ABSTRACT

Title of dissertation: MAGNETIZED PLANE WAVE

AND STRIPE-ORDERED PHASESIN SPIN-ORBIT-COUPLED BOSE GASES
Andika Putra, Doctor of Philosophy, 2018
Dissertation directed by: Professor Ian Spielman
National Institute of Standards and Technology and
Joint Quantum Institute, Department of Physics
at the University of Maryland

Quantum degenerate gases have provided rich systems to simulate engineered
Hamiltonians and to explore quantum many-body problems in laboratory-scale experiments. In this work, I focus on spin-orbit-coupled (SOC) Bose-Einstein condensates (BECs) of Rubidium-87 atoms realized using two-photon Raman coupling inwhich various novel phases are predicted to exist due to competing energies from theatomic internal structure, coupling strength, and many-body collisions.
BECs are observed primarily using the interaction between light and matter, where
it is common to probe the atoms with near-resonant light and image their shadow ona camera. This absorption imaging technique measures the integrated column densityof the atoms and it is crucial to focus the imaging system. I present a systematic
method to bring the ultracold atom systems into an optimal focus using the powerspectral density (PSD) of the atomic density-density correlation function. The spatial
frequency at which the defocus-induced artifacts first appear in the PSD is maximized
at the focus. The focusing process thus identifies the range of spatial frequencies overwhich the PSD is uncontaminated by finite-thickness effects.
Next, I describe magnetic phases which exist in spin-1 spin-orbit-coupled condensates at a near-zero temperature. I observe ferromagnetic and unmagnetized phases
which are stabilized by the locking between the spin and linear momentum of thesystem. Our measurements of both the first- and second-order transitions are inagreement with theory.
Finally, I discuss the stripe-ordered phase that occurs in SOC Bose gases favoring
the miscibility configuration. The stripe phase is theoretically predicted to have an
excitation spectrum analogous to that of a supersolid and to exhibit spatial density
modulation within specific regions of parameter space. I used optical Bragg scattering
to probe the small density modulation present in the atomic spatial distribution. I
present for the very first time observation of the stripe phase in a Raman SOC Bosegas and its phase diagram in various parameter space. Our observations of the phaseboundaries are consistent with theory and previous work.

MAGNETIZED PLANE WAVE AND STRIPE-ORDERED PHASESIN SPIN-ORBIT-COUPLED BOSE GASESbyAndika Putra

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
2018

Advisory Committee:
Professor Steven Rolston, Chair
Professor Ian B. Spielman, Advisor
Professor Frederick C. Wellstood
Professor Gretchen K. Campbell
Professor Mohammad Hafezic Copyright byAndika Putra
2017

Preface
The first question I asked myself when I started writing this dissertation was:“Why do I even bother doing this?” And, of course, the obvious answer was tograduate soon and obtain my Ph.D.! Finally, I am seeing the light at the end of thetunnel?! However, in the process of putting my ideas into written words, I found thiswas harder than expected and I needed to do serious planning.
The next step for me was to look up in the internet (which has become the verb
Google nowadays) and search for a good guideline for writing a doctoral thesis. Eventually, I stumbled upon an article by Patricia Gosling and Bart Noordam on “Mastering Your Ph.D.: Writing Your Doctoral Thesis With Style” [Science (2007), DOI:
10.1126/science.caredit.a0700183] and decided to stick with it. I found this article to be very helpful and I would like to share it with my readers. One of the manyreasons you might be reading this thesis is because you are in the process of writing
your own thesis. If that is the case, congratulations and good luck!
To quote from the article: “In a thesis, it is better to err on the side of being
too detailed than to risk leaving out crucial information. Be generous to the next
generation of researchers; a detailed description of your progress and failures williisave them a lot of time.” This gave me a sense of purpose in writing and helped me
answer the question “Why do I even bother doing this?”
Last but not least, I would like to mention Chapter 4 and 6 in this dissertation
are from previously published works: “Putra, A. et al., Optimally focused cold atom
systems obtained using density-density correlations. Review of Scientific Instruments
85, 013110 (2014),” and “Campbell, D. L. et al., Magnetic phases of spin-1 spin-orbitcoupled Bose gases. Nature Communications 7, 10897 (2016).” The former publishedpaper on optimally focused cold atom systems was a result of my research project asa requirement for advancement to doctoral candidacy. The latter paper was a Ph.D.
research project led by a senior grad student, Dan Campbell. Both published articles
have a Creative Commons (CC) license, which grants people the rights to share thearticles with proper citation. I acknowledge Dan for responding my email and giving
me permission to include his published paper in this dissertation.
As final words, I hope I did a decent job here and happy reading!

iii

For my mother andin the memory of my fatheriv

Acknowledgements
The very first person I would like to thank here is Dr. Ian Spielman for being
such an “awesome” awesome1 advisor. I express my gratitude for his guidance andsupport throughout research in my graduate school.
My first chapter in doing research work was spent in RbLi lab at UMD. I would
like to thank the lab members Daniel Campbell, Ryan Price, and Ana Valdés-Curiel
for all the good and bad times we spent together in the lab. They helped me grow asa person and without them this work would not have been possible.
I also thank my neighboring lab members at UMD: Varun Vaidya, Creston Herold,
Jiraphat Tiamsuphat, Sean Xiao Li, Daniel Barker, Ben Reschovsky, Neal Pisenti,
Matt Reed, Zach Smith, Aftaab Dewan for letting me borrow (or steal) their equipment (or the other way around). Besides, I really appreciate the exchange of ideas
during our interaction times as well.
The next half of my research work was spent in the RbChip lab at NIST. I thankFrancisco “Paco” Salces-Cárcoba for being the best labmate and soccer buddy (if I
have to choose one); Seiji Sugawa, the awesome post-doc who moved out last year, formaintaining the experiment in top condition and for developing many pieces of the

1Yes, this is not a typo. The word “awesome” is overused and underrated, and hence I would like toemphasize the awesomeness by squaring it!

v

lab software (the magnetic field control system is beyond amazing!); Abigail Perry, aprevious grad student, for building the original experimental apparatus. One thing I
almost forgot which Paco reminded me is Seiji’s famous quote: “Everyday is Friday! ”.
Also, I want to thank Yuchen Yue, the latest grad student who picked up the batonof RbChip lab; Chris Billington, a master Python programmer; and Emine Altuntaş,a newly-minted Ph.D. who just joined our lab. I wish all of you many successes.
I also acknowledge the RbK people next door at NIST: Dina Genkina, Hsin-I
Lu, Ming Wu Lu, Lauren Aycock, and Alina Escalera, for supplying me with readily
available lab equipment. On the contrary, the bias field from your Zeeman slower while
in operation is a real annoyance! Somehow, we managed to conduct our experimentswith little to no problem (Seiji and Hsin-I knew this better...).
My last two years working at NIST were energized by playing pick-up soccer. I
don’t think I can list all of the names here because there are a lot of people and someof them go by initials. Nevertheless, particular thanks go to Richard “Dick” Gates fororganizing it three times a week in all types of weather and all four seasons.
I would like to express my gratitude to my previous roommates and physicist
friends: David Green for his two lovely cats (one sadly now resting in peace); Matthew
Reed (and his best companion Rufus) for organizing and teaching Hapkido; Jonathan
Hoffman for his spontaneous laughters; Kristen Voigt for being the best “Mafia”;
Brian Calvert for being the cool particle physicist; Eric Kuo, Anjor Kanekar, andRangga Budoyo for the fun times playing “Seven Wonders”. Special thanks go to
Rangga for drawing me into the world of “Dark Souls”.

viI would also thank other grad students who arrived at UMD the same year as me:
Jake Smith, Cody Ballard, Kale Johnson, Jasper Drisko. I wish them the best of luckin their future endeavors.
I thank my roommates during my stay at NIST: Chandra Shahi, Chen Zhe, Claudia Salazar, Andy Cheong Tam, and Hasan Latief for providing supporting environments. My special appreciation goes to Jeremy Lee for being a responsive landlord.
Next, I want to thank Edwin Sucipto, Freddy Witarsa, Nicholas Kuo, and Pak
Andang Purnama for providing vibrant Indonesian communities in College Park area.
My personal recognition goes to my previous teachers who had gradually shaped
my knowledge in physics: Suhunan Nainggolan, Manaek Nababan, Yohanes Surya,
Rachmat Widodo Adi, Alexander Iskandar, and Tjia May On, to mention but a few.
Many thanks go to Steve Rolston, Fred Wellstood, Gretchen Campbell, and Mohammad Hafezi for agreeing to be in my dissertation committee. I would also thank
Gail Griffin-Ferris, Kim Emswiler, Jesse Anderson, Melissa Britton, Kelly Hickman,
Pauline Rirksopa, Paulina Alejandro, Jane Hessing, Jessica Crosby, Peter Shawhan,
Linda O’Hara and the staff both at JQI and NIST for their administrative supports.
They have made my time in graduate school very smooth.
Lastly, I would like to thank my family: my parents, Suwardi and Hou Lie Tjhing;
my elder brother, Adi Putra; and my younger sister, Ayu Anggreni. I express mygratitude for their constant support and tremendous patience with me while I was
finishing graduate school, despite them having little ideas of what I am doing.

vii

Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvi

List of Scientific Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1. Overview of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
6

Chapter 2. Bose-Einstein condensation in laser cooled rubidium-87 atoms . .
2.1. Interaction of atoms with the electromagnetic field . . . . . . . . . . . . .
2.1.1. Atomic structure and notation . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2. Hyperfine structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3. Dipole matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Laser cooling and trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1. Zeeman slower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2. Optical molasses – Doppler limit and sub-Doppler cooling mechanism
2.2.3. Magneto-optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4. Magnetic trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5. Evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. Bose-Einstein condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1. Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2. Thomas-Fermi approximation for Bose gases in the harmonic trap .
2.3.3. Expansion of partially condensed ultracold gases . . . . . . . . . . . . . .
2.3.4. Spinor Bose-Einstein condensates . . . . . . . . . . . . . . . . . . . . . . . .

9
9
9
12
16
22
23
24
27
28
31
34
34
36
38
43

Chapter 3. Experimental apparatus and procedure . . . . . . . . . . . . . . . . . .
3.1. Apparatus for laser cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1. Master and cooling lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49
49
49

viii

3.1.2. RF driving electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3. Laser for red-detuned optical dipole trap . . . . . . . . . . . . . . . . . . .
3.2. Green laser and Ti:sapphire laser . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3. The main experiment chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1. Optical layout on the upper part of the main experiment chamber .
3.3.2. Relay assembly for optical Bragg scattering . . . . . . . . . . . . . . . . .
3.4. Experimental sequences to produce 87 Rb BECs . . . . . . . . . . . . . . . .
3.5. Control of magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1. Bias field calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2. Cancellation of magnetic field gradient . . . . . . . . . . . . . . . . . . . . .
3.5.3. 60 Hz noise cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.4. Combined feedforward plus feedback control of the magnetic field .

56
57
58
60
60
65
68
74
74
76
81
82

Chapter 4. Optimally focused cold atom systems obtained using densitydensity correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1. Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1. Wavefield propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2. Beer-Lambert law and the paraxial approximation . . . . . . . . . . . .
4.3.3. Absorption imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4. Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4. Optimal focusing of elongated Bose-Einstein condensates . . . . . . . . .
4.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85
85
86
88
89
90
93
96
101
104

Chapter 5. Coherent manipulation of atoms with electromagnetic fields . . .
5.1. Adiabatic rapid passage with RF-dressing . . . . . . . . . . . . . . . . . . . .
5.2. Microwave-assisted partial-transfer absorption imaging . . . . . . . . . .
5.3. The ac-Stark effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1. Scalar light shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2. Light-induced effective magnetic field . . . . . . . . . . . . . . . . . . . . . .
5.4. Engineered Hamiltonians in 87 Rb Bose gases . . . . . . . . . . . . . . . . . .
5.4.1. Spin-dependent lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2. Spin-1/2 spin-orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.3. Spin-1 spin-orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5. Calibration techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1. Minimizing circular polarization of the optical light . . . . . . . . . . . .
5.5.2. Lattice depth calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.3. Coupling strength and degenerate minima in an SOC system . . . . .

105
106
110
113
114
117
118
118
120
124
127
127
129
131

ix

Chapter 6. Magnetic phases of spin-1 spin-orbit-coupled Bose gases . . . . . .
6.1. Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1. Critical line of phase transitions . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2. Metastable states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.1. System Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.2. Frequency selective Raman coupling . . . . . . . . . . . . . . . . . . . . . .
6.5.3. The physical basis for SOC using Raman lasers . . . . . . . . . . . . . .
6.5.4. Floquet and polynomial shift . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.5. Measurement details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.6. Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.7. Wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.8. Free energy and phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.9. Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.10. Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

134
134
135
139
140
144
146
148
148
148
149
150
151
152
152
153
155
155

Chapter 7. Probing atomic structure with optical Bragg scattering . . . . . . .
7.1. Theory of optical Bragg scattering . . . . . . . . . . . . . . . . . . . . . . . .
7.1.1. The semi-classical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.2. Quantization of the electric field . . . . . . . . . . . . . . . . . . . . . . . . .
7.2. Geometrical properties of optical Bragg scattering . . . . . . . . . . . . . .
7.2.1. The static structure factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.2. Beam waist of the Bragg probe . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.3. Angular width of the scattered Bragg signal . . . . . . . . . . . . . . . . .
7.3. Observations of optical Bragg scattering in engineered Hamiltonians .
7.3.1. Probing atoms in a one-dimensional lattice . . . . . . . . . . . . . . . . . .
7.3.2. Precessing magnetic field in SOC condensates . . . . . . . . . . . . . . . .
7.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

158
159
161
167
171
171
175
177
178
179
185
190

Chapter 8. Stripe-ordered phase in spin-orbit coupled condensates . . . . . . .
8.1. Theory of the stripe-ordered phase . . . . . . . . . . . . . . . . . . . . . . . . .
8.2. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3. Observations of stripe-ordered phase in the full phase diagram . . . . .
8.3.1. Metastability of the stripe-ordered phase . . . . . . . . . . . . . . . . . .
8.3.2. Coherence revival of the stripe-ordered phase . . . . . . . . . . . . . . . .
8.3.3. Stripe-ordered phase as functions of coupling strength Ω andtemperature T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193
195
201
206
211
213

x

215

Chapter 9. Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

220
220
223

Appendix A. Matrix elements of the hyperfine Hamiltonian in the presenceof a magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

226

Appendix B. Gross-Pitaevskii equations for F = 1 spinor Bose-Einsteincondensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

230

Appendix C. Machine drawing of camera mounts . . . . . . . . . . . . . . . . . . .

237

Appendix D. Floquet theory for periodic time-dependent Hamiltonian . . . .

240

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

246

xi

List of Tables
3.1

Reference frequencies for the laser cooling transitions . . . . . . . . . . . . .

53

3.2

Calibration parameters for each pair of Helmholtz coils . . . . . . . . . . . .

76

A.1

Magnetic interaction parameters for the D line transitions of 87 Rb . . . .

227

xii

List of Figures
2.1

Atomic structure of 87 Rb atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2

Hyperfine structure of 87 Rb atoms in an external magnetic field B . . . .

17

2.3

Examples of transition matrix elements in the D2 line of 87 Rb . . . . . . .

21

2.4

Setup of a magneto-optical trap (MOT) . . . . . . . . . . . . . . . . . . . . . . .

29

2.5

Temperature of a 87 Rb Bose gas released from a harmonic trap . . . . . .

43

2.6

Magnetization Mz (r) of two-component 87 Rb BECs . . . . . . . . . . . . . .

46

87

2.7

Time evolution of magnetization Mz (z) of two-component

Rb BECs .

47

3.1

Schematic of laser frequencies we use in the D2 line of 87 Rb . . . . . . . . .

50

3.2

Optical layout of the master (repump) laser . . . . . . . . . . . . . . . . . . . .

51

3.3

Optical layout of the cooling (cycling) laser . . . . . . . . . . . . . . . . . . . .

52

3.4

RF electronics drive of the AOMs . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.5

Laser system for the 1064 nm optical dipole trap . . . . . . . . . . . . . . . . .

58

3.6

Optical layout of the Ti:sapphire laser system . . . . . . . . . . . . . . . . . . .

59

3.7

The main experiment chamber where BEC is created and experiments
are performed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

3.8

Optical layout on the upper part of the main experiment chamber . . . .

62

3.9

Relay assembly for optical Bragg scattering experiments . . . . . . . . . . .

64

3.10 Bias field calibration for each pair of coils in the Helmholtz configuration75

3.11 Configuration of compensation coils used to cancel ambient magnetic
field gradient ∇Bz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

3.12 Images of spinor Bose gases in |F = 1, mF = 0i and |F = 1, mF = −1i inthe xy−plane. The spatial density distributions n0 (r) and n−1 (r) areoverlapping when the stray field gradient is cancelled . . . . . . . . . . . . .

80

3.13 Flowchart of the combined feedforward plus feedback control system forthe DC field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

3.14 Magnetic field measured by the magnetometer over hours of experimental
runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

xiii

4.1

Dependence of BEC images on image plane position . . . . . . . . . . . . . .

95

4.2

Spatial PSDs of the intensity produced by a 1 µm thick layer of randomly
distributed scatterers, a 100 µm thick sheet of random columnar
scatterers, and a 100 µm thick sheet of random scatterers . . . . . . . . . .

98

4.3

Optimal focus in TOF absorption imaging of elongated BEC with density
fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.1

Transfer of atomic populations between the |F = 1, mF = −1i and|F = 1, mF = 0i states with RF-dressing at coupling strength Ω . . . . . .

5.2

109

Illustration of microwave-assisted PTAI to provide feedback correction to
the field drift δDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.3

Scalar light shift ∆E (0) and scattering rate Γ for wavelengths in between
the D1 and D2 transitions of 87 Rb atoms. . . . . . . . . . . . . . . . . . . . . . . 116

5.4

Three cases of engineered Hamiltonians: (a) spin-dependent lattice, (b)spin-1/2 SOC, and (c) spin-1 SOC. . . . . . . . . . . . . . . . . . . . . . . . . . .

119

5.5

Alignment of the Raman quarter-wave plate angle θWP for Raman A andRaman B laser beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.6Time evolution of the 0-th momentum order P0 for lattice depth
V0 = 3ER and V0 = 40ER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

130

Energy dispersion of spin-1/2 SOC condensates, absorption imaging of
the bare momentum and spin states in TOF, and the two degenerate
minima q0 of the Raman dressed states . . . . . . . . . . . . . . . . . . . . . . .

132

6.1

Experimental system to realize spin-1 SOC Bose gases . . . . . . . . . . . .

137

6.2

Measured phase transition of spin-1 SOC Bose gases . . . . . . . . . . . . . .

142

6.3

Quenching dynamics through the first-order transition . . . . . . . . . . . .

145

6.4

Metastable states of spin-1 SOC Bose gases . . . . . . . . . . . . . . . . . . . .

147

6.5

Mean field phase diagram of spin-1 SOC Bose gases . . . . . . . . . . . . . .

156

7.1

Two-dimensional illustration of real space lattice and reciprocal
space scattering from periodic scatterers with direct lattice vector
Rn = n1 a1 + n2 a2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159

Bragg signal as a function of the incident probe angle θi . . . . . . . . . . .

173

5.7

7.2
7.3
7.4

87

Beam waist of the Bragg probe measured by releasing Rb atoms fromthe dipole trap while the lattice potential is kept on . . . . . . . . . . . . . .

176

Scattered Bragg signal as a function of the traveling distance of a slit
placed in front of the EMCCD camera . . . . . . . . . . . . . . . . . . . . . . . .

178

xiv

7.5

Bragg signal NBragg as a function of lattice depth V0 . . . . . . . . . . . . . .

180

7.6

Coherence revival after the condensates are released from the latticepotential at V0 = 5ER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

182

Optical Bragg scattering of atoms loaded into the 1D optical lattice as afunction of temperature T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183

7.7
7.8

Coherence revival of atoms in optical lattice below and above thetransition temperature TC ≈ 380 nK . . . . . . . . . . . . . . . . . . . . . . . . .

184

Bragg signal NBragg as a function of the Raman coupling strength Ωf forloading rates of 0.23ER /ms, 0.11ER /ms and 0.08ER /ms . . . . . . . . . . .

186

7.10 Bragg signal NBragg from SOC condensates as a function of revival time τ

188

7.9

7.11 Bragg signal NBragg as a function of Rabi pulsing time tpulse of the Raman
lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Theoretical prediction of stripe-ordered phase in (Ω, δ) parameter
space for an SOC 87 Rb Bose gas (Vext = 0) with chemical potential ofµ = 0.2ER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

200

Experimental setup to detect scattered Bragg signal from the stripe
phase in spin-1/2 SOC system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

202

8.3

Stripe-ordered phase in (Ω, δ) parameter space . . . . . . . . . . . . . . . . . .

207

8.4

Stripe-ordered phase as functions of magnetization hσz i and couplingstrength Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

210

8.5

Metastability of the stripe-ordered phase . . . . . . . . . . . . . . . . . . . . . .

212

8.6

Periodic revival of the scattered Bragg signal from the stripe phase . . .

214

8.7

Stripe-ordered phase in (Ω, T ) parameter space . . . . . . . . . . . . . . . . . .

218

C.1

Andor EMCCD camera (iXon Ultra 888) mounted on translational stages
with custom-made plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

C.2

Machine drawing (not to scale) of the front mounting plate for iXon
Ultra 888 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.1

8.2

C.3

238

Machine drawing (not to scale) of the base mounting plate for iXon Ultra
888 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

D.1 Quasi-energy dispersion relation (q) of spin-1 SOC Hamiltonian withcoupling strength Ω1 = ER and periodic driving ~ω = 12.7ER . . . . . . .

xv

245

List of Abbreviations
The following table explains the abbreviations used throughout this dissertation.

Abbreviation

Full meaning

AC, ac
ACF
AOM
ARP
BCS
BEC
BL
BRF
CCD
CMOS
CW, cwDC, dcDDS
EFL, efl
EMCCD
FPGA
FWHM
GPE
HDD
LIAD
MOT
OD
PID
PLL
PSD

alternating current
auto-correlation function
acousto-optic modulator
adiabatic rapid passage
Bardeen-Cooper-Schrieffer
Bose-Einstein condensate
Beer-Lambertbirefringent filter
charge-coupled device
complementary metal-oxide-semiconductor
continuous-wave
direct current
direct-digital-synthesized
effective focal length
electron-multiplying charge-coupled device
field-programmable gate arrayfull-width at half maximum
Gross-Pitaevskii equation
hard-disk-drive
light-induced atom desorption
magneto-optical trap
optical depth
proportional-integral-derivative
phase-locked loop
power spectral densityxviAbbreviation

Full meaning

PTAI
RF, rf
RHS, rhs
RMS, rms
ROI
RWA
SOC
SSFM
TEC
TF
TOF
TTL
UHV
UV
1D
2D
3D

partial-transfer absorption imaging
radio frequency
right-hand sideroot-mean-squareregion-of-interest
rotating-wave approximation
spin-orbit-coupled
split-step Fourier method
thermoelectric cooler
Thomas-Fermi
time-of-flighttransistor-transistor logic
ultra-high vacuum
ultraviolet
one-dimensional
two-dimensional
three-dimensionalxvii

List of Scientific Constants
The following table shows the physical and mathematical constants used throughout this dissertation [121].
Symbol
c
µ0
0
h
~
e
aB
µB
me
mp
µN
kB
i
e
π

Quantity

Numerical value

Unit

speed of light in vacuum
299 792 458m s−1
magnetic constant
4π × 10−7
N A−2electric constant 1/µ0 c2
8.854 187 817 . . . × 10−12
F m−1
Planck constant
6.626 070 040 (81) × 10−34
Js−34
reduced Planck constant h/2π 1.054 571 800 (13) × 10Js
−19
elementary charge
1.602 176 6208 (98) × 10
C
−10Bohr radius
0.529 177 210 67(12) × 10m
−26
Bohr magneton e~/2me
927.400 9994 (57) × 10
J T−1
electron mass
9.109 383 56 (11) × 10−31
kg−27
proton mass1.672 621 898 (21) × 10
kg
−27
nuclear magneton
5.050 783 699 (31) × 10
J T−1
Boltzmann constant
1.380 648 52 (79) × 10−23
J K−1√
unit imaginary number−1
Euler’s number
2.718 281 828 459 045 . . .
pi, Archimedes’ constant
3.141 592 653 589 793 . . .

xviii

CHAPTER 1

Introduction
The quest to understand the nature of light and matter has long been at the very
core of physics. For example, efforts to understand the spectrum of electromagnetic
radiation emitted by a “blackbody” object led to the discovery of Planck’s constant,the conception of spontaneous and stimulated emission, and the quantization of light.
These pursuits eventually revolutionized the theory of statistical mechanics and led
to the birth of quantum mechanics.
In 1924, Satyendra Nath Bose developed the theory of the statistical mechanics ofphotons which allowed a theoretical derivation of Planck’s law. Bose sent his work toAlbert Einstein who realized the import of it and supported Bose’s ideas, leading to
the formulation of a new class of particles that obeyed Bose-Einstein statistics [132].
Such particles are named bosons and they have integer values of total spin. Bose
predicted that a collection of such non-interacting particles can occupy the samestate and most of the particles will “condense” into the lowest accessible quantum
state at nearly zero temperature, which was known as Bose-Einstein condensation.
Evidence for Bose-Einstein condensate (BEC) was first linked to the observed
superfluid properties in helium-4 below the temperature 2.17 K at 1 atm (below
the lambda point) [105]. However, due to the strong atomic interactions in 4 He,

1

the original BEC theory must be modified in order to describe the emergence ofsuperfluidity [132]. It was not until 1995 that a nearly pure condensate was observedin a dilute vapor of rubidium-87 atomic gases [7], and separately in sodium-23 atoms[38], where the measured occupancy number of bosons in the ground-state provided
a direct evidence for BEC.
Since then, advances in technology have enabled the BEC state to be observed inmany more isotopes and particles at temperatures from 10 to 100 nK [31, 56, 65, 146].
These quantum degenerate gases have provided rich systems to simulate engineered
Hamiltonians and to explore quantum many-body problems in laboratory-scale experiments, which serve as benchmarks to validate or eliminate theoretical models.
Experimental observations of various quantum many-body phenomena include thesuperfluid-Mott insulator transition in the Bose-Hubbard model [58], condensation offermionic atom pairs in the BCS-BEC crossover regime [145], and antiferromagnetic
correlations in the Hubbard model [65], to mention but a few.
In my research, I was interested in studying spin-orbit-coupled (SOC) BECs of87

Rb atoms. The spin-orbit coupling scheme used counter-propagating lasers to in-duce two-photon Raman transitions between different spin states of the atoms, causing spin angular momentum of the atoms being locked to their linear momentum[102, 103]. In this scheme, I was able to control the detuning δ between coupledspin states by changing the Raman laser frequencies, and I could tune the couplingstrength Ω by varying the laser intensities. In this system, due to competing energies between the many-body interaction and single-particle Hamiltonian, it was2

predicted that there exist three different phases: the stripe-ordered phase, the magnetized plane-wave phase, and the unmagnetized zero-momentum phase [69, 98]. In
this dissertation, I present my observations of these phases.
In particular, I experimentally explored the magnetic phases present in spin-1
SOC condensates [22, 90]. This system exhibits magnetized and unmagnetized phases
in two-dimensional (2D) parameter space. The behavior is mainly governed by thesingle-particle Hamiltonian. Phase transitions between the magnetic phases are welldescribed by the Landau phenomenological theory [129]. Across one dimension of the2D parameter space, the free energy dispersion of the SOC condensates has three local
minima (with two of them being degenerate), which becomes a single global minimum
exhibiting a first-order phase transition. Meanwhile, along the other parameter space,
the free energy dispersion has two degenerate local minima which eventually merge
into a single minimum, displaying a second-order phase transition.
In SOC Bose gases, where two-body collisions within the s−wave approximation
favor miscibility [177], there exists a stripe-ordered phase in the ground state of the
many-body Hamiltonian within certain regions of parameter space [69, 98]. The stripe
phase has drawn interests because it has properties that are analogous to supersolidity.
Specifically, the excitation spectrum of the mean-field energy exhibit a roton-maxon
like feature [97, 109]. This roton-maxon character has recently been observed by Ji
et al. [79] for the magnetized phase, where the roton-mode is softening near thetransition to the stripe phase, in agreement with theoretical calculations.

3

Another important feature of the stripe phase is the existence of density modulations. For typical SOC 87 Rb atoms realized in the Raman coupling scheme, direct
observations of these density modulations in the ground state have proven challenging
for various reasons [108]. The spatial periodicity of the density modulation is relatively small, on the order of 0.4 µm, which is beyond the resolution of most imaging
systems. Besides, the density contrast is predicted to be less than 5% and experimental observation requires a precise control of the magnetic field, within 10 µG.
The main objective of my Ph.D. research was to observe the long-sought stripeordered phase experimentally. To overcome the above challenges, I first prepared theatoms in an equal population of mixed states since the stripe phase favors miscibility.
I then loaded the spinor condensates into the SOC dressed states and let the systemevolved into the metastable states, which gave a detectable region of the stripe phase.
I used optical Bragg scattering [17, 174] and a high-sensitivity camera to detect the
scattered photons by the very small density modulation of the stripe phase. I also
implemented a combined feedforward plus feedback control system to compensate formagnetic field drift.
Although there have been many efforts to enhance the stripe phase, at present
there is only one experimental group that has managed to directly observe the density
modulation of the stripe phase. The stripe-ordered phase was first detected by Jun-Ru
Li et al. [95] in SOC Bose gases. They realized a spin-orbit coupling scheme using a4

“superlattice” 1 potential [4, 5, 96], where the interaction strength between spin states
can be artificially tuned to enlarge the region of the stripe phase. Unfortunately,in their work, the presence of the density modulation was only observed in a narrow
region of coupling strength Ω. It was not obvious that the density modulation existed
within the predicted 2D parameter space of the SOC system and their observations
do not provide clear signatures of the phase boundaries of the stripe phase.
In this work, I present my observations of the stripe phase over a range of parameters in the phase diagram. I first investigated the stripe phase in the parameter
space of coupling strength and detuning (Ω, δ). My results were in agreement withtheoretical predictions [69, 98] and previous measurements [95, 103]. I explored thestripe phase for various hold times thold and studied the metastability condition of
the system [103]. I observed the scattered Bragg signal in the stripe phase to be decaying with a time constant of about 0.1 − 1 s, supporting arguments that the stripe
phase formation was due to a many-body spin-dependent interaction. To confirm
the long-range phase coherence of the stripe phase, I did a time revival scan of theobserved Bragg signal using Talbot interferometry [120, 150]. Finally, I examined thefinite-temperature phase diagram of the stripe-ordered phase as a function of couplingstrength and temperature (Ω, T ). My observation in the (Ω, T ) space had a profile
consistent with previous theoretical and experimental reports [28, 78, 178].

1The superlattice potential was created by superimposing two standing waves (lasers), with one laser

wavelength λ1 was an integer multiple of the other nλ2 = λ1 . This resulted in a spatially periodic
staggered (double-well if n = 2) potential.

51.1. Overview of the dissertation
1.1. Overview of the dissertation
Chapter 2 describes the essential part of the theory of atomic physics related tothis work. First, I give an overview of atomic structure, laser cooling and trapping
techniques used to achieve BECs. Next, I discuss the properties of BECs within themean-field Gross-Pitaevskii equation (GPE) and stationary solutions of the wavefunction within the Thomas-Fermi approximation. I also discuss a set of coupled
GPEs governing spinor BECs and atomic physics nomenclature associated with such
system, particularly for multi-component BECs in the F = 1 hyperfine manifold of
the 5S1/2 electronic ground states of 87 Rb atoms.
Chapter 3 describes the setup and apparatus for my experiments. I discuss the
hardware I used and the experimental sequences to produce 87 Rb BECs. I also discuss
the control of magnetic field in our laboratory, including the calibration of the field, its
measurement, and the feedback control to compensate for ambient field drift, which
is important in the detection of the stripe phase.
Chapter 4 describes the technique I used to optimally focus cold atoms systems
using the power spectral density of the atomic density-density correlations. This
technique is especially useful in finding the optimal focus of time-of-flight absorption
imaging, which requires no hardware changes and minimum computational effort.
This chapter is taken from previous published paper [140]: “Putra, A. et al. Optimally
focused cold atoms systems obtained using density-density correlations. Review of
Scientific Instruments 85, 013110 (2014).”6

1.1. Overview of the dissertation
Chapter 5 describes coherent manipulation of atoms with electromagnetic waves.
This chapter covers the method I used to prepare different mixtures of the spinor
F = 1 Bose gases using a radio frequency (RF) source. I discuss the microwaveassisted partial-transfer absorption imaging technique we used to provide feedbackfor our dc magnetic field control. I also review the ac-Stark effect within the framework of perturbation theory. Lastly, I discuss how I used the vectorial light shift torealize various engineered Hamiltonians in our systems, along with the accompanying
calibration techniques.
Chapter 6 describes observations of magnetic phases of spin-1 SOC Bose gases. I
discuss the experimental setup to realize the spin-1 SOC system and its corresponding
Hamiltonian. I present our measurements of the first- and second-order phase transitions between a polar BEC and a ferromagnetic BEC. This chapter is from Ref. [22]:“Campbell, D. L., et al., Magnetic phases of spin-1 spin-orbit-coupled Bose gases,
Nature Communications 7, 10897 (2016).”
Chapter 7 describes the optical Bragg scattering technique I used to probe atomic
structure. I start by considering the theory of optical Bragg scattering and its relation
to condensed matter physics and crystallography. I then discuss the geometricalproperties of the Bragg signal, including the incidence angle dependence, beam waistof the probe, and the angular spread of the scattered Bragg signal. Finally, I present
measurement results of optical Bragg scattering from density modulation in onedimensional lattice and precessing spin in SOC condensates.

7

1.1. Overview of the dissertation
Chapter 8 describes my observations of stripe-ordered phase in spin-1/2 SOC
condensates. I first present theoretical background of the stripe-ordered phase within
mean-field theory. Next, I briefly discuss the experimental setup and sequences I used
to create and observe the stripe phase. Ultimately, I demonstrate the observations ofstripe phase in various parameter space and display the phase diagram.
Finally in chapter 9, I conclude with a summary of my main results and discussprospects for future investigations.

8CHAPTER 2

Bose-Einstein condensation in laser cooled rubidium-87 atoms
In this chapter, I review the essential theory of atom-light interactions, laser cooling and trapping techniques to achieve Bose-Einstein condensation (BEC). First, I
discuss the important points related to the atomic structure, hyperfine levels, andoptical transition selection rules within the dipole approximation. Next, I give a briefsummary on laser cooling and evaporative cooling techniques that I used to produce
BECs. Finally, I discuss the properties of BECs within the mean-field theory.

2.1. Interaction of atoms with the electromagnetic field
2.1.1. Atomic structure and notation. An electron moving in a sphericallysymmetric potential has coordinates (r, ϑ, ϕ) and energy E. Canonical quantizationof these variables leads to conserved observables associated with discrete quantum
numbers [19]:r, E → n = 1, 2, 3, . . . ∞(principal quantum number)ϑ→ l = 0, 1, 2, . . . (n − 1)ϕ

→ ml = 0, ±1, ±2, . . . ± l (magnetic quantum number)(orbital quantum number)

In addition, the electron has an intrinsic spin of s = ±1/2. These four quantum
numbers are used to describe single electron configurations in central potentials.

92.1. Interaction of atoms with the electromagnetic field
The above single-electron description can be expanded to describe electronic configuration of an atomic system with many electrons. In general, the electrons arearranged in closed and open shells, and the low energy atomic spectroscopic properties are determined by the outer electrons in an open shell. For the outer electrons,
the total orbital angular momentum of the partially filled subshells are designated by
the convention L =Pli , the total spin is described by S =Psi , and the total an-

gular momentum is J = L + S. The eigenvalues of the angular momentum operators
are always positive or zero, and they are designated by capital letters L, S, and J. As
an example, a helium atom with two electrons can have two configurations: S = 0(singlet) and S = 1 (triplet). In alkali atoms, there is a single electron in the outershell (s = 1/2) and consequently the angular momentum operators are often written
in lowercase letters l, s, j = l + s [107].
The spectroscopic notation I use in this work has the form:

n2S+1 LJ

(2.1)

In this notation, the L quantum number is represented by its corresponding alphabetic character where integer values of L = 0, 1, 2, 3, 4, 5, 6, etc. are denoted by
S, P, D, F, G, H, I, etc.1 For alkali metals with a single outer electron, the factor
2S + 1 = 2 is often dropped for brevity.

1Due to historical reason in atomic spectroscopy, the first 4 letters stand for descriptors of lines inalkali spectra: sharp, principal, diffuse, fundamental. The rest of the letters are continuation ofalphabetical order [163].

10

2.1. Interaction of atoms with the electromagnetic field

Figure 2.1. Atomic structure of 87 Rb atoms.
The spin-orbit interaction between the electron spin S and the orbital angularmomentum L gives rise to the fine structure with total angular momentum J. Incorporating the spin of the nucleus, the coupling between the nuclear spin I and effective
magnetic field due to the orbiting electron splits the energy levels further into hyperfine structure. This gives the total angular momentum F = I + J with quantum
numbers F which describe the hyperfine splitting.
An example of the resulting atomic structure is given in Fig. 2.1. Here I consider
a 87 Rb atom, the alkali species I used in my ultracold atom experiments, which has
nuclear spin I = 3/2 [162]. The electronic ground state in 87 Rb has quantum numbers
L = 0 and J = 1/2. Its first excited state with L = 1 is split into two energy levels

112.1. Interaction of atoms with the electromagnetic fieldwith J = 1/2 and J = 3/2. The transitions between the ground state and the twolowest excited states are labelled as the D1 line for the 5S1/2 → 5P1/2 transition andthe D2 line for the 5S1/2 → 5P3/2 transition. These two separate D−line components
have transition wavelengths λD1 = 794.979 nm and λD2 = 790.241 nm, respectively,
which are resolvable by commercial diode lasers.

2.1.2. Hyperfine structure. Hyperfine splitting due to interaction between the
nucleus and electrons can be calculated using perturbation theory in terms of multipole expansion of the field potentials. However, the formulation of this theory isquite involved. A summary of the hyperfine calculation based on Ref. [151] is given
below.
I assume the nucleus as a point source and I consider an observation point outside
the nuclear matter in a spherical coordinate (r, θ, ϕ). The electrostatic potential V
and the magnetic vector potential A set up by the nuclear charges, currents, andspins can be written as

V

=

∞
X
k=0

A =

r−k−1 C(k) · Q(k) ,

∞
X
−i

k
k=0

(2.2)


r−k−1 −ir × ∇C(k) · M(k) ,(2.3)

where the functions C(k) are spherical tensor operators of rank k with parity (−1)k ,(k)

Q(k) are tensor operators for the electric multipole moments Qµ , and M(k) are tensor(k)operators for the magnetic multipole moments Mµ . The scalar product of two

12

2.1. Interaction of atoms with the electromagnetic field
spherical tensor operators of rank k are defined as

T

(k)

·U

(k)

=

k
X

(k)

µ=−k

(−1)µ Tµ(k) · U−µ .

(2.4)

The components of the spherical tensor C(k) are given by

Cµ(k) (θ, ϕ) =r

4π
Ykµ (θ, ϕ) ,2k + 1(2.5)

where Ykµ (θ, ϕ) are the spherical harmonics.
The nuclear electric and magnetic multipole moments of order k in Eqs. 2.2-2.3(k)can be expressed in terms of the spherical components Cµ (θ, ϕ):
ˆ
Q(k)
µ

= e

Ψ∗ glnuc rk Cµ(k) (θ, ϕ) Ψ d3 r,
ˆ

Mµ(k)

= µN

∗

Ψ



k



∇r Cµ(k) (θ, ϕ)

(2.6)


· glnuc

2
Lnuc + gsnuc Snuck+1



Ψ d3 r, (2.7)

where Ψ is the nucleus wavefunction, and µN is the nuclear magneton. The nucleus
orbital angular momentum operator Lnuc and its g-factor glnuc are defined by thesummation over the orbital angular momentum of each nucleon Li :2glnuc µN Lnuc =

A
X
i=1

gli

e~
Li ,
mi c

(2.8)

where mi is the mass of the i-th nucleon (either the mass of a proton or a neutron),
A is the number of nucleons, and gli are the orbital g-factors (gli = 1 for a proton,and gli = 0 for a neutron). The nucleus spin operator Snuc and its g-factor gsnuc are

13

2.1. Interaction of atoms with the electromagnetic fielddefined by the summation over the spin of each nucleon Si :gsnuc µN Snuc =

A
X
i=1

gsi

e~
Si ,
2mi c

(2.9)

where gsi are the spin g-factors (gsi = 5.5857 for a proton, and gsi = −3.826 for aneutron) [151] .
The interaction of a single electron with the nuclear potentials can be describedby the relativistic Dirac equation [151, 152]. Solutions of the interaction Hamiltonianin the multipole series lead to several energy shifts in the atomic structure [163].
The k = 0 monopole term leads to the fine-structure of the atoms. The k = 1
magnetic-dipole term and the k = 2 electric-quadrupole term leads to the hyperfine
Hamiltonian3
3(I · J)2 + 2 (I · J) − I (I + 1) J (J + 1)
4
I·J
2~
Hhfs = Ahf 2 + Bhf ~,~2I (2I − 1) J (2J − 1)

(2.10)

with Ahf is the magnetic-dipole hyperfine constant (applicable for I, J > 0) and Bhf
is the electric-quadrupole hyperfine constant (applicable for I, J > 1/2). Higher
order expansion terms are generally omitted since they are very small and difficult toobserve in alkali atoms.
In the presence of an external static magnetic field B along the z−direction, there
is an additional Hamiltonian

HB = − (~µL + µ~S + µ~I) · B =14

µB
(gL Lz + gS Sz + gI Iz ) Bz ,~(2.11)

2.1. Interaction of atoms with the electromagnetic field
where gL , gS , and gI are respectively the electron orbital, electron spin, and nuclear“g−factors” which account for their corresponding magnetic dipole moments µ~ L, µ~S,
and µ~ I . In Eq. 2.11 we have followed the convention that the signs of the g−factors
are opposite to the signs of the associated magnetic moments µ~ [8]. For alkali atoms,
the magnetic field contribution to the hyperfine structure splitting is typically small
compared to the fine-structure splitting, which allows us to describe the interaction
Hamiltonian HB in terms of the good quantum number J of the electrons’ total
angular momentum
HB =µB(gJ Jz + gI Iz ) Bz .
~(2.12)In order to calculate the hyperfine energy structure, one must diagonalize thetotal Hamiltonian Htot = Hhfs + HB , as given by Eqs. 2.10-2.12 (see Appendix A
for the derivation of matrix elements of the total Hamiltonian Htot ). In general,the degeneracy in each hyperfine manifold is lifted by the magnetic field and theeigenstates become superpositions of the |J, mJ , I, mI i basis states.2 For a very weakmagnetic field, the interaction Hamiltonian HB can be treated as a perturbation to
the hyperfine Hamiltonian Hhfs and the basis states |F, mF ; I, mI i closely describe theeigenstates of the system. The energy shift in this weak field regime is also known as
the “linear Zeeman effect” with (F, mF ) being the good quantum numbers. For verylarge magnetic field, where HB dominates Hhfs , the basis states |J, mJ , I, mI i closely
describe the eigenstates of the total Hamiltonian. The energy shift in this regime is
2Other choices of basis states are |F, mangular momentum operators I and J.

F ; I, mI i and |F, mF ; J, mJ i since F is the summation of the15

2.1. Interaction of atoms with the electromagnetic field
called the “Paschen-Back effect” and (mJ , mI ) are the good quantum numbers. In theregime of intermediate magnetic fields, the eigenstates of the total Hamiltonian Htot
are mixtures of the basis states |J, mJ , I, mI i. The energy shift in this intermediate
regime is called the “incomplete Paschen-Back effect” and mF are the remaining goodquantum numbers.
The hyperfine structure of 87 Rb in the presence of a magnetic field is shown in
Fig. 2.2. Here, I consider the 5S1/2 ground-state level and 5P3/2 excited level of theD2 transition. Figure 2.2 shows the hyperfine energy levels ∆E/h for magnetic fieldB < 500 G, typical values used in our laboratories. For the values of magnetic field
being considered, the energy shifts of the 5S1/2 level is within the linear Zeeman effectregime, as shown in Fig. 2.2(a). Meanwhile, the energy shifts of the 5P3/2 level span
the whole range of the linear Zeeman effect to the Paschen-Back effect, as shown in
Fig. 2.2(b).

2.1.3. Dipole matrix elements. In the dipole approximation, electromagnetic
radiation with wavelength λ is assumed to have a negligible variation over the spatial
extent ratom of a single atom, i.e. λ  ratom . In this limit, the interaction between
an atom and a photon is given by the Hamiltonian

Hdip = −d · E = −er · Eˆ,(2.13)

with the dipole moment d = er and electric field E = Eˆ. Since the atomic wavefunctions Ψ are written in terms of the angular momentum basis states, which transform

16

2.1. Interaction of atoms with the electromagnetic field

(a)
3000

F =2

1000
0
−1000
−2000

∆E/h [MHz]

2000

−3000
−4000

F =1
0

100

200

300

400

500

−5000

magnetic field B [G](b)mJ = 3/2

1500mJ = 1/2
F =3
F =2
F =1
F =0

0

mJ = −1/2

mJ = −3/2
0

500

100

200

300

400

−500

∆E/h [MHz]

1000

−1000
−1500

500

magnetic field B [G]Figure 2.2. Hyperfine structure of 87 Rb atoms in an external magnetic field B. Here, we consider the fine structure levels involved in theD2 transition: (a) 5S1/2 ground level and (b) 5P3/2 excited level.

