resolved spin states m. The horizontal axis is momentum
along the ex direction. Note that for each site m the distribution of momenta kx issymmetric. Figure 5b shows the fractional population in each site m, summed over
all momenta kx . In the case of rf coupling, φAB = 0 and the effective magnetic fluxΦAB /Φ0 = 0. Therefore, the fractional population along the spin direction looks
simply like a discretized ground state probability distribution of the square wellpotential.
Figure 5c-h shows analogous data with Raman coupling along the synthetic
direction. Figure 5d,g are the time-of-flight image and corresponding fractional
populations of atoms adiabatically loaded from the mF = 0 spin state, corresponding
to the central minimum (q = 0) of the lowest band in Figure 2b. There are two key
differences between this case and the rf case in Figure 5a-b. First, the momenta ofthe different spin states are no longer symmetric, as explained in sec. 6.2.3. Second,
the fractional populations in Figure 5g are no longer simply the discretized ground
136

(b)

Fractional population, nm

(a) 2
1
0
-1
-2

(c) 2

(f)

1
0
-1
-2

Lattice site, m

Fractional population, nm

(g)

(d)2
1
0
-1
-2

(h)

(e) 2
1
0
-1
-2

-2

0

2

-2

-1
0
1
Lattice site, m

2

Figure 5: Eigenstates of the synthetic dimensions lattice. Left column: time-offlight images, with position along es on the vertical axis and momentum along
es on the horizontal. Right column: fractional populations in each site m. (a,b)
rf coupling, resulting in φAB = 0. (c,f) Raman coupling, resulting in φAB > 0,
adiabatically loaded from the mF = 1 state. (d,g) Raman coupling, resulting in
φAB > 0, adiabatically loaded from the mF = 0 state. (c,f) Raman coupling,
resulting in φAB > 0, adiabatically loaded from the mF = −1 state.

137

state probability distribution of the square well potential: they are a narrowed
version of it, more strongly concentrated in the m = 0 site.
This can be understood by analogy with a 2D electron system in a perpendicular magnetic field, confined in one dimension with hard walls. Along the confined
direction, the wavefunction is localized to the scale of the magnetic length lB =ph̄/qB, with the center position at kx lB2 in the bulk state, where h̄kx is the electronspcanonical momentum. In our system, the magnetic length lB = a2 Φ0 /2πΦAB , orpin units of the lattice period a, lB∗ = 3/2π; this explains the narrowing of the bulk
state in Figure 5g.
In the 2D electron system, at large |kx |, the electron becomes localized near theedges, lifting the degeneracy of the otherwise macroscopically degenerate Landaulevels. In our case, stable edge states appeared as the additional minima in Figure
2b, at q ≈ ±0.66kL . We loaded these edge states by starting in the mF = ±1
states before adiabatically turning on the synthetic dimensions lattice to obtain theeignestates displayed in Figure 5c,f and Figure 5e,h respectively. These edge states
predominantly occupy the edge sites in the synthetic direction, and are strongly
confined there due to the narrow magnetic length. These localized edge states are
the analog to the current-carrying edge states in Fermionic integer quantum Hall
effect systems [114].

6.4 Chiral edge currents
The same pulsing procedure that was used for calibration (sec. 6.2.3) can
also be interpreted by analogy with the 2-D electron system. Figure 6a shows
schematically what happens when atoms are loaded from the m = 0 site into thelattice and tunneling along the synthetic dimension is pulsed on. Atoms begin
analogues of cyclotron orbits, tunneling out into the edge m = ±1 sites and tunnelingback to the bulk m = 0 state. The fractional populations in the three m sites as a138

function of time are shown in Figure 6b.
We performed this experiment for three different magnetic flux values: with rf
coupling giving ΦAB /Φ0 = 0, with Raman coupling giving ΦAB /Φ0 ≈ 4/3 and withinverted Raman coupling giving ΦAB /Φ0 ≈ −4/3. The inverted Raman coupling
was accomplished by switching the roles of the two Raman beams (see Figure 1a):the right going beam frequency was changed to 2π(ω + ∆ω) and the left going beamfrequencdy was change to 2πω, resulting in the opposite recoil momentum for thesame spin flip, flipping the direction of the effective magnetic field.
We define the current Im=±1 = nm hvm i, where nm is the fractional populationin site m and hvm i is the expectation value of velocity along ex for atoms in sites
m, as depicted in Figure 6a. The velocity is derived from the momentum measured
in time-of-flight images. The chiral current of the system is then defined as I =I1 − I−1 . We calculate this chiral current for data in Figure 6b, with ΦAB /Φ0 ≈
4/3, displayed in red dots in Figure 6c. Atoms in the edge sites m = ±1 exhibit
chiral motion, therefore the resulting chiral current is directly proportional to thefractional population in those sites and oscillates as a function of time in concert
with the oscillation in Figure 6b. in Figure 6c also includes data for the ΦAB /Φ0 ≈−4/3 (solid black dots indicate data and solid gray lines are from theory) andΦAB /Φ0 = 0 (empty black dots). As seen in the figure, reversing the direction of theeffective magnetic flux reverses the direction of the chiral currrent, and turning off
the magnetic flux results in no net chiral current. The chiral current I is normalized
here by the tunneling velocity 2tx /h̄kL .
As the chiral current I is proportional to the edge state population, we plot
it as a function of the expectation value of the absolute value of m, h|m|i, in Figure 6d. As expected, the chiral current is linear, with slope S. S is positive forΦAB /Φ0 ≈ 4/3, negative for ΦAB /Φ0 ≈ −4/3, and zero for ΦAB /Φ0 = 0. We thenstudy the dependence of the chiral current on the strength of tunneling along the

139

(a)

(c)

1

3
2

0.8

1

0.6
in units ofFractional population, nm

(b)

0.4

0
-1
-2

0.2

-3

0
0

1

2
Time,

(ms)

3

4

(d)

0

1

2
Time,

(ms)

3

4

(e)
3

3

2

0
-1

2

6

in units ofin units ofin units of

1

1
-2

4
2
0

-3
0

0.1

0.2

0.3

0.4

0.5

0

0

1

2

0

1

2

3

3

4

4

Figure 6: Measuring chiral currents in synthetic dimensions. (a) Schematic of theformation of chiral currents when the system is loaded into the bulk m = 0 site andtunneling along es is turned on suddenly. They start to tunnel both towards them = 1 sites (pink arrows), moving towards the right along ex , and the m = −1
sites (blue arrows), moving towards the left along ex . They then return to m = 0,
completing cyclotron orbits. (b) Fractional population in each spin state (m = 0,
+1, and −1 in green, pink, and blue respectively) as a function of time for a systemwith φAB > 0. Dots represent data and lines represent theory calculated from the full
Hamiltonian, eqn. 6.2.1, with parameters h̄Ω = 0.5EL , V0 = 6EL , h̄δ = 0.001EL ,and h̄ = 0.05EL . (c) Chiral current I as a function of time for φAB > 0 (red)φAB = 0 (empty black dots) and φAB < 0 (solid black). (d) Chiral current I asa function of h|m|i for the three values of φAB . Solid lines calculated from theory,
with the same parameters as in (b) for φAB 6= 0, and with parameters h̄Ω = 033EL ,V0 = 6EL , h̄δ = −0.01EL , and h̄ = 0.05EL for φAB = 0. (e) Peak chiral current
Imax as a function of tunneling asymmetry ts /tx , as measured experimentally (reddots) and calculated from the full Hamiltonina (red line). Inset: slope of best fit linesof current I as a function of h|m|i (as in (d)) as a function of tunneling asymmetry
ts /tx : nearly independent.
140synthetic dimension, in units of the real axis tunneling ts /tx . We refer to this as
the tunneling anisotropy: the asymmetry between the two dimensions. As shownin the inset to Figure 6e, the slope S of the chiral current as a function of h|m|i
is practically independent of the tunneling anisotropy. The small deviation from aflat line is explained by the deviation of our system from the tight binding model.
However, the maximal chiral current attained during the pulsing experiment, Imax ,depends strongly on the tunneling anisotropy (see Figure 6e). This is because themaximum fractional population in the edge states h|m|i increases with increased ts .
The increase is approximately linear at first, and then saturates at large ts /tx when
the fractional population in the edge states m = ±1 approaches 1.

