ABSTRACT
Title of dissertation: TOPOLOGICAL DISPERSION RELATIONS IN
SPIN-ORBIT COUPLED BOSE GASES
Ana Valde?s-Curiel,
Doctor of Philosophy, 2019
Dissertation directed by: Professor Ian Spielman
Joint Quantum Institute,
National Institute of Standards and Technology and
the University of Maryland College Park
Quantum degenerate gases have proven to be an ideal platform for the simu-
lation of complex quantum systems. Due to their high level of control it is possible
to readily design and implement systems with effective Hamiltonians in the labora-
tory. This thesis presents new tools for the characterization and control of engineered
quantum systems and describes their application in the realization of a topological
system with Rashba-type spin-orbit coupling.
The underlying properties of these engineered systems depend on their un-
derlying energies. I describe a Fourier transform spectroscopy technique for char-
acterizing the single particle spectrum of a quantum system. We tested Fourier
spectroscopy by measuring the dispersion relation of a spin-1 spin-orbit coupled
Bose-Einstein condensate (BEC) and found good agreement with our predictions.
Decoherence due to uncontrolled fluctuations of the environment presents fun-
damental obstacles in quantum science. I describe an implementation of continuous
dynamical decoupling (CDD) in a spin-1 BEC. We applied a strong radio-frequency
(RF) magnetic field to the ground state hyperfine manifold of Rubidium-87 atoms,
generating a dynamically protected dressed system that was first-order insensitive
to changes in magnetic field. The CDD states constitute effective clock states and
we observed a reduction in sensitivity to magnetic field of up to four orders of magni-
tude. We additionally show that the CDD states can be coupled in a fully connected
geometry and thus enable the implementation of new models not possible using the
bare atomic states.
Finally, I describe a new realization of Rashba-type SOC using Raman coupled
CDD states. Our system had non-trivial topology but no underlying crystalline
structure that yields integer valued Chern numbers in conventional materials. We
validated our procedure using Fourier transform spectroscopy to measure the full
dispersion relation containing only a single Dirac point. We measured the quantum
geometry underlying the dispersion relation and obtained the topological index using
matter-wave interferometry. In contrast to crystalline materials, where topological
indices take on integer values, our continuum system reveals an unconventional half-
integer Chern number.
TOPOLOGICAL DISPERSION RELATIONS IN
SPIN-ORBIT COUPLED BOSE GASES
by
Ana Valde?s Curiel
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2019
Advisory Committee:
Professor Alicia Kollar, Chair
Professor Gretchen Campbell
Professor Mohammad Hafezi, Dean?s representative
Professor Norbert Linke
Professor Trey Porto
Professor Ian Spielman, Advisor
?c Copyright by
Ana Valde?s Curiel
2019
Preface
A few weeks after I had started writing this thesis Ian came to me and asked
if I had learned all the physics I wish I had known while I was doing my PhD. I felt
a bit puzzled. I had been operating the lab, analyzing data and writing papers for
a while, of course I already knew the physics relevant to my research! It finally hit
me when I was writing the introductory chapters how many subtleties I had missed
and how much I still didn?t know. As experimental physicists being in the lab can
give us some physical intuition and a sense of understanding but sometimes that is
not enough. This was a very striking and unexpected side effect of the thesis and I
invite experimentalists reading this to challenge their lab intuition.
In the end, I actually found it very enjoyable to look back at the history of our
field and to put my research into a bigger context. I never thought the words thesis
and enjoyable could go well together! My advice for a graduate student starting to
write a thesis is try to enjoy the ride, it will definitely be stressful and overwhelming
but it is also a great learning opportunity.
A PhD thesis is not just a service for oneself but also a service to other students
and new members of the lab. I had this in mind during the process of writing and
I hope that future students find this document helpful.
ii
For my parents and in the
memory of Dr. Ernesto Valde?s Krieg
iii
Acknowledgments
I was very lucky to be surrounded by an amazing group of people throughout
my PhD. None of this would have been possible without them and I feel infinitely
lucky and grateful for that.
I would like to start by thanking my advisor Dr. Ian Spielman for giving
me the opportunity of joining his research group. Ian is one of the most brilliant
scientists I have had the pleasure to meet and I was very fortunate to be able to
work and learn with him. I am very grateful for all of his support, teaching, and for
always being so patient and good-humored. Ian?s unique advising style combining
top-quality science with cheerful and laid back interactions makes the excitement
for physics contagious!
I was exposed to a cold atoms lab for the first time when I spent a summer
as an undergrad working in the cavity QED lab of Dr. Luis Orozco. I thank him
for giving me this opportunity that would later define the course of my PhD. He is
a great mentor who cares very deeply about his students. I am extremely grateful
for his support, for all the delicious meals and bread, the concerts and plays, and
the good stories. I also thank him for carefully reading my thesis and providing
comments.
I would like to thank Dr. Bill Phillips for always being warm and welcoming
and for taking the time to participate in ?paper torture? sessions with me. I learned
a lot of physics (and history of physics) from these interactions. I would also like to
thank him for being an endless source of chocolate.
iv
My research was performed at the RbLi lab at UMD, where I was lucky that
my path crossed with many wonderful physicists and also now good friends. First I
would like to thank Daniel Campbell, Ryan Price and Andika Putra for welcoming
me to RbLi and patiently teaching me about the lab. A very special thank you
goes to our former postdoc Dimitri Trypogeorgos for taking me under his wing and
becoming my lab brother. He is a brilliant scientist who really helped to shape
my PhD. I am extremely grateful for all his physics and life teachings as well as
his friendship. I would also like to thank and acknowledge my current lab mates.
Qi-Yu Liang for her brutal honesty and for the physics discussions that tended to
start with confusion and end with enlightenment. I also thank her for the rabbit
pictures and videos. I admire her determination and her ability to stand up for what
she believes in. Mingshu Zhao and Junheng Tao for their enthusiasm and diligently
taking over the tasks of building a new BEC apparatus. I am also grateful for their
encouraging words when I was writing this thesis and my last paper. They are very
smart physicists and I am sure there will be great things to come in the new lab.
Our lab was also very fortunate to have a couple brilliant visitors. I thank
Nathan Lundblad for all his help when the lab was feeling a bit understaffed. I would
also like to thank Russ ?Mr. Broadband? Anderson for all the helpful experimental
diagnostics gadgets, the very thorough help on calculations and comments on papers,
and for always being in such a good mood. It was great fun to have him around.
Other great visitors not directly working with me but that had an impact on
my PhD include Marcell Gall, Pierre Dussarat, Max Schemmer, Daniel Ohl de Melo,
and Pierre Bataille. I thank them for their friendship and for the good times we
v
shared.
I have been away from the lab many months now and instead spending a great
deal of time in my office. I would like to thank my office mates for making it all
more fun. Hector Soza Mart??nez for being absolutely ridiculous. His sense of humor
reminds me that we shouldn?t take ourselves too seriously. Joao ?the champ? Braz
for being a great theory buddy. I really enjoyed our physics discussions and am
thankful for all the thorough comments on the Rashba paper. I am also thankful
that he made me leave my desk to get a beer when he thought I needed a break
from working too hard. Dina Genkina for always listening and offering encouraging
words, also for the constant flow of Kit Kats.
I was very lucky to find a great community of Latin American physicists (and
non-physicists) at UMD (thanks again Luis) who became some of my closest friends
and kind of a family away from home. Having them around made everything more
fun and helped me stay sane throughout my PhD. A big thank you to Paco, Ale,
Pablo, Dalia, David, Israel, Kristel, Andres, Hector, Maggie, Carlos, Monica and
Carla for their friendship. A special shout out to Dalia who has been like my
academic sister ever since we both joined Luis? labs. Having a good friend like her
sharing this PhD path really makes things better.
Outside of the university, I was also very fortunate to run into a wonderful
group of people, especially those affiliated to the extended International House of
Physicists (IHoP) family: Juraj, Kanu, Yigit, Shanon, Anna, Marko, Aydn, Seyit,
Daniel, Rory, Dina, Chris, Ankit, Ray. Having them around made life more fun and
enjoyable!
vi
Besides the people I met in grad school, I had a network of good friends back
home who have always been very supportive throughout the years. I thank Andrea,
Adriana, Elsa, Guillermo, and Bernardo for always cheering me and making me feel
like I was doing something cool.
A common denominator in many aspects of my PhD has been my partner
Paco. He has always been there for me through the best and worst times and I
am very grateful for that. Paco is a very kind and caring person and also a great
physicist. I also thank him for listening, making me laugh, and for all his help
with physics and non-physics issues. I was very lucky to find a partner, friend, and
colleague in him.
I was deeply touched by the support I received for my thesis defense. Friends
getting together to cook delicious tacos, surprise visits from Europe, family members
visiting from various places, emails and text messages from friends and colleagues
with good wishes. A special thanks to Kanu for also being my presentation coach.
Her advise and ?cups of tea? really helped me calm down when the nerves were kicking
in hard. I don?t want to sound repetitive but I am very lucky to be surrounded by
such a nice community.
Finally and most importantly, I would like to thank my family. My parents
always made my education a priority and none of this would have been possible
without their unconditional support. I also thank my brothers for their support
and all the good (and bad) times that we have shared. They try hard to annoy me
sometimes but I wouldn?t change them for anyone else.
vii
Table of Contents
Preface ii
Dedication iii
Acknowledgements iv
List of Tables xii
List of Figures xiii
List of Abbreviations xvi
1 Introduction 1
1.1 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Overview of Bose-Einstein condensation 7
2.1 Bose-Einstein condensation of an ideal gas . . . . . . . . . . . . . . . 8
2.1.1 Critical temperature . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Condensate fraction . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.3 Bose gas in a harmonic trapping potential . . . . . . . . . . . 12
2.2 Bose-Einstein condensation with atomic interactions . . . . . . . . . . 13
2.2.1 Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Multiple component BECs . . . . . . . . . . . . . . . . . . . . 17
2.2.3 Thomas-Fermi approximation . . . . . . . . . . . . . . . . . . 18
2.3 Density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Manipulation and detection of ultra-cold atoms 25
3.1 Electronic structure of 87Rb . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Interaction between atoms and magnetic fields . . . . . . . . . . . . . 28
3.2.1 Magnetic trapping . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Interaction between atoms and electric fields . . . . . . . . . . . . . . 32
3.3.1 Scalar polarizability . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3.1.1 Optical trapping . . . . . . . . . . . . . . . . . . . . 36
viii
3.3.2 Vector polarizability and effective magnetic fields . . . . . . . 38
3.3.2.1 Raman coupling . . . . . . . . . . . . . . . . . . . . 39
3.3.2.2 Spin-orbit coupling . . . . . . . . . . . . . . . . . . . 42
3.4 Detection: Resonant absorption imaging . . . . . . . . . . . . . . . . 43
3.4.1 High intensity absorption imaging . . . . . . . . . . . . . . . . 46
3.5 Coherent manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.1 Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.5.2 Adiabatic rapid passage . . . . . . . . . . . . . . . . . . . . . 51
3.5.3 Magnetic field stabilization with microwave assisted partial
transfer absorption imaging . . . . . . . . . . . . . . . . . . . 53
3.5.4 Ramsey interferometer . . . . . . . . . . . . . . . . . . . . . . 55
3.5.5 Floquet theory . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 Making BECs in the Rubidium Lithium apparatus 62
4.1 Overview of the RbLi apparatus . . . . . . . . . . . . . . . . . . . . . 63
4.1.1 Laser systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.2 Magnetic field control . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Experimental sequence to make BECs . . . . . . . . . . . . . . . . . . 67
4.3 Upgrades to the RbLi Machine . . . . . . . . . . . . . . . . . . . . . 71
4.3.1 Master laser system . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.2 Raman laser system . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.3 High power RF system . . . . . . . . . . . . . . . . . . . . . . 74
4.3.4 6.8 GHz microwave system . . . . . . . . . . . . . . . . . . . . 77
4.4 Computer control and data acquisition . . . . . . . . . . . . . . . . . 79
5 Fourier Transform Spectroscopy 83
5.1 Operating principle of Fourier spectroscopy . . . . . . . . . . . . . . . 84
5.2 Measuring the SOC dispersion with Fourier transform spectroscopy . 87
5.2.1 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2.2 Tunable SOC . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2.3 Application of Fourier spectroscopy . . . . . . . . . . . . . . . 92
5.2.4 Effective mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2.5 Measured dispersions . . . . . . . . . . . . . . . . . . . . . . . 99
6 Synthetic clock transitions through continuous dynamical decoupling 103
6.1 Basic principles of CDD . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.2 CDD of a spin-1 system . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3 The |xyz? states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.3.1 State decomposition . . . . . . . . . . . . . . . . . . . . . . . 107
6.3.2 Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.3.3 Transition matrix elements . . . . . . . . . . . . . . . . . . . . 109
6.4 |xyz? state preparation . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.5 Initial characterization of ? . . . . . . . . . . . . . . . . . . . . . . . 113
6.6 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.7 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
ix
6.7.1 Optimal response to noise . . . . . . . . . . . . . . . . . . . . 117
6.8 Driving dressed state transitions . . . . . . . . . . . . . . . . . . . . . 118
6.9 Concatenated CDD . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7 Topological order in quantum systems 128
7.1 Topology in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.2 Topological order in condensed matter . . . . . . . . . . . . . . . . . 131
7.3 Berry phase and Berry curvature . . . . . . . . . . . . . . . . . . . . 133
7.3.1 Aharonov-Bohm phase as an example of a Berry?s phase . . . 136
7.3.2 Chern number . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.4 The bulk-edge correspondence principle . . . . . . . . . . . . . . . . . 140
7.5 Example: two-level model . . . . . . . . . . . . . . . . . . . . . . . . 140
7.6 Monopoles and Dirac strings . . . . . . . . . . . . . . . . . . . . . . . 142
7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8 Unconventional topology in a quantum gas with Rashba SOC 145
8.1 Rashba spin-orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . 146
8.2 Rashba SOC for neutral atoms . . . . . . . . . . . . . . . . . . . . . . 148
8.3 Experimental implementation of Rashba SOC . . . . . . . . . . . . . 152
8.3.1 Raman coupling the |xyz? states . . . . . . . . . . . . . . . . 153
8.3.1.1 Floquet and off-resonant coupling effects . . . . . . . 155
8.3.1.2 Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . 156
8.3.2 Measuring quasimomentum distributions . . . . . . . . . . . . 157
8.3.2.1 Correcting shears from Stern-Gerlach gradient . . . . 159
8.4 Fourier spectroscopy of the Rashba dispersion . . . . . . . . . . . . . 161
8.5 Quantum state tomography with Ramsey interferometer . . . . . . . 163
8.5.1 Wave function evolution in Ramsey interferometer . . . . . . . 166
8.5.2 Combining phases from different measurements . . . . . . . . 170
8.5.3 Measuring the topological index . . . . . . . . . . . . . . . . . 171
8.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
9 Conclusions and outlook 175
A The good, the bad and the ugly of RbLi 178
A.1 The good . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
A.2 The bad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
A.3 The ugly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
B New apparatus 187
B.1 Water cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
B.2 Electrical installation . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
B.3 Coil winding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
B.4 Rb source and ?oven? . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
B.5 Table enclosures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
x
C Full derivation of the Raman coupled |xyz? states 195
Bibliography 205
xi
List of Tables
4.1 List of AOM driver components . . . . . . . . . . . . . . . . . . . . . 74
4.2 6.8 GHz microwave system components . . . . . . . . . . . . . . . . . 78
4.3 Camera technical specifications . . . . . . . . . . . . . . . . . . . . . 81
6.1 Summary of CDD energies . . . . . . . . . . . . . . . . . . . . . . . . 122
xii
List of Figures
1 The Bose-Einstein distribution . . . . . . . . . . . . . . . . . . . . . . 9
2 The Thomas-Fermi density distribution . . . . . . . . . . . . . . . . . 20
3 Momentum distribution of atoms near Tc . . . . . . . . . . . . . . . . 24
1 87Rb level structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Zeeman splitting of the 52S1/2 manifold of 87Rb . . . . . . . . . . . . 30
3 Raman coupling with two-photon transitions . . . . . . . . . . . . . . 40
4 Electric realizabilities and scattering rates as a function of wavelength 41
5 The Beer-Lambert law . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6 Resonant absorption imaging . . . . . . . . . . . . . . . . . . . . . . 45
7 High intensity absorption image . . . . . . . . . . . . . . . . . . . . . 47
8 The Rabi cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
9 Adiabatic rapid passage . . . . . . . . . . . . . . . . . . . . . . . . . 53
10 Magnetic field stabilization using microwave assisted PTAI . . . . . . 55
11 A Ramsey interferometer . . . . . . . . . . . . . . . . . . . . . . . . . 57
12 Magnetic phases of a spin-1 SOC system . . . . . . . . . . . . . . . . 61
1 The RbLi vacuum system . . . . . . . . . . . . . . . . . . . . . . . . 64
2 Laser cooling frequencies . . . . . . . . . . . . . . . . . . . . . . . . . 66
3 Diagram of the RbLi apparatus . . . . . . . . . . . . . . . . . . . . . 67
4 Magnetic coil geometry in the RbLi apparatus . . . . . . . . . . . . . 68
5 Master laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6 Ti:Sapphire laser output as a function of pump power . . . . . . . . . 73
7 Optical layout of Raman and optical lattice lasers . . . . . . . . . . . 75
8 AOM driving electronics . . . . . . . . . . . . . . . . . . . . . . . . . 76
9 High power RF system . . . . . . . . . . . . . . . . . . . . . . . . . . 77
10 Impedance matching of high power RF antenna . . . . . . . . . . . . 78
11 Schematic of 6.8 GHz microwave system. . . . . . . . . . . . . . . . . 80
12 Comparison of probe images for Flea3 and Mako cameras . . . . . . . 82
1 Operating principle of Fourier spectroscopy . . . . . . . . . . . . . . . 85
2 Experimental setup for engineering a tunable system with equal con-
tributions of Rashba and Dresselhaus-type spin-orbit coupling . . . . 88
xiii
3 Floquet quasi-energy spectrum of a three-level system with spin-orbit
coupling and periodic coupling strength . . . . . . . . . . . . . . . . . 90
4 Time evolution and Fourier transforms of a SOC system . . . . . . . 94
5 Power spectral density of the time dependent occupation probability
for each state in the measurement basis for three coupling regimes . . 95
6 Dipole oscillations of a spin-orbit coupled BEC in a dipole trap . . . 98
7 Converting energy differences into absolute energies . . . . . . . . . . 100
8 Spin-dependent SOC dispersion for three different coupling regimes . 101
1 Implementing CDD. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2 State decomposition of the |xyz? states . . . . . . . . . . . . . . . . . 107
3 Experimental CDD protocol . . . . . . . . . . . . . . . . . . . . . . . 112
4 Calibration of ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5 Spectroscopy of the |xyz? states . . . . . . . . . . . . . . . . . . . . . 116
6 The |z? ? |x? transition as a function of ?RF . . . . . . . . . . . . . 117
7 Optimal response of the zx transition . . . . . . . . . . . . . . . . . . 118
8 Coherent driving of dressed state transitions . . . . . . . . . . . . . . 119
9 Loss of contrast of coherent oscillations . . . . . . . . . . . . . . . . . 120
10 Concatenated CDD protocol . . . . . . . . . . . . . . . . . . . . . . . 123
11 Concatenated CDD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
1 The Gaussian curvature of a surface . . . . . . . . . . . . . . . . . . . 130
2 The quantum Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . 132
3 The Aharonov-Bohm experiment . . . . . . . . . . . . . . . . . . . . 137
4 Graphical representation of Chern number . . . . . . . . . . . . . . . 142
5 Graphical representation of Chern number . . . . . . . . . . . . . . . 143
6 Different topological invariants . . . . . . . . . . . . . . . . . . . . . . 144
1 Rashba dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . 147
2 The Rashba ring coupling . . . . . . . . . . . . . . . . . . . . . . . . 149
3 Rashba ring coupling eigenenergies . . . . . . . . . . . . . . . . . . . 150
4 Rashba ring coupling ground state dispersion . . . . . . . . . . . . . . 151
5 Experimental setup for implementing Rashba spin-orbit coupling . . . 153
6 Effect of neighboring Floquet manifolds on Rashba dispersion . . . . 156
7 Lifetime of Raman dressed states . . . . . . . . . . . . . . . . . . . . 157
8 Mapping momentum into quasimomentum . . . . . . . . . . . . . . . 159
9 Correcting shears in momentum distribution from magnetic field gra-
dients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
10 Fourier spectroscopy of a Rashba SOC syste: Experimental protocol
and representative data . . . . . . . . . . . . . . . . . . . . . . . . . . 162
11 Sample power spectral densities for all states used in the measurement
of the Fourier spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 164
12 Predicted dispersion relation and Fourier spectrum . . . . . . . . . . 165
13 Experimental protocol for three-arm Ramsey interferometer (not to
scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
xiv
14 Dragging the Dirac point through the atoms . . . . . . . . . . . . . . 168
15 Representative data of the probabilities measured after the three out-
put ports of Ramsey interferometer . . . . . . . . . . . . . . . . . . . 170
16 Weight arrays used for combination of interferometer phases . . . . . 171
17 Topological invariants obtained from quantum state tomography . . . 172
1 New MOSFET bank . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
2 The RbLi oven chamber . . . . . . . . . . . . . . . . . . . . . . . . . 181
3 Water cooling manifold schematic . . . . . . . . . . . . . . . . . . . . 183
1 Water cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
2 Configuration of the welding cables connecting the power supplies to
equipment in the lab. . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
3 Electrical installation . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
4 Lathing of magnetic transport coils . . . . . . . . . . . . . . . . . . . 192
5 Rubidium oven assembly . . . . . . . . . . . . . . . . . . . . . . . . . 193
1 Raman laser layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
2 Raman coupling of the |xyz? states . . . . . . . . . . . . . . . . . . . 200
xv
List of Abbreviations
Abbreviation Full meaning
BEC Bose-Einstein condensate
SOC spin-orbit coupling
RbLi Rubidium-Lithium
DD dynamical decoupling
CDD continuous dynamical decoupling
CCDD concatenated continuous dynamical decoupling
TOF time of flight
OD optical depth
TF Thomas-Fermi
MOT magneto optical trap
RF radio frequency
ARP adiabatic rapid passage
RWA rotating wave approximation
PTAI partial transfer absorption imaging
PSD power spectral density
TTL transistor-transistor logic
CCD charge-coupled device
AOM acousto-optic modulator
SG Stern-Gerlach
DDS direct digital synthesizer
1D one-dimensional
2D two-dimensional
3D three-dimensional
DC direct current
VNA vector network analyzer
FWHM full width at half maximum
NIST National Institute of Standards and Technology
JQI Joint Quantum Institute
xvi
Chapter 1: Introduction
Bose statistics were first developed in 1924 to describe the quantum behavior
of photons [1] and generalized by Einstein to include massive particles [2]. Today,
we routinely produce atomic Bose-Einstein condensates (BECs) in the laboratory
and even use them as a platform for the analog simulation of complex quantum
systems. The field has certainly come a long way!
A connection between Bose-Einstein condensation and the onset of superfluid-
ity in liquid 4He was first made in 1938 [3]. However, due to the strong interactions,
the occupancy of the lowest energy state is dramatically reduced, leading to the
search of this phase for weakly interacting Bose gases. Bose-Einstein condensation
in dilute atomic gases was observed for the first time in 1995 in vapors of 87Rb [4],
23Na [5] and 7Li [6]. The dilute nature of this gases required cooling down to tem-
peratures never achieved before and it wasn?t until the experimental development
of laser cooling and trapping together with evaporative cooling that bosonic atoms
were cooled the critical condensation temperature1. The experimental realization of
this new phase of matter opened new possibilities for studying macroscopic quan-
tum phenomena such as the propagation of collective modes [8, 9] and interference
1For a more in depth story of the BEC field I highly recommend reading [7].
1
of coherent matter waves [10] and jump-started an ever-growing field of research.
For this achievement, Eric Cornell, Carl Wieman, and Wolfgang Ketterle received
the 2001 Nobel Prize in physics.
It has been more than 20 years (and many more in the learning of atomic
physics) and the field of quantum degenerate gases has expanded to include degen-
erate Fermi gases [11] and spinor gases [12]. As the field continues to grow, new
control and detection techniques are being constantly developed, enabling the use
of BECs and ultracold atomic systems in general not just as an object of study
by themselves but as tools for a wide range of scientific endeavors, from precision
measurements [13] to the analog simulation of complex systems.
Quantum degenerate gases are an ideal platform for quantum simulation [14].
A straightforward example comes from the use of optical lattices, where the periodic
potential imparted by standing waves of light serves as an analogue to the crystal
structure in a solid. Perhaps the first iconic realization of a quantum simulation was
the study of the Bose-Hubbard model in three-dimensional optical lattices [15], the
bosonic analogue of a model which is believed to be relevant to high-Tc supercon-
ductors.
The development of light-induced gauge-fields [16] has been another important
milestone in the field of quantum simulation. Such fields can be used to mimic
the effect of magnetic [17, 18] and electric [19] fields with potential applications to
the realization of quantum Hall materials with large magnetic fluxes [17, 20] and
fractional quantum Hall states [21]. Furthermore, light-induced gauge fields can
be used to engineer spin-orbit coupling (SOC) interactions [22] as those present in
2
two-dimensional materials, a necessary ingredient for the spin quantum Hall effect
and certain kinds of topological insulators [23].
The precise level of control and tunability of ultracold atomic systems allow
us to readily implement important physical models in the laboratory. Furthermore,
we can go beyond conventional materials existing in nature, and create new ex-
otic atomic ?materials?, with interaction-dominated or topologically non-trivial band
structures that can help deepen our understanding of the physical consequences of
these effects on materials.
This thesis focuses on the development of new tools for the characterization
and control of engineered quantum systems and their application for creating and
characterizing a topological system with Rashba-type [24] SOC.
The creation of new engineered materials requires the ability to characterize
their single-particle energies. We developed a Fourier transform spectroscopy tech-
nique which allows us to probe the single particle spectrum, and verify our quan-
tum engineering, by only looking at the quantum coherent evolution. We have used
Fourier transform spectroscopy to characterize the spectra from variety of engineered
quantum systems with SOC [25,26] and fractional period adiabatic superlattices [27].
Atomic systems are susceptible to coupling to the environment, leading to
undesired effects such as the loss of coherence. In particular laboratories such as
ours greatly suffer from noise in ambient magnetic fields and go through great ef-
forts to diminish their effects. We implemented continuous dynamical decoupling
(CDD) on a set of internal atomic states which renders them first order insensitive
to magnetic field changes, effectively turning them into clock states. These CDD
3
states are a robust basis for performing experiments, however, the most important
feature is the fact that they can be coupled in a fully connected geometry unlike
the hyperfine atomic states. The CDD states gave us access to new matrix elements
which were essential for the engineering of Rashba SOC [26] as well as other novel
lattice systems [27] not presented in this thesis.
Rashba-type SOC is a relativistic effect present in two-dimensional materials
subject to an intrinsic out-of-plane electric field. The dispersion relation of Rashba
systems is characterized by a Dirac point at zero momentum and an infinitely de-
generate ring-shaped ground state. The presence of the Dirac point gives rise to
non-trivial topology while the ring-shaped single particle ground state opens new
possibilities of studying strongly correlated phases in the presence of interactions
for systems of both fermions and bosons [28?30]. Using ultracold atomic systems to
engineer Rashba-type SOC has been a longstanding goal [22].
In the last part of this thesis we describe a new experimental realization of
Rashba SOC using Raman coupled CDD states. Our system is certainly condensed
matter inspired. However, part of the beauty of engineered quantum systems is
we can depart from conventional materials, for example, by considering a system
with Rashba SOC but without an underlying crystalline structure. We studied the
quantum state geometry of such a system with non-trivial topology in a Brillouin
zone that extends to infinity. We find that our system is characterized by half-
integer valued invariants, defying our conventional understanding of the topology of
Bloch bands.
4
1.1 Thesis overview
This thesis describes both the standard experimental control and measurement
techniques used to create BECs of 87Rb as well as new techniques developed in our
laboratory for the engineering of Rashba SOC.
Chapter 2 describes the basic theory of Bose-Einstein condensation in dilute
gases. I focus on the properties of gases confined to harmonic potentials and their
density and momentum distributions as they are most relevant to the experiments
presented here.
Chapter 3 describes the properties of Alkali atoms and their interactions with
magnetic and electric fields which are used as standard tools for the creation, ma-
nipulation and detection of ultracold atomic systems.
Chapter 4 summarizes the experimental apparatus and mentions the most
important upgrades that were not reported previously.
Chapter 5 describes a Fourier transform spectroscopy technique that exploits
the relation between quantum coherent evolution and the underlying spectrum of a
system and that was used to characterize experiments described later in the thesis.
Chapter 6 describes an implementation of continuous dynamical decoupling
using a strong radio-frequency magnetic field that helped to both make our system
more robust against environmental magnetic field noise. The implementation of
CDD allows us to couple the internal states of the atoms in new ways that were not
possible before, opening the path for new kinds of quantum simulations described
in Chapters 8.
5
Chapter 7 presents concepts of topology in physics and its application to the
band theory of solids. These concepts will be important for a better understanding
of the topological properties of our Rashba spin-orbit coupled system.
Chapter 8 describes a new experimental realization of Rashba SOC using a
combination of laser beams that couple a set of CDD states. This system has a
topological dispersion relation but no underlying crystalline structure which allows
for topological invariants to take non-integer values.
Appendix A summarizes the best and worst aspects of the experimental ap-
paratus as a guide for future generations working in BEC labs.
Appendix B describes my work related to the design and construction of a
dual species apparatus for the production of BECs of 87Rb and 39K.
Appendix C shows the derivation of the full time-dependent Hamiltonian de-
scribing the Raman dressing of the RF dressed states (Chapter 6) used to generate
Rashba-type SOC (Chapter 8).
6
Chapter 2: Overview of Bose-Einstein condensation
Bose-Einstein condensate (BEC) is a quantum state of matter in which parti-
cles with integer valued spin all tend to occupy or ?condense? into the ground state.
In dilute gases, condensation occurs when the temperature of the system goes bellow
a critical temperature where quantum statistics become relevant.
BECs enable the study, both theoretically and experimentally, of macroscopic
quantum phenomena. There have been a number of fascinating experiments study-
ing the properties of these systems, from measuring interference fringes of a macro-
scopic wave function to studying collective effects such as the propagation of sound [7].
In our experiments, however, BECs are not the primary object of study, instead,
they are used as a platform enabling the simulation of analog physical systems.
In this Chapter I give an overview of Bose-Einstein condensation in dilute
atomic gases and I describe the properties most relevant to our experiments. I start
by describing the case of an ideal gas and then consider the effects of interactions
and trapping potentials as are present in our case. A reader interested in learning
about Bose-Einstein condensation in dilute gases in more depth should read [31]
and [32].
7
2.1 Bose-Einstein condensation of an ideal gas
At low temperatures and in thermodynamic equilibrium, the mean occupation
number of non-interacting identical bosons in the state with energy E is given by
the Bose-Einstein distribution
n( 1Ej) = ( ? ) 1 (2.1)e Ej ? /kBT ?
where T is the temperature, ? is the chemical potential (the energy cost of adding
or removing a particle) and kB is the Boltzmann constant. In the limit of large tem-
peratures the Bose distribution approximates the Maxwell-Boltzmann distribution
n(E ) ? e?(Ej??)/kTj (2.2)
which applies to classical, distinguishable particles. The chemical potential is de-
termined by the condition that the total number of particles N is equal to the sum
over all states in the distribution ?N = j n(Ej) and is a function of N and T . Ad-
ditionally, in order for n(Ej) to be positive definite we must have ? ? E0 where E0
is the energy of the ground state. From the Bose distribution, we can see that the
occupation number of the ground state is unbounded when ?? 0 as in Figure 2.1.
The number of particles occupying the excited states is bounded and when that
number is reached, the remaining particles can occupy the ground state and thus
Bose-Einstein condensation occurs.
8
4
? =0.5
? =0.75
3
? =1
2
1
0
0 1 2
E/kBT
Figure 1: The Bose-Einstein distribution. Occupation number as a function of
energy for different values of fugacity ? = exp(?/kBT ). Condensation occurs when
? = 0 (? = 1) and the occupation number in the ground state diverges.
2.1.1 Critical temperature
Bose-Einstein condensation can be understood in terms of the de Broglie waves
associated with particles. The thermal de Broglie wavelength is
(
2 2
)1/2
?h?
?th = (2.3)
mkBT
and it characterizes the spatial extension of the wave packet an individual particle
at temperature T . Condensation occurs when ?th becomes comparable with the
inter-particle separation n?1/3, where n = N/V and V is the volume. The quantity
n?3th is known as the phase space density which describes the number of particles
contained in a box with volume ?3th.
An analytical expression for the critical temperature at which atoms condense
can be derived using the Bose-Einstein distribution. For closely spaced energy lev-
9
n(E)
els (compared to kBT ) the sum representing the total number of particles can be
replaced by the integral ? ?
N = n(E)g(E)dE (2.4)
0
where g(E) is the density of states and g(E)dE corresponds to the number of avail-
able states with energy between E and E+dE. For a free particle in three dimensions
the density of states is
3/2
g( V mE) = ? 2E
1/2, (2.5)
2?2h?
and in general the density of states can be expressed as a power of energy g(E) =
C E??1? .
The integral in Equation 2.4 is not analytically solvable, however, we can make
the simplifying assumption ? = 0. The critical temperature Tc is determined by the
condition that all particles are in the excited states
N = ?Nex(Tc, ? = 0)?
= g(E)dE
0 eE/kBTc?? 1?
= ( )? x
??1
C? kBTc
0 ex ? 1
= c?(kBTc)??(?)?(?) (2.6)
?
where x = E/kBTc, ?(?) = ? x??1e?x0 dx is the Gamma function and ?(?) =?? ??
n=1 n is the Riemann zeta function. From Equation 2.6 we find that the critical
10
temperature for Bose-Einstein condensation is
( )1/?
kBTc =
N
. (2.7)
C??(?)?(?)
If we compute the phase space density for free particles in 3D with density
of states given by Equation 2.5 in combination with the expression for the critical
temperature (Equation 2.6) we find that when T = Tc
(3)
n?3th = ? 2 ? 2.612, (2.8)
the inter-particle spacing and the thermal wavelength are comparable. In order to
experimentally produce BECs, we deploy a combination of laser and evaporative
cooling techniques such that we can increase the density while minimizing the tem-
perature, maximizing the phase space density. The densities for BECs of Alkali
atoms typically range between 1013 and 1015 atoms/cm?3.
2.1.2 Condensate fraction
The total number of particles is given by N = N0 + Nex, where N0 is the
number of particles in the condensate. The number of particles in the excited state
will be given by the integral in Equation 2.4. For g(E) = C E??1? and ? > 0 the
integral converges, we can then evaluate the integral in Equation 2.6 for T < Tc and
11
get
Nex = c?((k
?
BT)) ?(?)?(?)?
= TN , (2.9)
Tc
since for T = Tc the total number N = Nex. The number of particles in the ground
state is
N0 = N [?Nex( ) ]?
= N 1? T (2.10)
Tc
2.1.3 Bose gas in a harmonic trapping potential
Our experiments are performed in optical dipole traps that can be described
as harmonic potentials
m ( )
V (r) = ?2 22 xx + ?
2 2 2 2
yy + ?zz (2.11)
The density of states for this case is
( ) = E
2
g E 2 , (2.12)2h? ?x?y?z
12
which corresponds to ? = 3 and C = (2h?33 ?x?y?z)?1. Using Equation 2.6, we find
that the transition temperature is
= h???N
1/3
k T 1/3B c (3)1 3 ? 0.94h???N (2.13)? /
where ?? = (? ? ? )1/3x y z is the geometric mean of the oscillation frequencies. Similarly
we find that the condensed fraction is
[ ( )3]
N0 = N 1?
T (2.14)
Tc
Condensates in harmonic traps have some striking features that will be further
explored in more detail in the following sections. The confining potential makes
the BECs both finite-sized and inhomogeneous which means that the BEC can be
observed both in momentum space and in coordinate space. Another consequence
of the inhomogeneity of these systems is the role of two-body interactions, which
gets enhanced and leads to noticeable effects in measurable quantities [33, 34] such
as interaction driven expansion when they are released from the confining potential.
2.2 Bose-Einstein condensation with atomic interactions
Even though atomic BECs are made from very dilute gases, the system is
far from being an ideal gas and a complete treatment requires taking into account
interactions.
The collisional properties of particles at low energies, such as cold atoms in
13
a condensate, are dominated by s-wave scattering which can be described in terms
of a single parameter the scattering length a that determines both the scattering
cross-section ? = 4?a2 and the phase shift of the scattered wave function.
The magnitude of the scattering length is determined by the interatomic in-
teraction potentials. For Alkali atoms at large distances, the two-body interactions
are dominated by an attractive Van der Walls interaction U(r) = ?C6/r6 that arises
from dipole-dipole interactions. At smaller distances, the attractive potential is re-
placed by a strong repulsive electron-exchange interaction U(r)??. This minimal
model captures the most important properties of the inter-atomic potential and can
be solved analytically [35].
