Simulating Strongly-Coupled
Light-Matter Systems with Ultracold
Atoms

A Dissertation Presented
byJoonhyuk Kwon
to
The Graduate Schoolin Partial Fulfillment of the Requirements
for the Degree ofDoctor of Philosophy
inPhysics

Stony Brook University
January 2022

Stony Brook University
The Graduate School

Joonhyuk Kwon
We, the dissertation committee for the above candidate for theDoctor of Philosophy degree, hereby recommend
acceptance of this dissertation.

Dominik Schneble – Dissertation Advisor
Professor, Department of Physics and Astronomy

Thomas Weinacht – Chairperson of Defense
Professor, Department of Physics and Astronomy

Tzu-Chieh Wei
Associate Professor, Department of Physics and Astronomy

Robert Konik
Chair, Condensed Matter Physics and Materials Science Department
Brookhaven National Laboratory

This dissertation is accepted by the Graduate School.

Eric Wertheimer
Dean of the Graduate Schoolii

Abstract of the Dissertation

Simulating Strongly-Coupled
Light-Matter Systems with Ultracold
Atoms
by

Joonhyuk Kwon
Doctor of Philosophy
inPhysics
Stony Brook University
2022

Understanding and harnessing light-matter interactions is central
to the development of applications in quantum information science.
Using ultracold atoms in optical lattices as a quantum simulator
for such optical phenomena, we have recently implemented a platform of artificial quantum emitters that radiate matter waves into
an analogue of a photonic-crystal waveguide. In this dissertation,
we extended our studies of analogues of atom-photon bound states
to implement a new type of strongly-coupled quasi-particle in a
lattice geometry that can be viewed as the matter-wave equivalentof an exciton polariton. We spectroscopically accessed the dispersion relation of this matter-wave polariton and studied polaritonic
transport in the superfluid and many-body Mott insulator regimes.
Polaritons, a key feature of strongly-coupled light-matter interactions, are at the heart of emerging platforms for quantum iniii

formation processing and simulation. Unlike all other polaritonsin the optical domain, our matter-wave polariton is dissipationfree and fully tunable and thus can serve as an ideal testbed forpolariton physics.By combining the Bose-Hubbard and WeisskopfWigner models, the matter-wave polariton system naturally connects two separate areas, condensed-matter physics and quantum
optics.

iv

Dedicated to my motherv

Contents
List of Figuresviii

Acknowledgementsx

1 Introduction

1

2 Basic experimental methods
2.1 Making BEC . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Hyperfine state control . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Zeeman sublevels in magnetic bias fields . . . . . . . .
2.2.2 Rabi oscillations . . . . . . . . . . . . . . . . . . . . .
2.2.3 Landau-Zener sweep . . . . . . . . . . . . . . . . . . .
2.2.4 Two-photon transition . . . . . . . . . . . . . . . . . .
2.2.5 Stern-Gerlach separation . . . . . . . . . . . . . . . . .
2.3 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Image analysis using principal components . . . . . . . . . . .
2.5 State-dependent potential . . . . . . . . . . . . . . . . . . . .

5
5
9
9
10
13
14
15
16
18
21

3 Theoretical concepts
3.1 Band structure in optical lattices . . . . . . . . . . . . . . . .
3.1.1 The tight-binding limit . . . . . . . . . . . . . . . . . .
3.2 Bose-Hubbard model . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Mean-field treatment and quantum phase transition . .

26
26
28
30
32

4 The matter-wave platform
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Theory: Weisskopf-Wigner model . . . . . . . . . . . . . . . .
4.3 Experimental platform . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Magnetic field stablization . . . . . . . . . . . . . . . .
4.3.2 Preparation of atomic quantum emitters . . . . . . . .
4.4 Matter-wave decay in the continuum . . . . . . . . . . . . . .

36
36
37
39
39
41
42

vi

4.4.1 Spontaneous emission . . . . . . . . . . . . . . . . . . .
4.4.2 Non-Markovian regime and the bound state . . . . . .
Decay in a structured vacuum . . . . . . . . . . . . . . . . . .
4.5.1 Realization of single-band structure . . . . . . . . . . .
4.5.2 Transition from Markovian to non-Markovian dynamics
4.5.3 Bound-state in a band-gap . . . . . . . . . . . . . . . .

43
45
45
45
48
50

5 Formation of matter-wave polaritons
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Measurement of the onsite-interaction energy . . . . . . . . . .
5.4 Theoretical approach . . . . . . . . . . . . . . . . . . . . . . .
5.5 Measurement of transport behavior . . . . . . . . . . . . . . .
5.6 Experimental details . . . . . . . . . . . . . . . . . . . . . . .
5.6.1 System preparation . . . . . . . . . . . . . . . . . . . .
5.6.2 Atom detection . . . . . . . . . . . . . . . . . . . . . .
5.6.3 Resonance condition . . . . . . . . . . . . . . . . . . .
5.7 Theoretical details . . . . . . . . . . . . . . . . . . . . . . . .
5.7.1 Polariton band structure . . . . . . . . . . . . . . . . .
5.7.2 Polariton Bose-Hubbard Hamiltonian . . . . . . . . . .
5.8 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . .

52
52
54
55
57
59
62
62
63
64
64
64
66
67

6 Toward future experiments
6.1 Coherent dynamics in a matter-wave platform . . . . . . . . .
6.1.1 Collective dynamics . . . . . . . . . . . . . . . . . . . .
6.1.2 State purification . . . . . . . . . . . . . . . . . . . . .
6.2 Frictionless impurity motion in 1D . . . . . . . . . . . . . . .
6.2.1 Critical velocity . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Basic description of 1D system . . . . . . . . . . . . . .
6.2.3 Yang-Gaudin model toward frictionless motion . . . . .
6.3 Technical development . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Moving lattice system . . . . . . . . . . . . . . . . . .

69
69
69
72
73
73
75
76
76
77

4.5

Bibliography

79

A Appendix
A.1 Logging magnetic fields . . . . . . . . . . . . . . . . . . . . . .
A.2 Automated polarization control . . . . . . . . . . . . . . . . .
A.3 Accordion lattice . . . . . . . . . . . . . . . . . . . . . . . . .
A.4 Microwave amplifier . . . . . . . . . . . . . . . . . . . . . . . .

vii

97
97
99
100
104

List of Figures
2.1 BEC machine at SBU . . . . . . . . . . . . . . . . . . . . . . .
2.2 Rubidium-87 energy level scheme . . . . . . . . . . . . . . . .
2.3 Breit-Rabi formula . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Rabi oscillation spectra . . . . . . . . . . . . . . . . . . . . . .
2.5 Landau-Zener transition . . . . . . . . . . . . . . . . . . . . .
2.6 Two-photon transition . . . . . . . . . . . . . . . . . . . . . .
2.7 Stern-Gerlach separation . . . . . . . . . . . . . . . . . . . . .
2.8 Mechanism of principal components . . . . . . . . . . . . . . .
2.9 Image process steps for PCA . . . . . . . . . . . . . . . . . . .
2.10 Result of PCA . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11 Optical potential for diﬀerent polarizations . . . . . . . . . . .
2.12 Optical potential for diﬀerent hyperfine states . . . . . . . . .

6
7
10
12
13
15
16
19
20
22
23
24

3.1
3.2
3.3
3.4

Energy band structure . . . . . . . . . . . . . . . . . . . . . .
Bandwidth calculation . . . . . . . . . . . . . . . . . . . . . .
BHM schematic diagram . . . . . . . . . . . . . . . . . . . . .
Quantum phase transition . . . . . . . . . . . . . . . . . . . .

28
30
31
33

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9

Illustration of matter-wave quantum emitters . . . . . . . . . .
Magnetic field tagging . . . . . . . . . . . . . . . . . . . . . .
Experimental preparation of quantum emitters . . . . . . . . .
Realization of matter-wave platform . . . . . . . . . . . . . . .
Illustration of implementing band structure . . . . . . . . . . .
Realization of band structure . . . . . . . . . . . . . . . . . .
Single band spectroscopy . . . . . . . . . . . . . . . . . . . . .
Transition from Markovian to non-Markovian dynamics . . . .
Bound states in bandgap . . . . . . . . . . . . . . . . . . . . .

37
40
42
44
46
47
48
49
51

5.1
5.2
5.3
5.4

Experimental scheme and polariton formation . . . . . . . . .
Excitation spectra in the Mott regime . . . . . . . . . . . . . .
Polariton band structure . . . . . . . . . . . . . . . . . . . . .
Renormalization of hopping . . . . . . . . . . . . . . . . . . .

54
56
58
60

viii

5.5
5.6
5.7

Full experimental sequence for onsite-interaction energy measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Adiabaticity of polariton preparation . . . . . . . . . . . . . .
Full experimental sequence for quantum transport . . . . . . .

62
63
64

6.1
6.2
6.3
6.4

Coherent emission for various initial states . . . . . . . . . . .
Non-uniform atom population in harmonic trap potential . . .
Dispersion of Bogoliubov excitation . . . . . . . . . . . . . . .
Moving-lattice system . . . . . . . . . . . . . . . . . . . . . .

70
71
74
77

A.1
A.2
A.3
A.4
A.5
A.6
A.7
A.8
A.9

Field monitoring infrastructure . . . . . . . . . . . . . . . . .
DMM monitoring program . . . . . . . . . . . . . . . . . . . .
Step-motor polarizer . . . . . . . . . . . . . . . . . . . . . . .
Lattice depth vs. polarization . . . . . . . . . . . . . . . . . .
Accordion lattice system . . . . . . . . . . . . . . . . . . . . .
Accordion lattice beam realization . . . . . . . . . . . . . . . .
Accordion lattice beam profile . . . . . . . . . . . . . . . . . .
Microwave monitoring system . . . . . . . . . . . . . . . . . .
Microwave output power gain curve . . . . . . . . . . . . . . .

98
99
100
101
101
102
103
104
105

ix

Acknowledgements
I have thought many times about the moment of writing this acknowledgement
as a final step of my dissertation. However, how I feel at this moment is notlike what I imagined; looking back on my journey towards my PhD degree, I
have to say that everything I achieved in this period was with the help of all
the great people around me and I am incredibly lucky to have them in my life.
I first want to thank my advisor, Dominik Schneble. He has been the bestadvisor that I could possibly have dreamt about. From the first moment I
met him in his class, he has always been supportive and full of passion. It
is impossible to describe in this limited space, but I still would like to write
down what I can since he is the one that I truly revere and want-to-be both asa scientist and a person. Dominik guided me how the physicist should be byshowing and sharing his life with us. His endless enthusiasm toward the truthand high standard of self criticism is more than what I read from the textbook
for being a good scientist. I also enjoyed every moment I talked with him no
matter what the topic was, since he is a great person beyond being an advisor.
He is always considerate, warm-hearted and full of (German) humor (not asarcasm!) that I love. What I do miss already is the presence of Dominikat the corner of the lab oﬃce as I am leaving, but I believe that a bonding
between me and him in our life has just started.
I also thank the committee for my dissertation, Tom Weinacht, Tzu-Chieh
Wei, and Robert Konik for their eﬀort, support and time. I appreciate theircareful and thorough reading of my dissertation with insightful comments.
I am thankful to Robert Konik for kindly oﬀering to serve as an external
committee member and providing critical comments and advice that make mythesis more accessible. I also thank Tom and Tzu-Chieh for always bringing
up critical questions with great insight for giving me good advice. I have to
say that I greatly enjoyed the joint meetings and discussions with Tzu-Chieh’sgroup over many years that were always providing theoretical richness andstimulation.
It was an exceptional pleasure to work and collaborate with wonderfulpeople in the laboratory. During my early years, I learned everything that Ixneeded in a lab from Ludwig Krinner. He is an excellent physicist with great
mastery of details that I always admire, and he is also a great lab mate as aperson. I still miss our post-conference trip to Miami. I am also indebted toArturo Pazmiño, who taught me all the skills that I need in lab. His amazing
skills for dealing with equipment and electronics still benefit the whole lab
greatly. My life in lab can not be described without Michael Stewart, who is
not only a great physicist but also a heartwarming person. I enjoyed all thefriendly banter we had all the time, even while under heart-attack-inducing
machine troubles, and most of all, we went through all the things together.
Alfonso Lanuza is a very talented physicist and I am absolutely indebted tohis awesome theoretical abilities. It was a great pleasure of mine to work withhim, and I am also happy to see that he is now a form of experimentalist as
well. I have to say that I admire his great artistic skills as much as I do Mike’s,
which might actually mean that I have some kind of a preconception toward(semi-)theorists. I would like to thank Youngshin Kim, a great physicist anda wonderful lab colleague. He is absolutely reliable and steadfast, and also
provides fantastic insights. I enjoyed all the moments that we measured all
the data points with a superposition of concern and joy. He is now truly acore of our lab and I am looking forward to seeing his great work in the near
future. Hongyi Huang, who recently joined our group, has already become an
essential member of the lab. He is an excellent graduate student and absorbs
all the knowledge like a sponge. I have no doubt about his successful PhD life
in our group.
I am also grateful for being a part of the wonderful AMO community at
Stony Brook. I am thankful to professors Hal Metcalf, Tom Weinacht, Eden
Figueroa and Tom Allison. Under their leadership, all AMO graduate students
were always rooting for and helping each other inside and outside of the work,and this made me truly feel like I belong here. I will miss all the small talkand Friday lunches with our incredible basement members especially Yusong,
Chuan, Brian K, Spencer, Sam, Brian A, Sonali, Guodong, Bertus, Yifan,
Xiaoyang, Samet, Jay, Eli, Chris, Eunji and Max.
My life outside of the lab in Stony Brook was all about my Foxhill friends,
Junsik You, Jaehyeok Chang, Dongwon Han and Kyle Lee. From the beginning to the end, I shared all my life with them with great pleasure. All the
moments that we shared together were memorable and invaluable. I alreadymiss everything we’ve enjoyed including food, drinks, trips and all friendly
banter. I am so glad that we all indeed made it as we initially planned many
years ago. Each of you deserves more than a full page, but here I shortened it
down since I know that this is only the beginning of our story, and we are the
one group.

xiLast but the foremost, I could not have made it without the endless supportof my family. I have always tried to make my mother and father proud of me,
and I am now incredibly happy to say that I have made it this far with theirsupport. I could always feel their bottomless love that is encouraging me all
the time, and I should say that I am also proud of my parents. I thank mysister, and I have to say that she is the most promising person, and I canstay abroad for many years without any concern only because of her. Lastly, I
would like to thank my wonderful wife, Sohyun. I never thought that I might
have a chance to thank my wife in my thesis, and I am so lucky that I am
doing so. It is hard for me to describe how much I got encouraged by her
support and presence. I can’t wait for the incredible journey that lies aheadof us.

We are grateful for the generous support by the National Science Foundation of the United States of America (NSF).
xii

Chapter 1
Introduction
Since the early 1900s, quantum mechanics has given a new point of viewof the microscopic world where classical understanding does not work, andindeed this has greatly aﬀected the development of statistical physics: At low
temperature, the spin nature of the particle determines the behavior of an
ideal gas to either Bose-Einstein statistics for spin-integer or the Fermi-Diracstatistics for spin-half integer particles.
Bose-Einstein Condensation (BEC) was predicted in 1924 [1–3] but only
relatively recently observed in dilute atomic gases [4–7] in 1995. BEC is named
after S.N. Bose and A. Einstein, who showed that the particles with integer
spin (boson) can have a macroscopic occupation of a single quantum state.
In a trapped gas, this degenerate state exhibits macroscopic coherence whichenabled the groups of C. Wieman and E. Cornell; R. Hulet; and W. Ketterle
to first observe BEC in the real world. Since then, quantum gases using
ultracold and dilute gases paved a way toward studying ideal quantum systems
in atomic, molecular, and optical (AMO) physics. Experiments showed thatcooling fermionic atoms also leads to quantum degeneracy as degenerate FermiDirac gases (DFG) [8–11] in potassium atoms. Starting from rubidium [4],various atomic species have been used to realize degenerate quantum gases in
the last two decades; alkali-metals (lithium [5, 7], sodium [6], potassium [11, 12]
and cesium [13]) which are easy to laswer-cool, alkaline earth metals (calcium[14], strontium [15, 16]), hydrogen [17] and metastable helium [18], chromium[19], and even lanthanides (ytterbium [20], erbium [21], and dysprosium [22])have recently come into heavy use based on the experimental purpose.
Research with BEC has also expanded explosively. In the early days, coherence properties [23–28], collective excitations [29] and propagation of sound[30] in BEC were studied followed by works on solitons [31, 32] and vortices[33–35]. Moreover, advanced techniques that are popular today were developed
such as tuning interactions between ultracold atoms via Feshbach resonances1[36, 37] and Bragg spectroscopy [38]. Studies with Fermi gases also gave insights into long-lasting questions such as BEC-BCS crossover [39, 40] and therealization of the polar molecules [41–44].
Ultracold atomic systems became even more versatile in combination with
optical lattices [45, 46]. Optical lattices are an ideal platform to test theoreticalmodels that were developed for condensed-matter physics by virtue of the factthat they are defect-free and have an unparalleled degree of tunability. Thissystem provides analogous models to study fundamental physics underlying
solid-state phenomena, thus playing the role of an analog quantum simulator.
Quantum simulation, first suggested by Feynman [47] in 1982, is one of themain directions in the quantum information paradigm.
Realizing the Bose-Hubbard model [48] on a square lattice led to the remarkable observation of the phase transition between a superfluid and a Mottinsulator [49, 50]. Other geometries [51–54] have also been studied for various
models, such as Haldane [55] and Hofstadter models [56, 57] with the helpof the realization of spin-orbit coupling [58–60]. Also, optical lattices withfermions have been used to implement the Fermi-Hubbard model and have led
to the detection of anti-ferromagnetism [61–64]. Adding disorder with laser
speckle [65] or atomic disorder [66] to the Hamiltonian allows for studies ofAnderson localization [65, 67–69] and Aubre-André model [70] with interesting
features such as a mobility edge [71–73] and a topological Anderson insulator [74]. Present-day technology even enables single-site resolved imaging in
two-dimensional systems for bosons [75–77] and fermions [78–81].
In this dissertation, we will discuss experiments with BEC using a novel
platform for the study of quantum-optical phenomena using matter waves thatwas realized in our laboratory recently. In this platform, we use hyperfinestate dependent optical lattices, which have a long history in the Schneble
laboratory where they have led to demonstrations of collinear four-wave mixingof matter waves [82], superfluidity of interacting mixtures [83], glassy behavior[66], probing ultracold-atom crystal with matter waves [84], and non-adiabatic
diﬀraction of matter waves [85]. With the improved techniques and equipment
stability, we recently utilized matter-waves as a substitution for photons in twolevel quantum emitter systems [86, 87]. Very recently, we combined this with
the Bose-Hubbard model of condensed-matter physics to realize a stronglyinteracting polaritonic system [88] as will be covered in the main text.
In chapter 4, we briefly review the earlier work [86] on an open quantum
system in which a two-level quantum emitter is coupled to a continuum ofmodes. The Hamiltonian of this system is the Weisskopf-Wigner model [89],which describes the mechanism of spontaneous decay of the excited state underemission of a photon [86]. The work established our matter-wave platform, in2

which a two-level system is realized via a microscopic atom trap (a site of an
optical lattice) that is occupied (excited state) or empty (ground state). The
quantum electrodynamic (QED) vacuum field that couples to the atomic dipole
moment is replaced by a matter-wave vacuum that couples to an eﬀective
dipole moments generated via a classical microwave field.
We also review our extended experimental study of non-Markovian dynamics of a matter-wave quantum emitter in a structured vacuum [87], to which
the author of this dissertation contributed significantly. By providing a band
structure to an emitted matter-wave of which we isolated a single band, we
observed the transition of decay dynamics from spontaneous decay to vacuum
Rabi oscillations, depending on the ratio between the band width and thecoupling strength. This gives a connection between regimes ranging from the
Weisskopf-Wigner model to the Jaynes-Cummings model of cavity QED limit
where only one mode is available for emission. It also highlights the role ofbound states, which are formed by an evanescent matter-wave surrounding apartly decayed emitter, in analogy to the atom-photon bound state [90].
In chapter 5, the main focus of this thesis, we extend our understandingof the atomic bound states to a more general context, a polariton, which is aquasiparticle formed between photon and matter-excitation. In our case, thisexcitation is an atom hopping in a lattice, while being surrounded by a virtual
cloud of matter-wave excitations; forming what we will call a matter-wave
polariton.
The history of polariton research is very deep and broad over many fields,
and since its first description as linear superposition of light and matter excitation [91, 92], polaritons have been an as essential key to understanding
the nature of light-matter coupling [93]. The core feature of a polariton is
captured by the realization of microcavity polariton [94], which shows observation of coupled exciton-photon mode splitting that suggests quasi-particle
interpretation. There are more than 70 diﬀerent types of polariton [95] invarious subfields, ranging from semi-conductor excitons [96–98], coupled cavity arrays [99, 100] to circuit QED [101], with various applications [102, 103]
especially in the quantum information since it provides a way to control theinteraction of photon.
These various platforms of diﬀerent polaritons all share the feature of strong
light-matter coupling, which is often diﬃcult to achieve due to a finite Purcell
factor [104], i.e. the residual coupling to modes not participating in the polariton formation. This is not the case for our matter-wave platform, where
the Purcell factor diverges. With our matter-wave platform, we realized a polaritonic dispersion relation in an optical lattice. Due to the coupling of theheavy particle to lighter matter wave, it is essentially in analogy to an exciton-3

polariton system [96, 98]. The matter-wave polariton we introduce here canbe used as a new platform for fundamental studies of polaritonic systems,
since it provides full tunability of polariton properties while still containing
the features of conventional polaritons. By controlling those parameters in theregime of interest, the system can also mimic polaritonic systems in waveguide[105–108] and circuit QED [102].
The system we introduce has both features of radiative polaritonic behavior
and condensed-matter physics. The band structure that we implemented canbe described by a ‘polaritonic’ Bose-Hubbard hamiltonian, which describes aphase transition between superfluid to Mott insulator.
In this dissertation, prior to reviewing the platform and related experiment, we first introduce the experimental setup and related background theories briefly in following chapter 2. More detailed theoretical concepts aboutband structure and Bose-Hubbard model is discussed in chapter 3. After reviewing the matter-wave platform in chapter 4 and discussing the experiments
on matter-wave polaritons in chapter 5, we finally suggest some future experiments in chapter 6.