6.5 Observation of skipping orbits
Semi-classically, electrons in a 2-D material pierced by a magnetic field canbe described in terms of cyclotron orbits in the bulk, as described in the previous
section, and skipping orbits on the edge. Skipping orbits arise from electrons on
the edge beginning cyclotron orbits, but hitting the edge of the system and beingreflected and beginning the next cyclotron orbit. Due to the chirality of the cyclotron
orbits, this results in the skipping orbits traveling in one direction along the top edge
and in the opposite direction along the bottom edge.
We observed an analogue of these skipping orbits in our system. We performed
the same experiment, pulsing on tunneling along the synthetic dimension, but thistime initializing the system on the edge, as shown schematically in Figure 7a. To
populate these states, we initially applied a detuning h̄δ = ±0.087EL , tilting thepotential along the synthetic direction as shown in Figure 7b. This made the initialstate, m = −1 in the figure, a potential minimum. We then pulsed on the tunnelingand observed the resulting dynamics.
Figure 7c shows the expectation value of position along es as a function of time
141

(b)

Energy(a)(c)

mLattice site

Lattice site m

2
Time,

3
(ms)

+2

0.1

0.5
0

0

-0.5

(e)

+1

m

(d)

1

-1

0

-1

-2

-0.1
0

1

2
Time,

3
(ms)

4

−1

0
Displacement

0

1

4

1

0

-1

−2

1

2

Figure 7: Imaging skipping orbits. (a) Schematic of pulsing experiment when atoms
are initialized on the m = −1 edge. The atoms move towards the m = 0 site,while moving to the right along ex (blue arrows). They then continue their semicyclotron orbits back to m = −1, from where they cannot finish the circle, forming
skipping orbits. (b) Schematic of the tilted box potential applied along the synthetic
direction. (c) Expectation value of position along es , hmi, as a function of pulse time
for atoms initialized in the m = +1 (red) and m = −1 (blue) states. Dots represent
data and lines are from theory calculated from the full Hamiltonian, eqn. 6.2.1,
with parameters h̄Ω = 0.41EL , V0 = 5.2EL , h̄δ = ±0.087EL , and h̄ = 0.13EL .
(d) Expectation value of the group velocity along ex , hvx i, for the same data as in(c). (e) Expectation value of displacement along ex , hδji in units of lattice spacing,for the same data as in (c) and (d). The displacement was obtained by integrating
hvx /ai, where a is the period of the optical lattice. Atoms initialized in m = −1
performed skipping orbits to the left, while atoms starting in m = −1 traveled to
the right.
142during the pulsing experiment. This expectation value is obtained by calculatingP
the fractional population nm on each site m and summing hmi = m mnm . The red
dots were obtained from an experiment where the atoms were initialized in the m = 1
site. The blue dots were obtained by starting in the m = −1 site. The expected
position oscillated with time, as expected for Rabi oscillations. The same data wasPthen used to extract the expected group velocity along ex , hvi = m nm hvm i as afunction of time. This is shown in Figure 7d. The group velocity oscillated with the
expected position hmi, and was positive for experiments starting in m = −1 andnegative for experiments starting in m = 1.
We obtain the expected displacement in units of the lattice spacing a, hδji
along ex as a function of time by directly integrating the expected group velocity.
The resulting displacement is shown in Figure 7e. As seen in the figure, for experiments initialized in m = 1, the atoms began cyclotron orbits, but reflected off
the edge and performed skipping orbits towards the left. Likewise, atoms initialized
in m = −1 performed skipping orbits along the opposite edge and in the opposite
direction. This experiment presents the first direct observation of skipping orbit
motion.

143

Chapter 7: Measuring Chern Number in Synthetic Dimesnions

As discussed in Chapter 1, the 2D topological invariant as applied to band
topology, the Chern number, is well defined for an infinite 2D system. For any finite
system, the Chern number can be thought of as the Chern number of an infinite
system that locally looks like the bulk of the finite system. This begs the question,
how narrow can a system get for this extension of the definition of Chern number
to still be meaningful?
In this chapter, we describe our experiment in measuring Chern number in theeffectively 2D synthetic dimensional lattice, as described in Chapter 6. This lattice
was elongated along the real direction and extremely narrow along the synthetic
direction, only 3 or 5 sites wide. We performed a transport experiment to sample
the band structure of this system as a function of crystal momentum kx along thereal direction, and observed the resulting motion along the transverse, synthetic,
direction, as shown schematically in Figure 1.
Similarly to Chapter 6, our system was qualitatively well described by thetight binding limit Harper-Hofstadter Hamiltonian (see eqn. 6.10) [16, 115]Ĥ = −

Xm,j
tx eiφm |j, mihj + 1, m| + ts |j, mihj, m + 1| + H.c.,(7.1)

where j and m label lattice sites along ex and es , with tunneling strengths tx andts respectively. Figure 1a sketches the atoms loaded into the lowest band of thesynthetic dimensions lattice in the F = 2 manifold of 87 Rb, creating 5 sites along144

the synthetic direction. As in Chapter 6, there was a 1D optical lattice along thelongitudinal ex direction, with lattice tunneling tx between adjacent sites labelled by
j. Internal spin states of the atoms defined sites along the transverse es direction,with tunneling ts induced by either rf or Raman coupling. In the case of rf coupling,
no phase was imprinted, φAB = 0. In the case of Raman coupling, an overall phaseφAB 6= 0 was imprinted, and we choose the Landau gauge in which the phase is
written on the longitudinal tunneling coefficient tx = |tx |eiφAB m .
The histogram in Figure 1a shows the fractional populations nm in each sitem. An example hybrid TOF image (see sec. 6.2.3) is shown below in Figure 1c,with the central m = 0 order marked by red cross-hairs. Then, a force is appliedalong the long (real) dimension of the system for some time ∆t, and a transverse
response is observed, as seen in Figure 1b. The fractional population has become
maximized in the m = 1 site, and the sample hybrid TOF image in Figure 1d showsa displacement along the longitudinal momentum axis of ∆qx .
Due to the extremely narrow widths in the synthetic dimension, our measurement cannot be readily interpreted as a quantum Hall conductivity measurement,
and a meaningful Chern number is not readily extracted in that manner, as discussed in sec. 7.2. However, we leverage the TKNN Diophantine equation [6] toperform an alternative measurement of the Chern number, discussed in 7.3. Weshow how this equation arises naturally in our synthetic dimensional system, andclaim that with this metric we can extend the definition of the Chern meaningfully
to systems as narrow as ours.

7.1 Experimental procedure
The setup for this experiment followed closely that of the original synthetic
dimensions experiment, as described in sec. 6.1. In contrast with the experimentdescribed in Chapter 6, we performed this experiment in both the F = 1 and F = 2
145

-2

0
Wavevector kx [kL ]

2

-2

0
Wavevector kx [kL ]2Figure 1: Quantum Hall effect in Hofstadter ribbons. (a) 5-site wide ribbon withreal tunneling coefficients along es and complex tunneling coefficients along ex , creating a non-zero phase φ around each plaquette. (b) After applying a force along
ex for a time ∆t, atomic populations shift transversely along es , signaling the Hall
effect. (c,d) TOF absorption images giving hybrid momentum/position density distributions n(kx , m). Prior to applying the force (c), the m = 0 momentum peak is
at kx = 0, marked by the red cross. Then, in (d), the force directly changed qx ,evidenced by the displacement ∆qx of crystal momentum, and via the Hall effectshifted population along es .

146

hyperfine manifolds of 87 Rb, creating both 3-site and 5-site wide strips. We began
the experiment by adiabatically loading the full synthetic dimensional system forboth F = 1 and F = 2, with both Raman coupling (ΦAB 6= 0) and rf coupling(ΦAB = 0).

7.1.1 Loading procedures
The loading procedure for the Raman coupled case was as follows. We prepareda 87 Rb BEC (see sec. 3.3.3) in the mF = −1 hyperfine state of the F = 1 manifold,or in the |1, 0i state for Raman data in F = 1 only. We ramped on the 1D opticallattice along ex adiabatically in 300 ms. For experiments in the F = 2 manifold, wethen applied a microwave pulse for 0.55 ms to transfer |1, −1i → |2, 0i manifold whilealready in the lattice. This was done after lattice loading to minimize the amountof time the atoms spent in the F = 2 manifold and thereby limit the spin-changing
collisions and losses associated with that state as observed in our experiment. Then,the Raman coupling field was ramped on adiabatically in 30 ms.
Loading the ground state of the F = 2 manifold presented a unique restriction,
as illustrated by Figure 2. For the experiments described in Chapter 6, a bias
magnetic field Bz was chosen such that the quadratic Zeeman shift was h̄ = 0.05EL ,in lattice recoil energy units. However, in the F = 2 case at this field, with a lattice
depth of 4.4EL , the mF = 0 ground state does not adiabatically connect to theRaman-coupled ground state. This is because, as shown in the band structure in
Figure 2a, the mF = ±2 energies of the lattice-coupled system at qx = 0 are lower
in energy than the mF = 0 state. Therefore, to avoid this issue, we chose a bias
field such that the quadratic Zeeman shift was h̄ = 0.02EL . The band structure forthis case is shown in Figure 2b. Here, the mF = 0 sinusoid at crystal momentum
qx = 0 is still the lowest energy, and therefore connects adiabatically to the Raman
coupled ground state.
147

(a)

(b)
−0.25

−0.3

Energy [EL ]

2

−0.30

1

−0.35

−0.4

−0.40
−0.5

−0.50

−0.6

−1

−0.55

−0.7
−1.0

0

−0.45

−0.60
−0.5

0.0

q [kL ]

0.5

1.0

−1.0

−0.5

0.0

0.5

1.0

−2

q [kL ]

Figure 2: Band structure of the lattice-coupled system in F = 2. Here, lattice depth
V0 = 4.4EL , Raman coupling h̄Ω = 0, and detuning h̄δ = 0. (a) Quadratic shifth̄ = 0.05EL . At qx = 0, mF = 0 is not the ground state. (b) Quadratic shifth̄ = 0.02EL . At qx = 0, mF = 0 is the ground state.
For the case where tunneling along the synthetic dimension was provided byrf coupling (ΦAB = 0), the loading procedure was as follows. We started in the
F = 1, mF = −1 state and turned the 1D optical lattice on adiabatically in 300 ms,
same as in the Raman case. For F = 2, we then transferred to the F = 2, mF = −2state with a 0.55 ms microwave pulse. Then, we set the bias field to a large detuningh̄δ > 1ER . Implementing adiabatic rapid passage (see sec. 2.2.3), we then swept
the field to resonance in ≈ 50 ms, thereby loading the rf coupled ground state.
This was necessary in the rf case as opposed to the Raman case, because there is
no momentum shift in the band structure, and therefore turning on the rf coupling
opens up an avoided crossing at every point in the band, making it impossible to beadiabatic with respect to this turn-on while on resonance.