If the range of the interaction is much shorter than the mean inter-atomic dis-
tance the interaction can be approximated by an effective pseudo-potential Ueff(r?
r?) such that ?
a = m 2 U
?
eff(r? r )dr (2.15)4?h?
which determines
2
U (r? r?) = 4?h? aeff ?(r? r?) = g?(r? r?). (2.16)
m
This approximation allows us to model the scattering between atoms as a hard-
sphere scattering process instead of considering the more complicated inter-atomic
potentials. The sign of the scattering length determines the attractive or repulsive
nature of the interactions and it plays an important role in the experimental produc-
14
tion of BECs as it determines the rate at which atoms thermalize during evaporative
cooling.
2.2.1 Gross-Pitaevskii equation
The full Hamiltonian describing N identical bosons with contact interactions
is
?[N p2 ] ?
H? = i2 + V (ri) + g ?(ri ? rj), (2.17)i=1 m i<j
where V (r) is an external potential and pi = ?ih??i is the individual momentum of
each atom. We consider a normalized eigenstate of the Hamiltonian ?(r1, r2, ..., rN)
that satisfies the Schro?dinger equation. We can simplify this state by taking a mean
field approach; we assume that the system has undergone condensation so that the
majority of the particles share the same single particle ground state ?(r), then the
wavefunction can be approximated by a symmetrized product
?N
?(r1, r2, ..., rN) = ?(ri), (2.18)
i=1
where ? is normalized to unity. The energy of the state from Equation 2.18 is given
by the expectation value
?
E = ??? H?? d[r 2 ]
= h? (N ? 1)N ??(r) ?2 ?
2 + V (r) + 2 g|?(r)|
2 ?(r)dr, (2.19)
m
15
where N(N ? 1)/2 ? N2/2 counts the number of terms in the interaction energy.
We introduce the wave function of the condensate ?(r) = N1/2?(r), and inserting it
in Equation 2.19 makes the N factors cancel out. The optimal form of ? should min-
imize the energy, introducing a Lagrange multiplier ?, subject to the normalization
?
condition N = |?(r)|2 dr
( ? ) [ 2 ]?
?( ) E ? ? |?|
2 h?dr = ?2 ?
2 + V (r) + g|?(r)|2 ? ? ?(r) = 0, (2.20)
?? r m
and we find that the condensate wave function obeys a non-linear Schro?dinger equa-
tion known as the Gross-Pitaevskii (GP) equation
[
h?2
]
? ?22 + V (r) + g|?(r)|
2 ?(r) = ??(r), (2.21)
m
where ? plays the role of the chemical potential. The dynamics of the condensate
will similarly be described by the time-dependent GP equation
[ ]
? 2
ih? ?(r, t) = ? h?2 + V (r) + g|?(r, t)|
2 ?(r, t) (2.22)
?t m
The GP equation describes the relevant phenomena associated with the prop-
agation of collective excitations and the expansion of the condensate when released
from a trap. The crucial assumption when deriving these equations is the mean
field approximation. This should be valid for dilute BECs where the condensate
fraction is close to unity. The excitations of the system can be described by a set of
equations similar to those of classical hydrodynamics derived from the GP equation
16
or alternatively using Bogoliubov theory for weakly interacting bosons [31].
2.2.2 Multiple component BECs
So far the discussion has been limited to single component BECs but most
of our experiments are performed using a combination of multiple atomic inter-
nal states. In general, for a multiple component condensate the scattering lengths
characterizing the interactions depend on the internal states of the incoming and
outcoming scattering channels. Two spin-f 1 particles colliding particles will be char-
acterized by 2f scattering lengths aF . For bosons the total spin F takes even values
and in particular for 87Rb atoms in the f = 1 hyperfine ground state there are two
scattering lengths a0 and a2 corresponding to the two-particle total angular momen-
tum states of F = 0 and F = 2 respectively. The values of scattering lengths are
a0 = 101.8a0 and a2 = 100.4a0 [12] where a0 = 5.29 ? 10?11 is the Bohr radius.
From the scattering lengths we can calculate two interaction coefficients
= 4?h?
2 a0 + 2a 4?h?22
c0 3 = 100.84a0m m
4?h?2 a0 ? a2
c2 = 3 ? ?4.7? 10
?3c0. (2.23)
m
Here c0 represents a spin-independent interaction strength that depends only on the
total density while c2 is a spin-dependent energy that is relevant only where there
is non-zero density of both atoms in mF = ?1 and is much smaller than the spin-
1Here I use the symbol f to denote the angular momentum of the individual particles and F
to denote the total angular momentum of the two particles.
17
independent energy. Similar to the case of single component BECs, the dynamics of
multiple component BECs is governed by a spinor GP equation (see [12, 31]). The
spin-dependent interaction strength gives rise to processes like coherent spin-mixing
oscillations and domain formation and coarsening which was previously studied in
the our lab [36].
The time scale at which interactions become relevant is set by the interaction
energies n|ci|. The most noticeable effect of interactions in our system is the density
profile of the condensate and its anisotropic expansion after it is released from a
trap which I will describe in the following sections. For the typical densities and
timescales of our experiments as well as the relatively high magnetic fields that we
operate at, we do not observe noticeable effects from interactions in the dynamics of
the system and in the remaining chapters I will describe the dynamics of the BEC
using single particle physics (i.e. the regular time-dependent Schro?dinger equation).
2.2.3 Thomas-Fermi approximation
For systems with large N , the interaction term in the GP equation is very large
compared to the kinetic energy2. As the kinetic energy becomes less important we
enter the Thomas-Fermi (TF) regime where the energy of the system is given only by
the external potential and the mean field energy and the GP equation is considerably
simplified [ ]
V (r) + g|?(r)|?(r)|2 ?(r) = ??(r). (2.24)
2It can be shown that the ratio of kinetic energy to interactions scales as N?4/5
18
In the TF regime the density distribution of the condensate n(r) = |?(r)|2
reflects the shape of the external potential
n(r) = g?1[?? V (r)], (2.25)
when ? ? V (r) > 0 and is otherwise zero. For a harmonic confining potential
(Equation 2.11) as is typical in our experiments we find that the length scale that
characterizes the size of the condensate is the Thomas-Fermi radius
??? 2?
R ?j = 2 , j = x, y, z. (2.26)m?j
The density of the condensate is described by an inverted parabola
(
? x2 2
)
y z2
n(r) = 1? 2 ? 2 ? 2 . (2.27)g Rx Ry Rz
as is shown in Figure 2a. By integrating over Equation 2.27 we find that
8? ?
N = 15 RxRyRz, (2.28)g
which allows to determine the number of atoms in the condensate based on the
density profile. In practice, in-situ BECs are very dense which can lead to some
technical difficulties when trying to image directly their density profiles (see Sec-
tion 3.4) so instead our images are taken after the atoms are released from the trap
and allowed to expand for some time.
19
1.0
Rx
Ry
4
Rz
0.5 3
2
0.0 1
?2 ?1 0 1 2 0 1 2 3 4 5
x/Rx t [ms]
Figure 2: In the Thomas-Fermi regime where interactions are large compared to
kinetic energy the density profile is determined by the external potential. a.Density
profile along ex of a BEC in a harmonic potential. b. Interaction driven expan-
sion of a BEC in time-of-flight for a trap with trapping frequencies (?x, ?y, ?z) =
2?(42, 34, 133) Hz obtained by numerically integrating Equation 2.30.
2.3 Density profiles
Direct imaging of the atoms probes most ultracold atom experiments. (e.g.
with absorption imaging, Section 3.4). If the atoms are imaged in-situ we access
their spatial density profiles. If the atoms are released from the trap and allowed to
expand in time of flight (TOF) we image their momentum distribution.
For the case of a BEC confined in a harmonic potential at zero temperature
(no thermal fraction) and in the Thomas-Fermi regime discussed in Section 2.2.3,
20
n(x)/n0
Ri(t)/Ri(t = 0)
the in-situ density distribution is described by
( )
( ) = 1? x
2 y2 2
n r n0 ? ?
z
R2 2 2x( Ry Rz )
= 15N 1? x
2 y2 z2
8 2 ? 2 ? 2 . (2.29)?RxRyRz Rx Ry Rz
Even though the BEC is in the motional ground state, it will expand during
TOF as a consequence of interactions. The expansion can be determined using the
time dependent GP equation. A detailed account of the procedure can be found
in [34], the procedure relies on using the ansatz that the TF radii expand as
Ri(t) = ?i(t)Ri(t = 0), (2.30)
where the condensate is in the trap at t = 0 which implies that ?i(0) = 1 and it is
then suddenly turned off at t > 0. If we insert the condensate wave function with
TF radii given by Equation 2.30 into the time-dependent GP equation we find a
series of differential equations
d2? ?2i i
dt2
= (2.31)
?i?x?y?z
whose solutions determine the density profile of the BEC in TOF. Alternatively, if
the density profile of the BEC is known from an image, these relations can be used
to back-propagate the original TF radii of the confined condensate. This is helpful
for example to calculate the atom number in the condensate using Equation 2.28.
21
Figure 2b shows the scaling factors ?i as a function of TOF that were obtained
by numerically integrating Equation 2.30 for a harmonic potential with frequencies
close to those characterizing the optical dipole trap used in most of our experiments.
For a thermal gas in a harmonic potential at temperatures higher than the
level spacing kBT > h??x,y,z the density is given by [7]
nth(r) =
1
3 g3/2(z(r)) (2.32)?th
where z(r) = exp(?? V (r)/kBT , V (r) is given by Equation 2.11, ? is the chemical
potential and gj(z) =
?
zii /i
j is the Bose function. The Bose function introduces
effects of quantum statistics and compared to the Maxwell-Boltzmann distribution
of distinguishable particles, the peak density of a Bose gas increases by g3/2(z)/z, a
phenomenon known as Bose-enhancement.
The distribution after TOF is calculable considering that the trapped atoms fly
ballistically from their position in the trap. An atom starting initially at the point
r0 moves to the point r after a time t if its momentum is given by p = m(r? r0)/t,
then
1 ?3 ( [ ?3 (m ?2 )])
ntof = g3/2 exp ?? x2 i
? ith i=1 ( [ 2 i=1 ])1 + ?2 2i t
? 1
2
g3/2 exp ( ?
mr
? 2 2 )/kBT (2.33)?th t
where the approximation in the second line is valid for t ??1i . The temperature of
the atoms can be estimated by looking at the wings of the density distribution after
22
TOF. Even with the case of Bose enhancement, the density of the wings still decays
exponentially as exp(?x2i /2?2i ). The temperature of the cloud can be determined
using
( )
= m ?
2
k T i 2B 2 1 + ?i( )?2i t2
? m ?
2
i
2 (2.34)t
For partially condensed clouds the density profiles will be given by a combina-
tion of the thermal density profiles and the Thomas-Fermi density profile. Figure 3
shows the density distributions of atoms extracted from images taken after a 21 ms
TOF and therefore the position is mapped to momentum. The images summarize
some of the main concepts discussed in this Chapter. Above Tc the density profile of
the atoms corresponds to the thermal distribution (Equation 2.33). When T < Tc
a small peak in the center of the thermal distribution appears indicating conden-
sation. As the temperature decreases, the fraction of atoms in the condensed state
(and therefore the height of the peak) increases. Equation 2.29 gives density distri-
bution of the condensed atoms, where the TF radius increases due to interactions
and the scaling factors can be found using Equation 2.30.
23
Figure 3: Momentum distribution of atoms near Tc after a 21 ms TOF. As the
atoms cool below Tc a sharp peak in the momentum distribution appears indicating
condensation.
24
Chapter 3: Manipulation and detection of ultra-cold atoms
All of the experiments described in this thesis were performed using ultracold
clouds of 87Rb. In this Chapter I describe the techniques and interactions that
make our experiments possible. This Chapter is not an extensive survey of atomic
physics but rather covers the topics that are most relevant to my experiments. The
references I included are helpful if the reader is interested in learning the details of
the derivations or wants to expand on a given topic.
In Section 3.1 I start by describing the electronic structure of 87Rb. In Sec-
tions 3.2 and 3.3 I describe atomic interactions with external fields that are necessary
for the creation, manipulation and detection of ultracold atoms. First I review the
interactions of atoms with magnetic fields and its application to magnetic trapping.
Then I describe the foundations of atom-light interactions that make possible both
laser cooling and trapping of atoms and give rise to Raman induced transitions.
In Section 3.4 I discuss the resonant absorption imaging technique that we use to
detect atoms after all our experiments are performed. Finally, in Section 3.5 I dis-
cuss coherent processes that use the magnetic and electric dipole interaction and
are relevant to the experiments presented in Chapters 5, 6, and 8.
25
3.1 Electronic structure of 87Rb
Figure 1: Level structure of the two lowest electronic energy levels of 87Rb (not
to scale). a. Ground and first excited state electronic configuration of 87Rb given
by the {n,L} quantum numbers. b. The interaction between the orbital angular
momentum and the spin of the electron leads to the fine structure splitting of orbitals
with L > 0. The splitting of the 52P line gives rise to the D1 and D2 lines. c. The
interaction between the total angular momentum and the nuclear spin causes the
fine structure levels to split further into states characterized by the quantum number
F .
Rubidium is an Alkali metal. Alkali metals correspond to the first group
(leftmost column) of the periodic table and are characterized by having a single
valence electron, which makes the description of their internal structure simpler
than that of other elements. The state of an electron in an atom is described by
its angular momentum L? and its spin S?. Due to Pauli?s exclusion principle there
26
can not be two electrons with the same quantum numbers and in multi-electron
atoms they tend to fill ?shells? of different angular momentum values, historically
labeled by the letters S, P, D, F, ...1 (corresponding to L = 0, 1, 2, 3, ...). In
particular, Rb has 4 filled shells and one electron in the 5S shell, where the number
5 corresponds to the principal quantum number n. Figure 1 shows the energy levels
of the ground state 5S and its closest 5P orbital.
The relativistic treatment of the electron?s motion gives rise to an interaction
between the electron?s intrinsic magnetic moment (the spin) S? and the orbital angu-
lar momentum L?. This spin-orbit coupling interaction H?fs = AfsL ?S causes the fine
structure splitting of the electronic orbitals into levels with different total electronic
angular momentum J? = L? ? S?. Figure 1b shows the 52S 2 21/2, 5 P1/2 and 5 P3/2 elec-
tronic configurations that arise from this splitting, using the spectroscopic notation
n2S+1LJ that indicates the values of the relevant quantum numbers. For S (L = 0)
orbitals, J = 1/2 is the only possible value and the levels do not split. For the P
orbital (L = 1) and a single electron with S = 1/2, J can be 1/2 or 3/2 and the P
orbital splits into two levels. The 52S1/2 ? 52P1/2 is known as the D1 line and has
wavelength ? = 794.979 nm and 5S1/2 ? 5P3/2 transition is known as the D2 line
and has ? = 790.241 nm [37].
The atomic level structure gets further modified by the magnetic interaction
of the electron with the nuclear spin I. This is another kind of spin-orbit interaction
that gives rise to the hyperfine splitting of the atomic levels which can be described
1These terms were used to describe the lines in the emission spectra when they were first
discovered. S stands for sharp, P for principal D for diffuse and F for further noted
27
by the Hamiltonian H?hfs = AhfsI ? J. A complete derivation of H?hfs can be found
in [38]. The hyperfine levels correspond to different values of the total angular
momentum F? = J? + I?. For 87Rb I = 3/2 [37] which results in the level structure
shown in Figure 1c, where the ground state is split into two levels with total angular
momentum F = 1 and F = 2.
3.2 Interaction between atoms and magnetic fields
Atoms have an intrinsic magnetic moment that is given by the sum of nuclear
and electronic moments
?B
?? = ? (gSS? + gLL? + gI I?) (3.1)
h?
where ?B is the Bohr magneton and gS, gL and gI are the ?g-factors? corresponding
to the spin, orbital and nuclear angular momentum. In the presence of an external
magnetic field B, the internal levels of an atom get modified due to the Zeeman [39]
interaction
H?Zeeman = ??? ?B. (3.2)
If the energy shift due to the Zeeman interaction is small compared to the hyperfine
splitting then F is a good quantum number and we can write
= ?BgFH?Zeeman F? ?B (3.3)
h?
28
where gF is the hyperfine Lande? g-factor and is given by
= F (F + 1)? I(I + 1) + J(J + 1) F (F + 1) + I(I + 1)? J(J + 1)gF gJ 2 + gI ,F (F + 1) 2F (F + 1)
(3.4)
and
?= 1 + J(J + 1) + S(S + 1)? L(L+ 1)gJ 2 ( + 1) . (3.5)J J
is the Lande? g-factor associated to the total electronic angular momentum J . The
total energy shifts can be calculated by diagonalizing the full atomic Hamiltonian
including the fine and hyperfine structure terms. Figure 2 shows the energies of
the |mF ? levels in the F = 1 and F = 2 manifolds of the 52S1/2 electronic ground
state of 87Rb as a function of magnetic field. If the magnetic field is small then the
Zeeman term can be treated as a perturbation to the atomic Hamiltonian and the
energy split is linear with the magnitude of the field Em = gF?BmFB, what isF
known as the ?linear Zeeman regime? where F and |mF ? are good quantum num-
bers. In contrast, in the ?Pachen-Back regime? at large magnetic fields2 the Zeeman
term dominates over the fine and hyperfine terms and therefore the good quantum
numbers of the system are J and mJ . Our experiments typically operate in an in-
termediate regime (B ? 10 ? 30 G, the gray box in Figure 2) where the externally
imposed magnetic field starts to be comparable to the magnetic field produced by
the valence electron at the nucleus and the energy of |mF = 0? gets a small shift in
energy that is quadratic in B. For atoms in F = 1 we define this quadratic Zeeman
2A couple orders of magnitude larger than the fields we operate at.
29
shift as  = E0 ? (E+1 ? E?1)/2, where Em is the Zeeman shift for state |mF ?.F
7.5
5.0
2.5
0.0
?2.5
?5.0
?7.5
0 500 1000 1500 2000 2500 3000 3500 4000
B (Gauss)
Figure 2: Zeeman splitting of the 52S 871/2 manifold of Rb. At small magneric
fields F and |mF ? are good quantum numbers describing the system and at large
magnetic fields (Pachen-Back regime) the states are described by the J and mJ .
Our experiments operate in the regime marked by the small gray box (B < 35 G).
For the particular case of states with J = 1/2 (like the ground state of Alkalis),
the Zeeman energies can be found analytically using the Breit-Rabi formula [40]
?
Em = ?
1 ?BgImFB 1 4mF 2
F 2(2I + 1) + ? + 1 +Ehf 2 2 + 1
x+ x , (3.6)
I
where ?Ehf = Ahf(J + 1/2) and x = (gJ ? gI)?BBz/?Ehf .
30
E (GHz)
3.2.1 Magnetic trapping
The sign of the Zeeman energy for different |mF ? states can be used to create
state-dependent traps for atoms. We implement magnetic traps using quadrupole
magnetic fields
B = B?(xex + yey ? 2zez) + B0, (3.7)
where B0 is a constant magnetic field, for simplicity I will assume that B0 = B0ez.
The Zeeman Hamiltonian gives a trapping potential
U(r) = gF?BmFB ? ( )2
= g ? m B? x2 + 2 + 4 ? B0F B F y z( ?? ??) 2B
?
? g ?F?BmFB ?+ 2 ??z ? B0 ?2B? ? (3.8)
where ?2 = x2 + y2 and the approximation on the second line is valid for small
displacements from the trap center.
The sign of the magnetic moment determines which states can be trapped.
For 87Rb the |F = 1,mF = ?1?, |F = 2,mF = 2, 1? are magnetically trappable. The
state |F = 2,mF = 0? is also weakly magnetically trappable due to the quadratic
Zeeman shift.
In addition to generating trapping potentials, we use quadrupole fields before
imaging the atoms to generate state-dependent forces that separate the different
|mF ? states in a similar way as the Stern-Gerlach (SG) experiment [41].
31
3.3 Interaction between atoms and electric fields
In the presence of an electric field E, an atom can become polarized and its
energy levels get modified by the Stark effect [42]. If the electric field is spatially
uniform with respect to the atom?s size we consider the electric field as a classical
object and its effect on the atom can be described by the Hamiltonian [43]
H?dip = ?d? ? E, (3.9)
where d? = ? ?e j rj is the atomic dipole operator, e is the electron charge and r?j
are the position operators of the atom?s electrons relative to the center of mass of
the atom. This approximation, known as the dipole approximation, is valid for
electromagnetic radiation when the wavelength is much larger than the size of an
atom ? ratom [44].
Consider first the simplified case of a two-level system interacting with a co-
herent electromagnetic field E = E(+)e?i?t + E(?)ei?t, where E(?) = ?E(?) are the
positive/negative frequency components of the field, ? the polarization, and ? is
the angular frequency. The shift in energy of the ground sate using second order
perturbation theory is
2? | ?g| ? ? d |e? |2|E(+)|2? egEg = ?
h?(?2eg ? ?2)
= ?1?(?)E22 (3.10)
32
where ?eg = (Ee?Eg)/h? is the angular frequency associated to the energy splitting
of the two states and ?(?) is a dynamic polarizability. Things are a bit more com-
plicated with real atoms though, and we need to take into account all the atomic
levels. Furthermore, there are degeneracies associated to the different angular mo-
mentum states as well as different light polarizations so we have to be more careful
with the orientation of the atom and the field. To take these effects into account
one can introduce a generalization of the polarizability [44,45] which takes the form
?
???(?) =
2?jg ?g| d? |ej? ?ej| d? |gj?
h?(?2 ? ?2j )? jg
= 2?F
?F ?F,m ? ?F | d? |F ,mF ?? ?F ,mF ? | d? |F,mF ?
. (3.11)
? h?(?2 2F , m ? F ?F ? ? )F
Here |ej? represent the excited states and ?jg = (Ej ? Eg)/h? and the expression in
the second line corresponds to the polarizability of the hyperfine levels of an atom
in the ground state |F,mF ? coupled to excited states |F ?,mF ??. We can write an
effective Hamiltonian for the Stark shift as
H? = ?? (?)E(+) (?)Stark ?? ? E? . (3.12)
The polarizability is a rank-2 tensor operator and can be represented by 3
irreducible tensor operators (see [44] for a complete derivation). In the limit of
small magnetic fields so that F and |mF ? are good quantum numbers describing the
state of the atom |n, F,mF ? the dipole Hamiltonian in this representation takes a
33
convenient form
H? (0) (?)Stark =? (E ? E(+)) + i?(1)(E(?)( ? E
(+)) ? F? ) ]
+ (2) (?) (+) 1? Ei Ej 2(FiFj + )?
1
FjFi 3F?
2?i,j , (3.13)
where ?(0), ?(1) and ?(2) are the scalar, vector and tensor polarizabilities respectively
and F? is the total angular momentum operator. For all our experiments ?(2) is very
small so I limit the discussion to the effect of the first two terms. The scalar term
is responsible for the dipole force that allow us to trap atoms using off-resonant
light and the vector component is necessary for engineering spin-orbit coupling and
other spin-dependent potentials through two-photon processes. Finally, I want to
emphasize that it is because of the fine and hyperfine structure that the treatment
using a tensor polarizability is necessary and the vector and tensor polarizabilities
arise.
3.3.1 Scalar polarizability
The scalar polarizability takes the form
(0) ?= 2?jg| ?g|d ? ? |ej? |2?
h?(?2 , (3.14)j jg ? ?2)
where the matrix element can be expressed in terms of the Clebsch-Gordan coeffi-
cients and the reduced matrix element using the Wigner-Eckart theorem [46]. For
the ground state of an Alkali atom (J = 1/2) and if the detuning is large compared
34
to the hyperfine splitting the expression above gets simplified to
? 2
?(0)
?
? 2?JJ
?|?J = 1/2?d?J ?|
3h?(?2 2) . (3.15)
J ? JJ ? ? ?
Due to the second order-perturbation theory treatment, the scalar polarizability can
be interpreted as arising from a two-photon process where the atom absorbs an off-
resonant (virtual) photon and then returns to its initial state by emitting another
(virtual) photon.
The dipole matrix elements needed to compute the polarizability are related
to the transition scattering rate via Fermi?s golden rule [44,46]
2
? = ?JJ ? 2J + 1 ? 2JJ ? 3 3 2 ? + 1 |?J?d?J ?| , (3.16)?0h?c J
and combining this with the expression for the intensity of the electric field I(r) =
20c|E(r)|2 it can be shown that the energy of the ground state manifold is shifted
by
?
2 (r) ? (1? ) ( )( ?c I D1 qgFmF 1 1U ?, r) = ? ?2 ?3D1 ? + ??D1 ? ? ?D1 ?
+ 2?D2(1 + qgFmF )
( 1 ? 1 )?3 + , (3.17)?D2 ? ?D2 ? ? ?D2
where q = 0,?1 for linearly and circularly polarized ?? light and only the the
most significant contribution from the closest transitions (the D1 and D2 lines) are
included. Here U(r) is related to the real part of the polarizability which is a complex
35
valued number. So far I have only considered a real valued polarizability by assuming
the excited states have an infinitely long lifetime. However, in reality the atom can
spontaneously emit photons and decay. This exponential decay can be accounted
for by adding an imaginary contribution to the energies ?D ? ?D + i?D?3/?3D of
the D1 and D2 transitions [47]. The scattering rate is related to the imaginary part
of the polarizability and is given by
2 [ ]
?( ) = ?c I(r) ? ?
3 ( 1 1 )2 2? 3 (D1 D2? 1 1 )2
?, r 2 6 ? + ?h? ? 6D1 ? + ?D1 ? ? ?D1 ?D2 ? + ?D2 ? ? ?D2
(3.18)
The energy shift U(?, r) is a conservative term and is related to dipole trap-
ping, while the scattering term ?(?, r) is dissipative and is important for laser cool-
ing. In the context of engineering potentials for ultracold atoms with off-resonant
light, the scattering is translated into heating because every time an atom emits a
photon with angular frequency ?L it gets a recoil momentum h?kL. If the frequency
? satisfies the relation ? + ?D  ? ? ?D, as is often the case, we can neglect the
terms proportional to 1/(?+?D), an approximation typically known as the rotating
wave approximation (RWA). If the RWA is valid then the frequency dependence of
both the energy shifts and the scattering rates are given by the detuning from the
D1 and D2 transitions.
3.3.1.1 Optical trapping
One important application of the scalar light-shift is to create optical traps for
clouds of ultracold atoms. An optical field with non-uniform spatial intensity gen-
36
erates traps (and anti-traps) for the atoms which experience a force proportional to
the intensity gradient Fdip = ??U(r). The attractive or repulsive nature of the trap
depends on the sign of U(r) which is determined by the sign of the detuning (blue-
detuned traps are repulsive and red-detuned traps are attractive). The production
of BECs in our lab relies on the use of focused Gaussian laser beams of wavelength
? = 1064 nm. The intensity profile of a focused Gaussian beam propagating along
ez is given by
x2+y2
I(x, y, z) = 2P ? ?2(z)2( )e (3.19)?? z
wh?ere P is the total power of the beam and the 1/e
2 radius is given by w(z) =
w0 1 + z2/z2R where the minimum radius w0 is known as the waist and zR = ??20/?
is the Rayleleigh range3. If the extent of an atomic cloud is small compared to
the size of the beam we can perform a Taylor expansion around r = 0 to obtain a
harmonic trapping potential
(
x2
)
( ) = ? 1? 2 + y
2 z2
U r U0 2 ? 2 . (3.20)?0 zR
The oscillation frequencies of the trap along the radial direction are ?r = (4U0/m?2 1/20)
and along the axial direction ?z = (2U0/mzR)1/2. The beam waist is usually much
smaller than the Rayleigh range (?0 ? 50 ? 150?m for my experiments) and the
trap is much stronger along the axial direction. To get around this we use a ?crossed?
3If the waist of a Gaussian beam is comparable to the wavelength ?, a term with longitudinal
polarization appears. The waist of the Gaussian beams used in the lab typically exceed ? by at
least one order of magnitude
37
dipole trap which is formed by a combination of two cross-polarized4 and frequency
shifted focused Gaussian beams propagating along perpendicular directions, ensur-
ing that we get good confinement of atoms along all spatial directions.
3.3.2 Vector polarizability and effective magnetic fields
The general expressions for the vector polarizability are quite complicated and
depend both on reduced matrix elements and Wigner 6 ? j symbols (see [44] for
example for the complete expressions). For the particular case of Alkali atoms and
for large detuning compared to the hyperfine splitting, the vector polarizability takes
a simplified form [16]
(0)
?(1) = 2?fs? (3.21)
3(E? ? h??)
where ?fs = 3Afs/2 and E? = (2ED1 + ED2).
If we recall the Zeeman Hamiltonian introduced in Section 3.2, the term pro-
portional to the vector polarizability in Equation 3.13 looks very similar to Equa-
tion 3.2 for an effective magnetic field
ih?
B (1) ?eff = ? ? (E ? E). (3.22)
?BgJ
For the intensities that we typically operate at, the vector light shift is small and
can be treated as a perturbation so the Hamiltonian resulting from this effective
4The beams are cross-polarized to avoid interference between them
38
magnetic field can be written as
?BgF
H?eff = Beff ? F? (3.23)
h?
3.3.2.1 Raman coupling
The vector light shift enables the realization of various spin dependent-potentials
in the laboratory. The results of Chapters 5 and 8 use a combinations of cross-
polarized laser beams such that the total electric field E? ? E 6= 0 could induce
two-photon Raman transitions. A Raman transition is also a two-photon process
but instead, we consider two ground states and an intermediate state that is off-
resonantly coupled as is shown in Figure 3a. Due to the large detuning, the pop-
ulation transferred into the intermediate state is negligible and the state can be
adiabatically eliminated [48]. In our experiments, we typically couple the |mF ?
levels of the F = 1 manifold after applying a bias magnetic field such that  is
non-negligible.
Consider two laser beams counter-propagating along ex and with polarizations
along ey and ez as in Figure 3b. The electric field from the Raman beams is
E(x, t) = Ea cos(kax? ?at)ey + Eb cos(kbx+ ?bt)ez, (3.24)
and consequently
E? ? E = 2iEaEb cos(2kLx? ?a,bt)ex, (3.25)
39
Figure 3: a. A Raman transition is a two photon process that couples two ground
states through and intermediate far detuned state. b. We induce Raman transitions
using a pair of cross-polarized laser beams and we set the difference in their angular
frequencies close to the Zeeman splitting between two consecutive |mF ? states.
where ?a.b = ?a ? ?b. The Raman Hamiltonian is given by
H?R = ? cos(2kLx? ?a,bt)F?x (3.26)
?
where ? = ?(1)gFEaEb/gJ ? IaIb is the Raman coupling strength. The geometry
and wavelength of the Raman fields determine the natural units of the system:
the single photon recoil momentum kL = 2?/?R and its associated recoil energy
E 2 2L = h? kL/2m, as well as the direction of the recoil momentum kL = kLex. For
most experiments we tune to what is known as the ?magic wavelength? or tune-out
wavelength [49] ?R = 790.034 nm, at which the ground-state scalar polarizability
vanishes and the scattering rate is reduced (Figure 4a,c). We occasionally had
to tune away from the magic wavelength, for example when we were starving for
laser power and wanted to increase our Raman coupling strength. An important
metric for us is Raman coupling strength and Figure 4b shows its dependence on
40
wavelength; tuning closer to resonance allows us to decrease the laser intensity for
the same intensities but comes with increased scattering rates and reduced lifetime
as can be seen in Figure 4d which shows the decay in number of Raman dressed
atoms as a function of time for different wavelengths.
a. b.
200 103
0 102
?200 101
784 786 788 790 792 780 785 790 795
Wavelength [nm] Wavelength [nm]
c. d.
0.06 6000 ? = 786 nm
? = 792 nm
4000
0.04
2000
0.02
0
784 786 788 790 792 0 1 2 3 4 5
Wavelength [nm] Raman pulse time [s]
Figure 4: a. Scalar polarizability as a function of wavelength near the D1 and D2
lines of 87Rb. We typically tune our Raman laser beams near the magic wavelength
? = 790.034 nm. b. Raman coupling strength as a function of wavelength measured
for a pair of Raman beams with waist w0 ? 150?m and powers of 50, 10 mW. c.
Scattering rate as a function of wavelength, while it is not minimized at 790 nm
its value is kept relatively low. d. Decay in number of Raman dressed atoms as a
function of hold time for the same beam parameters as in b.. At ? = 786 nm the
1/e lifetime is ? = 1.64 s and for ? = 792 nm it is reduced to ? = 0.72 s.
In a frame rotating with angular frequency ?a,b corresponding to applying the
unitary transformation U?(t) = exp(?i?a,btF?z) and neglecting the fast terms rotating
41
?E/h [Hz]
Scattering rate [1/s]
number of atoms [arb. u.]
Raman coupling strength [EL]
at frequency 2?a,b (applying a RWA) the transformed Hamiltonian is
U? ?H?RU? ? ih?U? ?
? ?
?tU? = ?a,bF?z + 2 cos(2kLx)F?x ? 2 sin(2kLx)F?y, (3.27)
which describes a helically precessing magnetic field with period ?R/2.
3.3.2.2 Spin-orbit coupling
The Raman Hamiltonian from Equation 3.27 can be further arranged to make
it look like a spin-orbit coupled (SOC)5 Hamiltonian that is familiar to condensed
matter physicists. If we apply a spin-dependent momentum boost which is described
by the unitary operator U?(kL) = exp(i2kLxF?z) the full Hamiltonian including the
Raman coupling and the free particle energies is transformed into
( )
= h?
2 ( )2 ? F? 2
H?SOC 2 q?x ? 2k F? +
z
L z
m 2 F?x + ?F?z + h? 1? 2 , (3.28)h?
where ? = E?1 ? ?a,b. We can go from a 3 level system to an effective spin-1/2
system if we set ?a,b = E?1?E0 and consider a sizable quadratic Zeeman shift  so
the |mF = 1? state can be adiabatically eliminated [50] giving an effective two-level
system. After performing a global rotation ??z ? ??y, ??y ? ??x, and ??x ? ??z of the
Pauli matrices describing the pseudo-spin of the two-level system, the Hamiltonian
becomes
h?2= ( ? h? h?H?SOC 2 qx k ?? )
2
L y + 2???z + 2 ???y (3.29)m
5Not to be confused with the spin-orbit coupling giving rise to the fine and hyperfine structure
mentioned earlier, perhaps a better name could be spin-momentum coupling
42
which corresponds to a spin-orbit coupled Hamiltonian with an equal superposition
of Rashba-type [24] (? ??xky ? ??ykx) and Dresselhaus-type [51] (? ??xky ? ?ykx)
SOC and effective magnetic field ? ? in the ey ? ez plane [22, 50]. In Chapter 8 I
discuss the Rashba term in more detail and introduce a way of engineering a system
with only Rashba-type SOC using multiple internal levels and Raman transitions.
3.4 Detection: Resonant absorption imaging
Ultracold atom experiments rely on optical imaging as the main method to
probe and characterize the system. We use resonant absorption imaging by shining
a probe laser at the atomic cloud and then imaging it into a camera. From the
absorption of the light, we can infer properties from the atoms such as the number,
temperature, integrated column density, and momentum distribution if we allow the
clouds to expand.
Consider a laser beam with intensity I(x, y, z) and angular frequency ? prop-
agating along ez through a cloud of atoms with density n(x, y, z) as in Figure 5a.
We define a (frequency-dependent) scattering cross-section ?(?) which characterized
the probability of an atom absorbing a probe photon and is given by the Lorentzian
function
?2c2 1 ?
?(?) = 3Aeg
?2 20 2? ? + ?2 4
(3.30)
/
plotted in Figure 5b, where ? is the scattering rate, ?0 is the transition frequency, ? =
???0 is the detuning, and Aeg is the Einstein coefficient associated to spontaneous
emission. As the beam travels through the cloud is absorbed and its intensity
43
-4 -2 0 2 4
(? ? ?0)/?
Figure 5: a. A laser beam traveling along ez through a medium with density
n(x, y, z. The intensity decays exponentially with the integrated column density and
the scattering cross section ??. b. The scattering cross section has a Lorentzian
line shape with a full width half maximum equal to ?.
reduced at a rate given by
dI (x, y, z) = ?n(x, y, z)?(?)I(x, y, z). (3.31)
dz
In the limit of small intensities, we can integrate this expression over the
thickness of the cloud and find that the intensity decays exponentially with the
density and the scattering cross section
? z
( ) = ? ?I x, y, z I(x, y, 0)e? 0 n(x,y,z )?(?)dz , (3.32)
?
where OD = z0 n(x, y, z?)?(?)dz? is the optical depth (OD) of the medium. If we
measure the OD of the cloud it is then straightforward to obtain the integrated
column density n(x, y), a result known as the Beer-Lambert law.
44
?(?) [arb. u.]