4Chapter 2
Basic experimental methods
For the past two and a half decades, Bose-Einstein Condensates (BEC) havebeen studied in various contexts and the related experimental techniques are
now well-established. Depending on the target application, however, the details for an experimental setup can be very diﬀerent. In this chapter, we will
briefly review our experimental setup for BEC of rubidium-87 atoms and therelated theoretical background at the ultracold atom lab at Stony Brook, whichis universal over this whole dissertation.

2.1

Making BEC

In this section, we will take a short look at how we produce and use 87 Rb BoseEinstein condensates in the Schneble laboratory. Details of the experimental
setup have already been discussed in a number of previous dissertations [109–115] and a publication [116]. Thus in order to avoid redundant information,
we will discuss the mechanism of creating BEC rather qualitatively, and focus
more on the unique features related to the experiments in this dissertation.
Fig. 2.1 shows a photo of our BEC machine, which only includes the
crucial part where the atoms are present. To obtain BEC, we start from asolid sample of room-temperature 87 Rb atoms, inside our ‘oven’ attached tothe ultrahigh vacuum chamber that controls the temperature of the sample
and its vapor pressure. This control is done by a Peltier element and we are
maintaining the temperature at ∼ 15◦ C. Sublimated atoms are now in an
ultrahigh vacuum (∼ 10−9 Torr) cylindrical tube of glass ((1) in fig. 2.1). A
first cooling step involves laser cooling in a magneto-optical trap (MOT), which
consists of three orthogonal pairs of counter-proparating beams that are reddetuned from the D2 line (F = 2 → F ′ = 3) by 18.7MHz (cycling light) (cf.
fig. 2.2). In brief, atoms absorb near resonant light at 780nm from laser beams

5

(1)

(3)

(2)

Figure 2.1: BEC machine at SBU. This photo only shows the ultrahigh-vacuum
chamber of the equipment that atoms moves through. The movable large
pair of (brown) coils in (1) generate a magnetic field gradient with its antiHelmholtz configuration, and a magneto-optical trap is created in a cylindrical glass tube with the help of near-resonant laser beams. The magneticallytrapped atoms are moved through the diﬀerential tube (2) and arrive at thescience-cell (3), along the path depicted as a red-white dashed line.
in co- and counter-propagating directions that change their momentum andproduce a friction force (F ∝ −v) using the Doppler eﬀect [117]. Using thepresence of Zeeman-shifted magnetic sublevels, a spatially varying resonancefrequency due to a linear magnetic field gradient (∼ 10G/cm) created by aquadrupole magnetic coil causes a restoring force (F ∝ −x) to the atoms. We
use three diﬀerent laser beams; cycling light (F = 2 → F ′ = 3) aforementionedthat creates orthogonal 3-axis counter-propagating laser trap, repump light(F = 1 → F ′ = 2) that pumps inactive F = 1 atoms back into the F = 2
states, and depump light (F = 2 → F ′ = 2) to finally park all atoms into thefinal F = 1 ground states via optical pumping (cf. fig. 2.2).
In practice, we use a Toptica DL-pro-780 as a seed laser locked via Dopplerfree polarization spectroscopy to rubidium to generate ∼ 30mW of 780nm
light with narrow linewidth (< 500 kHz). Since we need relatively high laser
power (> 400mW) for the MOT, we amplify the light to ∼ 1W using a Toptica
BoosTA laser amplifier, which provides enough laser power for cycling, depump
light (> 10mW) and F = 2 imaging light (cf. sec.2.3). For the repump light6

(and F = 1 imaging light), we use a Toptica DL-100 to generate light of> 50mW that is separately locked to F = 1 → F ′ = 2 transition using
Doppler-free saturation spectroscopy. All beams except for the repump lightare delivered to the atoms using single-mode optical fibers, and can be switchedusing a combination of mechanical shutters and acousto-optical modulators.

87Rb

mF
-3

-2

-1

0

1

2

3

F=3

0.27 GHz

5 2P3 2F=20.16 GHz

F=1
F=00.07 GHz

Cycling

5 2P1 2
DepumpD1

D2
780.241 nm
384.230 THz

Repump794.978 nm
377.107 THz

5 2S1 2SDOL 790.0nm

F=26.83 GHz

F=1

Figure 2.2: 87 Rb energy level scheme. Transition data for D1 and D2 line
is taken from [118]. 52 S1/2 contains the ground states that are in use forthe experiments that consists of five F = 2 states and three F = 1 states.
When a bias magnetic field is applied, Zeeman splitting will happen between
degenerated mF levels (see fig. 2.6). Note that the sublevels for 52 P1/2 arenot shown. For cooling process, three specific transitions are in use optically;
cycling (F = 2 → F ′ = 3), repump (F = 1 → F ′ = 2) and depump (F = 2 →F ′ = 2) light. These transitions are also used for our imaging (see sec. 2.3).
At the zero magnetic field (and zero gradient) accomplished by the gradientcoil around the cylindrical vacuum cube, applying the detuned cycling light
gives an “optical molasses” providing sub-Doppler cooling [119, 120]. Then7

by turning oﬀ the repump light and actively turning on the depump light, allcooled atoms are now in F = 1 state with the low temperature ≈ 100µK.
Next, we transfer the atoms into a quadrupole magnetic trap [121], using asame coil that is used for creating MOT above. Note that the atoms in hyperfine states of low-field seekers are only trapped here, which is |F = 1, mF = −1⟩(since F = 2 states are depumped at this stage). We then transport the atoms
into the ‘science-cell’ (fig. 2.1 (3)), an ultrahigh-vacuum (∼ 10−12 Torr) quartz
glass cell (2cm ×1cm ×5cm), by physically moving the magnetic trap coils.
The pressure diﬀerence between two sides is maintained by the diﬀerential
tube (diameter ∼ 1cm) that gives a factor of 100 in pressure (see fig. 2.1(2)). The maximum axial field gradient is very stiﬀ ≈ 480 G/cm [114] tomaximize adiabatic compression and optimize the transport through the differential pumping tube. The vacuum is maintained using a combination of ion
and titanium sublimation pump.
In the science-cell, we use a series of evaporative cooling steps [122]. The
first step is a pre-evaporative cooling in a linear quadrupole trap using an‘RF knife’, which creates oscillating magnetic fields of adjustable frequency(1∼100 MHz) that changes the internal states of the atom in resonance andexpels the most energetic atoms, thus lowering the average temperature of theatoms in the trap and increasing the phase-space density (PSD). The minimum
temperature we here can achieve is about 100µK because of Majorana losses
that happens from the sudden spin-flip as the atoms pass the center of themagnetic trap where the field is zero and no quantization axis is provided.
Then we create a time-orbiting potential (TOP) trap [123] by adding a rotating
bias field in x − y plane to the quadrupole magnetic trap. The field-zero is now
rotating rapidly around the center, while the magnetic field at the trap-center
is oﬀset from zero thus avoiding Majorana losses. We apply another ramp of
the RF knife as the TOP trap cools atom further.
We then transfer atoms from the magnetic trap into the 3D optical trap,
which is made by intersecting two orthogonal laser beams at 1064nm (ω0 =140µm, ∼ 1mW) from our ytterbium fiber laser (YLR-20-1064-LP-SF/PL series). The optical trap gives us full freedom of using magnetic fields in theexperiments as required. In the optical trap, we apply the third evaporative cooling step by simply lowering the optical trap depth slowly, which lets
the high-energy atoms escape and re-thermalizes the rest of atoms to lower
temperature and higher PSD.
With the combination of cooling steps, we achieve BEC of typically orderof ∼ 105 atoms at temperatures ∼ 100nK. We generally prepare the atoms in
the F = 1, mF = −1 state with the free control of atom number by controllingthe depth of the optical trap. The relevant hyperfine states of 87 Rb are shown8

in fig. 2.2 [118]. Among the 8 hyperfine states (3 for F = 1, 5 for F = 2),
we typically use |F = 1, mF = −1⟩ (red atom in fig. 2.2) and |F = 2, mF = 0⟩(blue atom in fig. 2.2) states for the convenience in technical point of view,especially in regards of magnetic field sensitiveness. (Most of the experiments
done in the lab need a high level of stability and accuracy, so any possible
issue that can induce the unwanted fluctuations should be minimized. Details
are discussed in later sections.)
It should be stated that the moving-coil system that we have in the lab
is somewhat non-ideal as it requires lots of open spaces for the coils to movethat cannot be used otherwise. On the other hand, the large spatial separation
between the MOT and the science cell makes a life of an experimentalist very
easy in practice, especially in terms of trouble-shooting. Also, this large coilpair provides multiple functions throughout the experimental process. Starting from the magneto-optical trap, it traps and moves the atom to the highvacuum cell, and serves as the main coil for the TOP. Furthermore, at thestage of the detection, by pushing back about ∼ 1cm, the coil also provides
the magnetic field toward transverse axis, which is used as Stern-Gerlach separation for diﬀerent hyperfine states. In the past, we even used the coils - inHelmholtz configuration- to access a Feshbach resonance at ∼ 1007G [114].

2.2

Hyperfine state control

2.2.1

Zeeman sublevels in magnetic bias fields

Most of the work in our lab is based on the manipulation of atoms in diﬀerent
hyperfine states. For example, starting with BEC of atoms at |1, −1⟩ state,
as previously discussed, we typically implement a transfer into |2, 0⟩ state.
Historically the |2, −2⟩ state was generally used together with |1, −1⟩ in our
lab to achieve high Rabi-frequency for a given coupling field, but we used
|2, 0⟩ in this dissertation for reasons of magnetic field insensitivity and stateselectivity (see sec. 2.5).
For implementing transitions between states, it is of importance to precisely
control the corresponding energy gap. In our lab, the optical dipole trap
potential that traps BEC is practically state independent since it is very fardetuned, so that there is no diﬀerential energy shift between the states. The
bare energy gap between states is mainly dependent on the oﬀset magnetic
field. The magnetic field dependency of each energy level in the S1/2 ground9

state manifold can be calculated in the Breit-Rabi formula [124] as follows:
!
∆E
∆E
2mx
Em (x) = −gI µN mB0 −
±
1+
+ x2
(2.1)
2(2I + 1)
2
I + 1/2
in case of an atom with J = 1/2, where I = 3/2 is the nuclear spin of rubidium87, B0 is the applied magnetic field, ∆E is the bare energy separation between
hyperfine states, x = (ge µB + gI µN )B0 /∆E and m = mi + mj is the magnetic
quantum number.

Level Energy [GHz]F=1-3|2,+2>|1,-1>

-4

3

-5

2

|1,+1>

-6
-7

F=2
4

|2,-2>
0

200

400

600

800 1000

1

0

200

400

600

800 1000

B field [G]Figure 2.3: Breit-Rabi formula that gives the energy level of each hyperfine
state depending on the magnetic field.
Fig. 2.3 shows the magnetic field sensitivity of the energy level for eachstate. In our experiment, we tune the oﬀset magnetic field to ∼ 5G alongz-axis to provide the quantization axis. At zero magnetic field, the energy
splitting between F = 1 and F = 2 is 6.834 682 610 GHz [118], and the linear
Zeeman shift between neighboring mF levels is ±0.7 MHz/G. For spectroscopy
purposes, the second-order contributions are even important at small fields.
For very high magnetic field (> 1000G), |2, −2⟩ states moves along with otherF = 1 states.

2.2.2

Rabi oscillations

The manipulation of the internal hyperfine state in magnetic bias field is a key
feature in our lab that allows us to prepare the system in a desired initial state.
For the fixed energy levels, the most general technique for state transition isa Rabi oscillation.
10

In detail, it uses the oscillatory magnetic fields that drives magnetic dipole
moment resonantly. Consider a spin-1/2 particle in an external magnetic field
B(t) = B z +B rot (t), where |B rot | ≪ |B z | and B rot ⊥ B z . The spin precessingin the external field B z forms a two-level system |g⟩, |e⟩ with energy spacing!γBz , subject to the external oscillating field Brot (t) = B1 sin ωt. Startingfrom the Schrödinger equation
˙ = Ĥ |ψ⟩
i!|ψ⟩

(2.2)

Ĥ = µ · B(t)(2.3)

we then have the Hamiltonian

where µ = γ!/2σ̂ is a magnetic dipole moment, σ̂ is Pauli matrices andγ = gµB /!.
In the rotating frame, we can obtain the relation
" #
"
# " #
! −δ Ω
ċg
c
i!
=−
· g
(2.4)
ċe
ce
2 Ω δ
where δ = ω − ω0 is the detuning from the resonance frequency ω0 = γBz andΩ = γB1 is the Rabi-frequency.
Starting in |g⟩ (spin down), the time evolution of the final state population
is given as P|e⟩ (t) = ce c∗e , which is
"
#
Ω2
t√ 2
2
2
P|e⟩ (t) = 2
sinΩ +δ .
(2.5)Ω + δ22Note that full population transfer can only be achieved on resonance (δ = 0).
Fig. 2.4 (A) shows the time-dependent state transition with fixed detuning
and Rabi-frequency, called Rabi oscillation or Rabi flopping, between |1, −1⟩
and |2, 0⟩ driven by a σ + polarized magnetic field. Here the coupling strengthis ∼ Ω = 2π × 3.7kHz, which gives about 270µs of the oscillation period asshown in the figure. We can also vary the detuning while the coupling time
t is fixed, and it gives Rabi spectra as shown in fig. 2.4 (B). Here the widthof the spectrum depends on the Rabi frequency as well as the pulse-duration.
For the short-pulse regime, the higher the Rabi frequency is, the larger the
spectral width is.
These Rabi spectra can be encapsulated in 3D plot from eq.2.5, as plottedin fig. 2.4 (C) in detuning and time domain with fixed Rabi-freqeuncy. Eachfeature from figure (A) and (B) can be shown as a cross section of the plot
with zero-detuning and t = 200µs pulse duration, respectively.
11

(A)

|2, 0 >
|1,-1>
170

175

180

183

185

187

189

191

193

195

197

200

205

210

450

500

550

600

650

frequency (kHz)(B)|2, 0 >
|1,-1>
0

100

150

200

350

400

time (µs)

(E)

0.8
0.6
0.4

n

0.2
200

210

1.0
0.8

)

190

frequency (kHz)

det

0.6
0.4

e(
μs

180

unin
g

tim

170

Fraction of | 2, 0 >

300

1.0

0.0

(D)

250

io
PopulatFraction of | 2, 0 >

(C)

50

δ (k

H z)

0.2
0.0
0

200

400

600

time (µs)

Figure 2.4: Rabi oscillation spectra. (A, B) Stern-Gerlach separated time offlight image of Rabi spectrum (oscillation) (Ω = 2π × 3.7kHz, square pulse)
between |1, −1⟩ and |2, 0⟩ states in detuning (time) domain for fixed t = 0(δ = 0). Blue (Red) solid line is a fitted curve with eq. 2.5. (C, D) Rabi
spectrum (oscillation) of the relative population of |2, 0⟩ state in detuning(time) domain, data extracted from (A (B)). (E) Theoretically expected 3D
plotted Rabi spectrum. The Rabi frequency is set as (Ω = 2π × 3.7kHz), thesame with (A-D). Each feature from (C) and (D) is a cross section of the plot.
In our experiment, we use the Rabi-spectrum measurement to identify the
coupling strength Ω and the corresponding magnetic field from the fitting as
shown in fig. 2.4. When both two bare ground states are not in lattices, theresonance frequency ω0 is the same with the bare energy splitting from the12

Breit-Rabi formula (cf. eq. 2.1), but in the presence of the lattices, we needto take into account the zero-point energy shifts.

2.2.3

Landau-Zener sweep
E

(A)
|e>

|g>

ћωR

δ

δ=ατ

|g>

|e>

(B)

|2, 0>|1,-1>

τ
Figure 2.5: Landau-Zener sweep. (A) Schematic diagram. The incoming
excited (ground) state goes out as the ground (excited) state if the sweep isadiabatic, along the thick solid line above (below). The adiabaticity depends
on the ratio between the vertical gap (ωR ) and the horizontal sweep (δ = ατ )where α is a sweeping velocity in time τ . Under the condition ωR2 /α ≪ 1,
the sweep is adiabatic. (B) Experimental realization. ωR = 2π × 8.0kHz, andthe sweep time τ is varied from 0.1 to 10ms while δ = 1MHz is fixed. The
incoming state |1, −1⟩ is transferred to |2, 0⟩, and the transition rate depends
on the sweeping time τ . For longer sweeping time, the adiabatic condition is
fulfilled and most of the initial state is turned into the final state.
One of the most reliable ways to transfer one state totally to another stateis to use the Landau-Zener (LZ) sweep [125]. This mechanism has been discovered in 1930s by Landau and Zener [126, 127], that uses the avoided-crossing

13

mechanism in energy and detuning domain. Fig. 2.5 (A) shows the schematicof the Landau-Zener sweep. In the energy-detuning plane, there is an avoided
crossing at the resonance δ = 0 where the two bare states (excited and ground)cross√each other. The two dressed states (eigenstate of eq. 2.4) are separated
by ! δ 2 + Ω2 . Starting from the one state at eﬀectively infinite negative detuning, scanning the detuning that crosses the resonance decides its final state
depending on how fast the scan is. If the scanning speed is slow enough, thechange of the eigenstate is adiabatic and the corresponding eigenvalue is deformed to avoid the level-crossing as shown in the figure. Here the competingparameters are the slope from x and y axis, i.e., the speed of detuning scan(δ = ατ ) where α is a sweeping velocity, and the Rabi-frequency (ωR ). If thesweep is too fast, the system is no more adiabatic and it follows the dashedline in the figure, so that the final state is the same with the initial state.
The adiabatic condition can be estimated qualitatively. From the figure(A), δ ≈ ωR which gives τ ≈ ωR /α. Also from the uncertainty condition∆E = !/τ , this should be much smaller than the energy !ωR . This indeed
gives ωR2 /α ≪ 1 as an adiabatic condition for the LZ sweep.
Fig. 2.5 (B) shows the experimental realization of the LZ transition. Starting with the initial state |1, −1⟩, changing the detuning across the resonanceunder the transition time τ gives a state transition to the final state |2, 0⟩.
Here, the sweep rate depends on the sweep time τ as shown; for very shortsweeps, the state is unchanged which means that the system is fully nonadiabatic. As the τ increases, the system gets more adiabatic, and most of the|1, −1⟩ population are turned into the final state |2, 0⟩ for long enough sweep
time τ .