172.1. Interaction of atoms with the electromagnetic field
in a similar way to spherical harmonics, it is natural to expand the polarization vector
ˆ in terms of the spherical unit vectors. These unit vectors are given by√û−1 = (êx − iêy ) / 2,û0 = êz ,√û+1 = − (êx + iêy ) / 2,

(2.14)

where û±1 corresponds to right- or left-circularly polarized light (σ± -transitions) andû0 corresponds to linearly polarized light (π-transition). This allows us to express
the position operator in terms of spherical harmonic functions:uˆq · ~r =r

4π qrY (θ, φ)
3 1(2.15)

where q = 0, ±1. Using the contraction rule in Eq. 2.4, we see that the dipole
operator d = er is a spherical tensor of rank k = 1.
We then apply the Wigner-Eckart theorem for spherical tensor operators. For
example, the fine structure transition matrix elements induced by light with angularmomentum q are:hJmJ |dq | J 0 mJ 0 i = hJ kdk J 0 i hJmJ |J 0 mJ 0 ; 1 qir00

= hJ kdk J 0 i (−1)J −J+m −m

2J + 1 0hJ mJ 0 |JmJ ; 1 -qi ,2J 0 + 1(2.16)

where |hJ kdk J 0 i| are the reduced matrix elements, |JmJ i is the initial atomic state,|J 0 mJ 0 i is the final atomic state, and hJmJ |J 0 mJ 0 ; 1 qi are the Clebsch-Gordan coefficients for the angular momentum coupling. The reduced matrix elements are related18

2.1. Interaction of atoms with the electromagnetic field
to the empirically determined spontaneous decay rate3

ΓJJ 0 =

ω03 2J + 1
2|hJ kdk J 0 i| ,30
3π0 ~c 2J + 1(2.17)

where ω0 is the transition frequency of the coupled fine structure states J and J 0 .
Extending the Wigner-Eckart theorem to the hyperfine structure we obtainhF mF |dq | F 0 mF 0 i = hF kdk F 0 i hF mF |F 0 mF 0 ; 1 qi
r00

= hF kdk F 0 i (−1)F −F +m −m

2F + 1
hF 0 mF 0 |F mF ; 1 -qi (2.18)
0
2F + 1

which gives the ratio of transition strength between different |F mF i and |F 0 mF 0 i
states. The reduced matrix elements for hyperfine transition are connected to thefine structure transition by00

hF kdk F i = hJ kdk J i (−1)

F 0 +J+1+I

p(2F 0 + 1) (2J + 1)






J

J0 1



 F0 F

where the last term in curly brackets is the Wigner 6-j symbol.








I 

, (2.19)

The dipole approximation accordingly leads to three transition selection rules

F0 = For F 0 = F ± 1,

mF = mF 0 + q ⇒ mF 0 = mF

(2.20)or mF 0 = mF ± 1,

(2.21)

3In some definitions, either the factor (2J 0 + 1) or (2J + 1), or both, is absorbed into the reduced

dipole matrix element |hJ kdk J 0 i|.

192.1. Interaction of atoms with the electromagnetic field(2.22)

F 0 6= F if mF 0 = mF = 0.

Considering transition to a single excited fine-structure component, the above relations result in two symmetry cases:(1) The matrix elements that couple to any single excited state sublevel |F 0 mF 0 i
add up to a factor that is independent of the particular sublevel chosen:X
qF2|hF 0 mF 0 |dq | F (mF 0 + q)i| =2J + 1
2|hJ kdk J 0 i| .
2J 0 + 1(2.23)The interpretation of this symmetry is simply that all of the excited state
sublevels of F 0 energy level decay at the same spontaneous emission rate Γ,and the decaying population “branches” into various ground state sublevels.
(2) Another symmetry arises from summing the matrix elements for a single
ground-state sublevel to all the sublevels in a particular F 0 energy level:

SF F 0 =P

2
0
q |hF mF |dq | F mF 0 i|
0 2

|hJ kdk J i|0

= (2F + 1) (2J + 1)




 J

J0



 F0 F

2


1 



I 

.

(2.24)

The above sum SF F 0 is independent of the particular ground state sublevel
chosen, and also obeys the sum ruleP

F 0 SF F 0= 1. These factors SF F 0

provide a measure of the relative strength of each of the F → F 0 transitions.
20

2.1. Interaction of atoms with the electromagnetic field

Figure 2.3. Examples of transition matrix elements in the D2 line of
Rb. Symmetry cases satisfying Eq. 2.23 are shown for the |0, 0i and|3, 3i excited states. Symmetry cases satisfying Eq. 2.24 are shown for
the |2, −2i and |2, 0i ground states.

87

Following the transition rules described above for each distinct fine-structure component, Fig. 2.3 illustrates the strength of transition matrix elements|hF mF |dq | F 0 mF 0 i|2|hJ kdk J 0 i|2(2.25)

for the D2 line of 87 Rb. Some of these are the transitions to the |F 0 , mF 0 i = |0, 0i
and |F 0 , mF 0 i = |3, 3i excited states satisfying the first symmetry case in Eq. 2.23.
I also show examples for transitions from the |F, mF i = |2, −2i and |F, mF i = |2, 0i
ground states satisfying Eq. 2.24.

21

2.2. Laser cooling and trapping
2.2. Laser cooling and trapping
The main idea behind laser cooling techniques is to exploit the interaction of
atoms and electromagnetic field to cool down the atoms to near absolute zero [115,134]. Laser light is used to slow down the atoms through many cycles of absorption
and spontaneous emission. Photons from excited atoms are spontaneously emitted
in random directions with zero net contribution to the atom’s average momentum.
The net result is that the atomic motion is cooled by the absorbed photon recoil
momentum. Unfortunately, the “randomness” of the spontaneous emission also causes
unavoidable “heating” of the atomic motion, setting a threshold to the lowest velocity
spread that can be achieved with this technique. The temperature limit due to thismechanism is called the Doppler temperature [24].
Ideally, atomic transitions involving the same ground state for many cycles ofabsorption and emission are desirable since it allows the atoms to be laser cooledindefinitely. The particular transitions used in laser cooling are known as cyclingtransitions. Due to population branching to several ground-state levels, an atom inan excited state might not return to its initial ground state after absorbing photons.
This process, known as optical pumping, causes depletion of the number of atoms
being trapped. In some cases, the atoms end up in dark states – ground-state levels
that are transparent to the cooling light in a cycling transition. Since the linewidths ofthe transition (excited states) and commercial lasers used are generally much smaller
than the separation between the different ground state components, a solution to this22

2.2. Laser cooling and trapping
problem is to apply another repump laser with a different frequency that re-excites
the atoms out of the “wrong” ground states. Thus, it is a common practice to apply
both cooling and repump lasers in most stages of laser cooling and trapping [115, 134].
2.2.1. Zeeman slower. In the first stage of laser cooling with a fixed laser frequency, it is common to slow down the atomic vapor originating from an oven using
photon recoil from the laser beams. As the velocity decreases, the resonance conditionchanges due to Doppler shift. One solution to this problem is to apply a spatiallyvarying magnetic field to alter the atomic hyperfine structure so as to compensate
the Doppler shift. This technique is known as a Zeeman slower [115].
An example of a B-field profile for the slower [41] along the slowing direction z is

B(z) = Bb − Btp
1 − z/z0 ,(2.26)

where Bb is a constant bias field, Bt is the magnitude of the spatially varying component, and z0 is the distance over which slowing takes place. Due to the recoil
momentum ~kR imparted by the laser beam with intensity I, an atom with mass mmoving at velocity v experience deceleration

a=

~kR Γ
I/Isat,2m 1 + I/Isat + 4 [δ0 + kR v − µ0 B(z)/~]2 /Γ2(2.27)

where Γ is the spontaneous decay rate of the excited state, Isat is the saturation
intensity4, δ0 is the laser detuning, and µ0 is the magnetic moment of an atom.
4The saturation intensity is related to the reduced dipole matrix element of the correspondingtransition and light polarization, which can be obtained by solving the optical Bloch equation [163].

23

2.2. Laser cooling and trapping
To achieve the maximum possible deceleration to cool atoms with initial speed vi
to the final speed vf , the Zeeman slower must have a length z0 given by
vf2 − vi2.
z0 = mη~kR Γ

(2.28)

Here the parameter η takes into account the imperfections in the Zeeman field, laser
intensity and polarization, and spatial constraints. When choosing η, the designer
has to bear in mind two criteria [41]:(1) how much laser power is available
I
Isat

=

η
,
(1 − η)(2.29)(2) how well the field profile (including its gradient dB/dz) can be replicated tofulfill the resonance condition at each point along the slower
2mµ0 dB
η= 2 3~ kR Γ dz

 0
µB− δ0 .
~(2.30)

2.2.2. Optical molasses – Doppler limit and sub-Doppler cooling mechanism. Optical molasses use the simplest laser cooling mechanism: a pair of counterpropagating beam with wavevectors ±kR . A two-level atom moving with velocity v is
subject to the Doppler effect and observes the laser beams with detuning δl,r = δ∓kR v,
where δ is the laser frequency detuning relative to the atomic transition. In the limit
of low intensity I  Isat , where stimulated emission is not important, we neglect the24

2.2. Laser cooling and trapping
interference effect between the counter-propagating lasers and we obtain the total
scattering force as Fscatt = Fleft − Fright = −αv, where the damping coefficient α isgiven by [115]:

α=−4δ
2~kR2 I.
2 
Γ Isat 1 + (2δl /Γ) 1 + (2δr /Γ)2

(2.31)

With a restoring force F = −αv, the damping opposes the motion of the atom in
proportion to its speed, similar to drag forces in viscous media and hence this coolingtechnique is known as optical molasses. The cooling rate F · v is maximized for the
detuning condition δ = −Γ/2.
In optical molasses, spontaneous emission plays an essential role in dissipatingenergy by emission of fluorescence photons [59]. However, the random momentum
recoil of the atoms hp2 i after spontaneous emission is responsible for heating. The
heating can be described in an analogous way to diffusion in Brownian motion [33]:d 2p= 2D0 ,
dt i heatingi = x, y, z,(2.32)

where D0 is the diffusion coefficient. Meanwhile, the cooling by optical forces can be
described by:
d 2
2α 2
pi cooling = −
p ,
dt
m i

i = x, y, z,(2.33)

where m is the atomic mass. The steady-state (satisfying the Fokker-Planck equation) is reached when the heating and cooling are balanced. One finds the steady-state25

2.2. Laser cooling and trapping
temperature T as kB T = D0 /α, where the atomic motion is associated with the temperature using the equipartition theorem kB T = hp2i i /m. For very low saturation
parameter s = I/Isat  1, the diffusion coefficient can be interpreted as a sum of two
independent terms due to vacuum fluctuations (spontaneous emission) and fluctuations in the number of absorbed photon, and this gives the total diffusion coefficient
D0 = (sΓ/2) (~kR )2 . The minimum temperature achievable with the maximum cooling rate δ = −Γ/2 is then given by the Doppler limit
TD =

~Γ
.
2kB

(2.34)In a typical D2 −line transition used in laser cooling of 87 Rb atoms, the Doppler
temperature is TD = 146 µK [162].
During research on laser cooling and optical molasses, it was found that the lowest achievable temperature was colder than the Doppler cooling limit [134]. This
perplexing result eventually led to a new understanding of how laser cooling could
work [37]. Two counter-propagating lasers, either with orthogonal linear polarizations (lin ⊥ lin) or orthogonal circular polarizations (σ + −σ − ), create a polarization
gradient. In the first case, the lin ⊥ lin configuration, spatially rotating ellipticity of
the combined light causes spatially oscillating energy shifts in various sublevels of theatoms. Atoms which are climbing a potential hill are losing kinetic energy and havea probability to be optically pumped back to the bottom valley, transforming partof their kinetic energy into potential energy. This irreversible mechanism is known

26

2.2. Laser cooling and trappingas Sisyphus cooling [37]. In the second case, the σ + −σ − configuration, the combinedpolarization vector is spatially rotating with the same ellipticity. A combination of
optical pumping and light shifts produces a highly sensitive motion-induced atomic
orientation resulting in a large imbalance between the radiation pressures of the twocounter-propagating waves.
Polarization gradient cooling works at velocities lower than the velocity capture
range for Doppler cooling (kv < Γ). The equilibrium temperature TSD in the subDoppler cooling is proportional to the laser power TSD ∝ ~Ω2 / |δ|, where the dipole
transition Rabi frequency Ω is related to the laser intensity via I/Isat = 2Ω2 /Γ2 .
The one-photon recoil energy then sets a fundamental physical limit on the lowest
achievable temperature in sub-Doppler cooling to be the recoil temperature TR =~2 kR2 /mkB [37, 162]. As an example, for 87 Rb atoms in the D2 transition with λD2 =780.24 nm, the recoil temperature is TR ∼ 362 nK [162].

2.2.3. Magneto-optical trap. Doppler cooling enables a quick cooling of atoms,
typically in a timescale of a few ms (set by the Planck constant over the photon recoil
energy). However, the atoms have a spatially diffusive motion and eventually theyleave the cooling region in a timescale of a few seconds. In order to trap the atoms
within a small volume, while cooling them at the same time, additional magnetic field
gradients are applied. This setup is known as a magneto-optical trap (MOT).
In a MOT, quadrupole magnetic field gradient is added to the 3D standing wave
configuration of the optical molasses. With the presence of a magnetic field gradient27

2.2. Laser cooling and trapping
along x such that B = b0 x , the detuning δ 0 (r) seen by the atoms depends on theposition r and their internal sublevel structure, i.e.

δ 0 (r) = δ − m0ex gJ,ex − m0gr gJ,gr

 µB 0b x,~(2.35)

where m0ex gJ,ex and mgr gJ,gr are respectively the magnetic moment of the excited andground states used in the laser cooling transition. This leads to a position-dependent
light force and this gives a restoring force −κr. The restoring coefficient κ scales
linearly with detuning and intensity (similar to the damping coefficient α), and it ismaximum for red-detuned light with δ = −Γ/2. For cold atoms close to the centerof the trap with small speed |v|, the atoms experienced a total force F = −αv − κrequivalent to that of a damped oscillator.
An illustration of a MOT is shown in Fig. 2.4. This setup confines the atoms
both in position and momentum, yielding a large phase-space density of atoms being
trapped – the main requirement towards BEC. The main benefit of a MOT is its
flexibility to convert easily into a magnetic trap or an optical molasses configuration.
Also, the presence of a strong magnetic gradient is helpful for selectively trapping
atoms in a specific hyperfine sub-level.

2.2.4. Magnetic trap. Once atoms are cooled close to the Doppler temperature, they need to be moved to other trapping potentials to be cooled further. One
practical approach is to use a magnetic trap, where atoms are being transported byshifting the center location of the trap. One typically uses a simple magnetic field28

2.2. Laser cooling and trapping

Figure 2.4. (a) Setup of a magneto-optical trap (MOT). Three pairsof counter-propagating laser beams with opposite circular polarizations
provide the damping force, while magnetic field gradient from a pair ofcoils with opposite current provides the restoring force. (b) A simplified
picture of different hyperfine Zeeman sublevels being cooled by the
optical molasses. In common practices, both cycling and repump lasers
are used to maximize the number of atoms captured in the MOT.
configuration produced by a pair of opposing anti-Helmholtz coils, with the axis along
the z−direction, which has the form:

B=b0(x, y, −2z) ,2(2.36)

where b0 is the magnitude of the field gradient and the field has an axial symmetry along the z−direction. This trap configuration is also known as the magnetic
quadrupole trap.
In my experiments, other pairs of Helmholtz coils generate a controllable uniform
magnetic field in certain directions. For simplicity, here I focus on the particularcase where there is a spatially uniform magnetic field B0 = B0 ez which displaces the29

2.2. Laser cooling and trapping
zero location of the magnetic trap. The magnetic trap provides a state-dependent
conservative potential (in the linear Zeeman regime):b0
U (r) = −µ · B = µB gF mF
2q

x2 + y 2 + 4 (z − z0 )2 ,(2.37)

where z0 = B0 /b0 and the strength of the magnetic field gradient in the xy− planeis half the gradient along z (since ∇ · B = 0). The force experienced by the atoms
can be calculated using F = −∇U , and only specific atomic Zeeman sub-levels canbe trapped selectively depending on the atoms’ magnetic moment (depending on thesign of gF mF ).
In magnetic trapping, it is crucial that the trapping force is enough to counteract the gravitational potential energy. For 87 Rb atoms with atomic mass m =1.443 × 10−25 kg and gravitational acceleration g = 9.81 m/s2 along the z-direction,
the gravitational field gradient expressed in frequency over distance is dUgrav /dz =mg = h · 2.14 kHz/µm. Levitation of magnetically trappable spin states is thusachievable by selecting the appropriate field strength and position with respect to themagnetic trap center. For 87 Rb atoms in the |F = 2, mF = 2i state, a magnetic field
gradient of b0 ≥ 15.3 G/cm is needed to levitate the atoms against gravity. If thegravity acceleration is along x− or y−direction, twice of the field gradient is needed(i.e. b0 ≥ 30.6 G/cm).

30

2.2. Laser cooling and trapping
2.2.5. Evaporative cooling. Despite having very low temperature, quantum
phase transition towards BEC is not necessarily attainable unless high enough phasespace density is obtained.5 Additional step of evaporative cooling, combined withlaser cooling, is critical to achieve BEC. In evaporative cooling, energetic particles
escape, carrying away their larger share of thermal energy, allowing the remaining
atoms to cool down. This works for a wide range of temperatures and densities, but
the process slows down as the system loses energetic particles, and hence significant
number of atoms need to be trapped. Here, I summarize the evaporative coolingtechniques [61, 83] which are relevant to my experiments.
2.2.5.1. Radio-frequency induced evaporative cooling. In magnetic traps, evaporation is done through a radio-frequency (RF) induced cooling technique [83]. The
scheme uses a tunable RF radiation in conjunction with the magnetic trap to dress
the atoms. Different mF sub-level states of the atoms are mixed by the RF field,
leading to a flatter potential curve at the avoided crossing of the dressed-state eigenvalues. As the atoms move away from the trap center, traversing the avoided crossing
region and come into resonance with the RF field, the atomic state is adiabaticallytransformed into a non-trapped state (with opposite sign of magnetic moment) bystimulated emission of RF photons [83]. This process is identical to the “spilling” of
the most energetic atoms out of the magnetic trap. Evaporation is then progressively

5The phase-space density is the ratio between the particle spatial density n and the number ofsignificantly occupied (momentum or energy) states per unit volume. This is equivalent to thenumber of particles contained within a unit of volume in the thermal de Broglie wavelength, $ =3/2
n 2π~2 /mkB T.

31

2.2. Laser cooling and trapping
forced into the system by ramping down the RF frequency fRF adiabatically, with afurther option of also lowering the magnetic field gradient.
2.2.5.2. Optical dipole trap. Optical dipole traps rely on the principle that an
off-resonant laser beam attracts or repels atoms, depending on whether it is red- or
blue-detuned from the atomic transition [61]. Heating due to spontaneous emission
with lifetime Γ is avoided if the laser light is far-detuned from the atoms’ transition
frequency ω0 and a high enough laser power I is used. In the rotating-wave approximation, the dipole potential Udip and scattering rate Γsc are given by [61, 132]:
3πc2 Γ
Udip (r) =I (r) ,
2ω03 ∆

3πc2
Γsc (r) =
2~ω03



Γ
∆

2

I (r) ,

(2.38)

where ∆ = ω − ω0 is the detuning and ω is the laser frequency6. A simple relation
exists between the scattering rate and the dipole potential: ~Γsc = ΓUdip /∆. These
results show two essential points for dipole trapping. For red-detuned traps (∆ < 0),
the potential minima is located at positions with maximum intensity; while for bluedetuned traps (∆ > 0), the potential minima corresponds to minima of the intensity.
The dipole potential scales as I/∆, whereas the scattering rate scales as I/∆2 .
Therefore, optical dipole traps usually use large detunings and high intensities tokeep the scattering rate as low as possible for a required potential depth.
To exploit the ac-Stark shift effect to create a trapping potential, a spatiallydependent intensity in the form of a Gaussian laser beam is used. A Gaussian beam
6In the case of multilevel atoms, one must sum all the contribution from the excited states with

Clebsch-Gordan coefficients as the pre-factor, i.e. 1/∆ →

32

P

2
j cj /∆j .

2.2. Laser cooling and trapping
propagating in the y−direction with wavelength λ and power P has the intensity
profile [61]:"#−2 x2 + (z − z0 )2
2P
I=
exp
,
πw2 (y)w2 (y)(2.39)

where the spot size parameter is:

w (y) = w0

s

1+



y
yR

2

.

(2.40)

Here the Rayleigh length is yR = πw02 /λ and the minimum beam waist w0 is centeredat (0, 0, z0 ). For a far red-detuned beam (∆ < 0), the above intensity gives a trapping
potential in the form:#
−2 x2 + (z − z0 )2.
V = −U0 exp
w2 (y)"(2.41)

Evaporative cooling is achieved by lowering the total trapping potential, i.e. by
ramping down the total laser power. The introduction of an offset z0 in the dipole
trap location with respect to a magnetic trap center is beneficial to limit Majorana
losses during the transfer process of the atoms from the magnetic trap to the optical
dipole trap [104].
Since a single focused beam only provides weak confinement along the propagation
axis, it is common to cross two optical dipole beams with orthogonal polarization tocreate nearly isotropic atomic ensembles with tight confinement in all three dimensions. In a particular case of crossed-beam traps with 90◦ intersection angle in the33

2.3. Bose-Einstein condensation
xy−plane and equal waist w0 (where w0  yR ), the dipole potential near its minimum
can be approximated by the Taylor series with lowest order of expansion [61]

VCB ' −V0#x2 + y 2 + 2 (z − z0 )2
.
1−
w02

"

(2.42)

Consequently, external potential experienced by the atoms with atomic mass m canbe approximated as a harmonic oscillator with trapping frequencies

ωx2 =

2V0
,
mw02

ωy2 =

2V0
,
mw02

ωz2 =

4V0
.
mw02

(2.43)

2.3. Bose-Einstein condensation
BEC is a quantum phenomenon where an ensemble of particles with bosonic nature (integer spin) occupy the same ground-state energy level as they are cooled
down to temperatures near absolute zero. In the last few decades, extensive theories of BECs have been developed [36], providing predictions of physical phenomena
exhibited by these quantum gases and offering new insights. For a more comprehensive discussion, I recommend the text by Pethick and Smith [132], and the text by
Pitaevskii and Stringari [135] for more advanced (curious) readers. In this section,
I discuss the theory of BECs relevant to my work, particularly for neutral particles
with weakly repulsive interaction such as 87 Rb alkali atoms.ads

2.3.1. Gross-Pitaevskii equation. The field operator Ψ̂ (r, t) describing theatomic many-body wavefunction in the Heisenberg representation is34

2.3. Bose-Einstein condensation

i~

h
i
∂
Ψ̂ (r, t) = Ψ̂ (r, t) , Ĥ
∂t

 2 2
ˆ
~∇
†
0
0
0
3 0
+ Vext (r, t) + Ψ̂ (r , t) V (r − r) Ψ̂ (r , t) d r Ψ̂ (r, t) ,= −2m
(2.44)

where Vext (r, t) is the external potential and V (r0 − r) is the many-body interaction
potential [59]. At very low temperatures, a large number of bosonic atoms occupy asingle state allowing us to replace the field operator Ψ̂ (r, t) with a classical function
Ψ0 (r, t) which acts as the order parameter for the system. Assuming the range ofthe interatomic forces is much smaller than the average distance d = n−1/3 between
particles (set by the density distribution n), we need only to consider configurations
involving pairs of interacting particles (configurations with three or more particles
interacting simultaneously are neglected). The two-body interaction potential for
dilute gases at low temperatures is given by the effective s−wave scattering potential

V (r0 − r) = gδ (r0 − r) =4π~2 aδ (r0 − r) ,m

(2.45)

where a is the s−wave scattering length and m is the atomic mass. In other words, if
the condition of diluteness |a|  n−1/3 is satisfied, then the s−wave scattering lengthis sufficient to characterize all interaction effects on the atomic properties. With these35

2.3. Bose-Einstein condensation
substitutions, we obtain the Gross-Pitaevskii equation (GPE)
 2 2

~∇
∂
2
+ Vext (r, t) + g |Ψ0 (r, t)| Ψ0 (r, t) .
i~ Ψ0 (r, t) = −∂t
2m(2.46)

Stationary solutions for the GPE give a time evolution of the condensate wavefunction that follows the condition

Ψ0 (r, t) = ψ (r) exp (−iµt/~) ,(2.47)

with chemical potential µ fixed by the normalization of the atoms’ number. The
stationary GPE reduces to

~2 ∇2
2
+ Vext (r, t) + g |ψ (r)| ψ (r) .
µψ (r) = −2m
(2.48)For a dilute Bose gas of particles interacting with repulsive forces, or a > 0 as in87

Rb, the stationary GPE given by Eq. 2.48 describes the ground state of the system[99]. The BECs are stable and robust to increasing number of particles.
2.3.2. Thomas-Fermi approximation for Bose gases in the harmonic
trap. We now consider the stationary GPE in a time-independent harmonic trap

Vext (r) =1 Xmωj2 rj2 .
2 j=x,y,z(2.49)

If the chemical potential µ and interaction energy g |ψ (r)|2 is larger than the energy
spacing of the harmonic oscillator µ ' g |ψ (r)|2  ~ωj , we can neglect the kinetic
36

2.3. Bose-Einstein condensationenergy term in the stationary GPE. The limit in which one ignores the kinetic energy
term is called the Thomas-Fermi (TF) approximation and Eq. 2.48 reduces to:µTF ψ (r) ' Vext (r, t) + g |ψ (r)|2 ψ (r) .

(2.50)

The analytical solution to the atomic density distribution is:

nTF (r) = |ψ (r)|2 =1 Xmωj2 rj2µTF −2 j=x,y,z1g!

(2.51)

if µTF > Vext (r), and nTF = 0 elsewhere. In the TF limit for harmonic traps, theatomic distribution has an inverted parabolic profile and the atomic cloud has anelliptical geometrical shape bounded by the condition µTF = Vext (r). The TF radii
are given by
Rj =

s

2µ
,
mωj2

j = x, y, z,(2.52)and the chemical potential constrained by the total number of condensed atoms N is
accordingly [23]
15N g
~ω̄
µTF =
=
8πRx Ry Rz
2

15N a

r

mω̄
~

!2/5

,

(2.53)

where ω̄ = (ωx ωy ωz )1/3 . The momentum distribution of the condensate in the TF
limit can be found by taking the Fourier transform of the TF wavefunction [59],15nTF (p) = N
Rx Ry Rz
16~3

37



J2 (p̃)p̃2

2,(2.54)

2.3. Bose-Einstein condensation
where J2 is the Bessel function of order 2 and
1
p̃ =
~

s X

p2x Rx2 .

(2.55)

j=x,y,z

It is important to note that the TF approximation breaks down near the boundaryof the density profile where nTF (r) → 0 as the kinetic energy term is comparable tothe interaction energy. It is then natural to define a characteristic length connected to√the interactions in the system, namely the condensate healing length ξ = h/ 2mgn0 ,where n0 is the peak density evaluated at the centre of the trap n0 = nTF (r = 0) =µ/g. Within the condensate healing length region near the boundary, where rj 'Rj ± ξ, one must include the finite-size corrections to the TF limit [36, 132, 135].

2.3.3. Expansion of partially condensed ultracold gases. Measurements of
the atomic density distribution reveals the condensation nature of an ultracold gas.
In the laboratory, this is normally done by releasing the harmonic trap and letting theatomic cloud expands. The momentum distribution of the cloud is transformed intoa spatial distribution. Images of the atomic cloud can be captured by the cameras.
Further, the time-of-flight (TOF) expansion increases the size of the spatial structureso that it is large enough to be resolved by the imaging system.
Ultracold gases produced in the lab have a small fraction of non-condensed (thermal) atoms. While the condensed fraction of the cloud can be approximated with the
TF distribution, the thermal density nth (r) can be calculated using a semiclassical38

2.3. Bose-Einstein condensation
expression for the Bose statistics in the grand canonical ensemble [59]ˆnth (r) =d3 p
np (r) =(2π~)3

ˆ

d3 p
1
,
3
(2π~) exp [β {H (r, p) − µ}] − 1(2.56)

where β = 1/kB T and T is the temperature, H (r, p) = p2 /2m + Veff (r) is thesemiclassical Hamiltonian and µ is the total chemical potential.
In my work, for temperature T below the critical condensation temperature Tc ,I can apply the Hartree-Fock theory to the many-body Hamiltonian [50, 135]. Since
I was only interested in the expansion of thermal atoms near the tail-end of thecondensate TF distribution (nTF (r) ≈ 0), I can approximate the effective potential
experienced by the thermal atoms as Veff (r) − µ ' Vext (r) − µTF and I obtain therelation7
H (r, p) − µ ' |p|2 /2m + Vext (r) − µTF .

(2.57)

Performing the integration in Eq. 2.56 over the momentum space yields the spatial
distribution of the thermal cloud [59]1µTF − Vext (r)1nth (r) = 3 g3/2 exp= 3 g3/2 (Z exp [−βVext (r)]) ,
λT
kB T
λT
where λT =(2.58)p2π~2 /mkB T is the thermal de Broglie wavelength, Z = exp (µTF /kB T )is the fugacity, and gm (x) =P∞jmis the Bose function.
j=1,2,... x /j

7For T Tc with nTF  nth , the Hartree-Fock approximation gives Veff (r) − µ ' Vext (r) +2gnTF (r)−µTF = gnTF (r). This is equivalent to saying the thermal atoms experienced a “trapping”
potential with an inverted parabola (TF) profile, due to interaction with BEC atoms [50].

392.3. Bose-Einstein condensation
A TOF expansion of the thermal cloud occurs after releasing the harmonic trap.
The momentum distribution function in Eq. 2.56 evolves in time according to np (r, t) =np (r − pt/m). This modifies the thermal density to:

h
i
1
1 Y
q
nth (r, t) = 3
g3/2 Z exp −β Ṽext (r, t) ,λT j=x,y,z 1 + ω 2 t2

(2.59)

jfor the effective harmonic trap
X
ωj2
1Ṽext (r, t) = m
r2 .
2 j=x,y,z 1 + ωj2 t2 j

(2.60)

To calculate the TOF expansion of the condensed part, I now generalize thefrequencies of the trapping potential to be time-dependent ωj (t). Their initial static
values ωj0 fix the initial equilibrium configuration of the system corresponding to the
TF distribution. To solve the time-dependent GPE in Eq. 2.46, I use the ansatz [23]Rj (t) = λj (t) Rj0 = λj (t)

s

2µ
,
2
mωj0

j = x, y, z,(2.61)

where λj (0) = 1 and Rj0 is the stationary values of the initial TF radii. In thistreatment, the inverted parabolic TF profile is preserved throughout the expansionof the cloud. The scaling factors obeys λj (t) the second-order differential equations2
ωj0
d2 λj (t)=− ωj2 (t) λj (t) ,dt2
λj (t) λx (t) λy (t) λz (t)40j = x, y, z.

(2.62)

2.3. Bose-Einstein condensation
For the sudden release of a harmonic trap, ωj2 (t > 0) = 0, the above equation becomes2
ωj0
d2 λj (t),=dt2
λj (t) λx (t) λy (t) λz (t)j = x, y, z.

(2.63)The expansion of the TF radii is obtained by solving Eq. 2.63 for λj (t).
The total density, including both condensed and non-condensed parts, after TOF
expansion for time τ isntot (r) = nTF (r) + nth (r)
"
#!
X
rj2
15N
=
max 0, 1 −+8πRx (τ ) Ry (τ ) Rz (τ )Rj2 (τ )j=x,y,z
"
#!
2 2
Y
ωj0
rj
m X
1
q
.
g 3 Z exp −
2 2
2k
T
1
+
ω
τ
3
2 2 2
B
j0
1+ω τ
j=x,y,zj=x,y,z λ(2.64)j0

T

In common imaging techniques, such as absorption imaging, the density distribution
being probed is always integrated over the imaging axis, i.e. along z, and one canwrite the column density asρtot (x, y) = ρTF (x, y) + ρth (x, y) .

(2.65)The TF distribution of the condensed cloud integrated over z gives the column density

ρTF (x, y) = "5Nmax 0,2πRx (τ ) Ry (τ )

41

X

rj2
1−
Rj2 (τ )j=x,y#3/2 

.

(2.66)

2.3. Bose-Einstein condensation
Meanwhile, the column density for the thermal cloud is given by [26]1ρth (x, y) = 3
λT

s

"
#!
X rj2
2πkB T Y
1q
g2 Z exp −
,
2
mωzo
2σj2
1 + ω2 τ 2
j=x,y
j=x,y

(2.67)

j0

where the root-mean-square (RMS) widths of the thermal density profile are

σj =

s

2 2
1 + ωj0
τ kB T,2
ωj0
m

j = x, y, z.

(2.68)For a long enough time τ  1/ωj0 , the thermal cloud expansion becomes isotropic
and the temperature can be inferred from the distribution RMS width, namely kB T =m (σ/τ )2 . In general, to measure the temperature of the system, two different releasetime with durations τ1 and τ2 are used. The thermal wings of the released ultracoldgases are fitted to yield different values of RMS widths, which provide a measure of
the system temperature as
m
T =
kB

 2
2 
σj2 σj1
− 2 .
τ22
τ1

(2.69)

Figure 2.5(a) shows an example of column density profile ρtot (x, y) of a 87 Rb Bosegas released from a harmonic trap. The column density is proportional to the opticaldepth (OD) obtained by absorption imaging along the z−direction. The squared
RMS widths, σx2 and σy2 , are obtained by fitting the thermal portions of the atomic
cloud with Eqs. 2.65-2.67. Plotting σx2 and σy2 with respect to the square of TOF

42

2.3. Bose-Einstein condensation

Figure 2.5. Temperature of a 87 Rb Bose gas released from a harmonic
trap. (a) Fitting the column density profile ρtot (x, y) of a released 87 Rb
Bose gas from a harmonic trap gives the RMS widths, σx and σy . (b)Plot of σx2 and σy2 with respect to the square of TOF duration τ 2 . The
temperature of atomic cloud T = 165(2) nK is obtained using Eq. 2.68.

duration τ 2 , we can then use Eq. 2.68 to obtain the temperature of the atomic cloud
T = 167(2) nK, as shown in Fig. 2.5(b).

2.3.4. Spinor Bose-Einstein condensates. The GPE used to describe a single
component condensate in previous subsections can be extended to study the dynamics
of multi-component BECs. Here I consider three-component condensates in the F = 1
hyperfine manifold described by the wavefunctions ψ−1 , ψ0 , and ψ1 in an externalpotential V . A set of coupled GPEs for the mean-field dynamics is given by [68, 159]
(see Appendix B for the derivation of the three-component GPEs)

43

2.3. Bose-Einstein condensationi~∂t ψ1i~∂t ψ−1

 2 2


~∇
2
2
2
=
−
+ V + (c0 + c2 ) |ψ1 | + |ψ0 | + (c0 − c2 ) |ψ−1 | ψ12m∗
+c2 ψ−1
ψ02 ,(2.70)+c2 ψ1∗ ψ02 ,(2.71)


 2 2

~∇
2
2
2
+ V + (c0 + c2 ) |ψ−1 | + |ψ0 | + (c0 − c2 ) |ψ1 | ψ−1=−2mi~∂t ψ0 =





~2 ∇2
2
2
2
−
+ V + (c0 + c2 ) |ψ1 | + |ψ−1 | + c0 |ψ0 | ψ02m+2c2 ψ0∗ ψ1 ψ−1 ,(2.72)

where c0 is the spin-independent coefficient and c2 is the spin-dependent coefficient.
For repulsive gases, the spin-independent coefficient c0 is positive. In such system,
the sign of spin-independent coefficient c2 characterizes the “magnetic” nature of theground states of spinor Bose gases, i.e. a negative (positive) sign of c2 represents a
ferromagnetic (anti-ferromagnetic) phase.
The spinor condensates are characterized based on the miscibility and immiscibility of the different spin components. The terms used here are analogous to the mixingof liquid phases: immiscible means the two components are phase separated, as inoil and water; and miscible means the two components are phase mixed. In a simple
picture of a homogenous two-component system, labeled by a and b, the mean-field44

2.3. Bose-Einstein condensationenergy takes the form
1
E=
2

ˆ


d3 r gaa n2a + gbb n2b + 2gab na nb ,(2.73)

where gaa , gbb , and gab are the interaction coefficients between two identical or different
spins. Minimization of this energy gives the criterion for miscibility gab <immiscibility gab >√√gaa gbb or

gaa gbb [132]. In a ferromagnetic condensate with c2 < 0, one finds

the states mF = ±1 are immiscible since g1,1 = g−1,−1 = c0 + c2 and g1,−1 = c0 − c2 ;while the states mF = +1 (or mF = −1) and mF = 0 are miscible since g0,0 = c0
and g0,1 = g0,−1 = c0 + c2 . The opposite happens for anti-ferromagnetic condensates
where c2 > 0. The alkali species we work with has a ferromagnetic nature since c2 hasa negative value in the F = 1 hyperfine manifold of 87 Rb 5S 1/2 ground state [177].
In the presence of external magnetic fields and field gradients, the spinor dynamics
governed by Eqs. 2.70-2.72 exhibit a variety of phenomena, including coherent spin
mixing, spin textures, spin-domain formation and coarsening [27, 165]. The spin
interaction energy |c2 | n sets the timescale for the spinor dynamics, where |c2 | n/h '10 Hz or h/ (|c2 | n) ' 0.1 s in a typical 87 Rb condensate. In spinor condensates, apcharacteristic length associated with c2 is the spin healing length ξs = h/ 2m |c2 | n0 .
When the spin healing length is larger than the size of the condensate, the interactionenergy is not sufficient to create spin domains in the condensates and hence the spin
dynamics in the presence of external field is largely suppressed [26]. For this reason,studies of spin dynamics in Bose gases are generally conducted in elongated quasi452.3. Bose-Einstein condensation

20

0

0

100

200

40

-100

0

-200

20

-20

Figure 2.6. Magnetization Mz (r) of two-component 87 Rb BECs. (a)Schematic illustrating a spinor BEC with domains in an anisotropic
crossed-dipole trap. (b) Images showing the progression from a uniformly magnetized condensate (short times) in which domains appear(intermediate times), and then grow spatially (long times); during thisprocess the condensate slowly decays away. (c) Color scale indicating
the degree of magnetization (colors from blue to red), and the density(intensity from black to colored).

one-dimensional trap where spatial spin structures have a tendency to form only inthe direction of the weak trap axis.
Examples of spin domain formation and coarsening, based on our previous workin Ref. [40], are shown in Figs. 2.6-2.7. Here, we explore the time-evolving magnetization of two-component 87 Rb BECs in the 5S1/2 electronic ground state. Our
BECs are well described in terms of a spinor wave-function Ψ(r) = {ψ↑ (r), ψ↓ (r)},where the |↑, ↓i pseudo-spins label the |F = 1, mF = ±1i atomic spin states. The
46

2.3. Bose-Einstein condensationz position (µm)

200
100
0
-100

z position (µm)

-200
200
100
0
-100
-200

0.1

1
thold (s)10

Figure 2.7. Time evolution of magnetization Mz (z) of two-component
Rb BECs. (a) Experimental data and (b) finite temperature simulation using the stochastic-projective GPE method [18]. In both simulation and experiment, the spatial structure of Mz (z) coarsens after an
initial growth period as domains coalesce.
87

mF = 0 is not populated initially, and hence the time evolution of ψ0 (r, t) stays zeroall the time. The dynamics of the two-component BECs are governed by the spinor
Gross-Pitaevskii equation


~2 ∇2
Ω⊥
2
i~∂t ψ↑,↓ (r) = −+ V (r) + (c0 − c2 ) n(r) + 2c2 N |ψ↑,↓ (r)| ψ↑,↓ (r) +ψ↑,↓ (r),2m
2(2.74)a continuum analog to the transverse field Ising model [75]. Here m is the atomicmass; n(r) = N |ψ↑ (r)|2 + |ψ↓ (r)|2 is the total density; V (r) is a spin-independent
external potential; Ω⊥ describes the Zeeman shift of a “transverse” magnetic field (in47

2.3. Bose-Einstein condensationan RF-dressed state); and c0,2 are the spin-independent and spin-dependent interaction coefficients [69, 127]. This Hamiltonian has a Z2 symmetry describing a reversalof |↑i and |↓i, which is absent in most binary mixtures [63, 70, 114, 176].
In 87 Rb’s F = 1 manifold, c0 = (100.86) × 4π~2 aB /m vastly exceeds c2 ≈ −4.7 ×
10−3 c0 [170]. For a static density profile and when Ω⊥ = 0, each spin component in
Eq. 2.74 is separately described by an attractive single-component GPE. In our spinor
experiment, we initiated a 34 µs RF-pulse that puts each atom in an equal-amplitude
superposition of the |↑, ↓i spin states, which is the ground state of the Hamiltonianin Eq. 2.74 when Ω⊥ is large. The system then evolves according to Eq. 2.74 withΩ⊥ = 0. This procedure is equivalent to rapidly quenching Ω⊥ to zero.
The quenched binary mixture is held of a variable duration thold , up to 20 s, while
spin structure forms and evolves. After thold , we remove the confining potential andallow the atomic ensemble to expand (largely transversely) for 19.3 ms, during whichtime we apply a Stern-Gerlach gradient force to separate the spin components [51]. We
detect the resulting density distribution of each spin component, |ψ↑ (r)|2 and |ψ↓ (r)|2 ,by absorption imaging. We then study the dynamics of the spin domain formationand coarsening, where the domains are quantized by spatially varying magnetization
Mz (r) = |ψ↑ (r)|2 − |ψ↓ (r)|2 , as shown in Fig. 2.6. The time evolution of the domains
as they grow and coalesce is displayed in Fig. 2.7. In this binary spin system, weobserve the dynamical generation of spin-domains from an initially non-equilibrium
system followed by their subsequent relaxation of progressively larger domains, i.e.
coarsening.

48

CHAPTER 3

Experimental apparatus and procedure
Our experimental apparatus was originally built with a plan to conduct atom-chip
experiments, where a nano-fabricated chip with micron scale wires will be placed near
the atoms to generate artificial gauge field [6, 131]. However, we shifted our priority
as the prototype of the chip was being bench-tested in the lab. Instead, we performed
other experiments, such as observation of stripe phase in spin-orbit-coupled (SOC)
condensates and study of ultracold atoms in one-dimensional (1D) trap.1
In this chapter, I give an overview of our experimental setup and the apparatus
we used to laser cool 87 Rb and produce Bose-Einstein condensates (BECs) in the
RbChip lab. I first discuss the optical layouts for each laser cooling system. I describe
the green laser and Ti:sapphire laser we used for creating 1D Bose gases and SOC
condensates. I next discuss the setup in the main experiment chamber, where the“science” is happening. Lastly, I give details of the control sequences we used to make
BECs and the method we developed to calibrate and cancel magnetic field noise.

3.1. Apparatus for laser cooling
3.1.1. Master and cooling lasers. In laser cooling of 87 Rb, we use the D2line since it has a cycling transition (F = 2 → F 0 = 3) for cooling and a repump
1As this dissertation is being revised, other lab members finally installed the chip into our main

vacuum chamber!

49

MOT repump

Bragg probe

780.241 nm
384.230 484 THz

MOT cooling

200 MHz72.218 MHz

97 MHzImaging probe

80 MHz156.947 MHz

136 MHz170 MHz72.911 MHz

optical pumping

266.650 MHz

80 MHz3.1. Apparatus for laser cooling

Cooling laser(slave)2.563 GHz6.834 GHz

Repump laser(master)

Figure 3.1. Schematic of laser frequencies we use in the D2 line of 87 Rb.

transition (F = 1 → F 0 = 1, 2) for recycling the atomic population back to its coolingstates. The transitions used in our laser cooling setup are shown in Fig. 3.1. Here,two continuous-wave (cw) diode lasers provide the necessary power for laser cooling.
Acousto-optic modulators (AOMs) are used to shift the laser frequency, to switch on
and off the laser power in several 10’s of ns duration, and to control the laser power
being injected into optical fibers.

50

3.1. Apparatus for laser cooling

Toptica
DL ProFigure 3.2. Optical layout of the master (repump) laser. The master
laser is locked to the transition F = 1 → F 0 = 2 using saturation
absorption spectroscopy of 87 Rb. The master laser provides power for
the MOT repump, imaging repump, and the Bragg probe.