In the experiment we measure the optical depth of a cloud by imaging the
probe into a CCD camera under two different conditions: first in the presence of
atoms to measure the attenuated intensity If = I(x, y, z) and then without any
atoms to get a measure of the initial intensity Ii = I(x, y, 0). The optical depth can
then be computed as ( )
OD = ln If . (3.33)
Ii
Figure 6 shows the different images used to compute the OD. In practice we take
a third image of the background intensity Ibg and subtract it from the other two
images.
probe probe
atoms background
Figure 6: Resonant absorption imaging. An atomic sample is illuminated with a
resonant probe whose intensity is later recorded on a CCD camera. Two additional
images of the unabsorbed probe intensity and the background intensity are captured
in order to reconstruct the optical density of the atoms.
45
3.4.1 High intensity absorption imaging
The use of the OD to infer the atomic density works well if we assume that
the intensity of the probing laser is low such that the atoms mostly stay in the
ground state. However, at high intensities a significant fraction of the atoms can
become excited and effects such as stimulated emission of light have to be taken
into account. As a result of this the scattering cross-section gets an additional
dependence on intensity (see [52] for a complete derivation)
1
?(?, I) = ?(?)1 + , (3.34)I/Isat
where I 3sat = ?hc?/3?0 is the saturation intensity, and when I = Isat the population
in the ground and excited state are equal. Integrating Equation 3.31 using the
modified expression for ?(?, I) gives
n(x, y)?(?) = ??? ln(If/Ii) +
Ii ? If
, (3.35)
Isat
where I have also added an additional dimensionless parameter ?? which can account
for imperfections in the imaging process (see [53]).
It is hard to reliably measure atomic clouds at low intensity when the OD is
of the order of 3 or 4 (such as our BECs) and a significant fraction of the imaging
light is absorbed. Due to the limited dynamic range of CCD cameras, the measured
OD saturates resulting, for example, in imaging ?flat-top? BECs rather than the
46
a. b.
2
0
0 50 100 0 50 100
x [?m] x [?m]
Figure 7: a. A BEC imaged at low intensity shows a ?flat-top? density profile. b.
In order to recover the Thomas-Fermi profile it is necessary to image high density
BECs with intensities larger than Isat.
usual Thomas-Fermi distribution as shown in Figure 7. To get around this issue we
typically image using intensities I > Isat. In order to correctly compute the column
density including saturation effects we need a conversion of Isat from mW/cm2 to
counts per pixel on the CCD camera. We follow the procedure described in [53] to
find the values of ?? and Isat in counts per pixel. To learn about other effects such
as the recoil momentum from the imaging light that could affect absorption images
see [54].
3.5 Coherent manipulation
We rely on driven quantum coherent processes within the electronic grounds
state using magnetic and electric dipole interactions techniques both for state prepa-
ration and characterization of our system. In all of the cases I consider a system
47
OD
described by the Hamiltonian
H? = H?0 + H?I(t) (3.36)
where H?0 describes unperturbed atomic levels and H?I(t) is a time dependent inter-
action. For simplicity I consider only a two-level system
?? ?????g 0H? = h? ??0 ?? (3.37)
0 ?e
where with |g? and |e? are the unperturbed ground and excited states where h??i are
the energies of the unperturbed states.
3.5.1 Rabi oscillations
Consider an interaction term that oscillates with frequency ? close to the
transition energy ?ge = ?g ? ?e
?? ???? 0 ? cos(?t)H?I = h? ???? . (3.38)
?? cos(?t) 0
The coupling strength ? here could be related to an electric dipole transition ? ?
?g|d ? E |e?6 or magnetic dipole ? ? ?g|? ? B |e? transition matrix element. The
6For our system intensities ? ? and we do not observe Rabi oscillations from (single photon)
electric dipole transitions.
48
state of the system at any given time is given by
|?? = c ?i?gt ?i?etg(t)e |g?+ ce(t)e |e? , (3.39)
and substituting this expression into the time dependent Schro?dinger equation we
find that
( )
c? (t) =? ei(???ge)tg 2 + e
?i(?+?ge)t c
?? ( )
e
c? (t) = ei(???ge)t + e?i(?+?ge)te 2 cg. (3.40)
We can apply a RWA if the term ?+?ge is large compared to ???ge. If we assume
that at t = 0 the system is prepared in |g?, the population in |e? describes what is
known as a Rabi oscillation [55]
?2
(? )
2 2
|c (t)|2 = 2 ? + ?e ?2 + 2 sin 2 t (3.41)?
?
where ? = ???ge is a detuning and ?? = ?2 + ?2 is known as the generalized Rabi
frequency. The Hamiltonian after applying the RWA is
?? ??????/2 ?/2?= ?H?0 h? ?? , (3.42)
??/2 ?/2
and its eigenenergies correspond to E? = ???/2. Notice that the difference between
the eigenenergies E+?E?is exactly equal to the frequency at which the populations
49
a. Rabi spectroscopy b. Rabi oscillations
1.0
0.5
0.0
?30 ?20 ?10 0 10 20 30 0 20 40 60 80 100 120
? [kHz] Pulse time [?s]
Figure 8: The Rabi cycle. Population transfered from |F = 1,mF = ?1? into
|F = 1,mF = 0? using an RF magnetic field with ? = 7.1 kHz. The markers in-
dicate experimental data points and the lines correspond to fits to the model in
Equation 3.41a. Population transfered for a 60?s pulse as a function of detuning ?.
b. Population transfered as a function of time close to resonance.
in |g, e? oscillate, this will come up again in Chapter 5. Figure 8 shows an example
of this process where we coupled an initial state |g? = |F = 1,mF = ?1? to |e? =
|F = 1,mF = 0? using a radio-frequency (RF) magnetic field with ? = 7.1 kHz.
Figure 8a shows the population in |e? as a function of ? for a ? pulse of duration
? such that ?? = ?. The location of the peak in this curve is as a way to find
the transition frequency (we use this method in Chapter 6). Figure 8b shows the
population transfered into |mF = 0? from |mF = ?1? as a function of time for ? ? 0;
we typically look at the frequency of these Rabi oscillations to calibrate the coupling
strength of an effective two-level system.
50
|ce|2
3.5.2 Adiabatic rapid passage
The Hamiltonian in Equation 3.42 can in principle be used to transfer all the
population from the initial state |g? into |e? if we set the detuning of our oscillating
field to ? = 0 and apply a ? pulse. Unfortunately we are susceptible to noise in both
? and ? and pulsing is not the most reliable technique for state preparation. To
transfer atoms within different |F,mF ? states within the 52S1/2 hyperfine manifold
we use instead an adiabatic rapid7 passage (ARP) protocol which is based on the
Landau-Zener model [56].
ARP relies on preparing dressed states; eigenstates of the atomic Hamiltonian
(Equation 3.42) which I label using the symbols |?? and dynamically changing the
detuning ? = ?(t). We start at a large and negative detuning ?  ?? where the
ground eigenstate |?? ? |g? and therefore by slowly turning ? on we adiabatically
prepare |??. We consider the rate of change in detuning ?t? > 0, and as ? increases
the state decomposition of |?? changes. When ? = 0 the dressed states correspond
to equal superpositions of the bare states
|?? = 12 (|g? ? |e?) , (3.43)
and when ?  ? we have |?? ? |e?. If the change in detuning is slow enough that
the system can adiabatically follow the ground eigenstate |?? then at the end of
this process the state can be successfully transfered from |g? into |e? by sweeping
7The term rapid is with respect to the spontaneous emission rate of the excited state being
coupled.
51
the detuning. It can be shown that the fraction that does not adiabatically follow
the ground state is given by the Landau-Zenner tunnel probability [44]
( )
??2
Plost = exp ?2 . (3.44)|?t?|
In the limit of large coupling strength compared to the detuning sweep rate, or
slow detuning sweep rate compared to the smallest energy gap (?  |?t?|) all the
population adiabatically follows the ground state.
We mostly use ARP with RF magnetic fields to transfer between different
|mF ? states within the F = 1 manifold. We set the frequency of the field ?RF so
that it matches the Zeeman splitting between the |mF = ?1? and |mF = 0? states
for a target bias field B0ez. We start with atoms in |mF = ?1? and at a bias
field Bi ? B0 ? 380 mG (? ? ?30 kHz). We ramp an ? = 20 kHz RF field with
angular frequency ?RF in 50 ms. We then sweep the detuning by linearly changing
the bias field in 50 ms. Finally, the RF field is abruptly turned off, projecting the
RF eigenstates back into the |mF ? basis. In Figure 3.5.2 we set ?RF = 23 MHz and
when the Zeeman splitting between |mF = ?1? and |mF = 0? is equal to ?RF we
observe an equal superposition of both states and if the detuning is swept beyond
resonance we can reliably prepare the |mF = 0? state.
In general it is necessary to consider the eigenstates of the three-level RF
52
Hamiltonian for a full description of this process
?? ????? ?? ?RF/2 0 ???
H?RF = ?????RF/2 ? ? ?RF/2??? , (3.45)? ?
0 ?RF/2 ?
but for large quadratic Zeeman shifts as is usually the case in our experiments we
can only look at an effective two-level system.
a. b.
1.0
10
0 0.5
?10
0.0
?20 0 20 ?20 0 20
? [kHz] ? [kHz]
Figure 9: a. Eigenenergies and eigenstate decomposition of an RF dressed Hamil-
tonian for a two-level system (Equation 3.42) with ? = 10as a function of detuning.
The eigenstates are linear combinations of |mF = ?1? and |mF = 0? (red and blue
respectively). b. Population in the |mF ? sates for different values of detuning
3.5.3 Magnetic field stabilization with microwave assisted partial trans-
fer absorption imaging
Most experiments are performed in the F = 1 ground hyperfine manifold with
some bias field B0ez that shifts the energies of the different |mF ? states. Due to
the linear dependence of the energies of the |mF = ?1? and the constant changes
53
Energy [kHz/h]
Population
in the ambient magnetic field we use microwave assisted partial transfer absorption
imaging (PTAI) to monitor and stabilize the magnetic field.
The method relies on transferring a small fraction of atoms into the 52S1/2
F = 2 manifold using an oscillating magnetic field with frequency close to the
6.8 GHz ground hyperfine splitting. The atoms in F = 2 can be imaged without
the use of repump light and therefore minimally disturbing the remaining atoms in
F = 1. We apply two microwave pulses for a total time ? with frequency ?0 ? ??
where ?? = ?1/(2?). We typically set ?0 equal to the Zeeman splitting between
the |F = 1,mf = ?1? and |F = 2,mf = ?2? states at a target magnetic field and
we set the coupling strength ?0  1/? such that only about 5% of the atoms are
transferred by each pulse. We image the transferred atoms following each pulse using
absorption imaging and from the measured densities we calculate the imbalance or
error
= n(?+)? n(??)nimb
n( ) + ( ) (3.46)?+ n ??
signal that is both insensitive to fluctuations in the number of atoms and linearly
sensitive to changes in magnetic field8. We use this error signal both to monitor the
magnetic field before performing experiments and to cancel long term drifts in the
field. In most cases, we chose the states |F = 1,mF = ?1? and |F = 2,mf = ?2? as
their relative energies are the most sensitive to changes in magnetic field. Figure 10a
shows the number of atoms transferred by each microwave pulse for different values
of bias magnetic field and Figure 10b shows the imbalance. The microwave frequency
8A single pulse on resonance is quadratically sensitive to detuning (see Equation 3.41)
54
a. b.
? 0.5150 +
??
100 0.0
50 ?0.5
?5 0 5 ?5 0 5
? [kHz] ? [kHz]
Figure 10: Magnetic field stabilization using microwave assisted PTAI. a. Popula-
tion transfered into |F = 2,mF = ?1? from |F = 1,mF = ?1? as a function of bias
magnetic field (global detuning ?). Each microwave pulse was ? = 250?s and de-
tuned by ?? = ?1/(2?) transfer a small fraction of atoms from |F = 1,mF = ?1?
into |F = 2,mF = ?1?. b. Error signal calculated using the transfered atoms by
each pulse. We lock the magnetic field to the ? 5 kHz (? 7 mG) wide linear portion
of the signal.
?0 is on resonance with the |F = 1,mF = ?1? ? |F = 2,mF = ?2? transition when
both pulses transfer the same number of atoms.
In [57] we studied partial transfer absorption imaging as a minimally destruc-
tive technique for imaging ultracold atoms. See Chapter 6 for an alternative solution
for dealing with magnetic field noise.
3.5.4 Ramsey interferometer
A Ramsey interferometer [58] is a setup relevant to Chapters 6 and 8. The
interaction Hamiltonian is the same as in the previous section but rather than being
on continuously it is pulsed on for a time ? = ?/2? (a ?/2 pulse), the system
is then let to evolve only under H?0 for a variable time tdark and finally a second
55
n [arb. u.]
nimb
?/2 pulse is applied. Figure 11a illustrates this protocol: the ?/2 pulses can be
visualized as ?/2 rotations on the Bloch sphere along ex that transform the initial
?
state |g? ? (|g? + |e?)/ 2. For the duration of the dark time, the system rotates
along the equator of the Bloch sphere (about the ez axis in general) by an angle
?tdark and finally the second pulse rotates the state along the ex axis again. The
probability of measuring the excited state is related to the phase accumulated during
the dark time is given by
?? ?? [ ]2 ( )
|ce(2 + )|2 = ????? ? sin(??/2) 2 ?tdark? tdark 2 ?? 2 cos 2 . (3.47)??/
In a Ramsey interferometer the oscillation frequency is only determined by the de-
tuning unlike the generalized Rabi frequency where the detuning adds in quadrature
with ?. The coupling strength dependence in the generalized Rabi frequency sup-
presses the effect of the detuning ? becomes large compared to ?, something that
will be relevant in Chapter 6. Additionally in Chapter 8 I rely on a variation of the
Ramsey interferometer to perform quantum state tomography. Figure 11b shows an
example of a Ramsey fringe as a function of detuning ? that we measured using two
states coupled with ? = 1 kHz.
3.5.5 Floquet theory
The RWA has been used multiple times throughout this Chapter so that
the Hamiltonian describing a driven system can effectively be viewed as time-
independent. This approximation is valid most of the time for our experiments,
56
a. Ramsey interferometer b. Ramsey fringe
1.0
0.5
0.0
?40 ?20 0 20 40
? [kHz]
Figure 11: a. A Ramsey interferometer: Two ?/2 pulses are separated by a time
tdark. The phase accumulated in the interferometer is equal to the detuning multi-
plied by the dark time. b. A Ramsey interference fringe. Data (blue dots) and fit
(black line) obtained from coupling two levels using an RF field with ? = 1 kHz. We
applied a pair of ? = 25?s pulses separated by a 50?s wait and varied the detuning
by changing the bias magnetic field.
however, if we want to give a complete description of a time periodic system Flo-
quet theory can be helpful. I will give a brief overview of Floquet theory using a
matrix approach that is particularly useful for numerical computations.
An arbitrary time periodic Hamiltonian H?(t) = H?(t + T ) can be written in
terms of its Fourier components
??
H?(t) = exp[ij?t]H?j, (3.48)
j=??
with ? = 2?/T and because H? and H? ?j = H??j due to the Hermitian nature of
H?. The eigenstates of the Hamiltonian can be written in a terms of quasi periodic
57
|c 2e(2? + tdark)|
functions9
??
|?(t)? = exp(?it/h?) exp[?ik?t] |?,k? (3.49)
k=??
where the term  is known as the quasi-energy. Inserting this expression into the
time-dependent Schro?dinger equation gives
? ?
(+ h??k) exp[?k?t] |? ?,k? = exp[i(j ? j )?t]H?j? |?,j? . (3.50)
k j,j?
In order for the equality to be true we must have j? ? j = ?k because the complex
exponentials form an orthonorormal basis and we can write
?( )
 |?,k? = H?j?l ? h??k?j,k ? 1? |?,j? , (3.51)
j
where 1? is the identity matrix. The expression can be recast into a matrix form
?? ? ???? ? ? ? ???? ????
?? ?
H?0 + 2h?? H?1 H?2 ? ? ? ? ? ? ???????? ? ? ??? ?? ?? ???? ?
???
??| ??? ?? 1 + ? ? ? ???| ??? ?,?1 ? ? H?? H?0 h?? H?1 H?2 ? ?,?1??? ??? ??? ?????
??? ?
???
? ?? |?,0? ?
=
? ?? H??2 H??1 H?0 H?1 ? ? ? ???? |?,0? ?? ? ? ?? ??
.
?? ?? |? ? ? ? ? H? H? H? ? h?? H?? ,1 ???? ???? ?2 ?1 0 1 ????????
|? ?,1? ????
? ? ? ? ? ? ? ? ? H??2 H??1 H?0 ? 2h?? ? ? ?
(3.52)
The quasienergies  can be computed by truncating and then diagonalizing
the matrix, and they are grouped into repeating manifolds separated in energy by
9Very much like Bloch wave functions
58
h??. The quasienergies within a manifold can be interpreted as the eigenenergies of
an effective time-independent Hamiltonian H?Fl that describes the evolution of the
system sampled stroboscopically at an integer number of driving periods, with the
time evolution operator U?(t , t + T ) = e?iT H?0 0 Fl .
Floquet theory plays an important role in the engineering of different disper-
sion relations for atoms in Chapters 5 and 8. I will give an example based on the
experiments described in [59] (not included in this Thesis), where we considered
a pair of Raman beams driving transitions between the |F = 1,mF ? states of the
ground hyperfine manifold with two different frequencies ??1,0 and ?0,+1 set to the
mF = ?1 ? mF = 0 and mF = 0 ? mF = 1 transitions. By performing indepen-
dent RWAs with respect to each of these transitions we found that the system could
be described by a magnetic Hamiltonian
2
H? = h?k2 + ?(x) ? F? + ?2F?
(2)
zz (3.53)m
with helical effective magnetic field ?1(x)/?1 = cos(2kRx)ex ? sin(2kRx)ey, an ef-
fective quadratic Zeeman shift ? = (? ? ? )/2, and F? (2)h? = F? 2/h?22 ?1,0 0,1 zz z ? 2/3
an element of the quadrupole tensor. The competing contributions between kinetic
and magnetic ordering energies gave rise to different magnetic phases. Figure 12a.
shows the ground branch of the dispersion relation for small ?1 < 4EL (top) and
large ?1 > 4EL (bottom). As the value of ?2 was decreased, the magnetization
in the system changed along with the location of the global minima in the disper-
sion. The experimental parameters ?1 and ?2 spanned a two-dimensional phase
59
diagram shown in Figure 3.53b that we experimentally mapped. The eigenener-
gies of Equation 12 are plotted in Figure 12c. However, in order to get a good
agreement between the experiment and the phase diagram we had use the full Flo-
quet Hamiltonian which results in having modified parameters in Equation 3.53
?(eff)2 = ?2 +O(?21/) (red dotted line in Figure 12b). Figure 12d shows three man-
ifolds of Floquet quasienergies for this system, illustrating their periodic nature.
60
1.0
0.5
0.0
?0.5
?1.0
0 1 2 3 4 5 6 7 8
?1/ER
0 1 2 3 4 5 6 7 8
4
0.5
0.0 2
?0.5
0
0
?2
?2
?4
?4
?2 0 2 Three minima: Two minima: Single minimum
q, in units of kL metastable regime degenerate
25
10
5
20
0
15 ?5
?10
10
?15
5
?20
00.5?0..005
?2
?4 ?2 q, in un0its of kL 2
0 ?25
?2 0 2 ?2 0 2
q, in units of kL q, in units of kL
Figure 12: Magnetic phases of a spin-1 SOC system. a. Ground state energies of
a spin-1 SOC system for ?1 = 1.5 EL (top) and ?2 = 1.5 EL(bottom). By changing
?2 we moved the location of the central minima. b. Phase diagram of a spin-1
SOC system. They green line corresponds to a line of phase transitions where the
system goes from magnetized to unmagnetized. c. Dispersion relation of effective
Hamiltonian computed for ?1 = 2 EL, ?2 = 0. d. Floquet qasienergy dispersion
relation for the same parameters. The magnitude of ?1 effectively modifies ?2 in
the RWA Hamiltonian.
61
Energy in units of EL
|Mz |
?2/ER
Chapter 4: Making BECs in the Rubidium Lithium apparatus
All the experiments presented in this thesis were performed in the Rubidium-
Lithium (RbLi) apparatus at the University of Maryland. The apparatus was de-
signed to produce mixtures of quantum degenerate gases of bosons and fermions.
The original plan was abandoned because the cross-species scattering length was
found to be repulsive and small (as ? 20 aB) [60] and the nearest Feshbach reso-
nance was measured to occur at the unexpectedly large magnetic field of 1066 G [61].
All our experiments were performed using only 87Rb instead.
The RbLi apparatus is scheduled to be shut down and the construction of a
new dual-species apparatus for 87Rb and 39K is underway. The RbLi apparatus has
been thoroughly described in [62,63] so I only give a brief overview of the apparatus.
I additionally discuss in detail the new elements that have been added to the setup
and implemented changes not previously reported. In Appendix A I discuss the best
and the worst aspects of the apparatus and in Appendix B I discuss some aspects of
the construction of the new apparatus for those interested to look deeper into the
technical aspects of experimental apparatus.
This Chapter is divided into three sections. In Section 4.1 I give a brief
overview of the RbLi apparatus and describe its basic capabilities. In Section 4.2 I
62
describe the experimental sequence used to produce BECs. Finally in Section 4.3 I
describe changes to the RbLi apparatus that were not previously reported.
4.1 Overview of the RbLi apparatus
The RbLi apparatus is divided into two optical tables. One table contains
laser systems that are fiber-coupled into the main experiment optical table, shown
in Figure 1, containing a vacuum system where atoms are cooled to degeneracy. The
vacuum system can be divided into three regions: an oven region where Rb and Li
atoms are heated up, a Zeeman slower that acts as a differential pumping stage and
an ultra-high vacuum (UHV) region with a glass cell where all the experiments are
performed.
4.1.1 Laser systems
We use three lasers to perform laser cooling and imaging of atoms: a cooling
laser that addresses the F = 2 ? F ? = 3 transition, a repump laser that takes
atoms that have decayed into the F = 1 state back to F = 2 via the F ? = 1
state and a master laser that provides a frequency reference for both lasers. The
frequency of the master laser is locked using saturation absorption spectroscopy to
the F = 3 ? F ? = 3 and F = 3 ? F ? = 4 crossover of the D2 line of 85Rb. A
fraction of repump and cooling light is used in beam for performing laser-cooling in
the Zeeman slower and most of the remaining light is combined in six beams used to
create a magneto-optical trap (MOT). We have two additional beams with cooling
63
Rb oven
Glass cell
Lab coordinates Zeeman slower
Figure 1: The vacuum system of the RbLi apparatus is divided into an oven, a
Zeeman slower and an UHV region containing a glass-cell where all experiments are
performed.
light used for imaging the atoms along two different directions. The frequencies of
each laser with respect to the master laser frequency with respect to the transitions
in 87Rb can be visualized in Figure 2.
We have two additional lasers that are used to generate potentials for the
atoms. The first one is a 30 W 1064 nm IPG Photonics laser located at the main
experiment table and that we use to make a cross dipole trap for the atoms. The
two dipole beams come from the zeroth and first order of an acusto-optic modulator
(AOM) and the beams propagate along the ex + ey and ex ? ey direction (the lab
coordinate system is shown in Figure 1. The other laser system is a Ti:Sapphire
laser used to generate Raman transitions and will be described in more detail in
64
Section 4.3.2.
We can detect atoms using two different imaging systems. The first one is
used primarily for diagnostics and it images the yz plane of the atoms from the +ex
side of the glass cell. The second system looks at the xy plane from bellow the glass
cell and is the main system used for data acquisition. Figure 3 shows a simplified
diagram with a top and a side view of the apparatus including all the lasers that
are used for cooling, trapping, Raman coupling and imaging.
4.1.2 Magnetic field control
The precise control of magnetic fields is essential during the multiple stages in
our experimental sequence. The RbLi apparatus has multiple coils as is illustrated in
Figure 4. Three pairs of coils in the vicinity of the glass cell generate bias magnetic
fields B = (Bx, By, Bz) along ex, ey and ez. Once BECs are produced we typically
use bias fields along ez to change the Zeeman energy of the different |mF ? states.
One pair of anti-Helmholtz coils generates a strong quadrupole magnetic field along
ez that is used in the MOT, for magnetic trapping, and to separate the different
|mF ? states before imaging via the Stern-Gerlach effect. An additional set of coils
arranged in a ?cloverleaf? pattern generates small gradients along ez, ex + ey and
ez ? ey which allow us to cancel stray magnetic gradients in Bz near the atoms.
The experiment also has the capability of producing oscillatory magnetic fields.
A set of coils on a printed circuit board (PCB) and placed right above the glass cell
produce linearly polarized radio-frequency (RF) magnetic fields either in the ey or ez
65
a. 80 160
MOT Slower Repump lock
1005 5325
87Rb 2  3 Master lock 87Rb 1  1
85
Cooling lock Rb 3-4 x 80 160
Slower MOT
b.
266.650
156.947
72.218
6.8 GHz
Figure 2: a.Cooling and repump frequencies relative to the master laser lock. b.
Cooling and repump frequencies relative to the 87Rb D2 line transitions.
66
MOT repump
slower repump
MOT cooling
Slower cooling
a. Top view b. Side view
Raman D MOT
dipole trap dipole trap xy probe
?cross? zx probe ?tight?
MOT MOT
Zeeman slower glass cell
slower beam atomic beam 
atoms atoms
MOT MOT
Raman C Raman B
xy imaging
zx imaging MOT
flipper mirror cooling + repump
mirror cooling
polarizing beam splitter Raman 790 nm
quarter wave plate dipole 1064 nm
Figure 3: Diagram of the RbLi apparatus as seen from a. the top and b. the side.
direction that are used for RF induced evaporation and to drive transitions between
|mF ? states. There is an additional setup for producing high-power RF fields which
will be described in more detail in Section 4.3.3.
4.2 Experimental sequence to make BECs
The production of BECs starts in an oven where Rb atoms are heated to 120 C
to produce an atomic beam. The Rb atoms come from the same 5 g ampule that was
installed when the apparatus was first built almost 10 years ago. The atoms then
67
a. Bias control a. Gradient control
gradient 
cancellation
coils
glass cell
quadrupole coils
Figure 4: Magnetic coil geometry in the RbLi apparatus a. We use three pairs of
Helmholtz coils to produce bias fields along ex, ey and ez. b. We have a pair of
coils that produce strong quadrupole magnetic fields for the MOT and magnetic
trapping. Additionally we have a pair of ?cloverleaf? coils to control the gradients in
Bz.
travel down a Zeeman slower [64] where they are laser cooled and then captured in a
MOT. For the MOT we apply a 5.5 A current to the quadrupole coils corresponding
to a 15.62 G/cm gradient. The cooling light is blue detuned by 18 MHz ? ?3? from
the F = 2 ? F ? = 3 cycling transition and the repump light is 16 MHz below the
F = 1 ? F ? = 2 transition. We typically load the MOT for a total time of 1 ? 5 s
before we turn off the Zeeman slower currents. In preparation for the molasses stage
we do a 30 ms decompression stage where we ramp down the gradient to 10 G/cm
and reduce the repump power.
The atoms are cooled further down in an optical molasses stage due to the po-
larization gradient from interfering counter-propagating lasers [65]. For this stage we
completely switch off the quadrupole coils and adjust the bias fields in all three di-
68
rections so that they are canceled at the atoms. During this stage, the repump power
is kept low and the frequency of the cooling light is decreased to 140 MHz below
the MOT frequency for 10 ms. We then completely turn off the MOT repump light
allowing atoms to decay into the F = 1 manifold and use a small amount of repump
light from the Zeeman slower to optically pump atoms into the |F = 1,mF = ?1?
magnetically trappable state for a total of 1.5 ms.
Once the atoms are successfully pumped into |F = 1,mF = ?1? we capture
them in a magnetic trap with a gradient of 62 G/cm and compress them by increasing
the current in the coils until we reach a gradient of 160 G/cm in 300 ms to enhance
the elastic collisional rates and promote faster thermalization during evaporation.
We perform RF-induced evaporation in the magnetic trap by turning on an RF field
polarized along ey with a frequency of 24 MHz, which transfers the hotter atoms at
the edges of the trap into the mF = 0 state which is not magnetically trappable. We
then perform an exponential ramp from the initial frequency to a final frequency of
4.5 MHz in 1 s, and as the frequency is swept the most energetic atoms are selectively
transfered into untrappable states.
For the final stage of evaporative cooling, we transfer the atoms from the
magnetic trap into an optical dipole trap. We start by turning on only the ?tight?
arm of the trap at full power (about 11 W) and slowly decompressing the quadrupole
trap to 45 G/cm in 1.5 s. We then turn on the second ?cross? beam in 1 s, splitting
the power so that 70?30% is distributed between the tight and cross beams. As the
cross dipole beam is being turned on, we ramp the quadrupole field further down to
14 G/cm, slightly above the value necessary for the trap to suspend atoms against
69
gravity while simultaneously shifting the bias field along ez to align the center of
the quadrupole trap to the dipole trap.
We evaporate atoms in the dipole trap in two stages. First we exponentially
ramp down the power to about 20% of its initial power in 1.5 s (0.5 s 1/e time).
Before the final evaporation stage we completely turn off the quadrupole trap in
1 s. Finally, we perform a second exponential ramp where the power is dropped to
about 30% of the intermediate power in 2 s (1 s 1/e time). The slow ramps ensure
that there is enough time for the atoms to thermalize as they evaporate. During the
second evaporation stage the atoms reach the critical temperature for Bose-Einstein
condensation and we are able to produce BECs with about 4 ? 104 atoms in the
|F = 1,mF = ?1? state. To transfer atoms into different |F,mF ? states within the
ground 52S1/2 manifold we use the ARP protocol described in Section 3.5.2.
Our experiments are very sensitive to changes in the environmental magnetic
field. In the past we used flux gate sensors (Stefan-Mayer model FL1-100 f) in the
apparatus to feedback and stabilize the magnetic field (see [63]). This sensors are a
useful tool, however due to space constraints we were not able to measure the fields
close to the atoms and additionally the range of magnetic fields that they operate at
is small (only 1 G, we typically operate at B > 10 G). We built a 6.8 GHz microwave
system (see Section 4.3.4) so that we could use the atoms themselves as sensors of
magnetic field. Once BEC is achieved and before performing any experiment, we
typically use a protocol involving two microwave assisted partial transfer absorption
images described in Section 3.5.3 to monitor and stabilize the magnetic field at the
atoms.
70
To keep the bias magnetic fields as stable as possible we additionally syn-
chronize the timing of our experiments to the 60 Hz line; this step is performed in
different stages of the experiment but its biggest impact is right before performing
any magnetically sensitive experiment.
4.3 Upgrades to the RbLi Machine
4.3.1 Master laser system
Previously we used a New Focus Vortex II TLB-6900 extended cavity diode
laser as our master laser locked using a homemade saturation spectroscopy setup
using a Rb glass cell and an FPGA based PID circuit (see [62, 63]). The frequency
of this laser was not very stable and the laser would constantly get out of lock.
We replaced the old master laser with a Vescent photonics DBR Laser Module
System which uses a distributed Bragg reflector laser diode with no external cavity
and is therefore very mechanically stable. The frequency of the laser is stabilized
and controlled using the D2-210 spectroscopy module and D2-125 laser servo. The
master laser system is considerably simplified as can be seen in Figure 5 and the lab
no longer suffers from an unstable master laser.
4.3.2 Raman laser system
The RbLi apparatus has a laser system with wavelength close to 790 nm that
is used to generate Raman induced transitions and spin-dependent potentials (see
Section 3.3.2). The original Raman laser system consisted of a Toptica DL Pro
71
spectroscopy
Figure 5: Master laser system. We replaced the old Vortex II laser with a Vescent
photonics DBR Laser Module System that is considerably more stable.
laser seeding a tapered amplifier chip mounted on a homemade copper holder. This
laser system was replaced by an M squared Ti:Sapphire laser (SolsTiS-400-SRX-F)
that is pumped by a 532 nm IPG GLR30 laser. We typically operate the pump laser
at 14.5 W. A fraction of this light is redirected into the path of a 1D optical lattice
and the remaining power is used to pump the Ti:Sahpire laser. Figure 6 shows
the original (at setup) dependence of the Ti:Sapph output power as a function of
pump power. We switched to using a Ti:Sapph laser because of its wide range of
tunable wavelengths in the near infrared (725? 875 nm) and its high power output.
In addition to this, the TA system had considerable amplified spontaneous emission
(ASE) close to 780 nm which caused increased scattering of photons and reduced
lifetimes due to our inability to completely filter it.
The output of the laser is split into 3 different Raman beams. The frequency
and power of each beam is independently controlled using IntraAction ATM-801A2
AOMs centered at 80 MHz. We drive the AOMs using homemade drivers made from
72
to beatnote lock
3
2
1
6 8 10 12 14
Pump power [W]
Figure 6: Ti:Sapphire laser output as a function of pump power. This data was
taken when the laser was being setup for the first time. The alignment of the laser
cavity was optimized to produce maximum power at 15 W pump power. The output
power is proportional to the pump power Pout ? 0.364Ppump.
the Minicircuits components listed in Table 4.1 and Novatech Model 409B direct
digital synthesizer (DDS) to generate an RF signal at the desired frequency. Figure 8
shows the arrangement of the components: we control the amplitude of the RF signal
using a mixer connected to a DC signal and the switch can turn off the signal in less
than 1?s using a TTL signal. We fiber-couple the light using single-mode optical
fibers (non-polarization maintaining) with angle-cut FC/APC type connectors at the
input (laser side) and flat-cut FC/PC at the output (experiment side). We made
this choice so to implement a phase lock that would cancel phase noise added by
the fibers. The idea behind this method is that a small fraction of the fiber-coupled
light is reflected at the flat-cut edge of the optical fiber and coupled back where it
can be heterodyne probed with the input light (see Figure 7). Our implementation
of the fiber phase lock is based on Section 3.6.3 of [66]. Because the fibers are not
polarization maintaining we control the output polarization of the light using paddles
73
Ti:Saph power [W]
(Thorlabs FPC030 and Thorlabs FPC560) that produce a controllable amount of
stress in the fibers and change their birefringence. This method makes injecting light
into the fiber very straightforward and painless because, unlike with polarization-
maintaining fibers, the polarization of the incoming light can be arbitrary and does
not need to be aligned to any fiber axis. None of the experiments presented in
this thesis used the phase lock but the experiments described in Chapter 8 were
performed using the new Raman laser system. Figure 7 shows a diagram of the
Raman optics as well as the 532 nm optical lattice optics which are shared on the
same breadboard.
Table 4.1: List of AOM driver components
Part number Description
ZHL-1-2W 2 W amplifier
ZAD-3+ Mixer
ZYSWA-2-50DR Digital switch
4.3.3 High power RF system
The experiments described in Chapters 6 and 8 required the use of high power
RF magnetic fields to achieve coupling strengths at the atoms ? ? 100 ? 200 kHz.
After multiple attempts to build a resonant coil either on a PCB (similar to the
coil used for RF induced evaporation, see [62, 63]) or winding some wires with the
right dimensions we found that the product that worked best for our needs was
a wireless power charging receiver coil (Wu?rth Elektronik Digikey part number
732-5646-ND) shown in the bottom panel of Figure 4.3.3a. The coil has a self-
74
AC 254-105-B Solstis IPG GL30
Ti:Sapph
Phase lock Phase lock 
+1 +1 0 +1 +1
Optical lattice
mirror AOM
polarizing beam splitter Faraday isolator
wave plate beam dump
ND filter optical fiber
spherical lens aspheric lens
Figure 7: Optical layout of Raman and optical lattice lasers. The Ti:Sapphire
laser provides tunable infrared light that we use for three different Raman beams.
The beams labeled as ?Raman C? and ?Raman D? can be phase stabilized. A small
fraction of the green pump laser is used to produce a blue-detuned 1D optical lattice.
resonant frequency at 22 MHz and a Q-Factor of 45. It has an inner diameter of
1.62 cm and an outer diameter of 2.8 cm, just the right size for us to place it snugly
next to the glass cell (on the ?ex side) with minimal perturbations to the laser
beams in its vicinity (it only slightly clips one MOT beam).
The loop is mounted on the PCB shown in Figure 9. The board has two
connections: the top one in Figure 9 has a small loop used as a pickup antenna that
we attach to a power detector Minicircuits ZX47-40-S+ to monitor the power
and the bottom lines have pads that can be used to make an impedance matching
75
Raman D GBE02-B
Raman C
10-5-532-HP
Raman B
Cavity and wavemeter
L R
DDS Switch Amplifier To AOM
I
Analog input TTL
Figure 8: Setup used to drive the AOMs controlling the power and frequency of
Raman beams. A similar setup is used to drive a coil used to generate high-power
RF fields at the atoms.
network.
We used a vector network analyzer (VNA) to perform the impedance match-
ing. The VNA sends a small amplitude RF signal into the circuit and measures the
amplitude and phase of the reflected power as a function of frequency from which
the impedance can be inferred. Figure 10a shows the reflected power as a function
of frequency for a test circuit and Figure 10b shows the complex valued impedance
as a function of frequency displayed on a Smith chart. The Smith chart is a helpful
way to visualize the impedance of a circuit: the black circles correspond to con-
stant resistance, with the rightmost point corresponding to an open circuit (infinite
resistance) and the largest circle corresponding to a short circuit (zero resistance).