2.2.4

Two-photon transition

Sometimes it is required to make a transition in the system between the main
two bare states (typically called red and blue) which is not allowed by theselection rules for a one-photon transition. For example, the transfer |1, −1⟩
to |2, 1⟩ is not generally allowed due to the conservation of angular momentum.
In such cases, we use a two-photon Raman resonance, which uses a third state
as a (virtual) intermediate step.
For our case, we use both microwave and radio-frequency fields as sourcesof one photon each. For the case of coupling |1, −1⟩ and |2, 1⟩ as depicted in
fig. 2.6, we proceed via the |2, 0⟩ state as a virtual state at large detuning,and coupling with |1, −1⟩ is done by microwave photon and |2, 1⟩ by radiofrequency photon, respectively. This process should not populate |2, 0⟩, theeﬀective virtual state here, and one should use a detuning much larger thanthe coupling strength. The total Rabi frequency of the resonant two-photon
14

2

F=2

1

F=1
-2

-1

0

1

2

mF
Figure 2.6: Two-photon transition. |1, −1⟩ and |2, 1⟩ states are coupled by
two-photon process, with the detuned virtual state |2, 0⟩. Ω1 is the microwave
photon coupling strength and Ω2 is the RF photon coupling strength. Each of
coupling is detuned by ∆ from the resonance. Here the total coupling strength
is given by Ωtot = Ω1 Ω2 /2∆.
transition is given byΩ 1 Ω2(2.6)
2∆where Ωi is the Rabi frequency of the i-th transition and ∆ is the detuningfrom the intermediate state that is used as a virtual state.
Ωtot =

2.2.5

Stern-Gerlach separation

With the experiments that manipulate the internal hyperfine states of theatom, it is crucial to detect each of their hyperfine states separately. To
accomplish this, we use the Stern-Gerlach separation technique [122] (in combination with state-selective imaging) that gives a force at the beginning of
the time of flight by spatially varying magnetic fields.
In practice, we use the same pair of large coils that is already used for
MOT and the quadrupole magnetic trap (cf. sec.2.1). After the atoms aretransferred from the magnetic trap to the optical trap in BEC preparation
steps, we translate the coil about 1cm along the x-axis. Right after the timeof flight step, we first rotate the bias magnetic field from the z-axis to the x15

mF
-2

-1

0

1

2

F=2

F=1
1

0

-1

Figure 2.7: Stern-Gerlach separation. The magnetic field gradient that is
applied at the beginning of the time of flight step gives a spatial separation of
atoms depending on their hyperfine states. Blue and Red dotted box describes
the position of atoms for each mF state. Due to the diﬀerent sign of g-factor,
the arrange of sublevels for F = 2 and F = 1 is opposite.
axis using external Helmholtz coils around the science cell along x-axis. Then
we apply the current to the large coils so that they generate a magnetic field
gradient B = b0 xx̂ to x-axis, where b0 ∼ 100 G/cm [113]. This provides forces
depending on the internal hyperfine state since F = −∇V where the potential
is from the Hamiltonian Ĥ = −µ · B, which indeed gives
F = −µB gF mF b0 x̂.

(2.7)

This force gives a spatial separation of atoms in time of flight imaging
depending on their hyperfine states. Since the force (in eq. 2.7) is proportional
to the Landé g-factor, which has the opposite sign for F = 1 and F = 2 in87
Rb, the direction of the separation for each F state depending on sublevels(mF ) is also opposite of each other.

2.3

Detection

It is crucial to conserve the information for the hyperfine state when detecting
the atom at the end of each experiment. In our experiments, we are mainly
using absorptive imaging technique with the combination of time of flight.
We use resonant light on the F = 2 → F ′ = 3 D2 cycling transition toinitially detect atoms at F = 2 states first (see fig. 2.2 for the transition).

16

Photons incident on atoms in F = 2 are scattered by resonant absorption,and this process creates a shadow of the atoms in the forward direction thatis detected by a 16-bit CCD camera (Princeton PIXIS 1024B). By comparing
the shadow with the background image that is taken in a later part of thetime of flight sequence, we can extract the optical density of the atom with aproper calibration. Atoms in F = 1 are optically pumped into F = 2 statesusing repump light, and then imaged in the same way ∼ 3ms later.
In this imaging process, the CCD camera is taking 3 photos for the imagingof each state (F = 2 and F = 1); one image with the presence of the atomby shining in resonant light (Ia ), the next image again after all atoms arevanished (I0 ), and the last ‘dark’ image without illuminating the light (bkg),with ≤ 1% non-linearity. The optical density (OD) for each camera pixel isthen calculated as follows [122]:"#
Ia − bkg
OD = −ln(2.8)
I0 − bkg
which gives the atom number N = OD × σ where σ is a calibration that
depends on the measurement environment (pixel size, light polarization, field
quantization direction, resonant scattering cross section, etc.) which is ∼ 125
in our lab [113].
There are two modes that we can choose in our setup; these are ‘normal’ imaging and ‘kinetic’ imaging, and we typically use the ‘kinetic’ imaging
method with the CCD camera. The normal imaging gives bigger region ofinterest, which is useful for the initial alignment. It is desired, however, to use
the kinetic imaging for more accurate measurement since it provides better
time resolution and a more precise optical density. Above all, Stern-Gerlach
separation (cf. sec. 2.2.5) gives the same spatial separation for same mF statesfor F = 1 and F = 2 (with opposite sign), so that we need to temporally separate states.
On kinetic imaging mode, after imaging F = 2 onto a partly covered CCDchip, the CCD camera pushes the image upward into the dark region andthe same procedure is repeated for imaging F = 1 followed by another shift,
taking the background image, and finally read-out of the chip. As a result,
we obtain a set of 3 images for F = 2, F = 1 and background that areplaced horizontally. Detecting the F = 1 states follows the sequence in 2.7ms
additional flight. Since there is no resonant light that sends F = 1 to F ′ = 3,
we first repump the F = 1 atoms by shining F = 1 → F ′ = 2 laser. The
atoms then spontaneously decay into F = 2 state, which is now qualified tofollow the previous detection method. Here it is critical to fully separate two
detections (F = 2 and F = 1), so that any remaining F = 2 atoms do not17contaminate the signal of the other.

2.4

Image analysis using principal components

Obtaining an accurate optical density (atom number) from the measured image(cf. sec.2.3) is very crucial. As mentioned before, we are measuring the shadowof the atom compared with the no-atom image which eﬀectively minimizes the
background noise by itself. In practice, however, there are some unwanted
noises on the image, such as a circular fringes, which are generally caused by
dust particles on optical elements in the path of the imaging light. Most of
the noise that is static in time can be easily eliminated via our normal imaging
process (cf. eq.2.8), but time-dependent (between the two images taken in arow) noise can give errors in the calculation of the optical density, because
those fringes are also considered as the optical density if they are in the regionof interest. This eﬀect becomes more critical for the experiments that deal
with relatively small atom number (! 104 ) which gives about the order of a
few tens of atoms per pixel.
To overcome this issue, we use the technique of principal component analysis (PCA) which is described in detail in L. Krinner’s thesis [113]. The
authoritative description there is more principle-oriented and quantitative, we
thus here explain the technique in a more qualitative way.
As a start, we here first briefly review the mechanism of PCA [128, 129].
PCA is an analysis technique that is widely used in large data processing formany purposes like image processing and compression due to its dimensionality
reduction property. In our case, we use PCA to reconstruct clean images byremoving noises that are learned from the references. To do this, we use
principal components (PCs) that are extracted from a reference data set ofmany images taken without atoms, where each PC is a sequence of vectors ina real coordinate space. These principal components (pi ) are unit vectors thatare orthogonal to each other, and they are ordered by magnitude where eachof components is in a direction that maximizes the variance of the data set.
The first component p1 is the best fit to the data (in other word, minimizing
the projection residual (the separation of data from the principal component),
which maximizes the variance of the data that passes the center (cf. fig. 2.8(C) bottom), whereas the second components p2 is the next best-fit that is
orthogonal to p1 . In this way, we can make a set of PCs that contains themost information (fringe pattern in our case) in a finite (and relatively small)number of components.
The mechanism of obtaining PCs is depicted in fig. 2.8. In our case, we
use an empty image (reference image) as a data set, Rj , (cf. fig. 2.8 (A))
18

(A)

(B)

Ri

(C)

ri

projection residualpi
σ2

Figure 2.8: Mechanism of constructing principal component. (A) The
schematic of the reference image with fringe, Ri . $(B) Centered image bysubtracting the mean of the references r j = Rj − j Rj /N (C) Principal
component is given as the best fitting of the data that minimizes the projection residual. The bottom projection shows why it simultaneously maximizes
the variance and minimizes the projection residual.
which contains the fringes that the analysis will study and remove later. Since
our image is 2-dimensional, we make this into a 1-dimensional vector form bychanging the basis {x, y} → x + dimx × y, where dimx is a number of pixels of
x-axis in the reference image. For simplicity, this process is not displayed in
the figure.
From the set$of N reference images Rj , the mean reference image is extracted as R = j Rj /N . To perform PCA properly, we need to center thedata, i.e. the mean of data should be zero (cf. fig. 2.8 (B)). For this, we now
subtract the mean for the reference images to construct data set as
r j = Rj − R(2.9)

From this r j , we extract the set of principal components pi that follows
the conditions above (cf. fig. 2.8 (C)). The variance (that is $naturally max1
2
imized from the principal components) is given by σ = N i (di · pi )2 =T
1(DP )T (DP ) = P T DND P ≡ P T V P , where we use D as a componentNof data set in a convention of [113], N is a number of data vectors, andV is the covariance matrix with corresponding eigenvector P . In practice,
the construction of pi is easily done by the in-built function of Mathematica;
19

PrincipalComponents[table] where table is the optical density data table of
the reference image.
Imaging data set
empty images

Ratom

dimensionality reductionreference images

Ri

Ratom

construct
Principal Componentspi

apply PCA

RfinalFigure 2.9: Image process steps for PCA. We take empty (atom-free) reference
images between the measurements, typically more than ∼ 40 in total. From theset of reference images, we extract the principal components for each groundF states. These components are applied to the equation eq. 2.10 to process
the images we want to analyze (Ratom ), and we finally obtain the processed
image Rfinal . The number of principal components in use is typically smalln ∼ 6, to avoid over-fitting.
The next step is then applying the obtained principal components to theimage that we want to process, which will get rid of fringes from an image
with atoms by using the learned fringe pattern from the atom-empty images.
Following the convention, we here denote Ratom as an image with atoms that
we want to analyze. Since PCA requires the zero-mean data set, we here also
subtract the mean of the reference images as r atom = Ratom − R. Then theoverlapped part from the subtracted image is considered, only for first a few

20

principal components (typically n = 6 in this dissertation) as follows:
Rfinal = r atom −

n
%
i=1

(pi · r atom ) × pi

(2.10)

Here the subtracted image with atom is projected into the principal components. Note here that if an image contains a large number of atoms, the meanof the image with atoms is far from the zero which causes the error significantly. In this case, we need to consider the error subtraction. In short, one
should put the ‘mask’ to screen out the region with atoms (r matom = m × r atom
where the ‘mask‘ m is 1 (0) for the outside (inside) of the regions that containthe atoms), and process the analysis, that is discussed in detail in the thesis[113].
The practical flow of the image process is depicted in fig. 2.9. In practice,
the reference image (empty atom image) is taken before and after the measurement to minimize the temporal changes between the PCA references andthe processing data. We obtain these reference images simply by not loading atoms in MOT but repeating the rest of the same sequence as the othermeasurement, which is to make the same environmental eﬀect that can bepropagated from the sequence steps. Also for processing PCA, we re-adjust
the region of interest (ROI) of the image properly. This is to zero the reference
image that increases the validity of the process.
Fig. 2.10 shows the result of PCA with the actual measurement data usedin our publication [88] (cf. chap. 5).

2.5State-dependent potential

Making optical lattices in a standing optical wave is key for many of ourultracold atom experiments, which enables us to control the eﬀective dimensionof the system. In our lab, we typically make 1D systems that reveal muchinteresting physics not shown in 3D, by tightly confining the atoms with a fardetuned orthogonal 2D optical lattice in x-y dimensions. The D2 (52 S1/2 →52 P3/2 ) transition wavelength of 87 Rb is ∼ 780nm, so using 1064nm laser gives
practically identical lattice tubes for all ground states of Rubidium atoms.
For the remaining 1D axis (z-axis in our lab), we choose a wavelength thatis close to the resonance in order to provide the state-dependency. From theatom-light interaction, we can tune both amplitude and sign by choosing theproper combination of the laser and atomic hyperfine state pairs. In detail,this state-dependency is also related with the polarization of the laser and AC

21

(A)

Ratom

(B)

RfinalFigure 2.10: Image processed by PCA. (A) is the raw time of flight image
Ratom and (B) is the processed image with PCA, Rfinal . Red arrows in (A)marks the fringes, that are removed in the processed image (B). This data
is used for the measurement of the onsite-interaction energy of matter-wave
polariton (cf. chap. 5).
Stark shift [130, 131]. By ignoring the counter-rotating terms, we have [110]
⎡
⎤
li 2
li 2
2
%
%
|cp |
|cp |
3πc ⎣ ΓD1
2ΓD
⎦ I(r)U (r, i, p) =+ 3 2
(2.11)
32
ωD1 l∈P ω − ωli
ωD2 l∈P ω − ωli
1/2

3/2

where i indexes the initial hyperfine state, ΓD1,2 is the scattering rate of theD1,2 transition, clip is a Clebsch-Gordon coeﬃcient between the initial and final
hyperfine states, p denotes the polarization of the light, which may be σ ± or
π-polarized, and I(r) is the intensity profile of the optical potential.
Fig. 2.11 shows the the optical potential of three diﬀerent hyperfine states(|2, −2⟩, |1, −1⟩ and |2, 0⟩) for diﬀerent polarizations (σ − , π, σ + ) in the vicinityof the D1 and D2 resonance wavelength. Note that in order to obtain diﬀerent
lattice depths depending on the hyperfine states, circular polarization σ ± isdesired since the magnetic non-neutral hyperfine states (mF ̸= 0) experience
diﬀerent optical potential from the unmagnetic state (mF = 0). Another22

Optical Potential (100mW/cm2)

(A)

(B)

σ-

(C)

π

σ+

2

0

-2
780

785

790

795

780

785

790

795

780

785

790

795

Wave length λ (nm)Figure 2.11: Optical potential for various polarizations (σ − , π, σ + for (A,B,C))
is calculated with diﬀerent hyperfine states (|2, −2⟩ , |1, −1⟩ , |2, 0⟩ for blue,
orange and green curve, respectively). In a case of circular polarization (A,C),
the magnetic hyperfine states (|2, −1⟩ and |2, 1⟩) are tuned out in a diﬀerent
way at the wavelength between D2 and D1 line due to the opposite sign of thedetuning. The magnetically insensitive state |2, 0⟩ is always tuned out at thesame wavelength without any polarization dependence.
interesting point is that this magnetic neutral hyperfine state (mF = 0) has
the same energy shifts for all polarizations, and it does not matter whether
F = 1 or F = 2 is used. Thus for providing the tunability for various lattice
depths, it is natural to choose circular polarization, and we are using σ − forthe experiments in this dissertation.
From eq.(2.11), we can also study the wavelength dependence for diﬀerent
hyperfine states. The choice of the hyperfine state pair set is more criticalfor experiments with state-dependent lattices. With the proper choice of an
atomic state pair, it is possible to set a ‘tune-out’ wavelength that only oneof the two state sees the optical potential while the other does not. This ispossible by virtue of the cancellation of couplings to the D1 (52 S1/2 → 52 P1/2 )and D2 line of 87 Rb, 795nm and 780nm respectively. Our experimental range
is between this blue-detuned (795nm) and red-detuned (780nm) wavelength
so that we can control the the relative potential including its sign by shifting
the wavelength. Fig. 2.12 (A) shows the optical potential in units of recoil
energy depending on wavelength near the resonance for diﬀerent ground states.
Here we arbitrarily choose σ − polarization. As discussed in previous section(cf. sec.2.2.5), due to the opposite sign of Landé g-factor between F = 1 and23

Lattice depth V/Er

20

(A)

10
0
-10
-20

Scattering rate Γ(1/s)

10
8

(B)

6
4
2
0
780

785

790

795

800

Wave length λ (nm)Figure 2.12: Optical potential for various hyperfine states. Lattice depths = Vlatt /Er is calculated with the circular polarization σ − . Substates of
F = 1 (solid) and F = 2 (dashed) show the opposite response against thewave length due to the sign of Landé factor, which gives perfect overlap for|1, −1⟩ and |2, 1⟩ as well as |1, 1⟩ and |2, −1⟩. (B) Corresponding scattering
rate. The scattering rate diverges for most states for the wavelength < 785nm
and > 792nm (cf. eq.2.12).
F = 2 state, |1, −1⟩ and |2, 1⟩ states give the same potential as well as |1, 1⟩ and|2, −1⟩. In our experiment, we choose the two states |r⟩ = |F = 1, mF = −1⟩
and |b⟩ = |2, 0⟩, where |b⟩ is tuned-out at the wavelength of λ = 790.02nm.
This is consistent with the recent report for the precision measurement on
tuned-out wavelength of Rubidium [132], which claims 790.01859(23)nm for
mF = 0 states.
When we use the state-dependent optical potential, it is very important toconsider heating from spontaneous photon scattering. The scattering rate is24

calculated by ΓSC = ΓU/!∆ with the eq.2.11 as follows [110]:
⎡
⎤
li
2
li
2
2
2
2
|cp |
|cp |
2ΓD2 %
3πc ⎣ ΓD1 %⎦ I(r).
ΓSC =
+
3
3
2
ωD1 l∈P (ω − ωli )2ωD2 l∈P (ω − ωli )2
1/2(2.12)

3/2

The scattering length for the same polarization σ − is shown in fig. 2.12. The
scattering rate diverges for most states for the wavelength < 785nm and >792nm, due to the D1 /D2 transition. The scattering rate at 790nm is on theorder of one per second (ΓSC = 1.68s−1 for |1, −1⟩ and 1.35s−1 for |2, 0⟩ at theintensity of 150W/cm2 that gives sz ∼ 20Er ) which is negligibly long comparedwith our experimental time scale ! 10ms.
With all the techniques described above, we are now ready to explore more
specialized directions of ultracold atomic physics. Especially the summaryin chapter 4 and the research described in chapter 5 are works based on thetechniques above.

25

Chapter 3
Theoretical concepts
In this chapter, we discuss same fundamental concepts for studying interacting
quantum gases in optical lattices. These concepts are particularly relevant forthe description of our experiments with matter-wave polaritons (cf. Chap. 5).

3.1

Band structure in optical lattices

In periodic potentials, the motion of a particle is strongly influenced by quantummechanical tunneling. We here first see how we obtain a band structure in
optical lattices. Since most of the experiments in this dissertation deal with
1D geometries, we will simply solve the system in 1D.
We start with the Hamiltonian for a single atom in a 1D optical lattice:Ĥ = −!2 ∇ 2+ V0 sin2 (kz z)
2m

(3.1)

where k = 2π/λ is the wavenumber and λ is the wavelength of the lattice.
In case of a single particle in a periodic potential V (z + Z) = V (z), where
Z is an integer multiple of λ/2, the eigenfunction of the Hamiltonian is given
by a Bloch wave as the product of a plane wave and a periodic function [133]φq (z) = eiqz u(z)(3.2)

where q is the quasi-momentum and uq (z) is a function with the periodicityof the lattice. Sometimes, it is useful to use the eigenfunction in another form
as follows:φq (z + Z) = eiqZ uq (z)(3.3)
for every Z in a Bravais lattice. Here, from the properties of the optical lattice,
we can naturally define some basic parameters.
26

In a microscopic point of view, the confinement of the atom in the opticallattice arises from the continuous absorption and stimulated emission of thephoton momentum. Thus there is an energy scale that the atom in the opticallattice naturally brings, the so-called ‘recoil energy’,Er =!2 k 22m(3.4)

where k is the recoil momentum and d = λ/2 = π/k. Typically we express Er
in units of h× kHz.
The lattice depth can then be expressed in units of the recoil energy, ass = V0 /Er where V0 is the amplitude in eq. 3.1. When s ≫ 1, the system
is called ‘tightly confined’ so that there is very little natural tunneling fromlattice site to site, when measured on the time scale !/Er . The harmonic
oscillator length aho can be also defined in terms of s, as**!
! −1/4
aho =
=
s
(3.5)
mωho
2mωr
√
where ωho = 2ωr s and ωr = Er /! is the recoil frequency.
Since the system is periodic, we can solve the Hamiltonian by only looking
at the first Brillouin-zone [−k, · · · , k]. Depending on the properties of the single atom and its confining potential, the energy eigenvalue εn (q) is continuousin certain energy ranges which form the energy bands with index n. The Bloch(n)waves are then written as φq (z). The regions between the separated bands
where no eigenvalue for Hamiltonian exists are called band-gaps.
The general dispersion relation expressed in the first Brillouin-zone is shownin fig. 3.1. Two extreme cases are very easy to catch here. In case of a freeparticle where V0 = 0 as in fig. 3.1 (A), there is no bandgap and the energy
is continuous. In the case of a deep lattice where s ≫ 1 (see fig. 3.1 (C)),
the ground energy band is very flat, approaching the energy of the harmonic
oscillator ground-state, with a diverging eﬀective mass m∗ = !2 (∂ 2 E/∂k 2 )−1 .
For most cases where the confining potential is finite, however, a level splitting
exists at the band edge, in analogy to an avoided crossing, and a bandgap is
induced (see fig. 3.1 (B)).
When the lattice is deep enough, it is convenient to represent the systemusing localized wavefunctions, the so-called Wannier functions (discussed indetail at sec.3.1.1). This is particularly true when motion is confined to theground band, n = 1. In this case, the Wannier function localized at site j with27

Energy ( Er )

(A)

(B)

s=0

(C)

s=3

5

5

5

4

4

4

3

3

3

2

2

2

1

1

1

0

0

0

-1.

