Dynamics of Matter-Wave Quantum
Emitters in Engineered Reservoirs

A Dissertation PresentedbyMichael Stewart
to
The Graduate Schoolin Partial Fulfillment of the Requirements
for the Degree ofDoctor of Philosophy
inPhysics

Stony Brook University
August 2020

Stony Brook University
The Graduate School

Michael Stewart

We, the dissertation committee for the above candidate for the Doctor ofPhilosophy degree, hereby recommend acceptance of this dissertation.

Dominik A. 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

Ana Asenjo-Garcia
Assistant Professor, Department of Physics
Columbia University
This dissertation is accepted by the Graduate School.

Eric Wertheimer
Dean of the Graduate Schoolii

Abstract of the Dissertation

Dynamics of Matter-Wave Quantum Emittersin Engineered Reservoirs
byMichael Stewart
Doctor of Philosophy
inPhysics
Stony Brook University
2020

Bose-Einstein Condensates (BECs) confined in optical lattices provide a rich playground for studying the physics of complicatedquantum systems in an exquisitely well controlled manner. In
this dissertation, we experimentally study an array of matter-wave
emitters realized with ultracold 87 Rb atoms confined in an array of
one-dimensional tubes, and we report for the first time on emission
experiments into a band structure. By varying the bandwidth, weare able to demonstrate a transition from (mostly) Markovian behavior to the limit of a pure Rabi-oscillation, as in cavity quantumelectrodynamics, and we also characterize the structure of twobound states, above and below the band, whose spatial shape isstrongly modified by the underlying lattice band structure.
We develop a theoretical model for the quantitative understandingof these and earlier results on emission from an isolated emitter
into free space. The modeling treats the emitter as a simple openquantum-system in which the harmonic oscillator ground state of adeep well of a state-selective optical lattice potential is coupled toiii

a gapped continuum of momentum states. We solve this model bydirectly integrating the Schrödinger equation using Laplace transforms and the tools of complex analysis, and we make predictions
for deviations from Weisskopf-Wigner type Markovian decay forexperimentally realistic parameter regimes, most notably, for thecase where the emitter energy is small or else negative. We connect
these decay behaviors to the existence of bound states, in which
the emitted matter-waves are unable to fully escape the originating
emitter. The number and character of these bound states, as wellas their e↵ect on the observable dynamics is explored for the caseof a free-particle dispersion with a single energetic edge, as well as
for a sinusoidal dispersion with two energetic edges.
Our results show that a matter-wave emission platform can provide
insight into the physics of photonic band-gap materials, currently aburgeoning research topic in quantum information science. We also
propose experiments leveraging our emitter system to study e↵ective Hamiltonian engineering within a dissipative quantum system.

iv

Contents
List of Figuresvii

Acknowledgements

viii

1 Introduction

1

2 Overview of the experimental platform
2.1 Implementing a quantum emitter for matter-waves . . . . . . .
2.2 Implementing reservoirs for emitted matter waves . . . . . . .
2.3 Coupling quantum emitters to the reservoir . . . . . . . . . .
2.4 Outline of the remainder of this thesis . . . . . . . . . . . . .

5
5
6
7
8

3 Introduction to the theory of Open Quantum Systems
3.1 Time reversibility and unitarity: an invitation . . . . . . . . .
3.2 System plus reservoir models . . . . . . . . . . . . . . . . . . .
3.3 The Weisskopf-Wigner model . . . . . . . . . . . . . . . . . .

10
10
12
14

4 Atomic spontaneous emission into structured reservoirs
4.1 Emission near a band gap . . . . . . . . . . . . . . . . . . . .
4.2 Matter-wave emission: overview . . . . . . . . . . . . . . . . .
4.3 Emission near a single energetic edge . . . . . . . . . . . . . .
4.3.1 Introduction to the 1D model . . . . . . . . . . . . . .
4.3.2 Population dynamics . . . . . . . . . . . . . . . . . . .
4.3.3 Markovian limit . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Momentum distribution of the emitted matter-waves .
4.3.5 Matter-wave bound states . . . . . . . . . . . . . . . .
4.4 Emission near a band with two edges . . . . . . . . . . . . . .
4.4.1 Solution for emission into a 1D band . . . . . . . . . .
4.4.2 Bound state oscillations . . . . . . . . . . . . . . . . .
4.4.3 Deep lattice limit . . . . . . . . . . . . . . . . . . . . .
4.4.4 Weak-lattice limit . . . . . . . . . . . . . . . . . . . . .

19
19
20
21
21
24
29
32
34
40
41
46
48
51

v

4.5

4.4.5 Bound states above and below the band . . . . . . . .
Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . .

52
55

5 Experiments on spontaneous emission of matter waves into atunable band structure
56
5.1 Experimental tools and techniques . . . . . . . . . . . . . . .
56
5.1.1 Condensate production in brief . . . . . . . . . . . . .
56
5.1.2 Atom Imaging Scheme . . . . . . . . . . . . . . . . . .
58
5.1.3 State-selective Optical Potentials . . . . . . . . . . . .
59
5.1.4 Microwave hyperfine transitions . . . . . . . . . . . . .
61
5.1.5 Sample preparation . . . . . . . . . . . . . . . . . . . .
62
5.1.6 Determining the resonance condition . . . . . . . . . .
63
5.2 Experiments on free-particle emission . . . . . . . . . . . . . .
63
5.2.1 Spontaneous Emission near a Single Edge . . . . . . .
63
5.2.2 Matter-Wave Emitter Bound State . . . . . . . . . . .
66
5.3 Experiments on emission into a band structure . . . . . . . . .
68
5.3.1 Band Spectroscopy . . . . . . . . . . . . . . . . . . . .
68
5.3.2 Time Dynamics for Emission into a Band . . . . . . . .
72
5.3.3 Bound States Above and Below the Band . . . . . . .
74
6 Theoretical considerations for bound-state induced many-body
physics in optical lattices
76
6.1 Master equations . . . . . . . . . . . . . . . . . . . . . . . . .
77
6.2 Modified superfluid to Mott-insulator transition . . . . . . . .
80
6.2.1 Standard superfluid Mott-insulator transition . . . . .
80
6.2.2 The SF-MI transition with long-range tunneling . . . .
84
7 Conclusion

92

Bibliography

94

vi

List of Figures
2.1
2.2

Schematic of quantum emitter platform . . . . . . . . . . . . .
Experimental coupling scheme . . . . . . . . . . . . . . . . . .

7
8

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9

Atomic spontaneous emission model schematic . . . . . . . . .
Single edge excited state population . . . . . . . . . . . . . . .
Emitted momentum spetcral properties . . . . . . . . . . . . .
Bound state properties near a single edge . . . . . . . . . . . .
Emission into a band structure . . . . . . . . . . . . . . . . .
Characteristic Band Emission Behaviors . . . . . . . . . . . .
Markovian Decay to Oscillation Transition . . . . . . . . . . .
Bound state oscillation amplitude and frequency vs. J . . . . .
Computed band bound state profiles . . . . . . . . . . . . . .

22
28
33
39
42
47
48
49
54

5.1 Science cell with lattice and imaging optics . . . . . . . . . . .
5.2 Absorption imaging of a BEC . . . . . . . . . . . . . . . . . .
5.3 State selective optical potentials for 87 Rb . . . . . . . . . . . .
5.4 Free-emission parabola comparison . . . . . . . . . . . . . . .
5.5 Excited state population dynamics in three regimes . . . . . .
5.6 Single-edge bound states . . . . . . . . . . . . . . . . . . . . .
5.7 Band structure and emission spectrum for sb > 0 . . . . . . .
5.8 Emission spectrum for sb < 0 . . . . . . . . . . . . . . . . . .
5.9 Ground band emission spectrum detail . . . . . . . . . . . . .
5.10 Emission dynamics into a single band . . . . . . . . . . . . . .
5.11 Bound states above and below the band . . . . . . . . . . . .

57
59
60
65
66
67
69
70
72
73
75

6.1
6.2
6.3
6.4

81
85
89
90

Bose-Hubbard model schematic . . . . . . . . . . . . . . . . .
Phase diagram of the Mott-Insulator to Superfluid transition .
E↵ective Hamiltonian phase diagram . . . . . . . . . . . . . .
Purifying a Mott Insulator by emission . . . . . . . . . . . . .

vii

Acknowledgements
What may in hindsight surprise nobody is that I’m finding this section to be
the hardest to write in my whole thesis, and that’s because I have so many
people to whom I’d like to dedicate this work. It is unlikely for me to be
sparing with my word count in any endeavor, let alone thanking everyone who
helped to make this thesis possible, but I will do my best.
I wish to first thank my advisor, Dominik Schneble, whose mentorship andguidance have proven invaluable in my development as a researcher. From
the outset, Dominik has a↵orded me the freedom to explore and grow as anexperimental atomic physicist in ways that made sense to me and at the sametime never made me feel like my theoretical penchant needed to be set aside. In
addition to providing incredible clarity on a vast range of atomic physics andultracold atom concepts and techniques (not to mention a style of performing
pen-and-paper calculations which I use to this day), I can always count on
Dominik to be present in the lab, keeping me on track, and eager to discuss
results, even before they’ve been fully acquired. As much as I joke about beinga lapsed theorist, I couldn’t be happier that Dominik chose to take a chance on
this would-be theorist and helped me to grow into a talented experimentalist
with a theoertical heart.
Throughout my many years at Stony Brook, I’ve had the pleasure of working with and befriending some incredible individuals in the lab. I truly could
not have gotten to the point of writing this thesis without Ludwig Krinner.
The wealth of knowledge he has about all things optics, technology, and physics
more than once saved me when learning to run experiments on my own, andthat’s not to mention the Sachertortes and Schwarzwälder Kirschtortes we got
up to baking outside of the lab. Jeremy Reeves took me under his wing andhelped me to find stable footing when I started, and his incredible, deadpan
humor was a pleasure to work with. I am indebted to Arturo Pazmiño fortechnical support. There’s nobody I can think of more skilled at electronics,
and my experiments wouldn’t have been possible without his tireless e↵orts.
Joonhyuk Kwon has been a steadfast companion and an indispensable sourceof optimism, through broken lasers and triumphant data acquisitions alike.

viii

I’m very excited to see how his work builds on my experiments and expands
the playbook of what is possible with matter-wave quantum emitters. I would
be remiss not to thank Alfonso Lanuza for his incredible ability to both carryout detailed theoretical calculations and to distill their essence into understandable intuition, and Youngshin Kim, though he recently joined the group,
because he has already revolutionized our data analysis procedures. I am going to miss attending conferences with you all, and I hope you’ll pour one out
for me after you give incredible talks highlighting your awesome work!
I want to thank my Ph.D. defense committee for working with me to complete my degree in a historical fashion, being the first fully remote defense in
the Department of Physics and Astronomy at Stony Brook due to Covid-19.
Each of Thomas Weinacht, Tzu-Chieh Wei, and Ana Asenjo-Garcia have mythanks for attending my defense and for their careful reading and thoughtfulcomments on the text of my thesis. I have had the pleasure of working closelywith Tom on the Quality of Life Committee for about as long as I’ve been at
Stony Brook, and it pains me to say that there’s not a chance anymore that
I emcee for the holiday party. I hope that this thesis helps settle the longstanding debate between Tom and Dominik about how to understand atomic
spontaneous emission. In addition to my committee, I must thank Martin
Cohen and Hal Metcalf for all of the informal mentorship I’ve been privileged
to receive over the years. Marty especially has been a gracious and patient
reader of many a paper draft, and his knowledge of all things optics has been
essential to my experimental success.
Outside of my own research group, I have to express my profound love andappreciation to the atomic physics community, especially Brian, Ian, Bertus,
Mehdi, Yusong, Chuan, YiFan, XiaoYang, Samet, Guodong, and Sam. I could
share anecdotes about what you each mean to me, but I only have limited
space, so suffice it to say that each of you in your own way has made living
and working in the basement a bit happier despite the chronic lack of sunlight.
I wish you all success in your own futures.
Beyond the atomic physics crowd, I want to thank Lucie and Nick, some of
my earliest friends at Stony Brook and long term roommates. There’s a dinnerparty brewing, I just know it, and too many inside-jokes to count. I also would
like to thank Fernando, Ben, Ashley, Melissa, Sheridan, Lupe, Batu, Mark,
Adith, Emily, Jxhn, Michael, Alyssa, Melissa and Haylee, and Matt. We’ve
made some good memories, and I’m eternally grateful to have met you.
My family has supported me throughout the years and shared in both thehighs and lows of Ph.D. life. My parents, Lydia and John, have always been
there to o↵er support when I needed a shoulder to cry on, as well as to join
in my celebration when I achieved something I thought would be impossible.

ix

Additionally John and Cara have helped me to succeed, with advice on jobhunting and motivational pictures of the family dog, Emmie. I couldn’t be
more grateful or proud of you as you start a new job and get your own doctorate
in physical therapy. We made it.
Finally, I’d like to thank Himanshu from the bottom of my heart. You’vebeen a rock, anchoring me in reality when I was liable to lose my head in theclouds, and you’ve made me laugh when I thought everything was hopeless.
I’m truly a better person for knowing you, and I can’t wait to see what thenext chapter holds in New York City.

x

Chapter 1
Introduction
The seminal insight of Einstein in the field of statistical mechanics [1] is, arguably, that classical understanding of an ideal gas of particles breaks downwhen it is cooled below a certain temperature. Specifically, he showed, based
upon the fully quantum calculation of Bose for the case of photons [2], thatmatter, in addition to light, has to be understood as having modified statisticalproperties at low temperature, and that for particles that we now call bosons(having integer spin), it is possible at sufficiently low temperatures to have amacroscopic occupation of a single quantum state [3]. This macroscopicallyoccupied state is described by a single, coherent wave-function, but it can bethousands to millions of times larger than the typical atomic scale at whichquantum mechanics dominates the behavior. This makes Bose-Einstein condensates (BECs) ideal systems for studying quantum physics in a laboratory
setting.
Today, BECs [2–4] are routinely produced in the laboratory following thesame procedures that led to the first condensates in dilute alkali-metal gases
25 years ago [5–8]. These ultracold (⇠ 100 nK) samples have become a cornerstone of Atomic, Molecular, and Optical (AMO) physics research (though
by no means the only avenue of research in this vast and expanding field).
While this thesis focuses on BECs, we also wish to note that fermions canbe cooled to quantum degeneracy as well, resulting in degenerate Fermi gases(DFGs) [9–12], whose applications are just as varied and fascinating. In theintervening decades since the creation of the first laboratory BEC, research efforts have demonstrated their coherence properties, with applications to atom
optics [13–18], wave-like physics of matter including sound [19, 20] and solitons [21, 22], and studies of superfluidity [23, 24]. Degenerate quantum gasesof hydrogen and helium [25, 26], alkali metals (e.g. potassium [12, 27] and cesium [28]), and more recently transition metals (chromium [29]), alkaline-earth
metals (calcium [30] and strontium [31, 32]), and lanthanides (ytterbium [33],1erbium [34], and dysprosium [35]) have been brought to quantum degeneracy
as well.
Through suitable experimental techniques, the aspiring AMO physicist cancontrol just about every relevant parameter of an ultracold gas. For example,
utilizing Feshbach resonances [36, 37], the interaction strength between ultracold atoms can be tuned. With the advent of optical lattices [38, 39], it
became possible to simulate condensed matter Hamiltonians, such as the BoseHubbard model [40] on a square lattice [41], and more recently, in hexagonal,
tunable, or graphene like lattices [42–46], and even in quasicrystalline geometries [47, 48]. Fermions in optical lattices can realize the Fermi-Hubbard modeland its attendant anti-ferromagnetism [49–53]. Promisingly, we can engineere↵ective Hamiltonian terms, e.g. via Raman laser pairs for spin-orbit coupling[54–56] or time-periodically modulated system parameters (realize Floquet engineering of e↵ective Hamiltonian terms) allowing for studies of Haldane andHofstädter models [45, 57, 58] and even relativistic physics [59] or long-range
transport across a lattice [60]. Advances in optical technologies have allowed
for the resolution of atoms on a single site of an optical lattice, for both bosons[61–63] and even fermions [64–67]. Disorder can be added to a quantum simulation using laser speckle [68] or atomic disorder [69]. Many important models
of many-body physics have no known theoretical solution, but they can bedirectly accessed by quantum simulation [70–73]. Exotically, lattices in momentum space can be realized, o↵ering, e.g., access to studies of self trapping[74] and topological Anderson insulators [75]. In addition to studies of single
species, composite degenerate gases can be used to study exotic physics, forexample bosons composed of fermions [76, 77] and ultracold polar molecules[78–80].
In order to focus the following discussion, we briefly introduce the notionof quantum simulation [81]. Quantum simulation (either digital or analog) is
one of the four pillars of quantum information science, driving major researche↵orts. Quantum simulation [72] in its analog (or analogue) form is an umbrella term for using one quantum system that is easy to produce and tune
with physical properties that are similar or identical to a quantum system ofinterest, to study a system which is more difficult to study in this way. Ultracold quantum gases are ideal platforms for performing quantum simulation
because they are ultracold, which makes them correspondingly “ultra-slow”(vtyp ⇠ mm/s) and “ultra-large” (rtyp ⇠ µm), such that their dynamics can beeasily observed.
The Schneble laboratory has a storied history of quantum simulation of everything from four-wave mixing of matter-waves [82], the superfluid to Mottinsulator transition in atomic species mixtures [83], disorder in the Bose glass2

[69], the interplay of disorder and interactions on Bloch dynamics [84], andmatter-wave di↵raction from a matter wave crystal [85] and from a nonadiabatic “non-”potential [86]. Furthermore our lab is interested in using
BECs to study the physics of open quantum systems, with previous works
ranging from studies of a quantum to classical transition in a system of delta
kicked rotors [87] to polaron physics [83] in hyperfine state mixtures. Researche↵orts on polaron physics have also led to attempts to study the spin-bosonmodel [88–90], though so far, technical limitations have frustrated our attempts to realize it. Most recently, we have demonstrated a novel platform forthe study of spontaneous emission of matter waves [91].
In order to understand how quantum systems, which are in principle time
reversal symmetric, can undergo irreversible processes, a number of theoretical
approaches to these open quantum systems have been developed [92, 93]. The
central idea is that of a small quantum system coupled to a vast reservoirof quantum states into which information about the system is lost and overtime never returns to the system. To be specific, a typical open quantum
system consists of a small number of states, e.g. a two-level atom, coupled to(infinitely) many states, e.g. photons in free space, or the Bogoliubov modes
in a BEC. If we consider an excited atom in free space, it is known that sometime later, the atom will decay to its ground state and release a photon ofsome energy in some direction. This spontaneous emission serves as a toy
open quantum system. The original treatment by Weisskopf and Wigner [94]in 1930 sought to explain the phenomenon using Dirac’s radiation theory [10].
It was realized later by Purcell that the density of available states can have a
profound e↵ect on the emission behavior [95]. At one extreme, we have cavityQED [96] with only one available mode for emission, in which it would be
impossible to speak of the single cavity mode as a “large” reservoir. The otherextreme is the aforementioned Weisskopf-Wigner situation, in which the decay
is completely controlled by the infinite reservoir of photon states. In between
these two extremes is the regime of a bounded continuum of photon states.
If these boundaries are realized as the edges of a band in one-dimension, thedensity of states diverges at these points, and any attempt to treat the system
in a Markovian approximation is doomed to fail. Historically, consideration ofthis regime has been limited to photonic crystal materials [97–99] (also called
photonic bandgap materials). It is this regime of singly and doubly bounded
continua that we implement in a matter-wave context in this thesis. Here, theenergetic edge(s) in the density of states leads to a much richer physics in thisintermediate regime, including exotic decays and bound states. In particular,such systems provide an exciting platform for the engineering of strong photonmediated interactions, both in the optical [100] and microwave [99] regimes.