The arcs correspond to constant reactance; the horizontal axis corresponds to zero
reactance (Im(Z) = 0), the top arcs correspond to Im(Z) > 0, and the lower arcs
to Im(Z) < 0. The circuit is impedance matched when Z = 50 Ohm (the center of
the Smith chart), the standard value of RF transmission lines. We tested different
components on the pads until we found a peak in reduced reflected power at the
desired frequency. It is also important to note that it was essential that the circuit
was installed in its final location in the experiment when measuring the impedance
76
a. Circuit schematic
Pickup coil
1 3 To antenna
2
Impedance matching
Mount on 1? post
b. Antenna mounted on PCB
Figure 9: We use a commercial resonator with an impedance matching network to
produce high power RF fields. a. Diagram of the impedance matching network. b.
A picture of the resonator mounted on a PCB. We place this device as close to the
atoms as possible next to the glass cell.
as the other parts in the vicinity of the antenna can shift the resonant frequency.
The driving electronics are very similar to the AOM drivers described in Sec-
tion 4.3.2. The only difference is we use a 30 W amplifier (Minicircuits LZY-22+)
instead of the smaller amplifiers needed to drive the AOMs.
4.3.4 6.8 GHz microwave system
We built a 6.8 GHz microwave system to couple atoms between the F = 1
and F = 2 ground hyperfine manifolds of 87Rb. We mostly use this system to
stabilize the bias magnetic field along ez at the atoms using microwave assisted
77
a. Reflections b. Smith chart
Figure 10: Impedance matching of high power RF antenna. a. Reflected power as
a function of frequency. b. Impedance as a function of frequency, visualized on a
Smith chart.
PTAI (Section 3.5.3). Table 4.2 presents a list of components used in the setup is
presented and Figure 11 shows a diagram of the connections.
Table 4.2: 6.8 GHz microwave system components
Part number Description
SRS SG384 Signal generator
Narda 4014C-30 Directional coupler
Marki IRW0618 Mixer
Minicircuits VBFZ-6260-S+ Bandpass filter 6? 8 GHz
Herley D1956 Voltage controlled attenuator
MSI MSH-5727901 46 dB gain amplifier
Narda 4014C-30 Circulator
Minicircuits ZX47-40-S+ Power detector
Maury microwave 1819C Stub tuner
ZYSWA-2-50DR Digital switch
The SRS generator serves as a source of a fixed frequency and amplitude
signal. We control the frequency by mixing a programmable ? 100 MHz signal
from a Novatech into a double balanced mixer; the RF signal can be turned on
78
or off using a TTL switch. The amplitude is controlled by commanding a 0 ? 6 V
signal from the control computer into an attenuator. The signal is amplified by
+43 dB using an amplifier mounted on a water cooled plate. The microwave signal
is broadcast to the atoms using a horn antenna. In order to get a coupling microwave
coupling strenght of the order of 10 kHz or larger it was important to place the horn
as close to the atoms as possible and impedance matching the transmission line to
maximize the radiated power. We additionally use a circulator that prevents any
reflected power to go back into the amplifier and couplers at different locations to
monitor the performance of the system. The last coupler and the circulator of the
system are connected to a power detector that outputs a DC signal proportional
to the microwave power. The impedance is tuned with a stub tuner by changing
the length of the stubs until the reflected power measured at the exit port of the
circulator is minimized at the desired frequency of operation.
4.4 Computer control and data acquisition
There have been two main changes in our computer control and data acqui-
sition system. We have transitioned from using a LabVIEW based control system
to a Python based control system, The labscript suite [67]. With the previ-
ous control software the lab devices were programmed using a graphic interface.
Labscript instead uses a hybrid approach in which the experimental sequences are
text based scripts. The use of scripted programming has given us more flexibility
and modularity for programming experiments and additionally, it is now very easy
79
L R Bandpass
SRS SG384 Coupler filter
-30 dB I Novatech
Circulator Coupler Amplifier Attenuator
-30 dB 0-6 V
Power Power
detector detector
V DC V DC
atoms
Stub tuner Horn
Figure 11: Schematic of 6.8 GHz microwave system.
to do multi-dimensional parameter scans. Each experimental shot is saved in a Hi-
erarchical Data Format version 5 file (HDF5) that includes images from cameras,
oscilloscope traces, and analog inputs as well as copy of the script used in the exper-
iment and the values of all of the parameters used. This has been a great upgrade
as we no longer rely on the person running an experimental sequence pushing the
?save? button and thoroughly documenting the experiment in question1.
The other upgrade worth mentioning is replacing our old Flea3 (FL3-FW-03S1M-C
from FLIR, formerly Point Grey) CCD camera with a Mako G-030 camera from
Allied Vision. With this new camera the time between two consecutive shots can
be as short as 96?s (we used to wait ? 30 ms with the Flea3 camera), greatly reduc-
ing the effect of mechanical vibrations in the experiment that produce fringes in the
1As I have been digging into old data, I greatly wish we had this feature sooner.
80
absorption images. Table 4.3 summarizes relevant specifications of both cameras.
In our experimental sequence the probe and atoms images are separated by 150?s,
which is not enough time for the atoms in the first absorption image to be cleared
out. We therefore had to change the order in the absorption imaging protocol: first
we take a picture of the probe which is 6.8 GHz detuned and minimally disturbs the
atoms, we then apply repump light during the 150?s interval in between the images
to transfer atoms into F = 2 and then take the absorption image of the atoms. Fig-
ure12 shows the OD computed using both cameras with no atoms present. When
there is a long interval in between the two images the probe captured on the camera
changes, leading to the fringes shown in Figure12a. In contrast for probe images
captured within a short interval the main noise contribution is shot noise as can be
seen in Figure 12b. The addition of this camera was essential to get a better signal
to noise ratio in the experiments reported in Chapter 8.
Table 4.3: Relevant specifications of Flea3 and Mako cameras.
Flea3 Mako
Sensor type CCD CMOS
Quantum efficiency at 780 nm ? 40% ? 30%
Dark noise 19.94 e/s 12.9 e/s
Frame rate at max resolution 120 fps 309 fps
Pixel size 5.6? 5.6?m 7.4? 7.4?m
Resolution 648? 488 644? 484
Dynamic range 65 dB 56.7 dB
Analog to digital converter 12 Bit 12 Bit
81
a. Flea3 camera b. Mako camera
0.7
0.0
?0.7
Figure 12: OD computed using two consecutive probe images without atoms. a.
With the Flea3 camera images are spaced by ? 30 ms. Changes in the probe reult
in finges in the OD. b. With the Mako camera images are spaced by ? 150?s.
82
OD
Chapter 5: Fourier Transform Spectroscopy
The high level of control in ultracold atomic systems makes them an ideal
platform for analog simulation of materials and other complex systems. The prop-
erties of these engineered ?atomic? materials depend on the underlying single particle
energies and it is important to characterize them. We worked on Fourier transform
spectroscopy for this purpose.
Many spectroscopy techniques in atomic physics rely on using a source of
coherent electromagnetic radiation with a well known frequency that probes the
internal structure of a system (atom). For example, in absorption spectroscopy [68]
coherent light is sent through an atomic medium and if the frequency of the light
is resonant with an atomic transition it is absorbed and a reduced transmission
is measured. Other variants of spectroscopy (e.g. Rabi spectroscopy [55], spin-
injection spectroscopy [69]) work under a similar principle: atoms absorb and emit
photons with frequencies equal to the transition energies between internal states.
Fourier transform spectroscopy instead employs the connection between the
energy spectrum of a system and its dynamics. This connection has been exploited
to study the spectrum of both condensed matter [70] and cold atom systems [71,72]
alike. As opposed to other techniques, Fourier spectroscopy relies only on follow-
83
ing the unitary evolution of an initial state suddenly subjected to a Hamiltonian
of interest and measuring probabilities in a basis that does not diagonalize that
Hamiltonian.
The frequency resolution of Fourier transform spectroscopy is limited only by
the coherent evolution timescale of the system under study and can otherwise be
applied to any system. Other applications of this technique implemented in our
laboratory that are not included in this Chapter include measuring the dispersion
relation of a Rashba spin-orbit coupled gas (see Chapter 8) and the band structure
of a fractional period adiabatic superlattice [27].
This Chapter is organized as follows: First, I give a general description of the
Fourier transform spectroscopy technique in Section 5.1. In Section 5.2 I describe
a set of experiments where we engineered a tunable spin-orbit coupled system and
applied Fourier transform spectroscopy. This work was published in [25].
5.1 Operating principle of Fourier spectroscopy
We focus on a system where we can measure the occupation probabilities
of a set of orthonormal states {|?i?} that fully span the accessible Hilbert space
of the system. We then consider the time evolution of an arbitrary initial state
|? ? = ?0 ai |?i? as governed by a Hamiltonian H? ?({?i}) and observe the occupation
i
probabilities of the {|?i?} states of the measurement basis as a function of time.
When H? ? is applied, the evolution of the initial state is |?(t)? = ? a c e?iE?jt/h? |??i i,j j?,
i,j
where E ?j and |??j? are the eigenenergies and eigenstates of H? ?, and ci,j(t) = ??i|??j?.
84
Figure 1: a. Eigenenergies of a three-level system described by H? ?(?1,?2,?3). b.
The system is prepared in |? ?2? and subjected to H? at time ti. The three panels
show the occupation probabilities of the states |?1? (blue), |?2? (black), and |?3?
(red) in the measurement basis, for evolution times up to tf . c. Power spectral
density of the occupation probabilities from panel b. The three peaks in the Fourier
spectra correspond to the energy differences present in panel a.
The probability
?
Pk(t) =|?? |?(t)?|2 = | a c ? ?iE
?
jt/h? 2
k i i,jcj,ke | (5.1)
i,j
of finding the system in a state |?k? of the measurement basis can be expressed
as a sum of oscillatory components, with amplitude given by the magnitude of the
overlap integrals between the initial state and the eigenvalues of H? ?
?
Pk(t) = 1 + 2|a2i ci,jcj,kci,lcl,k| cos(2?fj,lt), (5.2)
i,j=6 l
where fj,l = (E ? ?j ? El)/h is the frequency associated with the energy difference
of two eigenstates of the Hamiltonian. Fourier spectroscopy relies on measuring
the populations on each state of the measurement basis as a function of time and
extracting the different frequency components fj,l directly by computing the discrete
85
Fourier transform. The bandwidth and frequency resolution of the measurement
are determined by the total sampling time and the number of samples. For N
samples separated by a time interval ?t, the highest resolved frequency will be
fbw = 1/2?t, with resolution ?f = 1/?tN . This resolution can be decreased if
the Fourier transform is calculated using certain types of windowing functions that
enhance signal to noise ratio. Any higher frequency f > fbw will be aliased and
measured in the Fourier spectrum as falias = |f ?m/?t|, where m is an integer. If
interactions are present in the system, the dynamics get modified in a time scale
given by the magnitude of the interactions, giving an additional constraint to the
smallest frequency components of a single particle Hamiltonian that can be resolved
with our technique.
Figure 1 illustrates the principle of Fourier spectroscopy for a three level sys-
tem, initially prepared in the state |?0? = |?2?, subject to the Hamiltonian
? ?
????E1 0 0 ???? ?
? ?
??? 0 ?1 ?2?
H? ? = ? ? ?
?
+ h? ?
???? 0 E2 0 ???? ?????1 0 ?3?
?
???? , (5.3)?
0 0 E ? ?3 ?2 ?3 0
where we measure the occupation probability as a function of time for each of the
{|?1? , |?2? , |?3?} states. The three eigenenergies E ?i = hfi that result from diago-
nalizing H? ?are displayed in Figure 1a. The three energy differences hfjj? between
the levels determine the oscillation frequencies of the occupation probabilities, as
can be seen in Figure 1b. Finally, Figure 1c shows a plot of the power spectral
86
densities (PSD) with three peaks corresponding to the three relative energies of H? ?.
5.2 Measuring the SOC dispersion with Fourier transform spectroscopy
We applied the Fourier transform spectroscopy technique to measure the dis-
persion relation of spin-1 BECs with an equal superposition of Rashba and Dressel-
haus SOC, and tunable SOC strength.
5.2.1 System
All of our experiments started with BECs containing about 4 ? 104 atoms
in the |f = 1,mF = ?1? hyperfine state. The experiments described in Sec-
tion 5.2.3 were performed in an optical dipole trap with frequencies (?x, ?y, ?z)/2? =
(42(3), 34(2), 133(3)) Hz. We later modified the trapping frequencies in the xy
plane to try to make them more symmetric for the experiments described in Sec-
tion 5.2.4. We broke the degeneracy of the three mF magnetic sub-levels by apply-
ing a 1.9893(3) mT bias field along ez that produced a ?Z/2? = 14.000(2) MHz
Zeeman splitting, and a quadratic Zeeman shift  that shifted the energy of
|f = 1,mF = 0? by ?h?(28.45 kHz). We transfered atoms into the |f = 1,mF = 0?
state using ARP (see Section 3.5.2) and then we monitored and stabilized the
magnetic field using partial transfer absorption imaging (Section 3.5.3) by apply-
ing a pair of 250?s microwave pulses, each of them detuned by ?2 kHz from the
|f = 1,mF = 0? ? |f = 2,mf = 1? transition.
We induced spin-orbit coupling using a pair of intersecting, cross polarized
87
Raman frequencies
Figure 2: a. Setup. A bias magnetic field B0ez, with B0 = 1.9893 mT splits the
hyperfine energy levels of the f = 1 manifold of 87Rb by ?Z/2? = 14 MHz. A pair of
cross polarized Raman beams propagating along ex+ey and ex?ey couple the atoms?
momentum and spin states. b. The Raman frequencies are set to ?A = ?L + ? and
?B = ?L+?Z . We add frequency sidebands to ?B, separated by???. The amplitude
modulation from the interference between the multiple frequency components results
in tunable SOC. c. SOC dispersion for Raman coupling strength ?0 = 12EL and
? = 0, on four photon resonance.
Raman laser beams propagating along ex + ey and ex ? ey, as shown in Figure 2a
and b. The beams had angular frequency ?A = ?L + ? and ?B = ?L +?Z , where 2?
is the, experimentally controllable, detuning from four photon resonance between
mF = ?1 and mF = +1.
Our system was well described by the Hamiltonian including atom-light inter-
action along with the kinetic contribution
h?2q2
H? x 2SOC = 2 + ?qxF?z + 4EL1? + h??RF?x + (4EL ? )(F? ? 1?) + h??F? ,
(5.4)
m z
z
where q is the quasimomentum, F?x,y,z are the spin-1 angular momentum matrices,
? = h?2kL/m is the SOC strength, and ?R is the Raman coupling strength, propor-
88
tional to the Raman laser intensity. The Raman field coupled |mF = 0, q = qx? to
|mF = ?1, q = qx ? 2kL?, generating a spin change of ?mF = ?1 and imparting a
?2kL momentum. The eigenstates of H?SOC were linear combinations of these states
and |mF = 0, q = qx?, and the set {|mF , q?} constituted the measurement basis for
Fourier transform spectroscopy.
Figure 2c shows a typical band structure of our spin-1 SOC system as a func-
tion of quasimomentum for a large and negative quadratic Zeeman shift ? > 4EL.
In this parameter regime, the ground state band had a nearly harmonic dispersion
with an effective mass m? = h?2[d2E(kx)/d2x]?1, only slightly different from that of
a free atom.
5.2.2 Tunable SOC
We engineered a highly tunable dispersion relation in which we could indepen-
dently control the size of the gap at qx = 0 as well as the SOC strength ? by adding
frequency sidebands to one of the Raman beams. The state of the system could
change from |mF = ?1, q = qx + 2kL? to |mF = 1, q = qx ? 2kL? by absorbing a red
detuned photon first followed by a blue detuned photon and vice versa, in a similar
way to the M?lmer-S?rensen entangling gate in trapped ion systems [73]. When we
set the angular frequencies of the sidebands to ? = ?A + ?Z ? ??, the Hamilto-
nian (Equation 5.4) acquired a time-dependent coupling ?R(t) = ?0 + ? cos(??t).
This periodically driven system was well described by Floquet theory [74] (see Sec-
tion 3.5.5). Figure 3 shows the spectrum of Floquet quasi-energies for a system
89
described by Equation 5.4 where ?R oscillated with angular frequency ??.
Figure 3: a. Floquet quasi-energies of a three level Hamiltonian with SOC and
time periodic coupling strength. The quasi-energies are grouped into manifolds
consisting of three levels that get repeated with a periodicity equal to h???. b.
Energy differences of the Floquet quasi-energies. Each color represents the energy
difference, separated by a fixed number of neighboring levels. When the number
of neighboring levels is a multiple of 3, the energy differences are straight lines, a
result of the periodic structure of the Floquet manifolds.
We found an effective time-independent Hamiltonian using a unitary trans-
formation U?(t) and applying a RWA. The time evolution of the transformed wave
function |??? = U? ? |?? was given by the time dependent Schro?dinger equation for
the Hamiltonian H? ? = U? ?H?U? ? ih?U? ??tU? . We used
U?(t) = exp[? ?i sin(??t)F?x] (5.5)
??
so that ih?U? ??tU? = h??R(t)F? . The transformed Hamiltonian H? ?x (t) had terms
proportional to sin(?/?? sin(??t)), sin2(?/?? sin(??t)), cos(?/?? sin(??t)) and
90
cos2(?/?? sin(??t)) which we simplified using the Jacobi-Anger expansion
??
cos(z sin ?) = J0(z) + 2 J2n(z) cos(2n?) ? J0(z)
? n=1?
sin(z sin ?) = 2 J2n+1(z) sin((2n+ 1)?) ? 0,
n=0
where Jn is the the nth order Bessel function of the first kind and we neglected the
?fast? terms proportional to cos(2n?) and sin((2n + 1)?), essentially performing a
RWA.
This approximation is valid for h??? > ||+ 12EL and |qx| ? 2kL so that quasi-
energy manifolds are well separated as in Figure 3a. The resulting Hamiltonian
retained the form of Equation 5.4 with renormalized coefficients and an additional
coupling term
H?Fl =H?SOC(q,?0, ??, ??, ?) + ??F? , (5.6)xz
where ?? = J0(?/??)?, ?? = 1/4( + 4EL)[J0(2?/??) ? 1], ?? = J0(?/??)?, and
? = 1/4(4EL ? )? 1/4(4EL + 3)J0(2?/??). F?xz is the ??4 Gell-Mann matrix that
directly couples |mF = ?1, q = qx + 2kL? and |mF = +1, q = qx ? 2kL? states. The
experimentally tunable parameters ??, ? and ?0 can be used to tune the SOC
dispersion.
91
5.2.3 Application of Fourier spectroscopy
We used Fourier transform spectroscopy to measure the spectrum of the SOC
Hamiltonian (Equation 5.6) for three coupling regimes: (i) ?0 6= 0 and ? = 0,
(ii) ?0 = 0 and ? =6 0 and (iii) ?0 6= 0 and ? 6= 0. We turned on the Raman
laser non-adiabatically, in approximately 1?s. We let the system evolve subject to
H?SOC for up to 900?s, and then turned off the laser while releasing the atoms from
the optical dipole trap. We resolved the spin and momentum distribution using a
Stern-Gerlach gradient and a 21 ms TOF which allowed us to measure the fraction
of atoms in each state of the measurement basis {|mF , q?}. We chose the density
of sampling points and the maximum evolution time so that the bandwidth of the
Fourier transform was comparable to, or larger than, the highest frequency in the
evolution of the system while maximizing resolution. Experimental decoherence
resulting in loss of contrast of the oscillations due to magnetic field noise and small
magnetic field gradients present in our apparatus, was an additional constraint that
became significant around 1 ms.
Working with a BEC with k = 0 gave us access to only a single point in the
dispersion relation. In order to map the full spin and momentum dependent disper-
sion relation of H?SOC, we measured the time dependent occupation probabilities at
a fixed Raman coupling strength and different values of Raman detuning ? for the
same initial state. The detuning corresponded to the Doppler shift experienced by
atoms moving relative to a light source with quasimomentum qx/kL = h??/4EL. We
controlled the frequency and the detuning of the Raman beams using two AOMs,
92
one of which was driven by up to three phase coherent frequencies (the carrier fre-
quency plus two sidebands). For each of the three coupling cases that we measured,
we applied the Raman beams at detuning values within the interval ?12EL which
corresponds to quasimomentum values ?3kL.
This approach of changing detuning rather than using atoms with non-zero
quasimomentum had the advantage that the state preparation was very reliable
(making BECs at rest is easy1!) and we got very good signal to noise ratios due to
the relatively high densities of the BECs. The downside was that if one is interested
in looking at a large range of different momenta it can take a long time to repeat
each experiment for different detuning values. In subsequent implementations of
Fourier transform spectroscopy (Chapter 8 and [27]) we sacrificed some signal to
noise ratio for speed and used the momentum distribution of non-condensed atoms
to parallelize our measurements.
We mapped the band structure of SOC atoms for three different coupling
regimes. Figure 4a shows representative traces of the measured occupation proba-
bilities for short evolution times along with fits to the unitary evolution given by
H?SOC with ?, ?0, and ? as free parameters. The fit parameters agree well with
independent microwave and Raman power calibrations. In the lower two panels,
where the Raman coupling strength was periodically modulated, the occupation
probabilities oscillated with more than three frequencies since the full description of
the system was given by a Floquet quasi-energy spectrum. Figure 4b,c shows the
occupation probabilities for the parameter regime (iii) for longer evolution times
1Well, nothing in the lab is really ?easy?...
93
1.0 1 |mF = ?1, qx = 2kL?
0
1 |mF = 0, qx = 0?
0.5
0
1 |mF = 1, qx = ?2kL?
0.0 0
0 400 800 0 45 90
Pulse time [?s] Frequency [kHz]
Figure 4: a. Occupation probability for the three states in the mea-
surement basis |mF = ?1, q = qx + 2kL? (blue), |mF = 0, q = qx? (black), and
|mF = +1, q = qx ? 2kL?(red), following unitary evolution under H?SOC for times
up to 100 ?s at different spin-orbit coupling regimes: (i) ?0 = 9.9EL, ? = 04,
? = 5.8EL, (ii) ?0 = 0, ? = 8.6EL, ? = ?0.7EL, ?? =  + 12EL, and (iii)
?0 = 1.5EL, ? = 8.4EL, ? = ?4.7EL, ?? =  + 17EL. b. Occupation probability
for long pulsing up to 800 ?s for parameters as in (iii). c. Power spectral density of
the occupation probability. We subtract the mean value of each probability before
taking the Fourier transform to remove peaks at f = 0. The peaks in the PSD then
correspond to the relative eigenenergies of H?SOC .
along with the PSD of the occupation probability of each spin state.
We used a non-uniform fast Fourier transform algorithm (NUFFT) on a square
94
Occupation probability
PSD [arb. units]
Figure 5: Power spectral density of the time dependent occupation probability for
each state in the measurement basis for three coupling regimes: (Left) ?0 = 9.9EL,
? = 0, (Center) ?0 = 0, ? = 8.6EL, ?? =  + 12EL, and (Right) ?0 = 4.9EL,
? = 8.4EL, ?? = +17EL. Each panel is normalized to peak amplitude to highlight
small amplitude features in the PSD of the periodically driven SOC, and the highest
value on the frequency axis corresponds to the FFT bandwidth.
window to obtain the power spectral density of the occupation probabilities since
our data points were not always evenly spaced because of imperfect imaging shots.
The heights of the peaks in the PSD are related to the magnitude of the overlap
integrals between the initial state and the Raman dressed states. Figure 4c shows
the raw PSD of the time evolution of the system under H?SOC for a given Raman
coupling strength and detuning. We put together all the PSDs for the three coupling
regimes in the spectra shown in Figure 5. Each column corresponds to a different
95
coupling regime and the colors represent the different spin states of the measurement
basis. The spectra show that some overlap integrals vanish near ? = 0, which is
manifested as missing peaks in the PSD. The periodic structure of the Floquet
quasi-energy spectrum gave rise to peaks at constant frequencies of ?? and 2??
independently of the Raman detuning, and a structure that is symmetric about the
frequencies 2?f1 = ??/2 and 2?f2 = ??. A reader is interested in seeing another
nice experiment where the Floquet quasienergy spectrum becomes can be visualized
is advised to see [75].
5.2.4 Effective mass
Fourier transform spectroscopy only gives access to the relative energies of
a Hamiltonian. If we want to recover the absolute energies we need to have an
additional energy reference. The particular Hamiltonian H?SOC had a ground state
with a nearly quadratic dispersion. We measured its effective mass to obtain the
ground state energy which we used as a reference to recover the absolute energies.
We measured the effective mass of the Raman dressed atoms by adiabatically
preparing the BEC in the lowest eigenstate and inducing dipole oscillations [31]. The
effective mass of the dressed atoms was related to the bare mass m and the bare and
dressed trapping frequencies ? and ?? along the Raman recoil direction by the ratio
m?/m = (?/??)2. We measured this ratio following [19]. To induce the oscillations
we started in |mF = 0, kx = 0? state and adiabatically turned on the Raman laser
in 10 ms while simultaneously ramping the detuning to ? ? 0.5EL which shifted the
96
minima in the ground state energy away from zero quasi-momentum. We suddenly
brought the field back to resonance causing an abrupt change in the dispersion
relation that excited the dipole mode of the BEC. To obtain the bare state ? we
only used the Raman beams to induce the oscillations and then subsequently turned
off the beams while the BEC oscillated in the dipole trap and to obtain ?? the
Raman beams were kept on the whole time. For this set of measurements, we
adjusted our optical dipole trap to give new trapping frequencies (?x, ?y, ?z)/2? =
(35.6(4), 32.2(3), 133(3)) Hz, nominally symmetric in the plane defined by ex and
ey.
The Raman beams were co-propagating with the optical dipole trap beams;
therefore, the primary axes of the dipole trap frequencies were at a 45? angle with
respect to the direction of kL. The kinetic and potential terms in the Hamiltonian
including the contribution of the Raman and optical dipole trap were
h?2q2x h?
2q2
= y mH? 2 ?2 ?2 ?2? 2 +m? 2 + [? ?x + ? y ]m 2 x y
h?2= 2 + 1 2 + m2 kx 2 ky 4 [(?
2 2 2 2 2 2
x? + ?y?)(x + y ) + 2xy(?x? ? ?y?)], (5.7)m? m
where x?
? ?
= (x+ y)/ 2 and y? = (x? y)/ 2 are position coordinates rotated by 45?
. For an axially symmetric trap with ?x? = ?y? , the frequency of oscillation along
the Raman recoil direction is
2 = m? 2 2x 2m? (?x? + ?y?). (5.8)
97
Our trap had a small 3.4 Hz asymmetry and there was some coupling of the
motion along the axis perpendicular to kL which becomes more significant at larger
values of the effective mass. The sampling times for the measurements were small
compared to the trap asymmetry and we can locally approximate the motion of
the atoms by a simple harmonic function with a frequency along ex given by Equa-
tion 5.8.
Figure 6 shows the dipole oscillations along the ex and ey directions for the
three different coupling regimes we explored, as well as the bare state motion. The
resulting mass ratios for the three coupling regimes are m/m? = (i) 1.04(8), (ii)
0.71(7), and (iii) 0.62(4).
Figure 6: Oscillation of the BEC in the dipole trap along the recoil directions ex
and ey for (top) bare atoms, and the three parameter regimes that we explored (i),
(ii), and (iii). We believe that the observed low amplitude oscillations along ey are
due to the initial detuning ramp not being fully adiabatic.
98
5.2.5 Measured dispersions
Figure 7 explains in detail the interpretation of the multiple peaks in the PSD
and the steps that were taken to recover the dispersion relation using the effec-
tive mass and the Fourier spectra. The red line in Figure 7a represents a level
within a Floquet manifold that has the largest overlap integral with the initial
|mF = 0, q = 0? state. The peaks in the PSD correspond to energy differences be-
tween the marked level and the levels in neighboring Floquet manifolds pointed
by the colored arrows. We show the theoretically computed energy differences on
top of the measured PSD in panel b. The lowest frequency dominant peaks of the
PSD correspond to energy differences with the adjacent lower Floquet manifold. To
properly recover the SOC dispersion we shifted the PSD by a negative quadratic
term ?h?2q2/2m?x as we show on panel c. We finally invert the frequency axis and
shift it by ??.
We obtained the characteristic dispersion of a SOC system after adding a
quadratic term to the PSD, proportional to the measured effective mass and rescal-
ing the detuning into recoil momentum units. We combined the PSD of the time
evolution of the three |mF ? states to look at the spin dependence of the spectra. Fig-
ure 8 shows the measured dispersion relations as well as the Floquet quasi-energies
calculated for the Hamiltonian parameters obtained from our calibrations. The spec-
tral lines that can be resolved with our technique depend on the overlap integrals
of the initial state with the target Hamiltonian eigenstates. Additional energies can
be measured by repeating the experiment with different initial states. The spectral
99
Figure 7: a Floquet quasi-energy spectrum of a SOC Hamiltonian with periodic
coupling strength. The red line represents the eigenstate that has the largest over-
lap with the initial |mF = 0? state. The arrows indicate the energies of the states
that have non-zero overlap with the initial state and can be measured with Fourier
transform spectroscopy. b PSD of the occupation probability and numerically cal-
culated energy differences between the levels indicated by the arrows on panel a. c
PSD shifted by a quadratic term ?h?2q2x/2m?. The red box indicates the region of
interest where we can recover the SOC spectrum. d We invert the frequency axis
and shift it by ??.
lines we were able to resolve are in good agreement with the calculated energies of
the Hamiltonian.
100
Figure 8: Spin-dependent SOC dispersion for three different coupling regimes. We
combine the PSD of the occupation probability of the states |mF = ?1, qx = ?2kL?,
and shift each frequency by an amount proportional to the squared quasimomentum
and the effective mass. The dashed lines are the calculated Floquet energies for the
Hamiltonian using our calibration parameters.
Conclusion
This chapter introduced the basic principles of the Fourier transform spec-
troscopy technique and used it to measure the spin and momentum dependent
dispersion relation of a spin-1 spin-orbit coupled BEC. We additionally studied
a periodically driven SOC system and found a rich Floquet quasi-energy spectrum.
Our method can be applied generically to any system with long enough coherent
evolution to resolve the energy scales of interest and could prove particularly useful
to study systems where it is harder to predict or compute the exact energies, such
as cold atom realizations of disordered or highly correlated systems [76]. In our lab,
this technique has been used to study the spectrum of a Rashba spin-orbit coupled
system [26] and of a fractional period adiabatic superlattice [27].
Our main initial interest was to create tunable spin-orbit coupling and Fourier
spectroscopy was conceived as a tool to characterize it. We realized that the use of
101
Raman transitions from multiple frequency beams was equivalent to another exper-
iment that achieved tunable SOC using amplitude modulated Raman coupling [77].
102
Chapter 6: Synthetic clock transitions through continuous dynami-
cal decoupling
Most of the experimental techniques described in this thesis so far have used
the hyperfine |mF ? states as effective spins dressed with an RF or Raman field.
However, due to the linear dependence of their energies with respect to magnetic
field, and our lack of control of environmental changes we always had to take special
care to stabilize the magnetic field in the laboratory (see Section 3.5.3). An alterna-
tive to doing active magnetic field stabilization is to use clock transitions which are
first-order insensitive to changes in magnetic field but unfortunately they are not
present in all systems or for arbitrary system parameters. However, under almost
all circumstances, clock transitions can be synthesized using dynamical decoupling
protocols. These protocols involve driving the system with an external oscillatory
field, resulting in a dynamically protected ?dressed? system.
The idea of implementing continuous dynamical decoupling (CDD) came from
a theoretical proposal to engineer Rashba type SOC (see Chapter 8) using Raman
beams and a strong RF field [78], the second being a necessary ingredient for CDD.
We initially worked in implementing CDD protocols to create ?synthetic clock states?
as an intermediate step towards our final goal of engineering Rashba SOC. Just like
103
with Fourier spectroscopy, CDD became a workhorse of the laboratory both for
the stability it provides against environmental fluctuations and because it has given
us access to non-zero matrix coupling elements that we otherwise would not have
when working with the bare |mF ? states. We have continued to use CDD not only
for engineering Rashba SOC (Chapter 8) but also to engineer subwavelength optical
lattices [27] and Hofstadter [79] cylinders (work in preparation). On the theory side,
we developed a proposal that uses them as a platform for emulating PT symmetric
Hamiltonians [80].
This Chapter discusses the implementation of CDD in the ground F = 1
hyperfine manifold of ultracold 87Rb. First I give a general overview of dynamical
decoupling and continuous dynamical decoupling. Then I describe the technical
details and characterization of our CDD protocol which produces a protected three-
level system of dressed-states whose Hamiltonian is fully controllable. Finally, I
discuss an implementation of concatenated CDD that renders the system first-order
insensitive to both magnetic field noise and noise in the control field. This work was
published in [81] and was done in parallel with [82].
6.1 Basic principles of CDD
Dynamical decoupling (DD) protocols consist in applying an external control
Hamiltonian, generally implemented by a series of pulses, which has the effect of
canceling out the dynamics that arise from ta quantum system coupling to the en-
vironment. DD was first introduced in the context of nuclear magnetic resonance
104
(NMR) with the discovery of spin-echoes [83], where a single ?refocusing? pulse was
applied to eliminate dephasing of spins resulting from variations in magnetic field.
These ideas were later generalized in [84] to protect a system from decoherence
induced by interactions with a quantum environment. Continuous dynamical de-
coupling (CDD) relies on the application of time-periodic continuous control fields,
rather than a series of pulses.
A number of dynamical decoupling protocols, pulsed or continuous, have been
shown to isolate quantum systems from low-frequency environmental noise [85?93].
Thus far, CDD has inoculated multi-level systems in nitrogen vacancy centers in
diamond, nuclear magnetic resonance experiments, and trapped atomic ions [94?
101], from spatiotemporal magnetic field fluctuations.
6.2 CDD of a spin-1 system
40
0.4 | 1
RWA
20 0.2 |y
|0
0 0.0 |x
20 0.2 |1|z
0.4
40 0 1 2 3 4 5 400 0 400
B (mT) /2  (kHz)
Figure 1: a. Setup for implementing CDD using a strong RF magnetic field. b. Left:
dependence of the 52S 871/2, F = 1 ground state of Rb on magnetic field, where the
quadratic dependence of the |mF = 0? state?s Zeeman shift has been exaggerated
so it is visible on the same scale. Center: energies of the |xyz? eigenstates, for
?/2? = 200 kHz (black curves) and ? = 0 (grey curves). Right: TOF absorption
image of |z? at ? = 0, showing the constituent |mF ? states.
105
E/h (MHz)
We implemented CDD using a strong RF magnetic field with strength ? which
linked the three |mF ? states comprising the F = 1 electronic ground state manifold
of 87Rb. The RF field was linearly polarized along ex, and had angular frequency ?
close to the Larmor frequency ?0 = gF?BB0 from a magnetic field B0ez; gF is the
Lande g-factor and ?B is the Bohr magneton. Using the rotating wave approximation
for the frame rotating at ? (which is valid for ? >> ?), the system was described
by
H? = h??F? + h?(F? 2z z ? 1?) + h??F?x, (6.1)
with detuning ? = ? ? ?0, quadratic Zeeman shift , spin-1 angular momentum
operators F?x,y,z, and the identity operator 1?.
6.3 The |xyz? states
The eigenstates of Equation 6.1 correspond to the CDD basis, and we denote
them the |x?, |y? and |z? states.
106
6.3.1 State decomposition
The CDD states are linear combinations of the |mF ? basis states, and for
? = 0 the (non-normalized) eigenvectors are:
|x? = |?1? ? |1? ,
|y? = |?1? ? ?+ ?? |0?+ |1? , (6.2)
2?
| ? ??z? = |?1? ? ? |0?+ |1? .
2?
Figure 2 shows the normalized full state decomposition as a function of ?. The |xyz?
states adiabatically map to the |mF ? states for |?|  ?: for positive (negative)
detuning |z? maps to |1? (|?1?); |y? maps in the exact opposite way to |z?; and |x?
always maps to |0?.
|z |x |y
1.0
0.8
0.6
0.4
0.2
0.0
200 0 200 200 0 200 200 0 200
/2  (kHz) /2  (kHz) /2  (kHz)
Figure 2: Decomposition of the |xyz? states on the |mF ? basis for ?/2? =
145(1) kHz. We adiabatically prepared the system in any of the |xyz? states and then
suddenly turned off the RF dressing field. The |mF = ?1, 0, 1? states correspond to
blue, orange, gray respectively.
107
population
We labeled our dressed states |xyz? since for ?? 0+ and ? = 0, they contin-
uously approach the |XY Z? states familiar from quantum chemistry [21]:
|X? = |1? ??|?1? ,
2
| ? = |1?+?|?1?Y i , (6.3)
2
|Z? = |0? .
which transform under the application of the spin-1 operators as jklF?j |k? = ih? |l?,
so that a resonant probe field can induce transitions between at least one pair of
states, irrespectively of its polarization.