0

Quasi Momentum k

1.

-1.

0

1.

Quasi Momentum k

-1.

s=10

0

1.

Quasi Momentum k

Figure 3.1: Energy band structure for 1D lattice system with diﬀerent lattice
depths (s = Vlatt /Er = 0, 3, 10 for (A, B, C)). The energy bands are plottedin the first Brillouin-zone. The dispersion is expressed in quasi-momentum
k = π/d where d = λ/2, and energy Er .
position xj can be calculated by the Bloch waves as
1 % (1)1 %w(x − xj ) = √φq (x − xj ) = √exp−iqxj φ(1)q (x − xj )
N q
N q

(3.6)

where N is a normalization constant (number of sites).
The tunneling between neighboring sites is determined by the overlap ofadjacent Wannier functions,t = ⟨w(z)| Ĥ |w(z + d)⟩(3.7)

where Ĥ is the system Hamiltonian and w(z) is the Wannier function at position z. This will be discussed in more detail below section. We note that t
can also be calculated from the width [ε1 (k) − ε1 (0)] = w1 of the ground bandas t = w/4.

3.1.1

The tight-binding limit

In our optical lattice, especially in the case of 1D, the tight-binding approximation is very useful for understanding the dynamics of localized atom. It
is generally applicable when the leakage of the Wannier functions into neighboring sites is not too large (i.e. the Wannier functions are approximately
28

harmonic-oscillator ground state) but also not too small to neglect the correction to the picture of isolated sites.
In detail, we first consider the single particle Hamiltonian of a full periodic
crystal is given as
H = Hat + ∆U (z)(3.8)
where Hat is the Hamiltonian of the fully localized single atom, and ∆U (z)contains all corrections to the potential due to its sinusoidal nature, beyond
the harmonic approximation. We can assume that ∆U → 0 at the center of
each site. Under the assumption that the full periodic Hamiltonian H can beapproximated as Hat [133], we have
Hat ψn = En ψn(3.9)

where ψn (z) is localized in each lattice site, En is the energy of the n-th energy
level of a localized atom. In this case, we have a very small overlap integral
γn ,+γ(Z)n = dz ψn∗ (z)Hψn (z − Z).
(3.10)
We now want to make the connection between these localized wavefunctions
to the Bloch waves. To do this, recall here the Bloch condition eq.3.3; we wantto satisfy the condition for N linear combinations, which gives%
φq (z) =eiqz ψ(z − Z)
(3.11)
Z

where ψ(z) is indeed a Wannier function of eq.3.6, which can be expanded asa sum of localized orbital wave functions%
ψ(z) =
bn ψn (z)(3.12)nwhich is not necessarily a stationary-state wave function. The validity of thisapproximation mainly relies on following two assumptions. The first one is an
assumption of small overlaps (γn ) as stated above, and the second one is thesmall correction of the potential at large distance from center [133], say#% "+∗ψm (z)∆U (z)ψn (z)dz bn ≪ (ε(k) − Em )bm(3.13)nwhere Bloch energy ε(k) is the eigenvalue of the total Hamiltonian H = Hat +∆U (z) and k is a wavevector.

29Bandwidth (Er)We now discuss the diﬀerence of the Wannier function from the tightbinding atomic function. The Wannier functions at diﬀerent sites are orthogonal which is not the case for the other, and the complete set of the Blochwave can be expressed as the set of Wannier functions and vise versa.

10-1
10-2
10-3

tight-binding approx.
exact band cal.

10-4
0

5

10

15

20

25

30

Lattice depth s

Figure 3.2: Calculated band width from the tight-binding approximation andexact band calculation (from Mathieu function). For low depth, the tightbinding model fails due to the invalid assumption. As the lattice gets deeper,
the model converges to the exact model.
In the tight-binding regime, we can calculate the dispersion of the lowest
Bloch band, where the bandwidth is given as [134],44J = √ s3/4 exp −2(s)1/2
π(3.14)

where J is the tunneling coeﬃcient and s = Vlatt /Er is the lattice depth. Thisis a good approximation in large order, but it is always better to calculate
the bandwidth from the exact band structure analytically from the Mathieufunction for example [134], see fig. 3.2.

3.2

Bose-Hubbard model

A BEC in the optical lattices is a great platform to study experimentally the
Bose-Hubbard Hamiltonian (BHM) [50]. Based on the experimental implementation of the BHM, the transition from superfluid (SF) to Mott-insulator(MI) [135] has been studied experimentally [49, 136] which opened up new
possibilities toward the use of ultracold atoms as test-bench for theoreticalmodels from solid state physics.
30

In case of interacting scalar bosons, with the atom-atom contact interaction
Vcol (x) = gδ(x), the Hamiltonian in second quantization can be expressed as
+
!2 ∇2
g
+ V )ψ̂(r) + ψ̂ † (r)ψ̂ † (r)ψ̂(r)ψ̂(r)],(3.15)Ĥ =d3 r[ψ̂ † (r)(2m2where g is the coupling constant from the atom-atom collision g = 4π!2 as /m,
as is the s-wave scattering length, and ψ̂ † (r) and ψ̂(r) are the creation andannihilation operators of the bosonic field.
Here we can expand the field operator ψ̂ † (r) in the basis of Wannier function as% †ψ̂ † (r) =âj wj (r)(3.16)
j=1

where wj (r) = w(r − r j ) is the localized Wannier function and â†j is thecreation operator for boson on site j.
Then by defining the tunneling coeﬃcient J and on-site interaction energy
U as
+
!2 ∇2
J = Ji,j = − d3 r wi∗ (+ V )wj ,(3.17)
2m+
U = g d3 r wi∗ wi∗ wi wi ,(3.18)
the Hamiltonian can be rewritten as% †U%Ĥ = −Jâi âj +n̂j (n̂j − 1),
2
<i,j>
j

(3.19)

where n̂i = â†i âi is the number operator.

Figure 3.3: Illustration of the Bose-Hubbard Hamiltonian schematic. J is thetunneling between the lattice sites and U is the onsite-interaction energy.
With the consideration of the chemical potential of the system, we indeed

31

have the Bose-Hubbard Hamiltonian as% †%U%
ĤBH = −Jâi âj +n̂j (n̂j − 1) − µ
n̂j
2 j
<i,j>
j

(3.20)

where J is the tunneling between the lattice sites and U is the onsite-interactionenergy. Note that µ fixes the atom number in a trap. Fig. 3.3 shows a BoseHubbard Hamiltonian schematic.

3.2.1

Mean-field treatment and quantum phase transition

The BHM carries interesting many-body physics emerging from the presence ofcollisional interactions. The quantum phase transition between the superfluidand Mott-insulator is one of the most significant observations that has beenachieved in the optical lattice context. To find an approximate phase diagram,
we first apply the assumption of mean field approximation [137–140], which
averages out the all eﬀects from each individual atom. Then the many-bodyproblem turns eﬀectively into a one-body problem.
In detail, let’s first take a look how the two extreme regimes for competingparameter ratios give each quantum phase. For the weakly interacting regime,
J ≫ U , the interaction is negligible and atoms are all in same phase. Hereatoms occupy the lowest kinetic energy state as
|ψ⟩ =

1 † N
(a ) |0⟩
N ! k=0(3.21)

where N is a total atom number, |0⟩ is a vacuum state, and the operator a†kcreates a particle with momentum k (eq.3.20). This eq. 3.21 is a coherent(superfluid) state, and we can express it alternatively as a sum of the atoms
at all lattice sites as
1 % †|ψSF ⟩ =a |0⟩(3.22)
Nsite i iwhere i is a site index.
For the strongly interacting regime, J ≪ U , we can neglect J so that the
Hamiltonian eq.3.20 gives
Ĥ =

%
U%
n̂j (n̂j − 1) − µ
n̂j
2 j
j

(3.23)

in which all lattice sites are isolated from each other. In each individual site32

with n atoms, the energy is then given as
En = −nµ +Un(n − 1).
2

(3.24)

Thus the energy states are separated and gapped, which gives a stable stateagainst small perturbation like small tunneling, which is a Mott state. Notehere that this Mott state only has an integer filling n. The eigenfunction of
the Mott state can be represented as|ψMI ⟩ ∝(A)

1 . † N/Nsite
(a )
|0⟩ .
Nsite i i

(3.25)

(B)
0.25

0.20

zJ/U

0.15

0.10

n=1
n=2

0.05

n=3
0.00
0.0

1.0

2.0

3.0

μ/U

Figure 3.4: Quantum phase transition. (A) Phase diagram plotted as a function of the chemical potential µ and a scaled tunneling coeﬃcient zJ where z
is a number of the nearest neighbor sites, both are scaled by the interactionenergy U . The colored areas are Mott-insulating regions whereas the whitearea is the superfluid region. Mott-insulating regions are limited to integer
n which is an atom number per site. (B) Quantum phase transition in theexperiment. By ramping up and down the confined lattice in all 3 axes, thesystem experiences superfluid-Mott insulator-superfluid transition. Red andblue dashed line above the time of flight image roughly remarks the regime
where the system is, corresponding to the phase diagram in (A). For the superfluid regime, the data (left and right) are taken 15ms before (and after)
the lattice depths reach the maximum (middle, s⊥ = 18Er , sz = 10Er ) which
gives about half of maximum lattice depths.
Fig. 3.4 (A) is a phase diagram plotted as a function of the chemical potential µ and a scaled tunneling coeﬃcient zJ, both are scaled by the interaction
33

energy U . Here z is a number of the nearest neighbor sites which is 2 in a
1D system. The Mott-insulator region is colored in grey, and each lobe corresponds to the the integer number n which is an atom number per each site.
The rest area of white represents the superfluid regime.
In the case of a trapped BEC, there is a mean-field potential that is calculated from the Gross-Pitaevskii equation as [141]
!ωho
µ0 =
2

"

15N asaho#2/5(3.26)

where N is the total atom/number, ωho = (ωx ωy ωz )1/3 is the mean harmonic
trapping frequency, aho = !/mωho is the mean harmonic oscillator length.
For a large number of atoms where the kinetic energy (∝ N ) is negligibly smaller than the interaction energy (∝ N 2 ), we can use ‘local chemical
potential ’
1µ(z) = µ0 − m(ωz)2(3.27)
2
where the chemical potential is given by the peak density of condensate, whichis called local-density approximation or Thomas-Fermi approximation. Thisapproximation is generally valid in the center region, but fails at the boundaryof the atomic cloud where sharp density profile exists.
Fig. 3.4 (B) illustrated a typical experimental observation of the superfluid → Mott-insulator → superfluid transition, which is done by changing
the eﬀective tunneling coeﬃcient J by ramping up and down the confining
lattice depths in all 3D axes. A time of flight image is taken which gives thecoherence of the system from the diﬀraction peaks. In the superfluid regime,
the diﬀraction pattern displays clearly visible peaks. In contrast, in the Mottinsulator regime, the incoherence of the wave function from each site averages
out the diﬀraction and what left is a diﬀuse blurb centered around 0 in momentum space. While the transition can also be seen by only changing thelattice depth in 1D, here we changed all 3 lattices to observe clear diﬀraction
in both horizontal and vertical axes, following the sequence in [83]. When thelattice depth is low in the beginning (B, left), the wave functions on each site
are still coherent and the system is in the superfluid regime, which is depicted
as a blue dashed line in a phase diagram fig. 3.4 (A). As the lattices get deeper
and all wave functions are tightly confined in each site, the time of flight image
becomes diﬀuse due to the loss of coherence (B, middle). This is the case of
the red dashed line in (A); a deep lattice decreases the tunneling coeﬃcient J
which displaces the blue line down to the red line as depicted. As the lattices
get shallower again, the system recovers coherence (B, right) and as a result,34

the time of flight pattern is again comparable with the initial state. Note herethat the system in phase diagram is represented as a line, which is due to theinhomogeneity of the system.
In general, it is relatively easy to control the total atom number, but it is
impossible to achieve uniform µ over the size of the sample due to the harmonic
optical trap. Thus depending on the structure of the chemical potential, thedensity distribution of the atom gives well-known ‘wedding-cake’ shape [50,142, 143]; highest density with integer n at the center and layered by lower nto the side and they give plateaus (Mott regime), meanwhile the space between
layers is filled by a smoothly varying density (superfluid regime).
The theoretical concepts described in this chapter are applied to analyze
experimental results in Chapter 5. Especially fig. 3.4 is directly related to thephase transition that is experimentally observed.

35

Chapter 4
The matter-wave platform
To discuss the main topic of this dissertation on the formation of matter-wave
polaritons in chapter 5, it is crucial to understand our matter-wave platform.
This chapter gives a short review of the approach and detail results detailed
in our publication Spontaneous emission of matter waves from a tunable open
quantum system, Ludwig Krinner, Michael Stewart, Arturo Pazmiño, Joonhyuk Kwon, Dominik Schneble, Nature, 559, 589 (2018) [86] (sections 4.3
and 4.4) and our publication Dynamics of matter-wave quantum emitters ina structured vacuum, Michael Stewart, Joonhyuk Kwon, Alfonso Lanuza, Dominik Schneble, Phys. Rev. Res. 2, 043307 (2020) [87] (sections 4.3 and 4.5).
Details can be found in the PhD theses of Ludwig Krinner [113] and Michael
Stewart [115].

4.1

Introduction

The spontaneous photon emission into the vacuum due to the decay of an
excited atom is very sensitive to the density of states of the radiation field.
Depending on whether the emission is into free space or structured vacuum,
the emission behavior can be dramatically diﬀerent. Here, one of the mostinteresting phenomena is the formation of the atom-photon bound state [107,144, 145] in photonic bandgap systems [90, 146, 147].
We demonstrated a platform that is analogous to an atom in a photonic
bandgap material, and in which the photon is replaced with a matter-wave.
With our ultra-cold atom system, we can tune the vacuum coupling and excitation energy freely, providing us with a direct measurement of the atomic
bound-state. Moreover, depending on the relative amplitude between thevacuum coupling and the coupled bandwidth, the system exhibits the transition from Markovian decay to the fully non-Markovian dynamics, which

36

Figure 4.1: Schematic illustration of the quantum-emitter platform [86]. The
quantum emitters (gray wells) contain |r⟩ (red) atoms that are tightly confined with the state-selective optical potential. With a microwave coupling(coupling strength Ω, detuning ∆), |r⟩ atoms undergo a transition to theemitted matter-wave states |b⟩ (blue), which are free to propagate along the
1D tube. The momentum distribution of emitted |b⟩ atoms is observed in timeof flight image as shown in figure, depending on the excitation state energy ofthe quantum emitter. There is a natural left/right symmetry in the momentum space.Illustration adapted from Krinner et al., Nature 559, 589 (2018)[113].
gives the demonstration of the transition from spontaneous decay as the Rabioscillation.

4.2

Theory: Weisskopf-Wigner model

Spontaneous emission, leading to the irreversible decay in quantum mechanics,
can be described by the Weisskopf-Wigner Model [89]. In quantum optics, weconsider a two-level system characterized by the excited and the ground statethat is coupled to the vacuum. The atom starts out with the excited state |e⟩and spontaneously decays into |g⟩. Here we review the standard description,
and we will make connection to our system. The detailed approach is also welldescribed in textbooks such as [148, 149] and in the PhD thesis of L. Krinner[113].
To begin with, we start with the Hamiltonian of a particle under the clas-

37sical electromagnetic field as(p− qA)2Ĥ =+ V (r)2m(4.1)

where q is the particle charge, A is the vector potential and V is the scalarpotential. With the assumption of the Coulomb gauge (∇ · A = 0), and for asingle-electron atom, i.e. q = −e, we have our HamiltonianĤ = Ĥ0 +e
A · p = Ĥ0 + ĤI
m

(4.2)

2

p
+ V (r) is the atomic and ĤI is the interaction Hamiltonian.
where Ĥ0 = 2m
Under the dipole approximation (k · r ≈ 0) and the assumption that only two
states are coupled, we can indeed obtain the transition matrix element of ĤI[149] asĤI = −i ⟨g| d̂ |e⟩ εk,s E(4.3)

in two-level system where εk,s is the polarization vector of the photon and∂E = − ∂t
A is the classical electric field.
Now we can apply the above with the Hamiltonian of two-level system withthe excited state |e⟩ and the ground state |g⟩ in a quantized electromagnetic
vacuum,%1
Ĥ = !ω0 |e⟩ ⟨g| +!ωk (b̂†k,s b̂k,s + ) + ĤI
(4.4)
2
k,s

where !ω0 is the excited state energy, !ωk is the energy of the photon with thewave-vector k, s is the polarization state and b̂† (b̂) is the creation (annihilation) operator of a single excitation (i.e. photon) with given wave-vector andpolarization. Here the last term of 4.4 is again the interaction Hamiltonainthat couples the excited atom and the excitation (photon) mode as%ĤI =(−!gk,s b̂†k,s |g⟩ ⟨e| + H.c.).
(4.5)
k,s

Here gk,s is the interaction strength between the excited state and the groundstate, which is given by [149]gk,s = −iE0 εk,s ⟨g| d̂ |e⟩(4.6)/
with the (zero-point) vacuum field E0 = !ωk /2ϵ0 V , where ϵ0 is the permittivity of free space and V is system volume.
The interaction Hamiltonian 4.5 provides the basic description of spon-

38

taneous decay under the Weisskopf-Wigner Hamiltonian; by de-exciting theexcited state into the ground state, an excitation distributed over all accessible modes is generated by the creation operators b̂†k,s . This can be seen moreclearly by the general solution of the time-dependent Schrödinger equationi!∂t |Ψ(t)⟩ = Ĥ |Ψ(t)⟩ as%|Ψ(t)⟩ = A(t)e−iω0 t |e, 0⟩ +Bk (t)e−iωk t |g, 1k,s ⟩
(4.7)
k,s

where |e, 0⟩ is the state having excited atom without photon and |g, 1k,s ⟩ isthe state having ground state with a single photon with given wave-vector
and polarization. Putting this ansatz into the Schrödinger equation gives thefollowing equation directly%∗Ȧ(t) = igk,se−i(ωk −ω0 )t Bk,s (t)
(4.8)
k,s

Ḃk,s (t) = igk,s ei(ωk −ω0 )t A(t)(4.9)

with the initial condition A = 0|t=0 and Bk = 0|t=0 which means that thesystem starts with the initially excited atom state.
Details for obtaining the analytic solution above is well-described in [150]and in the PhD thesis of Michael Stewart [115]. In the usual Markovian
approximation, we assume that A(t) in eq. 4.8 varies slowly in time whichenables to take A(t) out of the time-integral while solving analytically, and it
gives an exponential decay.

4.3

Experimental platform

Two of our prior publications that we will briefly review in the following sections (sec.4.4, sec.4.5) share almost the same experimental platform. Most of
the universal features of our matter-wave emission experiments were alreadydescribed in [86]. Here we review some of the special experimental techniquesthat make our measurements possible.