3

These can result from the formation of atom-photon bound states [97] in which
an excited atom is surrounded by an exponentially localized cloud of photonic
excitations, which may extend over a long range.
While the field of waveguide-QED is broad (see [101–105]), we highlightspecifically corrugated optical waveguides in one-dimension [106, 107], whichhave displayed superradiance and coherent dipole-dipole interactions [108,109], as well as microwave guides using transmon qubits [110] which have also
displayed coherent interactions between two atoms [111]. The reason that wehighlight these one-dimensional systems is that we can realize a cold-atomic
matter-wave analogue in the lab. First proposed in 2008 [112–114] as a candidate for the quantum simulation of the emission of photons near a photonic
crystal, ultracold atoms in state-dependent optical lattices o↵er a complementary view of the physics of such systems. Since then, a flurry of theoreticalresearch has predicted the behavior of spontaneous emission of matter waves
in various geometries and dimensions [115–118] exhibiting strongly modified
emission properties and bound states. On the experimental side, we recently
developed a corresponding platform utilizing strongly confining 1D tubes witha state selective lattice along the third direction for the direct quantum simulation of photonic crystal physics [91, 114, 119]. Utilizing a microwave toenact an internal state change, we realize an array of initially excited emitterswith tunable excited state energy and coupling strength, which we can use to
explore in principle all regimes of the system, including those that are difficult
to reach in a material sample, as well as gain access directly to complementary observables from the material case, such as the momentum of the emitted
matter-wave radiation and the time dynamics of the excited state population.
Due to the somewhat technical nature of much of the work described inthis thesis, we first introduce the experimental system in the following chapter.
Equipped with a full understanding of the quantum emitter platform, we shallproceed to describe our experiments and the theoretical considerations which
led to them.

4Chapter 2
Overview of the experimental
platform
In order to study the spontaneous emission of matter waves into structured
reservoirs (emission vacua) of interest, we must bring together several ingredients. First, we need an emitter of matter waves and a tunable set of statesthat can be emitted into. Secondly, we require a coupling between the twostates, and finally, we require experimental probes of the coupled emitter andemitted matter wave system. Our recent implementation [91], which forms
the platform for the experiments discussed in this thesis, is largely based ona theoretical proposal of I. de Vega, et al. [112], though we shall point out
where our implementation di↵ers from that proposal.

2.1

Implementing a quantum emitter for matterwaves

In order to implement a system of quantum emitters for matter-waves, we startwith a Bose-Einstein condensate (BEC) of rubidium-87 atoms in the F = 1,
mF = 1 hyperfine ground state (c.f. chapter 5 and [120].) At the heart of ourexperimental implementation is that we have access to and control over theentire ground state manifold of our rubidium BEC, and so we can choose one
hyperfine state |ri = |F = 1, mF = 1i to be the excitation in the quantum
emitter and another hyperfine state |bi = |F = 2, mF = 0i to be the emitted
matter-wave state.
For the sake of our experiments, an |ri atom must be confined in position,and unable to access other states without an applied coupling. This turns
out to be conveniently achieved by using an optical lattice potential. The
simplest optical lattice is formed by the interference of two laser beams to5

create a standing wave. By confining the BEC to a deep optical lattice, we
localize the condensate atoms to the ground state wave-functions of the latticewells. By making the lattice very deep, we suppress tunneling from one wellto its neighbors, which is necessary for creating a state which, when suitably
coupled, can serve as a quantum emitter for matter waves.
In our lab, we study one-dimensional physics, and we can achieve thisthrough the use of more than one optical lattice. Using two very deep, horizontal optical lattices crossed at ninety degrees, we first create an array of
isolated 1D tube potentials using = 1064 nm light. These tubes are isolated
in the following sense: the timescale for particle in one tube to tunnel to aneighboring tube is much longer than the experimental timescales of interest,and so neighboring tubes do not play a role in the evolution of the system. We
then apply a third optical lattice along the array of tubes, and thus we create
an array of quantum emitters which emit into 1D. We control the propertiesof the array such that each emitter contains one or less |ri atom. This isachieved by first creating a Mott-insulating state and then “thinning it out”,
c.f. chapter 5.

2.2

Implementing reservoirs for emitted matter waves

Having discussed how we create an array of quantum emitters in 1D, we now
turn our attention to the second ingredient of our platform. As briefly mentioned before, we will use a di↵erent hyperfine ground state of rubidium-87,|bi = |F = 2, mF = 0i, as our emitted matter wave state(s). Naively applying
an optical lattice along the tubes for the |ri atoms will in principle also create
a potential for the |bi atoms, since a typical optical lattice is created using faro↵ resonant light, which a↵ects all of the hyperfine states equally. The resolution to this problem is to take advantage of so-called state-dependent opticallattices (SDOLs), which are detailed in section 5.1.3. In brief, by choosing alaser wavelength between the strong D1 and D2 transition in rubidium [121],
which for us will be ⇡ 790 nm, the optical potential can be chosen in such a
way as to be small or else vanish outright for |bi atoms, while remaining deepfor |ri atoms.
The shallow or vanishing SDOL potential along the tube for |bi atoms
suggests the following interpretation of the quantum states in the tube: theavailable states are plane waves propagating to the left or right (along theaxis of the tube, strictly speaking up and down), indexed by the (continuous)
momentum of the emitted wave. If the SDOL is nonvanishing for the |bi atoms,
6

Figure 2.1: Cartoon illustrating our quantum emitter platform. The emitters(shown as individual gray potential wells) contain |ri atoms, and by a suitablecoupling (described in section 2.3) can transition to the emitted matter-wave
states |bi, which are then free to propagate along the 1D tube. A typical experimental absorption image of the propagating atoms in time of flight is shown
below the cartoon. Due to the left/right (up/down) symmetry of the situation, we observe two blobs of atoms in the image, corresponding to emission
at |k|/kr , as fixed by the emitter excited state energy.
we instead work with Bloch waves of definite quasimomentum.

2.3

Coupling quantum emitters to the reservoir

The primary di↵erence between the theoretical proposal [112] and our experimental implementation is the manner in which we couple the emitters to the
propagating states. Rather than using a pair of Raman lasers as in [112], we instead utilize microwaves to couple the states and allow for transitions between
them. In particular, our two chosen hyperfine ground states in Rubidium87 are separated by approximately 6.8 GHz, and by choosing the frequencyof our microwave coupling, we can select the energy of |ri relative to |bi, as
illustrated in Fig. 2.2(B). The strength of the microwave coupling ⌦ is also
a tunable parameter in the experiment. In combination with the SDOL potentials described above, we thus realize an array of quantum emitters withtunable excited state energy ~ coupled to freely propagating (or else Blochwave) modes at tunable energy and tunable coupling strength ⌦.
7

(A) Bare States(B) Rotating
Frame(C) S.D.O.L.

Figure 2.2: Experimental coupling scheme to realize a matter-wave quantum
emitter. (A) The bare ground states |ri = |1, 1i and |bi = |2, 0i at 5 G magnetic field along z are chosen as the emitter and emitted states, respectively.
(B) Microwave coupling !µ ⇠ 6.8 GHz, variable strength ⌦ is applied, andworking in the rotating frame, |ri is detuned from |bi by an amount . (Thefact that the bare |ri state is lower in energy than |bi makes no di↵erence
once the coupling is applied.) (C) Introduction of a state-dependent opticallattice (S.D.O.L.). The trapped |ri atoms are shifted upward in energy by!0 /2, which shifts to= + !0 /2. |bi atoms are free to propagate along
a 1D tube, thus resulting in a continuum of momentum states |bk i havingenergy "k / k 2 , coupled to with strength ⌦k = ⌦ k , c.f. equation 4.13 dueto the variable Franck-Condon overlap between the Wannier-state in |ri andthe variable momentum state |bk i. Illustration adapted from Krinner, et al.,Nature 559, 589 (2018) [91].
Given these ingredients, all we require is an experimental probe of thesystem. To this end, our lab utilizes state selective absorption imaging in timeof flight. For details of our imaging scheme, see section 5.1.2, but in brief,
we let the atomic sample expand and fall, separate the hyperfine components,
and then image their shadows on a CCD camera using resonant light.

2.4

Outline of the remainder of this thesis

Having described our experimental platform, we are faced with two questions:what can we do with our platform, and how can we understand what we do?
The remainder of this thesis will attempt to answer these two questions. As
for what we can do with our platform, chapter 5 will describe experiments onspectroscopy of our system, time dynamics including fractional decay of theemitter population, and the formation of bound states of quantum emitters8and matter waves, for both free particle and Bloch wave reservoirs. In orderto understand our results, a detour through the basic theory of open quantum
systems, chapter 3, and the specific model of atomic emission that we are interested in [112, 114, 122] is required, and we present our model and calculatede↵ects in detail in chapter 4. While the experiments detailed in this thesis
focus primarily on single-particle e↵ects, our platform is well suited to studies
of many-body e↵ects, and we give an overview of some such e↵ects in chapter 6, focusing especially on how the well-known superfluid to Mott-insulatortransition is modified in our system. We end in chapter 7 with a summary ofwhat we have done up to this point and an outlook towards future work in
our lab.

9

Chapter 3
Introduction to the theory ofOpen Quantum Systems
3.1

Time reversibility and unitarity: an invitation

Quantum mechanics is well known to be governed by unitary time evolution,or equivalently, by the Schrödinger equationi~@| i = Ĥ | i
@t

(3.1)

which has time-reversal as a symmetry. (We note here that we are assuming theabsence of magnetic fields, which break the symmetry.) Mathematically [123],
this means there exists an anti-unitary operator ⇥ which leaves the Hamiltonian invariant, ⇥ 1 Ĥ⇥ = Ĥ. In short: the laws of physics look the samerunning forward in time as they do if time were to run backward. However,this contrasts with our everyday experience of the classical world, in which aball rolling along a rough surface eventually comes to a stop, but without some
sort of kick, a stopped ball never spontaneously starts rolling. Implicit in thepreceding example is a notion of dissipation, in which some energy in the system is lost to an external environment from which it never returns. One might
ask then whether there is a similar process at play in Quantum Mechanics,
which after all is understood to describe the natural world on small scales.
It turns out that we have examples of quantum mechanical systems which
exhibit dissipative behavior. Perhaps the most fundamental such example isthat of spontaneous decay of (and emission of a photon from) an excited atom,which results in a ground state atom having lower energy and a photon prop-10

agating away in some direction. To extend the analogy with the classical ballabove a bit more, the ground state atom doesn’t spontaneously become excited(except if it receives a “kick” by absorbing an appropriate photon that just so
happens to pass by). However, in quantum mechanics such an irreversibility
should in principle not be possible, because (3.1) is invariant under time reversal. The answer to this puzzle is the introduction of a (sub-)quantum system
of interest (the initially excited atom, or else the same atom in its groundstate), and an (infinitely-)large reservoir of quantum states (possible photon
polarizations, energies, and propagation directions). The system plus reservoir
as a whole is governed by reversible, unitary quantum mechanics, but if we do
not look too closely at the reservoir, the (sub-)system of interest appears toundergo dissipative behavior as its initial excitation escapes into the reservoir.
As described in detail in [124], many systems are well described as open
quantum systems. From diverse applications to quantum information andcomputing, quantum biology and chemical physics, applications of open quantum systems are varied and it behooves us to understand well how to describe
them.
Generally speaking, solid-state physics is concerned with many models of
open-quantum systems, especially when the coupling to the environment isweak. Brownian motion, the random motion of a particle in a backgroundfluid, may also display non-Markovian e↵ects, and theoretical and experimental e↵orts are ongoing for such systems ranging from quantum dots lasercoupled to reservoirs [88, 125] or nanomechanical oscillators coupled to BECs
[126–128]. Furthermore, open quantum systems in solid state physics may
also couple to fermionic (rather than bosonic) degrees of freedom, or even spin
degrees of freedom [129–133]. Even in biological applications, quantum behaviors are seen to arise, and an open quantum system description is appropriate.
Much e↵ort studying coherences in photosynthetic complexes (molecues) has
shown that these systems must be considered as quantum systems interactingwith a phonon bath [134–137]. These phenomena are beyond the scope of thisdissertation, but we note they are often treated using the same techniques
as solid-state systems [138]. Chemistry as well contains examples of openquantum-system behavior, where, for example in a large molecule, a system
consists of a few bonds (or molecular modes) of interest, and the rest of thebonds serve as a reservoir.
In section 3.2, we will formalize the notions introduced in the preceding
paragraphs. Then in section 3.3 we will present what is perhaps the most wellknown dissipative quantum system: the Weisskopf-Wigner model of spontaneous emission. The reader is strongly encouraged to spend time with the
Weisskopf-Wigner model, as it serves as a leitmotif for the rest of this thesis.

11

3.2

System plus reservoir models

Much theoretical e↵ort has been devoted to describing the dynamics of dissipative quantum systems [92, 93, 139, 140]. One promising approach to understanding dissipation in quantum mechanics is to divide the Hilbert space into
two parts: a small system S consisting of a few states of interest, and a large
reservoir (or bath) R consisting of (infinitely-)many degrees of freedom. The
system and reservoir are assumed to be coupled due to an interaction. Furthermore, we will assume that the Hamiltonian of the system plus reservoir
can be neatly decomposed as Ĥ = ĤS + ĤR + ĤI , with ĤS and ĤR depending
only on system (reservoir respectively) degrees of freedom, and ĤI contains allof the interactions between the two. Note that such a system S is said to be
an open quantum system (OQS) since its interaction with the large reservoir
may introduce dissipation or decoherence.
The main idea, then, is to solve for the dynamics of the system of interest
without needing to fully solve for or specify what happens in the reservoir.
Specifically, if the density operator for the system plus reservoir model is ⇢ˆ(t),our goal shall be to try to understand the dynamics of the reduced density operator ⇢ˆS (t), which is obtained by tracing over the reservoir degrees of freedom,i.e.
⇢ˆS (t) = TrR [⇢(t)](3.2)
The time evolution of this reduced density operator can then in principle bedescribed by a formally exact quantum master equation:
dˆ
⇢
=
dt

h

i
i Ĥ, ⇢ˆ + D (ˆ
⇢)

(3.3)

explored in more detail in chapter 6. In practice, approximations are often
made to assist in solving the full equations of motion, though this need not be
the case.
Perhaps the most famous system plus bath model is the Caldeira-Leggett
model [139], in which a system consisting of a particle of mass m moving in one
dimension (coordinate x) is coupled to a bath of quantum harmonic oscillators
via a linear interaction. In particular the system and reservoir Hamiltonians
may be written
p̂2
ĤS =
+ V̂ (x̂)
2m 
X p̂2
1
↵
ĤR =
+ m↵ !↵2 x̂2↵
2m↵ 2
↵

12

(3.4)
(3.5)

If the coupling to the environment is weak, then the interaction between thesystem and the bath can be assumed to be a linear function of the environment
coordinates x↵ , and thus the interaction Hamiltonian has the formXĤI =F↵ (x̂)x̂↵ + V (x̂)
(3.6)↵

The second term in (3.6) is a counterterm which renormalizes the potential felt
by the particle. Our goal here is not to study the Caldeira-Leggett model ingeneral, however, we wish to note this model has found widespread applicationin the field of open quantum systems due to its generality. In fact, we shallshow that the systems we consider have reservoir Hamiltonians of the form
(3.5), and only slightly generalized system and interaction Hamiltonians.
The interaction Hamiltonian (3.6) couples states in the system of interestand the reservoir. As hinted at before, it leads to a quantum master equation(3.3) which describes the evolution of the density operator (i.e. populations
and coherences) of the quantum system after having traced over the reservoir
degrees of freedom. It can be shown generally [92, 124] that the commutator
term is nothing more than the usual Heisenberg equation of motion underan e↵ective Hamiltonian (having shifted energy levels), and thus represents
unitary evolution. The second piece of the equation is called the dissipator, and
it is this term which leads to a decrease of the population in the excited statesof the quantum system, as well as their coherences, by capturing non-unitary“jumps” induced by the reservoir. In short, dissipation is accompanied by ashift of the energy levels, which we shall see generally applies to the systems
we consider!
We stop here to take stock of what we have set out to do. By separating acomposite quantum model into a system of interest and a reservoir with which
it interacts, we can maintain full unitarity of the total quantum model (andthus time-reversal symmetry) while at the same time introducing apparent
dissipation into the small system of interest. At this point, we could proceedby computing the (Markovian or non-Markovian) [124] master equation for
the reduced density operator of the system, and in this way, learn about thedissipative dynamics of the model. (See chapter 6 for some details.) However,
it is sometimes possible to obtain explicit analytic results for an open quantum
system, and we turn our attention to one such model now.