Finally, when ??? they are independent of the driving field amplitude and
continuously approach the eigenstates of the F?x operator
|x? = |1? ? |?1? ,
?
|y? = |1?+ 2 |0?+ |?1? , (6.4)
?
|z? = |1? ? 2 |0?+ |?1? .
6.3.2 Energies
The clock-like nature of these states is determined by their eigenvalues which
are even functions with respect to ? as can be seen by the leading order expansion
108
of the eigenenergies Ei = h??i for ?? 0

? 2x =? ?2 ? +O(?
4),
=1(? + ??)? (+ ??)?  ?2y 2 +O(?
4), (6.5)
?2 ? 4?2 + ??
=1?z 2(?? ??) +
(? ??) ?2,
2 + 4?2 + ??
?
where we have defined ?? = 4?2 + 2. The energy differences h??xy, h??zy and h??zx
are only quadratically sensitive to ? for ?  ? 1 so that detuning fluctuations
?? are suppressed to first order, making these a trio of synthetic clock states. The
curvatures of ?x and ?z have the same sign and in principle there is a critical
value of ? where the quadratic term in transition energy can be made arbitrarily
small, making it quartic in ?. However, this cancellation does not take place when
we consider the dependence of  on ? using the Breit-Rabi expression. It is still
possible to find an optimal ? for which ?zx depends quartically on ?, but it does
not occur at ? = 0 as is predicted by Equation 6.5 for constant .
6.3.3 Transition matrix elements
Unlike the |mF ? basis, an oscillatory magnetic field with the right polarization
can drive transitions between all pairs of the |xyz? states with non-zero transition
matrix elements. The transition matrix elements between the |xyz? have a depen-
dence on both ? and ?. For the ? = 0 case they can be read from the representation
of the spin-1 matrices in the |xyz? basis
1The energies are quadratic in ? for ? ?, and linear for ? ? with a slope of 7 MHz/mT.
109
?? ??? 2? 0 ? ? ??? ?? ???
F? ?x ?
???? 0 0 0 ????
?
?  0 ?2??? ??
? ?
??? 0 ? ?i(???)? 0 ?? ? (???)2 ?? 2 +4 ??
? ??? ?i(???)F? ? 0 ? ?
??
i(??+) ??
y
? (???)2 (??+)2 ??? 2 +4 ? +4? ?2? ?
??
0 ?i(??+) ??0
? (??+)
2
2 +4?
??? ?
?

? 0 ? ?? +1? ?2 0? ?
?
? ? ??  ?F? ? ???
+1
???
2 0 ?? 2 ??z ? (??+ )
2
 ?? , (6.6)? ?2 +4? 0 ??
??
2 0 ??
(??+ 2)
?2
+4
where the states have been ordered by decreasing energy (|y?, |x?, |z?). We see
that a term in a Hamiltonian that is proportional to F?x can only drive transitions
between |z? and |y? and that a coupling term proportional to F?y and F?z can drive
both drive transitions between |z? and |x? or |x? and |y? with different strengths.
The matrices in Equation 6.6 show that when ? and  are comparable in magnitude
there exists at least one non-zero transition matrix element for each pair of dressed
states and they can all be coupled cyclically.
110
6.4 |xyz? state preparation
We implemented CDD to BECs with N ? 5? 104 atoms in the F = 1 ground
hyperfine manifold of 87Rb. For all of the experiments described in this Chapter the
dipole trap had trapping frequencies of (fx, fy, fz) = (42(3), 34(2), 133(3)) Hz. We
applied a B0 ? 3.27 mT bias field that lifted the ground state degeneracy, giving
an ?0/2? = 22.9 MHz Larmor frequency, with a quadratic shift /2? = 76.4 kHz.
We determined that the ambient magnetic field fluctuations were dominated by
contributions from line noise giving an RMS uncertainty ??/2? = gF?B?B/h =
0.67(3) kHz.
The state preparation consisted of two stages of ARP. On the first stage we
followed the usual protocol described in Section 3.5.2 to prepare the BEC in any of
the |mF = 0,?1, 1? states. On the second stage, we adiabatically transformed the
|mF ? states into the |xyz? states. We started with the bias field far from resonance
(?(t = 0)/2? ? ?450 kHz) and with all coupling fields off. Then we ramped on
? in a two-step process. We first ramped from ? = 0 to an intermediate value
?mid, approximately half its final value in 1 ms. We then ramped ? to zero in 3 ms
by increasing the magnetic field B0. After allowing B0 to stabilize for 30 ms, we
ramped the RF dressing field to its final value ? in 1 ms, yielding the dynamically
decoupled |xyz? states. It was important that we waited for the field to stabilize at
an intermediate ?mid as we found several times that the capacitors on the impedance
matching network of the antenna used to generate the RF field would burn if we
kept the power on for too long.
111
After performing any experiment with the |xyz? states we measured their pop-
ulations by adiabatically deloading them back into the |mF ? basis. We first ramped
B0 so that ? approached its initial detuned value in 2 ms, and then ramped off the
dressing RF field in 1 ms. A typical experimental sequence for ? and ? can be
visualized in Figure 3. As usual, we obtained the spin-resolved momentum distri-
bution using absorption imaging after TOF, with a Stern-Gerlach field to spatially
separate the spin components. The right panel of Figure 1b shows a TOF image
of the |mF ? state decomposition of the |z? state. For this image as well as for the
measurement of the dressed state decomposition shown in Figure 2 we suddenly
(not-adiabatically) turned the RF coupling off, thereby projecting the |xyz? states
back into the |mF ? basis.
Figure 3: Detuning and RF coupling strengths ramps (not to scale) performed to
adiabatically prepare the |xyz? states starting in the |mF ? states and vice versa.
112
6.5 Initial characterization of ?
Producing RF fields with large coupling strength was not a trivial task and
when testing different antenna designs it was important to have an easy and quick
way of characterizing them. We mostly relied on two different techniques to get an
initial estimate of ?: first, we prepared atoms in |mF = ?1? and pulse on the RF to
drive transitions between the three |mF ? states. We fit the populations in the three
states as a function of pulsing time to the time evolution given the time dependent
Scho?dinger equation for the RF Hamiltonian (Equation 6.1) using a least squares
minimizer with ? and ? as free parameters.
1.0
mf = ?1
0.5 mf = 0
mf = 1
0.0
0 5 10 15 20
1.0 Pulse time [?s]
mf = ?1
0.5 mf = 0
mf = 1
0.0
0 2 4 6 8 10
Pulse time [?s]
Figure 4: We prepared the system in the |mF = ?1? state and pulsed ? for variable
times. We used a least squares minimizer to fit the populations in the |mF ? states
as a function of pulsing time to get an initial estimate of ?. The top panel shows
the time evolution of ?/2? ? 76 kHz and the bottom panel shows the evolution for
?/2? ? 238 kHz
Alternatively, we followed the loading procedure described in Section6.4 but
113
Fractional populations Fractional populations
suddenly turned ? off for different values of ? to get the decomposition of the |xyz?
states in terms of the |mF ? states. We then fit the populations to the eigenstates of
the Equation 6.1 with ? and ? as free parameters. Figure 2 is an example of such
type of calibration.
For an antenna with a quality factor such as ours (q ? 20) we could not
?suddenly? turn ? on or off as it takes some time for power to build up and to die
out when the RF fields are turned on or off2. If we did not include this into the
model used to calibrate ? we could get some results that were slightly off. We only
used these measurements as initial estimates and once we found an antenna design
that could produce a large enough ? we used the spectroscopy techniques described
in the next section to fully characterize the system.
6.6 Spectroscopy
We confirmed our control and measurement techniques spectroscopically by
measuring the energy differences between the |xyz? states with an additional probing
field with angular frequency ? + ?p, coupling strength ?p and polarized along ey.
In the frame rotating with angular frequency ? and after using a RWA the system
was described by the Hamiltonian
H? = ?F?z + h?(F? 2/h?2z ? I?) + ?F?( x )
+ ?p sin(?pt)F?x + cos(?pt)F?y . (6.7)
2It is possible to get the desired response by using a different function for the turn on/off that
overshoots (not just a step).
114
In this rotating frame the probe field initially polarized along ey had components
along ex and ey, resulting in at least one non-zero transition matrix element for all
transitions between pairs of dressed states. If the probing field was polarized along
ez we would not be able to drive the zy transition as can be seen from the matrix
elements in Equation 6.6.
To probe the dependence of the |xyz? state energies on detuning, we performed
Rabi spectroscopy (Section 3.5.1) by pulsing ?p on for a constant time and scanned
?p for different values of ?. Figure 1b shows the spectroscopically resolved values
of ?xy/2?, ?yz/2?, and ?zx/2? for ?/2? = 194.5(1) kHz and the side panel shows
a sample spectra measured with coupling strength ?p/2? ? 1 kHz and ?/2? ?
9 kHz. The dashed curves were computed by diagonalizing Equation 6.1, and they
clearly depart from our measurements for the zx transition. This departure results
from neglecting the weak dependence of the quadratic shift  on bias field B0. In
near-perfect agreement with experiment, the solid curves from the full Breit-Rabi
expression account for this dependency.
6.7 Robustness
We focused on the robustness of the zx transition which can be made virtually
independent of magnetic field variations due to the similar curvature of ?z(?) and
?x(?) (see the middle panel of Figure 1b). We quantified the sensitivity of this tran-
sition to field variations with three methods corresponding to the different markers
in Figure 6b: (1) Triangles denote data using Rabi spectroscopy as in Figure 6a.
115
410 zy
400
390
236 zx
234
180 xy
160
60 40 20 0 20 40 60
/2  (kHz)
Figure 5: Spectroscopic data showing transitions between the |xyz? states for
?/2? = 194.5(1) kHz. The vertical scale of the center panel (zx transition) has
only 10% the range of the other panels. The dashed lines correspond to the Hamil-
tonian of Equation 6.1 while the solid lines include the dependence of the quadratic
shift on ?. Right: representative spectra.
(2) Squares denote data in which a detuned ?-pulse of the probe field transferred
approximately half of the atoms from |z? to |x?. This ?side-of-peak? technique over-
comes the limitation of Rabi spectroscopy being first-order insensitive to changes
in ?zx. (3) Circles describe data using a double dressing technique that will be de-
scribed in Section 6.9. In each case we measured the energy shift from resonance as
a function of detuning (magnetic field) and then used a fourth-order polynomial fit
to extract the RMS residuals ??zx due to the known detuning noise 3. The results
are not consistent with the simple theory from Equation 6.7 (dashed) and instead
require the Breit-Rabi expression (solid) to obtain full agreement 4.
Even at our smallest coupling ?/2? = 69(1) kHz the typical magnetic field
noise was attenuated by two orders of magnitude, rendering it essentially un-
detectable. Ideally, the radius of curvature of ?zx(?) changes sign at about
3Our procedure also quantifies the small fluctuations that survive for spectra that are flat
beyond second order, as in Equation 6.1.
4The fluctuations can be even smaller for a given ? if we allow for ? =6 0.
116
transition frequency (kHz)
p/2  (kHz)
275 /2 = 231.8 kHz
272
255 /2 = 212.2 kHz
252
237 /2 = 194.5 kHz
234
100 50 0 50 100
/2  (kHz)
100
10 2
10 4
10 6
reduced H
Breit-Rabi
10 8
0 1 2 3 4
/
Figure 6: a. Transition frequency ?zx/2? for three values of ?/2?. The dashed
curves correspond to Equation 6.7, while the solid curves use the Breit-Rabi ex-
pression. b. The change in energy from our experimental detuning fluctuations as
measured in the |mF ? basis is ??/2? = 0.67 kHz (red dashed line). Triangles corre-
spond to |xyz? spectroscopy data, squares to side-of-peak ?-pulse data, and circles
to double-dressed data. The black dashed (solid) curve was calculated using Equa-
tion 6.7 (the Breit-Rabi expression). The shading of the data points corresponds to
the Rabi frequencies in Figure 8.
?/2? = 220 kHz, leaving only a ?4 contribution, however, in practice the small
dependence of  on B prevents this perfect cancellation. Still it is possible to see
the changing curvature of ?zx(?) near ? = 0 for different values of ? in Figure 6a.
6.7.1 Optimal response to noise
The sensitivity of the zx transition to detuning fluctuations can be optimized
further by working at ? =6 0 as shown in Figure 7.
117
E/h (kHz)
zx/2  (kHz)
For small values of ?, the optimum value of ? corresponds to one of the
concave features of the zx transition energy that arises due to the asymmetry in-
troduced by the quadratic shift. As ? gets larger, these features merge into a single
one and the optimum value is ? ? 0. The deviation from ? = 0 is due to an overall
tilt of the transition energy coming from the dependence of the quadratic shift on
?. At the optimum point ?/ ? 3 the sensitivity of the synthetic clock transition
is 1.9? 10?7 kHz, c.f, the 87Rb clock transition which scales as 57.5 kHz/mT2 and
gives 5.8? 10?7 kHz.
100 100
10 2
80
60
10 4
40
10 6
20
10 8 0
0 1 2 3 4 0 1 2 3 4
/ /
Figure 7: Left: Optimum response (solid) of the zx transition to detuning fluctua-
tions allowing for finite ? compared to ? = 0 (dashed) for the full Breit-Rabi model
(note the logarithmic vertical scale). Right: The values of ? that correspond to the
minimum derivative of ?zx.
6.8 Driving dressed state transitions
We explored the strength of the probe-driven transitions between these states
by observing coherent Rabi oscillations (Figure 8a) where our BEC was prepared
in |z? and the probe field had strength ?p/2? ? 1 kHz. The top panel shows
118
E/h (kHz)
/2  (kHz)
/2  (kHz)
1
0 0
1
0 69
1
0 87
1
0 110
1
0 138
1
0 167
1
0 194
1
0 223
0.0 0.5 1.0 1.5 2.0
time (ms)
1.0
0.8 zx
0.6 xy
0.4
0.2 zy
0.0 0 100 200 300
/2  (kHz)
Figure 8: a. Rabi oscillations. Phase coherence is maintained throughout the
oscillations in the dressed basis, while it is quickly lost in the |mF ? basis. The
marker size reflects the typical uncertainties on the dressed basis oscillations. b.
Relative transition matrix elements measured by driving Rabi oscillations between
all the |xyz? states at constant probe power for different values of ?. The zx (blue)
and zy (orange) transitions decrease monotonically with increasing ? for ? = 0,
while they increase for xy.
Rabi oscillations between |mF = 0? and |mF = ?1? states for reference, and the
remaining panels show oscillations between |z? and |x?. The observed Rabi frequency
between dressed states decreased with increasing ? indicating a dependence of the
zx transition matrix elements on ?. We repeated this experiment driving all possible
pairs of dressed state transitions at fixed ?p for, and Figure 8b shows the dependence
of these matrix elements on ? for ? = 0.
The coherence of the Rabi oscillations for longer times was limited by gradients
119
relative matrix elements amplitude
in ? that lead to phase separation of the dressed states, and therefore loss of contrast
in the oscillations. This effect was faster for smaller frequency Rabi oscillations. For
example for ?p/2? = 5 kHz we observed coherent Rabi oscillations with almost full
contrast for more than 10 ms while for the ?p/2? = 870 Hz oscillation shown in
Figure 9 the contrast was significantly reduced after 5 ms. The loss of contrast was
even worse when we tried performing a Ramsey sequence where the time evolution
was most sensitive to the environment. One solution to this problem would be to
change the experimental setup to a double loop antenna to generate a more spatially
uniform magnetic field.
1.0
0.5
0.0
0 1 2 3 4 5
Pulse time [ms]
1.0
0.5
0.0
0 1 2 3 4 5
Wait time [ms]
Figure 9: Loss of contrast in coherent oscillations. A Rabi oscillation (top) between
the |z? and |x? states with ?p/2? = 870 Hz decays by 1/e in 4.6 ms and a Ramsey
oscillation (middle) with about 1 kHz frequency decays in about 3 ms. The gradi-
ents in ? lead to phase separation of dressed states and loss of contrast for longer
pulse/wait times.
120
Populations Populations
In comparison, we found that for both Rabi and Ramsey oscillations between
the |mF ? states the phase started deteriorating after a few hundreds of ?s, this is not
surprising due to bias magnetic field temporal noise. We canceled gradient magnetic
fields so that no phase separation of the bare states was observed for > 10 sec. As
a result, the system can in principle undergo coherent evolution without loss of
contrast for a long time but because of field fluctuations between shots what we
observed instead was full contrast noise.
6.9 Concatenated CDD
The driving field ? coupled together the |mF ? states, giving us the |xyz?
synthetic clock states that were first-order insensitive to magnetic field fluctuations.
However, the spectrum of these states is still first-order sensitive to fluctuations of
the driving field ??. Reference [89] showed that an additional field coupling together
with these |xyz? states can produce doubly-dressed states that are insensitive to both
?? and ??: a process called concatenated CDD (CCDD). In our experiment, the
probe field provided the concatenating coupling field. Because ?p  ?, we focused
on a near-resonant two-level system formed by a single pair of dressed states, here
|z? and |x?, which we considered as pseudospins | ?? and | ??. These states were
described by the effective two-level Hamiltonian
h???
H? = ?p 2 ??3 + h?? cos(?pt)??1, (6.8)
121
with energy gap ?? ? ?z,x (shifted by off-resonant coupling to the zy and xy tran-
sitions) and coupling strength ?? ? ?p, as set by the matrix elements displayed
in Figure 8b. Here ??1,2,3 are the three Pauli operators.
We performed a second transformation into a frame rotating with angular
frequency ?p and use a RWA to compute the eigenenergies of Equation 6.8. For
large values of ?? the energies take the values E ? ? 2 ??,? ? ?? /2 + (? ) /2? . Even
though E?,? are still first order sensitive to ? because ?? ? ?z,x ? ?, its effect is
suppressed by a factor of 1/??. Thus, the concatenated CDD field protects from
the fluctuations ??? of the first dressing field in a similar way that CDD provided
protection from detuning noise ??. Table 6.1 summarizes the dependence of the
|xyz? and |??? energies on ?, ? and ??.
Table 6.1: Energies of the CDD and CCDD states as a function of ?, ? and ??. The
dependence on parameters not relevant to the expansion is given by the functions
f1, f2, g1 and g2.
CDD concatenated CDD
? dependence f1(,?)?2
2
f2(?, )?[ ?? ]
?, ?? dependence ? + g1(?, ) 1? ?
2 + ? + g2(?, ) 1 1? ??
We produced doubly-dressed states by doing (one more!) ARP sequence. We
initialized the system in the | ?? state with RF coupling strength ?i. We set the probe
frequency to be ? 20 kHz off resonant with respect to the | ?? ? | ?? transition and
ramped it on in 10 ms. We then ramped ?i ? ?f in 30 ms, changing the detuning
of the probe with respect to the | ???. The experimental sequence can be visualized
in Figure 10. We chose the value of ?f such that it would bring ?p to resonance
122
at ? = 0, creating double dressed states that were equal superposition of | ?? and
| ??. We quantified the sensitivity of this transition to large changes in the detuning
? in terms of the fractional population imbalance ???3? = P?(?)? P?(?), shown in
Figure 11a for ?f/2? = 138.2(1) kHz 5. This signal is first-order sensitive to ??,?,
and provided our third measurement of sensitivity to detuning in Figure 6b denoted
by circles.
Figure 10: Experimental protocol for implementing concatenated CDD. We started
an initial RF coupling strength ?i and ramped on the probe field ?p in a few ms with
?p = ?z,x(?f ) so that it was initially slightly off resonant with the zx transition.
We then ramped the RF field to ?f , brining ?p to resonance.
We compared the fidelity of preparing a superposition of the | ?? and | ??
states to adiabatically preparing a similar superposition of the the |mF = 0? and
|mF = ?1? states using a single ARP (no dressed states involved), both with a probe
field strength of ? 1 kHz. Figure 11b shows the RMS deviation of the population
imbalance measured over a few hundred repetitions of the experiment. The RMS
deviation for the dressed basis is 0.024(1) and is an order of magnitude smaller than
5We chose the maximum value of ? such that the population of |y?, was negligible after de-
loading.
123
for the |mF ? basis 0.29(1), where it practically impossible to prepare a balanced
superposition for the parameters used here 6.
Figure 11c shows the response of the | ?? ? | ?? transition to small changes
in the RF coupling strength ?? for different values of ?p. We prepared an equal
superposition of | ?? and | ?? following the same procedure as before for ?f/2? =
138.2(1) kHz. We then measured how the population imbalance changes for small
variations of ? ? an effective detuning in the ?twice-rotated frame? ? for different
probe amplitudes ?p. We defined a sensitivity parameter d???3?/d?, obtained from
the linear regime of the population imbalance measurements (see inset in Figure 11c).
The robustness of the doubly-dressed states against ?? fluctuations increased with
?p, thus verifying the concatenating effect of CDD in the |xyz? basis.
However promising the application of multiple concatenating fields might seem,
this procedure has a fundamental limitation. Each time a new coupling field is
applied the energies of the dressed states are reduced to something on the order
of magnitude of the coupling strength from the applied concatenating field. For
example, in the experiments we described here we started with |mF ? states with
transition frequencies on the order of MHz. The transition frequencies of the |xyz?
states were reduced to hundreds of kHz (or in general the magnitude of ?). After
applying the second concatenating RF field the transition frequencies of the |???
were reduced to the order of ?p ? 10 kHz which needed to be smaller than ? for the
second RWA to be valid. Therefore we see that after applying multiple concatenating
6In Figure 11b, the noise in the |mF ? basis does not follow a Gaussian distribution. This feature
is typical of line noise in these experiments.
124
fields we are at the risk of having some very robust states that are also very closely
spaced in energy which might not be desirable for some applications.
6.10 Conclusions
We realized a three-level system that is dynamically decoupled from low-
frequency noise in magnetic fields, measured now-allowed transitions between all
three states, and demonstrated control techniques for creating arbitrary Hamiltoni-
ans. These techniques add no heating or loss mechanisms, yet within the protected
subspace retain the full complement of cold-atom coherent control tools such as op-
tical lattices and Raman laser coupling, and permit new first-order transitions that
are absent in the unprotected subspace. These transitions enable experiments re-
quiring a fully connected geometry as for engineering exotic states, e.g., in cold-atom
topological insulators, and two-dimensional Rashba spin-orbit coupling in ultracold
atomic systems [78,102].
The synthetic clock states form a decoherence-free subspace that can be used
in quantum information tasks where conventional clock states might be absent,
or incompatible with other technical requirements [103]. Moreover, their energy
differences are proportional to the amplitude of the dressing field, and hence tunable,
so they can be brought to resonance with a separate quantum system. The effective
quantization axis can be arbitrarily rotated so that the two systems can be strongly
coupled, pointing to applications in hybrid quantum systems [104,105]. Introducing
a second coupling field shields the system from fluctuations of the first, a process
125
that can be concatenated as needed. More broadly, synthetic clock states should
prove generally useful in any situation where fluctuations of the coupling field can
be made smaller than those of the environment.
126
(a) 1.0
0.8
0.6
/2 = 138.2 kHz
100 50 0 50 100
(b) /2  (kHz)
30
20
10
00.0 0.2 0.4 0.6 0.8 1.0
population ratio
(c) 1.0
1
0.8 |x
|z
0.6 0 5 0 5
0.4 /2  (kHz)
0.2
0.0 0 2 4 6 8 10 12 14
p/2  (kHz)
Figure 11: (a) The fractional population imbalance of the ?? transition for ?/2? =
138.2(1) kHz over detuning ?. The dashed curve is calculated using Equation 6.7
and the solid one using the full Breit-Rabi expression. (b) The fidelity of preparing
a balanced superposition of |?? and |?? (dark blue) states compared to |mF = 0?
and |mF = ?1? states (light blue). (c) The robustness of ?, ? transition against
fluctuations ?? for different probe field coupling strengths. The points represent
the slope of the fitted curves to the fractional population imbalance (inset).
127
d 3 /d (1/kHz) counts 3
Pzx
Chapter 7: Topological order in quantum systems
Topological order can be found in a wide range of physical systems, from
crystalline solids [23], photonic meta-materials [106] and even atmospheric waves
[107] to optomechanic [108], acoustic [109] and atomic systems [110]. Topological
systems are a robust foundation for creating quantized channels for transporting
electrical current, light, and atmospheric disturbances. These topological effects
can be quantified in terms of integer-valued invariants such as the Chern number,
applicable to the quantum Hall effect [111, 112], or the Z2 invariant suitable for
topological insulators [113].
The topology of Bloch bands defines integers that serve to both classify crys-
talline materials and precisely specify properties, such as conductivity, that are
independent of small changes to lattice parameters [23]. Topologically non-trivial
materials first found application in metrology with the definition of the von Klitzing
constant as a standard of resistance, which is now applied in the realization of the
kilogram [114]. Today, topological systems have found applications in the engineer-
ing of low loss optical waveguides [106] and present a promising path to quantum
computation [115].
We got interested in topology when working on a system with engineered
128
Rashba [24] type spin-orbit coupling. Our system had non-trivial topology but it
broke from the usual mold of topological materials as it didn?t have an underlying
crystalline structure that conventionally yields to integer Chern numbers.
Before describing our experiments to both engineer and characterize the un-
conventional topology of a Rashba spin-orbit coupled gas, in this Chapter I take
a step back to describe the basic concepts of topology and its applications to the
band theory of solids. The ideas of topology and how exactly one can connect donuts
with band structures might feel a bit obscure and complicated for non-experts in the
field. The concepts introduced in this Chapter will be necessary for understanding
the results presented in Chapter 8.
7.1 Topology in mathematics
Topology is a branch of mathematics that studies continuity [116]. The most
familiar example might be that of objects being continuously deformed into one
another. For example, a donut can be continuously deformed into a coffee mug but
if we want to deform it into a pretzel we need to poke more holes in it. This gives
us some intuition that the donut and the mug must share the same topology, which
is different from that of the pretzel. Topology also studies more abstract objects
but I will limit the discussion to closed two-dimensional surfaces (manifolds) in
three dimensions, which will be enough to provide some intuition when we define
topological invariants for band structures in the following sections.
The topology of 2D surfaces can be classified by the Euler characteristic, and it
129
is related to the local Gaussian curvature of a surface by the Gauss-Bonet theorem.
The Gaussian curvature can be interpreted in the following way: at any point x0 on
a surface we can find a normal vector n? that is orthogonal to the tangent plane of
the surface (red arrow in Figure 1). We can then define a family of planes containing
n? and the intersection between any of these planes and the surface defines a curve.
If we perform a Taylor expansion of the curve around x0 the quadratic term defines
what is known as the normal curvature ?. The orange and blue lines in Figure 1 are
two examples of such curves and their curvatures correspond to the minimum and
maximum value of the surface; they are what is known as the principal curvatures
?min and ?max. The Gaussian curvature at any point of a surface is defined in terms
of the principal curvatures as K = ?min?max [116].
Figure 1: The intersection of planes containing the normal vector n? at x0 (red
star) and the gray surface define a family of curves. The minimum and maximum
curvatures, corresponding to the orange and blue lines respectively, correspond to
the principal curvatures of the surface at x0.
The Gauss-Bonnet theorem states that the integral of the local Gaussian cur-
130
vature over a closed surface is equal to the integer valued Euler characteristic
= 1
?
? 2 KdA, (7.1)? S
which is related to the genus g (number of holes or handles in the surface) by
? = 2(1 ? g). The Gauss-Bonnet theorem is a very powerful result as it relates
the local properties of a surface,the Gaussian curvature, with a global topological
invariant, the Euler characteristic.
In the following sections I will introduce topological invariants in the context
of condensed matter physics, which even though might seem a bit more abstract,
their interpretation can be closely related to the concepts just defined in this section.
7.2 Topological order in condensed matter
Just like topology classifies properties of geometric objects, one important task
of condensed matter physics has been to classify phases of matter. Many of these
phases, for example, magnetic or conducting phases, can be described in terms of
order parameters related to spontaneously broken symmetries [117]. However, in
the past few decades, an increasing number of systems have been found where it is
only possible to understand their phases and properties in terms of the underlying
topology of their quantum states. This new paradigm of physics has been so im-
portant that in 2016 the Nobel prize in physics was awarded to David J. Thoules,
F. Duncan M. Haldane and J. Michael Kosterlitz for the theoretical discoveries of
topological phase transitions and topological phases of matter
131
The effects of topology in condensed matter systems were first observed when
von Klitzing and colleagues [118] measured the quantized Hall resistance in two-
dimensional electron gases subjected to a strong perpendicular magnetic field. The
effect can be understood semi-classically by thinking of the electrons? quantized
cyclotron orbits1 that give rise to Landau levels. If the Landau levels are filled then
there is an energy gap separating two consecutive levels and the material acts as an
insulator but if an electric field is applied the orbits drift and the electrons will be
?skipping orbits? in the edge as can be seen in Figure 2, giving rise to what is known
as edge states.
Figure 2: The quantum Hall effect. An electron gas is confined in a two-dimensional
material and a strong magnetic field is applied perpendicular to the plane. The
electrons on the bulk travel in cyclotron orbits while the electrons on the edge
travel ?skipping orbits?.
In a seminal paper Thoules, Kohomoto, Nightingale, and den Nijs [111] ex-
plained that the quantization of the Hall conductivity is determined by the under-
lying topology of the band structure. Just like the Euler characteristic defined in
1This is an intuitive but not very complete explanation of the quantum Hall effect, see [119] if
you want to learn more about this subject.
132
Equation 7.1 classifies 2D solids that can be continuously deformed without open-
ing or closing holes, there is a topological invariant that classifies band structures
that can be deformed into one another without opening or closing an energy gap.
This invariant, initially known as the ?TKNN invariant?, was later recognized by
the mathematical physicist Bary Simon as the ?first Chern class invariant from U(1)
fiber bundles?2 [121] and the TKNN invariant became what is known today as the
Chern number or Chern invariant. Another very valuable contribution from Simon?s
work was that he made the connection between the Chern number and the Berry?s
geometrical phase [122] which will be defined in the following sections and will allow
us to make a physical interpretation of this otherwise abstract seeming topological
invariant.
7.3 Berry phase and Berry curvature
A Berry or geometric phase is used to describe the phase acquired by a state3
as it moves through a closed trajectory in parameter space. It plays a key role in
topological band theory and can help provide a physical interpretation of the Chern
number.
Consider a Hamiltonian H? that depends on a set of parameters r = (r1, r2, ...).
If the parameters are slowly changed in time, the corresponding change in the sys-
tem can be described by a path in parameter space r(t). The state |?(t)? evolves
2See [120] if you want to dive into hardcore topology.
3Geometric phases are not unique to quantum systems and are also present, for example, in
optics [123].
133
according to the time dependent Scho?dinger equation and at any given time t there
is a basis that satisfies
H?(r) |n(r)? = En(r) |n(r)? (7.2)
for r = r(t). Suppose the system is initially in state |n(r(t = 0))?, if the parameters
are changed slowly such that the adiabatic theorem is valid, then at time t the state
of the system can be written as
{ ? t }
|?(t)? = exp ? i dt?En(r(t?)) exp(i?n(t)) |n(r(t))? , (7.3)
h? 0
where the first exponential term corresponds to a dynamical phase factor, and
the second term is a geometric phase. By imposing that |?(t)? satisfies the time-
dependent Schro?dinger equation one finds that
?n(t) = i?n(r)|?rn(r)? ? r?(t), (7.4)
where the term
An(r) = i?n(r)|?rn(r)? (7.5)
is usually referred to as the Berry connection4 or the Berry vector potential for
reasons that will become apparent. Because eigenvectors can only be defined up to
a global phase, A is a gauge dependent quantity. If we make a gauge transformation
such that |n(k)? ? ei?(k) |n(k)? then the Berry connection is also transformed as
4This is related to the connection defined in differential geometry that is used to describe things
like parallel transport.
134
An(k)? An(k)??k?(k). However if we integrate the Berry connection on a closed
loop ?
?n(C) = An(r) ? dl, (7.6)
C
we obtain the Berry phase which, unlike the Berry connection, is gauge independent
(modulo 2?).
An alternative way to compute Berry?s phase uses Stokes?s theorem from vector
calculus
? ?
An ? dl = ? ??An ? dSC S
= ?n ? dS, (7.7)
S
where the vector field ?n = ??An is known as the Berry curvature or Berry field.
By rewriting the Berry phase in this way, its resemblance with the definition of the
Euler characteristic from Equation 7.1 becomes apparent.
Using some vector calculus identities the Berry curvature can be rewritten as
?n =i[?r ?n|]? [?r |n?]
?
= i[?n| ?r |j?]? [?j| ?r |n?]
j=6?n
= ?n| ?rH? |j? ? ?j| ?rH? |n?i
j=6 n (
, (7.8)
Ej ? E 2n)
where ?n| ?r |j? was replaced with ?n| ?rH? |j? /(Ej ? En) by differentiating Equa-
135
tion 7.2. This expression shows that ?n is a gauge independent quantity as it does
not depend on the derivatives of a particular gauge choice for |n? but rather on ?rH?
which is gauge independent. Also we can see that ?n becomes singular when there
are degeneracies present in the Hamiltonian, and these degeneracies act as ?sources?
for the Berry curvature. Finally, even though the system may remain in state |n?
during the adiabatic evolution, this expression for the Berry curvature makes it ex-
plicit that other eigenstates of the Hamiltonian have an influence in the Berry phase
acquired.
7.3.1 Aharonov-Bohm phase as an example of a Berry?s phase
A familiar example of geometric phases is the Aharonov-Bohm phase [124]
gained by an electron moving along closed trajectories around a solenoid. This
phase was initially conceived as a way of showing that in quantum mechanics mag-
netic vector potentials, typically conceived only as mathematical objects, can have
a physical effect on the wave function. They considered a coherent electron beam
split into two paths around a solenoid that produces a magnetic field B and later
recombined as shown in Figure 3. Outside the solenoid the magnetic field B = 0,
but there can be a non-zero magnetic vector potential such that B = ??A. Even
though the electron?s trajectories are not modified by the presence of the solenoid,
when looking at the interference pattern one finds that the two paths acquired dif-
ferent phases, and their difference is remarkably equal to magnetic flux piercing the
area enclosed by the path of the electrons ?? = 2??B/?0, where ?0 = h/e is the
136
flux quantum. This Aharonov-Bohm phase can be interpreted as an example of a
Berry phase in real space.
Figure 3: The Aharonov-Bohm experiment. A coherent electron beam is split into
two paths surrounding a solenoid which produces a non-zero magnetic field B inside
the gray region and B = 0 outside. The two beams are later recombined and an
interference pastern reveals a phase difference ?? = 2??B/?0 equal to the magnetic
flux enclosed by the electron?s path.
For a charged particle in the presence of a vector potential, the momentum
dependence of the free-particle Hamiltonian is modified p? p?qA so that the wave
function will depend on the magnetic vector potential as well. Using Equations 7.6
and 7.7 it can be shown that the Berry phase associated with a closed path around
the solenoid is exactly equal to the Aharonov-Bohm phase:
?
?n(C) =
e
? A(r) ? drh? C
= e ??A ? dS
h? S
=e?B , (7.9)
h?
In this particular example, the Berry connection is exactly equal to the mag-
netic vector potential and the Berry curvature is the magnetic field. This gives us
a very physical intuition for interpreting the Berry phase in terms of the ?magnetic
137
flux? from abstract sources of ?magnetic fields? in parameter space.
7.3.2 Chern number
The Chern number is conventionally used to describe the topology of ma-
terials which have an underlying crystalline structure. According to Bloch?s the-
orem, the wave functions of a space periodic Hamiltonian can be written as
|?(k)? = eik?r |u(k)?5, where r is the position and k the crystal momentum. The
wave functions |u(k)? are periodic and therefore invariant under the displacement
operator D?(n d) |u(k)? = |u(k)? when d is the unit cell size and n an integer. If we
define the Bloch Hamiltonian
H?(k) = eik?rH?e?ik?r, (7.10)
their eigenvectors are given by |u(k)? and the eigenvalues define the band structure.
Translational symmetry implies that H?(k + a) = H?(k) where a is a reciprocal
lattice vector. The crystal momentum or quasimomentum (in analogy to the Floquet
quasienergy) is only defined within the periodic Brillouin zone and therefore can be
mapped into a torus in d dimensions if we glue the edges together.
The Chern number of the nth band is defined as
1 ?
Cn = 2 ?n ? dk, (7.11)? BZ
5Very much like Floquet theory in momentum space
138
where the relevant parameter space is crystal momentum and the surface of integra-
tion corresponds to the BZ (a torus). The definition of the Chern number is closely
related to the Berry phase from Equation 7.7. For our previous example of a quan-
tum Hall system, the integer proportionality factor in the quantized conductance is
exactly equal to the Chern number.
Just like two-dimensional surfaces are classified by the integral of their Gaus-
sian curvature, the topology of Bloch bands and quantum systems, in general, is
determined by the integral of the Berry curvature. Similarly, the integral connects
local properties of a quantum system, the Berry connection, with a global topological
invariant, the Chern number. One subtle difference is that the Euler characteris-
tic is only determined by the surface (and its intrinsic Gaussian curvature) while
the Chern number is defined both by a surface (the BZ) and an additional local
curvature (the Berry curvature). By considering different lattice Hamiltonians one
can obtain a different Berry curvature, but the geometry of the BZ and thereby
the surface of integration is typically defined by a torus6. This difference will be
important later on when we describe the experiments performed to study a system
with Rashba spin-orbit coupling where the unit cell size is taken to infinity (i.e. we
remove the lattice).