4.3.1

Magnetic field stablization

Magnetic field tagging [151]Changes of magnetic field directly aﬀect the spacing between diﬀerent mF
sublevels and the absolute detuning which indeed changes the excitation energy. We thus need to characterize and stabilize the field as much as possible.
39

(A)

|2, 0>|1,-1>

(B)
F=2

F=1
-2

-1

0

1

2

Figure 4.2: Magnetic field tagging. (A) Time of flight images of 2−way fieldtagging. Starting from |1, −1⟩ the target experiment transfer atoms to |2, 0⟩,
and the leftover population at |1, −1⟩ state is used as a field tag by transferring
to |2, −1⟩ and |2, −1⟩ states. (B) Schematic illustration of 2−way field tagging
in energy level diagram.
Generally, one of the most direct ways to characterize magnetic fields at theposition of the atoms is to use the atomic cloud itself as a field sensor. For an
optically trapped BEC, we measure the Rabi spectrum. By taking a numberof (typically 6) diﬀerent data points around the resonance frequency, we cancalculate and fit the Rabi frequency of the coupling and corresponding magnetic field. The shortcoming of this method is, however, that the measurement
is not on-time with the experimental data. Since each of this measurement is
destructive, there is a temporal gap about 30 × 6 = 180s between when thedata is taken and when the calibration is done. Thus we would better know
the exact magnetic field when the data has taken, and the magnetic field tagging technique (described in [151] and L. Krinner’s thesis in detail) provides
it.
Fig. 4.2 shows the mechanism of 2−way magnetic field tagging. When theexperiment (between |1, −1⟩ and |2, 0⟩) is done, the information we generally
want to obtain for the experiments in this section is mostly the populationof each state and the momentum distribution of the emitted “blue” (|2, 0⟩)
atoms. Thus we are free to control the “red” (|1, −1⟩) atoms as long as thetotal population information is conserved, and this is a good condition to
adopt a modified field-tagging method [151] that only uses a transition fromtwo states instead of 6. By sending |1, −1⟩ atoms into |2, −1⟩ and |2, −2⟩states with microwave coupling immediately before the time-of-flight imaging,
we can obtain the information of magnetic field at the moment (more precisely,40

about 100µs delayed) of the measurement. The assumption that we should
consider here is that this concise version of field-tagging is more fragile on thelarge magnetic field fluctuation, since the fitting basically assumes that themagnetic field moves with in the right shoulder of the Rabi spectrum for eachtransition (cf. Figure 1(d) of [151]). Thus once the assumption fails due tothe large scale of field fluctuation, it is easy to be misguided from the fieldtagging.
Magnetic field following
In general, it is very easy to overlook the noise that comes from the 60Hz110V power source for electronics. In our experiment, however, the magnetic
field that is induced by this 60Hz noise is critical. We typically maintain ourbias magnetic field of 5G in order to supply the quantization axis, and thefluctuation of input source 60Hz gives the fluctuation of the magnetic field up
to mG order.
One good thing that we have is that the noise is given with the harmonicof 60Hz and all the electronics in the lab share the same power sources, whichmeans that all those noises are in phase. Thus by triggering the noise so thatmatching the phase perfectly in 180◦ , the system always experience the same
fluctuation which is easy to handle by sending our the compensating magnetic
field from the small Helmholtz configuration coil. Again, details can be foundin the PhD thesis of Ludwig Krinner [113].

4.3.2

Preparation of atomic quantum emitters

This subsection will explain how we prepare the atomic quantum emitter thatis an analogue to a photonic two-level system that emits matter waves insteadof the photon. The schematic diagram is given in fig. 4.3. We start withtwo diﬀerent ground states, |r⟩ = |1, −1⟩ and |b⟩ = |2, 0⟩ of 87 Rb atoms. By
coupling them with the microwave of ∼6.8GHz, with the coupling strengthΩ and detuning ∆, we can control the relative energy level of two states in
the co-rotating frame as shown in fig. 4.3. We then apply the state-selective
potential that only traps the |r⟩ state, so that the red atoms are confinedin each site of the optical lattice along the z-axis while the blue atoms arefree to move. Then in terms of the energy levels, the microwave couples theconfined |r⟩ states to free |bk ⟩ states with momentum k and kinetic energy
ϵk = !2 k 2 /2m.
In practice, starting from an optically trapped BEC of |1, −1⟩, we ramp
up the optical lattice in all three dimensions simultaneously for 90 ms. For
x-y plane, we use far-detuned optical lattice with λ = 1064nm which confines
41

(A) Bare States(B) Rotating Frame(C) State-dependent O.L.

Figure 4.3: Realization of atomic quantum emitters (A) The bare hyperfine
states pair (ground states |r⟩ = |1, −1⟩ (red) and |b⟩ = |2, 0⟩ (blue)) of 87 Rb ischosen. Both states are confined in tubes along z-axis. (B) With ∼ 5G bias
magnetic field, microwave coupling ωµ ∼ 6.8 GHz, variable coupling strengthΩ with detuning δ is applied. In the rotating frame, |r⟩ is detuned from |b⟩ byan amount δ. (C) The state-dependent optical lattice is applied along z axis,
and only the red state is trapped with this lattice. The zero-point energy of |r⟩
atoms ω0 /2 shifts δ to ∆ = δ + ω0 /2. The microwave then couples trapped |r⟩states to free |b⟩ states with a continuum of momentum states |bk ⟩ and kinetic
energy εk with free dispersion relation. Illustration adapted from Krinner, etal., Nature 559, 589 (2018) [86].
both red and blue atoms with same potential, s⊥ = 40Er in our case, whereEr,⊥ = h2 /2mλ2 ∼ h × 1.98 kHz. For the z-axis, we use state-selective latticewith λ = 790.0nm that only traps red with sr = 30Er , here Er,z ∼ h × 3.66
kHz. Then the system is in a deeply confined Mott-insulating regime.
After loading the |1, −1⟩ atoms, we send most of the atoms (∼ 0.82) tothe |2, 1⟩ state by using two-photon process with RF-pulse, and then blow thetransferred atoms away by illuminating the resonance imaging light on the D2
cycling transition. This process gives a stochastically populated lattice withan average site occupation of < ni >! 0.5, which means that every next site is
emptied out. Thus the experiment is eﬀectively in a single-particle regime thatfor which the eﬀects from the mean field or collective motion are not relevant.

4.4

Matter-wave decay in the continuumThis section reviews the experimental results based on our publication Spontaneous emission of matter waves from a tunable open quantum system, Ludwig Krinner, Michael Stewart, Arturo Pazmiño, Joonhyuk Kwon, Dominik

42

Schneble, Nature, 559, 589 (2018) [86].

4.4.1

Spontaneous emission

Here, we will see how the system we describe here is an analogue to the spontaneous emission from the Weisskopf-Wigner Hamiltonian described in theprevious section. We start from the interaction Hamiltonian of vacuum Rabi
coupling of two states |r⟩ and |b⟩ under the state-selective potential as described above,%ĤI =!gk ei∆k t r̂b̂†k + H.c.
(4.10)kwhere !gk = !Ωγk /2, ∆k = !2 k 2 /m with the consideration of zero-energy shiftof ∆ = δ + ω/2, and γk = ⟨k|ψe ⟩ is the Frank-Condon overlap factor. Here wecan describe the annihilation operator r̂ as r̂ = |g⟩ ⟨e| since the trapped red
atom turns into the free-propagating blue atom. Then we have
ĤI =

% !Ω
k

2

γk ei∆k t b̂†k |g⟩ ⟨e| + H.c.

(4.11)

which represents the Weisskopf-Wigner Hamiltonian of spontaneous emission.
The experimental results in fig. 4.4 (A) confirm that our atomic quantum
emitter system reproduces the spontaneous emission as expected; the normalized population of the red atom decays exponentially in time-domain as itsupposed. There is an oﬀset, however, that the population decays not all theway down to 0 but to 0.5, which we think is the eﬀect of the neighboring empty
sites. The emitted blue atom can always be re-absorbed, which gives the average 0.5 red atoms in the end. This was tested by comparing the theoreticalmodels for 1-site and 3-site emitter cases as shown in an inset of fig. 4.4 (A) .
With the presence of empty neighbor sites that act as ground state emitters,
the simulation gives both the oﬀset and the small oscillations which is the signof re-absorption.
The time scale of the decay,/ dynamics as√ τ , is given by the Markovian2∼ 1/Γ = 2ms, where Γ = Ωk̄ / ω0 ∆. Here, Ωk̄ = Ωγk̄ and k̄ = 2m∆/!.
We can also see how the emitted blue atoms behave. As for a two-level
photonic system, the emitted “blue” atom in our case carries away the eﬀectiveenergy diﬀerence between the quantum emitter states, ∆. Fig. 4.4 (B) showsthat the observed momentum distribution of the emitted blue atom which
agrees with the single-particle dispersion relation that is parabolic.

43

(A)

(C)

(B)

(E)

(D)

Figure 4.4: Results of Krinner et al. [86] (A) Spontaneous decay in the Markovian regime. The time evolution of the lattice population is shown in red circles. Here the coupling strength Ω = 2π × 0.74 kHz and detuning ∆ = 2π × 1.9
kHz, which gives (quasi-)Markovian regime ((Ω/∆)2 ≪ 1). The red line is our
experimental fit for exponential decay and gray line is theoretical model from
the Markovian approximation (dashed) and the isolated emitter model from[150] (solid). (Inset) Simulated decay dynamics for 1-site and 3-site model.
(B) Momentum space profile of emitted blue atom for the same parametersas (A). The white dashed line is the single particle dispersion, based on theenergy conservation from the positive detuning. (C) Non-Markovian dynamics. Same measurement with (A) for diﬀerent parameters (∆ = −2π × 1.7
kHz, Ω = 2π × 3.0 kHz). The inset shows the results of a numerical simulation. (D) Bound state for negative excitation energy (left), and the separationof the bound state by using adiabatic/non-adiabatic preparation of the system using gravitational sag (right). The bound state is frozen in space, but
the non-adiabatically released fraction rolls down the potential hill along thetube with accelerated non-zero momentum. (E) Momentum distribution of |b⟩
atoms for adiabatic/non-adiabatic preparations. Open (filled) circles represent
the sudden (adiabatic) turn-on of the coupling, and triangles show the diﬀerence of the two datasets. The solid line represents the square of the Fouriertransform of the analytical evanescent wavefunction of the bound state.

44

4.4.2

Non-Markovian regime and the bound state

One interesting feature of our system is the tunability of the energy diﬀerence
between the excited and ground states of the atomic quantum emitter.
As the detuning becomes small compared with the coupling strength, ∆/Ω ≪1, the decay dynamics experiences the eﬀect of the edge of the continuum. Instead of the exponential decay as traditionally seen for weak coupling (Markovian: memory-less), the dynamics become oscillatory. This non-Markovian
oscillation qualitatively agreed with the theoretical prediction [150], and wewere able to observe a similar oscillation with higher frequency for negative
detuning as shown in fig. 4.4 (C).
Compared to the fact that it is very natural to visualize the case of positive
detuning case where the emitted matter-wave carries the excess momentum, it
is not straight-forward to imagine the case of negative detuning since emitting
something with the negative energy is not physically allowed physically. Instead of the emission, what is observed is the bound-state, which is a naturalform of the evanescent waves with a binding energy !ωB ≈ !∆, where the negative detuning ∆ = −2π × 1.7 kHz is used. Fig. 4.4 (D) shows the illustrationof the bound-state formation with negative excitation/energy, and this boundstate is localized with the localization length ξ = 1/ 2m|ωB |/! ∼ 142nm.

4.5

Decay in a structured vacuum

This section reviews experimental results from our publication Dynamics ofmatter-wave quantum emitters in a structured vacuum, Michael Stewart, Joonhyuk Kwon, Alfonso Lanuza, Dominik Schneble, Phys. Rev. Res., 2, 043307(2020) [87]. The author of this dissertation contributed equally to taking
experimental data for this publication alongside Michael Stewart, who in addition elaborated on important theoretical aspects in his thesis [115]. A moregeneral theoretical analysis can also be found in a very recent theory paper ofour group (A. Lanuza et al., [152])

4.5.1

Realization of single-band structure

In the previous section, we studied the atomic quantum emitter system that is
analogous to an atom coupled to a photonic band-gap material. The diﬀerenceis, however, that there is only one band-edge in our case (the continuum edge
at zero energy), which is not the same as a real ‘band’. To make a more
realistic case, we extended the previous research toward the single band case.
The experimental realization is a straightfoward modification from theprevious experiment; instead of a state-selective potential, we here apply a
45

Figure 4.5: Illustration of the experimental configuration for implementing aband structure. Here both |r⟩ (red) and |b⟩ (blue) states are confined alongthe 1D tube, but with diﬀerent lattice depths from the state-dependent opticallattice. Red atoms are tightly confined with sr = 20, while blue atoms see avery shallow lattice sb ≈ 2.5, and the coupling between two states brings thesystem into two-level atomic quantum emitter as sec.4.4 with coupling strengthΩ and detuning ∆. Illustration adapted from Stewart, et al., Phys. Rev. Res.
2, 043307 (2020) [87].
merely state-dependent potential by choosing a slightly diﬀerent wavelength(790.4nm), based on Sec. 2.5. This gives a shallow “blue” lattice sb = 2.5Er(where red is still tightly confined with sr = 20Er , and the ground band has
the width of W1 = 0.5Er ≈ h×1.8 kHz. Fig. 4.5 shows a schematic illustrationof the configuration.
Here with the shallow band structure εn,q , the dynamics is governed by the
Weisskopf-Wigner Hamiltonian now involving Bloch waves instead of plane
waves
%
Ĥ =
!gn,q ei∆n,q t |g⟩ ⟨e| b̂†n,q + H.c.
(4.12)
n,q

where n is a band index, q is a quasi-momentum, ∆n,q = ∆ − εn,q /! is theeﬀective detuning of the emitter (excitation energy !∆) from the Bloch state|n, q⟩ = b̂†n,q |0⟩, the eﬀective vacuum coupling gn,q = γn,q Ω/2 contains theFranck-Condon overlap γn,q = ⟨n, q|ψe ⟩.
Depending on the phase between the red and the blue lattice, the coupling
to the band structure that we can spectroscopically access is diﬀerent due
to the Frank-Condon overlap factor between the wavefunction in the deep
lattice and the momentum modes in question. As shown in fig. 4.6 (A, B),
46

(A)

(B)

(C)

in-phase(D)

out-of-phase

Figure 4.6: Realization of band structure for emission of matter waves. (A)Band structure εn,q and corresponding Frank-Condon overlap γn,q = ⟨n, q|ψe ⟩for the case that the periodicity of the red lattice and blue lattice are in-phase.
The higher band is suppressed from the Frank-Condon overlap factor due tothe parity. (B) Momentum spectrum of the emitted matter wave. (C) Samemeasurement as (B) but for the case of an out-of-phase lattice, as depicted inbottom. Here we set the wavelength λ = 789.8nm to put the blue lattice asdisplaced (see sec. 2.5). Strong emission is measured in the first excited band.
(D) Corresponding Frank-Condon overlap γ for (C). Illustration reproduced
from Stewart, et al., Phys. Rev. Res. 2, 043307 (2020) [87].
for sb = 2.5Er (in phase), the emission into the ground band is much stronger
than that into the excited bands, as the Frank-Condon factor γn,q for thosebands is strongly suppressed. The suppression of the first excited band (orhigher even n-th band) is due to the odd parity of the relevant Bloch states,
and the suppression of the next excited band is caused by the low density ofstates and finite momentum width of the excited wave function.
By shifting the sign of the state-dependent lattice potential, we can set
the two lattices out of the phase which gives the negative eﬀective lattice toblue. Fig. 4.6 (C) shows the momentum distribution and the corresponding
Frank-Condon overlap for sb = −2.6Er (out of phase), where the emission into
the excited band is significant compared with the other case. This also agrees
47

with the calculated γn,q as in fig. 4.6 (D).
(B)

(A)

q/k

q/k

Figure 4.7: Single band emission spectrum. (A) The same measurement offig. 4.6 (B) but with a smaller step size for better energy resolution (0.1Er ):the dashed line in the zoom in is theoretically calculated band structure. (B)Theoretical prediction of the momentum distribution with the considerationof experimental Gaussian blurs with σE = 0.1Er and σq = 0.15k for energy
and quasi-momentum, respectively. The results are compatible with the measurement in (A). Illustrations are reproduced from Stewart, et al., Phys. Rev.
Res. 2, 043307 (2020) [87].
In our experiment, by limiting the state-dependent lattice as positive for|b⟩ atoms, we were able to safely focus the system into the single-band casewithout the concern of a contamination from the higher band. The widthof our ground band is ∼ 0.5Er ≈ h × 1.8 kHz. Fig. 4.7 (A) shows theemission spectrum focused on the single band with finer resolution (0.1Er ).
From the theoretical calculation of |Bq (τ )|2 , we calculated the quasimomentum
distribution for the comparison as shown in fig. 4.7 (B). Here we blurred theGaussian for the better comparison with the real measurement, by σE = 0.1Er
in energy and σq = 0.15k in quasimomentum.

4.5.2

Transition from Markovian to non-Markovian dynamics

Since a single-band scenario is accessible now, we can play with the competingparameters, the coupling strength Ω and the bandwidth 2ω̄ for the sinusoidal
band ε(q) = −!ω̄ cos(qπ/k) + !ω̄. We measured the time dynamics of the
population as in the previous experiment but by tuning the excitation energy
to the center of the band and changing the vacuum coupling versus bandwidth
ratio g/ω̄. The decay here is incomplete due to the presence of the bound state.
Fig. 4.8 (A) shows that the decay dynamics display a transition from theMarkovian to non-Markovian limit. In the case of weak coupling g/ω̄ ≪ 1 as
48

(C)

(A)

(B)

Figure 4.8: (A) Transition from Markovian to non-Markovian time dynamics
at the band center ∆ = ω̄ for various couplings. (1) weak coupling case Ω =2π × 0.4 kHz (g/ω̄ = 0.18) (2) intermediate-weak coupling case Ω = 2π × 1.0
kHz (g/ω̄ = 0.43) (3) intermediate-weak coupling case Ω = 2π × 2.3 kHz(g/ω̄ = 1.0) (4) strong coupling case Ω = 2π × 2.2 kHz (g/ω̄ = 4.9). The error
bar represents the standard error of the mean. The gray curves are theoreticalprediction from isolated-emitter model, and the bold line is the regime where
the model is valid. (B) Illustration of the relation between g and ω̄. Here ω̄
is in the order of the tunneling between the blue lattice sites. (C) Groundband emission spectrum for intermediate coupling case (A(2)) for τ = 400µs,
obtained from fig. 4.7 (A) by summing over quasi-momenta. The dashed lineis the isolated-emitter model, and the solid line is the same model but reduced
by 40%. Illustration is taken from Stewart, et al., Phys. Rev. Res. 2, 043307(2020) [87].
shown in Fig. 4.8 (A) (1), the data approximates Markovian decay which isirreversible. As the coupling gets stronger g/ω̄ ∼ 1, we observed a damped
oscillation (see (A) (2, 3)), and the strong coupling g/ω̄ ≫ 1 gives undamped
oscillation (A) (4). The two extreme cases (weak and strong coupling) represents the spontaneous decay and vacuum Rabi oscillation in cavity-QED limit,
and the intermediate coupling is the crossover between those cases.
We also can predict the time dynamics of the excitation amplitude theo-49

retically byiA(t) =2π+ ∞dωG(ω + i0+ )ei(∆−ω)t(4.13)−∞

where G(ω)= 1/[ω − ∆ − Σ(ω)/!] is the Green’s function and Σ(ω) =/2−i!g / ω(2ω̄ − ω) is the self-energy. We can note here that the singularity lies on ω = 0 and ω = 2ω̄ where are the edges of the single band as
measured. More details can be found in the publication [87] and M. Stewart’s
thesis [115].
There is a qualitative agreement between the experiment and the singleemitter model (eq.4.13), which is drawn as gray line in figure. However, there
is still a deviation and it gets bigger as the system goes more to the nonMarkovian side. This is due to the re-absorption from the neighboring empty
sites. This deviation can be reduced by considering a multi-emitter model.
This was confirmed recently by our group’s new preprint [152] (work spearheaded by A. Lanuza) which considers array and bandstructure eﬀects andsuccessfully recovers most of the observed behavior.