1487.1.2 Application of force and measurement
After adiabatically loading the ground state at qx = 0 for all the configurations,
we applied a constant force to the atoms, inducing a linear evolution of the crystal
momentum given by
Fx = h̄

dqx
.
dt

(7.2)

To apply this force, we displaced the crossing beam of the ODT, such thatinstead of being held at the potential minimum, the atoms were on an edge ofthe Gaussian beam, with a locally linear optical potential. This displacement wasachieved by frequency shifting the AOM that controlled the split between the two
ODT orders (see fig. 7). Note that the ODT is far detuned and this frequency
shift of < 1 MHz had no effect on the trapping potential depth. This is shown
schematically in Figure 3. Figure 3a shows the atoms (indicated by a black dot)at the minimum of the Gaussian crossing beam (potential in blue), with no force
applied. Figure 3b shows the beam displaced to the right relative to the atoms,
and the locally linear potential experienced by the atoms results in a positive force.
Similarly, in Figure 3c the beam is displaced to the left resulting in a negative force.
An example of the resulting evolution of the system is shown in Figure 4.
Figure 4a shows the lowest three bands of the Raman coupled synthetic dimensions
band structure in the F = 2 manifold, for our approximate experimental parametersof V0 = 4.4EL , h̄Ω = 0.5EL , h̄δ = 0EL and h̄ = 0.02EL . The starting point of the
BEC at qx = 0 is indicated by the black dot in the lowest band. The color indicatesthe modal position m̄ along the synthetic direction. Modal position is a slightvariant on the magnetization used in sec. 6.2.2. The modal position was foundby taking the fractional populations nm in the different m sites at a given crystal
momentum, and fitting them to a Gaussian distribution. The peak of the Gaussian
distribution was taken as the modal position m̄. In this experiment, we use m̄ as the149

(a) F = 0(b) F = |Fx |(b) F = −|Fx |Figure 3: Application of a constant force by displacing a Gaussian beam potential.
(a) Atoms are at the minimum of the Gaussian potential, not experiencing a force.
(b) Beam is displaced to the right, atoms experience a local linear potential resulting
in a constant positive force. (c) Beam is displaced to the left, atoms experience a
local linear potential resulting in a constant negative force.
metric of location along es in favor of the more conventional magnetization given byP
hmi = m mnm to avoid the large uncertainties introduced in the magnetizationfrom number fluctuations in the extremal m = ±F sites.
We displaced the crossing ODT beam to one side to apply a positive force,
inducing motion to the right qx → qx > 0 in the band structure, for various amountsof time ∆t inducing various changes in crystal momentum ∆qx . We then followed
the same measurement protocol as described in sec. 6.2.3: we abruptly turned off
the lattice, Raman or rf, and trapping ODT beams and allowed the atoms to expandin time-of-flight for 16 ms, mapping initial momentum kx to position on the TOF
images. During TOF, we applied a 2 ms Stern-Gerlach gradient pulse of ≈ 39 G/cm
(80 A quad coil current), separating the atoms according to site m along an axis
perpendicular to ex . We therefore performed a hybrid measurment of momentum kx
along the longitudinal ex axis and single site resolved position m along the transverse
es direction.
150

−2

−1

Modal position m̄012

Modal position m̄
0

1

2

0

Energy [tx ]

−1

−2
−2

−1

Energy [tx ]

0
−3
−1
−4
−0.50

2

−0.25

0.00
Crystal momentum qx [kL ]

0.25

0.50

−0.25

0.00
Crystal momentum qx [kL ]

0.25

0.50

−2

1
0

−3

Modal position m̄

−1

−2
−4
2
−0.50
1
0

Figure−14: Band structure in a 5-site wide ribbon. (a) Band structure computed
using −2
full Hamiltonian for a 4.4EL deep 1-D lattice (λL = 1064 nm), 0.5EL Raman
coupling2 strength (λR = 790 nm), and quadratic Zeeman shift  = 0.02EL , giving
Φ/Φ0 ≈ 4/3, tx = 0.078(2)EL , ts = 2.3(1)tx . The color indicates modal position m̄.
1
The black dot indicates the initial loading parameters. (b) TOF absorption imagesn(kx , m)0 for varying longitudinal crystal momenta qx .
−1

−2
-0.5

-0.25
0
0.25
Crystal momentum qx [2kL ]

151

0.5

From these hybrid TOF images, we extracted the fractional population in eachsite m. In addition, from the change in position along the momentum axis kx we
detected the crystal momentum ∆qx for each image, as shown in Figure 1c,d. We
therefore observed the evolution in the fractional populations and modal position as
a function of qx . Sample TOF images at different values of the crystal momentum areshown in Figure 4, with different single-site resolved imaging along es represented
in the vertical direction, different m sites shaded in different colors, corresponding
to the colorbar in Figure 4a. Similarly, the ODT crossing beam was displaced to the
opposite side an equal amount, applying the same magnitude of force in the opposite
direction and including motion to the left qx → qx < 0 in the band structure. In thisway, we obtained a complete map of the fractional populations nm in each transversesite m for each value of the longitudinal crystal momentum qx .

7.1.3 Density reduction
When force was applied to the atoms, their crystal momentum qx evolved, andthey were no longer confined to a minimum of the band structure. This opened up
the possibility for two-body collisions between the atoms that conserved the overall
momentum and energy of the pair while changing the crystal momentum of each
atom [116]. This lead to a smearing of the atoms along the crystal momentum axis,
obscuring the measurement.
To mitigate this problem, it was necessary to reduce the atomic density asmuch as possible while retaining enough atoms to observe a signal. In addition to reshaping the ODT beam (see sec. 3.3.4.4), we also cut down the overall atom number
after creating the BEC before loading the synthetic dimensional lattice. Starting in
F = 1, mF = 0, we applied a microwave pulse resonant with the |F = 1, mF = 0i →|F = 2, mF = −1i transition for 355 µs, until ≈ 80% of the atoms were transferredto the F = 2 manifold. We then shined on the XZ probe beam to selectively blow
152

away the F = 2 atoms, leaving ≈ 1000 atoms in the condensate.

7.1.4 Rf correction
For rf-coupled experiments, our loading procedure into the lowest band at
qx = 0 resulted in some latent non-adiabaticity that we could not get rid of or fully
explain. This non-adiabaticity resulted in oscillations in the fractional populations
nm as a function of time, even though the rf-coupled band structure predicts nodependence of nm on qx . However, these oscillations were present as a function oftime even when no force was applied to the atoms, indicating that these variations
were dependent on time and on not qx . Therefore, we used data obtained with noforce applied to correct the time-dependent variation in data where the force wasapplied.
Our correction procedure is shown in Figure 5. Figure 5a-c show the raw
fractional populations as a function of time for the case where a force was applied
to the right, to the left, and not at all, respectively. Note that the observed oscillations are quite large, but consistent between the three cases. This implies thatwhatever mechanism is causing these oscillations is a time dependent one and not
a consequence of any change in longitudinal crystal momentum qx .
To correct the data and extract the change in fractional populations caused byvarying qx , we first calculate the change in fractional population as a function of time
for the data with no force applied. This is given by ∆nFm=0 (t) = nFm=0 (t) − nFm=0 (0).
We then subtract this change from the data with force applied via nFm6=0,corrected (t) =nFm6=0 (t) − ∆nFm=0 (t). These corrected fractional populations as a function of time
are shown in Figure 5d,e for positive and negative force, respectively. In Figure 5f,
we have combined the corrected positive and negative force data and plotted it as
a function of crystal momentum, effectively mapping out the fractional populations
everywhere in the lowest band.
153

(a) negative force(c) no force

mF=+2
mF=+1
mF=0
mF=-1
mF=-2

0.35

Fractional populations(b) positive force

0.30
0.25
0.20
0.15
0.10
0.05

Corrected fractional populations

0.000

0.001

0.002

0.003

0.000

0.001

0.002

0.003

0.000

0.001

0.002

0.003

Time [s]Time [s]Time [s](d) negative force(e) positive force(f) combined

0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.000

0.001

0.002

0.003

0.000

0.001

Time [s]0.002Time [s]

0.003

−1

0

1

crystal momentum [kL ]Figure 5: Correction of oscillations in rf coupled data. (a,b,c) Raw fractional populations nm observed as a function of time ∆t with (a) positive force applied, (b) negative force applied, and (c) no force applied. (d,e) Corrected fractional populations
ncorrectedwith (d) positive force applied, (e) negative force applied. (f) Correctedmdata as a function of crystal momentum for both force directions combined.

154

7.2 Quantum Hall Effect interpretation
The measurement we performed is similar to a quantum Hall conductivity
measurement: we pierced a 2D material with an effective magnetic field, applied
a force (an electric field in a conventional quantum Hall setup), and observed atransverse response. To draw a more direct analogy, we can describe the quantum
Hall effect(QHE) from the microscopic perspective.