13

3.3

The Weisskopf-Wigner model

We consider as our worked example of a system plus reservoir model in which
irreversible decay arises in quantum mechanics the Weisskopf-Wigner modelof spontaneous emission [94], which can be found in many textbooks (see, e.g.
[141]. The system, solved using Dirac’s radiation theory [10], in this case isan initially excited, two-level atom whose exited (ground) state we label as |ei(|gi respectively). We consider such an atom to be located at the origin of athree-dimensional space, and we assume further that the atom is coupled to
an infinite continuum of propagating photon modes, characterized by a wavevector k and a polarization labeled by s. For completeness we note that theHilbert space consists of those states having an excited atom and no photons
present, |e, 0i, and those containing a ground state atom and exactly one
photon with specific wave-vector and polarization, |g, 1k,s i. The Hamiltonianof the system can be written
X
ĤW W = ~!0 |ei he| +~!k (â†k,s âk,s + 1/2) + ĤI
(3.7)
k,s

where ~!0 is the excited state energy of the atom, ~!k = c|k| is the energyof a photon with wave-vector k, and âk,s annihilates a photon having wavevector k and polarization s. As usual, its conjugate operator â†k,s createsthe corresponding photon, and the usual bosonic ladder operator algebra issatisfied, i.e. [âk,s , â†k0 ,s0 ] = (k k0 ) s,s0 .
The final term in equation (3.7) is the interaction Hamiltonian which couples the excited atom and the photon modes. It is [141–143]
i
Xh
†
ĤI =
~gk,s âk,s |gi he| + H.c.
(3.8)
k,s

where we have introduced a shorthand notation of gk,s to denote the strength ofeach interaction term. Before discussing what gk,s is in this model, we note thatthe interpretation of (3.8) is clear: at the expense of transitioning the atomfrom the excited state to the ground state, we can create a photon of specified
wave-vector and polarization with strength gk,s , and we can also excite theatom from the ground state to the excited state at the cost of destroying aphoton. The interaction strength is derived from a quantum-optics treatment[142], and it can be shown to ber~!kgk,s = i"k,s hg| d̂ |ei
(3.9)
2✏0 V
14

where ✏0 ⇡ 8.854 ⇥ 10 12 F·m 1 is the permittivity of free space, V is system
volume, "k,s is the polarization vector of the given photon, and hg| d̂ |ei is the
dipole matrix element coupling the excited state to the ground state via theelectromagnetic field [142].
Originally, Weisskopf and Wigner posited that any physically relevant solution to the Hamiltonian (3.7) would exhibit exponential decay, and fromthis assumption, they derived what the exponential decay rate would be. The
virtue of this model as an instructive example of system plus reservoir models
lies in the fact that an explicit solution can be found by solving Schrödinger’s
equation, which we shall demonstrate now. Because we are considering a restricted Hilbert-space having only one excitation present, the general solution
to the time-dependent Schrödinger equation will be of the formX
| (t)i = A(t)e i!0 t |e, 0i +Bk (t)e i!k t |g, 1k,s i(3.10)
k,swhen working in the interaction picture [144]. We begin with an initially
excited atom, so the initial condition is A(t = 0) = 1 and Bk (t = 0) = 0 for
all k. Substitution into (3.1) yields the following system of equations:X⇤Ȧ(t) = igk,se i(!k !0 )t Bk,s (t)(3.11)
k,s

Ḃk,s (t) = igk,s ei(!k !0 )t A(t)(3.12)

Formally, we can integrate (3.12) and substitute it into (3.11), which yields
the intermediate result
Z t
X
0
2
Ȧ(t) =
|gk,s |
dt0 e i(!k !0 )(t t ) A(t0 )(3.13)k,s0

We note that this solution is formally exact. It is in principle solvable byrealizing that a di↵erential equation where the derivative of a function is proportional to its convolution with another function can be solved by the methodof Laplace transforms. However, we note instead that the exponential in the
integrand oscillates wildly if !k and !0 di↵er, except when t ⇡ t0 . We therefore make the approximate replacement A(t0 ) ! A(t) and remove it from theintegrand in (3.13), yielding
Z t
X
0
2
Ȧ(t) =
|gk,s | A(t)dt0 e i(!k !0 )(t t )(3.14)
0k,s15This approximation amounts to assuming that A(t) varies slowly in time, which
is a form of Markov approximation. Systems which are Markovian are said
to be “memory-less”, that is, their evolution does not depend on their pasthistory. Rather, the current state of a Markovian system fully determines its
evolution a short time later, and by removing A(t) from the integral in (3.13)
to yield (3.14), we manifestly remove the system’s “memory of its previous
trajectory”, hence justifying the name Markov approximation.
In order to proceed, we introduce some notation: let k = !k !0 (independent of the wave-vector direction) and let ⌧ = t t0 . Changing the variableof integration in (3.14) from t0 to ⌧ and integrating yields
Z t
Z 1
i
0
i k (t t0 )
dt e
⇡ lim
d⌧ e i k ⌧ "⌧ = lim(3.15)
"!0
"!0 0
i"
k
0
where we have introduced an epsilon regulator " in order to tame the integrand’s behavior at long times. Equation (3.14) now becomesXȦ(t) =k,s|gk,s |2 A(t) lim"!0i(3.16)i"k

and it becomes imperative that we treat the summation over wave-vectors andpolarizations. Since we are considering a three dimensional system, we may
replace the summand as
X

V
!
(2⇡)3
k,s

Z 1

2

k dk

0

Z ⇡

sin ✓d✓

0

Z 2⇡

d

0

2
X

(3.17)

s=1

If we convert the momentum integral to energy by swapping the variable ofintegration from k = |k| to !k = ck, and let dge = hg| d̂ |ei, then (3.16)
becomesȦ(t) =Z ⇡sin ✓d✓

0

⇥

Z 1
0

Z ⇡

d

0

2
X
s=1

|"k,s · dge |2 A(t)d!k !k2 V
!k
lim
2
3
c c (2⇡) 2✏0 V ~ "!0

i
k

i"

(3.18)In order to make progress in (3.18), we note that the polarization sum can beshown to be
2
X|"k,s · d̂ge |2 = |dge |2 sin2 ✓(3.19)
s=1

16

and noticing that the integral contributes a factor of 2⇡, and with (3.19),
the theta integral contributes
Z ⇡0

sin ✓d✓

2
X
s=1

2

|"k,s · d̂ge | = |dge |

2

Z ⇡
0

4
sin3 ✓d✓ = |dge |2
3(3.20)and thus (3.18) becomes
Z 1
1 |dge |2 4
i!k3
Ȧ(t) =
A(t)
lim
d!
k
"!0 0
2 4⇡✏0 3⇡c3 ~(!k !0 )Z 1i( k + !0 )3= Constants ⇥ lim
d k
"!0
i"
k
!0

i"
(3.21)

where we have written the energy integral in a suggestive form.
At this point, we recall the Sokhotski-Plemelj theorem [145], whose statement is as follows: let a < 0 < b. The following replacement holds:
Z b
Z b
f (x)
f (x)
lim+
dx = ⌥i⇡f (0) + P
dx(3.22)"!0x
a x ± i"awhere here P indicates the Cauchy principal-value. This is the formal underpinning of the common physicist’s replacement11⇡ ⌥i⇡ (x) + P .
x ± i"
x(3.23)(As an aside, the author was always uncomfortable using this replacement incourse-work until discovering this rigorous formulation in a research context.)
Returning to the matter at hand, we see that the integral in (3.21) satisfies
the hypotheses of the theorem since in order to describe something physicallyreal, !0 > 0. We can thus utilize the Sokhotski-Plemelj theorem to write
Z 1
Z 1
i( k + !0 )3( k + !0 )3
3
lim
d k
= i⇡ ⇥ i(0 + !0 ) + iP
d k
(3.24)
"!0
i"
k
k
!0
!0
We therefore finally arrive at the solution to (3.14) by utilizing (3.24) in(3.21), and we find the (perhaps expectedly) simple form✓◆Ȧ(t) =
t + i L t A(t)(3.25)
217

where we have introduced the decay rate=arising from the first term in (3.24):

1 4 !0|dge |2
4⇡✏0 3 ~c3(3.26)

which is exactly the predicted spontaneous decay rate found in standard atomic
physics references [146], and the Lamb-shift L which arises from the second
term in (3.24):
1 2 |dge |2=PL
4⇡✏0 3 ⇡~c3

Z 2me c2 /~
0

d!k

!k3
!k

!0

(3.27)

where in order to avoid a divergently large integral in (3.27), we have cut o↵ theupper limit of integration from infinity to the electron-positron pair production
threshold. The population in the initially excited state decays exponentiallywith rate , and the energy of the state is shifted by an amount L [147].
We see from (3.25) that the population in our initially excited atom decays exponentially with a rate determined by properties of the atom and theelectromagnetic field, namely, proportionally to |dge |2 . This exponential decay
appears to be a dissipative e↵ect, however, we can think of it instead as a fully
coherent evolution of a combined system and bath featuring an infinitely longbeat note between the initially excited atom and all possible photon modes.
If one is only concerned with the evolution of the system, then one can traceover the e↵ects of the bath and arrive at an e↵ective model featuring exponential decay. We note that without replacing the upper limit of integrationin (3.27) by the pair production threshold for electrons and positrons, theintegral diverges. This problem was originally resolved by Bethe [148] using
mass-renormalization of the electron.
We have worked out this simple example in great detail for two reasons.
First, the models presented in the following chapters will di↵er from the solution presented here in that the Markovian approximation will no-longer beapplicable. In order to fully appreciate the richness of the non-Markovian solution, one must first grapple with the Markovian model. Secondly, in solving theWeisskopf-Wigner model, we arrived at a decay rate, but interestingly, we alsoarrived at a frequency shift, the celebrated Lamb shift of atomic physics. Thisconnection between decay and energy shifts, a manifestation of the KramersKronig relations, is quite profound in physics, and we will also see it again
and again in the following work. We next turn our attention to a system plus
reservoir model where we cannot make the Markov approximation, and in sodoing, develop a model which is ripe for quantum simulation in an ultracold
atom lab.

18

Chapter 4
Atomic spontaneous emission
into structured reservoirs
This chapter is partly based on our publication Analysis of non-Markovian
coupling of a lattice trapped atom to free space, Michael Stewart, Ludwig Krinner, Arturo Pasmiño, Dominik Schneble, Phys. Rev. A, 95, 013626 (2017)[114] (section 4.3), and on our recent preprint Fractional decay of matter-wave
quantum emitters in a synthetic bandgap material, Michael Stewart, Joonhyuk
Kwon, Alfonso Lanuza, Dominik Schneble, arXiv:2003.02816 (2020) [149] (section 4.4).

4.1

Emission near a band gap

The spontaneous emission of photons from excited atoms is extremely sensitive
to the density of states of the radiation field, and the observed emission behaviors thus depend upon whether an atom emits into free space (as in section3.3) or else into a cavity or some kind of material structure. This modificationof emission due to the density of states was first pointed out by Purcell in 1946[95]. In the extreme case of an optical cavity, where the spectrum consists ofa single state, an excited atom performs coherent vacuum Rabi oscillations
for as long as the photon remains in the cavity, which is a stark deviationfrom an initially excited atom in free space [96, 150, 151]. In between these
two regimes is the regime of an emission vacuum with a (singly or doubly)
bounded, continuous spectrum of emission modes, for which stark deviations
from exponential decay occur close to the energetic edge(s), realized decades
later in the context of optical waveguides [96].
The latter scenario is realized in a photonic band-gap material, in which
a periodic modulation of the index of refraction on the scale of the optical

19

wavelength creates a series of energy bands and gaps in which photos canpropagate or are forbidden to propagate respectively. Such materials weretheoretically considered first in the 1970s [97], and later (and independently)
in the late 1980s [98, 99], and a range of emission phenomena beyond theWeisskopf-Wigner treatment were predicted, including modified spontaneousemission rates and spectral signatures [152–155], as well as atom-photon boundstates [97–99]. As first proposed in 2008 [112], it is possible to realize thephysics of modified spontaneous emission and bound states using ultracold
atoms in optical lattices. It was later demonstrated theoretically [115–118, 122]
that the original proposal for emission of free matter waves in 1D (mimicking
emission near a single band edge) could be modified to include lattices andgeometries leading to even more exotic fractional decays and couplings.

4.2

Matter-wave emission: overview

We have previously demonstrated that certain system plus reservoir models
can be solved analytically, and in particular, we have shown that spontaneous emission of light from an excited atom is one such model. However,
we made use of a Markovian approximation in deriving (3.14), which is perfectly reasonable for atoms emitting into free space. However, when such an
atom emits into a photonic band gap material, as discussed in section 4.1,
the density of states diverges near the band edges, and for emission near suchdivergences, it is impossible to make a Markov approximation. We will now
present a model of spontaneous emission of massive matter-waves [114] whichemulates the behavior of an atom emitting a photon near the bottom of aband in a PBG-material in one dimension, and we will show that the densityof states has a profound e↵ect on the observable dynamics. In particular, weshow explicitly the transition between Markovian decay to non-Markovian, oscillatory decays featuring an observable population in the emitter at infinite
times, called fractional decay. We will also demonstrate the existence of exotic,
dynamically-bound states in the non-Markovian regime, and analyze their spatial structure. We note that the results of [112] consider a 3D system, whereas
here we consider a 1D system. The stronger non-Markovianity in 1D can bedifficult to observe in a material system [101], making our atomic simulator aneven more attractive platform for such experiments. We then will generalize
to a study of emission into a full band, again in 1D, rather than a semi-infinite
continuum, highlighting the similarities and di↵erences with the single edgecase. We note finally that these models are experimentally realizable in an
ultracold 87 Rb lab, and we will present experimental details and results in 5.

20

4.3

Emission near a single energetic edge

(This section presents the results of [114].) Behavior analogous to that of
spontaneous emission in photonic band gap materials has been predicted for
an atom-optical system consisting of an atom confined in a well of a statedependent optical lattice that is coupled to free space through an internalstate transition [112]. Using the Weisskopf-Wigner approach and considering
a one-dimensional geometry, we analyze the properties of this system in detail,including the evolution of the lattice-trapped population, the momentum distribution of emitted matter waves, and the detailed structure of an evanescentmatter-wave state below the continuum boundary. We compare and contrast
our findings for the transition from Markovian to non-Markovian behaviors to
those previously obtained for three dimensions.

4.3.1

Introduction to the 1D modelAs first proposed in [112], ultracold atoms in state selective optical lattices canbe used to simulate spontaneous emission of photons in a PBG-material. Withan eye on experimental implementations [83, 156, 157], we set up the following
model: an atom in a tightly-confining well of a deep optical lattice (withnegligible tunneling to other wells) is coupled to unconfined states via a nearresonant coupling field of frequency !µ , cf. figure 4.1(A). The trapped atom, in
the internal state |ai, is assumed to be in the harmonic-oscillator ground statewith energy ~!a = ~!a0 + d ~!0 /2 where ~!a0 is the bare (untrapped) energy
of |ai, and d is the system dimensionality (we will set d = 1 after deriving the
Hamiltonian in full generality.) The wavefunction in the well is Gaussian,
 2
1
r
exp
(4.1)
0 (r) =
d/2
2a2ho
⇡ d/4 ahopwith aho = ~/m!0 the harmonic oscillator length.
The atoms in the unconfined state are assumed to be simple plane waves
d/2, having momentum k and kinetic energy Ek =k (r) = exp(ik · r)/L
2 2
~ k /2m, where L ! 1. These states sit on an energy floor associated with
the internal energy of |bi, namely ~!b = ~!b0 . The system Hamiltionian canbe written asXZĤ =dd r ˆ †j (r)(Hj + ~!j0 ) ˆ j (r) + Ĥab
(4.2)
j=a,b

21

Figure 4.1: Decay mechanism for an atom confined to a well of a deep, stateselective lattice potential with coupling to a continuum of momentum modes
through an internal state transition. (A) The population in the well can beviewed as an occupational spin represented by an excited state (contain 1atom) or ground state (contain 0 atoms). The coupling, with strength givenby the Rabi frequency ⌦ and frequency !µ , can be tuned to positive (B)or negative (C) detunings
around the (zero-momentum) boundary of themode continuum. In (B), k is the momentum of the resonantly coupled freely
propagating mode. In (C), an evanescent matter wave with decay length ⇠ isformed.
with the interaction~⌦1Ĥab =
2Z

dd re i!µ t ˆ a (r) ˆ †b (r)µ̂(r) + H.c.

(4.3)

The operator µ̂(r) annihilates a coupling field quantum. Assuming the couplingfieldp⌦ † ↵ µ̂(r) can be replaced by its expectation value hµ̂i =p to be classical,
N ⇡ N + 1 = µ̂ , and can be pulled out of the integral to re-scale thepsingle-photon Rabi frequency to the N -photon Rabi frequency, ⌦N = N ⌦1 ⌘⌦.
To proceed, the field operators in (4.3) are expanded in the basis of states22

discussed above:ˆ a (r) =i!a tâ0 (r)eXi(Ek /~+!b )tˆ b (r) =b̂kk (r)e

(4.4)
(4.5)

k

The bare-Hamiltonian terms do not contribute to the Schrödinger equation
governing the time evolution of the state amplitudes in the interaction picture.
Hence, the interaction Hamiltonian becomesZ~⌦ X†
Ĥab =
exp[ i(!µ + !a )t + i(Ek /~ + !b )t]âb̂k dd r 0 (r) k⇤ (r) + H.c.
2 k
X ~⌦
=
ei k t k âb̂†k + H.c.
(4.6)
2
k
where

k is a k-dependent detuning and

k =
k =

~k 2
Z2m

;

k is a Franck-Condon overlap:= !µ

dd r 0 (r) k⇤ (r)

(!b

!a )

(4.7)
(4.8)

The atom can either be in the trapped state with no freely propagating modes
occupied, |1a , {0}k i, or it can be in the untrapped state with a freely propagating excitation present, |0a , 1k i, and hence we represent â = |0a i h1a |. (Notethat this reduced Hilbert space has exactly the same form as the Hilbert spaceof the Weisskopf-Wigner model discussed in section 3.3.) The interaction
Hamiltonian is then reduced to the formX ~⌦i kt †Ĥab =b̂k |0a i h1a | + H.c.
(4.9)
ke
2
k
By identifying |0a i and |1a i with the ground and excited states of an occupational spin as defined in fig. 4.1(A), and upon introducing gk = ⌦ k /2,(4.9) exactly reproduces the standard Weisskopf-Wigner Hamiltonian for thedescription of spontaneous photon emission from a two-level atom [141] in theinteraction picture, albeit with di↵erent momentum dependences in the gk andk terms. The di↵erences are caused by the quadratic dispersion relation of
the matter-waves, which coincides with that of light in a PBG material [153].
We note that, while the coupling to the continuum requires the introductionof the external drive, this does not a↵ect the structure of the Hamiltonian,23

in which the coupling is a simple constant in either case, albeit with di↵erent strength which can be modulated in general, and for which spontaneousemission arises for selected strengths.