An optical layout of the master laser which provides the repump light is shownin Fig. 3.2. The master laser Toptica DL Pro 100 is locked to the F = 1 → F 0 = 2transition in the D2 line of 87 Rb using saturation absorption spectroscopy [137] withan integrated DigiLock 110 PID module. In this spectroscopy, the frequency of thepump beam is shifted by 2f0 with respect to the probe beam, which is performed
using an AOM in the double-pass configuration [43]. The frequency commanded to
the AOM is f0 = 80 MHz, which gives a detuning of −80 MHz for the actual master
laser frequency from the F = 1 → F 0 = 2 transition frequency reference. This master
laser provides the repump power for the magneto-optical trap (MOT), absorption
imaging, and Bragg probe. Typically, the gross repump power needed is ∼ 30 mW
51

3.1. Apparatus for laser cooling

Toptica
DLC TA Pro

Figure 3.3. Optical layout of the cooling (cycling) laser. The “slave”
cooling laser is beat-note locked to the master laser. The cooling provides power for the MOT cooling, optical pumping, and imaging in thecycling transition.
with a laser linewidth of ∼ 1−10 MHz and hence a commercial diode laser is sufficient
to provide the repump light.
The second laser (slave) system, an integrated diode laser and tapered amplifier(Toptica TA Pro) is beat-note locked [139] with respect to the master laser. The
optical layout of the main cooling laser is shown in Fig. 3.3. Here, we used the analog

52

3.1. Apparatus for laser cooling
Table 3.1. Reference frequencies for the laser cooling transitions.
Laser transitions,
Master laser,
MOT repump,
Imaging (repump),
Bragg probe,
Beat-note cooling laser,
MOT cooling,
Imaging (cycling),
Optical pumping,F → F0
1→2
1→2
1→2
1→0
2→3
2→3
2→3
2→2

Frequency command (MHz)

Detuning (MHz)

80
80
80
200 or -200 (and -400)
102.875 × 6480
97
80 (beat-note at 104.65 × 64)-8000-51 or -451
-97
-17
0
136

driving electronics SYS DC 110 module to control the current and temperature of thelaser diode. This module also features the ability to control the laser frequency withina mode-hop-free tuning range of 20 GHz which is important in its application to laser
cooling. The tapered amplifier provides ∼ 2 W of power used for the MOT cooling,
imaging probe in the cycling transition (|2, 2i → |3, 3i), and optical pumping to thedark states. In addition, the unused monitor output of the MOT cooling fiber fused
couplers provides the main beam for very far-detuned Bragg probe for observationsof the stripe-ordered phase.
The frequency shifts for the laser cooling transitions are summarized in Table
3.1. The beat-note locked laser frequency has a factor of ×64 from the phase-locked
loop (PLL) frequency divider. The frequency command gives a red-detuned shift ofthe slave laser with respect to the master laser, a net shift of −97 MHz from theF = 2 → F 0 = 3 transition in the D2 line. In practice, for frequency scanning of
the cooling beams, we scan the beat-note frequency of the cooling laser where each53

3.1. Apparatus for laser cooling
AOM’s frequency is generally kept constant. For the example is given in Table 3.1, thebeat-note frequency is set at a different value to give the optical pumping a detuning
frequency of 136 MHz from the F = 2 → F 0 = 2 transition.
For the Bragg probe, I used a separate optical setup with double-pass AOM configuration to red-shift the frequency by −400 MHz. The single-pass AOM on the master
laser system in Fig. 3.2 can be switched to give either +200 MHz or −200 MHz frequency shift with respect to the master laser. This corresponds to a detuning of−51 MHz to −451 MHz from the closed transition F = 1 → F 0 = 0 in the D2 lineof 87 Rb [124]. Variations of the AOM frequencies then allowed me to do a nearresonant Bragg probe with the detuning range of ∆ = +10 MHz to −500 MHz for the
F = 1 → F 0 = 0 transition. For the purpose of doing very far-detuned Bragg probeof the F = 1 state, I used the monitor output port of the MOT cooling fiber fused
couplers / combiners. This gives a red-detuning of ∆ = 6.8 GHz for the F = 1 → F 0transition within the elastic Rayleigh scattering regime, where ∆/Γ = 1.1 × 103 and∆ is much larger than the energy splitting of the F 0 excited states.
For frequency-shifting using AOMs with typically low input optical power ∼ 1 W,
we used acousto-optic products from IntraAction Corp., i.e. the ATM-A2 series(780 nm optical wavelength). To drive the AOMs with a fixed frequency shift depending on the needs of the laser cooling processes, we used a stand-alone radiofrequency (RF) driver with a built-in RF oscillator (matching the center frequency of
the AOMs), i.e. the model DE series. However, in most applications, time-varying

54

3.1. Apparatus for laser coolingfrequency ramps of the lasers are required and hence we built a customized RF electronic system to drive the AOMs, as will be described in the next subsection. The
tunable frequency reference is supplied by a direct-digital-synthesized (DDS) signal
generator Novatech Model 409B which is programmable in 0.1 Hz steps from 0.1 Hz
to 171 MHz.
The capabilities of frequency shifting in this system are constrained by the AOMs’
crystal material. The maximum diffraction efficiency of the AOM at its operating
center frequency is ' 70−80%. Depending on the AOM specifications, the diffraction
efficiency usually drops to ' 40% if the RF driving frequency is off by 20% from theAOM operating center frequency. Frequency-shifted beams are then injected throughoptical fiber launch systems to the atoms in the main glass “science” cell. In thissystem, the laser powers are controlled by analog voltage sent to the electronic drivers
which drive the AOMs.
Although the acousto-optic devices offer switching capabilities at a timescale lessthan 1 µs, they do not provide a 100% transmission extinction ratio. Since pW laser
power of near-resonant light is enough to heat the atoms and disrupt the coolingprocess, additional mechanical shutters must be deployed to block leaking light fromthe AOMs. To do this, we used computer hard-disk-drive (HDD) shutters [106] placed
right before the optical fibers. The switching times of these shutters are typically inthe order of < 100 µs and they can completely block a beam size of ∼ 1 mm. The
HDD shutters have a 5 ms delay time before mechanical motions are generated oncethe driving transitor-transistor logic (TTL) signal is received.

55

3.1. Apparatus for laser cooling

Figure 3.4. RF electronics drive of the AOMs.
The HDD shutters have very low cost and are particularly advantageous for highpower applications. The shutter flags are easily modifiable, for example a 1/2”
mirror can be glued to the end of the HDD pivot arm to block high power > 1 W
laser beams. However, the HDD shutters often fail after > 30 thousand cycles (within
a year or two). For the latter reason, we opt for commercial shutter systems, such as
the SRS SR474 laser shutter system which has a lifetime of > 10 million cycles. We
measured the delay time of the SR474 shutters to be 3.7(1) ms and thus we had totake this delay into account in our experimental sequences when sending digital TTL
signals to close or open the shutters.
3.1.2. RF driving electronics. The typical customized RF electronics we used
to drive the AOMs are shown in Fig. 3.4. We used a frequency doubler (MK-3)
after the Novatech DDS output to increase the range of the oscillator frequency by afactor of 2. For example, we implemented the doubler to drive AOMs with operating
center frequency above 171 MHz. An RF amplifier (ZX60-8008E+) was added tocompensate attenuation from the frequency doubler. Note these two components
were not needed for applications with AOM’s operating frequency < 171 MHz in56

3.1. Apparatus for laser coolingmost of our experiments. To be able to control the RF output power via an analog(AO) voltage, we used RF mixer electronics (ZAD-1-1+) to combine an oscillating
RF source with a programmable analog DC voltage. We then used a TTL switch(ZYSWA-2-50-DR) for digital switching of the RF power within 100 s of ns. Lastly,
another RF amplifier (ZHL-1-2W) was used to raise the RF power to the operatinglevel of the AOMs, which is typically ∼ 1 W.

3.1.3. Laser for red-detuned optical dipole trap. The availability of commercial cw lasers has made evaporative cooling with a 1064 nm red-detuned optical
dipole trap a standard practice in the production of 87 Rb BECs. In our experiments,
we used a single-mode fiber amplifier IPG YAR-30K-1064-LP-SF to provide a continuous laser power of ∼ 10 − 20 W at a wavelength of 1064 nm. The amplifier isseeded by an NP Photonics FLS-25-1-1064 fiber laser source. The setup for thislaser system is shown in Fig. 3.5. It is important to note since this laser falls into theclass 4 category (based on the ANSI Z136.1 laser safety standard), appropriate safety
precautions must be taken. Hence we used beam dumps in any open-ended paths ofthe beam.
The 1064 nm laser power is divided into two parts, the main dipole and the weak
dipole (see Fig. 3.5). The “main” dipole power is split into two crossed-beam traps in
the main experimental chamber to create optical dipole trapping of BECs. Meanwhile,the “weak” dipole power was used in the 1D Bose gases project that was operated
by my colleague to provide a weak confinement in the longitudinal axis of a 1D trap.

57

3.2. Green laser and Ti:sapphire laser

Figure 3.5. Laser system for the 1064 nm optical dipole trap.
The 1D trap was created with a 532 nm blue-detuned optical trap with a LaguerreGaussian beam [91, 167].

3.2. Green laser and Ti:sapphire laser
The two main research projects going on in the lab as of Summer 2017 involved
1D Bose gases and SOC condensates. A green laser at 532 nm was used to create
the 1D blue-detuned optical dipole trap and a Ti:sapphire laser was used to deliver
the Raman transitions in SOC condensates. The green laser (25W Millenia eV)
provides ∼ 4W power to the 1D optical dipole fiber injection and ∼ 10W pump
power to the Ti:sapphire laser (SolsTiS-SRX-F 4000). The optical layout of thislaser system is shown in Fig. 3.6. We found the typical warm-up time for these lasers
to be about 15 − 20 minutes, after which they yielded nominally stable output power
within < 2%.

58

3.2. Green laser and Ti:sapphire laser

SolsTiS
Ti:Sapph

Millenia eV

Figure 3.6. Optical layout of the Ti:sapphire laser system. A green
laser at 532 nm is used to pump the Ti:sapphire laser and to provide
blue-detuned optical trap. The Ti:sapphire laser is tunable in nearinfrared regime, suitable for Raman applications to create SOC condensates and optical lattice experiments.
The Ti:sapphire laser is tunable in the near-infrared regime 725 − 875 nm, suitableto create SOC condensates involving Raman transitions and optical lattices. Tuningof the Ti:sapphire laser is done by first tweaking the intracavity birefringent filter(BRF) via the software user interface of the SolsTiS laser. The BRF tuning gives
a discrete change for the wavelength of the laser mode, which can take values from
0.15 to 0.80 nm. Next, changing the etalon tuning gives a more precise wavelength
in the order of 0.01 nm. The etalon tune control has to be set near the middle of its
adjustment range, i.e close to 50%, or else the laser may undergo mode-hopping to
the next etalon mode. The single mode operation of the Ti:sapphire laser with etalon

593.3. The main experiment chamberlocking gives a spectral linewidth of ∼ 10 MHz, which is sufficient for my experiments.
To further reduce the laser linewidth to sub-50kHz, a reference cavity could be used.
However, the SolsTiS SRX scanning reference cavity was not in operation (in needof repair) at the time of writing this dissertation.

3.3. The main experiment chamber
Figure 3.7 shows our main experiment chamber where BEC is created in an ultrahigh vacuum (UHV) environment. The main chamber consists of optical tables, UHV
glass cell, vacuum pumps, various electromagnetic sources, and cameras. The atoms
are manipulated using various lasers (as described in the previous sections), which aresent through optical fibers and focused to the atoms. Since the original plan for theapparatus was to conduct atom-chip experiments, the main chamber was designed
to be composed of the bottom and the upper parts. The bottom part contains cold
atoms in a MOT which are transported vertically to the location near the chip in theupper science cell [131], as shown in Fig. 3.7(b). In the upper cell, BEC is achieved
and science experiments are performed.
A full explanation on all components of the main experiment chamber requires anextensive description. In this section, my discussion will instead focus on the opticallayout on the upper part of the main chamber. Other details on the design of theapparatus can be found in previous dissertation from the group [131].
3.3.1. Optical layout on the upper part of the main experiment chamber. Figure 3.8 shows the optical layout on the upper part of the main experiment60

3.3. The main experiment chamber
(a)

upper
part

transport
towerbottompart(b)

Figure 3.7. The main experiment chamber where BEC is created andexperiments are performed. (a) Photograph of the RbChip main apparatus. (b) An illustration of the vertical magnetic transport system tomove the atoms from the bottom cell (MOT) to the upper science cell,as taken from Figure 4.1 of Perry’s thesis [131].

61

3.3. The main experiment chamber

Figure 3.8. Optical layout on the upper part of the main experimentchamber. The convention for the Cartesian coordinates used in ourlab is as displayed above, with gravity pointing along the ey direction.
The optical components located on the right side (+ex direction) of theglass cell is not shown here (see Fig. 3.9 instead).

chamber. The optical components consist of crossed-beam dipole trap, Raman lasers,
absorption imaging systems, and relay assembly for optical Bragg scattering. The
relay imaging system is not shown in Fig. 3.8 (see Fig. 3.9 instead) and it will be
discussed in the next subsection.
Our optical dipole trap is a crossed-beam setup [61] where one main beam is split
into two parts with an AOM. The zeroth order beam with higher power and tighter
focus is sent along ez . This is used to optically trap the atoms after being transferredfrom the magnetic trap. The +1 order beam with less power and slightly larger beam62

3.3. The main experiment chamber
waist, is sent along ex to serve as the “cross” dipole beam. Evaporative cooling is thenperformed by exponentially ramping down the overall power of both dipole beams andallowing the highly collisional thermal atoms to be evaporated away.
Typical trap frequencies of our dipole trap are (ωx , ωy , ωz ) = (93, 64, 39) Hz. Measurements of the dipole trapping frequencies are done by exciting the sloshing dipole
mode [118]. The atoms in the optical dipole trap are given an initial “kick” from apulsed magnetic gradient force. The frequency of the collective oscillations of theatoms’ center of mass is the optical trapping frequency.
Besides using the crossed-dipole beams to make BECs, we also have a weaker1064 nm optical dipole trap with beam path along ez . This beam was used to give a
lesser trapping frequency (∼ 10 Hz) in the x−direction for the 1D Bose gases project.
Two counter-propagating Raman lasers are used to create SOC condensates. The
Raman A beam along ex , as shown in Fig. 3.8, provides one of the Raman lasers.
Another Raman beam along −ex , labelled by Raman B, is located on the other sideof the glass cell and it will be discussed in the next section (see Fig. 3.9 instead).
Properties of the ultracold atoms are measured using the absorption imaging technique [81]. In our apparatus, we have three absorption imaging systems for different
purposes. Two imaging probe beams along ez are combined with a beamsplitter todo absorption imaging in situ and at ∼ 25 ms time-of-flight (TOF). These imaging
systems share the same first telescope pair for the objective and the eyepiece lenses.
Since it is important to minimize the aberrations in the in situ image, the optical63

3.3. The main experiment chamber

Relay lens assembly

Figure 3.9. Relay assembly for optical Bragg scattering experiments.
The bottom inset shows schematic of the relay lenses and the two
Fourier plane regions as mentioned in the text.

axis of the lens systems is aligned to the in situ location of the atoms. The verticallyspaced in situ and TOF images are then picked up by two separate telescope pairs,
giving a total image magnification of 5.22× for the in situ images and 3.11× for the
TOF images.
It is important to note the biggest problem we had in our experiment was backreflectance from the glass cell. Each surface of the glass cell gave about 8% reflectancefor light at 780 nm wavelength which was unavoidable unless we re-constructed thewhole apparatus. For this reason, careful alignment of the optics was crucial to avoid
interfering beam from back reflectance.

643.3. The main experiment chamber
3.3.2. Relay assembly for optical Bragg scattering. I used an optical Bragg
scattering technique to observe the stripe-phase in SOC condensates. Due to limited
optical access in our experiment, I had to come up with creative ideas to send our
Bragg probe beam in conjunction with the Raman lasers. In addition, I needed topreserve the abilities to do absorption imaging, to combine a 532 nm laser beam, andto dump the 1064 nm red-detuned optical dipole trap within the same optical axis.
To satisfy all these requirements, I constructed a relay imaging system consisting of
two pairs of f1 +f2 imaging telescopes (afocal system). The optical setup of the Bragg
probe with other laser beams is shown in Fig. 3.9.
The first pair of our relay imaging system was composed of lenses with focal length
f1A = 38 mm (EO49-104) and f2A = 250 mm (AC254-250-B). The second pair of lenses
had focal length f1B = 250 mm (AC254-250-B) and f2B = 150 mm (AC254-150-B). In
between these two pairs of lenses, I put an iris on the first image focal point to assist
with aligning the optics. This relay assembly had a total magnification of 3.9× andit allowed me to have an “arm extension” to send the Bragg beam within the Fourier
plane region of the second lens pair.
Similar lenses with focal length of f2B = 150 mm (AC254-150-B) were used tofocus the Bragg probe and the Raman B beam to the atoms. To collimate the Bragg
probe fiber output, an aspheric lens PAF-X-5-B FiberPort was used to give a 1 mm
output waist diameter. A C220TME-B collimating lens was used for the Raman beam
giving about 2 mm output waist diameter. With 3.9× de-magnification by the relay
imaging system, the beam waist diameters seen by the atoms were respectively 250 µm

65

3.3. The main experiment chamber
and 500 µm for the Bragg and Raman beams. To clean up the polarization of theRaman beams and Bragg probe, Glan-Laser GL10-B polarizers were used after thecollimated fiber output. Photodiodes were used to monitor the power of each Bragg
probe and the Raman B beam.
In a well-focused imaging system, a transverse displacement of the Bragg beam
automatically turns into a change in the Bragg angle with respect to the optical axis.
I used a D-shaped mirror mounted on two linear translation stages to send the Bragg
beam at λ = 780.24 nm. To calibrate the Bragg signal, I created periodic modulation
to the atomic density distribution by applying an optical lattice potential along ex[17, 77]. In my optical lattice system with periodicity of d = 395.01 nm, the Bragg’s
angle was θB = arccos (λ/2d) = 9.02◦ with respect to the optical axis. To satisfy the
Bragg’s condition, I first aligned the beam’s transverse displacement t to be always
equal to t = f1A · tan(θB ) = 6.03 mm along the Fourier plane region between lenses
f1A and f2A . Since the lens pair f2A −f1B formed a 1:1 imaging system, the 1st Fourier
plane in between lenses f1A − f2A mapped directly to the f1B − f2B region.
For alignment purposes, I made sure the Bragg signal was on the same spot with
the in situ BEC location, which was easily done with the relay imaging system. I
gradually displaced the Thorlabs PT1 xz-translational stages of the D-shaped mirror
using Newport Model 8302 piezo-controlled motor (Picomotor) actuators. I foundthe optimum Bragg angle by maximizing the measured in situ Bragg signal. To further
optimize and fine tune the Bragg signal, I also rotated the D-shaped mirror with a
Newport Model 8401 Picomotor actuator. I could do a scan of the Bragg angle with66

3.3. The main experiment chambera precision of ≥ 30 nm/f1A = 8 · 10−7 rad limited by the minimum incremental motionof the linear Picomotor actuator. I would like to point out that the Picomotors I used
experienced hysteresis and they had no feedback mechanism (open-loop) to measure
their exact movements. It was thus important for me to keep track on the scan
directions while stepping the translation stages.
The same relay imaging system was used to send the 790 nm Raman B beam anddetect the Bragg signal at the wavelength of 780 nm. I used laser clean-up filters toserve as dichroic mirrors for 780/790 nm light. An LL01-808-25 filter oriented at 33◦
incident angle acted as an ultra-narrow bandpass filter with 94% transmission at peakwavelength of 780 nm and bandwidth of ±3 nm. The laser line filter was placed at
the image focal point of f2A to give the best collimation angle for the Raman beam
being sent.
The diffracted Bragg signal was captured using an electron-multiplying chargecoupled device (EMCCD) camera, iXon Ultra 888, which has high photon sensitivity. This camera was mounted on custom-made plates to ensure mechanical stability(see Appendix C). The wavelength of the scattered Bragg signal was expected to be
780 nm. To reduce other spurious light making it to the camera, e.g. the Raman
laser with 790 nm wavelength, I introduced an additional filter before the camera toblock out any unwanted part of the spectrum from the detected light. I used an extra
FF01-786/6 filter which was aligned at 14.5◦ incident angle to transmit wavelengthsof light at 780±6 nm. When performing experiments using the Bragg probe, an easily
removable sheet was introduced in the half numerical aperture of the imaging plane67

3.4. Experimental sequences to produce 87 Rb BECs
behind the dichroic LL01-808-25. The purpose of this was to block most of the highpower Raman A beam coming from the opposite side and the atoms’ non-interfering
fluorescence signal, assuring the light being detected did satisfy the Bragg’s condition.
I noted that the dichroic mirrors acted as tilted parallel plates for 780 nm light.
They inevitably caused coma and spherical aberrations in our imaging system along
ex . However, since detection of the reflected Bragg signal mostly involved photon
counting, these aberrations did not have a significant effect on our optical Bragg
scattering measurements.

3.4. Experimental sequences to produce 87 Rb BECs
The following is a brief description of the standard experimental sequences weused to produce 87 Rb BECs.
• MOT loading / UV LIAD: Vapors of 87 Rb atoms were first captured in a
MOT at the bottom cell for a duration of 1.5 to 3 s. The loading procedure weused was based on the ultraviolet light-induced atom desorption (UV LIAD)
technique as described in Ref. [85]. An ampule provided the 87 Rb metallic
source and a custom designed adapter held a thermoelectric cooler (TEC)
to control the vapor pressure in the bottom cell. The TEC was usually kept
at < −0.5◦ C. The UV light came from digitally controlled 350 mW lightemitting diode sources (Mightex SLS-0309-B, with driver SLA-1200-2). The
duration of this MOT loading process typically took 2 − 3 s, depending on
the number of atoms we needed to produce.

683.4. Experimental sequences to produce 87 Rb BECs
In daily operations, we turned on the UV source for 15 − 20 mins tobuild up enough rubidium vapor pressure before starting our experiments.
For a typical experimental cycle with a total duration of 28 s, we found it
was best to keep the UV light on for about 8 s, even when we were notloading the MOT. We did this by turning on the UV light in other steps ofthe experimental cycle, such as during the dipole evaporation stage on the
top cell. It was crucial to maintain constant vapor pressure in the bottom
cell; since too short a duration of UV light caused the number of atoms in
the produced BECs to decrease gradually. On the other hand, too long a
duration of UV caused significant heating in the BECs.
• Compressed MOT : During this step, the phase-space density of theatoms was increased by compressing the MOT. Magnetic field gradient was
produced by a pair of opposing anti-Helmholtz (quadrupole) coils, with quantization axis along the z−direction. This was done by ramping up the currents in the quadrupole coils from 8.5 A to 20 A to produce stronger magnetic
field gradient within 0.5 s. In our MOT configuration, three extra pairs ofHelmholtz coils produced a bias field with adjustable orientation. In thisstage, the bias field was also changed to optimize the phase-space density.
• Optical molasses : After the compressed MOT stage, the system parameters were adjusted to create optical molasses for a duration of 19 ms. To

69

3.4. Experimental sequences to produce 87 Rb BECs
do this, the quadrupole field was changed to zero (no magnetic field gradient) and the polarization gradient from interfering counter-propagating lasers
cooled the atoms further [37]. The residual bias field was zeroed out in thisstep because cooling mechanism in optical molasses was very sensitive tostray magnetic field. In this step, the cooling and repump laser powers were
kept high enough (several times larger than Isat ) to optimize the rate of optical pumping, and the detunings were far enough from the atomic transitions
to optimize the cooling process [115].
• Pre-optical pumping : In this stage, a small bias field in the z−direction
was applied by changing the currents on the Helmholtz coils for 1 ms (enoughtime for the bias field to settle).
• Optical pumping : The optical pumping process transferred most of theatoms to the |F, mF = 2, 2i state within 0.8 ms. A laser beam with σ +polarization in resonance to the F = 2 → F 0 = 2 transition was shone on
the atoms. This eventually pumped most of the atoms to the dark states|F, mF = 2, 2i via decay by spontaneous emission. We specifically chose thelow-field seeking state |F, mF = 2, 2i due to its highest sensitivity to themagnetic field (the magnetic moment is |µ| = h × 2.1 MHz/G). This allowed
us to use smaller currents in our magnetic trap coils to transport the cold
atoms vertically.

70

3.4. Experimental sequences to produce 87 Rb BECs
• Magnetic trap recapture : During this step, the quadrupole current was
turned on for 0.1 s to recapture atoms in the magnetic trap. We typically ran
the quadrupole coil current at 34 A. Next, the magnetic trap was compressedby ramping up the quadrupole current to 90 A in 0.1 ms. This provided alarge enough magnetic field gradient of b0 ≈ 60 G/cm (see Eq. 2.36) to holdthe atoms agains gravity in the y−direction.
• Vertical magnetic transport :The |F, mF = 2, 2i atoms were trans-

ported vertically (y−direction) in 2.2 s by spatially shifting the magnetic
trap center. The transport field was generated using eleven pairs of overlapping coils in anti-Helmholtz configurations with quantization axis along z.
The geometries of the coils were fixed and the trap gradient force was kept
nominally constant to minimize heating of the atoms during transport. This
was achieved by maintaining three constraints during the transport:
δBz= 120 G/cm,
δz

δBz= 47 G/cm,
δyA=

δBz /δx= 1.28,δBz /δy(3.1)

where δBz /δz, δBz /δy, and δBz /δx are respectively the magnetic field gradients along the strong (bias field) axis, transport axis, and the non-elongatedweak axis. In each step of the transport sequence, three adjacent coil-pairs
were energized (except in the very first and last step) sequentially to movethe trapped atoms from the bottom MOT cell to the top cell in 2.2 s at an

71

3.4. Experimental sequences to produce 87 Rb BECs
average speed of 200 mm/sec. Details on the design of the vertical magnetictransport can be found in Perry’s thesis [131].
• RF evaporation : RF induced evaporation was performed for a duration of
4 s [39]. Forced evaporative cooling was accomplished by linearly ramping the
RF frequency from 20 MHz to 3 MHz in a fixed magnetic field with optimum
sweep (multiple steps) to keep up with collisional thermalization.
• Decompression : The magnetic trap was adiabatically decompressed fora total of 5 s. The decompression of the quadrupole field was done in twostages, each with a duration of 2.5 s. In the first stage, the RF power was
kept on at the frequency of 3 MHz and the field gradient was ramped down
to a small value below b0 = 15.3 G/cm (see Subsection 2.2.4) to allow transfer
of atoms from the magnetic trap to the optical dipole trap. In the second
stage of the decompression, the gradient field was gradually brought to zero
to completely load the atoms into the optical dipole trap.
• Dipole evaporation : During this step, several stages of evaporative cooling were done successively with exponential ramps of the optical trap power.
The atoms were cooled adiabatically via collisional thermalization, reachinga phase density necessary to achieve the phase transition towards quantum
degenerate states. In between these evaporation stages, either microwave
or RF-dressing could be performed to transfer the atoms from their initial|F = 2, mF = 2i state to other sub-levels in the F = 1 or F = 2 manifold.

72

3.4. Experimental sequences to produce 87 Rb BECs
In our typical experimental sequences, we performed the first evaporative
cooling of the atoms for 3 s to reach the degeneracy temperature and to produce a small fraction of BEC. We transferred the atoms from |F, mF = 2, 2i
state to |F, mF = 1, 1i state with microwave dressing. BECs in the F = 1
electronic ground states of 87 Rb were preferable since they have smaller threebody recombination loss rates [21, 157]. We performed a second evaporation
stage for 3 s to produce N ≈ 2.2 × 105 condensed atoms with negligible thermal fractions. We used RF-dressing to transfer the atoms from |F, mF = 1, 1i
state to other mF states. Finally, we performed a third evaporation stage for2.5 s to reduce the number of atoms and slightly lower their temperature.

In RbChip lab, we used several hardware devices to conduct our experimental sequences. A master hardware device based on a field-programmable gate array (FPGA)
card served as the master clock to control the timing sequences of other devices. We
used a SpinCore PulseBlaster to provide TTL pulses with programmable timing
sequences. Devices from National Instruments, NI PCI-6733 and NI USB-6343, provide programmable analog voltages and digital signals to our apparatus. Other hardware devices consist of Novatech 409B for generating programmable RF frequencies,
and Flea3 charged-coupled device (CCD) cameras for imaging the atoms.
The experimental sequences above were implemented with software interface written in LabVIEW or labscript suite [161] (an open-source Python based control system). The LabVIEW software allowed us to program devices in a serial way and it73

3.5. Control of magnetic field
was convenient because most of our commercial hardwares came with LabVIEW-based
libraries. In recent few years, our lab migrated our software control system fromLabVIEW to labscript. This allowed us to use scripted programming on our devices[161]. With labscript, a fuller exploration of the parameter spaces was feasible
and close-loop optimization of experimental parameters could be done online whilerunning the experiment. However, since it was time-consuming to translate LabVIEWbased libraries to Python compatible language, I decided to use the LabVIEW-based
software for most of my experiments.
3.5. Control of magnetic field
3.5.1. Bias field calibration. We used three pairs of Helmholtz coils to generate DC bias field in our experiments, with the coil quantization axes lie along the x,y, and z directions. The total field Btot near the center of the coil configuration as afunction of current Ii of the i−th pair is given byq
2
Btot [Ii ] = C 2 · (Ii − Ioff )2 + B⊥,

i = x, y, z,(3.2)

where C is the proportionality constant for the coil strength, B⊥ is the field perpendicular to the coil quantization axis, and C · Ioff is the offset due to the backgroundDC field along the coil quantization axis.
The calibration of the bias field in the science cell was performed by measuring thetotal magnetic field experienced by the atoms. To do this we applied an RF-coupling
to the atoms and measured the resonance frequency fRF for each value of current

74

(a)

2.0

8
C = 0.803 G/V
Ioff =1.511 V

1.5

4

1.0

2

0.5

0
−2

0

2

4

Ix [V]

(b)

2.5

12

C = 2.017 G/V
Ioff = -0.547 V 2
2
Btotal
G

10

2.0

8
6

1.5

4

1.0

2

Zeeman frequency [MHz]2
Btotal[G2 ]6Zeeman frequency [MHz]3.5. Control of magnetic field

0
−1

0

1

Iy [V](c)

3.0
C = 1.018 G/V
Ioff = 0.669 V 2
2Btotal
G

15

10

2.0

5

1.5
1.0

0
−2

0

2

Zeeman frequency [MHz]−24Iz [V]

Figure 3.10. Bias field calibration for each pair of coils in theHelmholtz configuration. Here the coil current is expressed in termsof the scaled currents Ix , Iy , and Iz .

75

3.5. Control of magnetic field
Table 3.2. Calibration parameters for each pair of Helmholtz coils
x-coil y-coil z-coil
C [G/V] 0.803 2.017 1.018
Ioff [V] 1.511 -0.547 0.669

Ii being sent to the coil. The measured frequency was converted to magnetic fieldstrength by calculating the corresponding Zeeman splitting of the hyperfine structure.
For example, RF-coupling between |F = 1, mF = −1i state and |F = 1, mF = 0i state
within the linear Zeeman regime gives the relation fRF = 0.7 MHz/G × Btot . We thenfit the measurement result to Eq. 3.2 to obtain the coil strength and the offset of thebias field, as shown in Fig. 3.10. In practice, to include all scaling factors from thedevices being used, the coil currents were expressed in terms of the applied voltage(V). From these measurements, we found each coil strength C in the units of G/Vand each offset current Ioff to give the zero bias field condition in our experiment cell,as summarized in Table 3.2.

3.5.2. Cancellation of magnetic field gradient. In experiments with spinor
BECs, control of the magnetic field gradient is very important. The spatial gradientof a vectorial magnetic field is a tensor quantity with 9 matrix elements. Gauss’s
law for magnetism ∇ · B = 0 reduces the tensor to 8 independent elements and incurrent free regions the tensor becomes symmetric with 5 independent components.
In practice, we cancelled the gradient for the bias field Bz along the quantization axis

76

3.5. Control of magnetic field
ez of the atoms, i.e. ∇Bz = 0. In order to do this, we used three additional pairs ofcompensation coils, as shown in Fig. 3.11.
The first pair uses anti-Helmholtz coils to cancel ∂Bz /∂z. The other two pairs arearranged in a clover-leaf configuration, as shown in Fig. 3.11(a). These latter pairsof four coils create a dipole-like magnetic field pattern if viewed along x, with zero
bias field at the origin. This gave us the ability to compensate the gradient ∂Bz /∂x(or the gradient ∂Bz /∂y if the coil setup shown in the right panel of Fig. 3.11(a) isrotated 90◦ in the z−axis).
Due to spatial constraints in the apparatus, we had to modify the setup of theclover-leaf coils. The two clover-leaf pairs, each consisting of four coils, were mounted
in such a way they cancel field gradients ∂Bz /∂x0 and ∂Bz /∂y’ in ex0 and ey0 , where
the new Cartesian axes were defined as1
ex0 = √ (ex + ey )21
ey0 = √ (ey − ex ) .
2

(3.3)

Further, we had to deform the planar orientation of the clover-leaf coils to make itpossible to mount them in our apparatus. The placement of the coils in practice isillustrated in the right panel of Fig. 3.11(b).
To keep things straight, we labelled the currents that go to the coils as: IA for theset of coils which gave ∂Bz /∂x0 , IB for the set of coils which gave ∂Bz /∂y 0 , and IC for
the anti-Helmholtz pair which gave ∂Bz /∂z 0 . In the calibration of the compensation
gradient ∂Bz /∂x, we typically scanned IA or IB while keeping IA + IB constant to77

3.5. Control of magnetic field(Helmholtz coils for bias field)

Figure 3.11. Configuration of compensation coils used to cancel ambient magnetic field gradient ∇Bz . (a) Conceptual design of the compensation coils. A pair of anti-Helmholtz coils (left) is used to cancel∂Bz /∂z, while a set of four coils (right) is used to cancel ∂Bz /∂x or∂Bz /∂y. (b) In practice, the placement must be adjusted and the coils
are deformed to meet the space constraints in the apparatus. IA , IB ,and IC are the currents on each set of the coils. (c) Photograph of thegradient compensation coils in our apparatus.

783.5. Control of magnetic field
maintain a nominally constant value of ∂Bz /∂y. Similarly, for compensating thegradient ∂Bz /∂y, we scanned IA or IB while keeping IA − IB constant so ∂Bz /∂x was
not changing. Compensation of the gradient ∂Bz /∂z was trivially done by scanningthe current IC .
In order to cancel the field gradient, we used the atoms to observe the straymagnetic field gradient near the atoms’ location. The simplest method was to use thecompensation coils to apply Stern-Gerlach forces to the atoms during TOF imaging(with gravity along y). Atoms with non-zero magnetic moment (mF 6= 0) would
have a shifted location after TOF and the strength of the field gradient from thecompensation coils could be calculated. The stray field gradient was cancelled if
atoms with different spin components fell on the same spot in the TOF camera,
independent of their mF spin state. This method was convenient to measure andcompensate the background gradients ∂Bz /∂x and ∂Bz /∂z, which were present inour system. However, it was not very sensitive to ∂Bz /∂y along the TOF direction
itself, since the atoms had moved in the y−direction from their in situ trap location.
A more precise way to cancel the field gradient in situ was to measure the degree ofphase mixing in a spinor mixture of Bose gases inside the harmonic trap. For example,
the spinor condensates of 87 Rb in the |F = 1, mF = 0i and |F = 1, mF = −1i states
were miscible if there was no field gradient in the system. To quantify the miscibility, I
introduce the variables n0 (r) and n−1 (r) to represent the spatial density distributionof the mF = 0 and mF = −1 components, respectively. The degree of miscibility is79

3.5. Control of magnetic field

Figure 3.12. Images of spinor Bose gases in |F = 1, mF = 0i and|F = 1, mF = −1i in the xy−plane. The spatial density distributions
n0 (r) and n−1 (r) are overlapping when the stray field gradient is cancelled. The degree of miscibility is parameterized by a dimensionlessmetric η as in Eq. 3.4, which has a value η ≈ 1 if the two components
are almost perfectly overlapping (n0 (r) ∝ n−1 (r)).
parametrized by a dimensionless metric

where the operator h. . .i =´

hn0 n−1 i
η=q,
2 2n0 n−1(3.4)(. . .) d3 r gives the spatial average over the density prod-

ucts. The metric η takes the value η = 1 if the two spatial density distributions
are perfectly overlapping or n0 (r) ∝ n−1 (r), and η = 0 if the two components arecompletely phase separated or n0 (r) n−1 (r) = 0.
Next, I scanned the currents of the compensation coils to optimize the degree of
miscibility η ≈ 1. The three-dimensional density distributions of the spin components
were obtained from images along two axes, i.e. along ex and ez . Figure 3.12 shows80

3.5. Control of magnetic field
image examples in the xy−plane (image axis along ez ) for spinor Bose gases in mF = 0
and mF = −1 states. If there was a small magnetic field gradient presence, the two
spin component were phase separated (η < 1). This method gave an error for thecompensated field gradient on the order of the spin-interaction energy scale overthe cloud size, i.e. µB gF b0 . 7 Hz/10 µm or b0 . 0.1 G/cm. Calibration of thecompensation coils gave me the currents needed to cancel the gradients as IA =0.57(1) A, IB = 0.11(1) A, and IC = −3.18(1) A.

3.5.3. 60 Hz noise cancellation. The other main source of field noise in our
experiment is the power line hum, i.e. 60 Hz field noise and higher order harmonics associated with the alternating current (AC) from the power line. In order tocompensate this AC field, we first used a Schmitt trigger [71] to synchronize our experimental sequences with the phase of the 60 Hz power line. This guaranteed ourhardware electronics worked at the same phase as the AC power for each cycle of theexperimental sequence.
Next, we used the atoms to measure the background 60 Hz noise. Spinor mixtures,
e.g. |F = 1, mF = 0i and |F = 1, mF = −1i, were prepared with RF-sweep throughtheir Zeeman resonance to give nearly 50%/50% equal spin populations [116]. The
mixtures were held in their RF-dressed state for a specific duration before being
released from the trap and imaged in TOF. Any ambient 60 Hz noise would cause
the spin populations to oscillate around their balanced 50%/50% populations.

813.5. Control of magnetic field
Lastly, to compensate for the 60 Hz noise, we sent an additional oscillating current
to our z−bias coil with amplitude Iosc and phase φ to give a total AC magnetic field

Bac,tot = c · Iosc cos (φ + 120πt) + Bbg cos (φbg + 120πt) ,(3.5)

where c is the proportionality constant for the generated field, Bbg is the backgroundfield amplitude and φbg is the phase of the AC magnetic field noise. We then intentionally drove the quantization coil until they were in-phase with the ambient 60 Hz
field, φ ' φbg . We then gradually scanned the amplitude until it was of opposite
polarity to the background amplitude, i.e. c · Iosc ' −Bbg . This compensated the
60 Hz noise, where Btot ' 0. Although there are still higher order harmonics (atfrequencies of 120 Hz, 180 Hz, etc.), we were nevertheless able to bring the AC noise
RMS level down to > 100 µG from typical levels of 1 mG.
3.5.4. Combined feedforward plus feedback control of the magnetic
field. A crucial step in conducting our experiments, especially for those requiring
hours of experimental cycles, was to counteract drifts in ambient DC magnetic field.
To do this we used a combined feedforward plus feedback control system to sendcorrection current to the power supply generating the bias field. The flowchart forthis control system is shown in Fig. 3.13.
To measure the ambient DC field, we used a pair of Stefan-Mayer FL-100 fluxgate magnetometers, which had a sensitivity of 1 V/100 mG and a measurement rangeof 1 G. Ideally, we would place two magnetometers on opposite sides of the glass82

3.5. Control of magnetic field

Figure 3.13. Flowchart of the combined feedforward plus feedback
control system for the DC field.

“science” cell at the same distance from the atoms. The averaged reading of bothmagnetometers gave the field measurement at the atoms’ location. However, due tospace constraints in our lab, we only used one magnetometer near the glass cell andassumed the measured field was proportional to the field experienced by the atoms.
To reduce radiated noise from the fluxgate electronics and unwanted induced eddy
currents, we enclosed the sensors in custom-made aluminum shields.
The fluxgate readings provided measurement of the field disturbance in our system, which was fed forward to the system as a correction to the desired input value.
We typically did this reading near the end of each experimental cycle by sending a
TTL signal to the fluxgates. Figure 3.14(a) shows how the magnetic field drifted overhundreds of experimental runs, as measured by the magnetometer before the fed forward control system was performed. Each experimental run took about 28 s and thefeed-forward signal was updated to the next run after 5 s delay time in programming
the next experiment sequence.

833.5. Control of magnetic field
(a)

(b)

Time interval (hours)
024Time interval (hours)

6

0

2

800

1000

4

6

8

1200

1400

1600

Field noise (mG)

Field noise (mG)

4
3
2
1

0.1
0.0
−0.1

0
0

200

400

600

Experiment run no.

1800

Experiment run no.

Figure 3.14. Magnetic field measured by the magnetometer overhours of experimental runs: (a) before the field control system wasin operation, and (b) while the field control system was in operation.
Besides using the magnetometer, we also measured the DC field “output” experienced by the atoms with partial-transfer absorption imaging (PTAI) technique [142].
This technique compared the population of atoms transferred by the red- and bluedetuned microwave pulses, a few ms right before the main sequence of our magnetic
field sensitive experiment was performed. The PTAI sensed the “output” field bymeasuring the population difference of transferred atoms. This measurement took
about ∼ 10 ms and it was fed back to the field setpoint in the next experimental run.
The fluxgate readings were used to correct for long-term drift of ambient fieldin the experiment, while the PTAI corrected for short-term errors within each experimental cycle. With both of these “sensors”, we thus did a combined feedforward
plus feedback control, giving a control of DC magnetic field with a variation of lessthan ±200 µG over the course of 10 hours of data taking. Figure 3.14(b) shows themagnetic field variation while the field control system was in operation.

84

CHAPTER 4

Optimally focused cold atom systems obtained using
density-density correlations
In our experiments, properties of ultracold atoms were generally measured withabsorption imaging after the atomic cloud was released during time-of-flight (TOF).
In this chapter, I discuss a systematic technique to determine the optimal focal plane
in absorption imaging of cold atoms, which is a reprint of my previous work published
in Ref. [140]. This work was originally initiated as we were studying domain formationand coarsening [40] in spinor Bose gases (see Figs. 2.6-2.7), where we observed spurious noise in the Fourier transform of the images. We eventually found this problem
was due to defocus in our imaging system. As the first author of the published work,
I developed theoretical models and performed simulations of diffraction-limited wave
propagation to quantify the optimal focus. Measurements and analysis of the imaging
data (taken by the group) confirmed the models. Accordingly we implemented thistechnique to identify the optimal focus in our absorption imaging system.

4.1. Abstract
Resonant absorption imaging is a common technique for detecting the two-dimensional column density of ultracold atom systems. In many cases, the system’s thickness along the imaging direction greatly exceeds the imaging system’s depth of field,

85

4.2. Introduction
making the identification of the optimally focused configuration difficult. Here we
describe a systematic technique for bringing Bose-Einstein condensates (BECs) andother cold-atom systems into an optimal focus even when the ratio of the thickness tothe depth of field is large: a factor of 8 in this demonstration with a BEC. This technique relies on defocus-induced artifacts in the Fourier-transformed density-density
correlation function (the power spectral density, PSD). The spatial frequency at whichthese artifacts first appear in the PSD is maximized on focus; the focusing process
therefore both identifies and maximizes the range of spatial frequencies over which
the PSD is uncontaminated by finite-thickness effects.

4.2. Introduction
Since the most important technique for obtaining properties of ultracold atoms isdirect imaging, a well-designed and well-aligned imaging system is crucial for obtaining high quality data which is valid at all length scales. While large scale properties
such as the system’s width or peak density can be obtained with little effort, significant
care must be taken for experiments requiring very good spatial resolution [13, 154], or
those studying correlations [47, 73]. It is difficult to bring objects extended along theimaging axis, such as degenerate Fermi gases [56, 146], 3D Mott insulators [58], andBose-Einstein condensates (BECs) [7, 38], into focus particularly after time-of-flight(TOF) expansion because their spatial thickness often exceeds the imaging system’s
depth of field. Even for such objects, a high degree of accuracy in focusing is required864.2. Introduction
to minimize imaging artifacts. Understanding and minimizing these artifacts is particularly important when studying density-density correlations, where the artifacts
can be confused with the correlation signal under study [30, 40, 73, 92, 153]. Here we
describe a fairly generic technique for focusing on these extended objects which is far
more precise than simply optimizing the “sharpness” of imaged atom clouds.
Absorption imaging is a ubiquitous approach for measuring the density distribution of ultracold atom systems [81]. A probe beam illuminates the atomic system, andthe resulting shadow is imaged onto a scientific camera, typically a charge-coupleddevice (CCD) or complementary metal-oxide-semiconductor (CMOS) detector. Ideally, the fraction of light absorbed would be directly related to the two-dimensional
column density ρ2D (x, y) =´

dz ρ(x, y, z) of the atoms along the imaging direction

ez , where ρ(x, y, z) is the density of atoms. If the thickness δz along ez exceeds theimaging system’s depth of field, then some of the atomic distribution must necessarily
be out of focus, invalidating any simple relationship between absorption and column
density. Given this, it is a challenge to obtain the optimal focal plane of the extended
system that minimizes the artifacts resulting from this defocus, e.g., at the center ofa distribution symmetric along ez .
Typically a system is brought into a focus by minimizing the size or apparent
diffraction effects from a compact object such as a trapped BEC; in many cases, nosuch compact reference at the desired image plane is available. In this paper, we
present a technique for determining the optimal focus of absorption-imaged extended
objects. Using this technique, we identify the focal plane within an accuracy of87

4.3. Theory
2 µm for a δz = 150 µm thick object. Specifically, given an object with densitydensity correlations [73] with a spatial correlation length `, we show that observationsof correlations in the optical absorption as a function of camera position allow us
to bring the object into focus to within a fraction of the depth of field associatedwith `, even without knowing the details of the correlation function. This optimal
focus is the camera position where the imaged auto-correlation function (ACF) mostaccurately reflects the atomic density-density correlations, minimizing both defocusinduced artifacts and the resolution limiting effect of the system’s finite thickness[40, 92].
In this work, we review the basic theoretical formulation required to understandlight propagating through an absorbing dielectric medium. We then consider several
example images created by different idealized objects, in each case noting how todetermine their optimal focus. Lastly, we experimentally apply this technique toimages of BECs after TOF.