7.2.1 Microscopic view of QHE
In the quantum Hall effect, a longitudinal force Fk , induced by an electric field
Fk = eEk , drives a transverse ‘Hall’ current densityj⊥ = σH Ek ,(7.3)

where σH is the Hall conductivity. This transverse current density can be expressedasj⊥ = n2D v⊥ e,(7.4)

where n2D is the 2-D charge carrier density, v⊥ is the transverse velocity of thecharge carriers and e is the electron charge. Choosing some increment of time ∆t,
we can express v⊥ and Fk as
v⊥ =

∆x⊥
,
∆t

(7.5)

∆qk
,
∆t

(7.6)

and
Fk = h̄where qk is the crystal momentum along the direction of the force. Plugging this
into eqn. 7.3, we obtainn2D e∆qk σH
x⊥
= h̄
∆t
∆t e
155

(7.7)

Re-expressing n2D in number of carriers N per plaquette, defining ∆x⊥ as transverse
displacement in units of lattice periods, we plug the above definitions into eqn. 7.3
to obtainNG∆x⊥h= σH 2 ,
∆qk
e

(7.8)

where G is the reciprocal lattice vector.
In addition, we know that the quantum Hall conductivity can be expressed interms of the Chern numbers Cn of the occupied bands n as

σH =

e2 X
Cn .
h n

(7.9)

In the conventional case of Landau levels, where each Landau level has a Chern
number Cn = 1, this amounts to the number of filled bands ν, giving σH = νe2 /h.
In our case, where the Chern number is not necessarily 1 and only the lowest bandis occupied, this instead reduces to the Chern number of the lowest band which wewill just call C, giving σH = Ce2 /h. Plugging this into eqn. 7.8, we can write

NG

∆x⊥
=C
∆qk

(7.10)

Therefore, if we observe (as was done in our measurement) the ratio of transverse displacement ∆x⊥ to the longitudinal crystal momentum ∆qk , we should be
able to fit that to a line and directly extract the Chern number C.

7.2.2 Chern number from Hall conductivity
We used the prescription given in the previous section to extract a Hall conductivity and therefore a Chern number from our data. The results for a 5-sitewide strip (F = 2) are shown in Figure 6a,b,c (top) for all values of flux studied:Φ/Φ0 = 0, ≈ −4/3, and ≈ 4/3, respectively. We calculated the modal position
156

Modal position m̄

2

2

2

1

1

1

0

0

0

−1

−1

−1

Conductivity σ̃H

−2
−2
−2
-0.5 -0.25
00.25
0.5 -0.5 -0.25
00.25
0.5 -0.5 -0.25
0
0.25
0.5
Crystal momentum qx [2kL ]Crystal momentum qx [2kL ]Crystal momentum qx [2kL ]
0.5

1.0

0.0

0.0

0.5

−0.5

−0.5

0

0.25
Max |qx | in fit

0.5

0.0

0

0.25
Max |qx | in fit

−1.0
0.5
0

0.25
Max |qx | in fit

0.5

Figure 6: Hall displacement. Top: modal position m̄ is plotted as a function of qx
for the 5-site ribbon with flux a. Φ/Φ0 = 0, b. Φ/Φ0 ≈ −4/3, c. Φ/Φ0 ≈ 4/3.
Gray circles depict the measurements; black dashed lines are the prediction of oursimple σ̃H and red curves are the expectation from the band structure of our thinribbon. Bottom: Extracted conductivity from the slope of a line of best fit to thedata (gray circles) and theory (red lines) as a function of maximum |qx | included in
the fit range, for each flux value. As discussed in sec. 7.1.4, the Φ/Φ0 = 0 data wascompensated to account for non-adiabaticity in the loading procedure.
m̄ (see sec. 7.1.2) of our atoms along es as a function of longitudinal crystal momentum qx . The data is represented by gray dots, with uncertainty bars reflecting
the propagated standard uncertainty from averaging six identical runs. For zero
flux Φ/Φ0 = 0 (Fig. 6a), m̄ was independent of qx ; in contrast, for non-zero flux
Φ/Φ0 ≈ ±4/3 (Fig. 6b,c), m̄ depends linearly on qx with non-zero slope.
Here, the change in m̄ was a transverse displacement ∆x⊥ . We fit the modal
position as a function of qx to a line (black dashes in the figure). From the slope ofthe line, we used eqn. 7.10 to extract the Chern number. We obtained C = 0.01(1),0.87(3), and −0.85(3) for zero, negative and positive flux respectively. This shows
the correct qualitative behavior, but differs significantly from expected values of
0, ±1.
157

7.2.3 Inadequacy for narrow systems
The red curves in Figure 6(top) show the expected behavior for our 5-site wide
system for adiabatic changes in qx as calculated from exact diagonalization of thefull Hamiltonian (see sec. 6.2.1), always within the lowest band (Fig. 4a), i.e., Blochoscillations. This theory predicts a nearly linear slope for small qx sharply returning
to m̄ = 0 at the edges of the Brillouin zone. A linear fit to this theory producesσ̃H ≈ 0, 0.6, and −0.6 for zero, negative and positive flux respectively, far from the
Chern number. In addition, they differ significantly both from the data and fromthe linear fit at the edges of the Brillouin zone.
This discrepancy is resolved by recalling that Bloch oscillations require adiabatic motion, meaning no transition to higher bands can happen. This is possible
in the narrow system under study, but quickly becomes impossible as the size of thesystem grows. This is illustrated in Figure 7. Figure 7a shows the band structureof our 5-site (F = 2) strip, with parameters used in the experiment. Here, there
are non-negligible band gaps at the edges of the Brillouin zone between the lowest
and second bands. With a weak enough force, the atoms could traverse this region
slowly enough to be adiabatic with respect to this band gap. However, for a larger
system, such as the 41-site wide system whose band structure is shown in Figure
7b, these band gaps become negligibly small and adiabaticity is impossible.
The departure of the data in Fig. 6(top) from the adiabatic theory (red lines)
at the edges of the Brillouin zone indicates a partial break down of adiabaticity waspresent in our data. However, it is not a complete breakdown as the data also differsfrom the linear fit at the edges of the Brillouin zone. The data was somewhere in thepartially adiabatic regime, only possible for our narrow system. This made neither
the adiabtic theory, nor the linear fit assuming perfect non-adiabaticity, applicableto our system. Therefore, the linear fit is not a good measure of the Chern number

158

Figure 7: Band structure of the synthetic dimesnions lattice with flux Φ/Φ0 = −4/3.
(a) 5-site wide system with experimental parameters V0 = 4.4, h̄Ω = 0.5, h̄δ = 0,
and h̄ = 0.02. There is a band gap between the first and second bands at the edgeof the Brillouin zone. (b) 41-site wide system with parameters V0 = 4.4, h̄Ω = 0.5,h̄δ = 0, and h̄ = 0 and no Clebsch-Gordan coefficients included. The band gap
between the first and second bands at the edge of the Brillouin zone is negligibly
small.

159

for our system.
One might suspect that limiting the domain of the linear fit such that bandedge effects are excluded would still provide a good measure of the Chern number.
This is not the case, as can be seen in Fig. 6(bottom). Here, we plot the measured
Chern number for a linear fit to our data (grey) and the adiabatic theory (red) fora limited range of crystal momenta qx . The range of the fit is plotted on the xaxis, ranging from only the center to the entire Brillouin zone. The slope of thebest fit line for non-trivial topologies (Φ/Φ0 6= 0), and thus the measured Chern
number, depends highly on the selected domain for both the theoretical (red) andexperimental (black) data, and the appropriate choice of range is ambiguous. We
conclude that for an extremely narrow system such as ours, a conductivity measurement is insufficient for determining the Chern number at reasonable tunnelingstrengths [117].

7.3 Measuring Chern number via Diophantine equation
To better identify the Chern number in our system, following theoreticalwork [117–121], we leveraged the TKNN Diophantine equation (see sec. 1.1.5) todetermine the Chern number of our system. This equation states that for rational
flux Φ/Φ0 = P/Q (for relatively prime integers P and Q) the integer solutions s
and C to the Diophantine equation

1 = Qs − P C(7.11)

uniquely1 determine the Chern number C of the lowest band.
1

Subject to the constraint |C| ≤ |Q|/2 [6, 22]. The integer s has no bearing on our argument,
but has been interpreted as the charge transported when the periodic potential is adiabaticallydisplaced [122, 123].

160

1/2
1

1/4

0
−1
site mqs [2kL ]

0

-1/4

2

-1/2
1/6

1
0

-1/6
1/10
-1/10
-1/2

−1
−2
0
qx [2kL ]

1/2

-1/2

0
qx [2kL ]1/2

Figure 8: Chern number from the TKNN equation. (a). Lowest band energy within
the Brillouin zone in an extended 2-D system, where qx and qs are crystal momenta
along ex and es , respectively. Top. Φ/Φ0 = 0. Middle. Φ/Φ0 = 1/3: Brillouin zone
shrinks by a factor of 3 and becomes 3-fold degenerate, distance between adjacentenergy minima spaced by 2kL /Q is labeled. Bottom. Φ/Φ0 = 2/5. (b). Fractional
population in each spin state in the lowest band at qs = 0. Top. Φ/Φ0 = 1/3.
Bottom. Φ/Φ0 = 2/5. A momentum shift along ex of 2kL /Q is accompanied byan integer number of spin flips C. A line connecting magnetic states separated by
2kL /Q, with slope C = 1 (top) and −2 (bottom), is indicated.