4.3.2

Population dynamics

In order to contextualize what we have done, we briefly consider what roles
the trapped atom and freely propagating modes play. The states |1a i and |0a iof the trapped atom (occupational spin) are our small system S of interest,
and the propagating matter wave modes |1k i represent the reservoir R of oursystem plus reservoir model. In analogy with the Weisskopf-Wigner model,
we will attempt a direct solution of the model for a certain initial condition,
and we will demonstrate that the dynamics are strongly modified compared to
the free space case. Furthermore, without explicitly tracing over the reservoir
degrees of freedom, we recover a transition from mostly Markovian behavior
to strongly non-Markovian behavior.
To proceed, we follow the usual approach of expanding the initial state as[141]X| (t)i = A(t) |1a , {0}k i +Bk (t) |0a , 1k i
(4.10)kwhere, due to the choice of the interaction picture, the dynamical phases havebeen left in the Hamiltonian. Application of the Hamiltonian (4.9) to thisinitial state, left multiplication by h1a , {0}k | or h0a , 1k |, and cancellation ofterms arising from the bare Hamiltonian, then results in the following systemof di↵erential equations for the state amplitudes:
XȦ(t) = igk⇤ e i k t Bk (t)(4.11)kḂk (t) = igk ei k t A(t)(4.12)For concreteness, we note that the dimensionality d modifies the FranckCondon factor (4.8). We will be interested in experiments in one spatial
dimension, and so we specialize to the case of d = 1 now.
To determine the time-dependent amplitude A(t) of the population trappedin the well, we proceed in analogy to the treatment of an excited atom coupled
to a PBG material [153]. In our case, the excited (ground) state of the emitting
atom is replaced by the occupational spin in the well. A straightforward
computation for d = 1 shows thatr2⇡ 1/2 aho
1 2 2
exp
k aho
(4.13)k =L
224

Next, (4.12) is formally integrated and inserted into (4.11). With the assumption that the momenta are closely spaced, i.e. that L diverges (for a typicalexperiment, this assumption can be cast in terms of a limit on energetic resolution, c.f. section (REF)), the sum in (4.11) is replaced by an integral, andthe result for the excited-state amplitude with (4.13) isZZ taho (⌦/2)2 1
2 20pȦ(t) =
dk
dt0 e k aho ei k (t t ) A(t0 )(4.14)⇡
10
where the explicit dependence on the system sizeR
PL from the Franck-Condon
factor is canceled out by the usual replacement k ! (L/2⇡) dk.
It is here that our approach first di↵ers from that in section 3.3. Specifically,
due to the form of the matter wave dispersion relation, i.e. !k / k 2 , wecan analytically compute the k-integral first before attempting to computethe time-integral in (4.14). Said another way, we do not invoke the Markov
approximation here, and our solution going forward should be understood to
not depend on the separation of system and reservoir timescales which suchan approximation requires. In fact, the full solution to (4.14) will exhibit
regimes in which the reservoir acts back strongly on the system and exchanges
population for many cycles before damping.
The k integration is carried out first in closed form, as it is simply a Gaussian integral, giving
2 i (t t0 )p Z t 00 (⌦/2) eȦ(t) =
2dt A(t ) p(4.15)
2 + i!0 (t t0 )0which can also be re-written in the form
Z tȦ(t) =
dt0 A(t0 )G1D (tt0 )(4.16)

0with the correlation function of the continuum [122] (bath correlation function)(⌦/2)2G1D (⌧ ) = p
ei ⌧1 + i!0 ⌧ /2

(4.17)

Eq. (4.16) is easily solved by the Laplace transform method. Denoting byf˜(s) the Laplace transform of a function f (t)Z 1
˜
f (s) = L{f (t)} =f (t)e st dt(4.18)
025

and using its standard relations, specifically the convolution property⇢Z tLf (t0 )g(t t0 )dt0 = L {(f ⇤ g)(t)} = f˜(s)g̃(s)(4.19)
0

we can show that in Laplace space, eq. (4.16) results inÃ(s) =

1
s + G̃1D (s)(4.20)

where we have chosen the specific initial condition that the emitter is initially
excited, i.e. A(t = 0) = 1 and Bk (t = 0) = 0 for all k.
We can solve (4.20) by inversion, which for the case at hand has the formal
solution
Z ✏+i1
1
i teA(t) =dsÃ(s + i )est
2⇡i ✏ i1
Z ✏+i11
est=
ds(4.21)
2⇡i ✏ i1
s + i + G̃1D (s + i )with ✏ chosen such that all singularities of the integrand lie to the left of theintegration contour, which in this case is a vertical line (Bromwich contour).
In general, eq. (4.21) cannot be solved analytically, but progress can bemade, as in the 3D case [122], by making the assumption of strong confinement 1 !0s, , or equivalently |⌘| = |( + is)/!0 | ⌧ 1. Physically, this istantamount to assuming that the Wannier function for the trapped a atom ismade up of contributions due to many di↵erent plane waves. This assumption does not accurately capture times below !0 1 , so the model is expected tobreak down at very short times. The Laplace transform of the bath correlation
function (written in terms of the previously defined ⌘) isG̃1D (⌘) =

p

2⇡

⇣
p ⌘
⌦2
i⌘ 1/2 e 2⌘ i + Erfi( 2⌘)
4!0(4.22)

where Erfi is the imaginary error function defined with respect to the usual
error function Erf as follows:
Erfi(z) =iErf(iz)(4.23)In strong coupling to leading order in ⌘ (keeping only the constant and negative1

Note that we erroneously referred to this approximation as strong coupling in [114].
The name strong confinement accurately reflects the assumption we are making.

26

power terms), this becomesG̃1D (s) ⇡C(1 i)
i L+p
;
s i

p
⌦2
⇡ ⌦2
; C=
p
L =
!0
4
!0

and therefore one must evaluate
Z ✏+i11est
i te
A(t) =ds2⇡i ✏ i1
s + i ˜ + C(1

p
i)/ s

(4.24)(4.25)The quantity L = ⌦2 /!0 in (4.24) corresponds to a Lamb shift of the detuningto˜ =⌦2 /!0(4.26)L =The inversion of (4.25) is now a straightforward exercise in examining the
singularities of the integrand and deforming around a branch cut. The solution,using the notation of [153], can be written as"
X 2u2jA(t) = exp(i t)exp(iu2j t)
2
˜
3uj +
j
Z 1
D
⇣ 1/2 exp( ⇣t)d⇣+ e i⇡/4(4.27)⇡ 0 ⇣ 3 2i ˜ ⇣ 2 ˜ 2 ⇣ iD2pwith D = 2C. The uj ’s are the roots of
u3 + ˜ u

D=0(4.28)

such that 3⇡/4 < arg(uj ) < ⇡/4 to ensure that the roots uj do not lie on
the branch cut in s-space. We comment here that the dimensions of D are[D] = T 3/2 , which is a consequence of working in one spatial dimension withfreely propagating matter waves. For convenience, we define a frequency scale(equivalently timescale)˜ = D2/3 =
⌦

✓

⌦
⇡
!0

◆1/3

⌦
2

(4.29)

which sets the scale over which the dynamics occur. We shall later see that˜ ⇡ 0.3⌦ in experimental contexts.
⌦
Since (4.27) is analytic, it can be evaluated over any range of interest; an
example with the salient features is shown in fig. 4.2. In analogy to what
is observed in PBG materials near a band gap [153, 158], depending on the
detuning there is a transition from a (nearly) exponential decay to oscillatory27

population

1.0
0.5
0.0

0
0

tim 5
e D 2/
3
t

-10

-20
3

-2/

10 ng D
ni
20
u
t
10
de

Figure 4.2: Computed population |A(t)|2 in the potential well as a functionof drive time and detuning in characteristic time and frequency units. The
thick white curve corresponds to = 0 (see vertical solid line), and the Lamb˜ = 0.1 (see the dotted vertical line on the detuning axis). The
shift is L /⌦˜chosen frequency scale is ⌦/2⇡
⇡ 200 Hz. For the depicted plot, !0 and ⌦ are
fixed by matching to experimentally reasonable values (!0 ⇡ 2⇡ ⇥ 30 kHz and⌦/!0 = 0.03).
behavior, in which the atomic population decays slightly but remains trappedin the well.
For large positive detunings
0, the system is so far away from the continuum boundary that the density of levels looks essentially unchanged fromthe original situation considered by Weisskopf and Wigner. In this regime,
one may make the standard Markov approximation, the details and results ofwhich can be found in the appendix. Inpparticular, the population is found to
decay exponentially with a rate = L ⇡!0 /2 exp( 2 /!0 ), c.f. (4.41).
For detunings that are smaller than the Lamb shift, < L (this includes
arbitrarily large negative detunings), the population in the well does not decay
completely, even for arbitrarily long times. This is a consequence of the factthat eq. (4.28) has a real root for ˜ . 0, i.e. for . L , which signals thatthe solution has an imaginary pole at s = iu2j , corresponding to a long-lived
excitation. This excitation is a dressed state which evolves at the frequency
determined by the real root of eq.(4.28). (We will show in section 4.3.5 how
this mathematical object corresponds to a long-lived bound-state of an emitter28

dressed with a cloud of approximately exponentially localized matter-waves.)
The behavior for = 0, i.e. the resonant case, is reminiscent of a damped
Rabi oscillation, though the frequency appears to vary weakly with time.
(For much stronger couplings than considered here, the trap potential andthe flat continuum hybridize into dressed potentials, eventually leading to an
undamped Rabi oscillation [86]). We note that the case under considerationin fig. 4.2 corresponds roughly to the small coupling limit considered in [86],however with an initial state having a broad momentum spread.
In contrast to 3D results discussed in [112], the oscillations in the population in the 1D case persist for many characteristic times at negative detunings.
Furthermore, the decay rate is maximum at detunings around L , and it becomes slower as the detuning is increased. The pronounced oscillatory behavior
in 1D is consistent with the divergence of the 1D density of states at zero energy. While the fundamental di↵erences between 1D and 3D systems would
be difficult to measure in PBG materials [101] (see also [105]), the tunabilityof the atom-optical system makes it an ideal candidate for the exploration ofdimensional e↵ects.

4.3.3

Markovian limit

We note that while we do not have to resort to a Markov approximation tosolve (4.14), we can invoke it and compare the results to the full solution,
as well as make contact with the results from Fermi’s Golden Rule for timedependent transition amplitudes, and we find it instructive to do so. The
computation which follows here is essentially the same as that of 3.3, and assuch, we shall omit details which are essentially already presented.
The starting point for a Markov approximation treatment is again assumingthat in (4.14), the exponential term oscillates quickly except for times t t0 ⇡ 0,so we will replace A(t0 ) by A(t) and remove it from within the time integral.
We can then evaluate the exponential time integral at the cost of introducing
an epsilon-regulator, recalling (3.15). The expression for the excited emitter
population is thusȦ(t) =aho ⌦2p A(t) lim
"!0
4 ⇡

Z 1
0

2 2

2ie k ahodk
k + i"(4.30)

wherein we have used the fact that the integrand is an even function of k inorder to change the limits of integration and introduce a factor of two. We
recall that k = ~k 2 /2m, and thus we are motivated to introduce the29following change of variables
~k 2
~k
1
!=
; d! =
dk; aho dk =
2m
m
!0

r

!0
!
d!; a2ho k 2 = 2!
!0

and using (4.31) in (4.30), we rewrite the equation in the form
Z 1 r
!0 ie 2!/!0LpȦ(t) = A(t)
lim
d!
! !
+ i"
23 ⇡ "!0 0(4.31)(4.32)

with L = ⌦2 /!0 as in 4.3.2. It is now our intention to make use of the
Sokhotski-Plemelj theorem (3.22), however our integrand does not yet satisfy
the hypotheses of the theorem. To this end, we shift the integration variable
to a new variable ! 0 = !
, in terms of which (4.32) becomesr
Z 1
0!0 ie 2(! + )/!0L0Ȧ(t) = A(t) plim
d!
(4.33)!0 +! 0 + i"23 ⇡ "!0Now, we will apply the Sokhotski-Plemelj theorem, i.e. use the replacement
1/x+i" = i⇡ (x)+P(1/x), but we must be careful to understand what we aredoing. In particular, we must consider that is a signed quantity which may
be positive or negative (or zero), so we consider two cases. First, for positive
detunings> 0, the hypothesesR of the theorem are manifestly satisfied, so
we may write (calling the lim"!0 . . . terms in (4.33) I( , !0 ))"!#rr!0 2 /!0I( > 0, !0 ) = ⇡e
1 iErfi
2(4.34)!0
consisting of both a real and imaginary part. Secondly, we can likewise consider
the case of negative detunings, i.e.
< 0, for which= | | > 0. In
this case, the hypotheses of the Sokhotski-Plemelj theorem are not satisfied,
however, as the integrand does not contain any singularities over the region of
integration, we may simply evaluate the integral, yielding0s1
r!0 2 /!0| |AI( < 0, !0 ) = i⇡e
Erfc @ 2(4.35)| |!0with Erfc the complimentary error function Erfc(z) = 1 Erf(z). For negative
detunings, the result is purely imaginary. Denoting by ⇥(z) the Heaviside step30

function, we can condense our results and rewrite (4.33) as
("
!#
r
r
⇡!0 2 /!0LȦ(t) = A(t)e
1 iErfi2⇥( )2 2| |
!0
9
0s
1
=
| |A
iErfc @ 2
⇥(
)
;
!0

(4.36)The solution to (4.36) is exponential with an imaginary part corresponding toa frequency shift and a real part for positive detunings only which corresponds
to a decay. That is✓◆r⇡!0 2 /!02n(t) = |A(t)| = expe⇥( ) ⇥ t(4.37)
L2| |We note that this di↵ers from our non-Markovian solution (4.27), where evenfor negative detunings we have some decay of the population. For the regime
in which /⌦1, the Markovian and non-Markovian solutions converge.
The Markovian decay rate furthermore must also arise in a treatment due
to Fermi’s Golden Rule. Specifically, the decay rate dn(t)/dt ⌘ must be
reproduced by2⇡=| h0a | ĤI |1a i |2 ⇢(E = ~ )(4.38)~
where ⇢(E) is the density of states and | h0a | ĤI |1a i |2 is a squared transition
matrix element arising from the interaction Hamiltonian. For our system ofinterest, these are
r
r
L
m
!0
⇢(E = ~ ) =⇥( )(4.39)⇡~ 2~!0and~2 ⌦2 ⇡ 1/2| h0a | ĤI |1a i |2 =2Lr~e 2 /!0m!0Combining these results in (4.38), we recoverr⇡!0 2 /!0
= L
e
⇥( )
2| |

(4.40)(4.41)

in full agreement2 with (4.37).
2Equation 4.41 corrects a typo in the in-line equation for31

in section III of [114].

4.3.4

Momentum distribution of the emitted matterwaves

It is common in ultracold atom experiments to use time-of-flight imaging as anexperimental probe and thus gain access to the momentum distribution of theultracold atomic cloud. For this reason, it is highly interesting to consider the
momentum composition of the radiation which is emitted from our initially
excited emitter. In particular, the coupled di↵erential equations (4.11) and(4.12) contain the amplitudes of both the occupied well and the emitted matterwaves. Using eq (4.27) for A(t), we may formally integrate the eq. (4.12) forḂk (t), leading torZ⌦ 2⇡ 1/2 aho k2 a2ho /2 t i k t0
Bk (t) = ieeA(t0 )dt0(4.42)
2
L
0
The absolute square of (4.42) gives the time-dependent momentum distribution:
 ✓ 2
◆
Z t
2
p ⌦2
~k
2
k2 a2ho000
L|Bk (t)| = ⇡ aho e
exp it A(t )dt
(4.43)
22m
0which for long times defines the emission spectrum as a function of k:
S(!k ) = lim |Bk (t)|2 .
t!1(4.44)

Note that in the nomenclatureand formalism of [155], this matterp wave emitterp2
corresponds to a = 2⇡⌦ /!0 and F (!k ) = exp( 2!k /!0 ) !0 /!k ⇥(!k ), a
spectral weight function not previously considered.
Using the numerical solution for A(t) with parameters 0  t0  t, , ⌦,and !0 , the integral in (4.43) and the momentum distribution L|Bk (t)|2 canthen be computed numerically. An example for the Markovian limit is shownin fig. 4.3A. The doubly-peaked momentum structure is characteristic of asystem in 1D with left-right symmetry; while the individual peaks are notgenerally symmetric around their centers, their location varies with detuning
in a simple way. The detuning supplies a kinetic energy ~ ˜ = ~(L ) tothe transferred atoms, which corresponds to a particular momentumqk(Ekin )/krec = ~ ˜ /~!rec(4.45)
where we have introduced ~!rec = (~krec )2 /2m with krec = 2⇡/ latt and lattthe wavelength of the optical lattice. Fig. 4.3B shows the extracted peak loca-

32

0.5

0

-2

peak width k/krec

0.3

0.6
0.4
0.2
0

0
-1
1
2
momentum k/krec

(C)

0.2
0.1
0.0

1 (B)
0.8

0.3

0

5
10
15
~
detuning D-2/3

(D)

0.2

1

0.1
0

1 2 3 4 5 6time D2/3t

1.04

peak center k0/krec

peak center k0/krec(A)peak width k/krec|Bk|2 (arb. units)

1

0.96

0.04 0.08 0.12
/0Figure 4.3: Momentum distributions of emitted matter waves (positive detunings, Markovian limit). (A) Sample momentum distribution taken at time˜ ⇡ 1.5 and positive detuning ˜ /⌦˜ ⇡ 17. The width and position of the⌦tpeaks depends on time and detuning, but the shape is representative for alarge range of parameters. (B) Calculated momentum peak position k0 (in˜ for ⌦t˜ ⇡ 4.0. The solid
units of krec , see text) as a function of detuning ˜ /⌦curve is a guide to the eye for the expected momentum based on exact energy
˜ ⇡ 17,conservation (see text). (C) Momentum peak width vs. time for ˜ /⌦saturating at a value set by the decay rate (see text). (D) Momentum peakwidth (green, filled circles, left vertical axis) and momentum peak position˜ ⇡ 1.5(orange, unfilled circles, right vertical axis) vs. Rabi frequency ⌦ for ⌦t˜ ⇡ 17. The peak width is obtained as the standard deviation aboutand ˜ /⌦
one peak in (A).
tion, k0 , given by the computed center-of-mass in fig. 4.3A from the numerical
data, as well as a no-free-parameter fit to the data of square-root form. The
close agreement with the simulated data degrades at small detunings, where
the doubly-peaked structure is lost due to edge e↵ects near the boundary, suchthat one characteristic momentum in this regime cannot be identified. Thisis one indicator of the strongly non-Markovian character of emission near theband edge where the density of states diverges.
It is also seen that the momentum distribution evolves in time, startingfrom wide peaks and tending to a tightly confined value, cf. 4.3C. To understand this behavior, consider that in the Markovian regime, the decay is33

exponential, and so will have an (approximately) Lorentzian emission spectrum S(!) [155]. The width of the momentum distribution is limited at earlytimes by the corresponding spectral Fourier width ( t ! ⇡ 1), but at longtimes should tend to a finite value set by the line width of the excited state,
given by (4.41). Indeed, computation of the emission spectrum S(!) (notshown) by an appropriate change of variables (i.e. momentum to energy), andextracting its width, c.f. fig. 4.3(C), yields a value of 1.25 for long times, ingood qualitative agreement with the expectation.
So far, the discussion of fig. 4.3 has focused on a fixed Rabi frequency⌦. When ⌦ is varied, both the decay rate and the Lamb shift L change
proportionally to ⌦2 . Thus, the width in kinetic energy (i.e. the width ofmomentum squared) at long times should vary quadratically with ⌦, and indeed, the extracted momentum width increases approximately linearly with⌦. Furthermore, the extracted peak separation decreases with increasing Rabi
frequency as the growing Lamb shift L pulls the lattice-trapped state to alower energy.