4.3. Theory
Monochromatic light of free-space wavelength λ and wavenumber k0 = 2π/λ propagating through an object with complex relative permittivity ε(r) = /0 and relative
susceptibility χ(r) = ε(r) − 1, where  is the permittivity, and 0 is the electric
constant, is described by the vectorial wave equation for the electric field E(r):∇2 E(r) + k02 ε(r)E(r) = −∇ [E(r) · ∇ ln ε(r)] .
88

(4.1)

4.3. Theory
In a medium where ε(r) is slowly varying, the right-hand side (rhs) of Eq. 4.1 canbe neglected, reducing Eq. 4.1 to separate scalar wave equations∇2 E(r) + k02 ε(r)E(r) = 0,

(4.2)

for each vector component of E(r), e.g., we might have E(r) = E(r)ex for linearlypolarized light along ex .

4.3.1. Wavefield propagation. Here, we cast the above scalar wave equation
into the form−
∂ 2 E(r)= ∇2⊥ + k02 E(r) + k02 χ(r)E(r),
2
∂z

(4.3)

suitable for light predominantly traveling along ez . For a known field configuration
at E(r) (such as the probe laser before it interacts with the atoms), Eq. 4.3 has theformal solutionE (r + ∆zez ) = exp ±i∆zq∇2⊥ + k02 + k02 χ(r) E (r) ,(4.4)

describing the field propagated a distance ∆z along ez .
Wave propagation in free space [i.e., χ(r) = 0 in Eq. 4.3] is solved exactly in the
angular spectrum representation [126]ˆEfs (r + ∆zez ) = P(∆z)E(r) =hid2 k2D P̃(k2D , ∆z)Ẽ(k2D , z) eik2D ·r2D ,
89

(4.5)

4.3. Theory
for a forward going wave, with the 2D position r2D = (x, y) and wavevector k2D =(kx , ky ); the Fourier-transformed wavefield Ẽ(k2D , z) =´

d2 r2D exp(−ik2D · r2D )E(r);and the transfer function for propagating a distance ∆z in free spaceh1/2 i2
2P̃ (k2D , ∆z) = exp i∆z k0 − k2D
.

(4.6)

The transfer function behaves differently in two regions of spatial frequencies: for22
k2D< k02 , P̃ is oscillatory (propagating regime), and for k2D> k02 , it is exponentiallydecaying (evanescent regime).
Meanwhile, considering only χ(r) [neglecting the first term in the rhs of Eq. 4.3],
the absorption and refraction of light traveling a distance ∆z is described by ˆ z+∆zp
EBL (r + ∆zez ) = Q(∆z)E(r) = exp ik0
dz χ(r) E(r).

(4.7)

z

Unlike the usual Beer-Lambert (BL, discussed in Sec. 4.3.2), this expression alone
does not reflect a good approximation to beam propagation for systems of any significant thickness.
4.3.2. Beer-Lambert law and the paraxial approximation . To better understand the independent influence of the beam’s propagation and its interaction withmatter, we apply the paraxial approximation to Eq. 4.3, allowing us to draw an analogue between the paraxial wave equation and the Schrödinger equation, which canbe solved numerically using a split-step Fourier method (SSFM) [76]. To understandthe difference between Eq. 4.7 and the usual BL law, we again turn to Eq. 4.3,

90

4.3. Theory
now assuming that the electric field can be written as E(r) = exp (ik0 z) E 0 (r), where
E 0 (r) is a slowly varying envelope along ez . Inserting this form into Eq. 4.3 gives theparaxial wave equation−2ik0∂E 0 (r)= ∇2⊥ E 0 (r) + k02 χ(r)E 0 (r),
∂z

(4.8)

where the assumed weak z dependence of E 0 (r) allowed us to drop ∂z2 E 0 (r). Like
above, the spatial evolution of an initial E 0 (r) can be partitioned into a spectral partP̃0 (k2D , ∆z) and a coordinate part Q0 (∆z), with



2
k2D
P̃ (k2D , ∆z) = exp −i
∆z ,
2k0
0



k0
Q (∆z) = exp i20ˆ z+∆zχ(r)dz .
z(4.9)(4.10)For the paraxial approximation to be valid, the condition |χ(r)|  1 must also hold:otherwise the Q0 (∆z) evolution would lead E 0 (r) to depend strongly on z.
We numerically evolve the paraxial wave equation [Eq. 4.8] along ez using a
split-step Fourier method (SSFM) [46, 87], where the operators in the rhs of Eq. 4.8
are split into two: one operator represents wave propagation in a uniform medium
using Eq. 4.9 and the other operator takes into account the effect of refractive index
variation using Eq. 4.10. In the SSFM, we alternately apply the two evolution
operators with steps of size ∆z. For each step, the complex amplitude E 0 (r) is91

4.3. Theory
propagated first by P0 (∆z/2), then by Q0 (∆z), and then again by P0 (∆z/2). The
resulting symmetrized split evolution

E 0 (r + ∆zez ) = P0 (∆z/2) Q0 (∆z)P0 (∆z/2) E 0 (r) ,(4.11)

has its first correction at order ∆z 3 .
The paraxial equations allow us to introduce the depth of field

ddof =

2
lmin
2k0
=
,
2
kmax
πλ

(4.12)

2is the largest k2D of interest and lmin = 2π/kmax is the corresponding minwhere kmax

imum length scale [these might be specified by: the maximum significant wavevectorin χ(k2D , z); the resolution of the physical imaging system; or at most by k0 ].
We obtain the BL law by assuming that the system is thin along ez , i.e., bothδz  ddof , and P̃0 (k2D , ∆z) may be neglected. For purely absorbing materials whereχ(r) ∝ iσ0 ρ(r), this gives the usual BL lawˆ z+δzI(r + δzez ) = exp −σ0
ρ(r)dz I(r)(4.13)zdescribing the attenuation of the free space optical intensity I(r) = c0 |E(r)|2 /2 byabsorbers of density ρ(r) and scattering cross-section σ0 . This BL result can also
be obtained without the paraxial approximation by first neglecting the ∇2⊥ term in
Eq. 4.3 (valid when kmax δz  1: a more strict requirement than in the paraxial

92

4.3. Theory
approximation where we had δz  ddof ) and again assuming |χ(r)|  1, a small
relative susceptibility.1
In experiment, the BL law is generally applied by comparing the intensities I(r2D )and I0 (r2D ) measured with and without atoms present, respectively. This relates the
optical depth
OD(r2D ) ≡ − lnI(r2D )
= σ0 ρ2D (r2D )I0 (r2D )(4.14)

to the 2D column density. In cold atom experiments, this column density is theprimary observable in experiment.

4.3.3. Absorption imaging. Here we consider systems of ultracold atoms illuminated by laser light on a cycling transition, where the atom-light interaction is
described by a complex relative susceptibility
i − 2δ/Γσ0
ρ(r).
χ(r) =
k0 1 + I/Isat + (2δ/Γ)2(4.15)

ρ(r) is the atomic density; δ is the laser’s detuning from atomic resonance; σ0 = 6π/k02
is the resonant scattering cross-section; Γ is the atomic linewidth; and Isat is thesaturation intensity [81]. The standard BL law is valid for dilute (ρ  k03 )2, spatiallythin systems (k0 δz  1), illuminated by low intensity (I0  Isat ) probe beams.
1Moreover, the gradient term ∇ ln ε(r) in Eq.

4.1 cannot be safely neglected for systems where|χ(r)| is large or sufficiently rapidly varying, although this would generally imply a breakdown of
the paraxial approximation as well.
2A dilute regime, with density ρ  k 3 , can be achieved by letting the atomic cloud to ballistically0expand.

93

4.3. Theory
The I0  Isat requirement can be lifted by introducing the intensity-corrected
optical depth
ODcor (r2D ) ≡ − lnI(r2D )
I0 (r2D ) − I(r2D )+,I0 (r2D )Isat(4.16)

which is related to the column densityρ2D (r2D ) =ODcor (r2D ),σ0(4.17)of dilute (ρ  k03 ), spatially thin systems (k0 δz  1). Due to the limited dynamic
range of the camera’s pixels [147] and the presence of background light, it is technicallydifficult to reliably detect uncorrected optical depths, larger than ≈ 4. Thus, wedeliberately select I0 > Isat , saturating the transition with I0 such that ODcor < 3.
In addition, the spatial thickness of many cold atom systems exceed the depthof field leaving parts of its distribution along imaging direction inevitably out offocus, thereby invalidating Eq. 4.17. Even for dilute clouds (after sufficient TOF),
images taken an equal distance above and below the focal plane can differ. This lackof symmetry makes a straightforward determination of the optimal focus difficult(lensing effects from even slightly off-resonance imaging beams and aberrations in theimaging system can complicate the situation further.)
To illustrate this difficulty, we consider images of BECs with the focal plane
displaced a distance d = −54 µm, 0 µm, and 54 µm from the BECs’ center (see Fig.
4.1). Because the BEC is thick compared to the depth of field, Eq. 4.17 does nothold; in addition lensing effects cause the cloud’s peak ODcor to behave asymmetrically94

4.3. Theory!"d = −54 µm

#& %&

(a)
!#

'%& '#& &$%&d = 0 µm
#& %&

!"

$

'%& '#& &

ODcor

!#

%
&

d = 54 µm
#& %&

!"
!#

'%& '#& &$%&'%& '#& &Peak ODcor

(b)

#& %&

'%& '#& &

#& %&

!%)(

!%)&

!*)(

!*)&
'(&

'#(

&

#(

(&

Figure 4.1. Dependence of BEC images on image plane position. (a)Intensity corrected optical depth measured d = −54 µm, 0 µm, and54 µm from the optimal focus: images (right) and line cuts at y = 0(left). (b) The peak optical depth depends only weakly on d; is not
maximized at d = 0; and has no structure on the 2 µm scale.

954.3. Theory
when the focus is behind or in front of the cloud. In these images, there are no sharp
features that identify the optimal focus at the micron level. Owing to the weak
dependence of large-scale parameters such as peak-height or width on defocus, such
precise focusing is not required in many experiments. As we see below, experimentsthat study correlations within such images are extremely sensitive to defocus andnew methods are required. Our technique brings images such as these into focus,
identifying an optimal focal plane at the ≈ 2 µm level.
4.3.4. Modeling. To obtain a basic understanding of our approach, we firstconsider the defocused image of a 1 µm thick absorbing medium, inhomogeneous
ex -ey plane, bounded above and below by vacuum, with, χ(r) = ig(x, y) for z ∈(−0.5 µm, 0.5 µm), where g(x, y) ≥ 0 is a Poisson distributed random variable. Like
atoms illuminated on resonance, this medium has a purely imaginary susceptibility.
The illuminating light is modeled by a plane wave with wavelength λ = 780 nm
suitable for imaging our 87 Rb Bose-Einstein condensates.3 While this object has no
visible structure, by virtue of its spectrally flat density-density correlation function,
it can be brought into focus.
The imaged intensity pattern I(x, y) from this 1 µm layer appears random atvarious distances from focus, but its correlations become oscillatory. To reveal thisinformation, we turn to its spatial power spectral density: the magnitude squared ofI(x, y)’s Fourier transform.4 The power spectral density (PSD) is circularly symmetric
3In our SSFM simulation of light traversing this medium, we used a ∆z = 1 µm step size.
4The Wiener-Khinchin theorem states that the spectral decomposition of the autocorrelation func-tion is equal to the power spectral density.

964.3. Theory
in the spatial frequency k2D = (kx , ky ) plane. Fig. 4.2(a) shows the PSD in thisk = |k2D | “radial” direction as a function of distance from focus d. This PSD has afringe pattern; the wavevector of the first minimum exceeds the maximum imaged
wavevector only near the image’s focus at d = 0 µm.
The physical origin of this structure can be understood by turning to the paraxial wave equations [Eqs. 4.9 and 4.10], and by first studying a single absorber at
r = 0 illuminated by a plane wave E00 (r2D , 0− ) = E0 . Equation 4.10 shows that athin absorber simply changes the amplitude of the field, leaving its phase untouched,
and for simplicity, we assume this absorber has a Gaussian profile in the ex -ey planewith width w0 . Thus the electric field just following the absorber is changed by
δE 0 (r2D , 0+ ) = −δE exp [−r2 /w02 ], with r2 = x2 + y 2 . The propagation of such agaussian mode by a distance d along ez can be solved exactly in the paraxial approximation, and in the spectral basis this is0δ Ẽ (k2D , d) = −πw02 δE exp

2
w02 k2D−42i
1+ 2 d
w0 k0



.

(4.18)The total field from an absorber located at a different location r0 in the ex -ey plane
simply acquires an overall phase factor exp [−ik2D · r0 ]. We now compute the experimentally relevant optical depth by taking the reverse Fourier transform of the fullelectric field, computing the intensity, then the optical depth, and taking the Fourier

97

4.3. Theory

0.10

0.15

0.20

100

0.05

0

Intensity PSD
0

0.25

0.30

(a)

k (µm−1 )

2.0
1.5
1.0
0.5
0

(b)

k (µm−1 )

2.0
1.5
1.0
0.5
0.0

(c)

k (µm−1 )2.0

Color scalefrom 0 to 0.03

1.5
1.0

400

300

200

-100

-200

-400

0.0

-300

0.5

d (µm)
Figure 4.2. (a) Spatial PSD of the intensity produced by 1 µm thicklayer of randomly distributed scatterers showing that fringes diverge infocus. (b) PSD produced by a 100 µm thick sheet of random columnar scatterers. (c) PSD produced by a 100 µm thick sheet of random
scatterers. The dotted lines are functional forms of the lowest curvedfringes, and in each case d is measured from the objects’ center.

984.3. Theorytransform to obtain (retaining terms of order δE/E0 )


 2 
2
w02 k2D
k2D d
2πw02 δE
g
exp −
cos
,
OD =
E0
4
2k0

(4.19)

with the same overall phase factor depending on the initial position. Averaging over√N randomly placed absorbers therefore gives an overall signal scaling as N with arandom overall phase. Taking the magnitude squared gives the PSD
 2 
k2D d
PSDthin = PSD0 × cos
,
2k0
2

with
PSD0 = N

2πw02 δE
E0

2



w2 k2
exp − 0 2D
2This quantity has zeros located at kzero [n] =(4.20)

.

(4.21)p2π(n + 1/2)k0 /d for integer n. In

our numerical simulation, the minima follow the functional form kzero [n] = A[n]|d|−1/2

as shown by the dotted lines in Fig. 4.2, with A[0] ≈ 5.08 and A[1] ≈ 8.76 for the
first and second zeros: the expected values for A[n]. Thus, for this thin apparently
structureless system, fringes in the PSD allow us to identify the focal plane.
To demonstrate the technique of finding optimal focus of an extended object, we
now consider a second disordered scattering potential with a columnar structure, now
100 µm thick, i.e., χ(r) = ig(x, y) for z ∈ (−50 µm, 50 µm), where again g(x, y) ≥ 0 isa Poisson distributed random variable. This object’s PSD is plotted as a function of
distance d from its center in Fig. 4.2(b); in addition to the same fringe pattern as for99

4.3. Theory
the 1 µm thick case, the PSD now vanishes at specific spatial frequencies independentof d. To model this, we note that the absorbers can now be located at a distance z
from the symmetry plane, so in Eq. 4.19, we replace d → d − z and integrate z from−δz/2 to δz/2, which ultimately gives the PSD
 2
 2 

k2D δz
k2D d
2
sinc
.
PSDcol = PSD0 × cos
2k0
4k0
2

0
[m] =
This predicts the appearance of additional zeros located at kzero(4.22)
p
4πmk0 /δz for

non-zero integer m (this is an artifact of the box-like density distribution of atoms, andwould be greatly softened in real systems where the density drops smoothly to zero).
0[1] = 1.00 µm−1 .
In our example, the lowest order horizontal fringes is located at kzero

Here again, we easily determine the optimal focus, d = 0 µm, from the diverging
curved-fringes.
Next, we consider a scattering potential fully disordered in 3D, again with a 100 µm
thickness, i.e., χ(r) = ig(x, y, z) for z ∈ (−50 µm, 50 µm), where g(x, y, z) ≥ 0 is a
Poisson distributed random variable. In this case, the independent random scatterers
along imaging direction causes the PSD to rapidly loose structure with increasing k2D(see Fig.4.2(c)). Here too, our random scatter model can be applied, giving22
cos (k2D
d/k0 ) sinc (k2Dδz/2k0 ) + 1
PSDrnd = PSD0 ×.
2(4.23)

This reduces to our earlier result when δz → 0 for a thin system and shows that,while the same fringes exist, they are rapidly attenuated for larger spatial frequencies,100

4.4. Optimal focusing of elongated BECs
where the signal approaches a constant background value. However, in principle thecurved-fringes still allow the optimal focus to be identified.

4.4. Optimal focusing of elongated Bose-Einstein condensates
Using on our model, we now consider absorption imaged BECs and implement
the technique presented in Sec. 4.3 to find the optimal focus.
We prepared N = 7×105 atom 87 Rb BECs in the 5S1/2 , f = 1, mF = 0 electronic
ground state in a crossed-dipole trap with frequencies ωx,y,z = 2π × (3.1, 135, 135) Hz.
In situ, the BECs were javelin shaped owing to the extremely anisotropic confining
potential. After a 17 ms to 21 ms TOF, we repumped into the f = 2 manifold, andresonantly imaged on the 5S1/2 , f = 2, mF = 2 to 5P3/2 , f = 3, mF = 3 cyclingtransition with a λ ≈ 780.2 nm probe laser.
The imaging system consisted of a CCD camera and two pairs of lenses functioning
as a compound microscope, magnifying the intensity pattern at the object by a factorof ≈ 6 at the image plane. The first pair of objective lenses, with effective focal length(efl) fˆ1 = 53.6 mm, collimated the light diffracted by the cloud and were separated
by a distance D = fˆ1 + fˆ2 from a second pair of lenses with a fˆ2 = 325 mm efl. The
resulting 0.23 numerical aperture implies that a 10.6 µm diffraction-limited spot on
our CCD sensor is larger than its 5.6 µm pixel size. The associated 1.7 µm spot-sizeon the cloud gives a ddof = 18.6 µm depth of field in our imaging system [74].
Instead of varying the distance from focus by physically moving imaging lenses
or the CCD, we changed the time during which the BEC fell along ez and obtained101

4.4. Optimal focusing of elongated BECs
absorption images with TOF times tTOF from 17.0 ms to 21.0 ms. At these TOFs, thecondensates’ radii were Ry,z ≈ 75(5) µm and Rx ≈ 210(10) µm. Initially, the cloud
was elongated in the harmonic trap with aspect ratio 43 to 1. The initial 43 : 1 aspectratio was reduced to 2.65 : 1 after TOF, and the transverse size of the cloud exceeded
the imaging depth of field by a factor of 8.
Figure 4.3(b) shows the 1D PSD of the atoms’ corrected optical depths along
ez , which is directly related to the absorption intensity through Eq. 4.16. The
fluctuations in the BEC’s density distribution behave like the randomly modulated
χ(r) in our example systems, creating a recurring fringe pattern in the PSD spectrum
as obtained in Fig. 4.3(c). The fringes are quite pronounced for quasi one-dimensional
BECs, where initial phase fluctuations map into pancake-shaped density fluctuations
arrayed along the initially long axis after TOF [42]. Despite the decreased contrast
at high spatial frequencies due to the BEC extent along ez , we clearly observe fringes
curving as a function of tTOF in Fig. 4.3(c). This allows us to determine the optimal
focus of the system.
From the above experimental data, we fit the two lowest order fringes to km ×[(d − z0 )2 /δz 2 + 1]

−1/4

, a peaked function with the expected d−1/2 behavior awayfrom z0 . The fits give an optimal focus location of z0 [0] = 1836(2) µm using thezeroth order fringe or of z0 [1] = 1837(2) µm using the first order fringe. These values
correspond to a TOF of 19.36(1) ms. We are thus able to determine the optimal focus
within ≈ 2 µm or equivalently ≈ 10 µs in TOF. Comparing the experimental data to
the theoretical forms, we notice that the fringes are slightly asymmetrical with their102

4.4. Optimal focusing of elongated BECs

(a)

ey

ex

ez
z (µm)

(b)

!%$$

#$$$

!(

#$

$"%
$"&
$

k (µm−1 )

!"#

!)$$

!'

!%

#!

k (µm−1 )(c)$"$ $"# $"& $") $"% !"$ !"#tTOF (ms)

!)$$

!%$$

#$$$

z (µm)

Figure 4.3. (a) Absorption imaged elongated BEC with density fluctuations. (b) 1D PSD of column density along weakly trap direction ex
as a function of tTOF . (c) Values of k where the 1D PSD is minimum.
The two lowest such k-fringes are depicted. Symbols denote the fringe
locations extracted from (b) plotted along with Lorentzian fits (dottedlines), determining the optimal focus. The solid curves depict theoretical functional forms for the two lowest order fringes. In (b) and (c) the0
dashed line marks k = kzero[1] = 0.82 µm−1 for our√ condensate thick0
ness of 150 µm; the dotted line marks k = kzero [1]/ 2, below which the
ACF of the focused images reliably reflects the ACF n2D (r2D ).

103

4.5. Summary
locations slightly below theoretical ones for larger TOF. Based on our simulations,this likely results from the z dependent magnification of our imaging system, whichchanges by about 10% as the atoms fall from 1420 µm to 2150 µm (17 ms to 21 ms
TOF).
4.5. Summary
We presented a systematic method to bring clouds of ultracold atoms, particularly
initially elongated BECs, into an optimal focus. The density fluctuations in the BECs
after TOF acted like random scatterers, creating diffraction pattern which changed
predictably as a function of distance from the optimal focus. Using TOF absorption
imaging, we demonstrated this method, pinpointing the optimal focus of the BEC towithin 2 µm for a 150 µm thick BEC. This robust technique is easily implemented,
requires no hardware changes, and uses a minimum of computation.

104

CHAPTER 5

Coherent manipulation of atoms with electromagnetic fields
Over the past decade, ultracold quantum gases have offered numerous examplesof new phases of matter. Control and manipulation have been primarily achievedby coherent interaction between light and matter. Within the semi-classical regime,
the classical optical fields perturb the quantized atomic structure where the external
electromagnetic waves are treated using Maxwell equations and the atoms are treatedquantum mechanically. In this discussion, I consider the electromagnetic waves to be
monochromatic (single-frequency) and coherent (having a definite phase relation).1
Under the influence of light-matter interactions, the atoms behave coherently in asense that mechanisms which cause dephasing of the atomic wavefunctions can beneglected. For this reason, quantum degenerate gases can have negligible fluctuationsin their collective atomic wavefunctions and this makes them potentially useful tosimulate various quantum phenomena.
In this chapter, I discuss the coherent manipulation of atoms with electromagnetic
waves from the radio-frequency (RF) range to the optical regime. Here, I discuss threedifferent regions of electromagnetic spectrum I used to engineer cold atom systems.
First, I used RF waves (with frequency ∼ 1 − 20 MHz) to transfer atoms between
1Quantization of the electromagnetic waves leads to the concept of optical “coherence” in quantum

optics. Interested readers may find Ref. [52] to be useful.

1055.1. Adiabatic rapid passage with RF-dressing
different mF states within a specific F manifold. Next, I used microwave radiation(with frequency ∼ 6.8 GHz) to transfer 87 Rb atoms among the F = 1 and F = 2
manifolds of their 5S1/2 electronic ground states, enabling me to do minimally destructive partial transfer absorption imaging [142]. Third, I used light from lasers
operating in the optical frequency range (with frequency ∼ 380 THz). Exploiting thevectorial light shift of the ac-Stark effect, I used two counter-propagating lasers tocreate a spin-dependent lattice and spin-orbit coupling to the atomic Zeeman sublevels. This configuration allowed me to engineer various effective Hamiltonians withminimal changes to the experimental setup. Lastly, I will discuss relevant calibration
techniques that I used to investigate the effective Hamiltonian under study.

5.1. Adiabatic rapid passage with RF-dressing
RF radiation with MHz frequencies are used to transfer different mF spin states
within the F = 1 (or F = 2) manifold of the 87 Rb atoms electronic ground state.
The simplest method to transfer atoms to different states is to do a Rabi oscillation
where the atomic states evolve during the pulse duration. However, this approach isvery sensitive to detuning and phase noise in the system. A better way to transfer
atoms is using the adiabatic rapid passage (ARP) techniques [130, 171] based on theLandau-Zener model [179].
Here, I give an overview of the ARP technique used in our lab to transfer 87 Rb
atoms among their 5S1/2 electronic ground states. Within a two-level system, the106

5.1. Adiabatic rapid passage with RF-dressing
Hamiltonian can be put in the form


δ (t)
 −
2
Ĥ = 

Ω







  −λt Ω =,δ (t)  
Ω λt
2
Ω

(5.1)

where δ (t) is the detuning between the two states coupled by the RF field withcoupling strength Ω. The atoms are brought into RF-dressed states and the time
evolution is governed by the dynamics of the detuning. In the Landau-Zener model[179], the detuning is ramped with a linear rate λ. The instantaneous eigenenergies√of the Hamiltonian in Eq. 5.1 are given by ξ(t) = ± λ2 t2 + Ω2 . Diagonalization of
the Hamiltonian matrix in Eq. 5.1 leads to the adiabatic form


†
† ∂
Û H Û − i~ÛÛ ΨA (t)
∂t








 −ξ(t) 0 
 0 −iλΩ 
~




= 
 + 2 (λ2 t2 + Ω2 )  ΨA (t)





0
ξ(t)
iλΩ
0

∂
i~ ΨA =∂t

(5.2)

where ΨA is the two-component wavefunction in the adiabatic basis. The measure ofadiabaticity is determined by comparing the off-diagonal coupling to the energy level
and requiring that~λΩ
[2 (λ2 t2 + Ω2 )]3/2

 1.

(5.3)

Equation 5.3 guarantees the atomic states can “follow” the time changing Hamiltonian.
The ratio in Eq. 5.3 has its highest value at t = 0 and hence the condition ~λ/2Ω2  1
is met by selecting a strong coupling Ω and a low rate of the detuning ramp λ.

107

5.1. Adiabatic rapid passage with RF-dressing
Although it is non-trivial, the time-evolving Landau-Zener model in Eq. 5.2 isexactly solvable. In the long time limit (starting from t → −∞), the probability forthe system to remain in its initial eigenstate is given by

Pinitial = exp (−πΛ) ,(5.4)

where the adiabaticity parameter is Λ = Ω2 /~λ. A full transfer between the twoatomic states (Pinitial ' 0) thus is easily achievable by selecting large Λ. The adiabaticity of the transfer is easily understood by comparing the bare uncoupled energies±λt to the adiabatic energies ±ξ(t), as shown in Fig. 5.1(a). Although the bare energies have a cross over at t = 0, the coupled energies have an avoided crossing andhence a small fractional population 1−Pinitial gets transferred to the others states, unless the ramping rate is very slow. Since the process of population transfer must occur
in a timescale shorter than the radiative lifetime of the excited state, this technique
is often known as “rapid” adiabatic passage as well.
In many practical cases, we would like to partially transfer atomic populations,i.e. Pinitial ∼ 0.2 or Λ ∼ 0.5, while still satisfying the adiabaticity condition 1/Λ  1.
To achieve this, we can apply specific sequences for the detuning ramp δ (t) and thecoupling strength Ω. In our experiment, we apply a half-Gaussian ramp to the detuning parameter δ (t); this involves a slow ramp of δ (t) in the beginning, followed by aspeeding up near the avoided crossing to give enough population transfer, and finallyending with a slow change in δ (t) to let the atoms follow the adiabatic eigenenergies.

1085.1. Adiabatic rapid passage with RF-dressing

4
3
2
1
0
1
2
3
4

3

2

1

0

1

2

3

Figure 5.1. Transfer of atomic populations between the|F = 1, mF = −1i and |F = 1, mF = 0i states with RF-dressing
at coupling strength Ω. (a) Avoided crossing in coupled two-levelsystem. (b) Experimental sequence to ramp the detuning δ (t) in time.
(c) Transferred population of the coupled two-level system.

For example, I consider ARP between the |F = 1, mF = −1i and |F = 1, mF = 0i
states with RF-dressing at coupling strength Ω. The time sequence of the ARP isshown in Fig. 5.1(b). To change the detuning δ (t) = µBdc (t) /~ − ωrf , we ramp
the bias field Bdc (t) instead of the RF frequency ωrf (t) . We operate at low field
Bdc . 20 G in the linear Zeeman regime where the magnetic moment of the state
under consideration is µ = 0.7 MHz/G. The sigmoid functional ramp in Bdc (t) to its
final value takes a duration of 15 ms; we hold the system in the RF-dressed state for
10 ms; and we finally turn off the RF-coupling and project the atoms to their barespin states. A “controlled” population transfer between the two states is then achievedby controlling the final value in the detuning ramp δ (t → ∞) = δf as displayed inFig. 5.1(c). I note this approach can be extended to three-level system as well [116],
such as coupling between the mF = −1, 0 and +1 states of the F = 1 manifold.
109

5.2. Microwave-assisted partial-transfer absorption imaging
5.2. Microwave-assisted partial-transfer absorption imaging
Electromagnetic radiation at ∼ 6.8 GHz can be used to transfer 87 Rb atoms among
the F = 1 and F = 2 manifolds of their 5S1/2 electronic ground states, as shown in
Fig. 5.2(a). This radiation is in the microwave band. In our experiments, microwave
radiation is broadcast by an antenna horn. For details on the experimental setup of
the microwave source, see Ref. [131].
In this section, I discuss the application of microwave-assisted partial-transfer
absorption imaging (PTAI) to provide feedback in our dc magnetic field control. A
small fraction of atoms in the F = 1 manifold is transferred to the F = 2 manifold,
and the atoms are imaged in the cycling transition F = 2 → F 0 = 3 using absorption
imaging. This technique is non-destructive to the atoms in the initial F = 1 stateas they are unaffected by the imaging light [142]. If there is any detuning present
between the two successive microwave pulses being applied, there is imbalance in thetransferred populations of the atoms. Within an experimental cycle, we then use
two successive PTAIs to provide a real-time feedback of magnetic field error from thesetpoint.
To control the magnetic field to be better than mG precision, I transfer a small
fraction of 87 Rb atoms from the ground state |1, −1i to the excited state |2, −2i. I
use this transition since it has the highest sensitivity to the magnetic field.2 Pulsing
the microwave for a duration τ , I partially transfer the atoms to the mF 0 = −2
2The difference in the magnetic moment is maximum for either |1, −1i → |2, −2i or |1, +1i → |2, +2itransition with ∆µF /h = 2.1 MHz/G in the linear Zeeman regime.

1105.2. Microwave-assisted partial-transfer absorption imaging
state. The coupling strength of the pulse was chosen to be Ω0  1/τ to guarantee
the Rabi evolution is less than a π/2 rotation on the Bloch sphere. I used two
subsequent microwave pulses with specific red- and blue-detuned frequencies ∓δ0from the transition resonance (see Fig. 5.2(a)). The Rabi pulse between the two-levelsystem yields the fractional population being transferred as:3
!
p
2
2
τ
+
∆
Ω
Ω20
±
0
P (∆± ) =sin2
Ω20 + ∆2±
2

2

 q
Ω0 τ
1
2
2
2
=
sinc
Ω0 + ∆ ± τ
2
2π

(5.5)

where ∆± = 2π (δDC ± δ0 ) is the combined detuning due to the magnetic field driftδDC and intentionally chosen microwave detuning δ0 . Figure 5.2(b) shows the fractional population transfer for blue- and red-detuned pulses as a function of δDC .
I then measured the normalized population imbalance between the two successive
microwave-assisted PTAIs:

Nimb =P (∆+ ) − P (∆− ),P (∆+ ) + P (∆− )(5.6)

which was converted to provide feedback correction to the bias field detuning δDC .
In my experiments, I used τ = 400 µs and δ0 = 1.25 kHz (the combined blue- andred-detunings are ∆± /2π = δDC ± δ0 ). Figure 5.2(c) shows experimental measurements of P (∆± ) and Nimb as a function of intentionally introduced field detuning
3The cardinal sine function has two definitions, the unnormalized sinc function sinc(x) = sin(x)/xand the normalized sinc function sinc(x) = sin(πx)/πx. Here I use the latter definition as commonlyused in information theory [164].

111

5.2. Microwave-assisted partial-transfer absorption imaging

Figure 5.2. Illustration of microwave-assisted PTAI to provide feedback correction to the field drift δDC . (a) Atoms are transferred with
red- and blue-detuned microwave pulses to the |2, −2i state and thenumbers are counted in absorption imaging. (b) Fractional populationtransfer P (∆± ) as given by Eq. 5.5 (top) and the normalized population imbalance Nimb as given by Eq. 5.6 (bottom). (c) Experimental
measurements of P (∆± ) and Nimb .
112

5.3. The ac-Stark effectδDC . With PTAIs, the measured population imbalance was fedback to the field control system to keep Nimb in the range of −0.75 < Nimb < 0.75. This gave corrections
to the drifting magnetic field in the range corresponding to −500 Hz to 500 Hz. I
note that the sensitivity to the field drift δDC is only determined by δ0 , if δ0 ∼ 1/τ
and Ω0 τ  1 are satisfied. In practice this gave me a field control within the rangeof ±200 µG from the setpoint of the resonant field condition at 20 G (as previously
mentioned in Subsection 3.5.4).

5.3. The ac-Stark effect
The interaction between laser-light and atoms within the dipole approximation is
described by the ac Stark shift [163]. A coherent electromagnetic field E (r, t) withfrequency ω can be written as

E (r, t) := Ẽ (r; ω) e−iωt + Ẽ∗ (r; ω) e+iωt ,(5.7)

where Ẽ (r; ω) and Ẽ∗ (r; ω) are the negative and positive frequency components of
the Fourier transform of E (r, t). Since the electric field is a real physical quantity,Ẽ (r; ω) is the complex conjugate of Ẽ∗ (r; ω) and vice versa. In this representation,
the intensity of light is given by I = 0 c |E (r, t)|2 t = 20 c Ẽ2

. The effective

Hamiltonian for the ac Stark shift is given by∗
ĤStark (F ; ω) = −α̂µν (F ; ω) Ẽ0µẼ0ν ,

113(5.8)

5.3. The ac-Stark effectwhere the spherical tensor polarizability operator α̂µν is commonly written in its
irreducible components [163]α̂µν (F ; ω) = α̂(0) (F ; ω) δµν + α̂(1) (F ; ω) iσµν
+α̂

(2)

Fσ
F



3
1
1 2
(F ; ω)(Fµ Fν + Fν Fµ ) − F δµν ,F (2F − 1) 2
3

(5.9)

where µ, ν are the indices for the chosen basis vectors and summation over the repeated indices is implied; F is the total angular momentum operator for the F hyperfine level; and σ is the quantization axis for the good quantum number mF . The
polarizabilities have scalar, vector, and tensor reduced components, which are respectively α̂(0) (F ; ω), α̂(1) (F ; ω), and α̂(2) (F ; ω). In my work, the term α̂(2) (F ; ω)is generally very small compared to the other terms and hence it can be neglected.
The scalar and vector components are discussed in the following subsections, where I
concentrate on the F = 1 manifold of the 87 Rb 5S1/2 electronic ground states.

5.3.1. Scalar light shift. For most laser frequencies, the largest contribution
to the F = 1 hyperfine level is from the D1 and D2 transitions. For large laser
detunings compared to the hyperfine splittings, I can express α(0) (F ; ω) in terms of
the fine-structure dipole matrix elements |hJ kdk J 0 i|2 :α(0)(F ; ω) ≈X |hJ kdk J 0 i|2 
J0

3~

114

1
1
−
ω + ωJ 0 J
ω − ωJ 0 J

,(5.10)

5.3. The ac-Stark effectwhere ω is the laser frequency, and ωJ 0 J are transition frequencies between the atomic
ground state J and the multiple excited states J 0 . Replacing the reduced dipole matrix
element with the spontaneous decay rate ΓJJ 0 , I can write:
2J 0 + 1 π0 c3|hJ kdk J 0 i|2
ΓJJ 0 .
=
3
3~
2J + 1 ωJJ
0(5.11)

Equations 5.8-5.11 then give the scalar light shift for the F = 1 manifold as

∆E

(0)






πc2 I 2ΓD2
1
1
1
ΓD1
1
(ω) = −
−
−
+ 3
,
3
2
ωD2
ω + ωD2 ω − ωD2ωD1 ω + ωD1 ω − ωD1(5.12)

where I is the laser intensity; ωD1 and ωD2 are the D1 and D2 transition frequencies;
ΓD1 and ΓD2 are the D1 and D2 transition linewidths.
In this calculation of the scalar light shift, I implicitly assumed α(0) (F ; ω) tobe real in Eq. 5.10. To include heating due to spontaneous decay, I can make a3
to the D1 and D2 transition frequencies which
substitution ωD → ωD + iΓD ω 3 /ωDthen gives the imaginary part of the polarizability [61]. Assuming there is no heating
due to the induced dipole fluctuations from gradients in the electric field, I obtain the
scattering rate (heating rate) as [54]πc2 I
Γ (ω) =
2~"

2Γ2D2 ω 3
6
ωD2



1
1
−
ω + ωD2 ω − ωD22

Γ2 ω 3
+ D16
ωD1



1
1
−
ω + ωD1 ω − ωD1

2 #

.

(5.13)

By tuning the laser frequency ω to be between the ωD1 and ωD2 transitions, I cancancel the scalar light shift in Eq. 5.12. The scalar light shift plotted as a function of115

5.3. The ac-Stark effect
(b)
0.04
500

Heating rate Γ [Hz]Scalar light shift ∆E (0) [Hz]

(a)

0

−500

0.03
0.02
0.01
0.00

784

786

788

790

792

784

786

Wavelength λ [nm]

788

790

792

Wavelength λ [nm]Figure 5.3. (a) Scalar light shift ∆E (0) and (b) scattering rate Γ forwavelengths in between the D1 and D2 transitions of 87 Rb atoms. The
scalar light shift is zeroed out at λzero = 790.027 nm and the heating
rate has a local minimum at λ = 788.43 nm. I use λzero to create ourengineered Hamiltonians with minimal heating.

λ = 2πc/ω is shown in Fig. 5.3(a). The wavelength λzero where ∆E (0) → 0 is called themagic wavelength (or tune-out wavelength in some references [9]). Substituting therelevant parameters for the D1 and D2 transitions (i.e. λD1 = 794.979 nm and λD2 =780.241 nm) into Eq. 5.12 gives λzero = 790.027 nm. In engineering the Hamiltonians,
tuning to λzero is beneficial since then the atomic internal energy-level spacing is notaffected by the light intensity. Furthermore, the heating rate at the magic wavelengthis minimal (see Fig. 5.3(b)), allowing me to manipulate the atoms at the magic
wavelength for a few seconds before significant loss occurs.
In above, I discussed the scalar polarizability due to the induced dipole momentby taking into account atom-light interaction with just the valence electron. However,there are correction terms due to the contribution from the core electrons (in the innershell) and a core modification by the valence electron when the atoms are interacting

116

5.3. The ac-Stark effectwith an ac electric field. Including these corrections give a more precise value forthe magic wavelength, λzero = 790.034(7) nm [9]. For this reason, I generally usedλzero = 790.034(4) nm in my experiments.

5.3.2. Light-induced effective magnetic field. I next consider the vectorial
light shift from the ac Stark effect to the F = 1 manifold. Recall that the hyperfine
Hamiltonian in the presence of an external magnetic field Bdc is given byĤhfs,B = Ahfs

I · J µB+
(gJ J + gI I) · Bdc ,~2~(5.14)

where Ahfs is the magnetic-dipole hyperfine constant, I is the nuclear spin operator,
J is the electron total angular momentum operator, gI and gJ are the g−factors of
the nuclear spin and the electron total angular momentum. The main idea of thevectorial light shift is to treat the effect of the vectorial light polarizability α̂(1) (F ; ω)as an effective magnetic field [53, 64]

Beff = −~iα̂(1) (J; ω) Ẽ∗ × Ẽ ,µB gJ

(5.15)

in the linear Zeeman regime. In this regime, the vectorial light shift is very small
compared to the Zeeman splitting (around 10’s of kHz compared to 10’s of MHz). I
can then treat the vectorial ac Stark shift as an effective perturbing HamiltonianĤeff =µB gF
Beff · F~

117(5.16)

5.4. Engineered Hamiltonians in 87 Rb Bose gases
to the hyperfine structure governed by Ĥhfs,B . This effective Hamiltonian allowed me
to engineer various potentials in spinor condensates as described in the next section.

5.4. Engineered Hamiltonians in 87 Rb Bose gases
I used different coupling schemes to create three specific engineered Hamiltonians.
Each scheme utilized two counter-propagating lasers at the magic wavelength λzero in
the cross-polarized (lin ⊥ lin) configuration, as depicted in Fig. 5.4. The vectorial
light shift from the two-photon Raman processes enabled me to generate an effective
magnetic field that perturbed the Zeeman structure in the F = 1 manifold of the
spinor Bose gases. I realized three different effective Hamiltonians by changing theorientation of the quantizing bias field and adjusting the “colors” of the counterpropagating lasers, as described below.

5.4.1. Spin-dependent lattice. The first case is shown in Fig. 5.4(a). In thiscase there is a bias magnetic field Bdc = Bx x̂ that is parallel to the propagation
direction of the laser along x. The two lasers have the same frequency ω which
is tuned to the magic wavelength ω = ωzero with zero scalar light shift. The laser
propagating along êx is linearly polarized in the y−direction with field amplitude EA
and the laser propagating along −êx is linearly polarized in the z−direction with field
amplitude EB . The total electric field at the location of the atoms has the form

E (r, t) = êy EA eikR x e−iωt + êz iEB e−i(kR x+φ) e−iωt + c.c.,118(5.17)

5.4. Engineered Hamiltonians in 87 Rb Bose gases

Figure 5.4. Three cases of engineered Hamiltonians:
dependent lattice, (b) spin-1/2 SOC, and (c) spin-1 SOC.

(a) spin-where kR = ω/c is the wavenumber of the laser and the phase difference between the
two lasers is chosen to be φ + π/2. The cross product of the two mutually conjugate
Fourier components of the total field isẼ∗ × Ẽ = EB EA e−i(2kR x+φ) + ei(2kR x+φ) êx = 2i cos(2kR x + φ)EB EA êx .

(5.18)

This produces an effective magnetic field of the form

Beff =

2~α̂(1) (J; ω) EB EA
cos(2kR x)êx .
µB gJ

119

(5.19)

5.4. Engineered Hamiltonians in 87 Rb Bose gases
Without loss of generality, I have omitted the relative phase φ in Eq. 5.18 by appropriate choice of coordinate origin.
In this configuration, the spin quantization axis is along ex (defined by the bias
field Bx x̂) and the effective Hamiltonian can be written as0 cos (2kR x) 0



V0 

Ĥeff =

,
0
0
0

2 


0
0 − cos (2kR x)(5.20)

where V0 = −4α(1) (gF /gJ ) EB EA is the depth of the spin-dependent lattice. The
lattice depth V0 can be tuned by changing the geometric mean of the laser intensity
since V0 ∝√IA IB . The lattice potential only affects the spin components mF = −1

and mF = +1 (with anti-phase relation between them), while the spin state mF = 0
is unaffected by the coupling light. This scheme provides a way to create a spindependent lattice in the F = 1 manifold.

5.4.2. Spin-1/2 spin-orbit coupling . In the second configuration, the two
Raman lasers have different frequencies ω + and ω − (or different wavenumbers k + =ω + /c and k − = ω − /c), as shown in Fig. 5.4(b). Each laser frequency is tuned close to
the magic wavelength ω + , ω − ' ωzero with a small frequency difference |ω + − ω − | ωzero . The value of δω = ω + − ω − is adjusted to be close to the Zeeman split ωZ
introduced by the bias magnetic field Bdc = Bz ẑ. In my experiments, the typicalbias field strength is Bz . 20 G which corresponds to a linear Zeeman split of ωZ .