7.3.1 TKNN Diophantine equation in synthetic dimensions
Surprisingly, the TKNN equation (Eqn. 7.11) has a direct interpretation in
the physical processes present in our system. Although the Hofstadter Hamiltonian
in eqn. 7.1 is only invariant under m-translations that are integer multiples of
Q , a so-called “magnetic-displacement” by ∆m = 1 accompanied with a crystal
momentum shift ∆qx /2kR = P/Q leaves Eqn. 7.1 unchanged. Together, thesesymmetry operations give a Q-fold reduction of the Brillouin zone along es , and add
a Q-fold degeneracy, as illustrated in Fig. 8a for Φ/Φ0 = 0, 1/3, and 2/5. Recallingthat the Brillouin zone is 2h̄kL periodic along ex , it follows that a displacement161

by 2kL /Q to the nearest symmetry related state involves an integer C magnetic
displacements, shown in Fig. 8b for Φ/Φ0 = 1/3 and 2/5, given by solutions to
2kL s − 2kR C = 2kL /Q, where s counts the number of times the Brillouin zone was“wrapped around during the C vertical displacements. Because this is exactly the
TKNN equation (7.11), we identify C as the Chern number.
Both C and s directly relate to physical processes. First, each time the Brillouin zone is wrapped around implying a net change of momentum by 2h̄kL apair of photons must be exchanged between the optical lattice laser beams. Second,each change of m by 1 must be accompanied by a 2h̄kR recoil kick imparted by
the Raman lasers as they change the spin state. This physical motivation of the
TKNN equation remains broadly applicable even for our narrow lattice, providing
an alternate signature of the Chern number.

7.3.2 Prescription for identifying Chern number
The prescription we used to identify the Chern number from our data andtheoretical calculations through the Diophantine equation argument is detailed in
Figure 9. Figure 9a shows the fractional populations in each m site as a function ofqx for a 3-site wide strip (F = 1 data) with flux Φ/Φ0 ≈ −4/3. Each of the fractional
populations (red, green, and blue representing m = −1, 0, and 1 respectively) was fit
to a parabola to extract the peak. These peak locations were interpreted as the band
structure minima corresponding to each m site. These points were then plotted as in
Figure 9c, with site m as a function of its maximizing crystal momentum qx . These
points were then fit to a line, whose slope was identified as the Chern number C. The
same prescription was used to obtain the theoretical predictions, as shown in Figure
9b,d. The theoretical fractional populations were obtained from the eigenvectors
corresponding to the lowest band of the full Hamiltonian, eqn. 6.2.1.

162

Fractional populations

(a)

(b)

1.0

0.8

0.8

0.6

0.6
0.4

0.4

0.2

0.2

0.0

0.0

site m

(c)

(d)
1

1

0

0

C =0.976±0.0403

−1
−1.0

−0.5
0.0
0.5
crystal momentum [kL ]C =0.967±0.0004

−1
−1.0

1.0

−0.5
0.0
0.5
crystal momentum [kL ]1.0

Figure 9: Calculating Chern number. (a) Fractional populations in m = −1,0 and1 sites (red, green, and blue respectively) as a function of qx for data taken in a
3-site wide strip (F = 1 data) with flux Φ/Φ0 ≈ −4/3. Dots represent data, lines
represent parabolic fits. (b) Theoretically calculated fractional populations for thesame system. (c) Site m as a function of maximizing qx from fits to data in (a).
Best fit line to the three points has a slope corresponding to the Chern number C.
(d) Site m as a function of maximizing qx from fits to theory in (b). Best fit line to
the three points has a slope corresponding to the Chern number C.

163

7.3.3 Properties of the method
This prescription for identifying the Chern number in narrow system can only
be considered meaningful if it converges to the more traditional exact integer value
in the limit of an infinite system. We study the behavior of the Chern number, as
identified through the prescription defined in the previous section, as a function ofwidth along the synthetic direction es . We solve this in the tight binding limit inmomentum space along ex , eqn. 6.12. We also set the detuning h̄δ = 0, quadratic
shift h̄ = 0, allowing us to drop the last diagonal term in eqn. 6.12. Additionally,
we neglect the variability in ts as a function m due to Clebsch-Gordan coefficientsby setting them to 1, assuming uniform coupling as in the Harper-Hofstadter Hamiltonian.
The resulting Chern number dependence on synthetic dimension size is presented in Figure 10a. Here, the flux Φ/Φ0 = −4/3, leading to an expected Chern
number of 1. We used a value of tunneling tx = 0.5EL and synthetic direction coupling h̄Ω = 0.5EL . As seen in the figure, for narrow lattices the measured Chern
number differs from unity, but converges to unity as the system size grows. Therealso appears to be a three distinct convergence curves, caused by the slightly differing
band structures depending on the number of sites modulo q.
We also study the dependence of the measured Chern number on the coupling strength along the synthetic direction h̄Ω for lattice widths relevant to ourexperiment—3 and 5 sites. We used the same Hamiltonian and parameter values
listed above. The results are shown in Figure 10b. In the limit of vanishing tunneling, both the 3 and 5-site wide Chern number converge to the exact integer valueof 1. This supports the hypothesis that deviation from unity at non-zero couplingstrengths is a consequence of the hybridization of edge states, which is facilitatedby stronger couplings.

164

(a)

(b)
1.10

3 sites
5 sites

1.05
1.08

Chern number

1.04
1.03

1.06

1.02

1.04

1.01

1.02

1.00

1.00
0

20
40
60
Number of sites along es

0.0

0.5
1.0
1.5
Coupling strength h̄Ω [EL ]2.0

Figure 10: Chern number dependence on (a) system size and (b) coupling strength.

7.4 Results and conclusion
Figure 11 shows the full evolution of fractional population in each m site asa function of crystal momentum qx in the lowest band, for all preparations studied
in the experiment: 3- and 5-site wide strips, with fluxes Φ/Φ0 = 0, ≈ ±4/3. The
black circles locate the peak of the fractional population in each spin state. Fromthese data, we extract the Chern number through the procedure described in theprevious section.
For the 3-site wide ribbon (F = 1 data), we measured a Chern number of0.99(4), −0.98(5) for negative and positive flux respectively2 , in agreement with theexact theory which predicts ±0.97(1), with uncertainties reflecting fit uncertainty ofpeak locations. For the 5-site wide ribbon, we measured 1.11(2), −0.97(4), close to
the theoretical prediction of ±1.07(1). The deviation from unity results from Φ/Φ0 −4/3 ≈ 0.01, a non-zero quadratic Zeeman shift, and ts > tx allowing hybridizationof the edge states [117].
2
Our Chern number extraction scheme fails for the rf case as the fractional populations are flatand there is no peak. We therefore assign a Chern number of 0 to flat distributions.

165

2site m

1
0
-1
-2

2
site m10-1-2-0.5 -0.25
00.25
0.5 -0.5 -0.25
00.25
0.5 -0.5 -0.25
0
0.25
0.5
Crystal momentum qx [2kL ]Crystal momentum qx [2kL ]Crystal momentum qx [2kL ]Figure 11: Chern number measurement. Lowest band fractional population measured as a function of crystal momentum in the ex and position in the es . Darker
color indicates higher fractional population. In the Raman-coupled cases, the points
represent the fitted population maxima and the Chern number is extracted from thebest fit line to those points. (a). 3-site (left) and 5-site (right) systems with positive flux. (b) 3-site (left) and 5-site (right) system with zero flux. (c). 3-site (left)
and 5-site (right) systems with negative flux. The parameters for 3-site data wereidentical to those for 5-site data, see Fig. 6 a, except ts = 2.880(1)tx .

166

Our direct microscopic observations of topologically driven transverse transport demonstrate the power of combining momentum and site-resolved position
measurements. With the addition of interactions, these systems have been shownto display chiral currents [124], and with many-body interactions are predicted togive rise to complex phase diagrams supporting vortex lattices and charge density
waves [125–127]. Realizations of controlled cyclic coupling giving periodic boundary conditions [61] along es could elucidate the appearance of edge modes as thecoupling between two of the three states is smoothly tuned to zero. In addition,
due to the non-trivial topology as well as the low heating afforded by synthetic dimensional systems, a quantum Fermi gas dressed similarly to our system would bea good candidate for realizing fractional Chern insulators [128].