4.3.5

Matter-wave bound states

The decay of the population in the initially excited emitter decays to a non-zerovalue for a finite range of detunings , most notably, for negative detunings,
where the decay of the excited state population is small and oscillatory, c.f.
fig. 4.2. This is the result of the presence of a bound state that the system
supports and which does not decay in time. We now elucidate the origin of
this bound state and demonstrate that it has a simple form.
First, we can write the di↵erential equations for the emitter and the emitted
momentum components, (4.11) and (4.12) using ! rather than k as our label,in Laplace transformed space as+ G̃(s)] 1(4.46)i ⇤ (!)Ã(s + i )s + i!

(4.47)Ã(s + i ) = [s + i
B̃! (s) =where in general G̃(s) is the Laplace transform of the bath correlation function(4.17). In general, this quantity is computed as
Z 1
G(!)
G̃bath (s) =d!
(4.48)s !
0
again, having switched to an energy viewpoint rather than a momentum
viewpoint at the cost of introducing a density of states ⇢(!). The function

34

G(!) = | (!)|2 ⇢(!), with (!) the Franck-Condon factor (4.13) expressed asa function of energy instead of momentum. For completeness, in this model,
these quantities take the formrp⇡aho ⌦2 2!/!0
L 2m2; ⇢(!) =| (!)| =e(4.49)
L2⇡ ~!
Now, in order to learn about the excitation spectrum of the system, weanalyze (4.46), and we realize that the singularities in the denominator willcorrespond to stable or unstable excitations. In particular, suppose that there
is a purely imaginary pole of the form s = i!B . We will discuss under which
conditions this is possible in what follows. However, for the moment if we justassume that such a solution exists, then from the Cauchy residue theorem, it
must correspond to a non-decaying exponential term in the solution for A(t).
To see this, it is enough to first note that the pole will lie to the left of the
Bromwich integration contour, and thus schematicallyZ ⇣+i1hi
1A(t) =
Ã(s)est ds = Res i!B Ã(s)est + Ac (t)(4.50)
2⇡i ⇣ i1
where the residue may be computed ashi
Res i!B Ã(s)est =est@s (s + i + G̃(s)) s= i!B⇣⌘ 1= 1 + @s G̃(s)|s= i!B
e i!B t= cB e i!B t(4.51)

justifying the preceding claim that solutions s = i!B correspond to nondecaying exponential terms in the excited state population. Hence (4.50) takes
the form
A(t) = cB e i!B t + Ac (t)(4.52)The term Ac (t) is the complement of our non-decaying solution. We are notinterested in its precise details here, though we will note that in the present
model, it decays to zero as t ! 1, since there will only be one such solutions = i!B . In principle, if there were more such solutions, then we could
separate out each such solution using the residue theorem and leave behind
Ac0 (t) which tends to zero as described.
The result from (4.52) is directly inserted into the equation for B̃! (s) and,defining ˜ (s) = (s + i!) 1 , using the convolution property of Laplace trans-

35forms to solve for B! (t):B̃! (s) =) B! (t) =i ⇤ (!)Ã(s)˜ (s)
Z t
⇤
i (!)
(t ⌧ )A(⌧ )d⌧(4.53)

0

The convolution in (4.53) can be evaluatedZ t⇤
B! (t) = i (!)
e i!(t ⌧ ) (cB e i!B ⌧ + Ac (t))d⌧0==i ⇤ (!)icB (e i!t e i!B t )0+ B!,c(t)! !B⇤(!)cB e i!B t+ B!,c (t)! !B(4.54)

where B!,c (t) is again the complementary piece to our non-decaying solution
whose precise form is irrelevant. Again, the exact form of B!,c (t) has no closed
form. Now, combining (4.52) and (4.54) into the state expansion for the systemyields the state
Z 1| (t)i = A(t) |1a , {0}i +B! (t)⇢(!) |0a , 1! i d!
0

= cB e i!B t |1a , {0}i + Ac (t) |1a , {0}i
Z 1 ⇤
(!)
i!B t
cB e
⇢(!) |0a , 1! i d!
! !B
0
Z 1
+
B!,c (t)⇢(!) |0a , 1! i d!
0

= | B i e i!B t + |with✓| B i = cB |1a , {0}iand|c (t)iZ 1
0c (t)i = Ac (t) |1a , {0}i +◆(4.56)

B!,c (t)⇢(!) |0a , 1! i d!

(4.57)g ⇤ (!)⇢(!) |0a , 1! i d!
! !B

Z 1
0

(4.55)The state | B i has no time dependence outside of the phase e i!B t , which we
now identify with the energy of the bound state, ~!B . The first term in eq.
(4.56) corresponds to the excited state of the lattice well, whereas the second
term yields an evanescent matter-wave cloud, including the ground state of36

the well. The state |c (t)i satisfies,|c (t)|

2=1

cB(4.58)

because | B |2 = cB . Since cB is the probability to remain bound, the interpretation of | c (t)i becomes clear. It represents the atomic population in thewell that is released into propagating modes and does not return in time. The
existence of this term is understandable if one considers that a sudden turn-onof the coupling at t = 0, as in the preceding treatment, represents a transient,
resonant coupling to many di↵erent momentum modes. (This e↵ect vanishes
if the coupling is turned on adiabatically.)
The above described state (4.56) is manifestly a bound state of emitted
matter waves “stuck” to the emitter. This is in direct analogy to the so-called
atom-photon bound state in PBG materials [97, 153, 158–161] as a “dynamic
state in a superposition of the excited and ground states with an admixture ofa photon “cloud”, which surrounds the atom”[97]. They are of experimental
interest as, for example, mediators of tunneling beyond the nearest neighborin lattice models and for the engineering of resonant dipole-dipole interactions
between emitters.
Returning now to the question of when such a bound state exists, we recall
that we are looking for purely imaginary roots of the denominator of (4.46).
In other words, we are seeking to solve the defining equation [153]i!B + i+ G̃( i!B ) = 0(4.59)

Now, if we recall (4.49), we can integrate (4.59) and arrive at the transcendental equation
0s
1
r
✓
◆
r
⇡
!0
2|!B |
2|!B | A!B =exp
Erfc @(4.60)
L
2
|!B |
!0
!0

with Erfc again the complimentary error function. The assumption of strong
lattice confinement, !0 being much larger than any other frequency scale in
the problem, allows forp the Taylor expansion of this result, and keeping theleading order term / !0 /!B and the sub-leading order term which does notdepend on !B , results in!B = (

L)

iD
p
!B

(4.61)

Comparison with the denominator in eq.(4.25) reveals this to be the same37

form upon rotating s = i!B . Thus the dynamical equations in sec. 4.3.2
have been recovered in a slightly di↵erent formalism.
The notable feature of this state is that it is made up of contributions
from all possible emitted matter waves, but they evolve with a fixed phase!B rather than according to their own dynamical phases. If we focus on theemitted wave part of (4.56) only, then we find the following expression for thebound state wave function
Z 1 ⇤
(!)
1/2
mw
⇢(!) hx | 0a , 1! i d!
(4.62)
B (x) = cB
! !B0
The matter-wave of eq. (4.62) contains contributions
p from all possible eigenstates hx | 0, 1! i with frequency !. Using k(!) = 2m!/~, the eigenstates
are '! (x) = hx | 0a , 1! i / cos[k(!)x]. In this way, the spatial profile of the
wave can be constructed by solving for !B given at fixed detuning, and thencomputing (4.62), c.f. fig. 4.4A.
As is particularly apparent from fig. 4.4A, B (x) is exponentially localized.
Note that the decay is di↵erent from the 3D case, where a Yukawa-type profile
is seen to arise away from the center of the bound state [112]. Fitting an
exponential to the wings of the computed states and extracting this decay fora large range of negative detunings allows for a comparison of the decay length
to an evanescent-wave model of the formr!0
⇠ = aho(4.63)
2| ˜ |The resulting fit decay lengths are shown in fig. 4.4B (red circles) [We plot
three sample curves in units of d0 = latt /2 with latt = 790.4 nm the lattice
wavelength as the characteristic length scale of the problem. We do so looking
ahead to the experimental implementation, c.f. 5 and [91].] It should be notedthat eq. (4.63) matches the expectation in which one naı̈vely assumes a matterwave with negative energy and a corresponding imaginary wave-vector  set
by the detuning only for large negative detunings ˜ . To properly capture thephysics of the system, ~!B , the bound state energy must be used instead ofthe detuning, with the associated characteristic lengthr!0⇠ = aho(4.64)
2|!B ( )|To determine !B ( ), the soft cuto↵ in the integral in eq.(4.48), exp( 2!/!0 )can be approximated by a sharp cuto↵ of the integral at ! ⇡ !0 . Expanding
the integrand to leading order in the, assumed small, quantity !/!0 and inte-

38

1

(A)

10-1

2
|mw
B |

|Bmw |2 (arb.)

100
10-2

0

-4

0
position x/d0

02
position x/d0
1.

4

10-3

4

10-4
10-5

-4

-2

1.
(B)
.8

(C)

state fractions

decay length /d0

.8

.6

.6
.4

.4

.2
0.

-1 -0.5-0.1 ~
detuning D-2/3

-2

0
-1
~
detuning D-2/3

Figure 4.4: Characteristics of the evanescent matter-wave. (A) Density dis˜ =tribution as a function of distance in lattice spacings d0 = latt /2 for ˜ /⌦-27 (purple, dot-dashed), -2.7 (orange, dashed), and -.27 (green, solid). (B)Extracted decay length in lattice spacings as a function of the detuning (reddots). The slope of the black line is 1/2 (after reversing the horizontal axes),
the scaling expected in 1D for large detunings. The purple (dotted) curve is themodel based on eq. 4.65, and the green (dot-dashed) curve is a more sophisticated model (see text). (C) The relative populations of the well (i.e. excitedstate of the well) (red, solid), and the evanescent matter wave (blue, dashed)
as a function of detuning. We also depict the population that is radiated away
(purple, dot-dashed).
grating, leads to an e↵ectively quadratic equation for !B . This equation has
a non-trivial negative root
!B =

2

+

1p
2

2 + ⌅⌦2(4.65)

which for small ⌦ coincides with the set detuning from the edge of the continuum.
The second term has the form of a generalized Rabi frequencyp
2 + ⌅⌦2 (⌅ is a numerical prefactor of order 1), and suggests that forintermediate detunings, the correction to the decay length is due to an AC
Stark shift induced by the coupling (note that it is independent of the lattice39

potential). This expression (4.64) with (4.65), however, underestimates the
decay length for small detunings, c.f. fig. 4.4B (purple, dotted). In orderto recover the proper behavior in this limit, eq. (4.60) must be solved. For
the regimes of validity of this work, this amounts to solving eq. (4.61), ane↵ectively cubic equation for the bound state energy, whose real root, wheninserted into eq. (4.64), gives the true decay length of the system. This curveis also shown in fig. 4.4B (green, dot-dashed), giving much better agreementwith the fit decay lengths.
As a final consideration of this dynamically bound state, we further analyze
the internal-state composition of the state | B i. The constant, cB , which
depends on the bound state energy ~!B , has the full form⇢p
exp(2|⌫B |) hL
cB = 1 + p
2
exp(
2|⌫
|)
2⇡|⌫B |
B
2⇡!0 2|⌫B |3/2
⇣p
⌘io 1
+ ⇡(1 4|⌫B |)Erfc
2|⌫B |(4.66)

where the parameter ⌫B is defined to be !B /!0 . This constant gives the probability amplitude to measure the system in the lattice-well atomic matter-wave
bound state, which as noted, is a superposition of the excited occupational
spin state and the evanescent wave. The probability to measure the atom inthe bound state, starts at 0 for a detuning of L and then increases monotonically with decreasing detuning before saturating at a value of 1, c.f. fig. 4.4C
(red, solid).
The relative proportion of the free space modes which make up the evanescent wave in | B i initially rises with increasing (negative) detuning before
reaching a maximum and then dropping o↵ to essentially zero (blue, dottedcurve in fig. 4.4C). For larger and larger negative values of the detuning,
the bound state energy ~!B sinks farther and farther below the continuum
boundary, and in this regime, it is impossible for the free-particle modes toparticipate in the formation of the bound state in a significant way, leading to
the above-noted drop-o↵ behavior.

4.4

Emission near a band with two edges

We recall that our motivation for studying spontaneous emission of matter waves was as a simulator for the behavior of an atom emitting a photon into a photonic band gap material. However, our previous model considering emission of freely propagating matter waves is an approximation,i.e. near the bottom of a band, we can take the quadratic approximation,

40

"(k)/~ ⇡ ~k 2 /2m⇤ ,and treat the case of essentially massive photons of effective mass m⇤ . However, real photonic band gap materials feature a band
structure, so it becomes reasonable to ask whether we can refine our treatment to consider a band structure. Remarkably, one can make progress for avariety of band structures and lattice geometries, and this is still an area ofactive theoretical research today [115, 117, 118]. Here, we consider the case ofa 1-dimensional sinusoidal energy band. We will see that the new singularities
in the density of states (at the upper and lower band edges) introduce quantitative di↵erences with the single edge case in section 4.3. Specifically, we
will find even stronger non-Markovianity resulting in persistent oscillations forlong time, which we understand to be a beating between two bound states now
supported by the system. The theoretical approach in this section is largely
inspired by the resolvent operator approach of [117] who considers emission
into a 1D sinusoidal band, but we feel that it considers regimes which are hereto-fore unexplored and makes satisfying connections with well-known atomic
physics results.

4.4.1

Solution for emission into a 1D band

Our starting point is to consider the simplified single-emitter Hamiltonian for
spontaneous emission as in [91, 114].
we consider Ĥ = Ĥr + Ĥb +P Specifically,†Ĥint with Ĥr =ee and Ĥb =q !(q)b̂q q̂ and the interaction HamiltonianisXĤint = gb̂q eg + H.c.
(4.67)q

Note that we are now working in a frame where the excited state energy of
our single-emitter is , and we will use the terms detuning and excited stateenergy interchangeably throughout what follows. We also have introduced thenomenclature “red” for our emitter state and “blue” for our emitted matterwave states with an eye towards the experimental implementation describedin [91] and section 5.3.
This model di↵ers from section 4.3 in two ways: first, we take for ouremitted matter-wave (henceforth “blue”) states the Bloch waves of an optical lattice having bandwidth 4J and dispersion !(q) = 2J cos(qd), whered = /2 (which is equivalent to having a state dependent lattice for boththe emitter (henceforth “red”) and blue atoms, albeit with di↵erent depths),
and second, following [115], we assume that all of these modes are coupled toequally with strength g = ~⌦/2. (This approximation holds when the lattice
for the red state is much deeper than other energy scales in the problem, so
the momentum width of the inital emitter excited state is very large. This is

41

(B)
DoS (num./energy)

(A)
r


b

Gap

0

Gap

Band

-3 -2 -1 0 1
Energy/J

2

3

Figure 4.5: Cartoon displaying the scheme for emission into a band structure.
(A) Red |ri atoms are confined to the wells of a deep, state-selective opticallattice potential, coupled to blue |bi atoms (di↵erent internal states) which only
feels a shallow optical potential. (Cartoon shows a Wannier interpretation forboth states, but in the model, we will treat the |bi states in a Bloch wave
formalism.) (B) Schematic of the density of states in the tight-binding limit
for the |bi atoms. In this case, there are two divergences in the density ofstates, resulting in a richer physics compared to the single edge case.
the same as the strong confinement approximation made in section 4.3.)
As before, we take for our state ansatz
X| (t)i = Cr (t) |er , 0b i +Cq (t) |gr , 1q i
(4.68)
qsubject to the initial condition that Cr (0) = 1 and Cq (0) = 0. Substitution
into the Schrödinger equation yields the coupled set of di↵erential equations
XĊr = i Cr ig
Cq(4.69)qĊq =i!(q)CqigCr(4.70)

We can now make a rotating frame transformation in the expansion coefficients
to simplify the equations of motion by taking Ar = Cr ei t and Aq = Cq ei!(q)t .