120

5.4. Engineered Hamiltonians in 87 Rb Bose gases
2π · 14 MHz. A careful choice of δω ' ωz (δω/ωzero ∼ 10−8 ) allows me to couple two
specific Zeeman sublevels to create a spin-1/2 spin-orbit-coupled (SOC) Hamiltonian,as described in more details below.
For this case, I can write the total electric field experienced by the atoms as

E (r, t) = ey EA exp[i(k + x − ω + t)] + ez iEB exp[−i(k − x + ω − t + φ)] + c.c.,(5.21)and consequentlyẼ∗ × Ẽ = iEA EB ei(2kR x−δω t+φ) + c.c ex= 2iEA EB cos(2kR x − δω t + φ)ex ,(5.22)

where the single-photon recoil momentum is defined as kR = (k + + k − ) /2.
The effective Hamiltonian in our system then becomes

Heff =√
2Ω cos(2kR x − δω t)Fx ,(5.23)

where the spin-1 matrices are 0 1 0 ~ Fx = √  1 0 1  ,

2


0 1 0





 0 −i 0 

~ 


Fy = √  i 0 −i  ,

2


0 i
0

and the Rabi coupling strength is Ω = α̂(1)

121

√





 1 0 0 Fz = ~  0 0 0  ,
0 0 −1√
2gF /gJ EB EA ∝ IA IB .

(5.24)

5.4. Engineered Hamiltonians in 87 Rb Bose gases
To get rid of the time dependence in the effective Hamiltonian (Eq. 5.23), I canapply a unitary transformation Ût = e−iδω tFz /~ and make the rotating-wave approximation (RWA), where the term oscillating with frequency 2δω is omitted, to obtainΩΩÛt† Heff Ût ' √ cos(2kR x)Fx − √ sin(2kR x)Fy .
2
2(5.25)The effective Hamiltonian in the rotating frame of the spin space, written in thematrix representation of the F = 1 spin states, becomes [69]ĥeff = Ût† Heff Ût − i~Ût†


∂ Ût
∂t

δω



= ~  Ω exp(−i2kR x)
 2

0

Ω
exp(i2kR x)
2



0




Ω
.
0
exp(i2kR x) 2Ωexp(−i2kR x)−δω2(5.26)

This effective Hamiltonian can be treated as a perturbation to the bare hyperfine spinstates, with the Zeeman eigenenergies are defined by the diagonal matrix0 −ωZ + /2 0



δ


†
ĥhfs,B = Ût Hhfs,B Ût = ~ 
 + ~ Î.
0
0
0


2


0
0 ωZ + /2(5.27)

Here I have introduced the quadratic shift  to take into account the slight asymmetry
between the linear Zeeman split ωZ and the zero energy reference, which is shifted by
δ/2. In other words, the energy levels of the mF = +1, 0, and mF = −1 states are122

5.4. Engineered Hamiltonians in 87 Rb Bose gases
chosen to be ~ (−ωZ + /2 + δ/2) , ~δ/2, and ~ (ωZ + /2 + δ/2), respectively. Choosing the laser frequency difference to be δω = ωZ +δ +/2, the combined single particle
Hamiltonian becomes [103] + 3δ/2ĥhfs,B + ĥeff = ~  Ω exp(−i2kR x)
 2

0

Ω
exp(i2kR x)
2

0






Ω
.
δ/2
exp(i2kR x) 2Ωexp(−i2kR x)−δ/22(5.28)The two spin states |↑i = |F = 1, mF = 0i and |↓i = |F = 1, mF = −1i can be coupled together by tuning δ, e.g. δ = 0 will bring the two spin states into resonance. In
the case of   δ and   Ω, coupling to the |F = 1, mF = 1i state can be neglected
and I obtain the Raman coupled spin-1/2 HamiltonianΩ
exp(i2kR x) 
δ/2

2
.
ĥhfs,B + ĥeff = ~  Ωexp(−i2kR x)
−δ/2
2


(5.29)The off-diagonal terms in Eq. 5.29 are responsible for the “spin-momentum locking”
in this Raman coupling scheme, where any change in the spin state ∆mF = ±1 isaccompanied by a change in the momentum ∆k = ±2kR . In typical experiments, thecoupling strength Ω and detuning δ are on the order of a few kHz while the linear
Zeeman splitting is on the order of a few MHz. This justifies both the RWA and thetreatment of ĥeff as a perturbation to the spinor hyperfine structure.
I now include the kinetic energy of the atoms and consider only the motion along x.
I apply another unitary transform Ûk = eikR xσz to the Hamiltonian in Eq. 5.29, which123

5.4. Engineered Hamiltonians in 87 Rb Bose gases
acts as a momentum translation operator to the spin states in the initial lab frame
1/2χ~ lab . In the new frame, the spin states become χ~ SO = Ûk Ût χ~ lab = ei(kR x−δω t)σz χ~ lab .
Finally, I arrive at the spin-1/2 SOC Hamiltonian

1/2ĥSO =δΩ~2(qx − kR σz )2 + ~ σz + ~ σx2m
22(5.30)

where σx,y,z are the Pauli matrices for spin-1/2 system:

 0 1 ,
σx = 


1 0





 0 −i ,σy = 


i 0





 1 0 .
σz = 

0 −1(5.31)

This configuration couples the bare atomic states |↑, kx = qx − kR i and |↓, kx = qx + kR i,
realizing an SOC Hamiltonian in the quasi-momentum space qx [158].

5.4.3. Spin-1 spin-orbit coupling. In the third scenario, as shown in Fig.
++
5.4(c), I introduce bi-chromatic frequencies ω−1and ω+1
to the laser beam propagating++++in the +x direction. This laser beam has wavenumbers k−1= ω−1/c, k+1= ω+1/c,

and field amplitude EA . The other Raman laser propagating in the −x direction has asingle frequency ω − . This laser beam has wavenumber k − = ω − /c and field amplitude
EB . In order to characterize the beat-note frequency of the bi-chromatic laser, I define++/2, the wavenumber
new quantitites: the average wavenumber k + = k−1
+ k+1



+
+
+
+
difference ∆k + = k+1− k−1/2, the average frequency ω + = ω+1+ ω−1/2, and++the frequency difference ∆ω + = ω+1
− ω−1
/2.
124

5.4. Engineered Hamiltonians in 87 Rb Bose gases
The total electric field for the counter-propagating laser geometry is given by
E =ey EA cos ∆k + x − ∆ω + t exp[i(k + x − ω + t)]+ ez iEB exp[−i(k − x + ω − t + φ)] + c.c.,(5.32)

which gives rise to the time-dependent effective Zeeman coupling termẼ∗ × Ẽ = iEA EB cos ∆k + x − ∆ω + t ei(2kR x−δω t+φ) + c.c ex= 2iEA EB cos ∆k + x − ∆ω + t cos(2kR x − δω t + φ)ex ,(5.33)

where δω = ω + − ω − and kR = (k + + k − ) /2. This gives an effective HamiltonianĤeff = 2Ω1 cos ∆k + x − ∆ω + t cos(2kR x − δω t)F̂x .

(5.34)To bring the three independent mF state into resonance to realize a spin-1 SOC+
system, I choose the bi-chromatic laser frequencies to be ω+1= ω− + ωZ + /2 and+ω−1= ω− +ωZ −/2. The linear and quadratic Zeeman shifts, ΩZ and , are defined as

before in Eq. 5.27. For a typical bias field strength of Bz . 20 G, the quadratic shiftis  . 2π ·100 kHz, which gives ∆ω + /2π ∼ 50 kHz and ∆k + /2π ∼ 10−4 m−1 . Since thesize of the atoms is typically in the order of 10 µm (and I am not interested in the shorttime dynamics of the Hamiltonian with time scale of . 10−13 s), I could safely neglect
the term ∆k + x inside the cosine function, where cos (∆k + x − ∆ω + t) ' cos (∆ω + t).
The effective Hamiltonian in Eq. 5.34 then becomes1255.4. Engineered Hamiltonians in 87 Rb Bose gases

Ĥeff = 2Ω1 cos ∆ω + t cos(2kR x − δω t)F̂x .

(5.35)

I apply a unitary transformation Ût = e−iδω tFz /~ (where δω = ωZ ) to the Hamiltonian in Eq. 5.35 and make the RWA to obtain a time-harmonic Hamiltonianĥhfs,B + ĥeff = Ω1 cos ∆ω + ti  F̂ 2hzcos (2kR x) F̂x − sin (2kR x) F̂y +.
2 ~(5.36)

with ∆ω + = /2. The Hamiltonian in Eq. 5.36 is a periodic function of time and canbe solved using Floquet theory, as discussed in Appendix D. Similar to the applicationof Bloch’s theorem in position-momentum space, the Floquet theory leads to theconcept of “quasi-energy” Floquet periodically defined within the first Brillouin zoneof the frequency space, i.e. 0 < Floquet < ~∆ω. This allows the Hamiltonian in Eq.
5.36 to be cast into the same form as a spin-1 SOC system [22]ĥ1SO =~2 k2
(2)
+ Ω1 [cos (2kR x) êx − sin (2kR x) êy ] · F + Ω2 F̂zz,
2m

(5.37)(2)

where the quadrupole tensor operator is given by F̂zz /~ = F̂z2 /~2 − 2/3 and there
is an additional Zeeman-like tensor coupling with strength Ω2 . Here Ω2 includes
the quadratic energy shift of the bare Zeeman sublevels and energy correction to
the lowest band of the Floquet quasi-energy spectrum. The Hamiltonian in Eq. 5.37
allows us to realize magnetic phases of spin-1 SOC Bose gases 5.36. This Hamiltonian
has a rich phase diagram as will be discussed in Chapter 6.

126

5.5. Calibration techniques
In the above SOC systems, it is important to note the offset ±kR from the recoil
momentum in quasi-momentum space is equivalent to the presence of an artificial
gauge field. As in the quantization of electromagnetic field, the gauge field A is
responsible for the presence of magnetic and electric fields, for example B = ∇ × A
and E = −∂A/∂t [100, 101]. Up to this point, I have treated the detuning δ in
the SOC Hamiltonian to be spatially uniform. Introducing a spatially dependent
detuning to the system δ (y), i.e. by applying a magnetic field gradient Bz êz →(Bz + b0 y) êz , a synthetic magnetic field can be realized. Different spin states of the“neutral” atoms act like charged particles moving in the synthetic magnetic field. In
this scenario, we could create vortices in the condensate and the vortex nucleation
exhibits a structural phase transition above a threshold artificial field strength. [93,101, 138]. Detailed discussions on the research work to explore vortex nucleation in a
non-uniform synthetic magnetic field can be found in my previous colleague’s (RyanPrice) thesis [139] and his published work in Ref. [138].

5.5. Calibration techniques
5.5.1. Minimizing circular polarization of the optical light. The engineered Hamiltonian schemes, as mentioned in the previous section, are realized using
the vectorial light shift governed by the term Ẽ∗ × Ẽ. It is thus important to makecertain on the degree of linear polarization of the light to produce the desired effective
magnetic field configuration.

127

5.5. Calibration techniques

1.00

Fractional population

(b)

1.0

Fractional population

(a)
0.8
0.6
0.4
0.2
280

300

320

340

360

0.75
0.50
0.25

140

θWP (degree)

150

160

170

180

190

θWP (degree)

Figure 5.5. Alignment of the Raman quarter-wave plate angle θWP
for (a) Raman A and (b) Raman B laser beams. Vectorial light shift
from circularly polarized light causes a shift in the resonance conditionof the RF-dressed states with 0.5 fractional population. Observed data(circles) are fitted to the logistic functions (solid curves) to determine
θWP which does not affect the resonance condition.

In order to minimize the circular polarization of the counter-propagating Raman
lasers, I make use of the vectorial light shift and measure the shift in the atomic resonance condition in an RF-dressed state. I utilize a combination of half- and quarterwave plates to realize polarization in the lin ⊥ lin configuration for the aforementioned
engineered Hamiltonians. The orientation of the linear polarization is easily aligned
by using the half-wave plate to maximize the coupling strength experienced by theatoms. To minimize any circularity in the laser polarization, the quarter-wave plateis used to compensate any unwanted elliptical polarization introduced by the opticalcomponents.
To align the quarter-wave plates, I prepared an equal 50%/50% population of|F = 1, mF = −1i and |F = 1, mF = 0i atoms in the RF-dressed states. Any lightshift alters the resonance condition of the RF-dressed states. With bias field oriented

128

5.5. Calibration techniquesalong the laser propagation direction ex , I adiabatically turned on a single Raman
beam with its wavelength tuned to λzero = 790.034(4) nm. Next, I gradually tweaked
the quarter-wave plate angle θWP and monitored the shift in resonance due to thevectorial light shift from circularly polarized light, as shown in Fig. 5.5. The circularpolarization was minimized when the atoms remained at resonance in the RF-dressedstate (with 0.5 fractional population) without being affected by the Raman beam.
This gave an accuracy within 1◦ on the quarter-wave plate reading.

5.5.2. Lattice depth calibration. To calibrate the lattice depth V0 , I applieda square pulse of standing-wave light with variable duration τ . The optical latticediffracts the condensate into multiple momentum orders ±2nkR , where n are integers
and kR = 2π/λ is the wavenumber of a single laser beam. The energy scale in thissystem is set by the (single-photon) recoil energy ER = ~kR2 /2m. In pulsing thelattice, I used a short interaction time τ  1/ER to observe the Kapitza-Dirac effectin the scattering of atoms by the standing-wave light [48, 55].
Time evolution of the condensate in the lattice can be modeled by solving thesingle-particle Hamiltonian (neglecting mean-field interactions)∂~2 ∂ 2
2− i~ ψ (x, t) = −+ V0 cos (kR x) ψ (x, t) ,∂t
2m ∂x2using the plane-wave basis ψ (x, t) =P

(5.38)

n cn (t) exp (i2nkR x). The highest momentum

order which can be populated is determined by the cutoff value nmax .

p

V0 /4ER .

√For a short pulsing duration τ  ~/ 4V0 ER [72], the fractional population Pn of the129

5.5. Calibration techniques
1.0
V0 = 3ER
V0 = 40ER

0.8

P0

0.6
0.4
0.2
0.0
0

20

40

60

80

100

Pulse time τ [µs]Figure 5.6. Time evolution of the 0-th momentum order P0 for lattice
depth V0 = 3ER (circles) and V0 = 40ER (crosses). Markers with dottedlines are numerical solutions of Eq. 5.38. Solid lines are approximate
solutions based on Eqs. 5.39-5.40.

n-th diffracted order has the form [48]2Pn = |cn (τ )|

' Jn2



V0
τ
2~



,

(5.39)

where Jn is the Bessel function of the first kind for an integer n. This approximation
is known as the Raman-Nath regime [48]. For shallow potentials V0 . 4ER , the time
evolution of the n = ±1 diffraction order has a sinusoidal form, similar to that ofRabi oscillation in a two-level system. The Rabi-like solution of P±1 is given by
1
P±1 =
4
where ~Ω =1 − 2P±1 .



V0
~Ω

2

sin

2



Ω
τ
2



,

(5.40)

p
V02 /2 + 16ER2 , and the 0-th order fractional population satisfies P0 =130

5.5. Calibration techniques
The time evolution of the 0-th momentum order is shown in Fig. 5.6. I determined
the lattice depth V0 by fitting the population in different n-th diffracted orders with
the numerical solution of Eq. 5.38.

5.5.3. Coupling strength and degenerate minima in an SOC system. To
calibrate the coupling strength for SOC cases, I applied a square pulse of the Raman
beams. The atoms then exhibit Rabi oscillation between the different coupled spinstates. For simplicity, I consider a spin-1/2 system in the following discussion. Sincea change in spin is always accompanied by a recoil momentum kick 2kR , I haveto include a factor of 4ER in the SOC energy dispersion during the Rabi pulsing,as shown in Fig. 5.7(a). For an initial state |↓i = |mF = −1i prepared at rest,
corresponding to the quasi-momentum qx = −kR , this couples the bare atomic states|↓, qx + kR i = |mF = −1, kx = 0i and |↑, qx − kR i = |mF = 0, kx = −2kR i. It is thentrivial to obtain the Raman coupling strength Ω by fitting the Rabi evolution of thetwo coupled states to the relation P↑ (t) = sin2 (Ωt/2).
Another important observation in the SOC system is the doubly-degenerate energy
minima. The energy dispersion in the quasi-momentum space qx (for zero detuningδ = 0) has two local minima, as illustrated in Fig. 5.7(a). For coupling strengthΩ < 4ER , the energy dispersion consists of two degenerate minima located at

q0 = ±kR

s

1−

131



Ω
4ER

2

.

(5.41)

5.5. Calibration techniques

4.8

300

5

4.2

4

200

3.6

3

3.0

100

2
1

0

0

100

2.4
1.8
1.2
0.6

1

0.0
100

2
1.5

1.0

0.5

0.0

0.5

1.0

1.5

1.5

1.0

0.5

0.0

0.5

1.0

1.5

0

100 200 300

0
1
2
3
4
5
6
7

Figure 5.7. (a) Energy dispersion of spin-1/2 SOC condensates forΩ/ER = 0, 2, 4, and 6 (as indicated by the red, magenta, blue, andcyan curves, respectively). (b) Example of absorption imaging where
the bare momentum and spin states are distinguished after TOF release while applying Stern-Gerlach gradient force. (c) Tracking the two
degenerate minima q0 of the Raman dressed states.
As Ω increases, the two dressed spin states merge into a single minimum at q0 = 0. To
observe the degenerate minima, I prepared atoms in the dressed state by adiabaticallyincreasing the coupling strength to a final value Ω (i.e. by ramping it up linearly
within 10’s of ms). I then abruptly (t < 1 µs) turned off both the Raman lasers andthe dipole trap to project the cloud into the bare spin and momentum states. Aftera few ms of time of flight (TOF), I did absorption imaging of the clouds to determine

132

5.5. Calibration techniques
their locations and the corresponding minima location q0 . The different spin states
are separated by applying Stern-Gerlach force during TOF, as shown in Fig. 5.7(b). I
note that the agreement between the imaging data and the expected minima location,
as depicted in Fig. 5.7(c), is consistent with the existence of SOC produced by the
Raman lasers.

133

CHAPTER 6

Magnetic phases of spin-1 spin-orbit-coupled Bose gases
In this chapter, I present our observation of magnetic phases of spin-1 spin-orbitcoupled Bose gases. This chapter is a reprint of our work published in Ref. [140]. As
one of the authors, I was involved with the data collection, analysis, and writing of themanuscript. My particular role was to collect the data on the quenching dynamics
through the first-order transition from the unmagnetized phase to the magnetizedphase (shown in Fig 6.3 below). While analyzing the data for the measured phase
diagram, we found a mismatch between our theoretical prediction and experimental
results. It turned out that there was a mistake in our theoretical model. We forgot
to take into account the laser beat-note frequency which we used to realize the spin-1
spin-orbit coupling system. My other contribution was then to include this correction
and apply Floquet formalism to our model Hamiltonian. Our group effort eventuallyobtained measurement and data analysis that were accurate and consistent.

6.1. Abstract
Phases of matter are characterized by order parameters describing the type anddegree of order in a system. Here we experimentally explore the magnetic phasespresent in a near-zero temperature spin-1 spin-orbit-coupled atomic Bose gas and the1346.2. Introduction
quantum phase transitions between these phases. We observe ferromagnetic and unpolarized phases which are stabilized by spin-orbit coupling’s explicit locking between
spin and motion. These phases are separated by a critical curve containing both firstand second-order transitions joined at a tricritical point. The first-order transition,with observed width as small as h × 4 Hz, gives rise to long-lived metastable states.
These measurements are all in agreement with theory.

6.2. Introduction
Most magnetic systems are composed of localized particles such as electrons [3],atomic nuclei [1], and ultracold atoms in optical lattices [57, 65, 89, 155], each with amagnetic moment µ. By contrast, itinerant magnetism [133, 149] describes systems
where the magnetic particles, here ultracold neutral atoms, can themselves movefreely, and for which magnetism is generally weak. Our spin-orbit-coupled BoseEinstein condensates [66, 90, 103, 141] (BECs) constitute a magnetically ordereditinerant system in which – unlike more conventional spinor BECs [159] – the atoms’kinetic energy explicitly drives a phase transition between two different ordered phases[90]. While the coupling between spin and momentum afforded by spin-orbit coupling(SOC) is insufficient to stabilize ferromagnetism in itinerant fermionic systems [181],
in bosonic systems it leads to magnetic phases that are not present in spinor BECs
without SOC [66, 141].
Phase transitions can generally be described in terms of a free energy G(Mz ) –including the total internal energy along with thermodynamic contributions that are135

6.2. Introduction
negligible for our nearly zero temperature T = 0 system – that is minimized for an
equilibrium system. Here the magnetization Mz = hŜz i/~ is an order parameter,
associated with the spin Ŝ, which changes abruptly as our system undergoes a phasetransition. Figure 6.1(c) shows typical T = 0 free energies: a first-order phase transition (top panel) occurs when the number of local minima in G(Mz ) stays fixed, but
the identity of the global minima changes; and a second-order phase transition (bottom panel) occurs when degenerate global minima merge or separate. These defining
features are true both for T > 0 thermal and T = 0 quantum phase transitions.
For spin-1/2 systems (i.e, total angular momentum, f = 1/2) like electrons, ferromagnetic order can be represented in terms of a magnetization vector M = hŜi/~.
This is rooted in the fact that the three components of the spin operator Ŝ transform
vectorially under rotation. More specifically, any Hamiltonian describing a two levelsystem may be expressed as H = ~Ω0 + Ω1·Ŝ, the sum of a scalar (rank-0 tensor) anda vector (rank-1 tensor) contribution. The former, described by Ω0 gives an overall
energy shift, and the latter takes the form of the linear Zeeman effect from an effective magnetic field proportional to Ω1 . Going beyond this, fully representing a spin-1(total angular momentum f = 1 with three mF sublevels: |−1i, |0i, and |+1i) Hamiltonian with angular momentum F̂ requires an additional five-component rank-2 tensor
operator – the quadrupole tensor – and therefore there exist “magnetization” orderparameters that are not simply associated with any spatial direction[16, 159, 165].
Pioneering studies in GaAs quantum wells [86, 160] showed that material systems with equal contributions of Rashba and Dresselhaus SOC described by the term

136

6.2. Introduction(a) Physical implementation
BEC

Eﬀective tensor ﬁeld

Eﬀective magnetic ﬁeld(b) Phase diagram(c) Free energies
0.8|M z |
0.5

Ω2/ER

0
−1
−2

Polar
BEC

Unmagnetizedﬁrst order

2
1

Cr

os

so

second order−3−4 Ferromagnetic
0

2

4

0.6

1.0

6

ve

r

First order

0.4

Free energy G/ER

0.0

0.2
0.0
−0.2
0.8
0.6

Second order

0.4
0.2

Spin Helix
BEC

0.0
−0.2

8

10

12

−1.0 −0.5

0.0

0.5

1.0

Magnetization MZΩ1/ER

Figure 6.1. Experimental system (a) Schematic and level diagram.
The |−1i ↔ |0i and |0i ↔ |+1i transitions of the f = 1 ground state
manifold of 87 Rb were independently Raman coupled, giving experimental control of Ω1 and Ω2 . (b) Phase diagram. The ferromagnetic
order parameter |Mz | is plotted against Ω2 and Ω1 . The solid (dashed)
red curve denotes the first-order (second-order) transition from themagnetized phase. (c) Free energies. Top: near the first-order phasetransition at Ω1 /ER = 1 for Ω2 /ER = −0.35, −0.1 and 0.15 for theblack, blue and red traces respectively, as marked by the red flags in(b). Bottom: near the second-order phase transition at Ω2 /ER = −2.5for Ω1 /ER = 4.5, 5.5, and 6.5 for the black, blue and red traces respectively.

137

6.2. Introduction
αkx F̂z , subject to a transverse magnetic field with Zeeman term Ω1 F̂x , can equivalently be described as a spatially periodic effective magnetic field. Our experimentswith spin-1 atomic systems use “Raman” lasers with wavelength λ to induce SOCof this form [29, 49, 69, 103, 172, 173, 180] with strength α = 2~kR /m, where thesingle-photon recoil energy and momentum are ER = ~2 kR2 /2m and ~kR = 2π~/λ.
This atomic system can therefore be described by the magnetic Hamiltonian

Ĥ =

~2 k2
(2)
+ Ω1 (x)· F̂ + Ω2 F̂zz
,
2m

(6.1)

describing atoms with mass m and momentum ~k interacting with an effective Zeeman magnetic field Ω1 (x)/Ω1 = cos(2kR x)ex − sin(2kR x)ey helically precessing in
the ex -ey ; and an additional Zeeman-like tensor coupling with strength Ω2 . Here,(2)

F̂zz /~ = F̂z2 /~2 −2/3 is an element of the quadrupole tensor operator. The competing
contributions between kinetic and magnetic ordering energies (interactions select between different nearly degenerate ground states, but only weakly perturb the locationof the phase transitions, see Methods) make ours an archetype system for studying
exotic magnetic order and understanding the associated quantum phase transitions,of which both first- and second-order are present in our experiments (Fig.6.1(b)).
We can easily understand the first-order transition in the limit of infinitesimal Ω1
where the tensor field favors either: a polar BEC for Ω2 > 0 (mF = 0: unmagnetized,
Mz = 0), or a ferromagnetic BEC for Ω2 < 0 (mF = +1 or −1: magnetized, |Mz | =1). As with spinor BECs [148], these phases are separated by a first-order phase1386.3. Resultstransition at Ω2 = 0. The ferromagnetic phase spontaneously breaks the Z2 symmetry
associated with the Hamiltonian’s invariance under the exchange |−1i ↔ |+1i. The
second-order transition can be intuitively understood by considering the large Ω1limit where the system forms a spin helix BEC (with local magnetization antiparallelto Ω1 : unmagnetized, Mz = 0). This order increases the system’s kinetic energy,
leading to the second-order phase transition into the ferromagnetic phase shown in
Fig.6.1(b) as Ω1 decreases. Analogues to this second-order phase transition are present
in other systems with effective spin-degrees of freedom such as double-leg ladders [11]or engineered optical lattices [128, 166].
These two extreme limits continuously connect at the point (Ω∗1 , Ω∗2 ), the green starin Fig. 6.1(b), where the small-Ω1 first-order phase transition gives way to the largeΩ1 second-order transition, and together these regions constitute a curve of criticalC
points {(ΩC
1 , Ω2 )}. Here we realized spin-1 spin-orbit-coupled BECs and varied themagnetic coupling fields using externally applied fields. By directly measuring thesystem’s magnetization, we studied the associated quantum phase transitions present
in the phase diagram, all in quantitative agreement with theory.

6.3. Results
As shown in Fig. 6.1(a), we realized this magnetic system by illuminating 87 Rb
BECs in the f = 1 ground state manifold with a pair of counter propagating and orthogonally polarized Raman lasers that coherently coupled the manifold’s mF states.
Physically, the spatial interference of the orthogonally polarized laser beams give rise139

6.3. Results
to the helical effective magnetic field (see Methods) with period λ/2. As we first
showed [103] using effective f = 1/2 systems, this introduces both a spin-orbit and a
Zeeman term into the BEC’s Hamiltonian, equivalent to Eq. 6.1. Here the quadratic
Zeeman shift from a large bias magnetic field B0 ez split the low-field degeneracy of the|−1i ↔ |0i and |0i ↔ |+1i transitions, and we independently Raman coupled thesestate-pairs with equal strength Ω1 . We dynamically tuned the quadrupole tensor fieldstrength Ω2 by simultaneously adjusting the Raman frequency differences; as shownin Fig. 6.1 we selected frequencies differences where the detuning from the |+1i to|0i and |−1i to |0i were both equal to Ω2 (see Methods). Without this technique,only the upper half-plane of the phase diagram (Fig. 6.1(b)) would be accessible:
containing only an unmagnetized phase, therefore lacking any phase transitions.
In each experiment, we first prepared BECs at a desired point in the phase diagram, possibly having crossed the phase transition during preparation. A combinationof trap dynamics [100, 102] collisions, and evaporation [78] kept the system in or near(local) thermal equilibrium. We then made magnetization measurements directlyfrom the Bose-condensed atoms measured in the spin resolved momentum distribution obtained using the time-of-flight (TOF) techniques described in Ref. [100].

6.3.1. Critical line of phase transitions. Our experiment first focused onthermodynamic phase transitions. We made vertical (horizontal) scans through thephase diagram by initializing the system in the unmagnetized phase at a desired valueof Ω1 (Ω2 ) with Ω2 & 0 (Ω1 . 10ER ), and then ramping Ω2 (Ω1 ) through the transition1406.3. Results
region. (As discussed in the methods our nominally horizontal scans of Ω1 followed
slightly curved trajectories through the phase diagram, such as the red dashed curvein Fig. 6.2(c)). Following such ramps, domains with both ±Mz can rapidly form,(2)

and we therefore focus on the tensor magnetization Mzz = hF̂zz i/~ + 2/3 which issensitive to this local magnetic order.
Using horizontal scans, we crossed through the second-order phase transition(Ω2 < Ω∗2 ) where the free energy evolves continuously from having one minimum(with Mzz = 0, for large Ω1 ) to having two degenerate minima (with Mzz > 0, forsmaller Ω1 ). As shown in Fig. 6.2(a), Mzz continuously increases with decreasing Ω1 ,reaching its saturation value as Ω1 → 0. Repeating these processes for Ω∗2 < Ω2 < 0,
we found a sharp first-order transition. In each case, data is plotted along with theory
with no adjustable parameters. Using data of this type for a range of Ω2 and fitting
to numeric solutions of Eq. 6.1, we obtained the critical points plotted in Fig. 6.2(c).
Because horizontal cuts through the phase diagram are nearly tangent to the transition curve for small Ω2 , this produced large uncertainties in ΩC
1 for the first-orderphase transition.
We studied the first-order phase transition with greater precision by ramping Ω2through the transition at fixed Ω1 (Fig. 6.2(b)) and found near perfect agreementwith theory. For all the experimentally measured critical points, see Fig. 6.2(c)top, separating the unmagnetized and ferromagnetic phase, we also measured thecorresponding transition width defined as the required interval for the curve to fallfrom 50% to 20% of its full range. This width ∆ decreases sharply at Ω∗1 , marking the141

Tensor magnetization M zz

Tensor magnetization M zz

(0)
Ω
2
ER

0.8

= −3.75

0.6
(0)
Ω
2
ER

= −1.0

0.2

0.6
0.4
0.2

−0.4

0

−0.3

−0.2

−0.1

0.0

Ω2/ E R−1

Ω2/ E R

1.
01
(3
0. )
74
(
0. 2)
41
(1
)

0.8

0.0

0.0

(c)

Ω1 /E R =
1.0

1.0

0.4

1.
86
(6
)

(b)

(a)

1.
48
(4
)

6.3. Results

(e)

−2
−3

Width, ∆ / ER

(d)
0.6

M zz = 0.95

0.4

M zz = 0.75

M zz = 0.12

0.2
0.0

0

1

2

3

4

5

6

7

8

Ω1 / ER

Figure 6.2. Measured phase transition (a) Tensor magnetization Mzz measured as a function of Ω1 , showing both second-order[Ω2 (Ω1 = 0) = −3.7500(3)ER ] and first-order [Ω2 (Ω1 = 0) = −1.0ER ]phase transitions in comparison with theory. These curves followed thenominally horizontal trajectories (see Methods) marked by red dashedcurves in (c). (b) Tensor magnetization measured as a function of Ω2 atΩ1 /ER = 1.86(6), 1.48(4), 1.01(3), 0.74(2), and 0.41(1), plotted alongwith the predicted critical Ω2 . In (a) and (b) the light-colored region
reflects the uncertainty in theory resulting from our ≈ 5% systematic
uncertainty in Ω1 . (c), (d) Phase transition. Black (red) symbols depict data obtained using vertical (nominally horizontal) cuts throughthe phase diagram. (c) measured phase transitions plotted along withtheory: solid (phase transition), and green vertical line (tricritical point,Ω∗1 ) Horizontal error bars correspond to one standard deviation on Ω1
and vertical error bars are the 95% confidence intervals from the fitting
function that determines the critical point.

1426.3. Results

Figure 6.2. (Continued) (d) 20% to 50% transition width showing theclear shift from first- to second-order with increasing Ω1 . (e) Domain
formation for Ω1 = 0.74(2) showing interaction-driven spin structurenear the first-order phase transition. In all images, red corresponds tospatial regions with local Mz ≈ −1, the green regions correspond toMz ≈ 0 and the blue regions correspond to Mz ≈ +1. In the polar
regime at Mzz = 0.12 only the mF = 0 cloud is visible; near the firstorder phase transition at Mzz = 0.75 all three mF = 0 clouds are visible
and have partially phase separated; and in the ferromagnetic regime at
Mzz = 0.95 only mF ± 1 clouds are visible and they have completely
phase separated.

crossover between second- and first-order phase transitions (see Fig. 6.2(c) bottom).
In these data, the width of the first-order transition becomes astonishingly narrow: as
small as 0.0011(3)ER = h × 4(1)Hz at Ω1 = 0.41(1). This narrowness results from theenergetic penalty associated with condensation into multiple modes for repulsively
interacting bosons. In addition we find that ramps through the 1st order transition
are hysteretic, and very slow ramps (See Methods) for the system to equilibrate.
During the long equilibration times required to study this transition, spin-domains
formed in our system, shown in Fig. 6.2(e). For Bose-condensed atoms, our TOF
procedure expanded the initial spin-distribution allowing us to reconstruct any insitu spin structure. Figure 6.2(e) shows that domains form as the systems enters into
the magnetized phase; these domains mark the role of interactions in spontaneously
breaking the Hamiltonian’s Z2 symmetry (see Methods for a comparison with sodium
where the sign of the spin-dependent interactions is reversed, and the Z2 symmetry
remains unbroken for a wide range of parameters). Figure 6.2(b) shows that additional
tripartite spin structure is present very near the first-order phase transition, which1436.3. Results
was not anticipated in our initial single-particle description. This tripartite mixture,
predicted in Ref. [125] is an in-plane ferromagnetic phase with no analogue in spinor
BECs or effective spin-1/2 SOC BECs.

6.3.2. Metastable states. We observed that scans crossing the second-ordertransition typically required 50 ms to equilibrate, while for scans crossing the firstorder transitions we allowed as long as 1.5 s for equilibration. Systems taken througha first-order phase transition can remain in long-lived metastable states. Here ametastable state with Mz = 0 persists in the ferromagnetic phase, and a pair ofmetastable states with Mz 6= 0 persists in the unmagnetized phase. We began ourstudy of this metastability by quenching through the first-order transition at Ω1 =0.74(8)ER with differing rates from −0.5 to −0.2 ER s−1 , as shown in Fig. 6.3. We
observed the transition width continuously decreases with decreasing absolute value
ramp rate (inset to Fig. 6.3), consistent with slow relaxation from a metastable initialstate.
We explored the full regime of metastability by initializing BECs in each of the mF
states, at fixed Ω2 , then rapidly ramping Ω1 (t) from zero to its final value fast enoughthat the system did not adiabatically follow into the true ground state, yet slow
enough that the quasi-equilibrium metastable state was left near its local equilibrium.
We found that the rate . 200ER s−1 was a good compromise between these two
requirements. For points near the first-order phase transition three metastable states

144

6.3. Results

0.8
10

0.6

Width, ∆/(ER ×10−3 )Tensor magnetization Mzz

1.0

8
6

0.4

4
2

0.2

0.0
−0.20

0
0.0

0.2

0.4

0.6

Ramp rate, |Ω̇2 |/(ER
−0.15

s−1 )

−0.10

Ω2 /ER

−0.05

0.00

Figure 6.3. Quenching dynamics The system was prepared in theunmagnetized phase with Ω1 = 0.74(8)ER and Ω2 was ramped throughthe phase transition at ramp-rates dΩ2 /dt = −0.2, −0.3, −0.4, and−0.5ER s−1 (blue, black, red, and green symbols, respectively). The
curves are guides to the eye. The inset shows the decreasing width,
defined as the required interval for the curve to fall from 50% to 20%
of its full range, of the first-order transition as the ramp-rate decreases.
Vertical error bars correspond to one standard deviation of up to 10
measurements, and horizontal error bars correspond to a systematic
error of 0.005ER from fluctuations in the bias field. Error bars on theinset panel correspond to the 95% confidence interval on the fitting
function of the quenching data.

exist (Fig. 6.4); near the second-order transition this count decreases, giving two
local minima which merge to a single minimum beyond the second-order transition.
We experimentally identified the number of metastable states by using Mz andits higher moments, having started in each of the three mF initial states. A small
variance in Mz , less than 0.25, indicates the final states are clustered together –associated with a single global minimum in the free energy G(Mz ) – and it increases

1456.4. Discussion
when metastable or degenerate ground states are present. We distinguished systems
with two degenerate magnetization states (Mz ≈ ±1) from those with three statesby the same method, since when Mz ≈ ±1, the variance of |Mz | is smaller than 0.25,
and it distinguishably increases beyond 0.25 as a third metastable state appears with
Mz = 0. In this way we fully mapped the system’s metastable states in agreementwith theory, as shown in Fig. 6.4

6.4. Discussion
We accurately measured the two-parameter phase diagram of a spin-1 BEC, containing a ferromagnetic phase and an unmagnetized phase, continuously connectinga polar spinor BEC to a spin-helix BEC. The ferromagnetic phase in this itinerantsystem is stabilized by SOC, and vanishes as the SOC strength ~kR goes to zero.
Our observation of controlled quench dynamics through a first-order phase transition
opens the door for realizing Kibble-Zurek physics [84, 182] in this system, where therelevant parameters can be controlled at the individual Hz level. The quadrupole(2)

tensor field ∝ F̂zz studied here is the q = 0 component of the rank-2 spherical tensor(2)

operator F̂q , with q ∈ {±2, ±1, 0}. The physics of this system would be further
enriched by the addition of the remaining four tensor fields. The q = 0 term weincluded is the most simple of the tensor fields to deploy, as it only required control
over frequencies. The q = ±1 components are relatively simple to incorporate byrf-coupling the |mF = −1i to |mF = 0i and |mF = +1i to |mF = 0i transitions withdifferent phases. The q = ±2 components require direct coupling between |mF = +1i
146

6.4. Discussion

1.0
|Mz |

0.5
0.0
−0.5
−1.0
0

1

2

3

4

5

6

7

8

5

6

7

8

Ω1 /ER
0

1

2

3

4

4

Ω2 /ER

2

0

−2

−4

Three minima:
metastable regime

Two minima:degenerate

Single minimum

Figure 6.4. Metastable states Top, Measured magnetization plotted along with theory. The system was prepared at the desiredΩ2 = −2ER ; Ω1 (t) was then increased to its displayed final value; during this ramp Ω2 also changed, and the system followed the curved
trajectory in the bottom panel. Each displayed data point is an average of up to 10 measurements, and the colored region reflects theuncertainty in theory resulting from our ≈ 5% systematic uncertainty
in Ω1 . Circles/crosses/stars represent data starting in mF = +1, 0, and−1 respectively. Bottom, state diagram: theory and experiment. Blue:two states; black: three states; white: one state. Colored areas denotecalculated regions where the color-coded number of stable/metastablestates are expected. Symbols are the outcome of experiment. Eachdisplayed data point is an average of up to 20 measurements.

147

6.5. Methods
and |mF = −1i which is straightforward using two-photon microwave transitions, but
is challenging to include with significant strength.

6.5. Methods
6.5.1. System Preparation. We created N ≈ 4 × 105 atoms 87 Rb BECs in
the ground electronic state |f = 1i manifold [104], confined in the locally harmonic
trapping potential formed at the intersection of two 1, 064 nm laser beams propagating
along ex and ey giving trap frequencies of (ωx , ωy , ωz )/2π = (33(2), 33(2), 145(5)) Hz.
The quadratic contribution to the B0 = 35.468(1) G magnetic field’s ≈ h × 25 MHz
Zeeman shift lifts the degeneracy between the |f = 1, mF = −1i ↔ |f = 1, mF = 0i
and |f = 1, mF = 0i ↔ |f = 1, mF = 1i transitions, by  = h × 90.417(1) kHz. We
denote the energy differences between these states as ~δω−1,0 , ~δω0,+1 , and ~δω−1,+1 .
6.5.2. Frequency selective Raman coupling. We Raman-coupled the three
mF states using a pair of λ = 790.024(5) nm laser beams counter-propagating along++
ex . The beam traveling along +ex had frequency components ω−1and ω+1, whilethe beam traveling along −ex contained the single frequency ω − . These beams werelinearly polarized along ez and ey respectively. The frequencies were chosen such++and δω0,+1 ≈ ω − − ω+1
independently Raman
that the differences δω−1,0 ≈ ω − − ω−1

coupled the |mF = −1, 0i and |mF = 0, +1i state-pairs, respectively. Furthermore++
we selected 2ω − − (ω−1+ ω+1
) = δω−1,+1 such that, after making the rotating wave

approximation (RWA) the |mF = ±1i states were energetically shifted by the same
(0)

Ω2

+
+
= [(δω−1,0 − δω0,+1 ) + (ω−1
− ω+1)]/2 energy from |mF = 0i, thereby yielding148

6.5. Methods
the frequency-tuned tensor energy shift depicted in Fig. 6.1(a). In addition, Ω2 in(0)

Eq. 6.1 differs from Ω2 by a small shift ∝ Ω21 / resulting from off-resonant coupling
to transitions detuned by 2, which we computed directly using Floquet theory (see(2)

Eq. 6.6 in Methods). This tensor contribution Ω2 F̂zz to the Hamiltonian mightequivalently be introduced by a quadratic Zeeman shift alone, giving Ω2 ∝ B02 > 0.

6.5.3. The physical basis for SOC using Raman lasers. As explained inRef. [80], the spin-dependent (vector) part of the light-matter interaction can be
written in terms of an effective Zeeman fieldΩ = iuv (E∗ ×E)(6.2)

with overall strength given by uv as defined in Ref. [80], where E is the total opticalelectric field from the Raman lasers. This field then enters into the Hamiltonian as
an effective magnetic field
Hv = Ω · F̂.

(6.3)

The electric field for the laser geometry depicted in Fig. 6.1(a) is E = E + ez exp[i(kR x−++ω−1t)] + E + ez exp[i(kR x − ω+1t)] + iE − ey exp[i(−kR x − ω − t)], giving rise to the time-dependent effective Zeeman coupling term
Ω = uv E + E − ex e2ikR x e−iδω−1 t + e−iδω+1 t + c.c= 2Ω1 [cos(2kR x − δω−1 t) + cos(2kR x − δω+1 t)] ex

149

(6.4)
(6.5)

6.5. Methods+
where we defined δω±1 = ω − − ω±1.

This gives the Hamiltonian term

Hv = 2Ω1 [cos(2kR x − δω−1 t) + cos(2kR x − δω+1 t)] F̂x .

(6.6)

In our experiment the ωZ /2π ≈ 25 MHz linear Zeeman shift is large compared to allother energy scales, so we make the rotating wave approximation (RWA) to arrive at
the low-frequency Hamiltonian

HRWA = Ω1 {cos[2kR x − (δω−1 − ωZ )t] + cos[2kR x − (δω+1 − ωZ )t]} F̂x− Ω1 {sin[2kR x − (δω−1 − ωZ )t] + sin[2kR x − (δω+1 − ωZ )t]} F̂y .

(6.7)

We selected our frequencies to be in four-photon resonance with the |−1i to |+1itransition, giving (δω−1 − ωZ ) = −(δω+1 − ωZ ), in which case the Hamiltonian istime-periodic. Equation 6.1 in the manuscript is obtained by making independent
RWAs on the |−1i → |0i and |0i → |+1i transitions separately, giving the helically
precessing coupling described in the manuscript.

6.5.4. Floquet and polynomial shift. The frequency differences δω−1,0 ≈++ω − − ω−1and δω0,+1 ≈ ω − − ω+1
nominally Raman dress both |−1, 0i and |0, +1i

state pairs independently. In practice, the cross coupling may be substantial andthe adjusted eigenenergies may be computed exactly from Floquet theory. For our + Ω02 = (δω−1,0 − δω0,+1 )/2 separation between Floquet bands [in our experiment150

6.5. Methods
 = h×90.417(1) kHz] we find the relationship between Ω1 , Ω2 and Ω02 is well describedby the polynomial

Ω2
Ω0
= 2 +
ER
ER



2 

Ω1
ER

−4

+ 5.80 × 10

−2

− 4.90 × 10

Ω1
ER

3

−2

+ 1.56 × 10
−5

− 2.76 × 10



Ω1
ER



4 

.

Ω1
ER



−3

− 4.41 × 10



Ω1
ER

2
(6.8)

6.5.5. Measurement details. In the horizontal scans in Fig. 6.1 of the manuscript, we ramped Ω1 at a rate of ≈ −40ER /s, allowing the system to adiabaticallytrack the ground state, and allowed 50 ms for equilibration before the measurement
process. In contrast, for our vertical scans we found the system required between
0.2 ms and 2 s to equilibrate.
We studied the metastable states by performing three separate experiments foreach raw data point: one each for a system initially prepared in mF = 0, ±1 state at
the desired value of Ω2 . We then adiabatically ramped Ω1 from zero to its final value.
For each resulting (Ω1 , Ω2 ) pair, we obtained the magnetization Mz for each initialstate. In all of the described procedures Ω1 was turned off immediately, at the startof our 28 ms time-of-flight imaging process which included a Stern-Gerlach gradient
to separate the spin components.

151

6.5. Methods
6.5.6. Operators. The total angular momentum f = 1 spin operators in Eq.
6.1 of the main manuscript take the explicit form












 0 1 0 
 0 −i 0 
 1 0 0 






~ 
~ 




F̂x = √  1 0 1  , F̂y = √  i 0 −i  , and F̂z = ~  0 0 0 




2
2






0 1 0
0 i
0
0 0 −1(6.9)

in the basis of the magnetic sublevels |−1i, |0i, and |+1i; together these comprise thevector operator F̂ = F̂x ex + F̂y ey + F̂z ez . Likewise the quadrupole tensor operator isexpressed as





0
0 
 1/3




F̂zz = ~  0 −2/3 0  .