6In the next chapter we consider a case where this breaks down.
139
7.4 The bulk-edge correspondence principle
Earlier I mentioned that topological systems provide very robust channels for
transporting things like electrical current and light. Transport phenomena typically
arise when there is a spatial interface between two topologically distinct phases. The
electrons skipping orbits at the interface of a (topological) quantum Hall material
and (trivial) vacuum are one example of this. Notice that for this particular example
the modes propagate along a given direction, they are chiral. In general, one can
expect to have modes moving along two directions, and the difference between the
number of these modes NL?NR is fixed and determined by the topology of the bulk
states. The bulk-edge correspondence principle relates the difference in the number
of these modes with the bulk topology of the materials at the interface:
?C = NR ?NL (7.12)
where ?C is the difference of Chern number on the interface.
7.5 Example: two-level model
Many of the concepts introduced in the previous section can be readily applied
and understood using a two-level model
H?(k) = h(k) ? ?? (7.13)
140
where ?? = (?x, ?y, ?z) are the Pauli matrices and h(k) = (hx(k), hy(k), hz(k)) are
functions of k. This model has been used to describe a number of physical systems
like graphene [112] and spin-orbit coupled systems [24,51]. Let us now consider the
simple case h(k) = k, for which ?kH? = ? and using Equation 7.8 it can be shown
that
h
? = ?2 3 (7.14)h
which can be recognized as the field of a Dirac monopole [125] with charge ?1/2.
The degeneracy in the energies that gives rise to the monopole is known as a Dirac
point as the energies in that vicinity resemble the dispersion of a massless Dirac
particle. In 2D materials where k is only defined within 2D plane hz corresponds
to the mass of a Dirac particle but its effect on the Berry curvature is equivalent
to that of moving the Dirac monopole along a fictitious kz dimension in a direction
determined by the sign of hz.
It follows from Equation 7.14 that the Berry phase gained by moving along a
closed path C is equal to the flux from the monopole in the surface enclosed by C as
is shown in Figure 4. This connects nicely with our intuition from the Aharonov-
Bohm effect. For a closed surface enclosing the Dirac point, the Chern number is
an integer equal to 1.
For a Hamiltonian with arbitrary h(k) we can define a normalized vector
h? = h/|h| and the Chern number takes the form
?
C = 14 (?kxh?? ?
2
k
? y
h?) ? h? d k (7.15)
141
Figure 4: For a two-level system, the Berry curvature from a Dirac point can be
viewed as a Dirac monopole in momentum (parameter) space. The Chern number
can be interpreted as the flux from the monopole on the solid angle subtended by
the vector h?(k) or alternatively as the number of times h?(k) wraps around a unit
sphere.
which can be interpreted as the number of times that the vector h?(k) wraps around
a unit sphere [126], a quantity that is known as the winding number. Depending
on the sign of hz the vector h?(k) will wrap around the north or south hemisphere,
so to have integer valued Chern numbers, Dirac points must come in pairs. Luckily
for lattice Hamiltonians this is guaranteed by the fermion doubling theorem [127].
In Chapter 8 I describe an engineered system that has a single Dirac point.
7.6 Monopoles and Dirac strings
We just gained some intuition about interpreting the Chern number as the
flux from Dirac monopoles. But if we stick to our knowledge of electromagnetism
we might remember that magnetic monopoles are forbidden since
? ?B = ? ? (??A) = 0. (7.16)
142
So how is it possible to keep a vector potential and have ??B 6= 0? The solution to
this problem was envisioned by Dirac [125] and is now called a Dirac string. If we
consider a semi-infinitely long and infinitesimally thin solenoid, the magnetic field
in the finite end will resemble that of a monopole as can be seen in Figure 5. This
tiny solenoid corresponds to the Dirac string. A more mathematical interpretation
Figure 5: For a two-level system, the Berry curvature from a Dirac point can be
viewed as a Dirac monopole in momentum (parameter) space. The Chern number
can be interpreted as the flux from the monopole on the solid angle subtended by
the vector h?(k) or alternatively as the number of times h?(k) wraps around a unit
sphere.
of these strings comes from the fact that in order to have ? ? B 6= 0 the vector
potential of a monopole must have ?lines? where it becomes singular. For example,
we can write for a particular gauge
A( ) = ?yer x + xeyg
r(r + ) (7.17)z
which is singular at the Dirac string located at the negative z axis where z = ?r .
The orientation of the Dirac string is gauge dependent, something that should not
surprise or bother us at this point. However, the physical effects of the Dirac string
143
should be gauge independent, or in other words, the Aharonov-Bohm phase gained
by a charged particle moving in a path that encloses the string should be an integer
multiple of 2?. This is argument gives rise to the Dirac charge quantization [125],
and in the context of topology, it guarantees that when we calculate the Berry phase
by integrating the Berry connection (vector potential) along a path that encloses a
Dirac string, its effect will be indistinguishable.
7.7 Conclusions
Topology plays a very important role both in mathematics and in physics. As
a closing remark, Figure 6 summarizes the main concepts that were introduced and
is a reminder that topological invariants are global properties defined in terms of
integrals of local properties. Furthermore, we can use our intuition from electro-
magnetic theory to interpret topological invariants in quantum mechanics.
Figure 6: The Euler characteristic and the Chern number are topological invariants
defined by integrals of local curvatures. The Aharonov-Bohm phase gives us physical
intuition to interpret the Chern number as the flux from a ?Berry field?.
144
Chapter 8: Unconventional topology in a quantum gas with Rashba
SOC
Topological order is present in a wide range of physical systems and is quan-
tified by integer valued topological invariants such as the Chern number. In this
Chapter, I describe a system with Rashba-type spin-orbit coupling whose topolog-
ical invariants can take half-integer values. If the concept of half-integer invariants
does not sound odd, think of a donut with half a hole.
Ultracold atomic systems are an emerging platform for engineering topological
lattices, from the Harper-Hofstadter model [20,128], the Haldane model [129], to the
Rice-Mele model [130,131] as well as assembling spin-orbit coupled lattices without
analogues in existing materials [132, 133]. However, experimental realizations of
topological materials have mostly focused on engineering different topological bands
(with different Berry curvatures) in lattice systems, where the BZ is always a torus.
Here I show that by eliminating the lattice potential and changing the BZ from T2
to R2, i.e. from a torus to a Cartesian plane, it is possible to create topological
branches of the dispersion relation with half-integer Chern number. This work has
been submitted for publication and is under review. A preprint of the manuscript
is available in [26].
145
These experiments combine the techniques from Chapters 5 and 6, additionally
I use key concepts of topology described in Chapter 7. This Chapter is organized
in the following way: First I give a general overview of Rashba SOC and describe
theoretical proposals for engineering this type of coupling in ultracold atom systems.
Then I describe our experimental implementation of Rashba SOC in the lab using
a trio of Raman coupled CDD states and validate our quantum engineering using
Fourier transform spectroscopy. Finally, I describe a quantum state tomography
procedure to measure the eigenstates of our system, from which we can directly
obtain the Chern number.
To avoid confusion between dressed state xyz labels and Cartesian coordinates
indicating the Raman geometry, in this Chapter I use the numbers 1, 2, 3 to label
experimental coordinates and the letters x, y, z to label clock state parameters.
8.1 Rashba spin-orbit coupling
Rashba SOC [24] appears in condensed matter systems where electrons are
confined in a 2D plane and experience an intrinsic out-of-plane electric field. If
the electron?s momentum is given by h?k = h?(kxex + kyey) and the electric field
is E = Eez, in the electron?s moving frame there will be a momentum dependent
magnetic field BSOC = ?h?k/m ? E/c2 = h?E/mc2(?ky, kx, 0). The interaction
between the electron?s spin with this field through the magnetic Zeeman interaction
146
?? ?BSOC gives rise to the SOC Hamiltonian
= 2?H?SOC (ky??x ? kx??y) (8.1)
m
where ? = g? 2BE/c , g is the electron gyromagnetic ratio, ?B is the Bohr magneton
and ??i are the Pauli matrices.
Figure 1 shows the Rashba dispersion relation which is characterized by having
a Dirac point located at k = 0 (see Section 7.5) and a degenerate ground state
that is described by the ring k2 + k2x y = ?2. If we combine Equation 8.1 with the
free particle Hamiltonian, the system is described by H? = (h?k ? A?)2/2m where
A? = ?(??yex ? ??xey) can be interpreted as a (matrix-valued) non-Abelian gauge
potential [134] whose elements do not commute. This non-Abelian gauge potential
is closely related to the Berry connection discussed in Section 7.3 and in combination
with the Dirac point hints at us that a system with Rashba SOC has non-trivial
topology.
1.0 1
0.5
0
0.0
?1
?1 0 1 ?1 0 1
px in units of ? px in units of ?
Figure 1: The Rashba dispersion relation has a Dirac point located at k = 0 and a
degenerate ground state that is described by the ring k2x + k2 2y = ? .
147
2
Energy in units of ?2m
py in units of ?
8.2 Rashba SOC for neutral atoms
Proposals for engineering Rashba type SOC in neutral atoms consist of using
lasers to link internal states of an atom with its linear momentum. In order to
achieve non-trivial gauge potentials it is necessary to couple N ? 3 levels (see [16]).
I will describe the proposal by [135] which considers a ?ring coupling? which is shown
in Figure 2 for the case of N = 3. The states |j? represent internal atomic states
and they are linked to each other with complex valued matrix elements ?j eikj ?x2 ,
where kj is a momentum transfer associated with the |j? ? |j + 1? transition and
? = ei?ii |?| represents the coupling strength. We require that
?ki = 0 so that no
momentum is transfered when a closed loop |1? ? |2? ? ? ? ? |N? ? |1? is completed.
For this case, the ki momenta vectors can be written as kj = Kj+1 ?Kj and we
make Kj = kL sin(2?j/N)ex + kL cos(2?j/N)ey, corresponding to the vertices of an
N sided regular polygon. We can further make a gauge transformation such that we
can replace the phases ?i associated to each coupling with ?? =
?
i ?i/N .
The Hamiltonian describing this coupling along with the kinetic term is
h?2= k
2 ?
H i(??+kj ?x)j,j? 2 ?j,j
? + 2 (e ?j,j
?+1 + h.c.), (8.2)
m
and after applying the unitary transformation Uj,j? = exp[iKi ? x]? 1j,j? it gets trans-
formed to
2
Hj,j? =
h?
2 |q + Kj|
2 + ?? (ei??j,j? ?j,j?2 +1 + h.c.), (8.3)m
1This transformation is equivalent to applying a state dependent momentum boost k? k+Kj
148
Figure 2: The Rashba ring coupling. To generate Rashba SOC in a system of
cold atoms it is necessary to cyclically couple N ? 3 inte?rnal states such that thetransition |j? ? |j+? 1 has a momentum transfer kj and j kj = 0 such that there
is no momentum transfer for a closed loop |1? ? |2? . . . |N? ? |1?. The ring coupling
combined with the free particle Hamiltonian give rise to a 2-level subspace that can
be described to first order by the Rashba Hamiltonian
where I replaced the momentum k by the quasimomentum q. The off diagonal terms
of Equation 8.3 is related to a 1D periodic tight-binding Hamiltonian with hopping
elements ?/2 where the internal states |j? represent lattice sites and completing one
loop corresponds to gaining a ?flux? of N??. To visualize how the Rashba Hamiltonian
emerges from this coupling scheme, it is helpful to write the Hamiltonian in a basis
that is conjugate to the index j2
1 ?N|l? = ? ei2?jl/N |j? (8.4)
N j=1
where the index l is analogous to the crystal momentum index for a Bloch Hamilto-
nian. In this new basis, terms in the diagonals are displaced to the off-diagonal and
likewise, terms in the off diagonals are displaced to the diagonal. The Hamiltonian
2Just like position and momentum are conjugate variables related by Fourier transforms, the
|j? and |l? basis are related by a discrete Fourier transform.
149
transformed to the Fourier basis starts looking very much Rashba-like
[ 2 ] 2
Hl,l? =
h?
2 (q
2 + k2L) +
h? kL
El ?l,l? + [(iqx + qy)?l?1,l? + h.c] , (8.5)
m m
where EL = 2h?? cos(2?l/3 + ??) correspond to the eigenenergies when q = 0. The
phase ?? can be tuned such that a pair of states with consecutive l index become
degenerate, indicating the presence of a Dirac point at q = 0. Figure 3 shows the
energies El for N = 3 and ?? = 0.
2?
0
?2?
0 1 2 3
l
Figure 3: Eigenenergies of Equation 8.5 for q = 0 for N = 3 and ?? = 0. For
this particular choice of phase, the energies of the l = 1 and l = 2 states become
degenerate
We consider the degenerate states as pseudospins which are described to zeroth
order by the Rashba plus free particle Hamiltonian
h?2q2 h?2kL
H?(0) = 2 + (??xqy ? ??yqx), (8.6)m m
with spin-orbit coupling strength given by ? = h?2kL/2. The zeroth-order Hamilto-
150
El
nian has continuous rotational symmetry while the proposed ring coupling only has
discrete rotational symmetry. The symmetry of the Hamiltonian is recovered when
higher-order corrections are added to the Hamiltonian. The complete expressions
for the higher-order terms for N = 3 and N = 4 can be found in [135], and they are
reminiscent of quadratic and cubic Dresselhaus SOC [136]. The largest leading order
term for the N = 3 case is inversely proportional to ? so that this ring-coupling
scheme results in a more ?Rashba-like? Hamiltonian as one goes to higher coupling
strengths. Figure 4 shows level curves of the ground state eigenenergies of Equa-
tion 8.5 for N = 3 and ?? = 0 for increasing ?. At low ? the dispersion has discrete
rotational symmetry and is characterized by three local minima3. As ? is increased
the local minima start merging into each other and in the large ? limit we recover
the characteristic Rashba ring-like dispersion.
1.5
0.0
?1.5
?1.5 0.0 1.5 ?1.5 0.0 1.5 ?1.5 0.0 1.5
qx, in units of kL qx, in units of kL qx, in units of kL
Figure 4: Ground state dispersion relation of Equation 8.5 for N = 3 and ?? = 0
for ? = 1.75 EL (left), ? = 3.5 EL (middle) and ? = 175 EL (right). Higher order
corrections to H?(0) decay as 1/?2 and in the large ? limit we recover the Rashba
ring dispersion.
3or... an alien face?
151
qy, in units of kL
8.3 Experimental implementation of Rashba SOC
We implemented the ring-coupling scheme and engineered Rashba SOC by
resonantly coupling the |xyz? states from Chapter 6 using two-photon Raman tran-
sitions [78] as depicted in Figure 5. The engineered system consisted of an effective
spin-1/2 Rashba subspace, along with a topologically trivial high-energy branch.
Our engineered Rashba system had a single Dirac cone near q = 0, where the
two lower dispersion branches become degenerate and the Berry curvature becomes
singular. Each of these branches extend to infinite momentum, making the sup-
porting manifold a plane rather than a torus. We characterized this system using
both spectroscopy and quantum state tomography. This allowed us to measure the
dispersion branches and directly observe the single Dirac point linking the lowest
two branches as well as to reconstruct the Berry connection to derive the associated
Chern numbers.
All of our experiments started with about with N ? 1 ? 106 87Rb atoms in
a crossed optical dipole trap [137], with frequencies (f1, f2, f3) ? (70, 85, 254) Hz,
just above the transition temperature for Bose-Einstein condensation. We initially
prepared the atoms in the |F = 1,mF = ?1? state of the 5S1/2 electronic ground
state and transfered atoms to the mF = 0 and mF = +1 states as needed using
ARP. A bias field B0e3 gave a ?0/2? = 23.9 MHz Larmor frequency along with a
quadratic shift of /2? = 83.24 kHz. The RF field used to generate the |xyz? states
and implementing CDD had strength ?RF = 1.41(2). We adiabatically prepared
the |xyz? states starting from the mF states following the procedure described in
152
14
10
6
2
?2
?1 21
? 00 ? 12
of kL
units
qx in
Figure 5: a Our engineered dispersion consisted of a two-level Rashba subspace (red
and blue) with a single Dirac point linking the lowest two branches and a topo-
logically trivial higher branch (gray). b We generated |xyz? states by combining a
bias magnetic field along e3 with an RF magnetic field oscillating along e1. These
states were coupled by three cross-polarized Raman laser beams propagating along
e1, e2 ? e1 and ?e1 ? e2. c Each pair of Raman lasers was in two-photon reso-
nance with a single transition between the |xyz? states which we coupled strengths
(?zx,?xy,?yz)/2? = (12.6(5), 8.7(8), 10(1)) kHz.
Section 6.4.
8.3.1 Raman coupling the |xyz? states
We Raman-coupled atoms prepared in any of the |xyz? states using the three
cross-polarized Raman laser beams shown in Figure 5b, tuned to the ?magic zero?
wavelength ?L = 790 nm where the scalar light shift vanishes (see Section 3.3.2.1).
We arranged the Raman lasers into the tripod configuration shown in Figure 5c,
bringing each pair into two-photon resonance with a single transition with strengths
(?zx,?xy,?yz)/2? = (12.6(5), 8.7(8), 10(1)) kHz. The geometry of our experimental
implementation differed from [78] where all Raman lasers are perpendicular. We had
to go away from the ideal configuration because we needed all of the Raman recoil
153
Energy in units of EL
f k Lo
nit
s
in
u
qy
vectors to lie on the imaging plane (spanned by e1 and e2) to image all the Raman
induced k dependent dynamics. As a result, the dispersion relation no longer has
discrete rotational symmetry, however, the Dirac point is still present in our system.
The only way to break the degeneracy of this system is to change the phase ?? and
for our laser configuration, we always have ?? = 0. ?
The energies of the |xyz? states were ?x = 0 and ?z,y = ?(? 4?2RF + 2)/2.
We set the frequencies of the Raman lasers to ?x = ?L + ?0 + ?xy, ?y =
?L + ?0 and ?z = ?L ? ?zx, where ?L = 2?c/?L and (?zx, ?xy, ?zy)/2? =
(166.47, 83.24, 249.71) kHz are the transition frequencies between pairs of dressed
?
states and are integer multiples of  for our coupling strength ? = 2.
The Raman coupled states can be described by the combined kinetic and light-
matter Hamiltonian
? ( 2 2 ) ?
H?R(
h? k
k) = 2 + h??i |j? ?j|+ h??
i(k
?e j,j? ?x?i?j,j? t)j,j |j? ?j?| , (8.7)
?{ } mj xyz j? 6=j
where kj,j? is the recoil momentum from an |j? ? |j?? transition and ?ij is the Raman
coupling strength between a pair of RF dressed states. The Hamiltonian above only
includes the matrix elements associated to resonant couplings. After applying the
unitary transformation U?j,j? = exp(?ikj ? x? ?jt)?j,j? |j? ?j?| the Hamiltonian takes
the familiar form of the ring coupling scheme
? ( 2 2 )
= h? (q ? kj)
?
H?R 2 + h??j |j? ?j|+ h??
?
jj? |j? ?j | , (8.8)
j?{xyz} m j=6 j?
154
where kj are the Raman wave vectors and ?j is the detuning from Raman resonance.
This coupling scheme simultaneously overcomes three limitations of earlier ex-
periments [138, 139] that performed the ring coupling using states in both 5S1/2
hyperfine manifolds of 40K : (1) working in the same hyperfine manifold elimi-
nates spin-relaxation collisions [140]; (2) unlike mF states, the |xyz? states can
be tripod-coupled with lasers far detuned relative to the excited state hyperfine
splitting greatly reducing spontaneous emission [21]; and (3) CDD renders the |xyz?
states nearly immune to magnetic field noise.
8.3.1.1 Floquet and off-resonant coupling effects
We operated in a regime where the transition energies between the |xyz? states
were integer multiples of ?xy: ?zx = 2?xy and ?zy = 3?xy, making the system
suitable for a full description using Floquet theory [141]. We observed that the
effective Raman coupling strengths for the driven three level system differed from our
calibrations which were performed looking at Rabi oscillations from individual pairs
of lasers because of the presence of nearby quasi-energy manifolds and off-resonant
coupling terms. This effect could be mitigated for larger values of ?xy as the spacing
between quasi-energy manifolds is increased and the off-resonant coupling terms less
relevant. Figure 6 shows the dispersion relation for both the effective and the full
Floquet Hamultonians. The energies from the effective time independent model
get closer to those of the full Floquet model when the spacing between Floquet
manifolds is increased (left panel). Appendix C has a full derivation of the Raman
155
Hamiltonian starting from the |mF ? basis in the laboratory frame including the full
time dependence and off-resonant coupling terms.
1.0
0.5
0.0
?1 0 1 ?1 0 1
qx, in units of kL qx, in units of kL
Figure 6: Solid lines: Dispersion relation from RWA Hamiltonian (Equation 8.8)
as a function of qx for ?i,j = 1.5 EL and ?i = 0. Dashed lines: Dispersion relation
computed for the full Floquet Hamiltonian. We considered ?zx = 2?xy and ?zy =
3?xy in both cases so the separation between Floquet manifolds is ?xy. In the
left panel ?xy = 83.24 kHz as in our experiments and in the right panel ?xy =
416.2 kHz. As the spacing between Floquet manifolds becomes we get a better
agreement between the spectra of the RWA and Floquet Hamiltonians.
8.3.1.2 Lifetime
The limited lifetime due to spontaneous emission has always been a concern for
Raman coupled systems. This was one of the reasons why we pursued the topology
direction rather than trying to measure a fragile many-body phase. The measured
spontaneous emission limited lifetime of our system was 320(17) ms, measured with
the Raman lasers applied to the mf states. However, it was reduced to 40(2) ms
when we Raman coupled the |xyz? states, which we attribute to technical noise in
the relative phase between the RF dressing field and the Raman laser fields, which
156
Energy in units of EL
has caused considerable consternation in ongoing experiments. All the experiments
reported here were short compared to this timescale so this decreased lifetime was
not an issue but it is a limitation that needs to be addressed in future experiments.
Figure 7 shows measurements of the lifetimes of Raman dressed atoms in both |mf?
and |xyz? states. We obtained the lifetime by fitting decaying exponentials to the
integrated OD from absorption images of thermal atoms where the Raman was
turned on in 1 ms and held on for up to 50?s4.
30000
20000
|xyz?
|mf ?
10000
0 10 20 30 40 50
Raman hold time [ms]
Figure 7: Lifetime of Raman dressed states. We Raman dressed atoms in the |mf?
and |xyz? states. The orange markers correspond to atoms initially prepared in
|mf = ?1? (no high power RF involved) and the blue markers correspond to atoms
|xyz? (three averaged traces). The lifetime of doubly dressed states is significantly
reduced as compared to the lifetime of the Raman dressed |mf? states, indicating
that of resonant scattering of photons is not our only loss mechanism.
8.3.2 Measuring quasimomentum distributions
Each pair of Raman lasers coupled states |i,k? ? |j,k + ki,j? where |i? and |j?
denote the initial and final |xyz? states, k is the initial momentum and ki,j = ki?kj
is the two-photon Raman recoil momentum. Dressed states with quasimomentum
4How long we could hold on the Raman was limited by the RF antenna heating up.
157
Integrated OD
q are comprised of three bare states |j,k? with momentum k = q ? kj. Figure 8
illustrates the mapping from bare momentum states into dressed quasimomentum
states. The eigenstates of our Rashba SOC Hamiltonian take the form
? ?
|? (q)? = a i?n,j(q)n n,j(q)e |j,k = q ? kj? , (8.9)
j?xyz
where the quasimomentum q is a good quantum number and the amplitudes are
parametrized by an,j(q) and ?n,j(q). We leveraged the wide momentum distribution
of a non-condensed ensemble (T ? 180 nK and T/Tc ? 1.1) to sample a wide range
of momentum states simultaneously. By starting separately in each of the |xyz?
states we sampled the range of quasimomentum states shown in Figure 8b, where
the momentum distributions of an initial state |j,k? is shifted from q = 0 by the
corresponding Raman wave vector kj.
Our measurement protocol consisted of abruptly removing the confining po-
tential and the Raman lasers, initiating a 21 ms TOF. During this TOF we adia-
batically transformed each of the |xyz? states back to a corresponding |mF ? state
following the procedure described in Section 6.4 and spatially separated them using
a Stern-Gerlach gradient. Finally, we used resonant absorption imaging to measure
the resulting spin-momentum distributions.
The FWHM of the cloud after TOF was about 700?m which is much larger
than the size of the in-situ cloud ? 50?m and the spatial density distribution
atoms imaged after TOF represented the momentum distribution of the atoms. For
the 7.4?m pixel size of our camera and the 3.283 magnification of our imaging
158
1 1 1
0 0 0
?1 ?1 ?1
?1 0 1 ?1 0 1 ?1 0 1
k1 in units of kL k1 in units of kL k1 in units of kL
1
0
?1
?1 0 1
q1 in units of kL
Figure 8: Mapping momentum into quasimomentum: a We used non-condensed
atoms with a broad momentum distribution (T ? 180 nK and T/Tc ? 1.1). b Atoms
in |j,k? are mapped to Raman dressed states with quasimomentum q = k+kj. The
black dots in the bottom panel represent the location of k = 0 for each of the |xyz?
states and the red dot corresponds to q = 0. We performed our experiments starting
separately in each of the |xyz? states, which allowed us to sample a larger range of
quasimomentum states.
system, the momentum resolution of our images was 0.018 kL and the momentum
distribution of a single state measured after TOF had a FWHM of ? 2.2 kL.
8.3.2.1 Correcting shears from Stern-Gerlach gradient
The magnetic field gradient used to separate the different mf states during
TOF additionally created a trap for atoms in the direction perpendicular to the
Stern-Gerlach separation, causing a compression (expansion) of the mf = ?1 (+1)
states (see Section 3.2.1). The faster moving atoms are subject to a stronger poten-
159
k2 in units of kL
q2 in units of kL k2 in units of kL
k2 in units of kL
60
58
56
?1 0 1
mf state
Figure 9: We measured the standard deviation of the momentum distribution along
the axis perpendicular to the SG for 10 shots on each mf state. From the slope of
the linear fit we obtain a shearing parameter ? ? ?0.067 for mf = ?1.
tial as they move further from the trap center. The projections of a given momen-
tum state k along and perpendicular to the Stern-Gerlach axis were transformed
as k? ? k? and k? ? (1 + ?)k?, where ? = 0 for mf = 0 and sign(?) = ?1 for
mf = ?1.
Since we were interested in mapping the momentum distribution of atoms
it was important for us to quantify and correct the effect of these shears in the
TOF images. We used two different methods: First, we prepared thermal atoms
in all three of the mf states and fit 2D Gaussians rotated by the angle of the SG
displacement (63.8 degrees for our images). Figure 9 shows the standard deviation
extracted from the Gaussian fits along the axis perpendicular to the SG direction as
a function of mf state. We performed a linear fit of ? and found that the mf = ?
states are expanded/contracted by about ?6.7% size of the compared to the mf = 0
cloud (? = ?0.067).
Alternatively we performed the Ramsey interferometer described in Section 8.5
160
? [pixels]
but coupling only 2 states, either |z? ? |x? or |x? ? |y? (mapped to |?1? ? |0?
and |0? ? |+1? after TOF). We looked at the oscillation frequencies of the density
for each pixel of the CCD camera (each sheared momentum state) and fit them to
Equation 8.17 for fixed value of the recoil momentum ki,j and with a free shear
parameter that modifies q. Using this method we extracted a shearing of the order
of 7%, in good agreement with the Gaussian fitting method.
The transformed momentum coordinates were described by a function
g(k) = (k?, (1 + ?)k?) (8.10)
and our observed data ( (shear)y ,k(shear)i ) was the density in the sheared coordinate
system at the ith pixel of the CCD sensor. We estimated the density in the unsheared
coordinate using the transformation
? e?xp[?( (k )? k(shear))2 2 2] (shear)g j / ? yy = i i ij (shear) , (8.11)2 2
i exp[?(g(kj)? ki ) /2? ]
where ? is the spacing between two consecutive pixels (? 0.018 kL. Prior to any
data analysis we applied this transformation to all of our images, where we used
different values of ? that define g(k) for each of the mf states.
8.4 Fourier spectroscopy of the Rashba dispersion
We directly measured the 2D dispersion relation using Fourier transform spec-
troscopy [25]. We considered the evolution of an initial state |i,k? suddenly sub-
161
1.0
1
0 0.5
?1
0.0
. . . 1.0 0 200 400 600
q1 in units of k1 L tp [?s]
0 0.5
?1
0.0
. . . 1.0 0 200 400 600
q1 in units of kL
1 tp [?s]
0 0.5
?1
0.0
?1 0 1 0 200 400 600
q1 in units of kL tp [?s]
Figure 10: a Fourier spectroscopy protocol. We applied the Raman lasers for a
variable time tp: a Rabi-type atomic interferometer analogous to a three-port beam
splitter. b Probabilities as a function of quasimomentum for a fixed Raman pulse
time tp = 420?s c Dynamics of the final populations of the |xyz? states with quasi-
momentum (q1, q2) = (?0.55,?0.18) kL (blue star in panels b) after initializing the
system in the |z? state.
jected to the Raman coupling lasers. This atomic Rabi-type interferometer is analo-
gous to the three-port beam-splitter depicted in Figure 10a. During a pulse time tp
we followed the dynamics of the populations in the |xyz? states which evolved with
oscillatory components proportional to ?j 6=n an,j(q) cos([En(q)?Ej(q)]tp /h?), with
frequencies determined by the eigenenergy differences En?Ej. Figure 10b shows the
momentum dependent populations for a fixed pulse time tp and Figure 10c shows
representative final populations as a function of tp for a fixed quasimomentum state.
We Fourier transformed the populations with respect to tp and for a given quasi-
momentum state for a total of 9 states accounting for each of the three initial |xyz?
states that were later split into 3 states, all of them with the same q. Figure 11
162
q2 in units of kL q2 in units of kL q2 in units of kL
Populations Populations Populations
shows the PSD computed for each of the 9 states for planes of constant q1. The
amplitude of the oscillatory components depends on the overlap integral between
the initial state and the Raman dressed states (see Equation 5.1) so sampling all
these states gave us access to a wider range of measurable frequencies. We produced
the spectral maps in Figure 12b by averaging the PSDs from the 9 different states
using n?, the mean population in tp, as a weight:
?
(mean) i,j P?SD (q)n? (q)PSD (q) = i,j i,j , (8.12)
i,j n?i, j(q)
where the indices i, j represent the different states of the grid shown in Figure 11.
The extrema in the spectral maps are the energy differences En ? Ej in the en-
gineered dispersion (Figure 10a) and together they show the presence of a single
Dirac point in the Rashba subspace, evidenced by the gap closing near q = 0 and
the photon-like lower branch. The dashed curves correspond to the energy differ-
ences computed for our system using the dispersions shown in Figure 12a, and are in
clear agreement with our experiment. This measurement directly confirms the ex-
pected set of energies, including the existence of a two-state subspace approximately
described by the Rashba Hamiltonian.
8.5 Quantum state tomography with Ramsey interferometer
The Fourier spectroscopy measurement confirmed our quantum engineering of
the Rashba Hamiltonian. However, the energies shed no light on the topology of the
different branches of the dispersion, which instead requires knowledge of the eigen-
163
40
20
0
?1 0 1 ?1 0 1 ?1 0 1 ?1 0 1
40 q1 in units of kL q1 in units of kL q1 in units of kL
20
0
?1 0 1 ?1 0 1 ?1 0 1 ?1 0 1
40 q1 in units of kL q1 in units of kL q1 in units of kL
20
0
?1 0 1 ?1 0 1 ?1 0 1 ?1 0 1
q1 in units of kL q1 in units of kL q1 in units of kL
Figure 11: PSD of all the analyzed states as a function of q2 for fixed q1 = 0.18 kL.
The different overlaps between the initial state, the Raman dressed states and the
measured final state result in peaks with different amplitudes.
states. The Berry curvature present in the definition of the Chern number (Equa-
tion 7.11) can be derived from the Berry?s connection An(q) = i ??n(q)| ?q |?n(q)?,
which as discussed in Chapter 7 behaves much like a vector potential in classical
electromagnetism. The Berry curvature ?n(q) = ?q ?A(q) is the associated mag-
netic field and the flux through any surface is the line integral of A(q) along its
boundary, after neglecting the contributions of Dirac strings which I will discuss
later. Using the expression for the Raman dressed eigenstates from Equation 8.9 we
obtain
?
An(q) = ? an,j(q)?q?n,j(q), (8.13)
j?{x,y,z}
164
Frequency [kHz] Frequency [kHz] Frequency [kHz]
8
6
4
100 100
2
0
0 0
8 ?1 0 1 ?1 0 1
102 2
qy in units of kL qy in units of k 10L
6
101 101
4
100 100
2
0 0 0
?1 0 1 ?1 0 1
qy in units of kL qy in units of kL
Figure 12: a Predicted dispersion relation as a function of q2 for fixed q1 = ?0.09 kL
(left) and 0.65 kL (right), computed for the experiment parameters. The energy
differences between the branches enclosing the vertical arrows appear as peaks in
the spectral maps below. b Power spectral density (PSD) for the same parameters
as above which we obtained by Fourier transforming the populations in the |xyz?
states with respect to tp. The dashed lines correspond to the energy differences
computed using the dispersion curves on the top panel.
which depends on both the phase and amplitude of the wave function. We obtained
an,j(q) and ?n,j(q) using a three-arm time-domain Ramsey interferometer (see Sec-
tion 3.5.4), implementing a variant of quantum state tomography [142, 143]. The
use of a multi-path interferometer allowed us to transduce information about phases
into state populations, which we readily obtained from absorption images.
Figure 13 shows our experimental protocol which I will describe in more de-
tail in the following section. We adiabatically mapped an initial |j,k? state into a
corresponding eigenstate |n,q = k + kj?, either in the topologically trivial highest
165
Energy diff. in units of EL Energy in units of EL
PSD [arb. u.] [PSD [arb. u.]
PSD [arb. u.] [PSD [arb. u.]
Detuning
Figure 13: Experimental protocol for three-arm Ramsey interferometer (not to
scale). (Top) We started with atoms in state |z, y,qi = k + kj? and with detun-
ing ?y = ?5 EL and ?z = ?5 EL. We ramped the Raman lasers on in 750 ?s and
then ramped the detuning to nominally zero. We let the system evolve in the dark
for times between 5 ?s and 400 ?s, followed by a 25 us Raman pulse. (Bottom)
The implemented experimental protocol was equivalent to a three-arm interferom-
eter that split an initial state into three final states with amplitudes related to the
initial wave function phases.
dispersion branch (n = 3) or in the topological ground branch (n = 1) by dynami-
cally tailoring both the Raman coupling strength and detuning. We suddenly turned
off the Raman coupling, allowing the three bare state components of the Rashba
eigenstates to undergo free evolution for a time tfree, constituting the three arms
of our time-domain interferometer. Finally, we applied a three-port beam splitter
using a brief Raman ?recombination? pulse to interfere the three arms.
8.5.1 Wave function evolution in Ramsey interferometer
Rashba dressed state preparation: We started with |xyz? states at dif-
ferent RF coupling strength ?0 = ?RF/?2 ? 20 kHz, such that the energies of the
166
Coupling strength
|z? and |y? states were shifted by about ?18.8 kHz. We used the same Raman fre-
quencies as described earlier and the change in the |z? and |y? state eigenenergies
corresponded to having non-zero values of ?z and ?y in Equation 8.8. We chose the
detuning such that the initial state had a large overlap with either the n = 1 or
the n = 3 eigenstates of Equation 8.8. We then ramped on the Raman coupling
in 750?s, adiabatically mapping the |z? and |y? states into the n = 1 or n = 3
eigenstates. Because our only experimental knob for dynamically changing the de-
tuning was ?RF we could not control ?x so when we initialized the system in |x? the
final dressed state always corresponded to the n = 2 branch. After turning on the
Raman, we ramped ?RF to its final value in 1 ms, effectively ramping ?z and ?y close
to zero. This detuning ramp had the additional effect of moving the location Dirac
point through the atoms when loaded in the n = 1 branch, creating a trajectory
where the state preparation was not adiabatic. This trajectory depended on the sign
of the detuning ramp so we combined data from different initial states that excluded
the Dirac point trajectories. Near the final location of the Dirac point, the state
preparation was not adiabatic regardless of the initial state or detuning used for the
ground state preparation. Figure 14a shows an example absorption image of atoms
initially prepared in the |y? state and with ?y such atoms are loaded in the n = 1
branch. The Dirac point was initially located near the lower-left edge of the cloud
and when ?i was ramped the location of the Dirac point was dragged across the
whole cloud leaving lines where non-adiabatic transitions occurred as can be seen in
Figure 14b. The location of the Dirac point as a function of ?i can also be directly
computed by numerically diagonalizing the SOC Hamiltonian from Equation 8.8.