4.5.3

Bound-state in a band-gap

Achieving a single-band structure provides a natural benefit; a new band edge
above the band, which didn’t exist in previous experiment. This is indeed
natural to discuss how is bound-state formed in each band gap and how theyaﬀect the system dynamics.
With the same steps as in [150], we found a bound state of the form
2
3
+ k
0 ±1
|g; 1, q⟩0ψ = N |e; 0⟩ + g
dq(4.14)
B
2k −k ωB± − ω(q)where N is a normalization constant, ωB± is the bound state energy, ω(q) =ε(q)/!. Here, |e; 0⟩ is a lattice site occupied by a red atom (zero momentum),and |g; 1, q⟩ is an emitted blue atom (and empty lattice site) with momenta q.
Note here that we now can apply the above-band bound state case. For theupper bound state, ωB± −ω(q) goes to zero near q = ±k so that the amplitude of
the quasi-momentum probability diverges at two finite quasi-momenta that are
not zero. Fig. 4.9 shows the experimental results which confirms the theoreticalprediction. Here, we adiabatically ramp up the microwave coupling in orderto minimize the contamination of the momentum space from the gravitationalsag eﬀect.
For the bound state below the band, the case is exactly similar with oneedge case in previous section; the momentum distribution of the bound state

50

(A)

(B)

Figure 4.9: Bound states in bandgap. (A) Momentum distribution of thebound state below the band edge, !∆− = −0.5Er . The bound state is preparedby adiabatically ramped up microwave coupling over 2ms with the maximum
coupling g/ω̄ = 0.43 (Ω = 2π × 1.0kHz). The colored bar is the density
plot that is averaged from the time-of-flight image. The gray curve is theGaussian blurred quasi-momentum distribution of the inset in a same way offig. 4.7 (B). (Inset) Calculated quasi-momentum distribution below the bandat !ωB− = −2!ω̄ ≈ h × −1.9kHz. (B) The same measurement but above
the band edge, !∆+ = 1.0Er . The bound state has two peaks unlike thebelow the edge case (A). (Inset) Same calculation as an inset of (A), but with
!ωB+ = 4!ω̄ ≈ h × 3.9kHz. Illustration is reproduced from Stewart, et al.,Phys. Rev. Res. 2, 043307 (2020) [87].
is centered at zero quasi-momentum and single-peaked. For the bound state
above the band, we observed a double-peaked momentum distribution, which
was not previously discussed in the literature. We simulated the momentum
distribution of the bound state in each side with our isolated-emitter model[87, 115] which is shown as inset for both fig. 4.9 (A) and (B). We applied thefinite volume eﬀect and imaging resolution to convolve the toy model into a
realistic prediction, and that is the grey curve in the main plots in fig. 4.9,
which shows qualitative agreements between the theory and the measurement.
In our paper [87], we also discussed the relationship between the boundstates and the observed dynamics. For very strong coupling g/ω̄ ≫ 1 with Rabi
dynamics, each atom oscillates between the “red” and the “blue” lattice, suchthat the oscillation can be interpreted as resulting from their dressed states,
where the spatial wavefunctions are the Wannier state of the corresponding
lattice potential. For weaker couplings g/ω̄ ≈ 1, the blue Wannier functions(which have an equal contribution of all Bloch states) are replaced by theevanescent waves of the bound states.

51

Chapter 5
Formation of matter-wave
polaritons
This chapter reproduces our publication Formation of Matter-Wave Polaritons
in an Optical Lattice, arXiv:2109.02243 (2021), with co-authors Y. Kim, A.
Lanuza, and D. Schneble [88]. Most text and figures are quoted verbatim fromthe publication, and the author of this dissertation gratefully acknowledges
the contribution of his co-authors. Additional figures are marked with anasterisk (*) in the figure caption. D. Schneble, the author of this dissertation,
and Y. Kim designed the experiments. The author and Y. Kim took themeasurements. Data analysis was performed by the author, Y. Kim and A.
Lanuza. Theoretical modeling was done by A. Lanuza. The results werediscussed and interpreted by all authors. Figures were created by the authorand A. Lanuza. D. Schneble supervised the project. The manuscript was
written by the author and D. Schneble with contributions from A. Lanuza andY. Kim.

5.1

Introduction

The polariton, a quasiparticle formed by strong coupling of a photon to amatter excitation, is a fundamental ingredient of emergent photonic quantum
systems ranging from semiconductor nanophotonics to circuit quantum electrodynamics [102, 153]. Exploiting the interaction between polaritons has led
to the realization of superfluids of light [96, 101] as well as of strongly correlated phases in the microwave domain [154], with similar eﬀorts underway for
microcavity exciton-polaritons [97]. Here, we develop an ultracold-atom analogue of an exciton-polariton system in which interacting polaritonic phases
can be studied with full tunability and without dissipation. In our optical-

52

lattice system, the exciton is replaced by an atomic excitation, while an atomic
matter wave is substituted for the photon under a strong dynamical coupling.
We access the band structure of the matter-wave polariton spectroscopicallyby coupling the upper and lower polariton branches, and explore polaritonic
many-body transport in the superfluid and Mott-insulating regimes, finding
quantitative agreement with our theoretical expectations. Our work opens up
novel possibilities for studies of polaritonic quantum matter.
Since their first description as a superposition of light and matter excitations [91, 92], polaritons have been an essential key for understanding thenature of strong light-matter couplings in solids. While the polariton concept[93] has been broadly expanded to various fields and contexts [95], its essenceis captured by the microcavity polariton, whose realization in a semiconductor[94] first revealed exciton-photon mode splitting as the signature of a mobile,
dynamically-coupled quasi-particle with a dispersion relation that hybridizes
those of its two constituents.
Polariton interactions, mediated by the excitation, open up novel possibilities for engineering eﬀective photon-photon interactions that can leadto Bose-Einstein condensation [96, 101] and strongly interacting many-bodystates [102, 154]. In this context, several polariton platforms including semiconductor microcavities [153], Rydberg polaritons [101], as well as waveguide[108] and circuit [102] quantum electrodynamics, provide breakthrough possibilities for quantum simulations [103, 155]. Here, it is an important challenge
to achieve strong coupling [145] and interactions [156] with tunable parameters
while managing dissipation [154] and assuring scalability [157].
In this chapter, using a recently developed experimental approach [86, 87]with ultracold atoms in an optical lattice, we introduce a scalable polariton
platform featuring full flexibility and no intrinsic losses, in which the photonic
constituent is replaced by a matter wave. While replicating the fundamental
features of conventional polaritons [95], the ratio of eﬀective mass and interactions of these novel matter-wave polaritons is fully tunable, and the system
is dissipation-free with an infinite Purcell factor. By controlling the parameters in the regime of interest, our system can simulate polaritonic manybody
properties in the regime of strong interactions [99, 100, 158].
In analogy to an exciton-polariton, we realize a quasiparticle that hybridizes the quadratic dispersion relations of two constituents of disparate
mass. While the light constituent playing the role of the photon in the microcavity is a free-space atomic matter wave, the other constituent, replacing
the exciton, is an atom with high eﬀective mass induced by an optical lattice,that is coupled to the matter wave via an eﬀective dipole moment. The coupling hybridizes the two dispersion relations into lower (LP) and upper (UP)

53

a

b
z

c

(

,

)

z

Figure 5.1: Experimental scheme and polariton formation. a,b, An|r⟩ (red) atom in the ground band of a deep state-selective optical lattice (red)is coupled through a microwave field (green) with strength Ω and detuning∆ < 0 to an unconfined state |b⟩ (blue) with energy below the edge of thecontinuum of motional states. The atomic ensemble is confined to a systemof tightly confining tubes created with state-independent transverse lattices(gray). c, In each tube, the coupling results in a polaritonic superposition oflattice-trapped atoms and matter-wave modes with quasi-particle hopping J¯and on-site interaction Ū .
polariton branches. The optical lattice confining the atom not only increases
its eﬀective mass but also provides an analog of strongly-correlated polaritonicarrays [153], in which the interaction between excitons is replaced by the natural collisional interaction between the heavy atoms in the lattice. We createa system of strongly-interacting polaritons, in which the competition between
hopping and interactions lead to a tunable phase transition from a polaritonic
superfluid to a Mott insulator, in agreement with expectations based on thepolariton band structure [152, 158].

5.2

Experimental setup

The scheme for our experiments is illustrated in Fig. 5.1. Using an opticallytrapped Bose-Einstein condensate of around 104 87 Rb atoms in the hyperfine

54

ground state |r⟩ ≡ |F = 1, mF = −1⟩, we load an array of 103 red-detuned,tightly-confining optical lattice tubes with depth s⊥ ≫ 1 measured in termsof the recoil energy Er⊥ = h2 /2mλ2⊥ , where λ⊥ = 1064 nm is the lattice laser
wavelength and m the atomic mass. Another lattice (“z-lattice”) with variable
depth s, in terms of a corresponding recoil energy Er = !ωr at wavelength λ,additionally confines the |r⟩ atoms along the vertical z axis aligned with thetubes. The |r⟩ atoms hop between sites of the z lattice with finite tunnel
coeﬃcient J, while transport across tubes is negligible on experimentally relevant time scales. The atoms are coupled to a second hyperfine ground state,|b⟩ ≡ |F = 2, mF = 0⟩ via a 6.8 GHz microwave field of strength Ω and negative detuning ∆ from the atomic resonance in the z-lattice, which is shifted by
the diﬀerence in zero-point energy. The wavelength λ = 790.0 nm and polarization (σ − ) of the z-lattice (for which Er = h×3.67 kHz) are chosen such that
it is fully state-selective and |b⟩ atoms do not experience its potential at all,and can freely move along the tubes (for times smaller than 2π/ωz ∼ 10 ms)
occupying a continuum of modes |p⟩ with momentum p.
As observed in [86], for the radiative decay of an atom from a lattice well,
the coupling to the free modes for ∆ < 0 induces the formation of/a bound state
containing an evanescent |b⟩ matter wave with decay length ∼ !/2m|∆|. If
now the evanescent wave starts to leak into neighboring sites, the coupling
between the two components can induce an eﬀective tunneling of the |r⟩ atom,and since the evanescent wave remains localized around the atom, this process
then corresponds the hopping of a quasiparticle. In a many-body system, suchmatter-wave polaritons will be characterized by an eﬀective tunneling matrix
element J¯ and an onsite interaction Ū . Here we explore the signatures ofpolariton formation in a Bose-Hubbard scenario in which the overall state of
the system is tuned via the ratio between tunneling and onsite interactions.

5.3

Measurement of the onsite-interaction energy

To explore the eﬀects of the vacuum coupling, we first measure the excitation
spectrum deep in the Mott regime with a small ratio J/U ∼ 5 × 10−3 between
atomic tunneling and onsite interactions. The procedure is summarized inFig. 5.2a and b. After preparing the system at sz = 14 and s⊥ = 40 (whereU = h × 1.7 kHz), we exponentially ramp up Ω at fixed ∆ on a time scaleτ ≫ 1/|∆| that is long enough to preclude nonadiabatic shedding of matterwaves [86]. With the coupling Ω applied, we sinusoidally modulate s⊥ atvariable frequency ω̄ in order to induce resonant excitations of the gapped

55

a

b

2π

π-shift

14

600μs

10ms

0

-

0

d
(i)

(a.u.)

TOF

15ms

0

0.4

0.5
0
1

(ii)

0
0
0

2
4
modulation frequency

6
(kHz)

polariton peak position (kHz)cpeak width

-

50μs

5ms

2.5ms

2

therm.

3
2.5ms

z

6

5

4

3
0.5 0.75 1 1.25

Figure 5.2: Excitation spectra in the Mott regime (s⊥ = 40, sz = 14).
a, Excitation: after ramping up Ω, s⊥ is modulated by ±30% at variablefrequency ω̄ for 10 ms. b, Detection sequence involving rethermalization anda gravitational π-phase shift for the |r⟩ atoms after the coupling is turnedoﬀ (details see sec. 5.7). The two panels display the |r⟩ and |b⟩ momentum
distributions measured after resonant excitation at detuning ∆/ωr = −1.00(7)and coupling Ω/ωr = 1.09(2). The width σ̃ is extracted through Gaussian fits
to the 1D density along z. c, Excitation spectra (i) for ∆, Ω as in b, with
the inset stripe showing the number of |b⟩ atoms; (ii) for the uncoupled case
Ω = 0 (red) and additionally sz =10 (gray). d, Detuning dependence of thethird excitation feature. Open red circles denote the |r⟩ peak position, and blue
dots denote the position of the maximum transfer of |b⟩ atoms, as extracted
from Gaussian fits, with error bars giving the spectral width of the excitation.
The green curve is the calculated polariton excitation energy ω̄(∆, Ω), where
the shaded areas include the width of the ground band (darker green) and theuncertainties in ∆ (±0.27 kHz) and Ω (±2%) (lighter green).

56

Mott phase [159] (we note that, while varying s⊥ changes the polariton onsite
interaction Ū , it does not directly aﬀect the single-polariton feature ∆). After
ramping Ω back down, we enter the superfluid regime in which we characterize
the eﬀects of the modulation, which are in the form of a reduction of coherence,
via time-of-flight (ToF) measurements.
The eﬀects of the coupling on the excitation spectrum are shown in Fig. 5.2c.
In a reference run without coupling, we observe excitation peaks near ω̄ = U/!
and 2U/!, as expected from resonant atom redistribution between sites [160].
The coupling has no noticeable eﬀect on the position of the peaks, suggestingthat for the parameters used (Ω, |∆| ∼ ωr ) the change of the on-site energy issmall. However, an additional feature appears, centered at a larger modulation frequency ω̄ not far from |∆| (see Fig. 5.2d), accompanied by excess |b⟩
population left after the coupling is ramped back down.

5.4

Theoretical approach

In order to discuss these observations further, we first develop a quantitativedescription of the expected polariton features, following the general approachof [158]. We first consider a |r⟩ atom coherently distributed over the sitesof the z-lattice, such that the system Hamiltonian takes the form of a seriesof coupled Weisskopf-Wigner models. By expressing the momentum of thevacuum modes pn,q in terms of a quasimomentum q ∈ [−k, k] and an integer
band index n ≥ 1, the non-interacting part of the Hamiltonian decouples as%Ĥ =εn,q ĉ†n,q ĉn,q ,
(5.1)
n,q

where
ĉ†n,q = αn,q r̂q† +%βn,n′,q b̂†n′ ,q

(5.2)

n′

creates a matter-wave polariton as a (Ω, ∆)-dependent superposition of a |r⟩
Bloch wave and all |b⟩ modes of the same q. The corresponding polariton
dispersion relation εn,q emerges from the equation%
εn,q − !∆q =!2 gn2 ′ ,q / (εn,q − En′ ,q )
(5.3)
n′

(see sec. 5.7 and [152]), where gn,q = Ω ⟨φ0 |n, q⟩ /2 is the coupling strength
between the originating lattice Wannier function |φ0 ⟩ and the free-momentum
mode |n, q⟩ normalized to a Wigner-Seitz cell, En,q = p2n,q /2m is the freeatom dispersion, and ∆q = ∆ + 4J sin2 (πq/2k) a q-dependent detuning from
57

a

(i)

1

d

(ii)

0

Fig. 4 c

4

UP
n=2

0

-1

n=1
1

-1

b

LP

-1

1

0

1

(i)

-1
1

0

-2
0

1

0
-1

0

1

-1

0

c

(ii)

0

1

Fig. 4 b

1

0.5

0

1

0

1

0

Figure 5.3: Polariton band structure, calculated for sz = 10. a, (i)Uncoupled band structure for lattice-trapped atoms in |r⟩ (red curve) andfree atoms |b⟩ (blue curve). (ii) Band structure for a coupling Ω/ωr = 1 anda detuning ∆/ωr = −0.5, with the two lowest polariton bands (n = 1, 2)
shown as purple curves. The green arrow denotes the separation ω̄0 from the
middle of the lower band to the bottom of the upper band. The dashed curves
reproduce the atomic band structure. b, Wannier functions obtained for thetwo scenarios of a, containing |r⟩ (red) and |b⟩ (blue) components, whered = λ/2. c, Fraction of the |b⟩ component |β|2 and polariton interactionenergy Ū as a function of Ω for ∆/ωr = −0.5. d, Polariton ground-bandtunneling coeﬃcient J¯ relative to J as a function of Ω and ∆. The dashedlines correspond to measurements shown in Fig. 5.4b and c.
the continuum edge that accounts for the bandwidth 4J of the z-lattice (fordetails see appendix). The n = 1 band is the LP branch of the matter-wave
polariton, while the bands with n ≥ 2 produce a series of UP branches.
Fig. 5.3a depicts the ground (lower) and first excited (upper) polariton
bands, ε0,q and ε1,q at a typical detuning. In real space, taking an atom local$
ized in site j, the coupling induces, via ĉ†n,j = q ĉ†n,q e−iqzj , the formation of amatter-wave polariton as a localized quasi-particle. Its ground-band Wannier
58

function ⟨z|ĉ†1,j=0 |0⟩ is shown in Fig. 5.3b. As a result of the coupling, the|r⟩ component acquires additional amplitude in neighboring sites, which canbe seen as being due to a re-coupling of the evanescent |b⟩ tail into the latticepotential.
We estimate the eﬀect of the coupling on the on-site interaction in the z4
4lattice by setting Ū ≈ ḡ ∥ ⟨z|ĉ†1,j |0⟩∥ dz, where we take the one-dimensional
collisional strength ḡ as state-independent (for 87 Rb, the diﬀerences are on thepercent level) , such that we capture the dominant eﬀects of the modified spatial overlap between the components. The magnitude of the shift δU = (U − Ū )depends on both the extent of the evanescent tail and its amplitude; resultsfor our parameters are shown in Fig. 5.3c. Consistent with our observation,
the maximum shift δU/U ∼ 0.1 is below what our spectroscopy method cansystematically resolve, as is also evidenced in a reference measurement without coupling in which we correspondingly lowered U by reducing the z-lattice
depth (see Fig. 5.2c).
Based on the polariton band structure, we now discuss the mechanism responsible for the resonant excitation of |r⟩ atoms in the lattice that is accompanied by the appearance of |b⟩ atoms. Polaritons in the lower band are mostly
in the |r⟩ state, which is tightly confined. As they are periodically squeezedin the orthogonal direction, they are subject to a strong perturbation of theirinteraction energy Ū . If this is done resonantly (ω̄ ≈ ω̄0 ), they can be excited
into the upper band, as it is mostly composed of the unconfined |b⟩ state andthus is less sensitive to such perturbations. The third peak in the excitation
spectrum (see Fig. 5.2c) located around the frequency diﬀerence ω̄0 between$
the bottom of the UP band and the center of the LP band, !ω̄0 = ε2,0 − q ε1,q
corresponds to a conversion from the deeply-bound Wannier polariton state toquasi-free states in the region where the density of states is highest. The band
gap increases with |∆|, requiring higher modulation frequencies ω̄ to excite
this resonance (c.f. Fig. 5.2d). Because the UP is dominated by |b⟩, the excitation from the LP leaves |r⟩ vacancies in the z-lattice, and atoms are insteadobserved as excess |b⟩ population around zero momentum, as seen in the inset
in Fig. 5.2b.