0
0
1/3

(6.10)In terms of these operators it is clear than any Hamiltonian involving only 1̂, F̂x , F̂yand F̂zz is invariant under the transformation that swaps |+1i and |−1i: a discreteZ2 symmetry.

6.5.7. Wavefunctions. The wavefunctions in the polarized and unpolarized
regimes are qualitatively different. In the unpolarized regime the wavefunction takes
the general form |ψi = A exp(2ikR x) |+1i −√1 − 2A2 |0i + A exp(−2ikR x) |−1i. The

value of A depends in detail on Ω1 and Ω2 , but two limits are clear. Firstly, when
Ω1 = 0 and Ω2 > 0 the system forms a spinor BEC in the polar phase, with A = 0:p
a BEC in mF = 0. Secondly, for Ω1 → ∞ the local spin follows Ω giving A = 2/5.
152

6.5. Methods
In the polarized regime the wavefunction has the general form

|ψi =A+1 exp[i(k0 + 2kR )x] |+1i −q

1 − A2+1 − A2−1 exp[ik0 x] |0i+ A−1 exp[i(k0 − 2kR )x] |−1i ,(6.11)

but with constraints: firstly Mz = A2+1 − A2−1 (the definition of magnetization), andsecondly Mz = −k0 /2kR (ensuring zero center-of-mass motion, see Ref. [100]). In
our experiment mF = ±1 are coupled at second order inΩ1 , and for |Mz | . 1, thesestates are ≈ 16ER detuned. Thus for Mz . +1, we have A−1 ≈ 0 with corrections atorder Ω21 , giving the wavefunction |ψi ≈ e−i2kR Mz x
in terms of the magnetization.

√

Mz ei2kR x |+1i −√1 − Mz |0i

6.5.8. Free energy and phase diagram. We obtained the free energy G(Mz )as a function of the magnetization Mz by first numerically solving the system’s Hamiltonian given by Eq. 6.1, obtaining the eigenenergies Eσ (k) and state ψσ (k), eachidentified by a momentum ~k and a “band” index σ ∈ {−1, 0, +1}. We then computed Mz for each of these states (dependent on kx , but independent of ky and kz ),thereby obtaining the internal energy E(Mz ) in the lowest band (σ = −1).
As our BEC is very near the ground state the free energy G(Mz ) = E(Mz ) − T S,
where T is the temperature and S is the entropy is well approximated by G(Mz ) ≈E(Mz ), and it is this free energy which is plotted in Fig. 6.1(c). We then obtained
the phase diagram in Fig. 6.1(b) by numerically computing the free energy for eachpair Ω1 , Ω2 and identifying its equilibrium magnetization.

153

6.5. Methods
For non-interacting systems, we found that the curves defining the phase transitions and also those bounding the region containing metastable states could all becomputed in closed form. Firstly, the critical point at which the first- and secondorder phase transitions meet is at
q
√
√
Ω∗1
Ω∗
= 8 10 5 − 22, and 2 = 52 − 24 5.
ERER(6.12)

The curve defining the first-order phase transition (for 0 ≥ Ω2 ≥ Ω∗2 ) is given by
v
s
u
2 

u
C
√
Ω1 (Ω2 )Ω2Ω2Ω2t= 2 2 72 + 10
−
+ 36
+4 ,
ER
ER
ER
ER

(6.13)and the curve defining the second-order phase transition (for Ω∗2 ≥ Ω2 ) is given byΩC
1 (Ω2 )
ER

v
u


s

2

Ω
Ω
1 u
Ω2 Ω2
Ω
2
2
2
t
= √
−16 −
−
− 12
− 88 − 112.
− 72
ER
ER
ER
ER ER
2 2

(6.14)The upper boundary of the metastable regime (in the unmagnetized phase) is givenbyΩ+
1 (Ω2 )=ER

 3/2 s
2(4 − Ω2 /ER )3.
3
4 + Ω2 /ER(6.15)and the lower boundary of the metastable regime (in the ferromagnetic phase) is given
by
v
u

2

s


u
Ω−
(Ω
)
1
Ω
Ω
Ω
Ω2 Ω2
2
2
2
2
t
1
= √
−16 −
− 72
±
− 12
− 88 − 112.
ER
ER
ER
ER
ER ER
2 2

(6.16)

154

6.5. Methods
This is the same equation defining the second-order phase transition with an added±, the full curve defining the boundary of the metastable regime crosses over between√
the + and − solutions at Ω2 /ER = 4(11 − 8 2).

6.5.9. Magnetic Fields. Because the free energy G(Mz ) is sensitive to unwanted detuning δ from the four-photon resonance near the phase transitions which
contributes an added symmetry breaking field δ F̂z to the Hamiltonian, controllingthe bias magnetic field and nulling its gradients is critical. A pair of flux-gate sensors measuring the ambient magnetic field along ez , allowed us to compensate forlong-term field drifts. We compensated any field gradients using four pairs of antiHelmholtz coils in a clover leaf configuration [117], and a conventional anti-Helmholtz
pair, all aligned along ez .

6.5.10. Interactions. We studied the impact of interactions at the level of mean
field theory using a variational approach, assuming an infinite homogenous system.
For each point in the phase diagram labeled by (Ω1 , Ω2 ), we first located the local
minima in the single-particle free energy described by the Hamiltonian Ĥ of Eq. 6.1
in the manuscript. The free energy had from one to three minima, with energies Ej
and eigenstates |ψj i.
We then considered an infinite system and minimized the mean field energy density
1
E=
V

ˆ

1
d xψ (x)Ĥ(x)ψ(x) +
2V
3

†

ˆ



d3 x (c0 + c2 )n2T − c2 n20 − 4c2 n+1 n−1 (6.17)
155

6.5. Methods
Rb|M z |

103 × (Ω2 − Ω∗2)/ E R0−1
Ω2/ E R(b) 3.0−2−3−4
0 1 2 3 4 5 6 7 8 9

87

2.5
2.0
1.5
1.0
0.5
0.0
−0.5
−1.0
0.0

0.5

Ω1/ E R−1
Ω2/ E R

23

1

|M z |
103 × (Ω2 − Ω∗2)/ E R

0

−2
−3
−4
0 1 2 3 4 5 6 7 8 9

1.5

2.0

Rb, expanded

2.0
1.5
1.0
0.5
0.0
−0.5
−1.0
0.0

0.5

Ω1/ E R

(e)

Na

1.0

87

2.5

0

(f)

Na, expanded|M z |

−1
−2
−3
−4
−5
−6
0.0

Ω1/ E R0.5

1.0
Ω1/ E R

1.0

1.5

2.0

Ω1/ E R

103 × (Ω2 − Ω∗2)/ E R23(d)(c) 3.0Rb, expanded|M z |

103 × (Ω2 − Ω∗2)/ E R

87

(a)

1.5

2.0

23

1

Na, expanded

0
−1
−2
−3
−4
−5
−6
0.0

0.5

1.0

1.5

2.0

Ω1/ E R

Figure 6.5. Mean field phase diagram (a)-(c) Rubidium. (d)-(f)Sodium. In each panel, the red curve marks the location of the phasetransition as computed excluding interactions. Panels (b), (c), (e),and (f) are all expanded views of the region around the critical curve.
Panels (a), (b), (d), and (e) plot the |Mz | order parameter: the plot isdark blue when |Mz | = 1 and there is a continuous gradation to whitewhen |Mz | = 0. Panels (c) and (f) plot the fraction of the wavefunction
in mF = 0: similarly, black color indicates when all the atoms are in
mF = 0 and white when none of the atoms are in mF = 0.

for an arbitrary linear combination of these single particle states with amplitudes αj ,where nσ (x) is the local density in a given spin state σ; nT (x) is the total local density;and c0 and c2 are the spin-independent and spin-dependent interactions respectively.
For 87 Rb these have the ratio c2 /c0 ≈ −0.005 and for 23 Na they are c2 /c0 ≈ +0.05.
In our minimization, we modeled our systems with a typical mean-field energy of(c0 + c2 )nT ≈ 1 kHz per particle.
156

6.5. Methods
Figure 6.5 shows the result of this calculation both for rubidium and sodium. In
both cases the overall phase diagram (Fig. 6.5 (a),(d)) is shaped by the single-particle
Hamiltonian; at this coarse level the rubidium phase diagram is hardly different fromthat predicted from single particle physics, but in the case of sodium a large swath ofthe expected ferromagnetic phase remains symmetry unbroken. This phase continuously connects to an equal superposition of |−1i and |+1i as Ω1 → 0.
The situation becomes more complex as we focus on Mz near the curve defining
the first-order phase transition (Fig. 6.5(b),(e)), for the case of rubidium a new
mixed phase appears at low Ω1 analogous to the striped phase in spin-1/2 systems,
but nothing new is apparent for sodium.
Lastly we consider the same region, but looking at the fraction of the variational
wavefunction in the mF = 0 spin state. For rubidium, this allows us to identify a
new state which is a three-way mixture of all three components considered in thevariational calculation (with no analogue in the Ω1 = 0 spinor limit), and we can see
the abrupt transition in sodium from a state connecting to the polar phase (Ω2 > 0)
and to the uniaxial nematic phase (Ω2 < 0). Each of these phases are as described inRef. [125].

157

CHAPTER 7

Probing atomic structure with optical Bragg scattering
Bragg scattering is a useful method to probe crystal structure and magnetic (spin)
structure in many solid state systems. In atomic physics, optical Bragg diffraction
has provided crucial insights into the localization of atomic wavefunction arranged
in an optical lattice [17, 174]. Interference of coherent light scattered by spatiallyperiodic atoms enhances the angular distribution of the diffracted light in the direction
satisfying Bragg’s law. Recently, this method has been extended to probe quantum
phase transitions in various engineered quantum systems, including the superfluid
transition to the Mott-insulator regime in a three-dimensional optical lattice [120],
and antiferromagnetic correlations in the Hubbard model realized with a two-spincomponent Fermi gas in an optical lattice [65].
Motivated by this work, I used optical Bragg scattering to map out the stripe
ordered phase in spin-orbit-coupled (SOC) Bose-Einstein condensates (BECs) realizedwith two-photon Raman coupling scheme. Complementary to this work, Jun-Ru
Li et al. has also observed the stripe phase in SOC BECs realized with opticalsuperlattices [96], as published in Ref. [95]. Before presenting my observations of the
stripe-ordered phase in Chapter 8, I first give a physical explanation of optical Bragg
scattering. In this chapter, I review the theory of optical Bragg scattering and discuss158

7.1. Theory of optical Bragg scattering

real spacelattice vector
reciprocal lattice vector

Figure 7.1. Two-dimensional illustration of (a) real space lattice and(b) reciprocal space scattering from periodic scatterers with direct lattice vector Rn = n1 a1 + n2 a2 . Bragg’s law is satisfied when the scattering wavevector q = ksc − kin follows the von Laue equation q = Gm in
the reciprocal space, where Gm = m1 b1 + m2 b2 is the reciprocal lattice
vector.

the characterization of the Bragg signal, with a focus on its geometrical properties.
In addition, I also present my observations of various atomic structures probed with
optical Bragg scattering.

7.1. Theory of optical Bragg scattering
Consider a system of scatterers arranged periodically with lattice vector Rn , where
the density distribution is a periodic function satisfying ρ (r) = ρ (r + Rn ), as shownin Fig. 7.1. The lattice vector describes the translational symmetry in the system:

Rn = n1 a1 + n2 a2 + n3 a3 ,

159

(7.1)

7.1. Theory of optical Bragg scattering
where a1 , a2 , and a3 are the primitive translation vectors, which span the smallest
volume of the lattice cell, and (n1 , n2 , n3 ) are integers. The corresponding reciprocallattice vectors Gm are defined by the relation Gm · Rn = 2πm where m are non-zero
integers. For an incoming wave with wavevector kin , the maximum intensity of thescattered wave with wavevector ksc in the far-field limit occurs when the scattering
wavevector q = ksc − kin satisfies the von Laue equation q = Gm [10]. The Bragg’s
law is equivalent to the von Laue equation in the case of elastic scattering where|ksc |2 = |kin |2 , specifically2kin · Gm = G2m(7.2)

2d sin θB = mλ,(7.3)

where d is the lattice spacing and θB is the angle between the incoming wavevector
and the lattice plane, as displayed in Fig. 7.1.
Needless to say, this textbook explanation of Bragg’s law assumes the scatterers
are arranged in a regular lattice. In this simple picture, the atomic distribution withina lattice site is typically assumed to be a δ−function and there are an infinite numberof lattice sites. The resulting scattered Bragg signal has a very sharp angular distribution and Eq. 7.3 determines the peak locations of the scattered signal. However,in real atomic experiments, the finite size of the samples must be taken into accountand this broadens the angular spectrum of the observed Bragg signal. Moreover,the nature of the atom-light interaction affecting the scattering properties has to be160

7.1. Theory of optical Bragg scattering
considered as well. In crystallography, one needs to include the interaction between
the crystalline matter with the scattered particles. This leads to the concept of theatomic form factor [10].
To study Bragg scattering in more detail, I present an overview of the relevant
theory in the following subsections. I consider the Maxwell-Bloch equations [25] whichgive a complete description of the atom-light interaction within the semi-classicalapproach. With suitable approximations, I then derive a differential cross-section, as
commonly used in the scattering theory. Next, I discuss the quantized nature of thelight and how this affects the scattered Bragg signal.

7.1.1. The semi-classical approach. In the semi-classical treatment of lightscattering by atoms [119], the electric field E (r, t) is treated classically and governedby Maxwell equations. For a system without free charges and free currents, Maxwell
equations lead to the general wave equation [25]:





1 2
1
1 2
∇ − 2 ∂t E (r, t) =−∇ (∇· ) + 2 ∂t P (r, t) ,
c
0
c
2

(7.4)

where the polarization density P (r, t) describes the “induced” density of charge andcurrent in the medium.
Here I assume the quantum mechanical model of an ensemble of identical atoms
and their behavior is governed by the master equation [12]:diρ (t) = L [ρ (t)] − [−d · E (R, t) , ρ (t)] ,
dt
~

161

(7.5)

7.1. Theory of optical Bragg scattering
where ρ (t) is the atomic density matrix, and L [ρ (t)] is the Liouville operator which
incorporates the atomic kinetic energy, internal energy, and radiative decays due tospontaneous emission and/or collisions. The equations of motions derived from themaster equation is also known as the optical Bloch equations. In Eq. 7.5, I have madethe usual electric dipole approximation. The interaction between light and atoms is
described by the dipole interaction −d · E, where E (R, t) is the electric field at thecenter of mass location of an individual atom and the field induces a dipole momentd in the atom.
As a counterpart to Eq. 7.5, the induced dipole moment creates polarization
vector field which is governed by the relation

P (r, t) = N Tr [ρ (t) δ (r − R) d] = N Trint [hr| ρ (t) |ri d] ,(7.6)

where N is the total number of atoms and the trace in Eq 7.6 is taken over the atomic
internal degrees of freedom.
The Eqs. 7.4-7.6 are the Maxwell-Bloch equations [25]. These coupled equations
can be solved in a self-consistent manner and the solution gives a comprehensive
picture of propagation of light through matter within the semi-classical approach.
For a physical understanding of Bragg’s law, I now consider a simplified version of
the Maxwell-Bloch equations. I assume the atomic system is illuminated by a weak
~ (r, t) = < [ˆE (r) exp (ik0 r − iωt)].
laser beam with a time-harmonic field of the form E

162

7.1. Theory of optical Bragg scattering
The wave equation then reduces to the scalar Helmholtz equation∇2 E (r) + k02 E (r) = −U (r) E (r) ,(7.7)

which describes wave propagation in a medium characterized by relative permittivity
ε (r). Here I have introduced the “scattering potential” U (r) = k02 [ε (r) − 1]. In thelimit of weak scattering, the scattered field in the far-field region behaves as

Esc (r) = F (q)exp (ik0 r)
,
r

(7.8)

where the scattering wavevector is q = ksc − kin and the scattering amplitude F (q)in the first Born approximation [34] is given by:
1
F (q) =4π

ˆ0

U (r0 ) e−iq·r d3 r0 .

(7.9)

In this approximation, the scattering is assumed to be elastic, where |ksc | ' |kin | = k0 .
To find the scattering of light by a periodic array of atoms, I use the definitionof susceptibility χ (r) = ε (r) − 1 and write the scattering amplitude as F (q) =(k02 /4π)´

0χ (r0 ) e−iq·r d3 r0 . At the detection point along the direction of the diffracted

wave vector k̂sc , the differential cross-section of the scattered light intensity is given
by [34]:πdP= |F (q)|2 = 2
dΩ
λ0

ˆ

163

3 0

0

−iq·r0d r χ (r ) e2.

(7.10)

7.1. Theory of optical Bragg scattering
Without loss of generality, I can express χ(r0 ) as a summation over each single latticesite local distribution χj r0j0

χ(r ) =

NS
X
j=1


χj r0j ,(7.11)

where the local coordinates of each j-th lattice site are r0j = r0 − Rj , the periodicityof the system is described by the lattice vector Rj , and the atoms are spread over Ns
sites of the lattice within a finite size. I then write the scattering amplitude asN

F (q) =S
π X
e−iq·Rj χ̃j ,2
λ0 j=1(7.12)´where the Fourier transform of χj r0j is given by χ̃j = d3 r0j χj r0j exp −iq · r0j .

Based on the Heisenberg uncertainty principle, the atom’s location can be thoughtof as having some fluctuations, i.e. r0j → r0j (t) = r0j + u (t). This effect can beincorporated by including the time displacement u(t) with respect to the lattice site
into the exponential term

e−iq·Rj → e−iq·Rj eiq·u(t) τ ,(7.13)

and using the Taylor expansion for the exponential of an operator, I obtain:
#


X
Y
1
1 2 2
iq·u(t)22e= exp −
q u=
exp − qi ui τ .
τ
2 i=x,y,z i i τ2
i=x,y,z"164(7.14)

7.1. Theory of optical Bragg scattering
Taking into account the spatial fluctuations, the elastic scattered intensity IdΩ ∝dP/dΩ of a finite-sized medium then becomesIdΩ ∝ e−2Wq (τ )NSXχ̃j χ̃∗k e−iq·(Rj −Rk ) ,(7.15)

j,k

where the Debye-Waller factor is defined by:e−Wq (τ )
1 2 2=
exp − qi ui τ .
2
i=x,y,zY(7.16)In the semi-classical treatment of Bragg scattering, the atomic structure is treatedwith the quantum master equation and this leads to the optical Bloch equations [163].
For a two-level system in the steady-state, the susceptibility is proportional to theatomic density ρat (r) and it can be written in the form [81]
i − 2∆σ0ρat (r) ,χ (r) =
k0 1 + s0 + 4∆2(7.17)

where ρat (r) is the atomic density, δ is the laser’s detuning from atomic resonance,
σ0 = 6π/k02 is the resonant scattering cross-section, Γ is the atomic linewidth, and s0
is the saturation parameter (as discussed previously in Chapter 4). Similar to Bragg
scattering in solid-state physics, I can now define the atomic form factor :
ˆ
fj =


0
d3 r0j ρl r0j e−iq·rj ,165(7.18)

7.1. Theory of optical Bragg scattering
where ρl r0j

is the local density distribution of the atomic electron cloud about

its center of mass located at the j−th lattice site. Substituting Eq. 7.17 for thesusceptibility into Eq. 7.15 gives−2WQ (τ )IdΩ ∝ eNSX
4∆2 + 1
fj fk∗ e−iq·(Rj −Rk ) .
(1 + s0 + 4∆2 )2 j,k(7.19)The properties of the scattered Bragg signal are governed by Eq. 7.19. It is useful
to define the static structure factor as

1

S (q) = PNS

2

j=1 |fj |NSX

fj fk∗ e−iq·(Rj −Rk ) =

j,k

1

NS
X

Natom j,k

fj fk∗ e−iq·(Rj −Rk ) .

(7.20)


For a Dirac-delta function in the atomic local density ρl r0j = δ r0j , the locationof the peak in the static structure factor coincides with that given by Bragg’s law(Eq. 7.3) for an infinite number of scattering sites NS  1. This is known as the
Kronig-Penney model [10].
In summary, Eqs. 7.19-7.20 describe the scattering from a periodic finite system
for a very-far detuned probe in the elastic Rayleigh scattering regime.1 The resultsin this subsection are limited to two-level systems where the structure factor is only
sensitive to the periodic density modulation (crystal structure). For a multi-levelsystem with two-ground states |↑i and |↓i, periodic spin structure could exist where
1Historically, Rayleigh scattering is named after the British physicist Lord Rayleigh, which is anelastic scattering of light by particles much smaller than the wavelength of light (equivalent to thedipole approximation) [169].

166

7.1. Theory of optical Bragg scattering
the internal structure of the atoms alternate between each lattice site, for example
due to the presence of a precessing magnetic field. Although there would be no spatial
modulation in the total atomic density, the spin structure or magnetic ordering could
lead to interference effects in the scattered signal [45].

7.1.2. Quantization of the electric field. In previous derivation of the MaxwellBloch equations using a semi-classical treatment, the electric field was treated as an
expectation value of its corresponding quantum operator and any incoherent component of the field due to radiative spontaneous decay was neglected. Although thesemi-classical approach has been successful in explaining many physical phenomena,
it does not account for radiative spontaneous emission due to atoms interacting withelectromagnetic vacuum fluctuations. To understand incoherent scattering due tospontaneous emission processes, one must treat the field in a quantized manner [119].
Here, I review the calculation of Bragg scattering from atoms acting as independent
scatterers, where the electric field is quantized and treated as an operator. The theory
presented in this subsection is based on Refs. [45, 65].2
Consider an ensemble of two-level atoms with ground state |gi, excited state |ei,dipole transition matrix element dge = hg| d |ei, and transition linewidth Γ. Whenbeing illuminated with a probe light at frequency ω0 and incoming wavevector kin ,2Correlations between the bosonic particles and multiple scattering in the atoms lead to a whole

new series of phenomena, such as superradiant effect and non-linear four wave mixing. However,this treatment is beyond the scope of my work. Interested readers could consult the review article
in Ref. [82] and references therein.

167

7.1. Theory of optical Bragg scattering
the electric dipole radiation emitted by an atom at a distance r is given by [35]


3 ~ckΓ
20 c 8π r2

1Ê+j (r) = √

1/2

Λ exp [iq · r̂j ] σ̂j− ,(7.21)

where Ê+j (r) is the positive frequency component of the electric field, q = ksc − kinis the scattering wavevector, r̂j is the position operator of the j−th atom, Λ =2|dge | is the polarization vector of the scattered field, and σ̂j− =ksc × (ksc × dge ) /ksc

exp (iω0 t) |gi he| is the off-diagonal matrix element of the atomic density matrix operator.
To obtain the total scattered intensity, I sum the field contributions from each of
the individual atoms and square the total field to get

I (q) = 20 c hΨ|

X
j

X

Ê+
k |Ψi

X

ge
−iq·r̂jρeg|uij hu| eiq·r̂k |uik +j ρk hu| e

Ê−
j ·

3 ~ckΓ
|Λ|2
=
2
8π r

"

k

j,k;j6=kHere I have used the product state |Ψi =

Q

X
j

#

ρee
.
j

(7.22)

j |uij |ψij to describe the wavefunctionof the array of atoms, with |uij represents the center of mass state of the j−th atom
(j)

(j)

and |ψij = cg |gi + ce |ei is the electronic states of the j−th atom. The density
matrix elements are represented by ρee , ρgg , ρeg and ρge .
I now define the translation vector Rj for each of the individual atom to describe
its center of mass, where the position operator satisfies hr̂j i = 0. The Debye-Waller1687.1. Theory of optical Bragg scatteringfactor is:hu| e−iq·r̂j−iq·Rj|uij = e
1 2 2
exp − qi ui τ = e−iq·Rj e−Wq (τ ) .
2
i=x,y,zY(7.23)

Solving for the steady-state solutions of the density matrix elements governed by the
optical Bloch equations, I obtain the scattered intensity as [45]:X s0 (2∆j + i)X(2∆k − i)
I (q)
s0 /2
−iq·(Rj −Rk )= e−2Wqe+
22A
2 1 + s0 + 4∆j (1 + s0 + 4∆k )1 + s0 + 4∆2jjj,k;j6=k(7.24)

where A = 3~ckΓ |Λ|2 /8πr2 , s0 is the on-resonance saturation parameter, and ∆ isthe probe detuning normalized to the linewidth Γ.
In light of the above derivation, it is helpful to make some comments on the
scattered intensity in Eq. 7.24.
• The last term in Eq. 7.24 captures the non-interfering parts of the intensity.
This term originates from radiative spontaneous emission and it becomes thedominant term if the saturation parameter satisfies s0  |∆|.3 I obtain thesame limit if the atomic sample has no correlation or no periodic behavior inits spatial arrangement, i.e. the first term is negligible.
• The first term on the right-hand side of Eq. 7.24 captures the interference
from spatial ordering of the atoms. For atoms with multi-level ground states,
3Here, I have implicitly assumed the frequency spectrum of the radiative spontaneous decay isdominated by that of the driving frequency ω0 (the coherent part). This subtlety originates fromthe solutions of the optical Bloch equations where periodic harmonic functions with frequency ω0
are imposed on all of the time-dependent variables.

1697.1. Theory of optical Bragg scattering
the interfering part also includes “magnetic ordering” of the system – where
the atomic ground states are alternating in a spatially periodic way. For avery large detuning ∆  s0 and ∆j ' ∆k = ∆, this term reduces to that ofEq. 7.19 if I replace the individual atoms with a continuum model for theatomic density.
• The non-interfering term scales as Natom , which describes the fluorescencefrom spontaneous emission of the excited state. Meanwhile, the interfering
2term scales as Natom, which can be interpreted as stimulated emission withdefinite phase relations between the scattered photons.
• When using optical Bragg scattering to probe periodic structure of 87 Rb
atoms, I typically used the D2 transition, which has a transition linewidthof Γ = 6 MHz. For a far-detuned probe and weak enough probe intensity,√or specifically δ  Γ 1 + s0 , the integrated scattered intensity depends
linearly on the probe pulse time τ . I observed the detected Bragg signal
did not saturate up to τ < 400 µs for the parameter values δ/Γ ∼ −103
and s0 ∼ 10, and this was before the atoms heated up to the point where
they were not trapped anymore. The detected Bragg signal also had a linear
dependence on incoming probe intensity s0 , and there was no saturation
effect for typical values of s0 < 100. Meanwhile, for near-resonant beam√with detuning δ < Γ 1 + s0 , the Bragg signal started to saturate pretty
quickly, on a timescale of τ > 5 µs for s0 ∼ 10.

170

7.2. Geometrical properties of optical Bragg scattering
7.2. Geometrical properties of optical Bragg scattering
7.2.1. The static structure factor. The simplest optical lattice can be madeby retro-reflecting a single laser beam or interfering two counter-propagating lasers
at wavenumber kd = 2π/λd , forming a standing wave with periodicity of d = λd /2.
In this one-dimensional (1D) lattice, let the x−axis lie along the laser propagation
direction kd and the xz−plane be the observation plane for the Bragg scattering (seeFig. 7.1). For an incoming Bragg probe with wavelength λ0 , I write the incoming
wavevector kin and scattered wavevector ksc as

kin = k0 cos θi ex + k0 sin θi ez ,andksc = −k0 cos θs ex + k0 sin θs ez ,(7.25)

where θi and θs are the incident and scattered angle with respect to the lattice axis
ex , and k0 = 2π/λ0 is the wavenumber of the Bragg probe. Here, I consider the probe
to be far-detuned with respect to the atomic transition and hence I am in the elastic
Rayleigh scattering regime where |ksc | ' |kin | = k0 .
I now calculate the static structure factor of the finite-size 1D lattice. Assuming
the atomic form factor to be identical for each lattice site j located at Rj = jdex ,the structure factor has the form [156]S (q) =

1

NS X
NS
X

Natom j=1 k=1

fj fk∗ e−iq·(Rj −Rk ) =

171

1
Natom

NS
X
j=1

2

e−iq·Rj|fj |2 .

(7.26)

7.2. Geometrical properties of optical Bragg scattering
The summation term in Eq. 7.26 gives [175]

2

|A| =
=

1
Natom

NS
X

2

e−ijqx d

j=1

1 sin2 [k0 (cos θs + cos θi ) Ns d/2]
sin2 (qx Ns d/2)=.
Natom sin2 (qx d/2)
Natom sin2 [k0 (cos θs + cos θi ) d/2]
1

(7.27)

I next assume the atomic density at each lattice site to have a Gaussian distribution
with identical sizes in y and z, i.e.:

ρl

r0j



 2

 2
y + z2x,= n0 exp − 2 exp −σxσr2(7.28)

where n0 is the peak density, σx is the axial RMS width along x, and σr = σy = σz isthe radial RMS width along y and z. In the harmonic approximation of the trapping
potential with radial beam waist wd , the thermal RMS size of the atomic layers can beprelated to the potential depth U0 and the temperature T , i.e. σx ≈ λd /2π · kB T /2U0p
and σr ≈ wd /2· kB T /2U0 . Consequently, the magnitude squared of the atomic form
factor becomes|fj |2 = n20 exp − qz2 σr2 + qy2 σr2 + qx2 σx2= n20 exp −k02 (sin θs − sin θi )2 σr2 − k02 (cos θs + cos θi )2 σx2 ,(7.29)

where qy = 0 with above choice of the plane of incidence (xz−plane). Note thedefinition of the static structure factor S = |A|2 |fj |2 in Eq. 7.26 is normalized to thetotal number atoms Natom , and hence S scales as S ∝ n20 Ns2 /Natom ≈ Natom . This also172

7.2. Geometrical properties of optical Bragg scattering
(b)
8

8

7

7

NBragg (×104 counts)

NBragg (×104 counts)

(a)

6
5
4
3
2
1

6
5
4
3
2
1

0

0
0.0

0.1

0.2

0.3

0.0

0.1

θi (rad)

(c)

0.3

(d)
8

8

7

7

NBragg (×104 counts)

NBragg (×104 counts)

0.2

θi (rad)

6
5
4
3
2
1

6
5
4
3
2
1

0

0
0.0

0.1

0.2

0.3

0.0

θi (rad)

0.1

0.2

0.3

θi (rad)Figure 7.2. Bragg signal as a function of the incident probe angle θi .
Here I show four different cases of ultracold atoms being loaded into
1D optical lattices and probed with optical Bragg scattering. (a) BEC
in a lattice with d = 395 nm, (b) BEC in a lattice with d = 405 nm, (c)thermal clouds in a lattice with d = 395 nm, and (d) thermal clouds ina lattice with d = 405 nm.

means that the Bragg intensity has a quadratic proportionality to either the number
2
of atoms Natomor the number of periodic layers Ns2 .

Based on this model for the structure factor, I can now discuss my observationsof the scattered Bragg signal NBragg from atoms in 1D lattice. An example of the173

7.2. Geometrical properties of optical Bragg scattering
scattered Bragg signal (integrated over the numerical aperture of our imaging system)
as a function of the incoming angle θi is shown in Fig. 7.2. Here I consider the cases ofa thermal cloud above the transition temperature TC and of Bose-Einstein condensed
atoms below TC . I also compare the results for two different lattice periodicities,
d = 395 nm [see Figs. 7.2(a) and (c)] and d = 405 nm [see Figs. 7.2(b) and (d)] .
Our collection optics had a capture angle α ∼ arctan (25/40) = 0.56 rad, centeredon the optical axis (along the lattice direction). This allowed me to scan the incoming
angle to a maximum value of 0.28 rad. For comparison, the Bragg probe had a beam
waist diameter of 0.25 mm or 5.6 × 10−3 rad, two order of magnitude smaller thanthe capture angle. Meanwhile, the lower limit of the incoming angle scan was set by
the edge of the D-shaped mirror which I used to send the Bragg probe. In Fig. 7.2,
notice there is a sudden drop in the Bragg signal at the start and end of the plots(θi < 0.08 rad and θi > 0.28 rad). This was due to the probe being clipped by the
optics, indicated by the gray shaded area. In this region, the incoming angle wasoutside the limit of my scanning ability.
I fit the observation results in Fig. 7.2 with the structure factor as given by
Eq. 7.26, as shown by the red curves in Fig. 7.2.4 The peak Bragg signal for theparticular case of infinite lattice is theoretically θB = arccos (λ0 /2d). This gives peak
Bragg angle at θB = 0.16 rad and 0.27 rad for d = 395 nm and 405 nm, respectively.
However, due to the finite size of the atomic cloud I observed small shifts in the peak
4In the fitting, I have used θ

s = θi for the sake of simplicity. A more precise model is to integrateθs within the capture angle of the collection optics and including the Gaussian spread of the Bragg
probe in θi .

174

7.2. Geometrical properties of optical Bragg scatteringvalue of θB . My observations are consistent to the structure factor model in Eqs.
7.26-7.29, where lattices with larger periodicity d have a larger peak Bragg angle. In
the fit, I also find that the parameter σx . d strongly affects the asymmetrical shapeof the Bragg profile.
In the BEC regime with T < TC , the diffracted signal has a broader spread in
the angular distribution than that of the thermal atoms, as shown by Fig. 7.2. The
reason is because the BEC is more confined spatially, and consequently the diffracted
signal becomes broader in reciprocal k−space [156]. In contrast, the hotter thermal
clouds have larger spatial size, and the atoms occupy more lattice sites Ns . The Bragg
signal has a sharper angular spread with larger Q−factor for thermal clouds and it
approaches that of the infinite lattice case. For this reason, I typically used thermal
clouds loaded into the 1D lattice to optimize the alignment of the Bragg probe.

7.2.2. Beam waist of the Bragg probe. In the Bragg setup (see Section 3.3.2),
I designed the beam waist diameter of the Bragg probe to be about 2w0 = 250 µm.
To verify the actual waist diameter, I measured the Bragg probe beam waist using
scattered light by the atoms. I first trapped Bose-condensed 87 Rb atoms in the 1064
nm optical dipole trap and adiabatically turned on the 1D lattice to a lattice depth
V0 & 20ER . I then suddenly turned off the dipole trap to let the atoms fall due
to gravity and move through the lattice standing wave created by the two counterpropagating lasers. I then pulsed the Bragg probe and measured the signal NBragg asa function of the vertical displacement dTOF during the time-of-flight (TOF).

1757.2. Geometrical properties of optical Bragg scattering

NBragg ( ×104 counts)

40
35
30
25
20
15
10
5
0
0

20 40 60 80 100 120 140
dTOF (µm)

Figure 7.3. Beam waist of the Bragg probe measured by releasing
87
Rb atoms from the dipole trap while the lattice potential is kept on.
My observations are shown in Fig. 7.3, for dTOF up to 160 µm (TOF duration
up to 5.6 ms). Since the atoms’ size (∼ 10 µm) was small compared to the designed
Bragg probe beam waist (∼ 250 µm), a change in the scattered Bragg signal NBraggby the displaced atoms resembled a change in the intensity profile of the Bragg probe
Iprobe ∝ exp (−2r2 /w02 ). This was within the assumption the atomic spatial distribution and coherence did not change as the atoms slide through the lattice potential. Fitting the measured data to a Gaussian profile (red curve in Fig. 7.3)
2
NBragg = N0 exp (−d2TOF /2σTOF) , I find that the width of the observed scattered in-tensity is σTOF ' 65 µm. Assuming that NBragg ∝ Iprobe , this implies a beam waist
diameter of 2w0 = 4σTOF ' 260 µm which is close to the designed value.
Note this measurement is only an estimate in a sense that I did not include theGaussian profile of the lattice depth confinement from the interfering laser intensity.
In my estimate, I assume the atoms’ spatial size and density modulation stay the same
during TOF; where the amount of scattered Bragg signal determines the intensity176

7.2. Geometrical properties of optical Bragg scatteringprofile of the Bragg probe. However, as the atoms slide through the lattice during
a few ms, the atomic spatial distribution and coherence might change as the lattice
depth gets weaker. For example, the product of the Bragg probe intensity with beam
waist w0 and the lattice laser intensity with beam waist wl gives a combined intensity22Itot with beam waist wtot
= w02 wl2 / (wB+ wl2 ). If the atoms follow the lattice depth

adiabatically NBragg ∝ Itot , I would then expect the observed Bragg signal in TOF tohave a smaller Gaussian RMS width since wtot < w0 .

7.2.3. Angular width of the scattered Bragg signal. To measure the angular extent of the scattered Bragg signal, I used an adjustable mechanical slit placed
at a distance of L ∼ 9.5 cm in front of the EMCCD camera (see Fig 3.9) whichcounted the number of scattered photons. The adjustable slit was a Thorlabs VA100
which gave 0.025” slit width adjustment per revolution of the micrometer drive. I
intentionally clipped the Bragg signal by closing the slit gradually, i.e. similar to theknife-edge measurement to determine the beam waist of a laser beam. Figure 7.4
shows the measured Bragg signal NBragg as a function of the slit traveling distance d.
The data points (red markers) are fitted to a logistic function (blue curve)f (d) =A+ c,
1 + exp [−r (d − d0 )](7.30)

where the fitting parameters are the function’s growth amplitude A, the offset c, thesteepness of the curve r, and the midpoint of the growth d0 .

177

NBragg (×104 counts)

7.3. Observations of optical Bragg scattering in engineered Hamiltonians

30

20

10

0
0

1

2

3

d (mm)

Figure 7.4. Scattered Bragg signal as a function of the traveling distance of a slit placed in front of the EMCCD camera.

The observations gave a traveling distance of dm ∼ 1 mm at the image plane region(with M = 3.9× magnification) before the Bragg signal was completely blocked.
This corresponded to an angular width of about θmeas = M · dm /L = 0.04 rad. For
comparison, the radial in situ cloud size was about D = 10 µm and the Bragg probe
wavelength was λ = 780 nm. This gave a predicted value for the angular extent of
the scattered signal to be θ ∼ λ/2D = 0.04 rad.

7.3. Observations of optical Bragg scattering in engineered Hamiltonians
In this section, I present my measurements of optical Bragg scattering in 1D latticeand spin-orbit-coupled (SOC) condensates. I created an optical lattice potential for
either |F = 1, mF = −1i or |F = 1, mF = +1i state using counter-propagating lasers
at the magic wavelength λzero = 790.034 nm with zero scalar light shift [9]. This
gave a periodicity of d = 395 nm in the atomic density distribution. I also examined1787.3. Observations of optical Bragg scattering in engineered Hamiltonians
spin-1/2 SOC condensates where I coupled the two states |F = 1, mF = −1i and|F = 1, mF = 0i with Raman transitions.
For the data presented below, I chose the frequency of the Bragg probe to be nearlyresonant with the atomic transition 5S1/2 , F = 1 → 5P3/2 , F 0 = 0 where the detuningwas δ ' −13Γ and the saturation parameter was s0 ' 10. The Bragg scattering fromthe optical lattice served as a calibration reference and the main measurements wereobservations of the spin-helix phase [22, 53] in SOC condensates at high couplingstrength Ω ∼ 1 − 10ER . The near-resonant Bragg probe is spin-sensitive and it allows
me to measure the precessing effective magnetic field in the SOC condensates.

7.3.1. Probing atoms in a one-dimensional lattice. Condensed atoms loaded
into an optical lattice will exhibit periodic modulation in their density distribution. I
prepared Bose-condensed atoms in an optical dipole trap (at 1064 nm) and adiabatically loaded them into the 1D lattice potential with lattice depth V0 . In the case of a
1D optical lattice, the atoms retain their long-range phase coherence (a characteristicof superfluidity) between different lattice sites even for large lattice depth V0 > 20ER
and there does not exist phase transition from the superfluid to the Mott-insulator
phase [77, 88]. Figure 7.5 shows the observed scattered Bragg signal NBragg counted by
the EMCCD camera. The signal increased monotonically for lattice depth V0 < 10ER
before it started to saturate at higher lattice depth confinement.
Based on Eq. 7.29 for the atomic form factor, I assumed the spatial extent of the√atomic wavefunction in a single lattice site scaled as σx ∝ 1/ V0 . I fit the observed
1797.3. Observations of optical Bragg scattering in engineered Hamiltonians
20
15
10
5
0

0

1

2

3

4

5

6

7

8

9

Figure 7.5. Bragg signal NBragg as a function of lattice depth V0 . Inset
shows the 68 pixels ROI for the Bragg signal imaged by the EMCCD
camera. Solid blue curve is a fit to Eq. 7.31, and the shaded gray line
shows the chosen threshold for a detectable Bragg signal (15% above
the offset value c obtained from the fitting).

Bragg signal with the function

NBragg (V0 ) = a exp (−b/V0 ) + c,

(7.31)

where a, b, and c are the fitting parameters. The fit gave an offset value of c =17.6 × 103 for a total of 68 pixels region-of-interest (ROI) in the imaged Bragg signal.
Accordingly, I set the lowest limit for a detectable Bragg signal to be 15% above theoffset (background) value c obtained from the fitting, as shown by the gray line in
Fig. 7.5. This gave a detection threshold V0 ≥ 0.48(2)ER for the lattice depth. Basedon numerical simulations of the Gross-Pitaveskii equation [14, 15], this thresholdcorresponded to ∼ 8% contrast in the density modulation for our typical 87 Rb BEC
with 2.2 × 105 atoms prepared in a harmonic trap with frequencies (ωx , ωy , ωz ) =180

7.3. Observations of optical Bragg scattering in engineered Hamiltonians
2π (93, 64, 39) Hz. For comparison, my BEC had a chemical potential of µTF /h ≈1.3 kHz and the lattice potential threshold in frequency unit was V0 /h ≥ 1.8 kHz.
In order to probe the coherence properties of the atoms, I initially prepared theatoms in an optical lattice with depth V0 , turned off the lattice suddenly and let theatoms expand for a duration of τ inside the harmonic dipole trap. I then used an
optical Bragg probe for a pulse time of ∼ 20 µs. The expanding atomic wavefunction
from each lattice site interfered with each other and after a certain expansion time,“rephasing” of the wavefunction yielded a maximum interference, similar to that of
the Talbot effect in optics [150, 168].
Figure 7.6 shows my observations of the atomic Talbot effect in a 1D lattice using
optical Bragg scattering [120]. The atoms were initially prepared in a 1D lattice with
V0 = 5ER , where ER = h2 /2mλ2d is the single photon recoil energy at wavelength
λd = 790.034 nm. The coherence revival was expected to occur once the width of
the atomic wavefunction expanded for a distance of d/2 = λd /4, or half of the latticespacing. Assuming the atoms expanded freely after being released from the latticepotential, the revival period Trev was predicted to be [120, 150] 5
md2
1Trev =
= ·
h
2



h
4ER



= 33.99 µs.

(7.32)

5The initial periodic density modulation d is recovered in the same spatial coordinates after 2T

rev .
Meanwhile, maximum interference occurs after Trev to yield similar periodic structures which areshifted by d/2. The Talbot time is defined as TTalbot = 2Trev [150].

181

7.3. Observations of optical Bragg scattering in engineered Hamiltonians

NBragg ( ×104 counts)

12
10
8
6
4
2
0
0

50 100 150 200 250 300 350
Revival time τ (µs)

Figure 7.6. Coherence revival after the condensates are released fromthe lattice potential at V0 = 5ER .

To obtain the revival period from the measurements displayed in Fig. 7.6, I fit thesedata to the functionf (τ ) = A exp (−λτ ) sin2 (2πτ /Trev + φ) + c ,(7.33)

where the term exp (−λτ ) was introduced to include the Debye-Waller factor.
My observations yielded a revival period of Trev = 32.9 (4) µs, slightly shorter
than the predicted value. I attributed the smaller measured Trev to the presence ofthe harmonic trap potential. This caused the collapse and revival dynamics of the
wavefunction to deviate from the predicted value in Eq. 7.32 [2]. In addition, due tothe interaction between particles, the collective atomic wavefunction has smaller effective mass. The energy dispersion  (q) for repulsively interacting particles with g > 0
has smaller curvature (larger second derivative) compared to that of free particles withno interaction g = 0. The effective mass is defined as 1/m∗ = (1/~2 ) [∂ 2  (q) /∂q 2 ].

182

7.3. Observations of optical Bragg scattering in engineered Hamiltonians

NBragg ( ×106 counts)

2.0
1.5

V0 =1.6ER
V0 =5.5ER
V0 =35.4ER

1.0

×2

0.5

0.0

×4

400

800

1200

Temperature T (nK)

1600

Figure 7.7. Optical Bragg scattering of atoms loaded into the 1D
optical lattice as a function of temperature. The temperature T was
measured using TOF release from the corresponding dipole trap without the lattice potential. The shaded area corresponds to the transition
temperature TC = 381 (30) nK when the BEC starts to appear.

Consequently, the effective mass became smaller m∗ ≈ 0.97m and the measured Trev
was shorter than its predicted value for free particles [88, 132].
I next observed the dependence of the Bragg signal on the temperature of theatoms, as displayed in Fig. 7.7. I prepared the atoms at temperature T by adjusting
the final power of the optical dipole trap (trap depth) after the evaporation stage.
The temperature T was measured using TOF release from the dipole trap (withoutlattice). I then loaded the atoms adiabatically into the optical lattice with potential
V0 and probed them with the Bragg beam. For a small lattice depth V0 = 1.6ER ,I observed a distinct kink where the Bragg signal was maximum at the transition
temperature TC (see Fig. 7.7). This kink gradually disappeared as I increased thelattice depth confinement.