42

In such a case, equations (4.71) and (4.70) reduce toXȦr = ig
Aq e i q t(4.71)q

igAr ei q t

Ȧq =(4.72)

where we have introduced q = !(q). These two equations (4.71) and(4.72) have manifestly the same structure as (4.11) and (4.12). We therefore
can solve the equations of motion in the same way.
Equation (4.72) may be formally integrated in time and substituted into
(4.71), and the sum over quasimomentum q in the resulting equation may
traded for an integral. After integration over quasimomentum, the equationof motion for Ar (t) takes the by now expected convolution form
Z tȦr (t) =Ar (t0 )Glatt (t t0 )dt0
(4.73)
0where we have introduced the bath memory function
Glatt (⌧ ) = g 2 ei ⌧ J0 (2J⌧ )(4.74)

where J0 (x) is a Bessel function of the first kind.
We proceed as in 4.3.2 and note that the Laplace transform of (4.73) subject
to the initial condition Ar (0) = 1 is analogous to (4.20)
Ãr (s) =1
s + G̃latt (s)withG̃latt (s) = p(sg2
i )2 + 4J 2

(4.75)(4.76)The inversion of this Laplace transform can be performed using the standard
Bromwich contour method
Z "+i11i t
e
Ar (t) =dsÃr (s + i )est
2⇡i " i1
Z "+i11
est dsp
Cr (t) =(4.77)
2⇡i " i1 s + i + g 2 / 4J 2 + s2
Since e i t Ar (t) = Cr (t), we can return to working with Cr (t). Next, werotate variables s = iz, essentially changing a Laplace transform into a Fourier

43transform, whence (4.77) becomes
Z 1
1
Cr (t) =dz
2⇡i 1 ze izt
+ ⌃1 (z)where we have introduced the self-energy ⌃1 (z):
Z
g2
dq
g2
p
⌃1 (s) ==2⇡ BZ z !(q)z 2 4J 2(4.78)(4.79)The self energy, well known in the context of condensed matter systems [162],
captures the influence of the environment acting back on the single emitter.
We note that the self energy and the Laplace transform of the bath correlation
function are related to each other by a shift and rotation:⌃1 (z) = iG̃latt ( iz + i )(4.80)At this point, we note that the definition of our self energy (4.79) matches
that of [115], and furthermore that our approach starting from the Schrödinger
equation eventually matches the approach of the authors of [115] starting fromthe resolvent operator, up to a minus sign in the denominator of (4.78) whose
origin we have thus far been unable to determine. As such, we follow theirapproach for the calculation of the Fourier integral. The integrand contains
two branch cuts running parallel to the negative imaginary axis located at z =±2J, and we must therefore analyticallycontinuepit into the region |Re(z)| p2
2J by making the substitution z
4J 2 !
z 2 4J 2 . Note that there
are three types of contribution [115] to the time evolution:• Bound states: These arise from real poles of (4.78), and they yield nonzero population in the excited state at long times. They physically correspond to bound states of matter waves evanescently localized to theatomic emitter. (Note that in section 4.3.5, the bound states corresponded to purely imaginary solutions. In rotating from a Laplace to a
Fourier picture, the real and imaginary axese have switched roles, whence
now real poles yield bound states.) There will in general be two suchpoles, corresponding to a bound state in the band gaps both below andabove the band. Their energies/frequencies are given as the solutions toz= Re (⌃e (z))(4.81)

which is quite complicated in general. We will deal with explicit expressions for it shortly.

44• Unstable pole: This pole has a negative imaginary part, and as such,it corresponds to an oscillatory excitation which decays over time. In
general, there will be one such pole solution, and it will occur in theanalytically continued region.
• Branch cut detours: In order to close the Laplace inversion contour, it
becomes necessary to go around branch cuts at nonanalytic points of theband. These contributions must be carefully computed by analyticallycontinuing onto the correct Riemann sheet and integrating. However,they are all transient and eventually decay with time.
Armed with this knowledge and the residue theorem, we define a few terms
and then schematically represent the excited state population as a function oftime. First, let the upper (lower) bound state frequencies (energies/~) bedenoted by ! ± . These are two purely real frequencies. Next, denote thefrequency of the unstable pole contribution by !
˜ = !U P + i U P /2 where hereisanegative
real
numberrepresenting
thedecay rate of the excitation.
UP±
Also, denote by R and RU P the complex residues at these poles. Lastly,
denote by the function BC (t) all of the contributions from the branch cut
detours. then the time evolution of an emitter starting in its excited state is+

Cr (t) = R+ e i! t + R e i! t + RU P e(i!U P | U P |/2)t +BC (t)(4.82)

In principle, (4.82) represents everything that we need to know about thesystem under consideration.
We can determine ! ± by solving the following polynomial equation (whichis equivalent to (4.81)):
1 4
z
g4

2 z3 + (2

4J 2 )z 2 + 8J 2 z

4J 2

2

=1

(4.83)

whose real roots for |Re(z)| > 2J determine the bound state energies. The
complex root of (4.81) having negative imaginary part is !
˜ . We note that as(4.81) is a quartic polynomial, there is another root, conjugate to !
˜ with positive imaginary part. However, this root lies outside the integration contour,
and thus does not contribute to the time evolution. The algebraic solution to(4.81) is quite complicated, and it does not grant us much physical intuition.
However, we note that it may be solved numerically for any chosen set ofparameters efficiently using modern numerical methods. The residues whichcorrespond to these poles may be computed as follows:
ResG (z0 ) =

@ (z

1
=
+ ⌃1 (z))|z=z045✓

g 2 z01+ 2(z0 4J 2 )3/2

◆ 1

(4.84)

where here, z0 may be any of ! ± or !
˜ , and due to the analytic continuation, thesecond term in parentheses in (4.84) takes a minus sign for !
˜ . Furthermore,with a little bit of e↵ort, we may write down an explicit form for the branch
cut detour contribution BC (t). We do not give its explicit form here, andinstead reference equation (10) in [118]. The important fact to note is thatfor long times t, the integrands both vanish, and thus the contributions fromthe branch cuts are incoherent, decaying behavior. We will later use this factthat limt!1 BC (t) = 0. With equations (4.83), (4.84), and (10) from [118],
we may numerically solve (4.82) for any range of parameters in which we areinterested. As an example, consider the di↵erent behaviors displayed in fig.
4.6, which displays nearly Markovian behavior across a whole band for weakcoupling (A) and strongly non-Markovian behavior across the band for strong
coupling (B). In particular, for large bandwidths, where the coupling to theedges in the density of levels is weak, we recover largely the Markovian decay
observed for large detunings in [91]. The other, more exotic e↵ects displayedin fig. 4.6 are the domain of the next sections.

4.4.2

Bound state oscillations

In numerically solving (4.82) for certain ranges of parameters, we observe
behavior in which the excited state population decays to zero before risingback up to some finite value and oscillating between this value and zero for allobserved times t thereafter, c.f. fig. 4.7.
These long-lived oscillations can be interpreted as a beat between the twobound states (above and below the band). We may demonstrate this as follows: Consider the last two contributions in (4.82). We have already indicatedthat limt!1 BC (t) = 0. Furthermore, “by inspection”, we may observe thatlimt!1 RU P e(i! | U P |)t = 0. Therefore, at late times, (4.82) becomes+Cr (t ! 1) = R+ e i! t + R e i! t

(4.85)and thus the observable excited state population is the absolute square:|Cr (t)|2 = |R+ |2 + |R |2 + 2Re(R+ R ) cos (! +! )t(4.86)

In other words, it is apparent from (4.86) that the population in the excitedstate at long times oscillates with a certain amplitude 2Re(R+ R ) around some“mean value” |R+ |2 + |R |2 at a frequency corresponding to the di↵erences ingenergy between the two bound states ! + ! .
The relative size of the residue terms determines how pronounced thise↵ect is, and as such, considerations of the residues can determine where in

46

(A)

(B)
1

Population

Population

1

0
2

2

Jt

0

4

-2

0
2

2

/J

Jt

6

0

4

-2

/J

6

Figure 4.6: Transition from nearly Markovian decay to Rabi-like oscillation.
Here we present the results of time evolution of the excited (|ri) population
as a function of time for and detuning with a coupling strength of (A) g =2⇡ ⇥ 0.43 kHz/2 and (B) g = 2⇡ ⇥ 1 kHz/2 and bandwidth J = 2⇡ ⇥ 0.45
kHz. In (A), the decay approaches Markovian exponential decay for a rangeof parameters within the band, only displaying significant oscillatory behavior
outside the band. In (B), the coupling strength and bandwidth are comparable,
and a more exotic set of behaviors is predicted. The decay is not Markovian
anywhere within the band, instead displaying a partial decay and oscillatory
behavior for all detunings. As we demonstrated in 4.4.2, this behavior is aconsequence of two bound states beating strongly for the chosen parameters.
Note also that at the level of a single band model, the time evolution behavior
is symmetric around the band center = 0. This results from the assumptionof a sinusoidal band.
parameter space we hope to observe such oscillations. We note that similar
behavior was considered for an array of coupled cavities [163] and in slow-lightwaveguide-QED [164].
Furthermore, as a test of this prediction, we have numerically evaluatedthe long-time behavior of the excited state population dynamics for varying
tunneling frequencies J (bandwidths 4J) at zero detuning and fit a sinusoid tothe results. The fit frequency and amplitude are then compared in 4.8 to the
expected bound state frequency di↵erence ! + ! and predicted oscillation
amplitude:
q
p
+
2+
!
!
=
2
2J
g 4 + 4J 4
(4.87)=047

Excited Fraction (#)

1.0
0.8
0.6
0.4
0.2
0.0

0

1

2
3
time (ms)45Figure 4.7: Emission dynamics into a band structure. Here we present theresults of time evolution of the excited (|ri) population as a function of timefor 5 ms with a coupling strength of g = 2⇡ ⇥ 1 kHz/2 and various bandwidths(computed in the tight-binding approximation). For bandwidths larger thanthe coupling strength, here ⇡ 4 kHz (blue, dashed), the decay into the band isapproximately a Markovian exponential decay. For bandwidths much less thanthe coupling, in the figure ⇡ 10 Hz (red, dash-dotted), the behavior exhibits
a nearly perfect Rabi oscillation. For intermediate bandwidths approximately
the same as the coupling strength, here ⇡ 1 kHz (green, solid), the behavior
exhibits first a complete decay of excited state population before a revival
settling to a long time oscillation which persists for all future times. Theseoscillatory e↵ects are discussed in detail in sections 4.4.2 (similar behavior was
reported in [163]).
Amp = |R+ |2 + |R |2 + 2Re R+ R
0
= 4 @1 + p( g 4 + 4J 4

2J 2 )qg1

2

4J 2 /(2J 2 +p
g 4 + 4J 4 )1 2A(4.88)The results agree exceptionally well, suggesting that the predicted behavior isindeed physically explained as an oscillation between these two states.

4.4.3

Deep lattice limit

The above observations suggest a connection to well-understood atomic physics,
namely, that of a Rabi oscillation. In particular, if the bath lattice becomesvery deep, then the tunneling rate in the lattice should tend to zero (J ! 0) as
the band gets narrower. In such a limit, one expects Rabi-oscillations between

48

(B)

Beat Frequency (kHz) Amplitude (fraction)

(A)

1.0
0.8
0.6
0.4
0.2
0.0
4
3
2
1
0

200 400 600 800 1000
Tunneling (Hz)Figure 4.8: Relation between the bandwidth of the emission band and thebound state beat frequency and amplitude at long times for fixed couplingstrength g = 2⇡ ⇥ 1 kHz/2 and detuning at the band center: (A) Oscillation
amplitude and (B) beat frequency. The dots are data points extracted froma sinusoidal fit to the numberical solution of the equations of motion for longtimes (15-20 ms) and the solid line is the expectation based on the model. The
shaded gray box indicates the region of bandwidths for which the oscillation
amplitude is less than 10% of full contrast, for which parameters the behavior
e likely not resolvable in an experiment.
the bath and the excited emitter. We now demonstrate that this is, in fact,
the behavior that occurs.
We note first that in the case where J ! 0, the bound-state energy defining
polynomial (4.83) becomes particularly simple
z

=

g2
z

(4.89)

or equivalently z 2
z g 2 = 0. This quadratic equation may be solved using
the quadratic formula, and the roots are found to be
!± =

1⇣
2

±

p

⌘
+ 4g 2 =
49

˜
±⌦
2

(4.90)

˜ 2 = ⌦2 + 2 , aswhere we have introduced the generalized Rabi frequency ⌦in typical atomic physics applications. For intuition, we fist consider the caseof resonant coupling, i.e.
! 0. In this case, we note that the bound state
frequencies are just ! ± = ±⌦/2 = ±g. The residues may be computed for thezero bandwidth case from✓
◆ 1
g2
ResG (z) = 1 + 2(4.91)z
and thus on resonance, ! ± = ±g, so R± = 1/2. We also note that since theband has zero bandwidth in this considered situation, there is no contribution
from the branch cuts or from an unstable pole. In such a case, (4.86) becomesexact for all times, and using the values we have computed for the residues,
we find
✓✓
◆ ◆
✓ ◆
1 1
⌦ ⌦
⌦
2
pe (t) = + cos+
t = cost(4.92)2 22
2
2

which is exactly the evolution we expect for a Rabi oscillation when the population starts in the excited state. Thus, we have recovered the regular Rabi
oscillation as a limit of emission into a deep lattice. We recognize the twobound states in this case as having converged to the dressed states of theinternal state evolution.
We next generalize to the case of finite detuning , guided by our intution from the resonant case. The bound state freqencies are (4.90). The
corresponding residues are computable as
R+ =

˜ 2
˜ 2
( + ⌦)(⌦), R =˜ 2 + ⌦2˜ 2 + ⌦2
( + ⌦)(⌦)(4.93)

One may show, after much tedious algebra, using (4.93) that the following
hold:
✓
◆
2
1
⌦2
+ 2
2
|R | + |R | =1+, 2Re(R+ R ) =(4.94)
˜2
˜2
2⌦
2⌦
Thus we find
! !
✓
◆
2
˜
˜
1
⌦2
+⌦
+⌦
pe (t) =
1+
+
cos
t
˜2
˜2
2
2
⌦
2⌦
!
2
˜
⌦2
⌦
=
+
cos2
t
(4.95)
˜2
˜2
2⌦⌦
We note that this is the well known result for the population of the initial

50

state in a detuned Rabi oscillation. As a check on our logic, we point out thatthe (more) well known expression for the population of the state which is notinitially populated is recovered by simply taking 1 pe (t). Specifically,!
˜⌦2⌦
1 pe (t) =sin2
t(4.96)
˜22⌦which confirms our intuition. A two-state Rabi oscillation arises as the deeplattice limit of our spontaneous emission system, as physical intuition says it
must when the density of levels of the target state becomes delta-like. We
note that another way of thinking about this limit of a very deep lattice is torealize that the coupling strength is so much larger than the bandwidth 4J
that we e↵ectively only couple a single emitter state to a single blue state,roughly corresponding to a cavity-QED limit [96].

4.4.4

Weak-lattice limit

We can also consider the case in which the lattice potential felt by the blue
atoms is vanishingly small. In this case, we expect to recover the free-particle
results of section 4.3. We shall now demonstrate that this is the case.
We note that for a very weak lattice, or no lattice at all, the assumption of
momentum independent coupling g breaks down. Instead, we must consider
a momentum dependent coupling g ! gq . The exact details of this coupling
will depend upon the problem at hand. We thus must refine the expression
for the 1D self energy in such a case toZ1|gq |2⌃(z) =
dq(4.97)
2⇡ BZ z !(q)For a shallow lattice, !(q) = !(k) ⇡ ~k 2 /2m⇤ , and thus we may write for(4.97)
1⌃(z) =
2⇡

Z ⇡/a

⇡/a

dk

z

|gk |21=
2⇤~k /2m
⇡

Z ⇡/a
0

dk

z

|gk |2~k 2 /2m⇤(4.98)

with the second equality following from the evenness of the integrand. We
have replaced q with k since in the limit of a vanishing lattice, quasimomentum
becomes indistinguishable from usual momentum.
It turns out to be useful to note that for the zero lattice limit, we havereintroduced the units of the lattice spacing a, which we take to zero, thus
sending the upper limit of integration in (4.98) to infinity. Next, we introduce51

the change of variables from momentum to energy at the cost of introducing adensity of states ⇢(!). Formally, dk ! ⇢(!)d!, and ! = ~k 2 /2m⇤ . Thus, theself-energy becomes
Z 1
Z 1|g(!)|2 ⇢(!)G(!)⌃(z) =
d!
=d!
(4.99)z !
z !
00where we have absorbed 1/⇡ into the definition of ⇢(!), and where G(!) isdefined3 after (4.48) in subsection 4.3.5.
We next note that if we takerp⇡aho ⌦2 2!/!0
L 2m⇤2|g(!)| =e, ⇢(!) =(4.100)L
2⇡~!
as is the case for a 1D free particle motion coupled to a deeply confinig harmonic oscillator well of frequency !0 with strength ⌦, then we findrZ
1 ⌦2 1!0 e 2!/!0⌃(z) = pd!
! z !
2⇡ !0 0
r
✓r
◆
⇡
1
2z
2z/!0
=
e
Erfc
(4.101)Lp
2!0z
as a function of a complex frequency z. After appropriately rotating (4.101),
we see that it is the same as the last term in (4.60). Therefore, we conclude
that in the limit of no lattice, we exactly recover the results and solutions fromsection 4.3.

4.4.5

Bound states above and below the band

In addition to examining the time dynamics across the varous limits of bandwidth and coupling strength, we can also study the bound states whose existence is indicated by the poles in the Laplace transforms of (4.69) and (4.70).
In particular, a purely imaginary pole in the Laplace transform sB = i!B(corresponding to a purely real pole in the Fourier transform inversion integral (4.78), c.f. discussion in 4.4.1) yields a non-decaying bound state. For

3

N.B.: it is NOT the Green’s function of the resolvent operator approach in [115].