5.5

Measurement of transport behavior

To further elucidate these features, we study the hopping in a polariton BoseHubbard model. Compared to the modification
onsite interaction Ū ,$of theiπq/k¯
the renormalization of J for the atoms to J = − q ε1,q e
for the polaritons
can be significant. In order to test the magnitude of this eﬀect, we compare thedegree of coherence of the |r⟩ component at diﬀerent lattice depths sz with that
59

a

b
2σ

tunn. coeff. (

)

2.5

0.1

c

2

2.5

2

2
0

1

2

3

0.01

2

1

peak width010lattice depth

20

1.5

0

1

0.5

1

2

1.5

1

1
0

2

1.5
0

2

4
6
8
lattice depth

10 12

0

1

2

0

Figure 5.4: Renormalization of hopping extracted from the coherence of
the |r⟩ component at s⊥ = 18. a, Scan over sz for ∆/ωr = −0.50(7), Ω/ωr =0.84(2) (purple), compared to a reference measurement with Ω = 0 (red).
Plotted is the fitted 1D Gaussian width σ along z of the central diﬀraction
peak in time-of-flight (see inset). The red curve is a piece-wise linear fit to
the reference, with slope set to zero left of the kink, which is used to establish
the relationship between σ and sz . The purple curve is obtained from the red
curve after shifting sz (J) (black arrow) according to the expected rescaling of¯ the shaded area accounts for the experimental uncertainties in ∆ and
J to J;Ω. Inset: tunneling coeﬃcients J (red) and J¯ (purple) as a function of sz . b,c, Parameter scans at sz = 10 according to the traces shown in Fig. 5.3d, withvariable ∆ at Ω/ωr = 0.84(2) (b), and with variable Ω at ∆/ωr = −0.50(7)(c). The theoretical prediction reflecting experimental uncertainties is shown
as a solid purple line surrounded by shaded areas; the insets in the bottom¯
half of b and in c show the corresponding ratios J/J.
The inset in the top
half of b shows the peak width for Ω/ωr = 0.28(1) (and !Ω/U = 1.01(1)).
Vertical error bars show the standard error of the mean; horizontal error bars
are less than the size of the data points. All data points are the average overat least 3 runs.
of the (coupling-free) atomic reference with known J ≡ J(sz ). Using the widthof the central peak of the |r⟩ diﬀraction pattern as a proxy [161], we record
the ToF distribution after switching oﬀ the optical potential and the coupling
and removing the |b⟩ component with resonant light (to eliminate four-wave
60

mixing [82]). The results for ∆/ωr = −0.5 and Ω/ωr = 0.8 are shown in
Fig. 5.4a. The peak width σ of the reference is well fit by a piecewise-linear
function with fixed zero slope in the superfluid region, and positive slope in the
Mott regime [66]. When we turn on the coupling, the measured peak width σ̄
is comparable in the superfluid region and then increases with sz , but not asmuch as the reference.
We use the fitted reference curve and the functional dependence J(sz )obtained from the uncoupled band structure to obtain an operational relationship σ(J) between peak width and hopping. Under the assumption thatthe onsite interaction remains unchanged, the same relationship should holdbetween σ̄ and J¯ obtained from the polariton band structure, and indeed wefind that the rescaled reference curve quantitatively reproduces the observed
peak widths for the experimental coupling parameters. In the |r⟩ basis, thepolariton formation eﬀectively renormalizes the depth of the applied lattice,leading to a horizontal shift of points of a given peak width towards increasingly larger lattice depths. Near the kink at sz = 4, the rescaling of hopping
implies both the presence of a superfluid-to-Mott transition of polaritons, andof a vacuum-coupling driven transition from the singly-occupied atomic Mott
lobe to a polariton superfluid.
The dependence of the hopping on the coupling parameters ∆ and Ω isshown in Fig. 5.4 (b, c). The measured peak width changes as expected from¯ ofthe rescaled reference curve, and the inferred polaritonic enhancement J/J
the hopping agrees well with the prediction of the polariton band structure. In
particular, unlike J, J¯ saturates for large sz as a result of matter-wave-induced
hopping.
In Fig. 5.4, there is good agreement between experiment and theory if
at least one of the coupling parameters is comparable to ωr . However, forweaker couplings near the continuum edge, a reduction of |∆| can lead toan increase of the peak width, as seen in the upper inset of Fig. 5.4b, a
behavior that cannot be explained by a violation of adiabaticity in applying
the coupling (1/τ ∼ 0.02 ωr ). For small Ω and |∆|, the |b⟩ component of the
LP has large spatial extent and low density, such that the collisional on-site
interaction between the tightly confined |r⟩ atoms can lead to an eﬀective
change of the detuning. Indeed, the divergence of the peak width occurs near|∆|/ωr ∼ U/Er = 0.28, consistent with a shift to eﬀectively positive detunings,
at which we expect radiative matter-wave decay to give rise to a quantum
Zeno eﬀect [162]. We note that similar interaction-activated dissipation that
inhibits transport in a Bose-Hubbard model has recently been observed using
photon-induced losses [163].

61

5.6

Experimental details

5.6.1

System preparations = 40E r

Orthogonalstate -independentlatticess z = 14E r
Ve r t i c a lstate -selective
lattice3E r∏-shiftmicrowave
coupling

TO F

200m s20m s

2. 5m s10m s

2. 5m s 5m s

Figure 5.5: (*) Full experimental sequence for onsite-interaction energy measurement. Fig. 5.2 shows the crucial part of this whole sequence.
Our experiments start with BECs of 1×104 |r⟩ atoms in an optical trap [86].
For the measurements of Fig. 5.2, the full experimental sequence is shown in fig.
5.5. We first ramp up the tubes over 150 ms to s⊥ = 40 using an exponential
ramp, followed by an exponential ramp of the z lattice to sz = 14 over 25 ms.
Following with the reversibly-applied microwave coupling (see Fig. 5.6) andlattice modulation spectroscopy, we detect excitations through changes in thecoherence back in the superfluid phase [66]. We first linearly ramp down thez-lattice to sz = 3 over 5 ms followed by 15 ms of thermalization. The tubes
and the optical trap are then switched oﬀ and the z-lattice depth is jumped
to sz = 14 (in 50 µs) and held for 600 µs to induce a gravitationally-induced πshift between the wells. The resulting diﬀraction pattern of |r⟩ atoms features
two zeroth-order peaks separated by 2!k in the z direction. We sum over thetransverse direction, fit the two peaks on top of a thermal background withthree Gaussians, and extract the width σ̄ as the average over the two peaks. We
note that the momentum distribution of the |b⟩ atoms remains unaﬀected by
the changes in sz , since resonant collisions during the rethermalization phase,
as well as collinear four-wave mixing in time-of-flight [82], are both suppressed
due to the mismatch of the |r⟩ and |b⟩ dispersion relations [84], and the strong62confinement in the tubes limiting coherence, respectively.

0. 25

2. 0

σ (a.u.)N b/ ( N r+ N b)

2. 2

1. 8

0. 0

Ω

Figure 5.6: Fraction of |b⟩ atoms (blue) and peak width of |r⟩ atoms (orange)
after a symmetric microwave ramp up and down (each over 2.5 ms) for theparameters of Fig. 5.4 at sz = 10. The dotted open blue circle is the blue atom
fraction for the sequence of Fig. 5.2 c without lattice modulation applied. We
suspect that the degradation comes from magnetic-field noise that is able todrive LP→UP transitions by directly aﬀecting ∆.

5.6.2

Atom detection

The full experimental sequence for the measurements in Fig. 5.2 is shown in
Fig. 5.5, we use a smaller tube depth, s⊥ = 18, and ramp up the z lattice more
slowly, over 80 ms (starting in the middle of the s⊥ ramp) in order to maximize
the system coherence over a large range of parameters. After switching oﬀ allpotentials, we fit the 1D diﬀraction patterns using a single background-freeGaussian to extract the width σ of the central peak after summing up over
the transverse direction. To eliminate possible eﬀects of collinear four-wave
mixing in ToF expansion, the |b⟩ atoms are removed using a short blast pulseof resonant cycling light [82].
For atom detection, we use standard absorption imaging on the F =2 → F ′ = 3 cycling transition after a ToF of 15 ms, preceded by SternGerlach separation of the |r⟩ and |b⟩ states. The F = 1 atoms are detected
after transferring them to F = 2 using a short repump pulse.

63s = 18Er
Orthogonalstate -independentlatticessz = 10Er

Ve r t i c a lstate -selective
latticemicrowave
coupling
80m s

2. 5m s

TO F

150m sFigure 5.7: (*) Full experimental sequence for quantum transport. Measurements in fig. 5.4 are based on this experimental sequence.

5.6.3

Resonance condition

In our experiments, we need to accurately determine and maintain the resonance condition (∆ = 0) between the |r⟩ in the z-lattice and the |b⟩ mode
continuum edge. We use lattice transfer spectroscopy [85] to adjust and monitor the resonance condition interleaved with a set of measurements, with an
extrapolated drift of maximally 300 Hz between shots. For this purpose, we
prepare the BEC in the |b⟩ state, and then apply 400µs long Rabi pulses(Ω = 2π × 1.0 kHz) to transfer a small fraction (< 30%) of atoms into the z
lattice, with calculated mean-field shifts well below 100 Hz.

5.7

Theoretical details

5.7.1

Polariton band structure

In the Schrödinger picture, our system Hamiltonian takes the form [150]% (r) †
%
(b) †
ĤS =
!ω0 r̂j r̂j +!ωn,qb̂n,q b̂n,q
j

+

%%
j

n,q

5

n,q

6
!gn,q e−i(ωµ t+qzj ) r̂j b̂†n,q + H.c. ,
64

(5.4)

(b)

(b)

where !ωn,q = !ω0 +En,q is the energy (internal plus kinetic) of the |b⟩ atoms,(r)!ω0 is the energy of the |r⟩ atoms, ωµ is the frequency of the microwave which(b)(r)
determines the detuning through ∆ = ωµ − (ω0 − ω0 ), zj = jπ/k is the$$$$∞ 4 +k dq
position of the j th site, j ≡ +∞
and
≡
j=−∞
n,q
n=1 −k 2k . The external
action of the microwave makes this Hamiltonian time-dependent. However, by
writing ĤS = Ĥ0,S + Ĥ1,S with
%
% (b)
(b)
Ĥ0,S =!(ω0 − ωµ )r̂j† r̂j +!ω0 b̂†n,q b̂n,q
(5.5)
n,q

j

one gets the time-independent interaction HamiltonianĤ ≡ Ĥ1,I = eiĤ0,S t/! Ĥ1,S e−iĤ0,S t/!
so thatĤ =%!∆r̂j† r̂j +

j

+

%%
n,q

j

%

5

(5.6)

En,q b̂†n,q b̂n,qn,q6!gn,q e−iqzj r̂j b̂†n,q + H.c. .

(5.7)

Following the approach of [158] (see [152] for an alternative), it is then possible to study the emergence of polaritons. Fourier-transforming the operators
%
%
r̂q =
e−iqzj r̂j ,r̂j =eiqzj r̂q
(5.8)
q

j

leads to
Ĥ =

%
q

+

%
n,q

!∆r̂q† r̂q +5%En,q b̂†n,q b̂n,q

n,q

6 %
!gn,q r̂q b̂†n,q + H.c. ≡
Ĥq

(5.9)

q

such that the Hamiltonian decouples into independent quasimomenta [Ĥq , Ĥq′ ] =0, where%Ĥq = !∆r̂q† r̂q +En,q b̂†n,q b̂n,q
+

$

%
n

5 n
6
!gn,q r̂q b̂†n,q + H.c. .

(5.10)

If we take ĉ†n,q = αn,q r̂q† + n′ βn,n′ ,q b̂†n′ ,q to create a quasiparticle (polariton) in
the band n with quasimomentum q, then the eigenvalue condition Ĥq ĉ†n,q |0⟩ =65

εn,q ĉ†n,q |0⟩ leads to the secular equations [158]7$(εn,q − !∆)αn,q = n′ !gn′ ,q βn,n′ ,q(εn,q − En′ ,q )βn,n′ ,q = !gn′ ,q αn,q(5.11)

which allow for the determination of the amplitudes αn,q and βn,n′ ,q (whichalternatively follow from Eq. (27) in [152]) and yield the equations for thepolariton dispersion relation
εn,q − !∆ =%n′

!2 gn2 ′ ,q.
εn,q − En′ ,q(5.12)

as given in the main text.

5.7.2

Polariton Bose-Hubbard Hamiltonian

Here we show how the band structure and interactions derived above leads to apolariton Bose-Hubbard Hamiltonian. Since the non-interacting Hamiltonianin lattice-momentum space is quadratic and has energies given by (5.12), it
reduces to%Ĥ =εn,q ĉ†n,p cn,p .
(5.13)
n,pThanks to the independence between quasimomenta, if there is some hopping J for |r⟩ atoms, this can be accounted for by simply making the detuning
8 92
q-dependent, ∆ → ∆q = ∆ + 4J sin2 πq.
2kReapplying (5.8) to the ĉn′ ,p operators, one can write the Hamiltonian inposition space as%Ĥ = −
J¯n,j−j ′ ĉ†n,j ĉn,j ′(5.14)
n,j,j ′

in terms of the hopping coeﬃcients

J¯n,j−j ′ = −%′εn,q eiπ(j−j )q/k .

(5.15)q

For negative detunings and modest microwave couplings (Ω/|∆| < 1) an
experimental run that starts with |r⟩ atoms remains dominated by |r⟩, corresponding to a ground polariton band (n = 1) that is roughly sinusoidal,
meaning that we can approximate as if there is only nearest$ neighbor hopping¯¯¯¯(J1,±1 ≡ J and J1,|n|>1 ≈ 0). The constant −J1,0 =q εn,q is the energyof a polariton Wannier function, which is relevant for the calculation of ω̄0in Fig. 5.2. This can be combined with the local energy shift generated by66the harmonic confinement of the experiment into a site-dependent energy ε̄j .
Omitting the band indices (ĉ1,j ≡ ĉj ), the above Hamiltonian then reduces to% †% †Ĥ = −J¯(ĉj+1 ĉj + ĉ†j ĉj+1 ) +ε̄j ĉj ĉj ,(5.16)
jj

which together with the polariton interaction Ĥint gives the Bose-Hubbard
Hamiltonian. The interaction itself is given byĤint =

Ū % † †ĉ ĉ ĉj ĉj
2 j j j(5.17)

where we calculate
1Ū ≡ ⟨0|ĉ0 ĉ0 Ĥint ĉ†0 ĉ†0 |0⟩
2 "+
+
4
= ḡ
dz|ψr (z)| + dz|ψb (z)|4#+22+ 2 dz|ψr (z)| |ψb (z)|(5.18)

in terms of the |r⟩ and |b⟩ components of the Wannier function, ψr (z) =⟨z, r| ĉ†0 |0⟩ and ψb (z) = ⟨z, b| ĉ†0 |0⟩, respectively. Here ḡ = 2!2 a/(ma0x a0y )depends on the harmonic confinement in the tubes (a0x and a0y ) and the(small) dependence of the scattering length a ≈ 5.3 nm on the internal stateof the atom is neglected.

5.8Conclusion and outlook

In this work, we focused on the lower polariton branch, corresponding to renormalized hopping of excitons in a material system [153], which has allowed us torealize a polaritonic quantum phase transition as well as a transition from anatomic Mott insulator to a polariton superfluid. By exploiting the lattice band
structure and applying specific couplings, exotic polariton band structures canbe implemented [152], including those featuring frustration. We can also focus
on the upper branch dominated by |b⟩ atoms, which corresponds to the renormalized transport of photons. Making the z-lattice state-dependent (insteadof state-selective [87]) will allow us to create the equivalent of coupled-cavity
arrays, and implement the analogue of photon blockade [164] for |b⟩ atoms,
with the strongly interacting |r⟩ state playing the role of a nonlinearity. Introducing couplings between more than two excitonic or photonic bands should67enable studies of analogues of multiexciton polaritons [165], multimode strong
coupling [166], and spin-orbit coupling [167]. This may open up possibilities
for studying topological polaritonic systems in higher dimensions [168].

68

Chapter 6
Toward future experiments
With our established platform, we can think about studies beyond the regime
that we have dealt with so far. In this chapter, we will discuss some ideas
for future experiments that should be feasible with only a small amount ofmodification.

6.1

Coherent dynamics in a matter-wave platform

Our matter-wave platform is very promising for studies of coherent dynamics in 1D waveguide-QED. In the following section, we will introduce somequalitative ideas that could possibly guide future experiments.

6.1.1

Collective dynamics

Arguably, one of the most interesting feature of 1D waveguide quantum electrodynamics is collective dynamics that produce super-radiant and sub-radiantphenomena. Our matter-wave platform has unique benefits since what is radiated are massive particles so that are easy to measure and control.
In general, the Dicke model [169] gives a superradiant phase in a photonic
system under the collective light-matter interaction [170] when the photons
share a single mode over the size of the sample. Given that the deBroglie
wavelength of our system is relatively short, the Dicke model is generally notdirectly applicable. Instead, we can apply superradiance in the timed-Dicke
model [171] that the radiation from one site is in phase with the radiation fromthe next site so that collective emission occurs. Then depending on the phaseof the initial state (red) and the momentum of emitted matter-wave (blue),
we can induce subradiant dynamics as shown in fig. 6.2.
69

(A)

s=4 (superfluid)
|b>

|r >
(B)

Δ=1Er

Δ=1.5Er

Δ=2Er

Δ=2.5Er

Δ=2Er

Δ=2.5Er

s=10 (Mott insulator)
|b>

|r >
Δ=1Er

Δ=1.5Er

Figure 6.1: Time-of-flight image of emitted blue atoms for diﬀerent initialstates. The system is prepared in the same way as the matter-wave polariton
experiment (see chap. 5) but the coupling here is set as positive (∆ > 0), which
is varied to check the momentum of emitted blue atoms. (A) For an initiallycoherent state (superfluid) with low red lattice depth (s = 4), the emission issuppressed and most of the population remains red. (B) For the incoherent
state (Mott-insulator) with strong confinement (s = 10), the emission is moresignificant. For all measurements, atom number is controlled n ∼ 1.2 × 104 .
In fig. 6.1, the fraction of emitted blue atoms changes whether the initial
red state is coherent or not. Here the coupling is weak (Ω/∆ < 1), and thedetuning is varied from ∆ = 1Er to ∆ = 2.5Er . As theoretically expected,
the emission from the initial coherent state is suppressed (fig. 6.1 (A)) which
is diﬀerent from the initial incoherent case ( fig. 6.1 (B)) similar to the one
studied previously (see sec. 4.4, 4.5). In case of the initial coherent state, we
also expect the superradient phenomena for a certain detuning (∆ = 4Er forthis case) which is not shown here. It will be interesting to study the collective
behavior more precisely, and also with controlling the superradiant condition.

70

Coupling scan
Still, there is a practical limitation to this experiment, which is the presenceof empty quantum emitters. This was already the case for the spontaneous
decay experiment in sec.4.4 that the empty quantum emitter (re-)absorbs the
propagating matter-wave from the neighboring sites. This reabsorption canbe very critical for the measurement of superradiant since it will limit any collective motion by eliminating the emitted matter-wave. This concern becomes
more serious if combined with the non-uniform lattice system in practice fromthe harmonic trap, see fig. 6.2 (A).
(A)

(B)

Figure 6.2: (A) Atoms located in the lattice under the harmonic trap potential.
The black solid line represents the harmonic trap potential, red line is an
optical lattice in 1D that confines red atoms. (B) Illustration of superradiantexperiment scheme. Three colored arrows represents the emitted matter-wave(radiant) where they occur in diﬀerent sequential time zone (dark blue → lightblue) to avoid re-absorption from the neighboring sites.
This non-box potential shifts up the lowest energy of the edge side, which
eventually induces the non-uniform atom population for each site. Thus any
Dicke-related model that requires a uniform site population (mostly < ni >=1) is not directly applicable (or hard to expect clean measurement), since thepresence of the empty sites (which is a ground-state quantum emitter) at theedge prohibits the analogy to the Dicke state that requires the same numberof emitters and atoms. A similar problem can also occur in any other cases oflattice with non-uniformly populated atoms.
Here, we propose a concept that might be able to solve the problem.
71

Fig. 6.2 (B) shows a schematic illustration of the idea. Instead of populating on average one atom per site, we start with relatively many atoms per sitenavg = 2 ∼ 3 with a very tight confinement from the lattice. Then each oflattice site will have a diﬀerent energy from the onsite-interaction energy shiftdepending on the populated atom number. To induce the collective radiation,
starting from the frequency that couples the highest atom number populatedsite, one can sweep the coupling resonance frequency (detuning) in a quantized
way with the corresponding energy shift (n = 3 to n = 2, and then n = 1).
With this quantized detuning scan, when the emitted matter-wave fromthe highly populated site reaches to the next site, the detuning is lowered sothat only the next populated site can emit without the re-absorption. The
key here is thus the control of changing the detuning of coupling on a proper
time corresponding to the emitted matter-wave, and this will be controlled bytuning the momentum of the matter-wave.
This proposed technique should be widely applicable since it does not necessarily assume the harmonic trap. The drawback is, however, that the time
window of the measurement is limited by the atom number (or the maximum
onsite energy) since once the coupling frequency reaches the n = 1 case, there
is no other way to lower the coupling frequency to prohibit re-absorption. In
practice, about 40,000 atoms will provide n = 4 atoms per site in the center ofthe harmonic trap, and this gives 4 emission cycles until the frequency sweep
reaches the lowest order, which is long enough and the technique is indeed
applicable. However this is not the case for the experiment that starts with arelatively small atom number (! 1.2 × 104 ).

6.1.2

State purification

Much of the work described in this dissertation is focused on the emitted blue
atoms, and red atoms are only used as a measure of the population dynamics
of quantum emitter. If we focus on the red state, however, we can also study
the coherent state of the untransferred red atoms. Especially since we knowthat the coherence of the initial state changes the emission of the blue atoms,
we can use this in a way to purify the initial state.
In detail, in case of the localized red atoms as in Ch. 4 where all wave
functions of red atoms are incoherent over the sites as a case of Mott-insulator,
the red atoms are free to be transferred to emitted blue state via spontaneous
decay. This is because the coupling between red and blue state in WeisskopfWigner model assumes a single, localized function as an emitter. If we startfrom a coherent red distribution, however, the situation is diﬀerent. The
coherent atoms form an extended polariton that cannot decay (unless thesuperraidant condition is met), but all incoherent parts should be able to be72

emitted. This might eventually work as a ‘purification’ of red state as pure
coherent system. This is basically what happens in fig. 6.1.
The idea is then pretty straightforward. By positively coupling the redstate with the blue state, the incoherent red state will be emitted and what isleft in red should only be the coherent state. By comparing the coherence of the
red state before/after this state-purification, we can experimentally confirm it.
There are some practical issues though. The first issue is the preparationof an adiabatic coupling. From our previous studies, we know that it is required to couple the red and the blue adiabatically for a system to exhibit thetheoretical prediction, and this requires non-negligible time (2.5ms for previous cases) which can induce additional collisional heating. This is also relatedwith the second issue that it is not easy to tell the incoherence of the Mottinsulator regime from the thermalization. It is therefore not trivial to checkthe coherence of the final state, but more carefully designed experiments inthe future will be able to resolve the current technical issue and generate aclear result as the author of this dissertation hope.