167

Appendix A: Brownian motion of solitons in a Bose–Einsten Condensate

168

Brownian motion of solitons in a
Bose–Einstein condensate
Lauren M. Aycocka,b , Hilary M. Hursta,c , Dmitry K. Efimkind , Dina Genkinaa , Hsin-I Lua , Victor M. Galitskia,c ,and I. B. Spielmana,1aJoint Quantum Institute, National Institute of Standards and Technology and University of Maryland, Gaithersburg, MD, 20899; b Department of Physics,
Cornell University, Ithaca, NY 14850; c Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, MD 20742;
and d The Center for Complex Quantum Systems, The University of Texas at Austin, Austin, TX 78712-1192
Edited by Alain Aspect, Institut d’Optique, Palaiseau, France, and approved January 10, 2017 (received for review September 7, 2016)

soliton | Brownian motion | Bose–Einstein condensate | diffusion |fluctuation dissipationWe studied the diffusion and decay of solitons in thehighly controlled quantum environment provided byatomic Bose–Einstein condensates (BECs), where density maxima can be stabilized by attractive interactions [i.e., bright solitons (1, 2)] or, as here, where density depletions can be stabilized by repulsive interactions [i.e., dark solitons such as
kink solitons (3, 4) and solitonic vortices (5)]. By contaminating these BECs with small concentrations of impurity atoms,
we quantitatively studied how random processes destabilize
solitons.
Our BECs can be modeled by the one-dimensional (1D)
Gross–Pitaevski equation (GPE): an integrable nonlinear wave
equation with soliton solutions as excitations above the groundstate. For a homogeneous 1D BEC of particles with mass
mRb with√ density ρ0 , speed of sound c, and healing lengthξ = ~/ 2mRb c, the dark soliton solutions



√
vs
ξ
z − vs t√ϕ(z , t) = ρ0 i +
tanh
[1]
c
ξs
2ξs

dynamics that yield the fluctuation–dissipation theorem for
positive mass objects—f (t) is responsible for Brownian motion,
whereas γ describes friction, but both have contributions from
impurity atoms.
Conventionally, the diffusion coefficient D is inversely proportional to the friction coefficient: D ∝ 1/γ. For negative mass
objects, we show that the diffusion coefficient is instead proportional to the friction coefficient D ∝ γ; this reflects thatfriction is an antidamping force for negative mass objects.
The interplay between friction and confinement drives diffusive behavior with linear-in-time variance in soliton position,
Var(z ) = Dt, the same Brownian motion present for positive
mass objects.
Solitons are infinitely long lived because of the integrabilityof the 1D GPE. Integrability breaking is inherent in all physical systems—for example, from the nonzero transverse extent ofquasi-1D systems. Indeed, the kink soliton in 3D—the direct analogue to the 1D GPE’s dark soliton solution—is only long-lived inhighly elongated geometries (7–9), where integrability breaking
is weak. Cold atom experiments have profoundly advanced ourunderstanding of soliton instability by controllably lifting integrability by tuning the dimensionality (5, 10). Here, we studied thefurther lifting of integrability by coupling solitons to a reservoirof impurities.
Experimental System
Our system (11) consisted of an elongated 87 Rb BEC, confinedin a nominally flat-bottomed time-averaged potential, createdby spatially dithering one beam of our crossed dipole trap. We
SignificanceSolitons, spatially localized, mobile excitations resulting froman interplay between nonlinearity and dispersion, are ubiquitous in physical systems from water channels and oceans to
optical fibers and Bose–Einstein condensates (BECs). From ourpulse throbbing at our wrists to rapidly moving tsunamis, solitons appear naturally at a wide range of scales. In nonlinear
optical fibers, solitons can travel long distances with applications for communication technology and potential for use inquantum switches and logic. Understanding how random processes contribute to the decay and the diffusion of solitons isessential to advancing these technologies.

are expressed in terms of time t,qaxial position z , soliton velocity vs , and soliton width ξs = ξ/

1 − (vs /c)2 . Such dark soli-

tons have a minimum density ρ0 (vs /c)2 and a phase jump
−2 cos−1 (vs /c), both dependent on the soliton velocity vs .
These behave as classical objects with a negative inertial mass
ms ≈ − 4~ρ0 /c, essentially the missing mass of the displaced
atoms. The negative mass implies that increasing the soliton velocity reduces its kinetic energy; thus, dissipation accelerates darksolitons (6). This can be seen from the soliton equation of motion−|ms |z̈ (t) = −γ ż (t) − ∂z V + f (t),[2]where −γ ż is the friction force and V is the confining potential due to the mean-field effects of the condensate. The
random Langevin force f (t) has a white noise correlator
hf (t)f (t 0 )i = 2γkB T δ(t − t 0 ), where T is temperature andkB is Boltzmann’s constant. The connection between the friction coefficient γ and f (t) derives from the same microscopic
www.pnas.org/cgi/doi/10.1073/pnas.1615004114

Author contributions: V.M.G. and I.B.S. designed research; V.M.G. proposed the initial
theoretical concept that I.B.S. refined into a working experimental concept; H.M.H.,
D.K.E., and V.M.G. developed the final theoretical framework; L.M.A. configured theapparatus for this experiment; L.M.A led the data acquisition and analysis effort towhich D.G. and H.-I.L. contributed; H.M.H. performed detailed simulations for our experiment; I.B.S. contributed to the understanding of experimental data; H.M.H. prepared
the theory discussions in the manuscript; and L.M.A., H.M.H., D.G., H.-I.L., V.M.G., and
I.B.S. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
1

To whom correspondence should be addressed. Email: ian.spielman@nist.gov.

PNAS | March 7, 2017 | vol. 114 | no. 10 | 2503–2508

PHYSICS

We observed and controlled the Brownian motion of solitons.
We launched solitonic excitations in highly elongated 87 Rb Bose–
Einstein condensates (BECs) and showed that a dilute backgroundof impurity atoms in a different internal state dramatically affects
the soliton. With no impurities and in one dimension (1D), these
solitons would have an infinite lifetime, a consequence of integrability. In our experiment, the added impurities scatter off themuch larger soliton, contributing to its Brownian motion anddecreasing its lifetime. We describe the soliton’s diffusive behavior using a quasi-1D scattering theory of impurity atoms interacting with a soliton, giving diffusion coefficients consistent withexperiment.

Coupling to Impurities
Having established a procedure for creating solitons, we turnedto the impact of coupling to a reservoir of impurities, thus further breaking integrability. Fig. 2 displays the soliton position vs.
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 fs of images with a soliton present and an increase in the sample variance Var(z ) =P(zi − hz i)2 / (M − 1) computed by using the number M of
measured positions zi at each time.
Reduced Lifetime. The addition of impurities had a dramatic

impact on the soliton lifetime. Although we lack a quantita-∗

In our system, number fluctuations increased at the lowest trap depth (Materials andMethods).

†

The soliton oscillation frequency was slightly shifted with impurities resulting froman unintentional change in the underlying optical potential. This change also slightly
reduced the BECs’ longitudinal extent.

2504 | www.pnas.org/cgi/doi/10.1073/pnas.1615004114

A

B
100
z [µm]z [µm]

100

0

-100

0

-100
-50

r [µm]50 0

2
OD [a. u.]-50

r [µm]50 0

2
OD [a. u.]Cz [µm]

100

0

-100

D
soliton position, z [µm]prepared N = 8(2) × 105 atoms∗ in the |F = 1, mF = 0i internal state at T = 10(5) nK. Our system’s ≈ 250 µm longitudinal extent was approximately 30 times its transverse Thomas–
Fermi diameter 2R⊥ set by the radial trap frequency ωr =2π × 115(2) Hz and chemical potential µ ≈ h × 1 kHz. We controllably introduced a uniform (12) 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
before evaporation to degeneracy (13). This gave impurity fractions NI /N from 0 to 0.062 in our final BECs.
We then launched long-lived solitonic excitations using aphase-imprinting technique (3, 4. Because our trap geometry had
a finite transverse extent, quantified by the ratio µ/~ωr ≈ 9, planar kink solitons could be dynamically unstable and decay into
3D excitations (14). Our soliton’s initial velocity vs ≈ 0.3 mm/s,
approximately 1/5 the 1D speed of sound c ≈ 1.4 mm/s (15),implies that it is in an unstable regime (6), where it will convert
from a planar kink soliton to a nearly planar solitonic vortex. For
highly anisotropic geometries such as ours, the density profile ofthese two types of solitons is nearly the same—as given by the 1D
GPE—reflecting that they become formally indistinguishable atlarge velocity (14).
We absorption-imaged our solitons after a sufficiently longtime-of-flight (TOF) that their initial width ξs ≈ 0.24 µm
expanded beyond our ≈ 2 µm imaging resolution. Fig. 1A shows
our elongated BEC with no soliton present, and, in contrast, Fig.
1B displays a BEC with a soliton taken 0.947 s after its inception.
The soliton is the easily identified density depletion sandwiched
between two density enhancements. We quantitatively identified
the soliton position as the minimum of the density depletionfrom 1D distributions (Fig. 1 B, Right). Our phase-imprinting
process launched several excitations in addition to the soliton ofinterest. After a few hundred milliseconds, the additional excitations dissipated, and the remaining soliton was identified. By
backtracking the soliton trajectory, we were able to identify thesoliton, even at short times.
Fig. 1E shows a series of 1D distributions taken from timet ≈ 0 s to 4 s after the phase imprint. These images show threesalient features: (i) The soliton underwent approximately sinusoidal oscillations; (ii) the soliton was often absent at long times;and (iii) there was significant scatter in the soliton position. Items
ii and iii suggest that random processes were important to thesoliton’s behavior. The solitons’ position zi —when present—isrepresented by the light pink symbols in Fig. 1D, and the darker
pink symbols mark the average position hzi i for each time t.