52

completeness, the Laplace transforms of Cr (t) and Ck (t) are1
s + i + ⌃e (z)ig C̃r (s)C̃k (s) =s i!(k)C̃r (s) =(4.102)(4.103)

Following the same procedure as in section 4.3.5, one can show that the polesB corresponds to a bound state having the structureZ ⇡gdq| B i = |er , vacq i|gr , 1q i
(4.104)2⇡ ⇡ !(q) !B
which has the same structure as encountered previously (up to a q independent
coupling constant g and a di↵ering dispersion relation !(q)), when we make aswitch to a frequency representation rather than a momentum representation.
For computational purposes, it is sufficient to work in momentum. The spatial
structure of the shallow lattice contribution to (4.104) is revealed by taking
the product with hx|, and we findZg ⇡q (x)dq(4.105)
B,mw (x) =2⇡!B⇡ !(q)
where we have tacitly chosen units in which qrec = 1 and q (x) is now the Blochwave corresponding to quasimomentum q in the band structure. As such, we
expect the bound states (4.105) to reflect the periodicity of the underlying
lattice. The first di↵erence to note from the case of free momentum waves isthat there are two bound states corresponding to a pole above and below theupper and lower edges of the band respectively.
Functionally, we can compute the Bloch waves q (x) numerically by exact
diagonalization for a number of discrete quasimomentum values q and thencompute the spatial distribution (4.105) to any desired degree of numericalaccuracy. The results for various lattice depths and detunings are consideredin fig. 4.9. We note that the bound states below the band exhibit a sort ofplateau into the neighboring wells, and the state above the band shows a sharp
modulation with a revival on the neighboring sites. We understand this to be
the result of the fact that below the band, the bound state is coupled moststrongly to the q/qrec = 0 Bloch waves, whereas above the band, it is coupledmost strongly to q/qrec = ±1. Indeed, a naı̈ve Fourier transform of (4.105)
Z 1
1'B (k) = p
e ikx B,mw (x)(4.106)
2⇡ 1
53

1

0
-2 -1 0 1 2
q/krec0-2 -1 0 1 2
k/krec1

0-2 -1 0 1 2
k/krec|+(z)|2 (arb.)

0-2 -1 0 1 2
q/krec1

10-2 0 2z/al|-(z)|2 (arb.)|-(q)|2 (arb.)(B)1|-(q)|2 (arb.)|+(q)|2 (arb.)(A)|+(q)|2 (arb.)shows that below the band, the momentum is most strongly concentrated
at k/qrec = 0 and above the band, this momentum decomposition is doubly
peaked near k/qrec = ±1, c.f. fig. 4.9.

10

-2 0 2z/alFigure 4.9: Computed momentum space (| B (q)|2 and |'B (k)|2 ) and position
space (| B (z)|2 ) wavefunctions for representative bound states above (A) andbelow (B) the single band. The representative bound states are computed for!B± = ±6J, i.e. one bandwidth away from the band edges. The bound state
above the band displays a maximum quasimomentum composition aroundq = ±qrec , and the bound state below the band is maximally populated by
q = 0. Note that compared to B (q), 'B (k) extends out of the first Brillouinzone. The position space wavefunctions are modulated at the periodicity of
the lattice, potentially resulting in observable e↵ects in an experiment.
We note that the Fourier transform is a decomposition into plane waves,
but it is also natural to ask what is the contribution of each Bloch wave tothe state B,mw (x). Indeed, we can compute the elements of a “Bloch” series,
since the Bloch waves form a complete basis, i.e.
Z ⇡/d0⇡/d0⇤
p (x) q (x)dx =(pq)

The resulting “Bloch” series coefficients 'B (p), defined by
Z 1⇤
B (p) =
p (x) B,mw (x)dx154(4.107)(4.108)

indicate how much of a contribution each Bloch wave, equivalently quasimomentm, makes to the bound state. We can compute this decomposition B (q)“by inspection” from (4.105), whereupon we findB (q) =g1
2⇡ !B !(q)(4.109)The results are shown in fig. 4.9. We note that for the case under consideration, in which !(q) is a single sinusoidal band that there are two possibilities
for the shape of B (q). In particular, for !B below the band, the denominator of (4.109) is largest for q/qrec = 0, whereas for !B above the band, thethe quasimomentum distribution is largest for q/qrec = ±1, confirming ourunderstanding of the distributions based on the simple Fourier transform. It
is further interesting to note that for !B+ = !B (i.e. if we choose boundstates located symmetrically around the band) the corresponding distributions (4.109) can be obtained from each other by shifting all quasimomenta
by half of the Brillouin zone. This exact symmetry will be lost if we consider
non-sinusoidal bands or multiple bands.
The di↵erence between the momentum and quasimomentum decompositions (or, in practice, their absolute squares, | B (k)|2 and |'B (q)|2 ) is not justa question of academic interest, for in an experiment, we may access both themomentum distribution of the matter wave component, as well as the quasimomentum distribution via a band map procedure. We shall discuss this further
in section 5.3.

4.5

Concluding remarks

In this section, we have explored in detail spontaneous emission behavior ofan atom trapped in a well of a deep optical lattice with variable coupling tofree space and to shallow lattice. The boundaries strongly modify the decay ofthe population in the lattice well, which displays crossovers from Markovian
to non-Markovian dynamics. For free space emission, the emitted matterwave spectrum at positive detunings is matched well by a simple model forfreely-propagating massive particles, and the evanescent wave state formed fornegative detunings decays exponentially away from the well. For emission intoa band structure, we expect an even stronger Markov to non-Markov crossover,as well as the presence of multiple bound states strongly modified by the lattice
periodicity.

55

Chapter 5
Experiments on spontaneousemission of matter waves into atunable band structureThis chapter discusses experimental results partly based on 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) [91] (sections 5.1 and 5.2) and partly basedon our preprint Fractional decay of matter-wave quantum emitters in a synthetic bandgap material, Michael Stewart, Joonhyuk Kwon, Alfonso Lanuza,
Dominik Schneble, arXiv:2003.02816 (2020) [149] (sections 5.1 and 5.3).

5.1

Experimental tools and techniquesExperiments in our lab are performed utilizing a moving-magnetic coil transporter apparatus to create Bose-Einstein condensates of Rubidium-87 [120].
Starting from a Magneto-Optical Trap (MOT) of ⇡ 1 ⇥ 109 87 Rb atoms, we
produce BECs having 1 ⇥ 103 atoms up to 2 ⇥ 105 atoms (depending on thespecifics of the experiment) with a duty cycle of approximately 20 seconds.
Specifics of the apparatus have been well documented [120, 165–172], and in
the following we provide a brief overview of the most important components
required to realize the models discussed in chapter 4.

5.1.1

Condensate production in brief

Here, we very briefly discuss condensate production in our moving-magnetic
coil transporter apparatus. (For more detail, see [120]) We utilize 87 Rb atoms

56

in their ground state 52 S1/2 (hyperfine states |F = (1, 2), mF i with two excitedstates 52 P1/2 and 52 P3/2 , of which the second one is separated from the groundstate by ⇡ 780 nm. Starting from a magneto-optical trap (MOT) consistingof three counter-propagating pairs of cycling light (cycling transition F = 2 !
F 0 = 3) and a gradient magnetic field, we capture and cool approximately
1 ⇥ 1010 atoms at a temperature of about 1 mK. (We note that in addition to
the cycling light, we apply repumping light, F = 1 ! F 0 = 2, to return atoms
which escape from the cycling transition back to cyclable states.) Following
the cooling in the MOT, we turn o↵ the magnetic potentials and subject theatoms to a phase of optical molasses cooling [173] to achieve sub-Doppler
cooling to about 50µK. The atoms are then pumped into the |1, 1i state
and transported to an ultra-high vacuum cell using our magnetic-transporter
apparatus [120, 172].

Figure 5.1: Picture of the science cell (ultrahigh vacuum cell) in which BECs
are produced (glass cell, upper left). Visible in front and just left of center isthe retro-reflection set-up used for creating one horizontal lattice.
After transport to the ultra-high vacuum cell, we evaporatively cool the57atoms in the quadrupole magnetic trap using a radio-frequency knife. Afterabout 10 s, the efficiency of this evaporation becomes limited, and we switch toa time-orbiting-potential trap (TOP trap) [174] and continue efficiently evaporating in the magnetic trap followed by an optical trap composed of two crossed
1064 nm beams (XODT, crossed optical dipole trap), ultimately reaching BEC
with a temperature of T ⇠ 100 nK and remaining condensate atom number
anywhere between 104 and 106 atoms, determined by the specific XODT evaporation parameters. We note that due to many technical improvements during
the author’s tenure in the lab [172, 175], the duty cycle of the experiment isapproximately 20 to 30 seconds, allowing for a factor of 2 or 3 increase in thenumber of data points which we can acquire in a given time.

5.1.2

Atom Imaging Scheme

We also briefly discuss our atomic detection scheme, for which more details
may be found in [120]. After creating a BEC in the XODT, we perform anexperimental iteration, and then quickly turn o↵ all trapping potentials in aband-mapping procedure [176]. The atoms thus fall in time-of-flight, during
which time they ballistically expand. This expansion during the approximately
15 ms time-of-flight used in the experiments described in this thesis results in
an atomic sample which is large enough to be imaged using absorption imaging
techniques [177] on a CCD camera (Princeton PIXIS 1024B). The time-of-flightalso essentially performs a Fourier transform operation on the atoms, and thus
our absorption images are approximately the momentum distribution of theinitial atomic cloud at the time of their release from the trap, albeit with only
density and not phase information.
After the atoms have fallen for an appropriate time-of-flight, we apply aStern-Gerlach separation protocol in order to spatially separate atoms having di↵erent hyperfine projections mF . The application of a gradient field
perpendicular to the atoms’ direction of motion separates the di↵erent spincomponents, thus resolving these states on the CCD. Since, however, there are
two possible values for F , Stern-Gerlach separation is not enough to recover
the full information of the atomic state, as, e.g., |2, 1i and |1, 1i have thesame mF times their respective Landé g-factors. In order to circumvent thisproblem, we first image atoms having F = 2 using resonant imaging light onto
approximately one third of the CCD chip operating in kinetics mode. After ashort time, the camera moves the illuminated pixels into an unused and shadedo↵ region, and we apply first repump light and then imaging light to image
the F = 1 atoms, which will have fallen by an additional 3 ms during thistime. In this way, we separate all possible hyperfine states of our atoms in our
imaging process, and thus we have access to the full atomic state population
58

(A)

(B)
F=1 BEC

Figure 5.2: (A) Cartoon illustrating the principle behind abosorption imaging
of a BEC. Resonant F = 2 light shines upon an atomic sample, which absorbs
and re-scatters the imaging light. This casts an e↵ective shadow on a CCD
camera. (B) Sample absorption image of an F = 1 BEC after converting
photon counts from the CCD camera into an optical density (OD). This image
is typical for a condensate only experiencing a crossed optical dipole (XODT)
trap, and for this image, the condensate is found to have ⇠ 100, 000 atoms.
information in our experiments.

5.1.3

State-selective Optical Potentials

Laser fields have long been used to create potentials for ultracold atom experiments, including the creation of periodic potentials (optical lattices) byretro-reflection of a laser beam to create a standing wave of light. In orderto perform the experiments discussed in chapter 4, we require a deep opticallattice potential for one atomic species and a shallow or vanishing lattice potential for the other. The answer can be found in the form of state-selective
lattice potentials, that is, potentials which have a di↵erent magnitude (andeven sign) for di↵erent atomic hyperfine states. The details of a state-selective
lattice have been throughly discussed in [125, 156, 157], and many experiments
in our research group make use of them to study phenomena from atomic 4
wave mixing, the Bose-glass transition in the presence of disorder, and matterwave di↵raction from an atomic or else from no potential [69, 82, 83, 86, 87].

59

(A)

V/Er

100
0

-100

770

780

770

780

790

800

790

800

 [nm](B) 6

 [1/s]420 [nm]

Figure 5.3: (A) Calculated optical potential for state selective light basedon equation (5.1). The potentials are shown for |ri = |1, 1i in red and|bi = |2, 0i in blue withpolarized laser light as a function of wavelength.
The overall scale is intensity dependent, and the selected curves are computedfor I = 0.38 kW/cm2 . The small-dashed line shows the wavelength used in
section 5.3,
= 790.4 nm, for which the lattice depths are sr = 20 andsb = 2.5. (B) Corresponding scattering rates for the same parameters. At= 790.4 nm, they are respectively r = 5 s 1 and b = 4 s 1
The basic idea behind a state-selective potential for 87 Rb is that due to thepresence of two strong ground state transitions (from the ground state 1 S1/2
to excited states 2 P1/2 and 2 P3/2 ), usually called D1 and D2 at ⇡ 795 nm and⇡ 780 nm respectively [121], the light shifts relative to each transition line cancancel for a certain hyperfine state of 1 S1/2 at a certain polarization. Thatis, if we tune the laser wavelength to be between the D1 and D2 lines, we60can select a wavelength for which the blue-detuned shift from D1 partially or
fully cancels the red-detuned shift from D2 . A full calculation can be found in[151,89], with the result
2
3
li 2
li 2
2
X
X
|c
|
|c
|
3⇡c 4 D1
2 Dpp
5 I(r)U (r, i, p) =
+ 3 2
(5.1)
3
2
!D1 l2P ! !li!D2 l2P ! !li
1/2

3/2

where i indexes the initial hyperfine state, D1,2 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 ± or⇡-polarized, and I(r) is the intensity profile of the optical potential.
We note that (5.1) depends on the choice of polarization and hyperfine
state. We plot in fig. 5.3(A) the optical potential for the state pair |ri =|F = 1, mF = 1i and |bi = |2, 0i. The tune-out wavelength for |bi atoms
occurs at = 790.0 nm independently of polarization, and forlight, thiswavelength produces a residual blue-detuned optical lattice for |ri atoms. By
choosing wavelengths appropriately around this point, we may create a differential lattice for |ri and |bi atoms having the same detuning or opposite
detunings, resulting in an e↵ectively shifted lattice for one state by /4.
For completeness, we note that near the D1 and D2 lines, the 87 Rb atoms
are subject to near resonant scattering, whose rate may be computed similarly
to (5.1) [178] with the result⇡c2 D2
1 g F mF q
2 + g F mF q
+ D2
I(r, t)(5.2)
i (r, t) =
D1
3
2
2
2~!0
D1
D2
The scattering rate is shown for the same parameters as the optical potentials in fig. 5.3(B). For the experiments we will describe in the following, thescattering rates for |ri and |bi atoms are approximately 4.8 s 1 and 3.7 s 1
respectively, which will not prove to be limiting for the experiments described
below.

5.1.4

Microwave hyperfine transitions

In order to perform our experiments, we require precise control over the hyperfine state of our Rubidium-87 BECs. Specifically, we will work completely
in the ground state manifolds F = 1 and F = 2 after preparing an initial|F = 1, mF = 1i BEC [120]. In order to perform emission experiments, wemust prepare atoms in |ri = |1, 1i and |bi = |2, 0i, and for magnetic fieldpost-selection, we require also the ability to the populate |2, 1i and |2, 2i
61

hyperfine states [175]. At a fixed magnetic field, applied in the laboratory by
three sets of Helmholtz-coils (one per spatial direction), the transitions within
the F = 1 and F = 2 manifolds are in the radio-frequency (RF) range, whereas
the transitions between the F = 1 and F = 2 manifold are in the microwave
regime, as can be calculated from the Breit-Rabi formula [179].
In order to generate microwave fields for hyperfine state transitions, we utilize a microwave synthesizer locked to an external Rubidium frequency standard, to which we mix in a smaller, RF signal created according to the experimental design needs using Agilent Tektronix arbitrary function generators.
This signal is ultimately fed into a rectangular microwave waveguide antenna,
which generates the microwave field at the position of the atoms. (For fulldetails of the microwave set-up, see [169].)

5.1.5

Sample preparation

The experiments described in this chapter begin by creating an opticallytrapped Bose-Einstein condensate, as discussed in section 5.1.1. Ultimately,
we seek to create a Mott-insulating phase having average site occupation number hni i . 0.5. The experiments all deal with 1D physics, for which we create
deep horizontal (state-independent) lattices such that the tunneling time between neighboring tubes is much larger than the experimental timescales. We
note that in ramping on optical lattice potentials, we incur a shift of the position of the trap minimum with respect to the un-shifted trap due to gravity,
known as gravitational sag. In the experiments of section 5.2, we turned on all
optical lattice potentials at the same time, resulting in an appreciable gravitational sag, which had consequences on our observations, described in moredetail in section 5.2.2. In order to minimize gravitational sag for the experiments of section 5.3, the horizontal lattices are first adiabatically ramped up in
80ms followed by the vertical state-dependent lattice (90 ms) to final depthsof 40Er,1064nm , 40Er,1064nm and 20Er,790.41nm so that the atomic cloud sits at
approximately the trap minimum potential, with a residual confinement along
the z-direction of !z ⇡ 2⇡⇥100 Hz. Here, Er, is the recoil energy of the lattice.
We determine the lattice depths using Kapitza-Dirac di↵raction [38, 180–184]of the atomic sample in the three directions of interest. This procedure createsa Mott-insulating sample deep within the Mott regime. With atoms loaded
into the lattice, a variable fraction f is transferred, at a bias field of 5 G, toan intermediate |2, 1i state using a two-photon microwave and radio-frequency
pulse of about 2 ms duration. The transferred atoms are removed using resonant light on the D2 cycling transition. After the pulse sequence (in which f
is adjusted between 0.6 and 0.85 to compensate for di↵ering initial atom number), the remaining sample has about 2.7(3) ⇥ 104 |ri atoms with an average
62

site occupation of hni i . 0.5 in the tubes.

5.1.6

Determining the resonance condition

The resonance condition= 0 is defined with respect to the transition between the band minimum "n=1,q=0 and the harmonic-oscillator ground state in
the emitter potential (with a residual bandwidth of 1⇥10 2 Er ). We use latticetransfer spectroscopy [86] to determine the resonance condition. An opticallytrapped BEC of |ri atoms is first transferred into the |bi state, after which
the state-dependent lattice potential is ramped on slowly. Microwave pulsesof duration ⌧ = 400µs are then applied at a fixed strength ⌦ = 2⇡ ⇥ 1.0 kHz
and variable frequency to transfer maximally 30% of population into the |ristate. The= 0 frequency for use in the experiment is obtained from a
fit of a Rabi spectrum to the data. Systematic residual mean-field shifts are
estimated to be between 150 and 270 Hz for all initial atom numbers used,based on a direct simulation of the 1D time-dependent Gross-Pitaevskii equation. The resonance condition (which depends on both optical and magnetic
fields) is stabilized using a post-selection magnetometry technique, yielding an
uncertainty of E ⇡ 350 Hz [91, 175].

5.2

Experiments on free-particle emission

In this section, we highlight some of the experimental results of [91], with a
special focus on where the data obtained therein do and do not match the models presented in sec. 4.3. We note that while material photonic crystal systems
have observed many predicted features of spontaneous emission near a continuum edge (such as modified spontaneous emission rates [101, 105] and Lamb
shifts [185], spectral signatures of non-exponential decay [186], and evidence
for photon bound states in photonic crystal waveguides [109] and corrugatedmicrowave guides [110]), a direct observation of many salient features was firstaccomplished using our cold atomic platform [91]. This experiment has been
described in detail in the PhD thesis of my colleague Ludwig Krinner [172],thus we shall only survey the results here.