6.2

Frictionless impurity motion in 1D

The mobility of impurities in ultracold atomic systems has been studied ina wide range with various platform, since it shows un-intuitive interesting
phenomena [172–175]. In this section, we will briefly discuss the plan foran experiment on frictionless motion of impurities in a 1D system. The basic
idea is based on [176], that dissipation can be suppressed by tuning the relative
scattering length between the impurity and the background atoms.

6.2.1

Critical velocity

Since we will discuss the motion of impurity, it is crucial to understand thecritical velocity. We here start from the very basic description of the BEC;
the Gross-Pitaevskii equation
2
3
!2 2
2
i!φ̇ = −∇ + Vtrap (r) + g|φ| φ(6.1)
2m
where g = 4π!2 a/m is an interaction parameter and a is the scattering length.
In the presence of the weak perturbation like density dip, φ = φ0 + δφ, we
can expand the eq. 6.1 with the ansatz of Bogoliubov transformation [141]δφ = e−iµt/! (u(r)e−iωt − v ∗ (r)eiωt )
73

(6.2)

εq

kinetic energy
dominant

interaction energy
dominant

0

momentum q

Figure 6.3: Dispersion of Bogoliubov excitation. Depends on the momentum q,
the ratio of corresponding kinetic energy (!q)2 /2m and the interaction energy
gn0 gives two regime in the dispersion. For small q, the dispersion relation islinear and the slope gives the speed of sound vS whereas large q gives parabolic
free-particle like dispersion.
and with the assumption of the perturbation as a running wave u(r) ∝ uq eiq·r ,which indeed gives the dispersion relation of the Bogoliubov excitation!
"#
1 (!q)2 (!q)2
ω(q) =+ 2gn0
(6.3)
!
2m
2m
as shown in fig. 6.3. The two terms inside the square-root gives two extreme
limits. For large q, the dispersion relation is more or less quadratic, corresponding to the excitation of a free particle. On the other hand, small q gives
the linear (sound-like) dispersion relation near 0 as*
gn0
ω(q) " q
,
(6.4)
m
/
from which we can define the speed of sound as cS = gn0 /m.
We now can apply the Landau criterion [177] to this dispersion relation,
the minimum (critical) velocity for excitation is given asv≤

ϵq= vminp(6.5)

from the resonant excitation (in moving impurity frame) ϵq (v) = 0 that gives74

ϵq − p· v = 0 where ϵq is the dispersion relation given by eq. 6.3. When thevelocity of the impurity is less than the minimum velocity (v < vmin ), themotion of impurity cannot excite the BEC. Note that for v > vmin in 3D, theexcitation comes oﬀ at an angle.
This critical velocity depends on the regime as discussed above. In a caseof large q, i.e. kinetic energy is dominant and the interaction is negligible2( (!q)≫ 2gn0 ), the critical velocity is proportional to p which means that the2meﬀective critical velocity is zero. Thus any motion in this regime will create
excitation, which is very intuitive. For small q regime, the strongly interacting
2case ( (!q)≪ 2gn0 ), the critical velocity is indeed given by the speed of sound2mvS from eq.6.4. Thus the impurity motion that is slower than vS will not create
excitation, i.e. there is no heating from the motion.
However, this is only the case for 3D system, unfortunately. 1D has atotally diﬀerent physics for the impurity motion, and we will discuss it infollowing subsection.

6.2.2

Basic description of 1D system

To discuss the specialty of 1D, let us first take a look the basic descriptionof 1D. In 1D system such as in a lattice tube, the Hamiltonian is given as[141, 178, 179]
% !2%
H1D = −
∂z2j +
g1D δ(zi − zj )
(6.6)
2m
j
i<j
: ;
!2 2a
where i, j are the particle indices, g1D = m a2 is the interaction parameter⊥and a is the related scattering length.
We can compare the two diﬀerent energies here:
5 6the interaction energy!
πEint = n1D g1D and the kinetic energy Ekin = 2mn1D
can extract the Lieb-Liniger parameter γ [180] as
" #
2 2a
2
Eint
=
n1D ≡ γ
2
Ekin
π a⊥
π
2

2

. From the ratio, one(6.7)

where n is the atom number. The system is in the superfluid regime whenγ ≪ 1, while γ ≫ 1 gives the Tonk-Girardeau (TG) regime [181, 182].
In TG regime, there are many interesting studies for the impurity suchas quantum transport phenomena [183] and induced Bloch oscillation [175].
However those studies are based on the property of impurity as an obstaclethat changes the dynamics of the environment, and this is generally true because any motion in 1D will eventually create an excitation no matter with the75speed of the impurity. At a glance, this looks like a contradiction with eq.6.5,
the critical velocity. However, this is due to the unique property of 1D system
which does not have enough degrees of freedom to resolve the quantum fluctuation. Based on [184], there is a drag force on impurity in 1D system evenbelow the critical velocity that induces the energy dissipation, the quantum
fluctuation. This force is due to the quantum fluctuations that is reflectedfrom the impurity for each side.

6.2.3

Yang-Gaudin model toward frictionless motion

There is a way, however, to overcome the inevitable excitation from the motionof the impurity. Yang and Gaudin’s work [185, 186] in 1960s gives us a cluethat in an integrable system with equal mass, the friction can be suppressed.
The recent research [176] calculated from the idea and claimed that the friction
constant κ is given by
"
#2
abb
4
κ∼T
−1
(6.8)
arb

where a is the scattering length between subscripts r and b, which denotes
each of the two constituents of the system respectively, and T is the systemtemperature. The integrability is naturally provided by the ultracold atom
system in 1D, since s-wave scattering is the only eﬀective collision in TG
limit.
Thus by controlling the scattering length between the impurity and thebath as equal to the one between bath atoms, one can make the friction negligible, i.e. dissipation-free impurity motion. Using the pair of |2, 0⟩ (bath)and |1, 1⟩ (impurity) might be the most appropriate choice since the scattering length of |1, 1⟩ ↔ |1, 1⟩ and |1, 1⟩ ↔ |2, 0⟩ are compatible (100.4 and
97.7, respectively [187]). The diﬀerence can be adjusted by using the Feshbachresonance that gives"#∆a(B) = aBG(6.9)
B − B0where B is the bias magnetic field and ∆ is the diﬀerence of magnetic fieldfrom the resonance [37]. From the calculation, ∼ 9G of bias magnetic fieldbrings the two scattering length to the same value.

6.3

Technical development

This section will explain the technical improvements and/or eﬀorts in the
direction of such experiments. The work in the following subsections describes

76

some earlier work of the author of dissertation that can be applied for future
experiment (see previous section, sec.6.2). The other minor technical works
that might be potentially useful in the future are noted in the appendix (A).

6.3.1

Moving lattice system

(A)

(B)

(under the table)

85MHz

-1

85MHz+δf
0

0

+1

AOM
λ/4 waveplate
λ/2 waveplate

mirrorpolarizing
beam-splitter

lens

Figure 6.4: Moving state-dependent lattice setup. (A) Illustration of the setup.
The structure is similar to that discussed in the thesis of B. Gadway (2012),
but the system has been re-built from scratch with modified details. (B)Experimental realization of (A). Red line with arrows describes the guided
beam path. Green arrows after each AOM shows the non-deflected path, andwe are taking the −1 and the +1 order of diﬀraction for each AOM (this is
the opposite order from the plan depicted in (A).
There are several ways to launch the atoms along the tube such as giving amomentum kick. In order to control the motion precisely, however, it is betterto control the confining lattice itself. In our target experiment, especially,
we only want to make a relative movement for the impurity, i.e. a certain
hyperfine state, we need to move the state-dependent lattice.

77

The mechanism of moving lattice system which follows an earlier implementation [111] is quite simple as shown in fig. 6.4 (A). Since the optical lattice
is based on the standing wave, if there is a slight frequency diﬀerence between
the beams generating the lattice, the node moves together. To achieve this,we used acousto-optic modulator (AOM) for shifting the frequency slightly.
When the light passes through AOM, the frequency for the diﬀracted order ischanged by absorbing the phonon that AOM is driven by, which is generally
around 80MHz (In our setup, we use 85MHz for optimizing the diﬀraction
eﬃciency). Since the retro-reflecting laser should have equal polarization andwavelength to be interfered with the forwarding laser, double-path AOM system is required; by choosing −1 and +1 diﬀraction order for each of AOM
path, the total wavelength is basically unchanged. Fig. 6.4 (B) shows the realized setup for the moving state-dependent lattice system. The choice of AOM
order is diﬀerent from fig. 6.4 (A) due to the practical reasons, which does not
aﬀect to the result.
We here apply the additional frequency δf which is negligibly small but
only large enough to make the optical lattice moving with the speed of v =δf λ/2 for fixed f . If we modulate the frequency δf in time domain, we caneven accelerate the moving lattice. This can be used to double or cancel out
the gravity by moving lattice in z-axis.

78

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96

Appendix A
Appendix
This is an appendix for technical improvements to which the author of thisdissertation contributed.

A.1

Logging magnetic fields

The importance of the precise control of the magnetic field in our lab cannot be
overemphasized. All transitions and related accurate measurements depend on
the magnetic field, and monitoring of it is absolutely important. There is an inmeasurement method called field-tagging (see sec.4.3.1), but this measurement
is narrow-windowed, also experiment-dependent. Moreover, magnetic field
drifts on a large scale due to sources outside of the lab are of concern as well.
This includes fields that are produced by other labs nearby.
To address this issue, we mounted a 3-axis magnetometer (Mag-3MS1000,
Bartington Instruments) near the science-cell as shown in fig. A.1 (A) (1). The
magnetic field along the vertical axis is measured so that we can extract any
large-scale magnetic field drifts that the atoms feel over the experiment.
Similarly, it is important to monitor and stabilize the current of our power
supply when accessing ∼ 1007G Feshbach resonance (see A. Pazmiño’s thesis[114]); for this purpose we use a current transducer (DS600IDSA, Danisense),
shown in fig. A.1 (A) (2).
The sensors are read out with 7-1/2 digit digital multi-meters (DMM)(Agilent). To extract the value of magnetic field/current at a certain time, i.e.
when the experiment happens, we need to trigger DMMs and the measured
values are read out via GPIB. The system configuration is displayed in fig. A.1(B).
Fig. A.2 shows our Labview-based DMM monitoring program. The digital trigger from the experiment-control system triggers the DMM and the

97

(A)
(1)

(2)

S-cell(B)
Experimental
sequenceer

Magnetometergtrig

7 1/2 digit DMM
7 1/2 digit DMM

TOF Image processing
GPI

B

Logging program

Current Transducer

Measurement

Figure A.1: Field monitoring infrastructure. (A) (1) Magnetometer (in thewhite dashed circle) is placed near (! 15cm) the science-cell. The verticaldirection of magnetic field is measured. (2) Transducer (marked as whitedashed circle) is placed to the input side of IGBT that acts as a switch of large
magnetic field coil. For both cases, the measured signal is connected to digital
multi-meter (DMM). (B) Schematic diagram of the system configuration.
measured value is then sent to the program and recorded. Depending on thesetting, we can record several measurements after a single trigger which is helpful to track the local stability. The program shows the real-time accumulated
curve for the convenience, and provides a log file with extra time-stamp so thatthe values can be matched with the corresponding experimental data. The log
is given in .lvm format which is compatible with both Excel and Mathematica.

98

Figure A.2: DMM monitoring program.

A.2

Automated polarization control

The polarization of our state-dependent lattice determines the strength of thecoupling for the diﬀerent magnetic hyperfine states. To control the polarization, we use a rotatable λ/4 plate in a linearly polarized beam. In most cases,
once the polarizer is set, it is never touched to preserve the target polarization,
unless it needs to be changed.
The problem is that, sometimes it is required to rotate the polarization ina very accurate manner. Furthermore, a dynamical change of the polarization
may be needed depending on experimental sequence. For example, if we want
to change the relative lattice depth of |r⟩ and |b⟩ states using state-dependentlattice frequently, the best way is to change the polarization since changing
the wavelength in a real-time measurement is not feasible, with our Coherent
899 Ti:Saph laser.
This motivated us to install a rotational mount with rotatable λ/4 plate.
Fig.A.3 shows the installed automated mount (PRM1Z8, Thorlabs), which
99

SDOL axis

Figure A.3: Step-motor polarizer (dashed yellow circle) is mounted on the wayof forwarding state-dependent laser in front (below) of the science cell. The
vertical axis that SDOL is elongated is marked as white line.
is connected to a brushed DC servo moter (KDC101, Thorlabs) that is notshown in figure. It is mounted below the science cell on the way for forwarding
state-dependent lattice in z-axis. This step-motor polarizer is controlled by aprogram (Kinesis, Thorlabs) which we connected to our experimental sequence
program (Cicero) to enable the execution of pre-set rotations by triggering in
the experiment.
Fig.A.4 shows results obtained with the installed system for calibrating
the optimal state-dependent light polarization with the rotation angle varied
between experimental runs.

A.3

Accordion lattice

The presence of gravity is sometimes very unhelpful in terms of a uniform
weakly confining system. We designed new optical lattice setup that gives100Lattice depth s (Er)

30
25
20
15
10
5
0
0

50

100

150

200

250

Waveplate angle (°)

Figure A.4: Lattice depth is measured for diﬀerent polarization using the
installed step-motor polarizer. The lattice depth is measured by KD (KapitzaDirac) diﬀraction, and the polarization is changed by automatic triggeringsystem.
transverse trap confinement in y ′ − z plane which provides eﬀective 1D tube
along x′ -axis that is irrelevant to the gravity. Here x′ and y ′ are the new axes
that are rotated 45◦ in the x − y plane.
science-cell

f=40mmmirrorimaging
light

14˚

‘

‘

Figure A.5: Design for accordion lattice setup. f = 40mm 1-inch lens is thebiggest lens that we can mount to the science cell as close as possible dueto the existing configuration. Two independent laser beams pass through thesame lens above/below with 8mm distance from the center, then cross at theposition of the atom in the science cell with the incident angle of 14◦ against
the orthogonal axis. The beam passes the same f = 40mm lens behind thescience cell to control the profile and reflected back from the mirror. The
mirror has a hole (diameter: ≈ 0.8cm) at the center to be compatible with theexisting light path for imaging.
The main point of concern for the setup design is to not destroy any existing101

setup for the lab consistency. As shown in previous setup for the current beampath, 45◦ against the science-cell is already occupied by existing x − y opticallattices. This causes a problem when it is combined with the fact that the wallof our science-cell in use is not specially coated for anti-reflection, thus we are
not allowed to put the laser beam for the optical lattice with the orthogonal
incident angle against the wall plane. Moreover, the space near the science-cell
is very restricted due to the presence of the moving coil as discussed in sec.2.1.
(A)

(B)

(C)

Figure A.6: Accordion lattice beam realization (A) Design of the accordion
lattice system in practice. (B) Illustration of the setup on the top of realexperimental configuration. (C) Calculated optical potential profile from theaccordion lattice
The setup of the accordion lattice came out to circumvent this restriction,
which is shown in fig. A.5. Instead of giving the angle of incidence withseparate systems, we can use one big lens that focuses down the two laser
beams for optical lattices at the same time. Two beams are separated enoughnot to interfere with each other and also not to touch the existing imaging
light path. At the same time, however, if the beams are located too far close
to the edge, the lens does not work as supposed due to the spherical aberration(using aspheric lens is helpful, but this requires more space in general).
In terms of the incident angle, the ideal angle is 45◦ which gives orthogonallattices. In our system, the largest available angle is 14◦ , which is marginal102

but still feasible to make 2D optical lattices. At the exit side of the glass cell,
we need an additional f = 40mm lens in combination with a retro-reflecting
mirror which is shown in fig. A.7. The reflecting mirror has a hole in the centerso that the imaging light can propagate through.
(A)s-cell
setup

(B)

(nm)
1

optical fiber
w/ coupler

lens(f=300mm)
lens(f=40mm)0.5

beam
profilemirrorw/ hole

0
20

40

60

80

(cm)

20

40

60

80

(cm)

-0.5
-1

(C)

(um)
50

curvature0
profile-50

Figure A.7: Accordion lattice beam profile. (A) The schematic diagram of
setup is drawn in scale to be compared with for beam profile below. (B)Calculated beam profile with the equipped parameters. (C) Corresponding
beam curvature. At the position of the atom where beam is focused, thecurvature is infinite.
Fig. A.6 shows the practical setup of the accordion lattice. Due to space
limitations, we decided to mount the final lens in an upside down way, and thisis done by putting the second layer which is depicted as a white box in fig. A.6(B). Fig. A.6 (B) shows the setup on top of the current configuration at theearly phase of the project, which are mostly in place now, except the whitebox zone which consists of a tube-like lens mount and a rotational mount. The
purpose of the rotational mount is to tilt the beam with a very small amountfor the final alignment.
Fig. A.7 shows the calculated beam profile for the designed setup. The
choice of fiber couplers and lens in earlier beam path are also made for optimizing the lattice profile at the science cell. The beam profile in fig. A.7(B) inside the science cell is not diverging or converging too much so that thelattice can capture atom uniformly. The waist at the focus is designed (andtested) as ∼ 100µm, and this can be adjusted down to ∼ 60µm within current
103setup. The calculation is done with ‘ABCD’ matrix method [141], and thisgives the beam curvature as well, shown in fig. A.7 (C). The curvature diverges
at the focus (where atom is supposed to be) as planned.

A.4

Microwave amplifier

We use microwave radiation at ∼ 6.8GHz to couple F = 1 levels and F = 2
levels.
(A)
microwave source(6.8 GHz)ff+fmodIQ mixer
I-3dB
Q

attenuation /phase split

Antenna

TTL signal+58dB
amplifierTriad
TA 1029-45d

fmod
function generator

B at
tenu

mini-circuitzx47-40-s+

ationpower metermonitoring
oscilloscope(B)sourcesidebandFigure A.8: (A) Schematic diagram of microwave amplifying system with
newly implemented monitoring feature. (B) Recorded output signal. The
first step in signal is from the microwave source (6.8GHz) that is not resonantwith the energy level of atom. The second step is from the sideband that ismixed with the power/frequency from the function generator, and this is thepulse that is in use for the experiment.
After a decrease in power of our existing system (cf. PhD thesis of D. Pertot[110]), we installed a new amplifier with ∼ 58dB amplification (TA1029, Triad104

RF Systems). The microwave source that provides 6.8GHz is mixed with RF
frequency from a function generator that we can arbitrarily control, and thenamplified [110]. The amplifier allows to monitor its output with −45dB ofnatural attenuation inside, which is useful to compensate any thermal drifts.
We connected this monitoring output to a Mini Circuits powermeter (ZX-4740-s+) whose output is displayed on an oscilloscope with a proper trigger at
an instance of measurement as shown in fig.A.8 (B). The first inverted plateau
is a pre-triggered signal from the main band of the microwave source which
is not resonant with the atom, and the second inverted peak is the resonant
microwave pulse that we are using in experiments.

output power (dBm)

40
35

30

25
-20

-15

-10

-5

0

5

10

15

function generator input power (dBm)

Figure A.9: Microwave output power gain curve. Function generator thatcreate fmod frequency controls the final output microwave power at antenna.
The output power is monitored with −45 dB attenuation from the monitoring
port, and then re-adjusted (see fig. A.8 (A)). There is 2 ∼ 3dB loss from acable, and this is also the case for the output side. The power seems saturated
around ∼ 40dBm (∼ 42dBm with a consideration of cable loss) , which is alittle bit lower than 44dBm (Psat specification).
Fig. A.9 shows the gain curve of the new amplifier. The output poweris monitored from the system described above (fig. A.8) and then converted
into units of power (dBm). We set the active linear region of input from thefunction generator side up to 20dBm, to protect our internal circuit configuration. The gain curve gives an amplification of 56dB which is a bit short ofthe specification, but still acceptable.

105