100

0

-100
0

1

2
time, t [s]34Fig. 1. Soliton oscillations. (A) An absorption image after a 19.3 ms TOFof an elongated condensate without a soliton and a longitudinal density
distribution obtained by averaging over the remaining transverse direction.
(B) An absorption image and 1D distribution at time t = 0.942 s with asoliton with ≈30% imaged contrast. (C) A subset of the data where each
1D distribution is a unique realization of the experiment plotted vs. time t.
Notice a soliton was often absent at longer times. (D) The axial position ziof the soliton (light pink) vs. time t for different realizations of the experiment. We repeated each measurement 8 times. Dashed lines represent theedges of the elongated condensate. The dark markers represent the average soliton position hzi i at each time t. In A and B the longitudinal densities
are given in arbitrary units (a.u.).

tive model of the soliton’s decay mechanism, there are several reasons to expect a finite lifetime. When dissipation ispresent, solitons accelerate to the speed of sound and disintegrate. Furthermore, numerical simulations show that, inanharmonic traps, solitons lose energy by phonon emission,accelerate, and ultimately decay (16). 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. 3A,
showing the measured survival probability fs vs. time for a rangeof impurity fractions. We fit to our data a model of the survival
probability− ln(t/τ )1√
fs (t) = 1 − erfc
,
[3]
2
2σ
essentially the integrated lognormal distribution of decay times,
suitable for decay due to accumulated random processes (17).
The survival probability fs (t) has a characteristic width parameterized by σ and reaches 1/2 at time τ , which we associatewith the soliton lifetime. Fig. 3B shows the extracted lifetime τvs. impurity fraction NI /N , showing a monotonic decrease. Our
maximum NI /N gives a factor of four decrease in lifetime τ .
Aycock et al.

include interactions between impurity atoms. A soliton (Eq. 1)
acts as a supersymmetric Pöschl–Teller (18, 19) potential for the
impurity atoms with exact solutions in terms of hypergeometric functions (20). Impurity scattering states with momentum kz
in the rest frame of the soliton are described by the reflection
coefficient
1 − cos(2πλ)
R(kz ) =,[5]cosh(2πkz ξ) − cos(2πλ)

1.2% impurities

0

-100

where λ(λ − 1) = g 0 /g. In 87 Rb, we have g ≈ g 0 , giving λ ≈ 1.5.
The scattering problem is fully characterized by R(kz ), and the
problem is reduced to that of a classical heavy object moving
through a gas of lighter particles.
To understand soliton diffusion over many experimental runs,
we studied their distribution function f (t, z , vs ). We used akinetic equation equivalent to Eq. 2 with a stochastic force due toelastic collisions with the impurity atoms and a harmonic√ confining potential V (z ) ≈ − |ms |ω 2 x 2 /2, where ω = ωtrap / 2 is theeffective frequency (21, 22). In the limit of small soliton velocity (vs /c)2  1, the time-dependent distribution function can becalculated exactly (Materials and Methods). The kinetic equation has no stable solutions: Eventually, all solitons accelerateand disappear. However, the timescale for acceleration is set by
Γ−1 = |ms |/γ, was many seconds in our experiment. In the limitof Γt  1 and Γ  ω, the variance in position grew linearly withtime and diffusive behavior emerges (i.e., Var(z ) ≈ Dt). We calculate the diffusion coefficientγ + γ 0 kB T
vi2
D≈
+
,
[6]
|ms |ω 2 |ms |2

3.1% impurities

100

0soliton position, z [µm]-1004.1% impurities

100

0

-100
4.7% impurities

1000where vi is the soliton’s initial velocity. The offset γ0 accounts forany diffusion present without impurities. The friction coefficient
γ is given byZ
2~ X ∞ dkz 2 ∂γ=kz
R(kz )n() [1 + n()], [7]
kB T
∂kz
−∞ 2π

-100

m,l

6.3% impurities

100

0

-100
0

1

2
time, t [s]34Fig. 2. Impact of impurities. Here, we plot the position zi of the soliton(light pink) vs. time t after the phase imprint for different impurity levels.
The dark pink markers represent the average position hzi i for each time t.
Dashed lines represent the endpoints of the condensate vs. t.

Soliton 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. 4B, thisscatter, quantified by Var(z ), increased linearly with time. We
obtained the diffusion coefficient D as the slope from linear
fits to these data and calculated D using a quasi-1D scattering
theory. The energy of the infinitely long 1D system is given by
the GPE energy functional
E [ϕ, ψ] =Z  2~ |∇ϕ|2~2 |∇ψ|2g
g0++ |ϕ|2 |ϕ|2 + |ϕ|2 |ψ|2 dz , [4]
2mRb
2mRb
2
2

describing the majority gas interacting with itself along with the
impurities with interaction coefficients g and g 0 , respectively.
The fields ϕ and ψ denote the condensate and impurity wave
functions. Because the impurities are very dilute, we do not
Aycock et al.

22
an extension of ref. 23. m,l (kz ) = ~2 kz2 /2mRb +~2 jm,l/2mRb R⊥is the impurities’ quasi-1D dispersion along with quantized statesin the radial direction, described by Bessel functions. We accountfor radial confinement by summing over quantum numbers mand l . n() is the Bose–Einstein distribution for impurity atoms‡ .
Fig. 4B plots D measured experimentally (markers) and computed theoretically (curves; colored for different temperatures)
as a function of NI /N . The theory provides rather accurate estimates of the experimentally observed diffusion coefficient, with asingle fitting parameter given by γ0 = 5.32×10−4 mm2 /s. γ0 is set
by the diffusion coefficient at NI /N = 0, where D is suppressedin agreement with our theory. Diffusion at zero impurity concentration could be due to a number of factors, including scatteringof thermal phonons from the soliton, as well as trap anharmonicity (6, 16). In our quasi-1D system, the soliton is not reflectionless
to phonons in the majority gas as it would be in 1D.

Conclusion and Outlook
Our data show that added uncondensed impurity atoms contribute to soliton diffusion; however, the soliton lifetime falls
monotonically with increasing impurity fraction, even when the
additional impurities all enter the condensate. We speculatethat this might arise from two independent effects: (i) A static
soliton forms a potential minimum for impurity atoms, implying that, after some time, impurities will congregate in these‡In our model, the condensed atoms do not contribute to the stochastic force underlying
diffusion because they are all in the ground state. The impurity atoms are condensed
for impurity fraction & 0.2%; thus, the number of thermal impurity atoms is constant,leading to constant friction coefficient.

PNAS | March 7, 2017 | vol. 114 | no. 10 | 2505

PHYSICS100

A 1.0B

No impurities

2.0

0.0
soliton lifetime, τ [s]survival probability, fs

0.5

1.0
3.1% impurities
0.5
0.0
1.0

4.7% impurities

1.5

1.0

0.5

0.5
0.0
0

1

2
time, t [s]

3

4

0.0

0.00

0.02
0.04
0.06
impurity fraction, NI/N

Fig. 3. Soliton lifetime in the presence of impurities. (A) Histograms of soliton occurrence probability fs vs. time t after phase imprint. The blue solid curves
are fits to the lognormal-based survival function, from which we extract the lifetime τ . For each impurity fraction, we stopped collecting data when fs fell
below approximately 0.2. (B) Lifetime τ extracted from fit to the survival fraction fs vs. impurity fraction NI /N.

Materials and Methods
BEC Creation. We created BECs in the optical potential formed by a pair ofcrossed horizontal laser beams of wavelength λ = 1064 nm (11). The beam
traveling orthogonal to the elongated direction of the BEC was spatiallydithered by modulating the frequency of an acoustic-optic modulator at a
few hundred kHz. This created an anharmonic, time-averaged potential. To
reach the extremely cold temperatures necessary to realize long-lived solitons, we evaporated to the lowest dipole trap depth in which our technicalstability allowed us to realize uniform BECs.

Temperature Measurement. We measured temperature below the majority atom’s condensation temperature Tc = 350 nK by removing the majority atoms and fitting the TOF expanded impurity atoms to a Maxwell–Boltzmann (MB) distribution (13). Once the temperature was below Tc forthe impurity atoms, MB fits systematically underestimated the temperature. Fitting the small number of impurity atoms to a Bose distribution wastechnically challenging due to low signal-to-noise ratio and the addition ofanother free parameter, the chemical potential. To limit the number of freeparameters, we performed a global fit on the different impurity fractions,where we constrained the chemical potential µ to be negative. This provided an estimate of the temperature with large uncertainties. We foundfor our usual operating parameters, and, based on information from bothtemperature measurements, T = 10(5) nK.

2506 | www.pnas.org/cgi/doi/10.1073/pnas.1615004114

Impurity Characterization. We used a Blackman enveloped rf pulse at a
∼ 9 G magnetic field to transfer the |F = 1, mF = 0i atoms primarily to the|F = 1, mF = + 1i internal state (25). We varied the impurity fraction 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 more effective evaporation of the minority spin state (13). We
characterized the impurity fraction through careful, calibrated absorption
imaging with a Stern–Gerlach technique during TOF to measure the relative
fraction of the impurity atoms after evaporation.