167
At the end of this stage, excluding the points of non-adiabatic transitions, the state
of the system was described by the eigenstates in Equation 8.9.
Figure 14: a We prepared atoms in the |y? state such that they are adiabatically
mapped to the ground dispersion branch. b We ramped ?i and dragged the Dirac
point through the atoms.
Free evolution: We suddenly turned off the Raman coupling, projecting the
Raman dressed states back into the |xyz? basis. Each state represented a different
branch of the interferometer and they acquired phases that are proportional to tfree
? ?
|? (q)? ? a (q)ei?n,j(q)e?iEj(q)tfree/h?n n,j |j,k = q ? kj? , (8.14)
j?xyz
where Ej(q) = h?2q2/2m is the free particle energy. The Rashba wavefunction
phases ?n,j(q) that we were interested in measuring were imprinted during the
loading procedure. The dynamical phases Ej(q)tfree/h? acquired in the different
interferometer arms do not contribute to our knowledge of the Rashba eigenstates
as they describe the evolution of the system in the absence of Raman dressing.
Recombination pulse: We applied a 25 us Raman pulse that acted as a
168
second beam splitter in our interferometer sequence. The wave function after the
pulse was
? ?
|?(q)? = an,j(q)ei(?n,j(q)?Ej(q)tfree/h?)Uj,j?(q) |j,k = q ? kj? , (8.15)
j,j??xyz
where Uj,j?(q) = |Uj,j?(q)| exp( (pulse)i?j,j? (q)) is the matrix element of the unitary trans-
formation exp(iH?R(q)tpulse) associated to the Raman pulse. At the end of this
procedure, the population in a final state |l,q? was
?
Pl(q, t) = |Ul,i||
? (pulse)
Uj,l| an,ian,j cos(?i,j(q)t+?n,i(q)??n,j(q)+?l,i,j (q)), (8.16)
i=6 j
which directly reads out the phase differences. Here (pulse)(q)?l,i,j is a smoothly varying
phase imprinted by the recombination pulse and is independent of q in the limit of
short, strong pulses and does not affect the topological index of the system. The
angular frequencies
?i,j(q) = h?q ? ki,j/m+ ?i,j (8.17)
result from the known free particle kinetic energy, the recoil momenta and detun-
ing ?i,j from the tripod resonance condition. Figure 13b shows the momentum-
dependent populations in each output port at fixed tfree = 160?s and Figure 13c
shows the populations as a function of tfree for a representative quasimomentum
state (q1, q2) = (0.55,?0.92) kL.
We obtained the relative phases ??n,i,j,l(q) = ?n,i(q) ? (pulse)?n,j(q) + ?l,i,j (q)
from Equation 8.16 by fitting the measured populations to the sum of three cosines
169
c
1 0.6
0
0.3
?1
0.0
. . . 0 100 200 300 400
1 q1 in units of kL 0.6 tfree [?s]
0
0.3
?1
0.0
. . . 0 100 200 300 400
1 q1 in units of kL 0.6 tfree [?s]
0
0.3
?1
0.0
?1 0 1 0 100 200 300 400
q1 in units of kL tfree [?s]
Figure 15: a Probabilities as a function of quasimomentum for the three output
ports of the interferometer at tfree = 160?s b Probabilities as a function of free evo-
lution time tfree for an input state with quasimomentum (q1, q2) = (0.55,?0.92) kL
indicated by the blue star on a and in the topological ground branch (n = 1)
with the known free particle frequencies but unknown amplitudes and phases.
8.5.2 Combining phases from different measurements
We combined the phases ??n,i,j,l(q) obtained from fits to six different states
(two initial states split into 3 states). Similar to the Fourier spectroscopy measure-
ments, we performed a weighted average of the fitted relative phases
?
? (q) = l ???n,i,j,l(q)wi,j,l(q)?n,i,j
l wi,j,l( )
, (8.18)
q
where the weights were determined using fit uncertainties and when loading atoms
in the topological branch we additionally accounted for the trajectory of the Dirac
170
q2 in units of kL q2 in units of kL q2 in units of kL
Population Population Population
point during the loading procedure. Figure 16 shows an example of two different
weight arrays used to combine the phase difference associated to the z ? x transition
i, j = z, x for the topological branch n = 1 (?1,z,x):
1 60
0 40
?1 20
?1 0 1 ?1 0 1
qx in units of qL qx in units of qL
Figure 16: Two sample weight arrays wi,j,l(q) for i, j = z, x and atoms prepared in
the topological disperion branch. We obtained the weights using the uncertainties
from the fits and the trajectory of the Dirac point during the state preparation.
We combined the weights and the phase differences ??n,i,j,l obtained from the time
dependent fits using Equation 8.18 to obtain the phase maps displayed in Figure 17a
The ?spokes? in the weight arrays correspond to high uncertainty regions. This
uncertainty comes from our inability to resolve the phases of low frequencies ?ij(q)
as well as when two different frequencies ?ij(q) and ?i?j?(q) are close to each other
which is limited by the largest value of tfree in the experiment. The phase maps
in Figure 17a represent fits to a combined a total of 120, 000 different time traces
(60, 000 for each dispersion branch).
8.5.3 Measuring the topological index
Figure 17a shows typical phase-maps for both the non-topological and topo-
logical branches. In the non-topological phase-maps, the momentum dependence
of the recombination pulse ?pl,i,j(q) causes a smooth variation of the phases along
171
qy in units of qL
wz,x,l(q) [arb. u.]
1
0
?1
?1 0 1 ?1 0 1
q2 in units of kL q2 in units of kL
1
 Topological
 Non-topological
0
1
0
1
0
0.0 0.5 1.0 1.5 2.0 0 1 2
Polar angle in units of ? Polar angle in units of ?
0.5
0.0
?0.5
0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5
Loop radius in units of kL Loop radius in units of kL
Figure 17: Topological invariants from quantum state tomography, for the non-
topological branch (n = 3, left) and the topological branch (n = 1, right). a Phase
differences as a function of quasimomentum from the the z ? x transition b Phase
differences as a function of polar angle for a loop radius 0.77 kL from the z ? x
(top), x? y (middle) and y ? z (bottom) transitions. The phases associated with
the topological branch are characterized by two ? valued discontinuities. Each row
of phases was shifted by a constant value so that the three rows of phases share the
same vertical axis. All phases shown here were binned and averaged using the fit
uncertainties as weights. c Inferred Chern number as a function of loop radius. For
loops with q > 0.4 kL we obtained an integrated Berry phase and asymptotic Chern
number of ?B/2? = 0.01(1) for the non-topological branch and ?B/2? = 0.5(5) for
the topological branch.
172
Berry phase in units of 2?
q2 in units of kL
Phase in units of 2?
the Raman recoil axes that does not affect the evaluation topological index of our
system. We recovered the phases ?n,j of the full spinor wave function from the
relative phases ??n,i,j(q) by choosing a particular gauge such that ?n,3 = 0. We
then used the values of an,i obtained from measuring the populations in the |xyz?
states at tfree = 0 in combination with the phases of the wave function to com-
pute the Berry connection [144]. Figure 17b shows the three phase differences as a
function of polar angle for a loop of radius q ? 0.77 kL. In addition to the smooth
variations induced by the recombination which are present in both columns, the
phases of the topological branch have two ? valued jumps that lead to non-zero
Berry phases when the Berry connection is integrated along a closed loop in mo-
mentum space. Figure 17c shows the integrated Berry phase as a function of loop
radius. As mentioned earlier the largest value of tfree limits how well we can resolve
the phases of small frequencies and this is reflected in the large variation in the
Berry phase depicted in the shaded region of Figure 17c near q = 0. For loops with
q > 0.4 kL we obtain an integrated Berry phase that suggests an asymptotic Chern
number of ?B/2? = 0.01(1) for the non-topological branch and ?B/2? = 0.5(5)
for the topological branch. However, Berry?s phase measurements like ours include
the (potential) contribution of any Dirac strings traversing the integration area. In
our system, these are possible at the Dirac point (red dot near q = 0), and each
contributes ?2? to ?B (see Section 7.6). Even with this 2? ambiguity we are can
associate a half-integer Chern number with the topological branch, which is possible
only for a topological dispersion branch in the continuum.
173
8.6 Conclusion
In conventional lattices ? for example graphene, or the topological Haldane
model ? it is well established that Dirac points each contribute a Berry?s phase
of ?B/2? = ?1/2 [145], but crystalline materials conspire for these to appear in
pairs [127], always delivering integer Chern numbers. In contrast, our continuum
system contains a single Dirac point, resulting in a non-integer Chern number. This
leads to intriguing questions about edge states at interfaces with non-integer Chern
numbers with non-integer Chern number differences. Initial studies in the context of
electromagnetic waveguides [146] and atmospheric waves [107] have applied Chern
invariants and the bulk-edge correspondence to continuous media.
While the true Rashba Hamiltonian features a ring of degenerate eigenstates,
our implementation including the quadratic and cubic Dresselhaus-like SOC lifts
this macroscopic degeneracy giving three nearly degenerate minima [135]. Already
these three minima could allow the study of rich ground state physics in many
body systems of bosons, for example the formation of fragmented BECs [28] when
the system does not condense into a single-particle state. Furthermore, the use of
additional spin states or larger Raman couplings can partially restore this degeneracy
allowing the possible realization of fractional Hall like states [147].
174
Chapter 9: Conclusions and outlook
This thesis presented new experimental techniques that have proven to be
useful in the control and characterization of ultracold atomic systems and applied
them in a new implementation Rashba-type spin-orbit coupling.
We developed a Fourier transform spectroscopy technique [25] which is based
on measuring the quantum coherent evolution of a single particle system under a
quench of a Hamiltonian of interest. This technique was successfully used to measure
the dispersion relation of a BEC with tunable 1D (equal combination of Rashba and
Dresselhaus) spin-orbit coupling. The use of this technique was extended to thermal
gases with broad momentum distributions to perform a parallelized measurement of
the dispersion relation of a system with Rashba-type SOC [26] as well as the band
structure of a fractional period adiabatic superlattice [27].
We implemented CDD in the ground hyperfine manifold of 87Rb by applying a
strong RF magnetic field [81]. The CDD states are first-order insensitive to magnetic
field fluctuations, making them effective clock states, and additionally, have non-zero
matrix coupling elements that allow cyclical couplings that are not possible in the
bare hyperfine |mF ? states. The clock states have made our experiments more robust
against magnetic field noise and were a necessary ingredient for the implementation
175
of Rashba spin-orbit coupling as well as the engineering of fractional period adiabatic
superlattice and an ongoing project involving Hofstadter cylinders
Finally, we showed a new implementation of Rashba spin-orbit coupling using
Raman induced transitions of the CDD states [78,135] and characterized the topol-
ogy underlying this system [26]. We presented a protocol for performing quantum
state tomography which involves a three-arm Ramsey like interferometer and used
it to reconstruct the momentum-dependent wave function and calculate topological
invariants. Unlike conventional materials with an underlying crystalline structure
where topological invariants take integer values, we found that our system in the
continuum was characterized by half-integer invariants. Our Rashba implementation
offers the possibility of studying new ground state physics at the nearly degenerate
minima like the formation of fragmented condensates [28] or possible realizations of
fractional Hall like states [147]. One open question for a half-integer Chern number
system like ours is what kind of edge states emerge at the interfaces where the Chern
numbers differ by a non-integer number.
All the current experimental efforts focus on understanding the physics of a
Hofstadter cylinder under different magnetic fluxes and the role that disorder can
play in driving phase transitions. Additionally, a considerable effort is going to the
fabrication of a new apparatus that will allow the creation of ultracold samples of
87Rb and 39K. The use of bosonic 39K atoms will allow the tuning of scattering
lengths using Feshbach resonances [148], enabling new kinds of experiments where
atomic interactions can play an important role. The new apparatus will feature
better optical access that will enable high resolution imaging and the projection of
176
arbitrary potentials.
177
Appendix A: The good, the bad and the ugly of RbLi
This appendix summarizes the best, the worst and the meh aspects of the
RbLi apparatus. Hopefully the items presented can help guide some decisions of
future students building experimental apparatuses for ultracold atoms.
A.1 The good
It is very easy to come up with a list of bad things that don?t work quite well
in the lab. Coming up with a list of good things that work well is harder as we do
not tend to think too much about the things that we are not currently fixing. The
list bellow summarizes what I believe are the outstanding good players in the lab.
Overkill transistor banks: Large currents in the lab (quadrupole and Zee-
man slower) are controlled with transistor banks formed by a group of MOSFETS
whose drain and source are connected in parallel and sharing the same gate voltage
that is controlled by a PI servo. The Zeeman slower always operates at a fixed
current but the current in the quadrupole coils is dynamically changed throughout
the experimental sequence and a fast response is desirable. In 2013 we replaced the
quadrupole MOSFET bank with a new unit that contains 20 IXFN 520N075T2 tran-
sistors rated for 75 Vand 480 A (left panel of Figure 1). Even though our currents
178
are well bellow the 480 A limit, the performance of the transistors really decays as
the drain to source voltage is increased as can be seen in the right panel of Figure 1.
The use of more transistors reduces the power dissipation of each individual tran-
sistor which allows us to operate the power supply at a higher voltage of 15 V that
helps counteract the inductive kickback of the coils. With the new transistor bank
the turn on time of the coils was reduced from 100 ms to 5 ms leading to improved
magnetic trapping and better Stern-Gerlach pulses for imaging, only with an un-
avoidable small number of blown off transistors. This was a one time problem that
happens typically when the turn-on voltage for one or a few transistors is slightly
lower than the rest and therefore few number of transistors end up dissipating all
the power when they are first turned on. Using transistors from the same batch is
advisable so that they operate as similarly as possible.
source
gate drain
Figure 1: Left: New MOSFET bank. Right: Safe operation regime of the IXFN
520N075T2 MOSFET. Even though they are in principle rated for up to 480 A the
maximum safe current is greatly reduced at larger drain to source voltages VDS. A
high VDS is desirable to reduce the inductive kickback during turn on.
179
Oven chamber including hand made in vacuum shutters: Before go-
ing into the Zeeman slower, the atoms that are heated in the Rb oven travel to
the main oven chamber that is pictured in Figure 2b containing a cold-cup and an
oven shutter. The cold-cup is a cylindrical shaped copper piece that is attached
to the cold end of a thermo-electric cooler (TEC) via a copper rod. We keep the
cold-cup temperature at ?30 C in order to capture excess Rb atoms in the chamber
and prevent damaging the ion pumps. The oven shutter allows us to block after
the MOT loading stage to prevent unwanted heating. We use a homemade (by
Ian) device, made from a re-purposed hard drive disk shutter with a metallic flag
attached to its end. The shutter is electrically connected to en electric feedthrough
with vacuum-compatible Kapton sealed wires. Other apparatus within the JQI [149]
have commercial shutters from Uniblitz and some of them have failed in the past.
Overall we have found this setup to be very reliable. The only problem we experi-
enced once was some accumulation of Rb on the cold cup that started blocking the
atomic beam. To remedy this we reversed the polarity of the TEC and heated the
cold cup barely enough so that the accumulated Rb atoms melted and moved away
from the aperture of the atomic beam.
Ultraviolet LEDs: We have two 3 W ultraviolet LEDs from Mightex placed
at the glass cell side of the vacuum system. One is aimed at the vacuum window
where the slower beam enters and the other is placed aiming at the glass cell. The
LEDs prevent Rubidium from depositing on the vacuum system and can conve-
niently be turned on and off with a TTL signal from the computer. We have found
that routinely turning them on (for example, leaving them on overnight) leads to a
180
To TEC
atomic beam
cold cup
oven shutter
Figure 2: The RbLi oven chamber. We use a cold cup to prevent excess Rb atoms
from damaging the ion pumps and a homemade in-vacuum shutter to block the
atomic beam after the MOT stage to prevent heating of the atoms at later stages.
smoother operation of the system.
Mirror mounts with picomotor actuators: We use 8816-6 picomotor
optical mounts from New Focus Optics whose deflection angle can be electronically
adjusted on the order of microradians. The addition of picomotor mounts has made
alignment of laser beams to the atoms significantly easier. We use this mounts on
the last tunable mirror before the atoms for beam paths whose alignment is critical,
for example in optical dipole trap and Raman beams.
Polarizers on MOT beams: This item is a bad thing disguised as a good
thing. The light of our MOT beams is coupled to polarization maintaining optical
fibers. We found that despite our best efforts to align the polarization of the incom-
ing light to the slow/fast axis of the fiber the fluctuations in the output polarization
could cause considerable instabilities in the BEC production. To keep the polariza-
tion clean we placed polarizers at the output of the fibers. We found that despite
the power hit we get from the changes in polarization, this solution leads to a much
181
more stable production of BECs.
Lab couch: When the experiment is functional enough that data can be
taken long hours in the lab are often required. If it gets late, the lab couch allows
the person running the experiment to take small naps as the data keeps coming
while still being close to the apparatus in case something needs to be fixed. A
slightly rested grad student tends to make less poor decisions than an exhausted
grad student.
Other elements already mentioned in the main text: The new master
laser from Vescent photonics has been very stable and reliable. The new Mako
camera has been very helpful to get rid of unwanted fringes in absorption images.
Labscript makes writing experimental sequences very straightforward.
A.2 The bad
The bad, these are elements of the apparatus that were constant sources of
pain and if considering a new experimental design should be avoided.
Water cooling shared between two labs: The quadrupole and Zeeman
slower coils as well as the transistor banks require water cooling due to Joule heat-
ing. Our lab space is shared with a Rubidium-Ytterbium ultracold mixtures ap-
paratus [150] and amongst the things we share is the water cooling system. The
schematic in Figure 3 illustrates a simplified layout of the water cooling system.
The water was filtered at two different points, first each line has a 440?m partic-
ulate filter from Swagelok and then the water returning to the heat exchanger is
182
filtered with a low-impedance cellulose cartridge (McMaster 7191K11). Both filters
only capture impurities in the water for one given flow direction. One of the failure
modes which occurs when one of the booster pumps is turned on before the heat
exchanger, causing water to flow from one experiment to the other and bringing a
collection of nasty things that escapes the filters into the coils. Over the years our
system has suffered of clogged filters, clogged coils and broken booster pumps. For
best operation it is highly recommended that the cartridge filter is changed and that
the Swagelok filters be cleanded at least once a year and that a 10% solution of an
anti-corrosive Optishield Plus in water is used as a coolant. Even when follow-
ing this practices, we managed to find lots of gunk and unidentified objects (sand?
glass? mud? oxide? dead bacteria?) in the water, just at a slower rate. Besides
this issue, just having to interrupt the operation of two experiments whenever any
plumbing works needs to be done on one experiment is quite inconvenient.
Back flow :(
Booster pump Booster pump
Correct flow
RbLi RbYb
manifold Heat exchanger manifold
Filter Filter
Figure 3: Simplified schematic of the shared water cooling manifold.
183
Flipper mirrors: The optical path of the MOT beams near the atoms is
very close to that of Raman, optical dipole trap and probe beams. Since the MOT
beams are only used at the early stages of the experiment it is tempting to use
flipper mirror mounts so that once they are no longer needed they can be moved
away to make space for other beams. This was the approach originally taken in the
lab and we used 8893-K motorized optical flipper mounts from Newport in multiple
locations. As they break over and over again, they have been slowly replaced by
more stable solutions such as periscopes or polarizing beam splitters wherever it is
possible. Flipper mirrors are always bound to break, it is only a matter of time
before it happens. Avoid using them unless you absolutely have no alternative.
Optical fibers right below air vents: The optical fibers connecting the
main experiment optical table and the laser optical table are routed close to a pair
of AC vents in the lab. The changes in air temperature result in polarization fluctu-
ations at the output of optical fibers (see previous section about MOT polarizers),
a constant cause of pain and instability in our BEC production. We have tried to
remedy this issue by partially blocking vents and enclosing the fibers in a large PVC
tube.
Free spaced dipole laser: The laser system providing 1064 nm light for the
optical dipole trap is not fiber coupled and is setup in the same optical table as the
vacuum system; we are not able to change the laser without destroying the alignment
of the dipole trap with the atoms. This issue became relevant while setting up a
1D optical lattice by retro-reflecting one of the dipole trap beams as we noticed
that the laser mode changes on a very fast timescale, leading to large fluctuations
184
in the optical lattice. In the original design of the laser the light was meant to be
fiber coupled using high-power photonic crystal fibers but they did not have built
in mode expanders which resulted in the tip of the fibers repeatedly getting burnt
after some time of use. In short, mode expanders are recommended in applications
involving large optical powers.
A.3 The ugly
The ugly elements are not quite bad but they don?t function flawlessly either.
If given the option to replace them with something better I definitely would.
Kepco bipolar power supplies: We use three Kepco BOP 20-20M bipolar
power supplies to provide the current for the bias coils. While it is nice to have
a commercially available power supply that can provide ?20 A they come with a
few drawbacks. First the current they provide has 60 Hz noise in it and in order
to suppress it and stabilize the currents we must use a PI feedback circuit. The
power supplies has multiple banks of NPN and PNP transistors inside mounted on
a big heat sink with fans attached to it making them quite noisy; it is not optimal
to place them close to the main experiment chamber and long connections open
the door to unwanted ground loops. Additionally they have a few failure modes.
All the problems we experienced whenever they stopped working (e.g. the output
current railing) were symptoms of malfunctioning transistors, something that we
experienced on multiple occasions.
185
Toptica?s BoosTA: Our cooling light comes from a Toptica DL Pro laser
that is amplified using a Toptica BoosTA tapered amplifier system. The output
power of this TA has been relatively stable over the years, in fact it has been
operating for over 7 years now without the need of replacing the TA chip, unlike a
homemade TA setup (see [62, 63]). The downside is it has a tendency to turn itself
off and on its worst days it would turn off so often that it would be impossible to
operate the experiment. We have not been able to identify the problem despite our
best efforts to look into the TA controller, the TA itself, multiple conversations with
Toptica engineers, etc.
Too many USB devices connected to the same computer: We use
multiple USB-6229 data acquisition (DAQ) devices from National Instruments.
They are located at different points of the lab and then connected to the control
computer through USB to optical fiber adapters that break the ground between the
computer and the rest of the lab equipment (a practice we always try to follow when
connecting things to the computer). We have a total of 6 NI devices in addition
to other equipment like oscilloscopes all connected to the computer through BNC
cables. Often times we struggled with the computer failing to detect one or multiple
devices and it would take a very special (and different every time) combination of
plugging and unplugging, turning off and turning back on things until all devices
were recognized by the computer. We observe that the problem occurs less often
when we don?t have as many USB devices connected to the computer.
186
Appendix B: New apparatus
As mentioned in the main text, the construction of a new apparatus for pro-
ducing BECs of 87Rb and 39K is underway. The design of the apparatus is intended
to be a bug fix version of the RbChip [151] lab at NIST. The new apparatus does
not have a Zeeman slower and instead will use magnetic transport coils to move
atoms from a MOT cell to the main science cell.
This Appendix describes some aspects of the design and construction of the
new apparatus where I was involved. Disclaimer: none of this things have been
tested yet so we don?t know yet if it will all work horribly. I have tried to include
an extensive list of the part numbers used so they can be used as a future reference.
B.1 Water cooling
The lack of a Zeeman slower in the apparatus greatly simplifies the water
cooling system compared to that of RbLi. Since we don?t anticipate to have any
coils with high flow impedance we expect that the pressure from a recirculating
chiller will be enough to provide water cooling to the transistor banks and the
magnetic transport coils.
Our choice of chiller was the TF1LN400-LN 1.4 kW recirculating chiller from
187
Thermo Fisher Scientific. The water is filtered both at the output and the re-
turn with a high-impedance filter with a cellulose cartridge filter(replacement filter:
Mcmaster 4422K3, filter housing: McMater 7191K11). A breakout manifold di-
vides the chilled water into 5 different lines, each one with a flow meter (Proteus
Industries 0101C110) that can be used to interlock the current of water cooled
applications to the flow of water. Based on previous experiences with plumbing (see
Appendix A) I highly encourage replacing the filters at least once a year and to use
a solution of 10% corrosion inhibitor (e.g. OptiShield Plus) and distilled water as a
coolant.
a. Service corridor side b. Experiment side
chilled house water
breakout manifold
water in
water out
filters flow meters
chiller
Figure 1: Water cooling
188
B.2 Electrical installation
We have two Agilent 6690A (440 A at 15 V max. current) to provide all the
necessary currents. The power supplies are located in the service corridor and are
connected to three copper bars corresponding to ?15 V and ground using welding
cable (McMaster 7818A17) that is laid on cable trays (McMaster 30065T11 and
similar). The copper bars serve as a hub for power distribution inside the lab where
lab devices such as transistor banks can be connected. The positive and negative
terminals of the power supples have two cables attached to them and they are all
arranged in the pattern shown in Figure 2 so that the magnetic field produced by
currents in them is closer to a magnetic quadrupole which decays faster than the
field of a magnetic dipole. We are trying to minimize unwanted magnetic fields at
the experiment and being nice to our neighbors as well. We are not planing to use
commercial bipolar power supplies in this lab (see Appendix A) and instead we will
be using a MOSFET based homemade supply that will draw current from the ?15 V
rails.
Figure 2: Configuration of the welding cables connecting the power supplies to
equipment in the lab.
189
2 2
1 3
power supplies 3
1
Figure 3: A roller coaster ride, from the power supplies in the service corridor to
three copper bars that distribute the power.
B.3 Coil winding
All the coils in the apparatus including magnetic transport, bias and gradient
cancellation will be made of ribbon wire (Laminax from Bridgeport Magnetics).
We followed the the coil fabrication process described in [151] which involves first
winding a fixed number of turns around a prefabricated form with a particular
geometry. The coils were then covered with a machinable epoxy (Stycast 1266)
to fill in any air gaps. A lesson we learned while doing this is that only room
temperature epoxy should be mixed. We keep the epoxy in a fridge to extend its
lifetime but if it is cold some tiny drops of water will condense in it as it is being
mixed and it will not properly be cured. To minimize the air bubbles inside the
epoxy we placed the coils on a vacuum bell and we pumped the air out using an
electric vacuum pump (McMaster 4396K21). After the epoxy has cured (overnight
if it is left at room temperature or less if it is left at higher temperature) the coils
190
are ready for lathing to remove all excess epoxy and kapton tape up to the surface of
the copper. After some trial and error (and lots of frustration) we found that using
a diamond tip cutter (McMaster 3316A32) and spinning the lathe not faster than
150 rpm gives the best results. Using a cutter that is not sharp enough or cutting
too aggressively close to the soft copper results in deformed traces that that merge
into each other causing unwanted shorts. The coils of the RbChip apparatus were
machined at NIST using a special aluminum form to mount the coils on the lathe
(see. The machinist at UMD considered this was not safe enough so I instead had
to mount the coils using a 6 jaw chuck as shown in Figure 4a. For anyone making
coils in the future: it is sort of an art to get it right and screwing up many coils at
the beginning is part of learning the art. That being said, an important aspect of
learning the ?art? right be as careful and consistent as possible from the start.
B.4 Rb source and ?oven?
The new apparatus will have a cold ?oven? to keep the vapor pressure low
connected to a glass cell where light induced atomic desorption (LIAD) [152] will be
used to increase the Rb vapor pressure for MOT loading using non-thermal means.
The Rb source consists of a 1.33? CF flanged bellow (Kurt J. Lesker MH-CF-A03)
with a Rb ampule. The bellow is housed in the ?oven? which is designed to keep the
source at temperatures near 1 C. The oven is made of hollow aluminum cylinder
with a slit on one side with tapped holes so that 1/4 ? 20 screws can be used to
tighten it and fix the oven to the source. The bottom of the oven attaches to the
191
a. Coil mounted on lathe b. Coils before and after lathing
c. Coil labeling system
Figure 4: a. I used a six jaw chuck and a diamond cutter on the lathe to remove
the excess epoxy and kapton on the coils. b. Coils before (left) and after lathing
(right). c. A good labeling system is important to keep to ensure uniformity of
coils. The ?Wartortle? coil shown here has 64 turns. During fabrication we keep
track of all the number of turns and resistance of all coils in a table.
cold end of a TEC that provides the cooling. The hot side of the TEC is attached
to a heat sink made of a hollow brass piece with tapped holes for 1/4? NPT pipes
that will be used to provide water cooling. Figure 5a shows CAD drawings of both
of this parts.
We tested the performance of a prototype without any vacuum systems at-
tached to it. We applied a 2 A current to two TECs (Digikey 102-1664-ND) sand-
wiched in between the heat sink and oven. The heat sink was water cooled using
? [18]C water. Figure 5b shows the temperature of both the oven side (top) and
the heat sink side (bottom) of the assembly. We did not use any insulation for this
192
test to prevent condensation (see Figure 5c right) which should be done once its
installed on the apparatus.
Our initial plan was to control the temperature using a linear temperature
controller designed at the JQI (the design is available at the JQI GitHub) inter-
faced to the lab computer and Labscript using a serial to ethernet adapter (WIZnet
WIZ107SR). This project is not completed to this date.
a. Rb ?oven? and heat sink b. Bench test temperatures
c. Testing prototype oven and heat sink
Figure 5: Rubidium oven assembly
193
B.5 Table enclosures
The enclosures of the optical tables are made of Alumalite from Laminators
Inc. mounted on frames made out of aluminum extrusions from 80/20 and slid-
ing tracks (2220 and 2210). Alumalite is a sandwich of a corrugated corrugated
polypropylen material in between two thin sheets of aluminum. We chose this ma-
terial because it is strong and lightweight. Its laser safety properties remain to be
tested but we anticipate it is better than the acrylic panels at the RbLi lab which
are essentially transparent at 1064 nm.
To the new and future members of the lab: I sincerely hope the things I
designed and built don?t cause you much pain!
194
Appendix C: Full derivation of the Raman coupled |xyz? states
In this Appendix I derive the full time-dependent Hamiltonian describing the
|xyz? states coupled by three Raman beams to produce Rashba type SOC (see
Chapter 8. It takes a bit of thinking to understand what the frequencies of the
different Raman coupling terms correspond to in the RF rotating frame where the
|xyz? states are defined. Hopefully this Appendix helps to clarify our specific choices
for the laser geometry and polarization.
Our system is based on the proposal described in [78] to engineer a system with
Rashba-like SOC. We consider 87Rb atoms in the ground hyperfine F = 1 manifold
subject to a constant magnetic field B0ez and an RF magnetic field BRF cos(?RFt)ex
as in Chapter 6. The system is described by the Hamiltonian
H?RF =

?0F?z ? (F? 2z ? 1) + 2?RF cos(?RFt)F?x, (C.1)h?
where ?0 = gF?BB0 is the Larmor frequency,  is a quadratic Zeeman1 shift that
breaks the degeneracy of the |mF = ?1? ? |mF = 0? and |mF = 1? ? |mF = 0?
transitions, ?RF = gF?BBRF/2 is the RF coupling strength and 1 is the identity
1the quadratic dependence of  on the applied bias field is not explicitly indicated on the symbol
but is important to take into account to get the detuning just right
195
matrix. We transform the Hamiltonian into the RF rotating frame using the unitary
transformation U?(t) = exp(?i?RFtF?z). The spin-1 operators under this transforma-
tion become
F?x ? cos(?RFt)F?x ? sin(?RFt)F?y
= ei?RFtF?+ + e?i?RFtF??
F?y ? sin(?RFt)F? + cos(?RFt)F? (C.2)x y
= 1(ei?RFtF? ? e?i?+ RFtF??)
i
F?z ? F?z.
The unitary evolution in the rotating frame is described by the transformed Hamil-
tonian U? ?(t)(H?RF? ih??t)U?(t), which after neglecting terms that are oscillating with
angular frequency 2?RF is
H?RWA = ?F?z ?
 (F? 2z ? I) + ?RF F?x (C.3)h?
The eigenstate of Equation C.3 are the |xyz? states described in Chapter 6. Now
we apply three Raman beams as shown in Figure 1. This configuration differs from
the one proposed in [78] for technical reasons as we wanted all the Raman recoil
vectors to lie within the imaging plane of the experiment which corresponds to the
xy plane.
The electric field at the atoms in the lab (not rotating) frame is
?3
E(x, t) = E e ei(ki?x??it)i i , (C.4)
i=1
196
Figure 1: Laser layout: We use a strong RF field and three linearly polarized Raman
beams propagating in the xy plane couple the |xyz? states and engineer the Rashba
Hamiltonian.
where Ei is the field amplitude, ?i is the angular frequency, ki is the wave vector and
ei is the polarization of each of the beams. In order to have the right coupling matrix
elements to cyclically couple all three states in a ring-like coupling as described
in [135] we need a Hamiltonian of the form
H?SOC = (?x,?y,?z) ? F? (C.5)
(see Section 6.3.3). This is possible if we choose two Raman beams to have in plane
polarization and one vertically polarized beam:
(k ,?k
e = 1y 1x, 0)1 ,||k1||2
e2 = (0, 0, 1),
(C.6)
= (k3y,?k3x, 0)e3 ,||k3||2
197
The Raman coupling strength is proportional to the vector polarizability (see
Section 3.3.2) and the Hamiltonian describing the atom-light coupling is
H?R = (iuvE? E?) ? F? , (C.7)
where uv is the vector polarizability. The expression for the cross product of the
electric field at the atoms is quite messy, lets rewrite it in a more convenient way:
E? E? = (E?e e?i(k1?x??1t) + E?e e?i(k2?x??2t)1 1 2 2 + E?e e?i(k3?x??3t)3 3 )? c.c
= E?E (e ? e )ei[(k2?k1)?x?(?2??1)t] + E?E (e ? e )ei[(k3?k1)?x?(?3??1)t]1 2 1 2 1 3 1 3
+ E?E (e ? e )ei[(k1?k2)?x?(?1??2)t] + E?E (e ? e )ei[(k3?k2)?x?(?3??2)t]2 1 2 1 2 3 2 3
+ E?E (e ? e )ei[(k1?k3)?x?(?1??3)t]3 1 3 1 + E?3E2(e3 ? e )ei[(k2?k3)?x?(?2??3)t][ 2
= 2i (e ? e )Im{E?E ei(k21?x??21t)1 2 1 2 }
+ (e ? e )Im{E?E ei(k32?x??32t)1 3 1 3 }]
+ (e ? e )Im{E?E ei(k32?x??32t)2 3 2 3 }
(C.8)
and using the definition of the polarization vectors (Equation C.6)
? = (?k1x,?k1y, 0)e1 e2 = ?k?||k1||2 1
? = (0, 0,?ke e 1yk3x + k3yk1x)1 3 = ez sin ?13 (C.9)||k1||2||k3||2
? = (k3x, k3y, 0)e2 e3 ||k ||2 = k?3,3
198
we obtain the the desired Hamiltonian describing the atom light interaction
[ ]
iu E?v ? E ? F? = ?2uv ? k?1Im{12}+ ez sin ?13Im{13}+ k?3Im{23} ? F?
(C.10)
= (?x,?y,?z) ? F?,
where
? = k1x Im{? i(k21?x?? k21t) 3x i(k32?x??32t)x ||k || 12e }+ Im{? e }1 ||k || 233
? = k1y Im{? kei(k21?x??21t)}+ 3y Im{? ei(k32?x??32t)} (C.11)y ||k || 121 ||k3|| 23
? = Im{? ei(k31?x??31t)z 13 },
and
?12 = 2u E?v 1E2
? = ?2u E?13 v 1E3 sin ?13 (C.12)
?23 = ?2uvE?2E3.
Now we need to transform Eq. C.10 into the rotating frame, this is where
things start getting fun. The ?slow? or ?fast? nature of a given term depends on the
specific choice of frequencies on each Raman beam which should be such that the
frequency differences ?ij are resonant with dressed state transitions in the rotating
frame as shown in Figure 2a.
I showed in Equation C.2 that in the rotating frame the F?x and F?y operators
get additional factors of exp(?i?RF) while F?z remains unchanged. We must there-
fore have the frequencies of beams giving rise to F?x and F?y coupling to differ in
199
frequency by an additional ?RF and the beams that give the F?z coupling to be close
in frequency. The two possible ways of doing so are shown in Figure 2b and they
determine weather ?21 and ?31 are positive or negative.
a. Resonant Raman coupling b. Possible Raman laser frequency configurations
(i) Blue detuned (ii) Red detuned
Figure 2: a The choice of laser frequencies should be such that in the frame ro-
tating with frequency ?RF we get resonant Raman coupling of the |xyz? states. b.
Possible laser frequency configurations: i) Blue detuned configuration: There are
2 frequencies smaller by about ?RF and one larger frequency. ii) The red detuned
configuration: there are 2 frequencies that are larger by about ?RF and one smaller
frequency.
Lets look at the firs term of the ?xF?x coupling to get an idea of how the RWA
200
RF eigenenergy
Laser frequency
will work here:
1 k1 ( ) ( )?(1) ? xF? ? ei(k21?x??21)t ? ?? e?i(k21?x??21t) ei?RFtF? + e?i?12 + RFtx x 4 12 F?i ||k ?1||
? 1 k1
( )
x ? ei(k21?x?(?21??RF)t)F? ? ?? e?i(k21?x?(?21??RF)t)4 F?i ||k 12 ? 12 ?1|| (
= 1 k1x2 |?12| sin[k21 ? x? (?21 ? ?RF)t+ ?||k || 12]F?x1 )
? cos[k21 ? x? (?21 ? ?RF)t+ ?12]F?y .