183

7.3. Observations of optical Bragg scattering in engineered Hamiltonians

T >TC
T <TC

NBragg ( ×104 counts)

4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0

50

100

150 200
Revival time τ (µs)

250

Figure 7.8. Coherence revival of atoms in optical lattice below andabove the transition temperature TC ≈ 380 nK. The atoms, initiallyconfined in a 1D lattice potential with V0 = 3.43ER , are released fromthe lattice and let expand in the harmonic trap to exhibit the atomic
Talbot effect.
I attributed the decreasing Bragg signal for T < TC to the decreasing number ofatoms Natom and smaller size of the BEC cloud (fewer lattice sites Ns are occupied)
as they become colder. On the other hand, thermal atoms have relatively larger
fluctuations in the atomic wavefunction at temperature T > TC , and I still observed
a significant amount of Bragg signal. This was expected since Natom was bigger andmore lattice sites Ns were occupied, compared to the BEC case.
I also studied the coherence revival of the atomic wavefunction for T < TC andfor T > TC (see Fig. 7.8). The atoms were loaded into an optical lattice with
V0 = 3.43ER after being prepared at temperatures of T ∼ 340 nK and T ∼ 420 nK(for the condensed and thermal phases, respectively). As before, I observed the atomic
Talbot effect for condensed atoms at T < TC , as shown by the blue triangles in Fig.
7.8. I fit the data to Eq. 7.33, as shown by the solid blue curve in Fig. 7.8. In

184

7.3. Observations of optical Bragg scattering in engineered Hamiltonians
contrast to the condensed case, the observed Bragg signal for the thermal clouds (redcircles in Fig. 7.8) decayed rapidly once the lattice potential V0 was turned off andthe atoms expanded in the dipole trap. This indicated that the thermal clouds hadstrong fluctuations causing any initial phase coherence to decay rapidly.
Finally, I would like to make a few minor comments on some experimental details.
As described previously, I applied a bias field Bdc to create the spin-dependent lattice.
There were always transient eddy currents in the system when I turned off the biasing
field, and it took about 10 − 100 ms before the bias field Bdc settled to a stable value.
For this reason, I chose to keep Bdc constant while performing Bragg scattering.
The bias field affected the resonance condition for the optimum Bragg signal andI have found this to be spin-dependent as well. I measured a detuning value ofδ ' −80 MHz to give the maximum Bragg signal, for a bias field of Bdc ∼ 5 G along
ex and saturation parameter s0 ' 10. I did not observe any dependence on thepolarization of the Bragg probe. This was expected since I chose the near-resonanttransition to be 5S1/2 , F = 1 → 5P3/2 , F 0 = 0 (which has the same transition matrix
elements between the different sublevels mF = ±1, 0 → mF 0 = 0). To find the
optimum parameters for the detuning δ and s0 in a multi-level system theoretically,
I needed to solve the Maxwell-Bloch equations numerically. Instead, I chose theparameter values which gave the optimum Bragg signal experimentally.

7.3.2. Precessing magnetic field in SOC condensates. In this subsection,
I discuss my observations of near-resonant optical Bragg scattering off of spin-1/2

185

7.3. Observations of optical Bragg scattering in engineered Hamiltonians

NBragg (×104 counts)

8dt Ω = 0.23ER /ms
dt Ω = 0.11ER /ms
dt Ω = 0.08ER /ms

6
4
2
0
0

2

4

6

Coupling strength Ω/ER

Figure 7.9. Bragg signal NBragg as a function of the Raman coupling strength Ωf for loading rates of 0.23ER /ms, 0.11ER /ms and0.08ER /ms. The dotted line shows NBragg is proportional to the transverse spin polarization |hσx i| of the SOC system.

spin-orbit-coupling (SOC) condensates. I prepared 87 Rb atoms in equal population
mixtures of spin-down |↓i = |F = 1, mF = −1i and spin-up |↑i = |F = 1, mF = 0i
states. I dressed the SOC Hamiltonian by ramping up the Raman coupling strength Ω
gradually from 0 to its final value Ωf with a linear rate of dt Ω. For adiabatic dressingof the atoms with dt Ω = 0.08ER /ms, I observed an increase of the Bragg signal as afunction of coupling strength, up to Ωf < 4ER . Beyond this point, the Bragg signal
started to saturate, as depicted in Fig. 7.9. I found the adiabaticity condition was
satisfied when the atoms were dressed with a rate below dt Ω . 0.1ER /ms.
In this engineered SOC Hamiltonian, the bias field Bdc = Bz ez and the counterpropagating lin ⊥ lin Raman lasers create an effective magnetic field of Beff ∝Ω cos (2kR x − δω t) ex , where kR is the single-photon recoil momentum and δω is thedetuning between the Raman lasers (see Chapter 5). For coupling strength Ω < 4ER ,186

7.3. Observations of optical Bragg scattering in engineered Hamiltonians

the system has two degenerate minima located at q0 = ±kR
dressed eigenstates are [69, 98] cos θ 
 exp (iq0 x)χ̃+ = − sin θq
1 − (Ω/4ER )2 and the sin θ 
 exp (−iq0 x) .
and χ̃− = − cos θ(7.34)The spinor wavefunction of the system then becomes Ψ (x) = n (C1 χ̃+ + C2 χ̃− ) which
has non-zero transverse spin polarizationpkR2 − q02Ω=−,
hσx i = −kR
4ERfor Ω < 4ER .

(7.35)

Near-resonant optical Bragg scattering, which is spin-sensitive, probes the precessing spin polarization. The precessing spin has a periodicity of dspin = π/kR = λzero /2,
identical to the density modulation in a periodic lattice case. For coupling strengthΩ ≥ 4ER , the minima merges at q0 = 0 and the spinor wavefunction is fully polarizedwith hσx i = −1, causing the Bragg signal to saturate. I attributed the observed
dependency of NBragg as a function of Ωf to the existence of the transverse spin polarization hσx i. For an adiabatic ramp, the measured NBragg is proportional to thetransverse spin polarization |hσx i|, as shown by the dotted line in Fig. 7.9. I notedthat the spin-sensitive Bragg signal ceased to exist as I detuned the Bragg probe
further away from the atomic transitions.
Figure 7.9 shows how the Bragg signal behaves as I shortened the loading duration
to be non-adiabatic, i.e. dt Ω = 0.11ER /ms and dt Ω = 0.23ER /ms. As I increased the187

7.3. Observations of optical Bragg scattering in engineered Hamiltonians(b)Ω = 0.42 ERΩ = 1.39 ER

20

40

60

80

Revival time τ (µs)

14
12
10
8
6
4
2
0

NBragg ( ×104 counts)

14
12
10
8
6
4
2
0

NBragg ( ×104 counts)

(a)

100

Ω = 2.85 ERΩ = 4.40 ERΩ = 6.02 ER

20

40

60

80

Revival time τ (µs)

100Figure 7.10. Bragg signal NBragg from SOC condensates as a functionof revival time τ . (a) I observed the atomic Talbot effect for initialdressing at coupling strength of Ω = 0.42ER and Ω = 1.39ER . (b)The periodic coherence revival ceases for higher coupling strength atΩ = 2.85ER , Ω = 4.40ER , and Ω = 6.02ER .

loading rate, the atoms did not have enough time to evolve and follow the minima
condition q0 adiabatically. This led to more dynamics in the Bragg signal.
I next measured the coherence revival in this Raman dressed system (see Fig.
7.10). After I loaded the atoms into the Raman dressed states, I quickly turned off theRaman lasers and let the atoms expand in the harmonic dipole trap. I then used the
Bragg probe to observe the atomic Talbot effect for low coupling strength Ω = 0.42ER
and Ω = 1.39ER . In contrast to the previous case, I did not observe a clear signatureof periodic coherence revival for Ω = 2.85ER , Ω = 4.40ER , and Ω = 6.02ER . At
higher SOC coupling strengths, the dynamics of the different dressed spin states in
the harmonic dipole trap might cause degradation in the coherence properties of theatoms. Although, the behavior shown in Fig. 7.10 was interesting, I did not makefurther investigations of this effect.

188

1.0

1.25

0.8

1.00

0.6

0.75

0.4

0.50

0.2

0.25
20

30

40

NBragg (×105 counts)

Fractional population N↑ /Ntot

7.3. Observations of optical Bragg scattering in engineered Hamiltonians

50

Pulsing time tpulse (µs)

Figure 7.11. Bragg signal NBragg as a function of Rabi pulsing time
tpulse of the Raman lasers. I observe the Bragg signal (red, right y-axis)
to oscillate with twice the frequency of the Rabi oscillation (blue, left
y-axis) in the coupled atomic states.

On the other hand, I further confirmed the origin of the scattered Bragg signal
was due to the (precessing) magnetic field by performing Bragg scattering after doinga Rabi pulse of the Raman coupling. Figure 7.11 shows Rabi oscillations of the SOC
states (indicated by circles) due to a square pulse with duration tpulse . The Bragg
signal (indicated by squares) oscillates with twice the frequency of the Rabi oscillation
between the coupled |↑i and |↓i states. Here I measured an optimum Bragg signal
for nearly equal mixtures of the |↑i and |↓i populations after Rabi pulsing. This
indicated the Bragg signal was related to the spin structure factor.
It is important to note here the angular alignment of the Bragg probe was optimized with respect to the corresponding spin-dependent optical lattice at the samelaser wavelength λzero . I expected the spin structure in the SOC dressed system tohave the same periodicity d = λzero /2 to the spin-dependent lattice case. I did not

189

7.4. Summary
find any spatially periodic structure corresponding to the minima in the SOC energyqdispersion, namely q0 = ±kR 1 − (Ω/4ER )2 . In other words, the atoms might havea modulation with different periodicity of"
2 #−1/2
λ
Ω
π
zero
d0 =
=
1−
.
2q0
2
4ER

(7.36)

However, within the range accessible to the the Bragg probe angle, I did not observe
any significant enhancement of the Bragg signal corresponding to the increasing periodicity d0 as a function of Ω. Based on my observations in the optical lattice potential
with a fix periodicity d = λzero /2 , the density-modulated BEC atoms had a variationof less than < 20% in the detected Bragg signal within the scan range of the probe
angle. In practice, I optimize the incoming Bragg probe angle with thermal clouds inthe corresponding lattice at periodicity d = λzero /2 and fix the Bragg angle in all of
the conducted measurements.

7.4. Summary
In this chapter, I present my measurements of various atomic structure using optical Bragg scattering. I discuss the static structure factor to describe geometrical properties of scattered Bragg signal due to periodic distribution of atoms. Measurementsof the Bragg counts as a function of the probe incident angle for different periodicities(d = 395 nm and d = 405 nm) are in agreement with the model. Condensed atoms,
which have less number of spatially periodic atoms, give broader angular distribution

190

7.4. Summaryin the scattered Bragg signal (more spread in the reciprocal k-space) compared to hotter atoms, which occupy more space and have more number of periodic layers (lessspread in the reciprocal k-space). I also measured the beam diameter of the Bragg
probe 2w0 ' 260 µm and angular width of the scattered Bragg signal θ ' 0.04 rad.
To calibrate the Bragg signal, I created spatial density ordering in the atoms byloading them adiabatically into a 1D optical lattice potential. Based on the atomic
form factor, I observed the Bragg signal to scale as NBragg (V0 ) ∝ exp (−b/V0 ) withrespect to the lattice depth V0 . To probe the phase coherence of the condensed atoms,
I demonstrated the atomic Talbot effect where the initial density modulation of theatoms was recovered after a certain Talbot time.
I observed a peak in the scattered Bragg signal at the BEC transition temperature
TC . Unfortunately, I did not conduct further investigations on this behavior. I would
speculate this might be caused by increasing spatial correlation between the atomic
wavefunction as the atoms reached the BEC phase [122]. However, a more carefulcalibration of the atoms’ number and their spatial distribution would need to be
performed before I could convincingly argue for the cause of this peculiar peak at TC .
Measurements of the atomic Talbot revival did confirm the atoms lost their coherence
properties as the temperature increased above TC .
The results being discussed here are for condensates in the mF = −1 state affectedby the spin-dependent lattice potential. For a mixture of spinor Bose gases in the
F = 1 manifold, the mF = ±1 states are bounded to the lattice potential while
mF = 0 atoms are free. With optical Bragg scattering, I could measure how the energy

191

7.4. Summary
exchange between atoms confined in a lattice potential coupled to an external bath oflattice-free atom might affect the spatial density ordering of the atoms. Unfortunately,
I did not conduct such experiment due to my time constraints. The optical Bragg
scattering is promising to study thermalization between a lattice and free Bose gases,
complementary to the work done by McKay et al. [110, 111, 113].
With near-resonant Bragg scattering, I probed the precessing magnetic field thatexist in the SOC condensates as the atoms were loaded adiabatically into the SOC
dressed states. I showed the Bragg signal was proportional to the transverse magnetization of the SOC system. The Bragg signal was optimized when 50%/50% mixtureof the atoms existed in the two degenerate energy levels of the SOC system. As
the Raman coupling strength Ω was increased, the two degenerate minima of the
SOC energy dispersion merge to form a single minimum. Consequently, the interference of the two opposite momenta vanished and the observed Talbot revival became weaker. The spin-structure ordering lost its coherence at high Raman couplingstrength Ω > 2.85ER , as confirmed by the Talbot revival data.
Besides loading the atoms adiabatically into the SOC dressed states, I also did a
short (Rabi) pulse of the Raman coupling strength and probed the atoms with nearresonant Bragg scattering. The frequency of the observed Bragg signal was twice thefrequency of the Rabi oscillation since the Bragg signal was maximized at 50%/50%
mixture of the atoms, confirmed the origin of the scattered Bragg signal was due tothe precessing magnetic field in the SOC system.

192

CHAPTER 8

Stripe-ordered phase in spin-orbit coupled condensates
Spinor Bose gases favoring the antiferromagnetic configuration (miscibility) [165]in the presence of spin-orbit coupling exhibit three phases: the stripe-ordered phase,the magnetized plane-wave phase and the zero-momentum phase [69, 98]. The stripe
phase occurs when each spin component, with two opposing momentum components±q1 , are interacting with each other due to the interatomic collisions. This results inthe formation of a spatial interference pattern with a period of π/q1 .
The stripe-ordered phase has properties that are analogous to supersolidity, firstdescribed by G. Martone and Y. Li et al. [97, 109] for atomic spin-orbit-coupled (SOC)systems. Within the Bogoliubov theory [59], the excitation spectrum of the meanfield energy in the stripe phase exhibit two gapless bands [97]. This gives a divergent
structure factor and static response function for wave vectors near the boundary of
the Brillouin zone. In addition, in the plane-wave phase of the SOC system, theexcitation spectrum is not symmetric and exhibits roton-maxon like structure where
the gap of the rotonic structure becomes smaller as one approaches transition to
the stripe phase [109]. The roton-mode softening is a precursor of a phase transition
which spontaneously breaks the continuous translational symmetry. This gives rise to193

a crystalline long-range phase order in a superfluid with weak short-range interaction.
This characteristic is a signature of “supersolid” phase of matter [136].
Most recently, Ji et al. [79] have observed roton-mode softening in the excitation
spectrum as the SOC Bose gas crosses the phase transition towards the stripe phase.
The roton-maxon excitation spectrum in the plane-wave phase was measured using
Bragg spectroscopy, and they found good agreement with theoretical predictions [109].
Besides in SOC Bose gases, the roton-maxon spectrum has also been observed in otherengineered ultracold atoms, such as in the shaken lattice [62], in BEC-cavity systems[94, 123], and recently in dipolar quantum gases [31].
Another predicted feature of the stripe phase is the existence of spatial density
modulation. This is challenging to measure because the expected density contrast isvery low and the modulation periodicity is on the order of 0.4 µm (or less). In theinitial work by Lin et al. [103], a signature of the stripe phase region was observed
indirectly using time-of-flight (TOF) imaging. Atoms in the dressed states were
projected into their bare spin and momentum states; the degree of phase-mixing
characterizes the region where stripe-ordered phase occurs.
Here I describe how I used optical Bragg scattering to detect the density modulation. Recently, similar technique was used by Jun-Ru Li et al. [95] to detect the
stripe phase in SOC system realized in a superlattice potential [96]. So far, this has
been the only experimental work to directly observe the density modulation of the
stripe phase. Unfortunately, in the work of Jun-Ru Li et al., the existence of the density modulation was only demonstrated as a function of detuning δ between the two1948.1. Theory of the stripe-ordered phase
SOC states in a limited region of coupling strength Ω. It was unclear if the observed
density modulation appeared within the predicted two-dimensional (2D) parameter
space of the SOC system [108].
In this chapter, I report my observation of the full phase diagram of the stripeordered phase. I realized an SOC system using two-photon Raman transitions for arange of coupling strength Ω [67, 102, 103]. My measurements of the stripe-orderedphase in (Ω, δ) phase space were in agreement with theoretical predictions [69, 98]. I
also investigated the metastable windows where the density modulation exists [103]and observed the atomic Talbot effect in the time-evolution of the density modulation[120, 150]. Finally, I explored the stripe-ordered phase in the finite temperatureparameter space of (Ω, T ) and my results were consistent with previous works [28,78, 178].
8.1. Theory of the stripe-ordered phase
The single-particle Hamiltonian of an SOC Bose-Einstein condensate realized withcounter-propagating Raman lasers is given by [69, 103]:

hSO
0 =


~2 
δ
Ω
2
(qx − kR σz )2 + k⊥
+ ~ σz + ~ σx + Vext ,
2m
2
2

(8.1)

where m is the atomic mass, kR is the single photon recoil momentum, k⊥ is thelinear momentum perpendicular to the SOC direction, δ is the effective detuning,Ω is the Raman coupling strength, and Vext is the external potential. The term
−2qx kR σz is analogous to Rashba and Dresselhaus spin-orbit couplings (with equal1958.1. Theory of the stripe-ordered phaselinear contributions) in condensed matter physics [44, 103, 144]. The spin-1/2 system
is described by the Pauli matrices σx , σy , and σz , and the coupling between the twodifferent spin states are locked to the linear momentum in the quasi-momentum spaceqx . The single photon recoil energy ER = ~2 kR2 /2m of the Raman coupling sets theenergy scale of this Hamiltonian.
I now include many-body interactions in the s−wave channels of the atoms anddescribe the mean-field energy functional of the system asˆE [Ψ] =dr"1
Ψ† hSO
0 Ψ+

X

2 α,β

gα,β

#
 †

Ψ Pα Ψ Ψ Pβ Ψ ,
†

(8.2)

where Ψ = (ψ↑ ψ↓ )T is a two-component wavefunction representing the spin-1/2 system; α, β are spin indices with projection operators to the two spin states given byP↑ = (1 + σz ) /2 and P↓ = (1 − σz ) /2; and gαβ = 4π~2 aαβ /m are the two-bodyinteraction terms corresponding to the s−wave scattering lengths aαβ .
To determine the ground states of the mean-field energy, I used the variational
method as described in Ref. [98, 108]. I chose the ansatz for the spinor wavefunction
as [69, 98, 108]:
















 ψ↑  √   cos θ+  ik x
 sin θ−  −ik x  = n̄ C+  e + + C− 
 e −  . (8.3)
Ψ (r) = 


 




ψ↓
− sin θ+− cos θ−Here the variational parameters are C+ , C− , k+, k− , θ+ and θ− , with n̄ is the atomic
density distribution. The parameter values for the lowest energy eigenstates were196

8.1. Theory of the stripe-ordered phasedetermined by minimizing the energy functional in Eq. 8.2 subject to the constraint´

d3 r Ψ† Ψ = N (or |C+ |2 + |C− |2 = 1) where N is the total number of atoms.
I first consider a uniform system with zero trapping potential Vext = 0 and neglect

the perpendicular kinetic energy k⊥ = 0. In the variational method, minimization of
E [Ψ] with respect to θ± yields the relation 2θ± = arccos (k± /kR ). Next, I consider
the special case of spin symmetric interaction g↑↑ = g↓↓ = g and zero detuning δ = 0,
which leads to the relation k+ = k− = q1 . For miscible condensates (g > g↑↓ ) in the
low density limit gαβ n̄  ER , the variational approach yields lowest energy eigenstates
which are compatible with the three following phases [98, 108]:(1) The stripe-ordered or spin-mixed phase.
For small values of Raman coupling Ω, the ground state is a linear combi√
nation of the plane-wave states e±iq1 x with equal weights |C+ | = |C− | = 1/ 2.
The magnetization of the system or the difference between the spin population is given by hσz i = |ψ↑ |2 − |ψ↓ |2 = 0. The most striking feature of thisphase is the appearance of density modulations in the form of stripes:~Ω
n (r) = n̄ 1 +cos (2q1 x + φ) ,(4ER + G1 )(8.4)

where φ is the relative phase between the complex variables C+ and C− . The
periodicity of the fringes π/q1 is determined by the wave vector

q1 = kR

s

1−

~2 Ω2
,
(4ER + G1 )2

197

(8.5)

8.1. Theory of the stripe-ordered phasewith interaction parameter G1 = n̄ (g + g↑↓ ) /2.
(2) The plane-wave or magnetized phase.
For larger values of Raman coupling, the atoms condense into a single
plane-wave or magnetized state in either the momentum state q1 (C− = 0) or−q1 (C+ = 0) whereq1 = kR

s

1−

~2 Ω2
,
(4ER − 2G2 )2(8.6)

with the interaction parameter G2 = n̄ (g − g↑↓ ) /2. In this phase, the density
is uniform n (r) = n̄ and the spin polarization is given by hσz i = ±q1 /kR . The
ground state of the system is doubly degenerate, where the choice between
the two eigenstates at qx = ±q1 is determined by spontaneous spin symmetry
breaking (if δ = 0). However, any non-zero detuning δ breaks the degeneracyof this magnetized phase.
In the low density limit G1, G2  ER , the transition between the stripeand plane-wave phases occurs at
4ER
Ω1→2
=
c
~

r

2G2
.
G1 + 2G2(8.7)(3) The single momentum or “zero momentum” phase.
At sufficiently large values of Ω, the two-degenerate minima at ±q1 merge
and the system enters the single-minimum phase where qx = 0 and hσz i = 0.
In this phase the gas is fully polarized along the x−direction, i.e. hσx i = −1.

198

8.1. Theory of the stripe-ordered phase
The transition between the plane-wave and the single-minimum phases issecond-order and it takes place at a critical value of Raman coupling:

Ω2→3
=
c

1
(4ER − 2G2 ) .
~

(8.8)

In Chapters 5 and 6, I discussed the transition between the plane-wave and single
momentum phases within the single-particle Hamiltonian hSO
0 . In this chapter, I focus
on the stripe-ordered phase. The existence of spin exchange due to the two-bodycollisions gives rise to the stripe-ordered phase in the SOC system if the interaction
2parameter G2 is positive or the miscibility condition g↑↑ g↓↓ > g↑↓is satisfied [98].

Now consider the two spin states |↑i = |mF = 0i and |↓i = |mF = −1i in the
F = 1 manifold of the 87 Rb atoms’ 5S1/2 electronic ground states. The s-wave
scattering lengths between the different spin states are a↑↑ = 101.41aB and a↓↓ =a↑↓ = 100.94aB where aB is the Bohr radius. In the presence of spin-asymmetricinteractions and non-zero detuning δ, I can extend the variational approach I used
above to include energy corrections from first order perturbation theory. For smallvalues of 0 < (g↑↑ − g↓↓ )  g↑↑ , the asymmetry in the intra-atomic interactions canbe compensated by the choice of detuning shift ~δcorr = −n̄ (g↑↑ − g↓↓ ) /2 [108].
Figure 8.1 shows theoretical prediction for the phase diagram of the stripe-orderedphase in the (Ω, δ) parameter space. Here I consider 87 Rb atoms with chemical potential of µ ≈ g↑↑ n̄ = 0.2ER . This corresponds to a uniform atomic density of199

8.1. Theory of the stripe-ordered phase

Figure 8.1. Theoretical prediction of stripe-ordered phase in (Ω, δ)parameter space for an SOC 87 Rb Bose gas (Vext = 0) with chemical
potential of µ = 0.2ER . Using variational method to solve the manybody ground state, I obtain the phase diagram of (a) magnetization and(b) density contrast of the stripe phase versus detuning ~δ and couplingstrength ~Ω. The stripe-ordered phase has non-polarized magnetizationand non-zero density contrast within the area bounded by the solid
curves.

n̄ = 0.94 × 1014 cm−3 , a typical value for 87 Rb condensates produced in our laboratory. Figure 8.1(a) shows mixing of the spin states in the stripe-ordered phase. This
leads to non-polarized magnetization −1 < hσz i < 1 within the detuning windown̄ |g↑↑ − g↓↓ | /h . 3.45 Hz for transition coupling strength Ω < Ω1→2
. At the detuningcvalue δ = δcorr , the critical transition from stripe-ordered phase to plane-wave phasepoccurs at the Raman coupling strength ~Ω1→2'
4E(g↑↑ − g↑↓ ) /2g↑↑ = 0.19ER .
R
cFigure 8.1(b) displays the density contrast of the stripe modulation, which scales as∝ ~Ω/ (4ER + µ). The maximum density modulation of the stripe phase near thecritical transition point is then given by ∼ 0.19ER / (4ER + µ) ' 0.045 or 4.5%.
200

8.2. Experimental setup
8.2. Experimental setup
I realized a spin-1/2 SOC system by applying two counter-propagating lasers
in lin ⊥ lin configuration, as shown in Fig 8.2. The lasers induced two-photon
Raman transitions between the hyperfine states in the F = 1 manifold of 5S1/2
electronic ground states of 87 Rb atoms. I tuned the Raman lasers’ wavelength tocancel the scalar light shift at λzero = 790.034 nm [9], which gave a single photon recoil energy of ER = h × 3.678 kHz. For this coupling scheme, I applied abias field of Bdc = 19.8595(4) G. Following Subsection 5.4.2, this gave a linear
Zeeman split of ωZ /2π = 13.948 MHz and a quadratic shift of /2π = 56.7 kHz.
Coupling to the other state |F = 1, mF = 1i was neglected, as it was detuned by
between56.7 kHz  ER /h. This setup realized the single-particle Hamiltonian hSO
0
the two states |↓i = |F = 1, mF = −1i and |↑i = |F = 1, mF = 0i as in Eq. 8.1. The
Raman laser frequencies were adjusted to control the detuning δ of the spin-1/2 SOC
system and the external potential Vext was the harmonic trapping potential.
The total mean-field energy of the system was given byˆ
E [Ψ] =

dr



1
1
2
Ψ† hSO(c0 + c2 ) n2↓ + (c0 + c2 ) n↑ n↓
0 Ψ + c0 n ↑ +
2

2



.

(8.9)

Here n↑ and n↓ are respectively the atomic densities of the spin-up and spin-down
states, and the interaction parameters are c0 /h = 7.79 × 10−12 Hz · cm3 and c2 /h =−3.61 × 10−14 Hz · cm3 . To allow comparison with the mean-field energy functional in
Eq. 8.2, I defined the interaction terms in Eq. 8.9 as g↑↓ = g↓↓ = c0 + c2 and g↑↑ = c0 .

201

8.2. Experimental setup

20
15
10
5
0

0

1

2

3

4

5

6

7

8

9

Figure 8.2. (a) Experimental setup to detect scattered Bragg signal
from the stripe phase, showing counter-propagating Raman lasers, optical Bragg probe, and a highly-sensitive camera. (b) Schematic torealize the spin-1/2 SOC system with two-photon Raman transition.
(c) Calibration of the Bragg signal using a spin-dependent optical lattice. Inset shows the region of interest for the Bragg signal imaged by
the camera.

In my experiments, I typically produced N ' 2.2 × 105 ± 15% condensed 87 Rb
atoms in a harmonic trap with trapping frequencies (ωx , ωy , ωz ) = 2π·(105, 67, 40) Hz.
This corresponded to a chemical potential µ ' 1.45 kHz and a peak density n0 =1.86 × 1014 cm3 . The stripe phase is predicted to exist within the detuning width
202

8.2. Experimental setupof δ/2π . |c2 n0 | /h ' 7 Hz for coupling strength values below ~Ω < 0.19ER . Thisrequires the ambient magnetic field to be controlled within < 10 µG precision. Besides
the small detuning window, the stripe phase also has a very low density contrast, on
the order of . Ω/ (4ER + µ). In other words, a maximum of only 4.3% of the atoms
are expected to be contributing to the periodic density modulation.
In my measurements, I used fluxgate magnetometers to measure drifts in theambient magnetic field and used partial-transfer absorption imaging (PTAI) in eachexperimental cycle to correct for magnetic field drifts in the system (as described in
Chapter 5). The combined feedforward plus PID feedback control system allowed me
to compensate field drift within < 400 µG over hours of data taking. The PTAI also
served as a sensor of the bias field value and gave an accuracy of ∼ 40 µG in the fieldmeasurement. In terms of the Zeeman frequency detuning, this limited my resolution
to δ ≈ 30 Hz. To calibrate the coupling strength Ω, I applied a square pulse of theRaman beams and obtained Ω from sinusoidal fit to the Rabi oscillation between
the different coupled spin states. I used highly sensitive and low-noise photodiodes
to measure the laser intensities in each experimental cycle. This gave a precision of∼ 0.02ER in the determination of Ω.
As I have noted several times already, the predicted stripe phase exists within avery small window of (Ω, δ) parameter space. In fact, the precision of the equipment
I used was barely sufficient. To overcome the aforementioned obstacles, I specifically
prepared the SOC dressed states to increase the detectable region of the stripe phase.
Based on our knowledge that the stripe-ordering is intertwined with the physics of203

8.2. Experimental setup
spin-mixing, I first prepared the atoms in an equal population of spin |↑i and |↓istates. I then adiabatically loaded the phase-mixed spinor Bose gases into the SOC
Hamiltonian by increasing Ω linearly for 50 ms. Next, I held the system at constantΩ for 1 s to reach a metastable equilibrium. Finally, I quickly ramped Ω up to > 6ER
within 200 µs to map the dressed states of the system into well-defined momentum
states at ±kR . The resulting density modulations in the spatial coordinates were
amplified as the wave packet of the atoms were enhanced in the corresponding momentum states.
To probe density modulations in the stripe phase region, I used far-detuned optical
Bragg scattering in the elastic Rayleigh scattering regime. The experimental setup is
depicted in Fig. 8.2(a)-(b). Here, the Bragg beam with wavelength λB = 790.2 nm
was red-detuned by 6.34 GHz from the F = 1 → F 0 = 0 transition in the D2 line.
The atoms was polarization insensitive to the Bragg probe since its detuning is waylarger than the transition linewidth, 6.34 GHz  6.06 MHz. In addition, I chose theF = 1 → F 0 = 0 transition for a specific reason that the different magnetic sub-levels
mF = −1, 0, and 1 in the F = 1 ground state have the same transition strength tothe mF 0 = 0 excited state [124] (see Fig. 2.3).
In order to calibrate the Bragg signal, I modified my setup to create a 1D spindependent lattice [110, 112] for spin |↓i state. To do this, I flipped the bias magnetic
field orientation to be along the laser propagation direction in ex and changed theRaman laser frequencies to be identical ω + = ω − . This allowed me to create density
modulation with periodicity λzero /2, similar to the stripe-ordered phase and on the2048.2. Experimental setup
same axis with the spin-orbit coupling direction, as mentioned previously in Section
5.4. Fig. 8.2(c) shows a typical Bragg signal from the 1D lattice system detected by
the electron-multiplying charge-coupled device (EMCCD) camera acting as photon
sensor in the relay imaging system. Empirical calibration using a 1D spin-dependentlattice with similar parameters for the atoms gave a limit of 8% on the minimum
observable density modulation contrast.
For an infinite crystal structure, the lowest order Bragg’s condition is met at
an angle of θB = 0.16 rad with respect to the lattice axis. However, due to thefinite transverse size of the atoms and wide acceptance angle of our imaging setup(numerical aperture of 0.3), the Bragg’s condition for the incoming angle was slightly
different and not very sharp. I observed a peak Bragg signal at an incoming angle ofθB ' 0.2 rad with a full-width at half maximum (FWHM) of 0.07 rad [156]. I could notdetect variations as small as 0.1% in the periodicity of the stripe-ordered phase density
modulation d = π/q1 as a function of Ω. Further, the Bragg scattering measurementsof the SOC states were limited to be within coupling strength of Ω < 0.65ER . I
did not observe significant deviations in the Bragg signal corresponding to spatial
structures with periodicities other than d ≈ π/kR .
During data collection, I pulsed the optical Bragg probe for a duration of 100 µs;
the atoms were greatly disturbed for pulsing duration of > 250 µs. I chose this value
because I did not observe any significant increase in the signal-to-noise ratio for alonger pulsing duration. Further, the 100 µs Bragg probe duration allowed me toimage the remaining atoms using TOF in the same experimental cycle. The TOF

205

8.3. Observations of stripe-ordered phase in the full phase diagram
images gave me an additional observation on the magnetization hσz i of the prepared
SOC dressed states. This gave a benchmark for comparison to the Bragg signal as afunction of the detuning δ.

8.3. Observations of stripe-ordered phase in the full phase diagram
Figure 8.3 shows the observed Bragg counts NBragg in (Ω, δ) parameter space. For
this data, I initially prepared the atoms in an equal population of the two miscible spinstates. I then loaded the atoms into the Raman dressed states with frequency detuningδ as discussed in the previous section. I found that the detection of the stripes was
possible for larger detuning window compared to the theoretically predicted value
because the population relaxation between the two miscible spin states is very slow.
The timescale for the initially phase-mixed spin states to eventually phase separate
in the magnetized state is estimated to be on the order of ∼ |δ/2π − c2 n0 /h|−1 .
The scattered Bragg signal from the stripe phase region in the full-phase diagram
is shown in Fig. 8.3(a). Figure 8.3(b) shows a cut through Fig. 8.3(a) that displays
NBragg as a function of detuning δ for four different values of Ω. The Bragg signal
shows a distinct double-peak structure with a dip in the center at high couplingstrength Ω ≥ 0.3ER . The observed double-peak structure is consistent with JunRu Li et al.’s measurements in Ref. [95]. In their work, the inter-spin interaction
strength g↑↓ was engineered to yield a larger detuning region and they observed an
additional peak in the Bragg signal near δ ≈ 0 Hz. However, in my data, this feature
was not resolvable due to the limitation in our ability to scan δ/2π to better than206

8.3. Observations of stripe-ordered phase in the full phase diagram
NBragg ( ×104 )

(a)
1

2

3

4

(b)
5

6

7
6
5
4
3
2
1
0
1000

NBragg ( ×104 )δ/2π (Hz)500

0.13ER
0.23ER
0.30ER
0.42ER

0
500
0.1

0.2

0.3

0.4

0.5

0.6

Ω (ER )

(c)

500

0

500

1000

δ/2π (Hz)

(d)

105

Apeak

FWHM (Hz)

103

104

0.1

0.2

0.3

0.4

0.5

102

0.6

Ω (ER )

0.1

0.2

0.3

0.4

0.5

0.6

Ω (ER )

Figure 8.3. Stripe-ordered phase in (Ω, δ) parameter space. (a) Phase
diagram of the stripe phase region in (Ω, δ) space, as observed by optical
Bragg scattering. (b) Bragg counts NBragg as a function of detuning δfor various coupling strength Ω. Solid lines are fitting to the differenceof two Lorentzian distributions as in Eq. 8.10. (c) Fitted amplitude of
the Bragg signal as a function of coupling strength Ω. (d) The FWHM
obtained from fitting as a function of coupling strength Ω. The grayarea shows the predicted transition coupling strength Ωc = 0.19ER ±15% from stripe-ordered phase to magnetized plane-wave phase.

30 Hz precision. The data in Fig 8.3(a)-(b) were taken with a step size of 35.8 Hz forthe detuning parameter δ/2π. Error bars in Fig. 8.3(b) represent 1σ standard errorof the mean with a sample size of 3 to 4 images. I note that the double-peak structure
might be blurred out for some values of Ω due to the noise in δ.

207

8.3. Observations of stripe-ordered phase in the full phase diagram
I fit the observed signal NBragg (δ) at some constant value of Ω with the differenceof two Cauchy-Lorentz distributions (Lorentzian profiles)

f (δ) =

A1
A0
+ c,
2 −
1 + [(δ − δ0 ) /σ0 ]1 + [(δ − δ0 ) /σ1 ]2(8.10)

with 6 fitting parameters: A0 , A1 , δ0 , σ0 , σ1 and c. The fitting function f (δ) is
depicted by the solid lines in Fig 8.3(b). The peak amplitude of the Bragg signal
Apeak was obtained from the maximum value of the fitting function. Figure 8.3(c)shows Apeak to decay exponentially as a function of Ω. The fitted detuning FWHM ofthe combined Lorentzian profiles as a function of Ω is shown in Fig. 8.3(d). I observed
a rapid decrease in the FWHM as a function of Ω below the critical transition couplingstrength ~Ωc ' 0.19ER . I note that there could be a case of overfitting NBragg (δ) to
the function f (δ) in Eq. 8.10; e.g. if there were only a few data points showing thedip between two peaks centered around δ0 . In this case, the observed data were closer
to a single Lorentzian distribution (A1 = 0). I attributed the double-peak structurein the observed NBragg (δ) might be caused by the presence of a harmonic noise inδ. Although I did not investigate this further, it would be interesting to develop a
numerical model to calculate NBragg (δ) and see how the density modulation in the
stripe phase was affected by the presence of a small ac component in δ during thesystem preparation.
In an ideal scenario, the presence of detuning δ 6= 0 breaks the symmetry of thetwo degenerate states in the plane-wave region for Ω > Ωc , where the clouds are208

8.3. Observations of stripe-ordered phase in the full phase diagram
spatially separated and exhibit no density modulation. However, due to the finite
spin healing length [27] of the condensates (even for δ 6= 0), there always exists a
region where the atoms occupy both spin states of the two local minima and they
are being phase mixed. The FWHM is largest for Ω < Ωc in the stripe phase region,
decreases as Ω approaches Ωc and it reaches an asymptotic value for Ω  Ωc . In Fig.
8.3(d), I fit the FWHM to the function FWHM(Ω) = A/Ω2 + c, and I obtained thebest fit values A = 6.32 Hz/ER2 and c = 215.99 Hz. I note that the results shown in
Fig. 8.3 demonstrate that my observations of the stripe phase using optical Bragg
scattering were consistent with previous measurements conducted in Ref. [103]. In
Ref. [103], the stripe phase region was characterized by the degree of phase separationof the two-component condensates in the SOC dressed states.
The density modulation in the stripe phase existed if there was a mixing of thetwo spin components. Any detuning noise would alter the population of the two spincomponents and their degree of mixing. Instead of having δ as a parameter space,
I could choose the magnetization or population imbalance between the two spins
hσz i = |ψ↑ |2 − |ψ↓ |2 as an equivalent parameter space. The magnetization hσz i was
obtained from TOF imaging of the remaining atoms after a 100 µs Bragg pulse in thesame experimental cycle. Since the Bragg pulse induced transfer between the SOC
states, I corrected this by also measuring hσz i in identical SOC system without the
Bragg pulse being applied.
Figure 8.4(a) shows the corresponding magnetization hσz i in (Ω, δ) parameter
space. The Bragg counts NBragg as a function of hσz i for four different values of2098.3. Observations of stripe-ordered phase in the full phase diagram

σz

­

(a)

0

(b)
1

8
7
6
5
4
3
2
1
0
1.0

500
0
500
0.1

0.2

0.3

0.4

0.5

0.6

Ω (ER )

(c)σgauss (Hz)

Agauss
0.1

0.2

0.3

0.4

0.5

0.0

σz

­

(d)

105

104

0.13ER
0.23ER
0.30ER
0.42ER

NBragg ( ×104 )δ/2π (Hz)

1

®

0.5

0.6

Ω (ER )

0.8
0.7
0.6
0.5
0.4
0.3
0.2

0.1

0.2

0.3

0.5

1.0

®

0.4

0.5

0.6

Ω (ER )

Figure 8.4. Stripe-ordered phase as functions of magnetization hσz iand coupling strength Ω. (a) Phase diagram of magnetization hσz iin (Ω, δ) space obtained from TOF imaging, complementary to Fig.
8.3(a). (b) Bragg counts NBragg as a function of hσz i for various couplingstrength Ω. Solid lines are fitting to a Gaussian distribution. (c) Fitted
amplitude of the Bragg signal as a function of coupling strength Ω.
(d) The gaussian width obtained from fitting as a function of couplingstrength Ω. The gray area shows the predicted transition couplingstrength Ωc = 0.19ER ± 15% from stripe-ordered phase to magnetizedplane-wave phase.

Ω is displayed in Fig. 8.4(b). I fit the observed NBragg as a function of hσz i witha Gaussian distribution, as indicated by the solid lines in Fig. 8.4(b). Similar toprevious case where I analyzed the dependency of Apeak to the detuning δ (see Fig
8.3(c)), the Gaussian peak of the Bragg signal Agauss also decayed exponentially as

210

8.3. Observations of stripe-ordered phase in the full phase diagrama function of hσz i as shown in Fig. 8.4(c). Figure 8.4(d) shows the σ-width for theGaussian fitting σgauss as a function of Ω. Here, again, I observed σgauss to scale as
σgauss (Ω) = A/Ω2 + c as shown by the solid line in Fig, 8.4(d). These measurements
supported the argument that the stripe phase strongly favors phase-mixing between
the spinor components in the SOC system.
8.3.1. Metastability of the stripe-ordered phase. I next investigated themetastability of the stripe-ordered phase. I adiabatically loaded the system into theRaman dressed states by linearly increasing Ω for 50 ms (with δ ≈ 0). I held theatoms for various hold times thold in the dressed states, and finally ramped up the
coupling strength Ω > 6ER in 200 µs before doing the optical Bragg probing. Figure8.5(a) shows NBragg in the (Ω, thold ) parameter space, where each grid represents morethan 3 averaged data points. Figure 8.5(b) shows that the Bragg signal decays as afunction of hold time thold . I fit the data in Fig. 8.5(b) with an exponential function
NBragg = A exp (−thold /τ ) + c, and plot the lifetime τ in Fig. 8.5(c).
In the data shown in Fig. 8.5(c), I observed a lifetime of τ & 180 ms for the
initially phase-mixed spin states in the stripe phase region Ω < Ωc , and a lifetime ofτ . 180 ms in the magnetized plane-wave states Ω > Ωc . The rate of change of τwith respect to Ω changed significantly at the critical point Ω = Ωc . I characterized
this behavior by fitting the measurements of τ (Ω) to the function

τ (Ω) =


A exp (−b1 Ω) ,for Ω ≤ 0.19ER , A exp (−b1 · 0.19ER ) exp {−b2 (Ω − 0.19ER )} , for Ω > 0.19ER ,211(8.11)

8.3. Observations of stripe-ordered phase in the full phase diagram
NBragg ( ×104 )

(a)
1

2

3

4

5

6

(b)
7

8

9

14
12
10
8
6
4
2
0

0.4
0.3
0.2
0.1

0.07ER
0.11ER
0.20ER
0.41ER

NBragg ( ×104 )Ω (ER )

0.5

50 100 150 200 250 300 350

thold (ms)

50 100 150 200 250 300 350

thold (ms)(c)τ (ms)

103

102
0.1

0.2

0.3

0.4

0.5

Ω (ER )
Figure 8.5. Metastability of the stripe-ordered phase. (a) Bragg signal as functions of coupling strength Ω and hold time thold . (b) Bragg
counts NBragg as a function of hold time thold for various couplingstrength Ω. (c) Lifetime τ of the metastable stripe-ordered phase as afunction of Ω. The gray area shows the predicted transition couplingstrength Ωc = 0.19ER ± 15% from stripe-ordered phase to magnetizedplane-wave phase.

as indicated by the dashed line in Fig. 8.5(c). I obtained the best fit values A =4.83 × 103 ms, b1 = 14.6ER−1 , and b2 = 3.6ER−1 . The lifetime rapidly decayed as Ω
approached the critical transition value Ωc , and it eventually settled to smaller valuesfor Ω  Ωc . Again, these measurements were consistent with Ref. [103] and showed212

8.3. Observations of stripe-ordered phase in the full phase diagramthat the metastable state of the stripe phase was a result of competing energy scale
between the SOC Hamiltonian and the spin-dependent interaction.

8.3.2. Coherence revival of the stripe-ordered phase. In order to confirm
the phase coherence of the stripe phase, I did a time revival scan of the observed
Bragg signal using Talbot interferometry, as in Refs. [120, 150]. After loading theatoms (see Section 8.2), I turned off the Raman lasers, let the atoms expand in theharmonic dipole trap potential for a duration of trev , and then probed the structurewith optical Bragg scattering. I expected the expanding coherent matter wave, which
initially had a periodic spatial pattern d, to interfere and exhibit the atomic Talbot
effect with a coherence revival period of Trev = md2 /h.1
Figure 8.6 shows my measurements on the atomic Talbot effect. The observed
Bragg counts NBragg as functions of (Ω, trev ) are displayed in Fig. 8.6(a), where eachdata mesh element is an average of more than 5 realizations. Figure 8.6(b) shows ahorizontal cut through the phase diagram in Fig. 8.6(a); i.e. NBragg as a functionof trev . The plot shows a coherence revival of the detected signal. The maximum
interfering signal eventually decayed due to the Debye-Waller factor as the atoms
heated up and had more spatial fluctuations.