Var(z) [mm2]A 0.004

0.002

0.000
0

B
D [10-4 mm2/s]

minima (7), broadening and destabilizing the soliton; or (ii)because the soliton moves in excess of the speed of sound for the
impurity atoms, even condensed atoms can reflect from the moving soliton. Although this coherent reflection process would not
add to diffusion, it would transfer momentum, thereby increasing the apparent damping coefficient and thereby reducing thesoliton lifetime. This latter model predicts a reduction of lifetime qualitatively similar to, but quantitatively in excess of, thatobserved in experiments.
Solitons in spinor systems with impurity scatterers is an exciting playground for studying integrability breaking and diffusionof quasiclassical, negative-mass objects. Our observed reduction
in soliton lifetime with increasing impurity fraction is in need of aquantitative 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, for example, due to non-Markovian effects,
as was recently discovered for bright solitons (24). Future experiments could study the impact of different types of impurities onsoliton dynamics by introducing impurities of a different atomic
mass. Lastly, mixtures with tunable interactions could freely tune
the amount of impurity scattering, offering an additional way tochange D.

1time, t [s]

20

2

5 nK
10 nK
15 nK

10

0
0.00

0.01

0.02
0.03
0.04
impurity fraction, NI/N

0.05

0.06

Fig. 4. Brownian diffusion constant dependence on impurities. (A) An
example for the linear fit of Var(z) vs. t for 1.2% impurities. Data are binned
into 0.36-s bins; the uncertainties are the sample SD. (B) The diffusion coefficient D vs. impurity fraction NI /N. The experimental results (markers) are
extracted from the slope of a linear fit of the sample variance Var(z) vs.
time t. The uncertainty in D is the uncertainty from that fit. See Materials and Methods for an explanation of uncertainty in NI /N. The theory
curves (solid and dashed curves) plot the calculated D for our measured
temperature.

Aycock et al.

Soliton Creation. We applied a phase shift to half of a condensate byimaging a back-lit, carefully focused razor edge with light red detunedby ≈ 6.8 GHz from the D2 transition for 20 µs.

gi (t, ω), with Gaussian form
f(t, z, vs ) =2πScattering Theory of Impurities. Minimizing Eq. 4 with respect to ϕ∗, ψ ∗gives the coupled equations of motion
~2
g0
2
2
2
∂ ϕ(z, t) + g|ϕ| ϕ +|ψ| ϕ,
2mRb z2~2g0
22i~∂t ψ(z, t) = −∂z ψ(z, t) +|ϕ| ψ.
2mRb
20∂ ψ(z )+∂z0 2"

γs2 λ(λ − 1)2
+ kz
cosh2 (γs z0 )#0ψ(z ) = 0.

[8]

g1 (t, ω) =
[9]

[10]

√
The
dimensionless parameters
are z0 = (z − vs t)/ 2ξ, kz2 = 4mRb ξ 2 /


2
0
2 0
~ Ez + mRb vs /2 − g ρ0 /2 , λ(λ − 1) = 2mRb ξ g ρ0 /~2 = g0 /g, and γs =q
1 − (vs /c)2 . g and g0 are the effective 1D interaction parameters after
integrating over the transverse degrees of freedom in ψ and ϕ. Because thetransverse wave functions are different, in general, g0 /g . 1. However, R(kz )is periodic in g0 /g (through λ), and small variations in this parameter do notstrongly affect the result. Solving for ψ(z0 ) and the scattering matrix thengives R(kz ) Eq. 5. For λ ≈ 1.5, this potential also has a single, shallow boundstate. Occupation of the bound state by an impurity atom can only occur
through three body collisions (two impurity atoms and soliton), scenariosthat we did not consider here.
Kinetic Theory of the Soliton. To define a diffusion coefficient, we studied
the distribution function of many solitons, f(t, z(t), vs (t)) (corresponding tomany experimental runs). The distribution function of solitons follows a
Boltzmann equation with a collision integral in Fokker–Planck form

Af + B


∂f
,
∂p



−Γvs f −


∂z V
2 ∂f
f + Γvth
,
|ms |
∂vs

[12]

2
where Γ = γ/|ms | and vth= kB T/|ms | is the thermal velocity. This equation can be solved analytically by using the method of characteristics inthe case of a harmonic potential V(z) = − |ms |ω 2 z2 /2. The solution is thetime-dependent distribution function f(t, z, vs ), parametrized by functions

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Aycock et al.

4g1 g3 − g22h2

g1 v s + g3 z

2

[13]

1 + 4ω 2 (et − 1) − et [cos(t ω̄) + ω̄ sin(t ω̄)]2ω 2 ω̄ 2

[14]

2et
[1 − cos(t ω̄)]
ω̄ 2

[15]

g2 (t, ω) = −g3 (t, ω) =1 + 4ω 2 (et − 1) + et [ω̄ sin(t ω̄) − cos(t ω̄)]22et/22ω̄t ω̄
g4 (t, ω) = −
sin
ω̄
2

g5 (t, ω) = −

et/2
ω̄

 



t ω̄
t ω̄
sin
+ ω̄ cos
,
2
2

[16]
[17]

[18]

where we work p
in dimensionless units t → t/Γ, ω → ωΓ, vs → vth vs , z →vth z/Γ, and ω̄ = 4ω 2 − 1. Eq. 12 does not have a stable solution where∂f/∂t → 0, due to the fact that the soliton is inherently unstable. The solution given in Eq. 13 is valid Rfor vsR c. Finally, we calculate the variance insoliton position, Var(z)(t) = dvs dz z2 f(t, z, vs ) = 2g1 + vi2 g24 , finding theexact expression (with restored units)


2
4vth
(eΓt − 1)
4vi2 eΓtt ω̄2Var(z)(t) =+sin
4ω 2 − Γ2
4ω 2 − Γ22v 2 Γ2 eΓtω̄Γt+ 2 th 2
1
−
e
cos(t
ω̄)
+
sin(t
ω̄)
[19]
ω (4ω − Γ2 )
Γpwhere ω̄ =4ω 2 − Γ2 . In the limits Γt  1, Γ  ω, we find diffusive
behavior Var(z) ≈ D(t)t, with the time-dependent diffusion coefficientv2 Γv 2 Γ 2 t ω̄D(t) ≈ th2 + i 2 sin
.
[20]
ω
ω
2
Setting sin2 (t ω̄/2) ≈ 1/2, we find the diffusion coefficient D presented in
Eq. 6. We note that in the limit Γt  1, ω → 0, we have Var(z) ∝ Γt 3 —thevariance has no linear in t dependence, and the soliton undergoes ballistic
motion, followed by exponential increase of Var(z) and soliton death§ .

[11]

where A and B are the drift and diffusion transport coefficients, and theleft-hand side is a total time derivative. For vs  c, we can write A ≈ γvs andB ≈ γkB T, where vs is the soliton velocity and γ is the friction coefficient
given in Eq. 7. Finally, we write the soliton momentum as p = −|ms |vs (23).
The kinetic equation then takes the form
∂f
∂f
∂
+ vs
=
∂t
∂z
∂vs

1

where vi is the soliton initial velocity and functions gi (t, ω) are given by

2

df
∂
=
dt
∂p

4g1 g3 − g22

(

exp −

+ g2 vs z + vi vs (g2 g4 + 2g1 g5 ) + vi z(g2 g5 + 2g3 g4 )
)
i
2
2
2
+vi (g3 g4 + g1 g5 + g2 g4 g5 ) .

In the experiment, we observed that the soliton remained stable for longtimes in the presence of impurities. Therefore, we neglect the last term of
Eq. 8, giving the well-known solitonic solution in Eq. 1. We seek a solution
for the impurity wavefunction ψ(z) in the soliton rest frame. In the radial
direction, the single-particle wave functions are the usual Bessel functions
for a particle in a cylindrical well. For ψ(z), we combine Eqs. 1 and 9 with0
ψ(z, t) = e−iEz t/~ eimRb vs z /~ ψ(z0 ). This gives a Schrödinger equation with aPöschl–Teller potential (18, 20),
2

1

ACKNOWLEDGMENTS. We thank Martin Link and Stephen Eckel for carefully reading our manuscript; and Joachim Brand for greatly beneficial conversations. This work was partially supported by the Army Research Office’s
Atomtronics Multidisciplinary University Research Initiative, by the Air Force
Office of Scientific Research’s Quantum Matter Multidisciplinary University
Research Initiative, National Institute of Standards and Technology, and the
National Science Foundation through the Physics Frontier Center at the Joint
Quantum Institute. H.M.H. received additional fellowship support from the
National Physical Science Consortium and National Security Agency. V.M.G.
was supported by the Simons Foundation.

§

Our trap was not strictly harmonic; however, the restoring force emerges for any trapwith nonzero slope ∂z V 6= 0, making our results qualitatively applicable to our experimental set.

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Venghaus H, Monemar B, Yamamoto Y (Springer, Heidelberg), 3rd Ed, Vol 2, p 205.
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11:06196.

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14. Mateo AM, Brand J (2015) Stability and dispersion relations of three-dimensional solitary waves in trapped Bose-Einstein condensates. New J Phys 17(12):125013.
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Aycock et al.

Appendix B: Geometrical pumping with a Bose–Einstein condensate

175week ending
20 MAY 2016

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;

ð1Þ

jwhere 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

200402-1© 2016 American Physical Society

PRL 116, 200402 (2016)

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 p¼ﬃﬃﬃ½ΩrfpcosðϕÞþ Ω̄ cosð2kR x̂Þ; −Ωrf sinðϕÞ−
ﬃﬃﬃ
δΩ 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 staggeredpﬃﬃﬃ
in 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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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 → exp½iγðq̂ÞjΨðqÞi, 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 agreement with conventional descriptions [18,24,27],
this predicts
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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