(C.13)
Here the upper sign corresponds to the ?21 > 0 case (blue detuned) and the lower
sign to ?21 < 0 (red detuned) and I performed a RWA approximation in the second
line by neglecting the terms oscillating with frequency close to 2?RF. Similarly, the
second therm of ?xF?x is
1 k3 ( ) ( )?(2) ? xF? i(k32?x??32)t ? ?i(k32?x??32t) i?RFt ?i?RFtx x 4 ?23e ? ? e e F?+ + e F??i ||k3|| ( 23
? 1 k3x2 |?23| sin[k32 ? x? (? ? ?||k || 32 RF)t+ ?23]F?x3 )
? cos[k32 ? x? (?32 ? ?RF)t+ ?23]F?y
(C.14)
where I used the same sign convention as before. It is important to keep in mind
that if ?21 is positive then ?32 must be negative and vice versa. Lets keep cranking
201
the algebra!
?(1) ? ?1 k1
( ) ( )
y
F? ? ei(k21?x??21t ? ?? e?i(k21?x??21t) i?RFt ?i? ty y 4 e F?||k || 12 12 + ? e
RF F??
1 ( )
? ?1 k1y ? ei(k21?x?(?21??RF)t)F? + ?? e?i(k21?x?(?21??RF)t)4 ||k || 12 ? 12 F??1 (
= ?1 k1y2 |?||k || 12| cos[k21 ? x? (?21 ? ?RF)t+ ?12]F?x1 )
? sin[k21 ? x? (?21 ? ?RF)t+ ?12]F?y ,
(C.15)
1 k3 ( ) ( )?(2)F? ? ? y ? ei(k32?x??32t ? ?? e?i(k32?x??32t) ei?23 RFty y 4 23 F?+ ? e?i?RFtF?||k ?3|| (
? ?1 k3y2 |?23| cos[k||k || 32 ? x? (?32 ? ?RF)t+ ?23]F?x3 )
? sin[k32 ? x? (?32 ? ?RF)t+ ?23]F?y .
(C.16)
The complete Hamiltonian in the rotating frame after doing the rotating wave ap-
202
proximation is then
?
= 1 |?12|H? ?( )2 k1x sin[k21 ? x? (?21 ? ?RF)t+ ?12]? k1y cos[k21 ? x? (?21 ? ?||k || RF)t+ ?12] F?x1 ( ) ?
+ ? k1x cos[k21 ? x? (?21 ? ?RF)t+ ?12]? k1y sin[k21 ? x? (?21 ? ?RF)t+ ?12] F? ?? y
1 |? |?( )23
2 k sin[k ? x? (? ? ? )t+ ?||k || 3x 32 32 RF 23]? k3y cos[k32 ? x? (?32 ? ?RF)t+ ?23] F?x3 ( ) ?
+ ? k3x cos[k32 ? x? (?32 ? ?RF)t+ ?23]? k3 sin[k32 ? x? (?32 ? ?RF)t+ ?23] F? ?y y
+|?13| sin(k31 ? x? ?31t+ ?13)F?z
(C.17)
In order to go from this rather complicated looking Hamiltonian to the effective
time independent Hamiltonian used in Chapter 8 we need to take two steps: first the
off resonant coupling terms need to be neglected. This can be more or less safely
done since they will be detuned by something on the order of tens to hundreds
of kHz. Second we need to go into a second transformed frame using the unitary
transformation
?
U? = ei(ki?x??it) (C.18)
i?{xyz}
and eliminate the ?fast? terms that are proportional to exp(i?ijt). The neglected
terms of the Hamiltonian in Equation C.17 have the effect of slightly shifting the
eigenenergies of the effective Hamiltonian from Equation 8.8. We interpret this shifts
in energy as coming from new effective Raman coupling strengths that slightly differ
from our calibrations performed by measuring the Rabi frequencies of individual
203
pairs of Raman beams.
204
Bibliography
[1] Bose. Plancks Gesetz und Lichtquantenhypothese. Zeitschrift fu?r Physik,
26(1):178?181, December 1924.
[2] A. Einstein. Quantentheorie des einatomigen idealen Gases. In Albert Einstein:
Akademie-Vortra?ge, pages 237?244. John Wiley & Sons, Ltd, 2006.
[3] F. London. On the Bose-Einstein Condensation. Physical Review, 54(11):947?
954, December 1938.
[4] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A.
Cornell. Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor.
Science, 269(5221):198?201, July 1995.
[5] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee,
D. M. Kurn, and W. Ketterle. Bose-Einstein Condensation in a Gas of Sodium
Atoms. Physical Review Letters, 75(22):3969?3973, November 1995.
[6] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet. Evidence of
Bose-Einstein Condensation in an Atomic Gas with Attractive Interactions.
Physical Review Letters, 75(9):1687?1690, August 1995.
[7] Ketterle W., Durfee D.S., and Stamper-Kurn D.M. Making, probing and
understanding Bose-Einstein condensates. Proceedings of the International
School of Physics &ldquo;Enrico Fermi&rdquo;, 140(Bose-Einstein Condensa-
tion in Atomic Gases):67?176, 1999.
[8] D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell.
Collective Excitations of a Bose-Einstein Condensate in a Dilute Gas. Physical
Review Letters, 77(3):420?423, July 1996.
205
[9] M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee,
C. G. Townsend, and W. Ketterle. Collective Excitations of a Bose-Einstein
Condensate in a Magnetic Trap. Physical Review Letters, 77(6):988?991, Au-
gust 1996.
[10] M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn,
and W. Ketterle. Observation of Interference Between Two Bose Condensates.
Science, 275(5300):637?641, January 1997.
[11] B. DeMarco and D. S. Jin. Onset of Fermi Degeneracy in a Trapped Atomic
Gas. Science, 285(5434):1703?1706, September 1999.
[12] Dan M. Stamper-Kurn and Masahito Ueda. Spinor Bose gases: Symmetries,
magnetism, and quantum dynamics. Reviews of Modern Physics, 85(3):1191?
1244, July 2013.
[13] Xibo Zhang and Jun Ye. Precision measurement and frequency metrology
with ultracold atoms. National Science Review, 3(2):189?200, June 2016.
[14] Immanuel Bloch, Jean Dalibard, and Sylvain Nascimbe?ne. Quantum simu-
lations with ultracold quantum gases. Nature Physics, 8(4):267?276, April
2012.
[15] Markus Greiner, Olaf Mandel, Tilman Esslinger, Theodor W. Ha?nsch, and
Immanuel Bloch. Quantum phase transition from a superfluid to a Mott
insulator in a gas of ultracold atoms. Nature, 415(6867):39?44, January 2002.
[16] N. Goldman, G. Juzeliu?nas, P. O?hberg, and I. B. Spielman. Light-
induced gauge fields for ultracold atoms. Reports on Progress in Physics,
77(12):126401, November 2014.
[17] A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B. Spielman, G. Juzeliu?nas,
and M. Lewenstein. Synthetic gauge fields in synthetic dimensions. Phys. Rev.
Lett., 112:043001, Jan 2014.
[18] Y.-J. Lin, R. L. Compton, K. Jime?nez-Garc??, J. V. Porto, and I. B. Spielman.
Synthetic magnetic fields for ultracold neutral atoms. Nature, 462(7273):628?
632, December 2009.
[19] Y.-J. Lin, R. L. Compton, K. Jime?nez-Garc??a, W. D. Phillips, J. V. Porto,
and I. B. Spielman. A synthetic electric force acting on neutral atoms. Nature
Physics, 7(7):531?534, July 2011.
[20] Hirokazu Miyake, Georgios A. Siviloglou, Colin J. Kennedy, William Cody
Burton, and Wolfgang Ketterle. Realizing the Harper Hamiltonian with
Laser-Assisted Tunneling in Optical Lattices. Physical Review Letters,
111(18):185302, October 2013.
206
[21] Nigel R. Cooper and Jean Dalibard. Reaching Fractional Quantum Hall States
with Optical Flux Lattices. Physical Review Letters, 110(18):185301, April
2013.
[22] Victor Galitski and Ian B. Spielman. Spin-orbit coupling in quantum gases.
Nature, 494(7435):49?54, February 2013.
[23] M. Z. Hasan and C. L. Kane. Colloquium: Topological insulators. Reviews of
Modern Physics, 82(4):3045?3067, November 2010.
[24] Yu A. Bychkov and E. I. Rashba. Oscillatory effects and the magnetic sus-
ceptibility of carriers in inversion layers. Journal of Physics C: Solid State
Physics, 17(33):6039, 1984.
[25] A. Valde?s-Curiel, D. Trypogeorgos, E. E. Marshall, and I. B. Spielman. Fourier
transform spectroscopy of a spin?orbit coupled Bose gas. New Journal of
Physics, 19(3):033025, 2017.
[26] A. Valde?s-Curiel, D. Trypogeorgos, Q.-Y. Liang, R. P. Anderson, and I. B.
Spielman. Unconventional topology with a Rashba spin-orbit coupled quan-
tum gas. arXiv:1907.08637 [cond-mat, physics:quant-ph], July 2019. arXiv:
1907.08637.
[27] R. P. Anderson, D. Trypogeorgos, A. Valde?s-Curiel, Q.-Y. Liang, J. Tao,
M. Zhao, T. Andrijauskas, G. Juzeliu?nas, and I. B. Spielman. Realization
of a fractional period adiabatic superlattice. arXiv:1907.08910 [cond-mat,
physics:physics, physics:quant-ph], July 2019. arXiv: 1907.08910.
[28] Tudor D. Stanescu, Brandon Anderson, and Victor Galitski. Spin-orbit cou-
pled Bose-Einstein condensates. Physical Review A, 78(2):023616, August
2008.
[29] Tigran A. Sedrakyan, Alex Kamenev, and Leonid I. Glazman. Composite
fermion state of spin-orbit-coupled bosons. Physical Review A, 86(6):063639,
December 2012.
[30] Hui Hu, Lei Jiang, Xia-Ji Liu, and Han Pu. Probing Anisotropic Superfluidity
in Atomic Fermi Gases with Rashba Spin-Orbit Coupling. Physical Review
Letters, 107(19):195304, November 2011.
[31] C.J. Pethick and H. Smith. Bose-Einstein Condensation of Dilute Gases.
Oxford University Press, Oxford, UK, 2005.
[32] Bose-Einstein Condensation. International Series of Monographs on Physics.
Oxford University Press, Oxford, New York, June 2003.
[33] Franco Dalfovo, Stefano Giorgini, Lev P. Pitaevskii, and Sandro Stringari.
Theory of Bose-Einstein condensation in trapped gases. Reviews of Modern
Physics, 71(3):463?512, April 1999.
207
[34] Y. Castin and R. Dum. Bose-Einstein Condensates in Time Dependent Traps.
Physical Review Letters, 77(27):5315?5319, December 1996.
[35] G. F. Gribakin and V. V. Flambaum. Calculation of the scattering length in
atomic collisions using the semiclassical approximation. Physical Review A,
48(1):546?553, July 1993.
[36] S. De, D. L. Campbell, R. M. Price, A. Putra, Brandon M. Anderson, and
I. B. Spielman. Quenched binary Bose-Einstein condensates: Spin-domain
formation and coarsening. Physical Review A, 89(3):033631, March 2014.
[37] Daniel Adam Steck. Rubidium 87 D line data. Available online,
http://steck.us/alkalidata, April 2018. revision 2.1.5.
[38] Charles Schwartz. Theory of Hyperfine Structure. Physical Review, 97(2):380?
395, January 1955.
[39] P. Zeeman. The effect of magnetisation on the nature of light emitted by a
substance. Nature, 55(1424):347?347, 1897.
[40] G. Breit and I. I. Rabi. Measurement of Nuclear Spin. Physical Review,
38(11):2082?2083, December 1931.
[41] W. Gerlach and O. Stern. Der experimentelle Nachweis des magnetischen
Moments des Silberatoms. Zeitschrift fu?r Physik, 8(1):110?111, December
1922.
[42] J. Stark. Beobachtungen u?ber den Effekt des elektrischen Feldes auf Spek-
trallinien. I. Quereffekt. Annalen der Physik, 348(7):965?982, 1914.
[43] Jacques Dupont-Roc Claude Cohen-Tannoudji and Gilbert Grynberg.
Atom?Photon Interactions. Wiley-VCH, Weinheim, 2006.
[44] Daniel Adam Steck. Quantum and atom optics. Available online at
http://steck.us/teaching, January 2015. revision 0.12.2.
[45] Ivan H. Deutsch and Poul S. Jessen. Quantum control and measure-
ment of atomic spins in polarization spectroscopy. Optics Communications,
283(5):681?694, March 2010.
[46] J. J. Sakurai. Modern Quantum Mechanics. Addison Wesley Publishing Com-
pany, Menlo Park, California, 1994.
[47] Rudolf Grimm, Matthias Weidemu?ller, and Yurii B. Ovchinnikov. Optical
Dipole Traps for Neutral Atoms. In Benjamin Bederson and Herbert Walther,
editors, Advances In Atomic, Molecular, and Optical Physics, volume 42, pages
95?170. Academic Press, January 2000.
208
[48] Rui Han, Hui Khoon Ng, and Berthold-Georg Englert. Raman transitions
without adiabatic elimination: a simple and accurate treatment. Journal of
Modern Optics, 60(4):255?265, February 2013.
[49] Bindiya Arora, M. S. Safronova, and Charles W. Clark. Tune-out wavelengths
of alkali-metal atoms and their applications. Physical Review A, 84(4):043401,
October 2011.
[50] Y.-J. Lin, K. Jime?nez-Garc??a, and I. B. Spielman. Spin-orbit-coupled Bose-
Einstein condensates. Nature, 471(7336):83?86, March 2011.
[51] G. Dresselhaus. Spin-Orbit Coupling Effects in Zinc Blende Structures. Phys-
ical Review, 100(2):580?586, October 1955.
[52] Christopher J. Foot. Atomic Physics. Oxford University Press, Cambridge,
UK, 2006.
[53] G. Reinaudi, T. Lahaye, Z. Wang, and D. Gue?ry-Odelin. Strong satura-
tion absorption imaging of dense clouds of ultracold atoms. Optics Letters,
32(21):3143?3145, November 2007.
[54] D. Genkina, L. M. Aycock, B. K. Stuhl, H.-I. Lu, R. A. Williams, and I. B.
Spielman. Feshbach enhanceds-wave scattering of fermions: direct observation
with optimized absorption imaging. New Journal of Physics, 18(1):013001,
December 2015.
[55] I. I. Rabi. Space Quantization in a Gyrating Magnetic Field. Physical Review,
51(8):652?654, April 1937.
[56] Clarence Zener and Ralph Howard Fowler. Non-adiabatic crossing of energy
levels. Proceedings of the Royal Society of London. Series A, Containing Pa-
pers of a Mathematical and Physical Character, 137(833):696?702, September
1932.
[57] Erin Marshall Seroka, Ana Valde?s Curiel, Dimitrios Trypogeorgos, Nathan
Lundblad, and Ian B. Spielman. Repeated Measurements with Minimally
Destructive Partial-Transfer Absorption Imaging. arXiv:1907.05372 [physics],
July 2019. arXiv: 1907.05372.
[58] Norman F. Ramsey. A Molecular Beam Resonance Method with Separated
Oscillating Fields. Physical Review, 78(6):695?699, June 1950.
[59] D. L. Campbell, R. M. Price, A. Putra, A. Valdes-Curiel, D. Trypogeorgos,
and I. B. Spielman. Magnetic phases of spin-1 spin?orbit-coupled Bose gases.
Nature Communications, 7:10897, March 2016.
[60] C. Silber, S. Gu?nther, C. Marzok, B. Deh, Ph. W. Courteille, and C. Zim-
mermann. Quantum-Degenerate Mixture of Fermionic Lithium and Bosonic
Rubidium Gases. Physical Review Letters, 95(17):170408, October 2005.
209
[61] B. Deh, C. Marzok, C. Zimmermann, and Ph. W. Courteille. Feshbach res-
onances in mixtures of ultracold 6Li and 87Rb gases. Physical Review A,
77(1):010701, January 2008.
[62] Daniel L. Campbell. Engineered potentials in ultracold Bose-Einstein conden-
sates. PhD thesis, University of Maryland College Park, Digital Repository
at the University of Maryland, 6 2015.
[63] Ryan Price. Phase transitions in engineered ultracold quantum systems. PhD
thesis, University of Maryland College Park, Digital Repository at the Uni-
versity of Maryland, 6 2016.
[64] William D. Phillips and Harold Metcalf. Laser Deceleration of an Atomic
Beam. Physical Review Letters, 48(9):596?599, March 1982.
[65] Paul D. Lett, Richard N. Watts, Christoph I. Westbrook, William D. Phillips,
Phillip L. Gould, and Harold J. Metcalf. Observation of Atoms Laser Cooled
below the Doppler Limit. Physical Review Letters, 61(2):169?172, July 1988.
[66] Christoph Becker. Multi component Bose-Einstein condensates From mean
field physics to strong correlations. PhD thesis, Universita?t Hamburg,
http://citeseerx.ist.psu.edu, 6 2008.
[67] P. T. Starkey, C. J. Billington, S. P. Johnstone, M. Jasperse, K. Helmerson,
L. D. Turner, and R. P. Anderson. A scripted control system for autonomous
hardware-timed experiments. Review of Scientific Instruments, 84(8):085111,
August 2013.
[68] Doppler-Limited Absorption and Fluorescence Spectroscopy with Lasers. In
Wolfgang Demtro?der, editor, Laser Spectroscopy: Vol. 2 Experimental Tech-
niques, pages 1?75. Springer Berlin Heidelberg, Berlin, Heidelberg, 2008.
[69] Lawrence W. Cheuk, Ariel T. Sommer, Zoran Hadzibabic, Tarik Yefsah,
Waseem S. Bakr, and Martin W. Zwierlein. Spin-Injection Spectroscopy of
a Spin-Orbit Coupled Fermi Gas. Physical Review Letters, 109(9):095302,
2012.
[70] David M. Jonas. Two-Dimensional Femtosecond Spectroscopy. Annual Review
of Physical Chemistry, 54(1):425?463, 2003.
[71] Bryce Yoshimura, W. C. Campbell, and J. K. Freericks. Diabatic-ramping
spectroscopy of many-body excited states. Physical Review A, 90(6):062334,
2014.
[72] Limei Wang, Hao Zhang, Linjie Zhang, Georg Raithel, Jianming Zhao, and
Suotang Jia. Atom-interferometric measurement of Stark level splittings.
Physical Review A, 92(3):033619, September 2015.
210
[73] Anders S?rensen and Klaus M?lmer. Entanglement and quantum computation
with ions in thermal motion. Physical Review A, 62(2):022311, July 2000.
[74] G. Floquet. Sur les e?quations diffe?rentielles line?aires a? coefficients pe?riodiques.
Annales scientifiques de l?E?cole Normale Supe?rieure, 12:47?88, 1883.
[75] Chunqing Deng, Jean-Luc Orgiazzi, Feiruo Shen, Sahel Ashhab, and Adrian
Lupascu. Observation of Floquet States in a Strongly Driven Artificial Atom.
Physical Review Letters, 115(13):133601, September 2015.
[76] J. Eisert, M. Friesdorf, and C. Gogolin. Quantum many-body systems out of
equilibrium. Nature Physics, 11(2):124?130, February 2015.
[77] K. Jime?nez-Garc??a, L. J. LeBlanc, R. A. Williams, M. C. Beeler, C. Qu,
M. Gong, C. Zhang, and I. B. Spielman. Tunable spin-orbit coupling via
strong driving in ultracold-atom systems. Phys. Rev. Lett., 114:125301, Mar
2015.
[78] D. L. Campbell and I. B. Spielman. Rashba realization: Raman with RF.
New J. Phys., 18(3):033035, 2016.
[79] Douglas R. Hofstadter. Energy levels and wave functions of Bloch electrons in
rational and irrational magnetic fields. Physical Review B, 14(6):2239?2249,
September 1976.
[80] D. Trypogeorgos, A. Valde?s-Curiel, I. B. Spielman, and C. Emary. Perpetual
emulation threshold of PT -symmetric Hamiltonians. Journal of Physics A:
Mathematical and Theoretical, 51(32):325302, June 2018.
[81] D. Trypogeorgos, A. Valde?s-Curiel, N. Lundblad, and I. B. Spielman. Syn-
thetic clock transitions via continuous dynamical decoupling. Physical Review
A, 97(1):013407, January 2018.
[82] R. P. Anderson, M. J. Kewming, and L. D. Turner. Continuously observing a
dynamically decoupled spin-1 quantum gas. Physical Review A, 97(1):013408,
January 2018.
[83] E. L. Hahn. Spin Echoes. Physical Review, 80(4):580?594, November 1950.
[84] Lorenza Viola and Seth Lloyd. Dynamical suppression of decoherence in two-
state quantum systems. Phys. Rev. A, 58(4):2733?2744, October 1998.
[85] I. Cohen, N. Aharon, and A. Retzker. Continuous dynamical decoupling uti-
lizing time-dependent detuning. Fortschr. Phys., 65(6-8):1600071, June 2017.
[86] F. F. Fanchini, J. E. M. Hornos, and R. d J. Napolitano. Continuously decou-
pling single-qubit operations from a perturbing thermal bath of scalar bosons.
Physical Review A, 75(2), February 2007. arXiv: quant-ph/0611188.
211
[87] N. Aharon, I. Cohen, F. Jelezko, and A. Retzker. Fully robust qubit in atomic
and molecular three-level systems. New J. Phys., 18(12):123012, 2016.
[88] Michael J. Biercuk, Hermann Uys, Aaron P. VanDevender, Nobuyasu Shiga,
Wayne M. Itano, and John J. Bollinger. Optimized dynamical decoupling in
a model quantum memory. Nature, 458(7241):996?1000, April 2009.
[89] J.-M. Cai, B. Naydenov, R. Pfeiffer, L. P. McGuinness, K. D. Jahnke,
F. Jelezko, M. B. Plenio, and A. Retzker. Robust dynamical decoupling
with concatenated continuous driving. New Journal of Physics, 14(11):113023,
November 2012. arXiv: 1111.0930.
[90] A. Bermudez, P. O. Schmidt, M. B. Plenio, and A. Retzker. Robust trapped-
ion quantum logic gates by continuous dynamical decoupling. Phys. Rev. A,
85(4):040302, April 2012.
[91] I. Baumgart, J.-M. Cai, A. Retzker, M. B. Plenio, and Ch. Wunderlich. Ul-
trasensitive magnetometer using a single atom. Phys. Rev. Lett., 116:240801,
Jun 2016.
[92] G. A. Kazakov and T. Schumm. Magic radio-frequency dressing for trapped
atomic microwave clocks. Phys. Rev. A, 91(2):023404, February 2015.
[93] L. Sa?rka?ny, P. Weiss, H. Hattermann, and J. Forta?gh. Controlling the
magnetic-field sensitivity of atomic-clock states by microwave dressing. Phys.
Rev. A, 90(5):053416, November 2014.
[94] Arne Laucht, Rachpon Kalra, Stephanie Simmons, Juan P. Dehollain, Juha T.
Muhonen, Fahd A. Mohiyaddin, Solomon Freer, Fay E. Hudson, Kohei M. Itoh,
David N. Jamieson, Jeffrey C. McCallum, Andrew S. Dzurak, and A. Morello.
A dressed spin qubit in silicon. Nat Nano, 12(1):61?66, January 2017.
[95] D. Farfurnik, N. Aharon, I. Cohen, Y. Hovav, A. Retzker, and N. Bar-Gill. Ex-
perimental realization of time-dependent phase-modulated continuous dynam-
ical decoupling. arXiv:1704.07582 [quant-ph], April 2017. arXiv: 1704.07582.
[96] Atsushi Noguchi, Shinsuke Haze, Kenji Toyoda, and Shinji Urabe. Genera-
tion of a Decoherence-Free Entangled State Using a Radio-Frequency Dressed
State. Phys. Rev. Lett., 108(6):060503, February 2012.
[97] D. Andrew Golter, Thomas K. Baldwin, and Hailin Wang. Protecting a Solid-
State Spin from Decoherence Using Dressed Spin States. Phys. Rev. Lett.,
113(23):237601, December 2014.
[98] N. Timoney, I. Baumgart, M. Johanning, A. F. Varo?n, M. B. Plenio, A. Ret-
zker, and Ch Wunderlich. Quantum gates and memory using microwave-
dressed states. Nature, 476(7359):185?188, August 2011.
212
[99] S. C. Webster, S. Weidt, K. Lake, J. J. McLoughlin, and W. K. Hensinger.
Simple Manipulation of a Microwave Dressed-State Ion Qubit. Phys. Rev.
Lett., 111(14):140501, October 2013.
[100] A. Barfuss, J. Teissier, E. Neu, A. Nunnenkamp, and P. Maletinsky. Strong
mechanical driving of a single electron spin. Nat Phys, 11(10):820?824, Octo-
ber 2015.
[101] S. Rohr, E. Dupont-Ferrier, B. Pigeau, P. Verlot, V. Jacques, and O. Arcizet.
Synchronizing the Dynamics of a Single Nitrogen Vacancy Spin Qubit on a
Parametrically Coupled Radio-Frequency Field through Microwave Dressing.
Phys. Rev. Lett., 112(1):010502, January 2014.
[102] Gediminas Juzeliu?nas, Julius Ruseckas, and Jean Dalibard. Generalized
Rashba-Dresselhaus spin-orbit coupling for cold atoms. Phys. Rev. A,
81(5):053403, May 2010.
[103] D. Bacon, J. Kempe, D. A. Lidar, and K. B. Whaley. Universal Fault-Tolerant
Quantum Computation on Decoherence-Free Subspaces. Phys. Rev. Lett.,
85(8):1758?1761, August 2000.
[104] Pablo Solano, Jeffrey A. Grover, Jonathan E. Hoffman, Sylvain Ravets,
Fredrik K. Fatemi, Luis A. Orozco, and Steven L. Rolston. Chapter Seven
- Optical Nanofibers: A New Platform for Quantum Optics. In Chun C. Lin
and Susanne F. Yelin Ennio Arimondo, editor, Advances In Atomic, Molec-
ular, and Optical Physics, volume 66, pages 439?505. Academic Press, 2017.
DOI: 10.1016/bs.aamop.2017.02.003.
[105] Ze-Liang Xiang, Sahel Ashhab, J. Q. You, and Franco Nori. Hybrid quantum
circuits: Superconducting circuits interacting with other quantum systems.
Rev. Mod. Phys., 85(2):623?653, April 2013.
[106] Tomoki Ozawa, Hannah M. Price, Alberto Amo, Nathan Goldman, Moham-
mad Hafezi, Ling Lu, Mikael C. Rechtsman, David Schuster, Jonathan Simon,
Oded Zilberberg, and Iacopo Carusotto. Topological photonics. Reviews of
Modern Physics, 91(1):015006, March 2019.
[107] Pierre Delplace, J. B. Marston, and Antoine Venaille. Topological origin of
equatorial waves. Science, 358(6366):1075?1077, November 2017.
[108] V. Peano, C. Brendel, M. Schmidt, and F. Marquardt. Topological Phases of
Sound and Light. Physical Review X, 5(3):031011, July 2015.
[109] Zhaoju Yang, Fei Gao, Xihang Shi, Xiao Lin, Zhen Gao, Yidong Chong, and
Baile Zhang. Topological Acoustics. Physical Review Letters, 114(11):114301,
March 2015.
[110] N. R. Cooper, J. Dalibard, and I. B. Spielman. Topological bands for ultracold
atoms. Rev. Mod. Phys., 91:015005, Mar 2019.
213
[111] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs. Quantized
Hall Conductance in a Two-Dimensional Periodic Potential. Physical Review
Letters, 49(6):405?408, August 1982.
[112] F. D. M. Haldane. Model for a Quantum Hall Effect without Landau Lev-
els: Condensed-Matter Realization of the ?Parity Anomaly?. Physical Review
Letters, 61(18):2015?2018, October 1988.
[113] C. L. Kane and E. J. Mele. Z2 Topological Order and the Quantum Spin Hall
Effect. Physical Review Letters, 95(14):146802, September 2005.
[114] D. B. Newell, F. Cabiati, J. Fischer, K. Fujii, S. G. Karshenboim, H. S. Mar-
golis, E. de Mirande?s, P. J. Mohr, F. Nez, K. Pachucki, T. J. Quinn, B. N.
Taylor, M. Wang, B. M. Wood, and Z. Zhang. The CODATA 2017 values of
h, e, k, and N A for the revision of the SI. Metrologia, 55(1):L13?L16, January
2018.
[115] Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar
Das Sarma. Non-Abelian anyons and topological quantum computation. Re-
views of Modern Physics, 80(3):1083?1159, September 2008.
[116] Eduardo Nahmad-Achar. Differential Topology and Geometry with Applica-
tions to Physics. 2053-2563. IOP Publishing, 2018.
[117] L. Landau. The Theory of Phase Transitions. Nature, 138(3498):840?841,
November 1936.
[118] K. v. Klitzing, G. Dorda, and M. Pepper. New Method for High-Accuracy
Determination of the Fine-Structure Constant Based on Quantized Hall Re-
sistance. Physical Review Letters, 45(6):494?497, August 1980.
[119] David Tong. Lectures on the Quantum Hall Effect. arXiv:1606.06687 [cond-
mat, physics:hep-th], June 2016. arXiv: 1606.06687.
[120] Mikio Nakahara. CRC Press, New York, NY, 2003.
[121] Barry Simon. Holonomy, the Quantum Adiabatic Theorem, and Berry?s Phase.
Physical Review Letters, 51(24):2167?2170, December 1983.
[122] Berry Michael Victor. Quantal phase factors accompanying adiabatic changes.
Proceedings of the Royal Society of London. A. Mathematical and Physical
Sciences, 392(1802):45?57, March 1984.
[123] S. Pancharatnam. Generalized theory of interference, and its applications.
Proceedings of the Indian Academy of Sciences - Section A, 44(5):247?262,
November 1956.
[124] Y. Aharonov and D. Bohm. Significance of Electromagnetic Potentials in the
Quantum Theory. Physical Review, 115(3):485?491, August 1959.
214
[125] Dirac Paul Adrien Maurice. Quantised singularities in the electromagnetic
field,. Proceedings of the Royal Society of London. Series A, Containing Papers
of a Mathematical and Physical Character, 133(821):60?72, September 1931.
[126] Ralph M. Kaufmann, Dan Li, and Birgit Wehefritz-Kaufmann. Notes on topo-
logical insulators. Reviews in Mathematical Physics, 28(10):1630003, Novem-
ber 2016. arXiv: 1501.02874.
[127] H. B. Nielsen and Masao Ninomiya. The Adler-Bell-Jackiw anomaly and Weyl
fermions in a crystal. Physics Letters B, 130(6):389?396, November 1983.
[128] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch.
Realization of the Hofstadter Hamiltonian with Ultracold Atoms in Optical
Lattices. Physical Review Letters, 111(18):185301, October 2013.
[129] Gregor Jotzu, Michael Messer, Re?mi Desbuquois, Martin Lebrat, Thomas
Uehlinger, Daniel Greif, and Tilman Esslinger. Experimental realization of the
topological Haldane model with ultracold fermions. Nature, 515(7526):237?
240, November 2014.
[130] H.-I Lu, M. Schemmer, L. M. Aycock, D. Genkina, S. Sugawa, and I. B.
Spielman. Geometrical pumping with a bose-einstein condensate. Phys. Rev.
Lett., 116:200402, May 2016.
[131] M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch. A Thou-
less quantum pump with ultracold bosonic atoms in an optical superlattice.
Nature Physics, 12(4):350?354, April 2016.
[132] Zhan Wu, Long Zhang, Wei Sun, Xiao-Tian Xu, Bao-Zong Wang, Si-Cong Ji,
Youjin Deng, Shuai Chen, Xiong-Jun Liu, and Jian-Wei Pan. Realization of
two-dimensional spin-orbit coupling for Bose-Einstein condensates. Science,
354(6308):83?88, October 2016.
[133] Wei Sun, Bao-Zong Wang, Xiao-Tian Xu, Chang-Rui Yi, Long Zhang, Zhan
Wu, Youjin Deng, Xiong-Jun Liu, Shuai Chen, and Jian-Wei Pan. Highly
Controllable and Robust 2d Spin-Orbit Coupling for Quantum Gases. Physical
Review Letters, 121(15):150401, October 2018.
[134] Frank Wilczek and A. Zee. Appearance of Gauge Structure in Simple Dynam-
ical Systems. Physical Review Letters, 52(24):2111?2114, June 1984.
[135] D. L. Campbell, G. Juzeliu?nas, and I. B. Spielman. Realistic Rashba
and Dresselhaus spin-orbit coupling for neutral atoms. Physical Review A,
84(2):025602, August 2011.
[136] Tudor D. Stanescu and Victor Galitski. Spin relaxation in a generic two-
dimensional spin-orbit coupled system. Physical Review B, 75(12):125307,
March 2007.
215
[137] Y.-J. Lin, A. R. Perry, R. L. Compton, I. B. Spielman, and J. V. Porto. Rapid
production of 87Rb Bose-Einstein condensates in a combined magnetic and
optical potential. Physical Review A, 79(6):063631, June 2009.
[138] Lianghui Huang, Zengming Meng, Pengjun Wang, Peng Peng, Shao-Liang
Zhang, Liangchao Chen, Donghao Li, Qi Zhou, and Jing Zhang. Experimental
realization of two-dimensional synthetic spin?orbit coupling in ultracold Fermi
gases. Nature Physics, 12(6):540?544, June 2016.
[139] Zengming Meng, Lianghui Huang, Peng Peng, Donghao Li, Liangchao Chen,
Yong Xu, Chuanwei Zhang, Pengjun Wang, and Jing Zhang. Experimen-
tal Observation of a Topological Band Gap Opening in Ultracold Fermi
Gases with Two-Dimensional Spin-Orbit Coupling. Physical Review Letters,
117(23):235304, December 2016.
[140] Satoshi Tojo, Taro Hayashi, Tatsuyoshi Tanabe, Takuya Hirano, Yuki
Kawaguchi, Hiroki Saito, and Masahito Ueda. Spin-dependent inelastic colli-
sions in spin-2 Bose-Einstein condensates. Physical Review A, 80(4):042704,
October 2009.
[141] N. Goldman and J. Dalibard. Periodically Driven Quantum Systems: Effective
Hamiltonians and Engineered Gauge Fields. Physical Review X, 4(3):031027,
2014.
[142] N. Fla?schner, B. S. Rem, M. Tarnowski, D. Vogel, D.-S. Lu?hmann, K. Seng-
stock, and C. Weitenberg. Experimental reconstruction of the Berry curvature
in a Floquet Bloch band. Science, 352(6289):1091?1094, May 2016.
[143] Cle?ment Godfrin, Rafik Ballou, Edgar Bonet, Mario Ruben, Svetlana Kly-
atskaya, Wolfgang Wernsdorfer, and Franck Balestro. Generalized Ramsey
interferometry explored with a single nuclear spin qudit. npj Quantum Infor-
mation, 4(1):53, October 2018.
[144] Takahiro Fukui, Yasuhiro Hatsugai, and Hiroshi Suzuki. Chern Numbers in
Discretized Brillouin Zone: Efficient Method of Computing (Spin) Hall Con-
ductances. Journal of the Physical Society of Japan, 74(6):1674?1677, June
2005.
[145] L. Duca, T. Li, M. Reitter, I. Bloch, M. Schleier-Smith, and U. Schneider. An
Aharonov-Bohm interferometer for determining Bloch band topology. Science,
347(6219):288?292, January 2015.
[146] Ma?rio G. Silveirinha. Chern invariants for continuous media. Physical Review
B, 92(12):125153, September 2015.
[147] Tigran A. Sedrakyan, Victor M. Galitski, and Alex Kamenev. Statistical
Transmutation in Floquet Driven Optical Lattices. Physical Review Letters,
115(19):195301, November 2015.
216
[148] Cheng Chin, Rudolf Grimm, Paul Julienne, and Eite Tiesinga. Feshbach
resonances in ultracold gases. Reviews of Modern Physics, 82(2):1225?1286,
April 2010.
[149] Roger Brown. Nonequilibrium many body dynamics with ultracold atoms in
optical lattices and selected problems in atomic physics. PhD thesis, University
of Maryland College Park, Digital Repository at the University of Maryland,
6 2014.
[150] Creston D. Herold. Ultracold mixtures of Rubidium and Ytterbium for open
system quantum engineering. PhD thesis, University of Maryland College
Park, Digital Repository at the University of Maryland, 6 2014.
[151] Abigail R. Perry. An apparatus for light-less artificial gauge fields and new
imaging techniques. PhD thesis, University of Maryland College Park, Digital
Repository at the University of Maryland, 6 2015.
[152] B. P. Anderson and M. A. Kasevich. Loading a vapor-cell magneto-optic
trap using light-induced atom desorption. Physical Review A, 63(2):023404,
January 2001.
217