1The initial periodic density modulation d is recovered in the same spatial coordinates after 2T

rev .
Meanwhile, maximum interference occurs every time period of Trev to yield similar periodic structures
which are shifted by d/2.

213

8.3. Observations of stripe-ordered phase in the full phase diagram
NBragg ( ×104 )

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

0.5

1.0

1.5

(b)
2.0

2.5
1.5

1.0

0.5

0.0

50 100 150 200 250 300 350 400

trev (µs)

50 100 150 200 250 300 350 400

trev (µs)(c)(d)
Trev (µs)

Arev ( ×104 )

1.4
1.2
1.0
0.8
0.6
0.4
0.2

0.09ER
0.31ER
0.39ER
0.65ER

2.0

NBragg ( ×104 )Ω (ER )

(a)0.1 0.2 0.3 0.4 0.5 0.6 0.7Ω (ER )

40
38
36
34
32
30
28
26

0.1 0.2 0.3 0.4 0.5 0.6 0.7Ω (ER )

Figure 8.6. Periodic revival of the scattered Bragg signal from thestripe phase. (a) Observed Bragg signal as functions of (Ω, trev ). (b)Coherence revival of the Bragg signal for various coupling strength Ω
as a result of the atomic Talbot effect. (c) Revival amplitude A of
the Bragg signal as a function of Ω (as obtained from a fit to Eq.
8.12). The gray area shows the predicted transition coupling strengthΩc = 0.19ER ± 15% from stripe-ordered phase to magnetized planewave phase. (d) Period of coherence revival Trev as a function of Ω(as obtained from a fit to Eq. 8.12). The dashed line shows the average value of observed Trev , while the solid line shows the theoreticalprediction of Tth (see Eq. 8.13).

I fit the time evolution of the Bragg counts to an exponentially decaying function
times the square of a sine:f (trev ) = A exp −λt2rev

 2
sin (2πtrev /Trev + φ) + c ,214(8.12)

8.3. Observations of stripe-ordered phase in the full phase diagram
where A is the revival amplitude and Trev is the revival period. Figure 8.6(c) shows
the revival amplitude A of the Bragg signal as a function of Ω. The revival amplitudedecayed rapidly once the coupling strength crosses the transition at ~Ωc = 0.19ER .
Figure 8.6(d) shows that the observed revival period Trev = 33.0(3) µs does not dependon the coupling strength Ω. This disagreed with the theoretically predicted value(based on a single-particle model) which was given by
m
Tth =
h



π
q1

2

mλ2zero
'
4h


−1
~2 Ω2
1−
16ER2

(8.13)The predicted revival period Tth was dependent on the coupling strength Ω, where
its values varied from Tth = 34.0 µs to 35.1 µs for ~Ω = 0ER to 0.7ER (see solid line
in Fig. 8.6(d)). I attributed this discrepancy due to the specific procedure I used tocreate and measure the stripe phase. At the end of loading the atoms in the SOC
dressed states, I ramped Ω up to > 6ER to map the dressed states of the system
into well-defined momentum states at ±kR . Thus, I did not observe an increasing
trend in Trev as the period of the density modulation d = π/q1 increases for higher
coupling strength Ω. Moreover, the collapse and revival dynamics were affected by
the presence of harmonic potential and the spinor condensate interaction energy.

8.3.3. Stripe-ordered phase as functions of coupling strength Ω andtemperature T . In this subsection, I investigated the stripe-ordered phase as functions of coupling strength Ω and temperature T . The atoms were first prepared in an
equal population of spin |↑i and |↓i states. I varied the initial temperature of the cold215

8.3. Observations of stripe-ordered phase in the full phase diagram
atoms by changing the final optical trap power in the evaporation stage. Next, thephase-mixed spinor Bose gases were loaded into the SOC Hamiltonian by increasing Ω
linearly for 50 ms and the atoms were held at constant Ω for 1 s to reach a metastableequilibrium. Finally, I ramped Ω up to > 6ER within 200 µs and performed the Bragg
scattering measurements. The presence of the SOC energy dispersion affected the momentum distribution of the atom, and hence the temperature T was determined for
the SOC dressed system in the metastable equilibrium. For each Bragg scattering
measurement, I repeated identical experimental run without pulsing the Bragg probe
and measured T as described by Ref. [78]2.
The 2D density distribution of the atoms ρ2D in the bare states was measured
using absorption imaging after TOF expansion time τ . A Stern-Gerlach gradient
force was applied to separate the two spin components with a distance ±d/2 along
the z−direction. The temperature T of the system was obtained by fitting the 2D
density distribution to the function [78]

ρ2D (x, z, T, τ ) = ρ2D,↑ (x, z, T, τ ) + ρ2D,↓ (x, z, T, τ ) ,(8.14)

where

 mx

d
x − vR τ, z − , T, τ a2(+)
− kR
2
~τ


 mx

d
(−)
2
+ ρSOC x − vR τ, z − , T, τ a(−)− kR ,
2~τ(+)
ρ2D,↑ (x, z, T, τ ) = ρSOC

2See Eqs. S.1-S.7 in the supplementary information of Ref. [78]216

(8.15)

8.3. Observations of stripe-ordered phase in the full phase diagram

 mx

d
x + vR τ, z + , T, τ b2(+)
+ kR
2
~τ


 mx

d
(−)
+ ρSOC x + vR τ, z + , T, τ b2(−)+ kR .
2~τ(+)
ρ2D,↓ (x, z, T, τ ) = ρSOC



(8.16)

Here m is the atomic mass, kR is the single photon recoil momentum, and vR = kR /mis the recoil velocity. The density distribution in the upper and lower branches of the
dressed state with quasi-momentum qx is given byrm (x02 + z 02 )m2 vR2 x02 ~2 Ω2 

∓
+
µ −

2
2
(±)
2τ
τ
4
0 0
 , (8.17)
ρSOC (x , z , T, τ ) = C g2 exp 



kB T











and the coefficients are given by

Ω
Ω
s
,
a2(±) (qx ) = s
2


2
2
~kR qxΩ2~kR qx ~kR qx
Ω

8
+
∓
+
m
4
m
4
m
b2(±) (qx ) =s2~kR qxΩ2 ~kR qx
∓
+
m
4
m
s
.
2
2
~kR qxΩ2+
m
4(8.18)(8.19)

Figure 8.7(a) shows my measurements of the Bragg counts NBragg in the finitetemperature phase diagram. Each data mesh element is an average of at least 4 data
points. Figure 8.7(b) shows slices of the Bragg counts as a function of Ω for various
temperature T . For the data presented in Fig. 8.7, the temperature T was variedwith a step size of 60 nK and the standard error of estimate from fitting was 30 nK.

217

8.3. Observations of stripe-ordered phase in the full phase diagram

1

2

3

4

6

400

119 nK
178 nK
238 nK298 nK

5

350

4

300

3

250
200

2

150

1

100
0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

400
350
300
250
200
150
100

Figure 8.7. Stripe-ordered phase in (Ω, T ) parameter space. (a)Phase diagram of observed scattered Bragg signal in the parameter
space of coupling strength and temperature (Ω, T ). (b) Bragg counts asa function of Ω for various temperature T of the system. (c) Estimatedphase boundaries for the BEC transition temperature TC (shaded grayarea), and the critical coupling strength Ωc (shaded blue area) below
which I observed an enhancement in the Bragg signal associated withthe stripe-ordered phase.

The BEC transition temperature TC was determined from the data when a condensed
fraction of the Bose gas started to appear, as shown by the shaded gray area in Fig.
8.7(c). Despite the phase diagram measurements had a low resolution, I estimated thephase boundary of the stripe phase as the critical coupling strength Ωc below which
I observed a monotonic increase in the Bragg signal as Ω became smaller, shown by218

8.3. Observations of stripe-ordered phase in the full phase diagram
the shaded blue area in Fig. 8.7(c). In this data, I observed the critical couplingstrength Ωc to be decreasing for higher temperature T of the SOC system. This is inagreement with previous experimental and theoretical work [28, 78, 178].
In addition, I also observed an increase in the Bragg signal in a small transientregion in the (Ω, T ) phase space beyond Ω > 0.19ER (see Fig. 8.7(c)). I attributed thisincrease in the Bragg signal as being due to the metastable nature of the phase-mixed
Bose gases, where thermal fluctuations at higher Raman coupling had a significant
role. Meanwhile, in the normal phase above the BEC transition temperature TC ,I observed a small increase in the Bragg signal for higher coupling strength Ω. I
associated this increase to the substantially larger spatial correlations of the atomic
wavefunction. It might be of particular interest to study the observed Bragg signal
associated with the density correlations from quantum and thermal fluctuations in
the phase space of (Ω, T ). Nevertheless, careful calibrations of the Bragg signal are
important since, as I have noted previously, the thermal clouds in general have more
atoms and larger size compared to those of the BECs.

219

CHAPTER 9

Conclusions and Outlook
9.1. Conclusions
The central thesis of my Ph.D. research work was to observe the stripe-orderedphase in a spin-orbit coupled (SOC) Bose-Einstein condensate (BEC). A signatureof the stripe phase is the existence of density modulation. I prepared atoms inmetastable states to enhance the stripe phase region and probed the density modulation using optical Bragg scattering. I was able to explore the full phase diagram of the
stripe phase, and my measurements of the phase boundaries were in good agreementwith theoretical predictions and previous experimental results [69, 78, 98, 103].
In Chapter 4, I demonstrated a systematic method to bring ultracold atoms into
optimal focus [140]. This technique relied on the behavior of the power spectral
density of the optical depth to eliminate defocus-induced artifacts. The density fluctuations in BECs, are mapped from the phase fluctuations after time-of-flight release.
They act like random scatterers to create diffraction pattern which change as a function of distance from the optimal focus. Using this approach, I was able to pinpoint
the optimal focus of the BEC to within 2 µm for a 150 µm thick BEC cloud.
In Chapter 5, I showed the techniques I used to manipulated ultracold atoms withelectromagnetic waves, ranging from radio frequency (a few MHz’s) to the optical220

9.1. Conclusions
band (hundreds of THz). I explained the preparation methods for different mixturesof the spinor F = 1 Bose gases and how I used microwave-assisted partial-transfer
absorption imaging to provide feedback for our DC magnetic field control. Further,
I exploited the vectorial light shift of the ac-Stark effect and used it to demonstratevarious engineered Hamiltonians [53]. This enabled me to realize spin-orbit coupling
using a two-photon transition Raman coupling scheme. With a few hardware changes,
I also created spin-dependent lattice [110] and used it to calibrate the optical Bragg
signal in later experiments.
In Chapter 6, I presented our measurements of the phase diagram of a spin-1 BEC,
containing a ferromagnetic phase and an unmagnetized phase [22]. We characterized
the first- and second-order phase transition from the ferromagnetic phase to the polar
spinor BEC and spin-helix BEC and our results were in agreement with theory. Our
observation of controlled quench dynamics through a first-order phase transition opens
the door for realizing Kibble-Zurek physics [84, 182] in this system.
In Chapter 7, I discussed my study of the optical Bragg scattering from ultracold atom systems. I gave detailed explanations of the various contributions to thescattered Bragg signal, which capture the non-interfering and interfering parts of theatomic distribution [45]. I demonstrated a new experimental technique to characterize the scattered Bragg signal, focusing on its geometrical properties [156]. Optical
Bragg scattering from one-dimensional lattice was displayed as a function of lattice
depth and I investigated the atomic Talbot revival [120, 150] in this system. Finally,
I studied many physical phenomena using the Bragg scattering technique, including:221

9.1. Conclusions• the Bragg counts show a peak near the BEC transition temperature TC at
low lattice depth,• there is no atomic Talbot revival for temperature above TC , consistent withthe thermal clouds having no phase coherence on the length scale of thelattice periodicity,• for a near-resonant Bragg probe, the scattered Bragg signal was observed
from the spin-helix BEC in the SOC dressed states prepared both diabaticallyand adiabatically,• a Rabi pulse of the Raman coupling was applied to create an SOC systemand the following Bragg scattering showed similar oscillating behavior withtwice the Rabi frequency, which confirmed the presence of precessing spinpolarization in the SOC system.

These findings open up new prospects to study the behavior of quantum degenerategases which possess spatial density modulations.
Finally, I studied the long-sought stripe-ordered phase in SOC condensates. As
described in Chapter 8, I explored the parameter space of Raman coupling strengthΩ and detuning δ between the two coupled spin components. My observation of the
Bragg signal showed a double-peak structure with a dip in the center as a function ofdetuning δ within the stripe phase region. The δ-width of the stripe phase decreases
as a function of Ω and it eventually reaches an asymptote above the critical transition
coupling strength ~Ωc ' 0.19ER . I examined the metastability condition of the stripe
2229.2. Outlook
phase and confirmed previous work [69, 98, 103]. The stripe phase is claimed to have a
crystalline order with long-range phase order and exhibits roton-maxon like structure
analogous to supersolidity [97, 109]. Our measurements of atomic Talbot revival in
the stripe phase do indeed show the expected coherence behavior. Last but not least,
I studied the finite-temperature T phase diagram of the stripe-ordered phase. My
results in (Ω, T ) parameter space based on the scattered Bragg signal support recent
theoretical work [28, 178] and they are in agreement with previous measurements [78].

9.2. Outlook
In the process of conducting this research, I would like to mention two things
which can cause unnecessary problems. First, the definition of frequency and angularfrequency are often overlooked by atomic physicists where energy is often expressedin terms of “frequency” (either the energy being divided by h or ~). I often found thisto be an unnecessary source of errors, which were time-consuming to hunt down. In
terms of technical issue in the laboratory, I found that the back-reflectance from thevacuum glass cell is very annoying. The glass cell acted as a plane-parallel plate which
introduced coma (or comatic) aberration for an off-axis beam propagating throughit. It is often desired to send laser beams to the atoms in a normal direction sincespherical aberration by glass plate can be conveniently corrected with lenses, andit eases optical alignment. However, undesired back-reflectance from the glass cell,
especially for near-resonant beams, causes an optical lattice potential from a normal

223

9.2. Outlook
angle beam. It is thus important to have a well-designed and well-coated glass cell to
reduce the back-reflectance as much as possible.
Regarding future outlook, optical Bragg scattering has proven to be useful to investigate density modulation present in quantum degenerate gases, particularly thespatial correlations of the atomic collective wavefunction. The temperature dependence shows a peculiar behavior near the threshold transition temperature of BEC
for low lattice depth. One question is whether it is possible to observe quantum statistical effects from the many-body atomic correlations as the atoms condense into
the degenerate energy level [122].
In the limit of weak elastic scattering, the optical Bragg probe maps the spatial
atomic distribution into Fourier space as the static structure factor. Hypothetically, if
we can scan the incoming Bragg angle and detect the scattered signal in all directions,
we could determine the atomic density in real space (extending the technique used in
X-ray crystallography to atomic physics).
In addition to the work presented in this dissertation, several other important
projects were going on in our laboratory. One project involved the production of
one-dimensional (1D) Bose gases using a tightly focused Laguerre-Gauss beam. The
realization of 1D Bose gases opens up new possibilities to study more elaborate engineered many-body problems, such as 1D spin-orbit coupling in 1D Bose gases, using
optical Bragg scattering to study the 1D density-density correlations, and investigating the stripe phase in a (quasi) 1D system. In (quasi) 1D condensates, where the spin
healing length is larger than the transverse size of the clouds, magnetic domains only224

9.2. Outlook
from in the longitudinal axis of the condensate. This should be readily observablein absorption imaging. Another interesting topic is the formation of stripe-orderedphase and its time-evolution in (quasi) 1D SOC condensates. One could use both
optical Bragg scattering and absorption imaging techniques to obtain information on
the density modulation and magnetic domains of such system simultaneously.

225

APPENDIX A

Matrix elements of the hyperfine Hamiltonian in the presenceof a magnetic field
In cold atom experiments, it is common to modify the atomic structure with an
external dc bias field. Here I discuss the matrix elements of the hyperfine Hamiltonian
operator, where diagonalization of the matrix operator gives eigenenergies of theatomic structure.
The atomic interaction with an external static magnetic field, where the bias field
quantization axis is chosen along ez , is given by [163]HB =µB(gS Sz + gL Lz + gI Iz ) Bz ,
~

(A.1)

where gS , gL , and gI are respectively the electon spin, electron orbital, and nuclear
g−factors. In alkali atoms, the energy shift due to the magnetic field is commonly
small compared to the fine-structure splitting. This allows me to describe the interaction Hamiltonian with the quantum number J of the electron’s total angularmomentum, namely
HB =µB(gJ Jz + gI Iz ) Bz ,
~

(A.2)

where the fine structure Lande g-factor is given by

gJ = gLJ (J + 1) − S (S + 1) + L (L + 1)
J (J + 1) + S (S + 1) − L (L + 1)
+ gS.
2J (J + 1)
2J (J + 1)
(A.3)

226

Table A.1. Magnetic interaction parameters for the D line transitionsof 87 Rb.
Physical constants

ValuesMagnetic dipole constant, Ahf

5S 1/2 h · 3417.34130545215(5) MHz
5P1/2
h · 407.25(63) MHz
5P3/2
h · 84.7185(20)MHz
5P3/2
h · 12.4965 (37)MHz
5S 1/22.00233113(20)
5P1/2
0.666
5P3/2
1.3362 (13)−0.0009951414(10)2.0023193043622 (15)0.99999369
3/2
h · 1.399624604 (35)MHz/GElectric quadrupole constant, Bhf
Fine structure Lande g-factor, gJ

Nuclear g-factor, gI
Electron spin g-factor, gS
Electron orbital g-factor, gL
87
Rb nuclear spin, I
Bohr magneton, µB

The hyperfine Hamiltonian in the presence of an external magnetic field becomes

Htot = Ahf

3(I · J)2 + 2~32 I · J − I (I + 1) J (J + 1) µB
I·J
~4
+B
+
(gJ mJ + gI mI ) Bz .
hf
~2
2I (2I − 1) J (2J − 1)~
(A.4)

Experimental values of Ahf , Bhf , and the g-factors for 87 Rb atoms are given in Table
A.1.
In order to solve the eigenenergies of the Hamiltonian in Eq. A.4, it is natural
to choose the basis states as |J, mJ , I, mI i where the quantum numbers describe theatomic total angular momentum consisting of nuclear spin and electron orbital angularmomentum. To simplify the notation in the remaining of this appendix, the basis
states |J, mJ , I, mI i are represented by |mJ , mI i. The Hamiltonian operator becomesa square matrix with dimension (2I + 1) (2J + 1). The expressions for the matrix
elements can be calculated using properties of the angular momentum operators. I

227

recall the angular momentum ladder operators are defined by

J+ = Jx + iJy

and J− = Jx − iJy ,(A.5)

which operations on a given angular momentum state |J, mJ i satisfy the relationsp
J+ |J, mJ i = ~ (J − mJ ) (J + mJ + 1) |J, mJ + 1i ,

(A.6)

p
J− |J, mJ i = ~ (J + mJ ) (J − mJ + 1) |J, mJ − 1i ,(A.7)

for bounded values of −J ≤ mJ ≤ J.
With the definition of the ladder operators, I can express the magnetic dipole
term as
M :=

1
1
I·J
= 2 Iz Jz + 2 (I+ J− + I− J+ ) .
2
~
~
2~

(A.8)

The relevant matrix elements for the first term on the right-hand side of Eq. A.4 areconsequently given by:

hmJ , mI |M| mJ , mI i = mJ mI ,hmJ , mI |M| mJ + 1, mI − 1i =

(A.9)

1p
(J − mJ ) (J + mJ + 1) (I + mI ) (I − mI + 1),
2(A.10)

hmJ , mI |M| mJ − 1, mI + 1i =1p(J + mJ ) (J − mJ + 1) (I − mI ) (I + mI + 1).
2(A.11)In a similar manner, I express the electric quadrupole term in Eq. A.4 as [143]Q=

3
3(I · J)2 + 2 I · J − I (I + 1) J (J + 1) ,
4
~
2~

228

(A.12)

and obtain the following relevant matrix elements (non-zero for I 6= 1/2 or J 6= 1/2):
hmJ , mI |Q| mJ , mI i =



1 2
3mI − I (I + 1) 3m2J − J (J + 1) ,2

3(2mJ − 1) (2mI + 1)
4p(J + mJ ) (J − mJ + 1) (I − mI ) (I + mI + 1),(A.13)

hmJ , mI |Q| mJ − 1, mI + 1i =

3(2mJ + 1) (2mI − 1)4p
(J − mJ ) (J + mJ + 1) (I + mI ) (I − mI + 1),(A.14)

hmJ , mI |Q| mJ + 1, mI − 1i =(A.15)

3p(J + mJ ) (J + mJ − 1) (J − mJ + 1) (J − mJ + 2)
4p
(I − mI ) (I − mI − 1) (I + mI + 1) (I + mI + 2), (A.16)

hmJ , mI |Q| mJ − 2, mI + 2i =3p(J − mJ ) (J − mJ − 1) (J + mJ + 1) (J + mJ + 2)
4p
(I + mI ) (I + mI − 1) (I − mI + 1) (I − mI + 2). (A.17)

hmJ , mI |Q| mJ + 2, mI − 2i =A special case exists for J = 1/2 where the Hamiltonian in Eq. A.4 can be solved
analytically. The eigenenergies in this case are given by
s


1
∆E
g
m
±
x
4 mI ± 12 x
∆Ehf
∆Ehf
hf I
I
2
+
±
+ x2 ,
E1/2,±1/2,I,mI = −1+
2 (2I + 1)
(gJ − gI )2
2I + 1(A.18)
where ∆Ehf = Ahf (J + 1/2) and x = (gJ − gI )µB Bz /∆Ehf . The solution in Eq. A.18
is also known as the Breit-Rabi formula which is particularly useful to describe thehyperfine splitting of electronic structures in alkali atoms with J = 1/2 (i.e. the 5S 1/2
electronic ground state and 5P 1/2 excited state in the D1 line of 87 Rb atoms).

229

APPENDIX B

Gross-Pitaevskii equations for F = 1 spinor Bose-Einsteincondensates
I consider F = 1 spinor Bose-Einstein condensates (BECs) within s-wave scattering limit between the two-body collisions. In the following treatment, I assume theatomic interactions do not change the rotational symmetry of the system, implying a
conservation of total angular momentum of the colliding pair of atoms. In the s−wavechannel, the relative center-of-mass orbital angular momentum of the two colliding
atoms (labelled by index i and j) is zero, and the total angular momentum of theincoming state is simply the sum of the two atomic hyperfine spins Fij = Fi + Fj .
Exchange symmetry of the colliding two identical bosons then restricts the quantum
number Fij to be an even integer, or it can only take the values Fij ∈ {0, 2, . . . , 2F }[159].
Within these assumptions, the two-body interaction is only applicable when theoutgoing state has the same total angular momentum with the incoming state, andhence the collisional interaction can be described by the potential operatorV̂int =

X 4π~2 aFij
1X 3
δ (ri − rj )P̂Fij ,
2 i,j
m
F

(B.1)

ij

where the first summation term is taken over all colliding particle pairs; aFij are thescattering lengths for collisions between the pair (i, j) with total spin Fij ; and P̂Fij are
the projection operators which project the two-atom wave function into the subspacewith total angular momentum Fij . To eliminate the summation over the possible
collisional channels, I consider the outer product of identity operators for the two230spin-1 particles
Iˆi ⊗ Iˆj =X

P̂Fij = P̂0 + P̂2 ,(B.2)

Fij =0,2

and the composition of the F = 1 spin operatorsX  Fij
F2ij − F2i − F2j
Fi · Fj(Fij + 1) − F (F + 1) P̂Fij = P̂2 − 2P̂0 .
=
=
~2
2~2
2
F =0,2ij(B.3)

This allows me to rewrite the interaction potential as
1X 3
Fi · FjˆˆV̂int =
δ (ri − rj ) c0 Ii ⊗ Ij + c2
,
2 i,j
~2

(B.4)

where the interactions are separated into the spin-independent part c0 Iˆi ⊗ Iˆj and spindependent part c2 Fi · Fj /~2 . The coefficients c0 and c2 are related to the scattering
lengths a0 and a2 (for collision channels with total spin 0 and 2) via the equations
c0 =4π~2 a0 + 2a2
,
m
3

c2 =

4π~2 a2 − a0
.
m
3

(B.5)

In the formalism of second quantization, the many-body Hamiltonian within thes−wave interactions for F = 1 spinor BEC becomes
ˆ

(


 2 2
1
~∇
Ĥ =
dr
−
+ Vext (r) Ψ̂α + c0 Ψ̂†α Ψ̂†α0 Ψ̂α0 Ψ̂α
2m
2
)
 

1 
+ c2 Ψ̂†α Fαβ Ψ̂β · Ψ̂†α0 Fα0 β 0 Ψ̂β 0
,
2
3

Ψ̂†α

(B.6)

where Vext (r) is an external scalar potential, the indices α, β, α0 , β 0 correspond tothe mF = −1, 0, +1 components of the F = 1 manifold, and repeated indices are
implicitly summed over. In Eq. B.6, I have simplified our notations by omitting the231position and time dependence of the field operators Ψ̂α (r, t) in the writing.1 Thebosonic field operators satisfy the commutation relationshiΨ̂α (r, t), Ψ̂†β (r0 , t)= δαβ δ(r − r0 ),hihiΨ̂α (r, t), Ψ̂β (r0 , t) = Ψ̂†α (r, t), Ψ̂†β (r0 , t) = 0,

(B.7)
(B.8)

and the spin-1 matrix operators are given by 0 1 0 ~ Fx = √  1 0 1  ,

2


0 1 0







 0 −i 0 

~ 


Fy = √  i 0 −i  ,

2


0 i
0



 1 0 0 Fz = ~  0 0 0  . (B.9)

0 0 −1The explicit form of the total Hamiltonian in second-quantized form becomes
ˆ

(


~2
†
†
Ĥ =
dr
∇Ψ̂α · ∇Ψ̂α + V (r)Ψ̂α Ψ̂α
2m



i
c0 + c2 h † †
†
†
†
†
†
Ψ̂1 Ψ̂1 Ψ̂1 Ψ̂1 + Ψ̂−1 Ψ̂−1 Ψ̂−1 Ψ̂−1 + 2Ψ̂0 Ψ̂0 Ψ̂1 Ψ̂1 + Ψ̂−1 Ψ̂−1
+
2
h
i
c0
+ Ψ̂†0 Ψ̂†0 Ψ̂0 Ψ̂0 + (c0 − c2 ) Ψ̂†1 Ψ̂†−1 Ψ̂−1 Ψ̂1 + c2 Ψ̂†1 Ψ̂†−1 Ψ̂0 Ψ̂0 + c.c. . (B.10)
2
3

In the mean-field approximation, I then replace the field operators by the complex
wave functions to describe the three-component spinor condensates:








 Ψ̂1 (r, t)  ψ1 (r, t) 








Ψ̂(r, t) =  Ψ̂0 (r, t)  → Ψ (r, t) =  ψ0 (r, t)  ,








Ψ̂−1 (r, t)ψ−1 (r, t)(B.11)

1For a single (spin) component of interacting particles in the s−wave approximation, the many-body

Hamiltonian becomes Ĥ =

´

h 
i
´
d3 r Ψ̂† −~2 ∇2 /2m + Vext (r) Ψ̂ + 21 d3 r0 Ψ̂† Ψ̂0† gδ 3 (r0 − r)Ψ̂0 Ψ̂

since a = a0 = a2 and g = 4π~2 a/m.

232

with the normalization to the total number of atoms N is given byˆ
d3 r Ψ † (r, t) · Ψ (r, t) = N.

(B.12)

I obtain the Gross-Pitaevskii mean-field energy for F = 1 spinor BEC as
ˆ
EGP =

ˆ

(


~2
2
∗
∇ψα · ∇ψα + V (r) |ψα |d rε =
dr
2m


 c0
c0 + c2 |ψ1 |4 + |ψ−1 |4 + |ψ0 |4
+
2
2


+ (c0 − c2 ) |ψ1 |2 |ψ−1 |2 + (c0 + c2 ) |ψ0 |2 |ψ1 |2 + |ψ−1 |2 ∗ ∗ 2
+c2 ψ1 ψ−1 ψ0 + c.c. ,(B.13)
3

3

where ε is defined as the energy density functional . The right-hand side terms on thefirst line of Eq. B.13 describe the single-particle physics of the system; the terms on
the second line describe the self-interaction or intra-state collisions between particlesof the same spin; the remaining terms describe the spin-exchange mechanism withthe third line representing the inter-state collisions between particles with different
spins, and the last term on the fourth line representing coherent spin mixing whichis responsible for the relaxation to the spinor ground state energy.
Next, I derive the time-evolution of the wave function Ψ (r, t) from the principleof least action2. I define the action of the system as [132]ˆS [Ψ (r, t)] =

ˆ
dt



d3 r L ψα , ψα∗ , ψ˙α , ψ̇α∗ , ∇ψα , ∇ψα∗ ,(B.14)and the Lagrangian is given byL=ii~ h ∗ ˙
ψα ψα − ψ˙α∗ ψα − ε.
22Other derivation method is to use the Heisenberg representation i~∂ Ψ̂(r, t)/∂t =233(B.15)hiΨ̂(r, t), Ĥ

Using the variational principle method to find the extremum value of S, I obtain theequation of motion:
∂L
∂ ∂L
+
−
−∇·
∂t ∂ ψ̇α∗
∂ψα∗



∂L
∂∇ψα∗



= 0,

(B.16)

(B.17)

which can be expressed in terms of ε as
∂ψα
δε
∂ε
i~
=
=
−∇·
∗
∂t
δψα
∂ψα∗



∂ε
∂∇ψα∗

.

Finally, I arrive at the time-dependent Gross-Pitaevskii equations (GPEs) describing
the F = 1 spinor condensate wave functions:i~∂t ψ1i~∂t ψ0i~∂t ψ−1




~2 2
2
2
2
∇ + Vext + (c0 + c2 ) |ψ1 | + |ψ0 | + (c0 − c2 ) |ψ−1 | ψ1=−2m∗
+c2 ψ−1
ψ02 ,



~2 2
2
2
2
=
−
∇ + Vext + (c0 + c2 ) |ψ1 | + |ψ−1 | + c0 |ψ0 | ψ02m

(B.18)+2c2 ψ0∗ ψ1 ψ−1 ,(B.19)



~2 2
2
2
2
∇ + Vext + (c0 + c2 ) |ψ−1 | + |ψ0 | + (c0 − c2 ) |ψ1 | ψ−1=−2m+c2 ψ1∗ ψ02 .

(B.20)

Note special cases exist in the dynamics of the spinor condensates if I only consider two
spin states. As an example, if there is no initial atoms in mF = 0 or ψ0 (r, 0) = 0, andthere is no spin coupling from the single-particle Hamiltonian and external potential,
the GPEs reduce toi~∂t ψ1i~∂t ψ−1


~2 2
2
2
=
−
∇ + Vext + (c0 + c2 ) |ψ1 | + (c0 − c2 ) |ψ−1 | ψ1 , (B.21)
2m


~2 2
2
2
=
−
∇ + Vext + (c0 + c2 ) |ψ−1 | + (c0 − c2 ) |ψ1 | ψ−1 . (B.22)
2m


234

Despite their complexities, the set of coupled GPEs in Eqs. B.18-B.20 can befurther simplified and understood by drawing analogies to other physical systems. In
a specific condition where the available spin interaction energy |c2 | n is less than that
required to create spatial spin structures in the condensates, or the spin healing lengthpscale ξs = h/ 2m |c2 | n is larger than the size of the condensate, I can separate thespatial dynamics and the internal spinor dynamics. This is known as the single-mode
approximation (SMA) [26, 27]. Separation of variables for the spinor wave functions
lead to









 ψ1 (r, t)  χ1 (t) 

 √





−iµt/~ Ψ (r, t) =  ψ0 (r, t)  = N φ (r) e χ0 (t)  ,








ψ−1 (r, t)χ−1 (t)(B.23)

where N is the total number of atoms, φ (r) is the common spatial mode function,and χα (t) are the position-independent spinor wavefunctions with the normalizaP2tion
α |χα (t)| = 1. The spatial mode function φ (r) describes the normalizedatomic density distribution |φ (r)|2 = n (r) and it satisfies the normalization con´
dition d3 r n (r) = 1. The chemical potential µ is determined by the stationary
spin-independent GPE for the spatial mode
~2 2
∇ + Vext + c0 n (r) φ (r) .
µφ (r) = −2m(B.24)In the SMA, the coupled spinor equations becomei~∂t χ1 = E1 χ1 + c (ρ1 + ρ0 − ρ−1 ) χ1 + χ20 χ∗−1 ,i~∂t χ0 = E0 χ0 + c [(ρ1 + ρ−1 ) χ0 + 2χ1 χ−1 χ∗0 ] ,i~∂t χ−1 = E−1 χ−1 + c (ρ−1 + ρ0 − ρ1 ) χ−1 + χ20 χ∗1 ,

235(B.25)(B.26)(B.27)

where c = c2 N´

d3 r n (r)2 is the integrated spin mean-field coefficient and ρα = |χα |2is the spin fractional population. In above, I have included energy shifts from the
Zeeman hyperfine structure, where I parameterize the linear and quadratic Zeeman
effects respectively as η = (E−1 − E1 ) /2 and δ = (E−1 + E1 − 2E0 ) /2. Besides thePtotal number of atoms α ρα = 1 is conserved, the equations of motion within the
SMA also guarantee the magnetization M = ρ1 − ρ−1 to stay conserved.

In above, I have discussed the dynamics of spinor Bose condensates in the magnetic
hyperfine structure where atomic interactions are governed by the two-body collisions.
This treatment can be extended to other engineered physical systems, i.e. the socalled “spin” states are realized in the optical lattice or double-well potential, and theinteractions can be tuned accordingly via laser-assisted tunneling [96].

236

APPENDIX C

Machine drawing of camera mounts
In Bragg signal detection, I used an iXon Ultra 888 electron-multiplying chargecoupled device (EMCCD) camera from Andor Technology. I designed custom-madeplates for mounting purpose and extra drift stability, which inspired by Fig. 3 in Ref.
[20]. The camera was mounted on two translation stages, as displayed in Fig. C.1.
Machine drawings of the mounting plates are shown in Figs. C.2-C.3.

Figure C.1. Andor EMCCD camera (iXon Ultra 888) mounted ontranslational stages with custom-made plates.

237

238

A

B

C

D

.375

8

.269

.590

.590

1.180

1.300
n

.590

7

1.831

R0.250

.590

3.200

.590
1.510

2.100

1.510

6

6

1.831

CBORE FOR 1/4-20
CAP SCREW

CBORE FOR M5
CAP SCREW

2.360

4.200

.590

CBORE FOR M5
CAP SCREW

Ø1.3" THRU HOLE

CBORE FOR M5
CAP SCREW

.500

1.510

1.510

.920

1.510

.150

7

.500

1.510

.269

.600

1.700

.331

.300

2.331

1.831

1.831

.257

5

CBORE FOR M5
MOUNTING CAP SCREW
Ø0.217" THRU
Ø0.394" 0.2" DEEP
4 PLACES

5

4

.175

.200

.438.266

6.550

.175

1.510

4

.217

.394

.200

.375

3

3

APPROVEDMFG

QA

CHECKEDAndika Putra

DRAWN

2

9/6/2016

2

SCALE

D

SIZE

2:1

PartA_Face

DWG NO

Face Plate iXonUltra888

TITLE

Figure C.2. Machine drawing (not to scale) of the front mounting plate for iXon Ultra 888.

CBORE FOR 1/4-20 (M6)
MOUNTING CAP SCREW
Ø0.266" THRU
Ø0.438" 0.2" DEEP
2 PLACES

2.360

.590

.590

.908

MOUNTING POINTS
4-40 (#4) UNC TAPPED
0.15" DEEP, 8 PLACES

8

1

SHEET

1

1 OF 1

REV

A

B

C

D

239

A

B

C

D

8

8

.100

.680

7

1.000

2.280

1.000

.921

3.200

4.560

2.000

1.000

1.079

6

6

.500

.680

1.201

.500

.500

.500

.500

.325

1.567

CBORE FOR 1/4-20 (M6)
MOUNTING CAP SCREW NORMAL FIT
Ø0.266" THRU, Ø0.438" 0.4" DEEP
15 PLACES THIS SIDE

5

1.000

4

4

CBORE FOR 1/4-20 (M6)
MOUNTING CAP SCREW NORMAL FIT
Ø0.266" THRU, Ø0.438" 0.4" DEEP
2 PLACES THIS SIDEMOUNTING POINTS
1/4-20 UNC TAPPED 0.4" DEEP,
4 PLACES ON BOTH SIDES

3.134.100 .625
.525

1.567

5

.266

.438

.400

.225

3

MOUNTING POINTS
1/4-20 UNC TAPPED 0.4" DEEP,
2 PLACES ON BOTH SIDES

3

APPROVEDMFG

QA

CHECKEDAndika Putra

DRAWN

2

SCALE

D

SIZE

2:1

PartB_BaseDWG NO

Base Part iXonUltra888

TITLE

9/6/2016

2

Figure C.3. Machine drawing (not to scale) of the base mounting plate for iXon Ultra 888.

.400

.400

1.567

.325

.500

1.359

7

1

SHEET

1

1 OF 1

REV

A

B

C

D

APPENDIX D

Floquet theory for periodic time-dependent HamiltonianIn this appendix, I discuss the Floquet formalism to treat a time-periodic Hamiltonian [32, 60]. I am interested in a Hamiltonian with time periodicity T satisfying
the relationĤ(r, t) = Ĥ(r, t + T ),(D.1)

where the driving frequency of the system is defined by ω = 2π/T . Similar to theapplication of Bloch’s theorem in a spatially periodic Hamiltonian, the application of
Floquet theory to the time-dependent Schrödinger equationĤ(r, t)ψ(r, t) = i~∂ψ(r, t),∂t(D.2)

gives solutions of the formψα (r, t) = exp (−iα t/~) φα (r, t),where φα (r, t + T ) = φα (r, t).

(D.3)

This leads to the concept of quasi-energy α which is a real parameter and is unique
within energy band ~ω (identical to the Brillouin zone in the Bloch’s theorem).
The eigenvalues of the quasi-energy is determined by the Hermitian operator Ĥ (t)∂Ĥ (r, t) φα (r, t) = Ĥ(r, t) − i~φα (r, t) = α φα (r, t),
∂t

240

(D.4)

where the choice of the eigenvalues and eigenvectors are non-unique. Specifically, theFloquet modes

φα0 (r, t) = exp (imωt) φα (r, t),m ∈ integers,(D.5)

are also solutions to the eigenvalue problem in Eq. D.4 with shifted quasi-energyα0 = α + m~ω.

(D.6)

Consequently, I obtain a whole class of solutions indexed by α0 = (α, m) with m ∈ Z,
and the eigenvalues {α } can be mapped into the first Brillouin zone defined by−~ω/2 ≤ α < ~ω/2.
It is then convenient to extend the initial Hilbert space R of square integrable
functions to the composite Hilbert space R ⊗ T , where I include the Fourier space
T of the time-periodic functions with period T describing the “band” index m. In theinitial Hilbert space R, the inner product of two square integrable functions f (r) andg (r) is defined byˆ∞
hf |gi :=d3 r f ∗ (r) g (r) ,

(D.7)

−∞

which form a complete set of orthonormal basis if the functions ϕα (r) and ϕβ (r)satisfy the relationhϕβ (r) |ϕα (r)i := δα,β .

(D.8)

Meanwhile, in the Fourier space T , the temporal part is spanned by the orthonormal
set of Fourier vectors ht|mi = exp (imωt) which satisfy the inner product relation:
1(n, m) =T

ˆT
dt exp [i (m − n) ωt] = δm,n .
0

241

(D.9)

In the extended Hilbert space, the eigenvectors of the operator Ĥ (t) with indicesα0 = (α, m) and β 0 = (β, n) obey the orthonormality condition1
hφα0 |φβ 0 iR T :=
T

ˆT

ˆ∞
dt

0

d3 r φ∗α0 (r, t) φ∗β 0 (r, t) = δα0 ,β 0 = δα,β δm,n ,(D.10)−∞

and fulfill the completeness relation
XXαm

φ∗αm (r, t) φαn (r0 , t0 ) = δ (r − r0 ) δ (t − t0 ) .

(D.11)

Thus, the Floquet modes are mutually orthogonal for each instant of time and they
form a complete orthonormal set.
I now consider a total Hamiltonian given by the form1

Ĥ (r, t) = Ĥ0 (r) + V̂ (r, t) ,(D.12)

where V̂ (r, t) = V̂ (r, t + T ) is the periodic perturbation with monochromatic frequency ω = 2π/T and the unperturbed Hamiltonian has a complete orthonormal setof eigenfunctions:Ĥ0 (r) |ϕα (r)i = Eα0 |ϕα (r)i ,(D.13)

with orthonormal relation satisfying Eq. D.8. I then expand the wavefunction asψα (r, t) = exp (−iα t/~) φα (r, t) = exp (−iα t/~)∞ X
X

n=−∞ αcnα |ϕα (r)i e−inωt , (D.14)

where the expansion coefficients cnα are time-independent. Here I have two indices on
the cnα −coefficients:(1) α from the initial Hamiltonian Ĥ0 (r) which sets the Hilbert space R, and1Without losing any generality, I define our time t = 0 when the overall Hamiltonian is describedby Ĥ0 (r) and its eigenvectors set the completeness relation in the initial Hilbert space R.

242

(2) n from the Fourier expansion of the time-periodic perturbation V̂ (r, t).

To solve for the quasi-energies of the periodic Hamiltonian, I use the matrix (linear
algebra) approach. Insertion of the expanded wavefunction in Eq. D.14 into the
Schrödinger equation in Eq. D.2 gives∞ XhX

n=−∞ αiĤ(t) − α − n~ω cnα |ϕα (r)i e−inωt = 0.

(D.15)´T
Next, I multiply Eq. D.15 by hϕβ (r)| eimωt and integrate over (1/T ) 0 dt to obtain:
∞ X
X



hϕβ | 1Tn=−∞ α

∞ X
X





ˆT

dt Ĥ(t)ei(m−n)ωt |ϕα i − (α + n~ω) δmn δαβ  cnα = 0,0

 Eα0 − n~ω δαβ δmn + hϕβ | 1Tn=−∞ αˆT
0(D.16)

dt Ŵ (t)ei(m−n)ωt |ϕα i cnα = β cm
β . (D.17)

The above form in Eq. D.17 gives the explicit matrix elements for the eigenvalue
problem to find the quasi-energies.
For a perturbation with harmonic time dependence, i.e. Ŵ (t) = Ŵ0 cos (ωt), theonly surviving indices in the integral term of Eq. D.17 are (m − n) = ±1. The
eigenvalue problem in Eq. D.17 accordingly can be written in the matrix form as:
..
.



..

..
.

.
..



 .







m = −1 
E0 + ~ωI W
0





Φ = α Φα ,
m=0 
W
E0
W
··· 
 ···
 α





m=1 
0
W E0 − ~ωI




..
.
.
.
.
.
.
.
.
.
.
· · · n = −1 n = 0 n = 1 · · ·243(D.18)

T−1 0
1where I have defined the eigenvectors as Φα = (. . . , c−2
α , cα , cα , cα . . .) . The diagonal

matrix E0 represents the eigenvalues of the unperturbed Hamiltonian and W represents the off-diagonal coupling with its matrix elements is given by hϕβ | Ŵ0 |ϕα i /2.
Finally, I give an example of Floquet formalism to calculate the quasi-energies ofa time-periodic Hamiltonian. Here I consider our engineered Hamiltonian to realize aspin-1 spin-orbit coupling system (see Subsection 5.4.3). The Hamiltonian operatorof our system after rotating-wave approximation consists of the terms

Ĥ0 → E0(q − 2kR )2 quad
+

2m
~

2
= ~ 
0


0


0

0

2

q
2m
0



0
(q + 2kR )2 quad+2m~

 0 Ω1 0 

1 


Ŵ (t) → W = √  Ω1 0 Ω1  ,2
0 Ω1 0 , (D.19)


(D.20)

where m is the atomic mass, kR is the single-photon recoil momentum, q is the quasimomentum space, quad is the Zeeman quadratic shift, and Ω1 is the Raman couplingstrength. Substitution of Eqs. D.19-D.20 into Eq. D.18 gives me the explicit form of
the eigenvalue problem to be solved. I then solve the quasi-energies of the spin-orbitcoupled (SOC) Hamiltonian  (q) as a function of q.
Figure D.1 displays the quasi-energy dispersion relation (q) of the spin-1 SOC
Bose gases, where I set the energy scale in the units of ER = ~kR2 /2m. The frequencyof the time-periodic perturbation is equal to the quadratic Zeeman shift, i.e ~ω =12.7ER , in our system. Here I consider the case where the coupling strength isΩ1 = ER . I observe the band diagram to be repeating itself in each Brillouin zone
with a period of ~ω, as bounded by the shaded region in Fig. D.1.

244

quasienergy  (ER )

20
10
0
−10
−2

−1

0

1

2

quasimomentum q (kR )Figure D.1. Quasi-energy dispersion relation (q) of spin-1 SOC
Hamiltonian with coupling strength Ω1 = ER and periodic driving~ω = 12.7ER .

245

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