5.2.1

Spontaneous Emission near a Single Edge

We briefly discuss how to use the techniques of sec. 5.1 in order to simulate
the physics of spontaneous emission near a single energetic edge. We create astrongly-confining array of 1-dimensional tubes using retro-reflected = 1064
nm light (depth V0 = sEr with s = 40 and Er = ~2 k 2 /2mRb the recoil energy,63related to the wavelength through k = 2⇡/ ). Along the third axis, we apply
state-selective light at = 790 nm, for which the state |bi = |2, 0i experiencesno lattice potential at all, sb = 0, and with power chosen such that the state|ri = |1, 1i feels a blue-detuned lattice of depth sr = 30. In this way, thered atoms |ri realize the ground and excited states of the emission model insec. 4.3 as an occupational spin, that is, an empty well of the deep lattice
is |gi, and a well with a single atom present is |ei. The atoms in state |bi
are free to propagate along the tube axis, and thus they form a continuum ofstates corresponding to the momentum of the atom: |0i is the state with noblue atom present, and |1k i is the state with one blue atom having momentum(equivalently, wavevector) k.
By applying a microwave coupling at a given strength ⌦ and detuning ,we realize the Hamiltonian of eq. (4.9), and we schematically present thescheme in fig. 2.2. It is important to note that the detuning
now plays
the role of the excited state energy of the atomic quantum-emitter (QE), andthat this value may be tuned arbitrarily above or below the edge of the mode
continuum (corresponding to = 0). We note in contrast that the QE excitedstate energy is fixed by a material’s properties in a photonic band-gap material,and furthermore that we may also tune the strength of our coupling gk / ⌦by varying the strength of our microwave signal.
Much like the Weisskopf-Wigner model discussed in sec. 3.3, there should
exist a regime well inside of the continuum (i.e. for ⌦/ ⌧ 1) for which
the emission of atoms from the excited state is Markovian, and whose rate isgiven by Fermi’s Golden Rule, c.f. sec. 4.3.3. We experimentally probe thisregime in fig. 5.5 for ⌦/ ⇡ 0.4 and⇡ 2⇡ ⇥ 2 kHz by creating a (mostly
empty array of) QE(s) and applying coupling at strength ⌦ for a time ⌧ andrecording the relative population of |ri and |bi atoms. Due to residual axial
confinement along the 1D tube, we restrict our attention to times shorter than
the inverse trapping frequency, ⌧exp < ⌧z = 2⇡/!z ⇡ 10 ms. The predicted
Markovian decay rate is = 2⇡ ⇥ 72 Hz, and for comparison with the theoryof chapter 4, we show the full solution. The theoretical model is seen to di↵er
dramatically from the observed behavior in that the model decays to zero
population for long times and the experimental data only partially decay inthe long time limit. We understand this e↵ect as a manifestation of the factthat we have more than one QE (in fact, a whole lattice of them at the startof the experiment), and this plateau is consistent with numerical simulations
displaying a phenomenon similar to radiation trapping [91, 172].
In addition to the decayed fraction of |ri atoms, we can measure the momentum of the emitted atoms. Conservation of energy dictates
p that for a fixed
QE energy ~ , the momentum will vary according to k̄ = 2m /~, and in64

2arb. u.

k/kr

(A)

.5

0

-2
-10

1

0

0

/2 [kHz]

20

30

(B)

Figure 5.4: (A) Theoretically predicted momentum space emission profiles fora pulse of ⌧ = 400 µs, ⌦ = 2⇡ ⇥ 1.5 kHz, and variable detuning . The
computation assumes a deep lattice potential of depth sr = 30. (B) Measured
momentum space profile for the parameters as in (A) (reproduced from [91]).
The white-dashed line shows the expectation based on energy conservation.
addition to this naı̈ve estimate, we may compute the distribution of the emitter radiation as a function of time and QE energy. We present a comparison ofboth theoretical approaches to the data, taken for ⌧ = 0.4 ms and ⌦ = 2⇡ ⇥1.5
kHz, in the lower half of fig. 5.4, with the theoretical expectation for |Bk |2 (c.f.
equation (4.43)), shown in the top half of fig. 5.4. We see that the qualitativeand quantitative features are generall well explained by the model presented
in section 4.3.
The strongest non-Markovian behavior is predicted for the regime where⇠ 0 kHz, where the 1D density of states diverges at the band edge. We plot
results from time-dependent population measurements near the edge ( =2⇡ ⇥ 0.1 kHz and ⌦ = 2⇡ ⇥ 3.0 kHz) and at negative QE energy ( =2⇡ ⇥ 1.7 kHz) in fig. 5.5. The theoretical model of sec. 4.3.2 (shown as a
black line in fig. 5.5) is found to display qualitative behavior similar to thatwhich is observed for approximately = 0, and good quantitative agreementwith the data is achieved for the model at negative detunings. The deviationsof the data from the model for⇡ 0 are consistent with numerical simulations featuring more than one emitter (see [91]), and when the QE energy isvery negative, the coupling to |bi modes is reduced to the point that the dynamics is e↵ectively governed by a single QE, thus recovering the quantitative65

Population

(A)

1

.6

Population

Population

02
4
Evolution time (ms)

(B)

1

.5
1

(C)

.75
0

.5
1
Evolution time (ms)Figure 5.5: Predicted (black line) and observed (red circles with error bars)
decay dynamics of the emitter population when emitting in the vicinity of asingle edge in 3 regimes. (A) (Quasi-)Markovian regime (when (⌦/ )2 ⌧1) displays an exponential decay. Data taken at ⌦ = 2⇡ ⇥ 0.74 kHz and= 2⇡ ⇥ 1.9 kHz, and the error bars are the standard error of the mean(s.e.m.) from at least three iterations. (B) Non-Markovian behavior near theedge,⇠ 0, with= 2⇡ ⇥ 0.1 kHz and ⌦ = 2⇡ ⇥ 3.0 kHz. Observed
behavior in (A) and (B) deviates from the model of section 4.3, and this canpartially be explained by presence of neighboring emitters [91]. (C) NonMarkovian behavior away from the edge ⌦/ < 0, with = 2⇡ ⇥ 1.7 kHz
and ⌦ = 2⇡ ⇥ 3.0 kHz. Here, where neighboring emitters play no role, thedata and model agree within the error bars. Figure partially reproduced from[91], see also [172].
predictions of the model.

5.2.2

Matter-Wave Emitter Bound StateThe fractional decay in the theoretical model of sec. 4.3.2 has its roots in thepresence of a long-predicted bound state in which the emitter dresses itself
with a coherent cloud of emitted modes [97]. In particular, the incomplete

66

(A)

(D)

-

=142(3)nm
 1.3

(B)(C)

Figure 5.6: (A) Schematic illustration of bound states when the quantum emitter energy is negative. (B) Histograms showing the asymptotic population of|bi atoms after ⌧ = 2.6 ms, ⌦ = 2⇡ ⇥ 3.0 kHz and = 2⇡ ⇥ 2.2 kHz. The
top (bottom) panel shows the results for a sudden (respectively adiabatic)
coupling of |ri to |bi, and the shift in their centers hints at the formation of abound state “smoothly” for the slow coupling case. Each histogram is taken
for 50 points. (C) Sample momentum distributions for sudden (left) and slow
(right) microwave couplings at parameters as in (B). The apparent asymmetry
in the sudden coupling case suggests that some portion of the population ismoving preferentially in one direction along the tubes. (D) Measured boundstate. Guided by the observation in (C), the sudden coupling case is asymmetric due to the system being located away from the trap center, and theemitted atoms being subject to the gravitational potential due to this gravitational sag [91] (top panel). We plot the atom number for sudden coupling(empty black circles), slow coupling (blue circles), and the di↵erence of thetwo (gray triangles). The gray triangles are distributed about ksag = 0.8kr ,as expected from the gravitational sag in the trap. The gray curve passing
through the blue data is a Fourier transform of (4.62) fit to f it = 2⇡ ⇥ 2.1
kHz (in good agreement with the experimental value). This wavefunction has
a spatial decay length of ⇠ = 142 nm. Inset: Comparison of the size of the
Wannier function on a site (red) and the evanescent matter wave cloud of thebound state (blue, shaded). Figure reproduced from [91], c.f. also [172].
decay is goverened by how strongly the bound-state excitation is coupled toin (4.27). This bound state has a wavefunction that is roughly exponentially67

localized around the QE, c.f. fig.
p 4.4, with a decay length ⇠ approximately
given by (4.63), namely, ⇠ / 1/ | |.
In order to study the bound state in detail, it is important to not populate
any propagating modes |1k i appreciably, for which we develop a protocol toramp up the microwave coupling slowly with respect to the bound state binding
energy !B ⇡ . This slow coupling strength ramp prevents propagating
modes from becoming populated, and thus all observed |bi atoms belong to
the bound state. The result is shown in fig. 5.6. We note that a Fouriertransform of the bound-state wavefunction (4.62) fits the data well with only
the QE energy
as a fit parameter, and the corresponding decay length ispredicted to be ⇠ ⇡ 142 nm. Further evidence is provided for the existence ofthe bound fraction resulting from a roll-down of the emitted population fromthe side of the harmonic potential, a discussion of which can be found in [91],and which was discovered and understood by L. Krinner independently of theconsiderations arising from the theoretical model of sec. 4.3.5.

5.3

Experiments on emission into a band structureThis section presents in large part the results of our preprint [149]. As discussed in section 4.4, distinctive features in emission dynamics and boundstate structures arise in the presence of a band-structure, specifically in thepresence of a band with two edges rather than a semi-infinite continuum. Sincematerial systems have a band structure consisting of multiple bands, the insights we gain in our ultracold-atomic simulator may be of relevance to diverse
applications in waveguide-QED.

5.3.1

Band Spectroscopy

The experiment proceeds as in 5.2, however, in the present experiment wechoose a non-vanishing sb > 0 (depth of the lattice for |bi atoms) such thatthe mode continuum acquires a band structure. The dynamics of an isolated
emitter (ground and excited states |gi and |ei) is then governed by the HamiltonianXĤ =~gn,q ei n,q t |gi he| b̂†n,q + H.c.
(5.3)
n,q

involving Bloch bands, where n,q ="n,q /~ is the e↵ective detuningof the emitter (excitation energy ~ ) from the Bloch state |n, qi = b̂†n,q |0iwith energy "n,q (band index n, quasimomentum q); the vacuum coupling68

gn,q = n,q ⌦/2 contains the Franck-Condon overlap n,q = hn, q| e i with theoriginating harmonic-oscillator ground state | e i in an emitter. We choose
sb = 2.5 (at = 790.4 nm) giving a ground-band width W1 = ("1,k "1,0 ) =0.5 Er ⇡ h ⇥ 1.8 kHz. The band structure and corresponding Franck-Condon
factors are illustrated in Fig. 5.7(B).
(B)

n,q

(C)

4

4

2

2

0

0
0

q/k

10

n,q

1

arb. units
0.0 0.5 1.0
1.0
0.5
0.0
-2

0

q/k

2

/Er

n,q/Er

(A)

-0.5
-1 0

1

Figure 5.7: (A) Experimental scheme. 87 Rb atoms in two hyperfine groundstates |ri (“red”) and |bi (“blue”) are confined in state-independent 1D lattice
tubes. A state-dependent longitudinal lattice provides strong confinementfor one hyperfine state (sr1) and a shallow band structure for the other(sb ⇡ 2.5). Coupling between |ri and |bi (strength ⌦, detuning ) leads toemission into the band structure; each well acts as an emitter characterized byan occupational spin (states |ei and |gi) and excitation energy ~ . (B) Band
structure and relative strength of the vacuum coupling n,q(") (Franck-Condon
factor computed for a selection of densely spaced Bloch waves) for sb = 2.5
and sr = 20. (C) Measured quasimomentum distribution versus emission
energy ~ , as seen with absorption imaging after 14 ms time of flight, andaveraged over at least 3 runs. The lattice parameters are as in (B); the couplingstrength is ⌦/2⇡ = 1.0(3) kHz and the pulse duration ⌧ = 400µs. The zoom
in is taken with a smaller step size of 0.1Er , and an average over at least 4
runs for each quasimomentum distribution, and the calculated band-structureis shown (white, dashed).
We first characterize the momentum distribution of the emitted |bi atoms
as a function of the excitation energy ~ (calibrated using lattice transfer
spectroscopy [172]). After loading the array with |ri atoms, we apply a rectangular microwave pulse of duration ⌧ = 400µs and Rabi frequency ⌦ = 2⇡ ⇥1kHz; following a 500µs-long rampdown of all three lattices for band mapping.
The emitted |bi atoms are detected after time-of-flight using state-selective
absorption imaging. The measured distribution, shown in Fig. 5.7(C), reveals
that emission into the ground band is much stronger than that into the firstand higher excited bands. The suppression results from the structure of the69

vacuum coupling gn,q , which for even-n bands is reduced due the approximate
parity of the relevant Bloch states for sb > 0; a further suppression of the relevant Franck-Condon factor for higher n is due to the finite momentum widthof | e i and the decrease in the density of states.
We note that it is also possible in our system to study the case where sb < 0,which we can accomplish by changing the state-selective lattice wavelength
using the tunability of our Coherent 899 Titanium-Sapphire laser. Specifically,by shifting the wavelength in the opposite direction from the tune-out point
for |bi atoms, we create a situation in which the lattice depth for the |bi atoms
is negative. We note that this results in an e↵ective shift by /4 of the latticewells for the emitters and the emitted matter-waves. This results in FranckCondon factors that are appreciable for excited bands n 2, c.f. Fig. 5.8(A),and in observed emission profiles with appreciable contributions from both theground and first exited bands, cf. Fig. 5.8(B).

(B) arb. units

4

4

2

2

/Er

n,q/Er(A)0

0.0 0.5 1.0

0
0

n,q

-2

1

0

q/k

2

Figure 5.8: (A) Franck-Condon factor n,q and observed emission profile (B)for sb = 2.6 at = 789.8 nm. The strongest emission signal occurs in thefirst excited band; the relatively strong percentage of atoms within the firstband gap results from the strong coupling to the first excited band, giving rise
to non-Markovian emission in the band gap. We note a residual asymmetry
in the momentum distribution, which we attribute to residual gravitationale↵ects.
While the case sb < 0 is certainly interesting in its own right, both for thedifficulty of realizing a similar system in a material as well as e↵ects arisingfrom the coupling to multiple bands, it is beyond the scope of this thesis.
We shall instead focus on the case sb > 0. As is apparent from fig. 5.7,70

coupling to bands other than the ground band is suppressed on experimentallyobservable scales, and this justifies using the single band model of section 4.4.
We therefore focus on the e↵ects of only the ground band in what follows.
We have found in section 5.2 that the decay dynamics in an array ofemitters is generally subject to propagation, reabsorption and collisional effects [91]; however a description for an isolated emitter should apply in the
short-time limit. Furthermore, due to the experimentally motivated choice of"1,0 = 0, the equations of section 4.4 are modified in the following way: theground band is approximately sinusoidal,⇣ q⇡ ⌘"(q) = ~¯
! cos
+ ~¯
!
(5.4)
k
(denoting "(q) ⌘ "1,q , and ~¯! ⌘ W1 /2) and the vacuum coupling g = hg1,q iq ⇡0.39⌦ is approximately constant over the band, with negligible coupling toother bands. We have re-introduced physical momentum units by taking k =⇡/d to be the recoil wave-vector. The excited-state amplitude A(t) then evolves
as the Fourier transform
Z 1i
A(t) =d! G(! + i0+ )ei( !)t
(5.5)
2⇡ 1
of the Green’s function
G(!) =

1!

⌃(!)/~(5.6)

in which the interaction of the emitter with the band is captured through theself-energyZ~2 g 2 kdq~g 2⌃(!) =
=p
(5.7)
2k
"(q)
!(! 2¯!)k ~!

whose singularities now lie at ! = 0 and ! = 2¯
! , which are the band edges
given (5.4).
The dynamics of the emitter is governed by the singularities of G(!) definedby !
⌃(!)/~ = 0 which can be determined using standard analyticaltechniques (see section 4.4). In contrast to the single-edge case of section 4.3,
there are now two bound states [116], one below the band and one in the gap
above the band, as well as a nearly Markovian decay resulting from coupling
to the band and additional, incoherent losses due to the divergence of the selfenergy at the band edges.
For a comparison with the model, we first consider the ground-band emission spectrum, which can be obtained from fig. 5.7(C) by integrating over the71

Population

1.0
0.8
0.6
0.4
0.2
0.0

-1

0
/Er

1

Figure 5.9: Ground-band emission spectrum for g/¯! = 0.43 and ⌧ = 400 µs,
obtained from the distributions in fig. 5.7 (C) by summing over quasimomenta.
The dashed curve is the prediction of the isolated-emitter model, and the solid
curve is the same prediction reduced by 40%. Deviations from the model havealready set in by ⌧ = 400 µs, as can be seen in fig. 5.10.
quasimomenta. The agreement of the spectrum with the prediction (fig. 5.9) is
good up to an overall scaling factor of order unity. This discrepancy arises fromthe fact that for the parameters chosen, the model of a single emitter alreadybreaks down, even for short pulses of 400 µs. This can be seen most apparently
when we consider the time dynamics of an initially excited population.

5.3.2

Time Dynamics for Emission into a BandIn addition to measuring emission dynamics as a function of excited emitterenergy ~ , we can also fix the emitter energy, e.g. at the band center ~ = ~¯!
and instead record the population of the excited emitter as a function of thecoupling pulse duration. We can distinguish three behavior regimes in the time
dynamics as a function of the ratio of coupling strength to bandwidth g/¯!,
cf. fig. 5.10(A): irreversible decay for g/¯! ⌧ 1 (weak coupling), damped oscillatory decay for g/¯! ⇠ 1 (intermediate coupling), and undamped oscillations
for g/¯!
1 (strong coupling).
A Wannier picture provides an intuitve interpretation of the model’s predictions via the quantum Zeno e↵ect [187, 188] (cf. fig. 5.10(B)): here, the atom
coherently cycles with Rabi frequency ⌦0 = 2g between the strongly confining
emitter well and a corresponding |bi well of the shallower lattice, where it issubject to tunnel escape at a rate ⇠ !
¯ that damps the coherent local evolution.
In the strong-coupling limit, the bandwidth is negligible, and the population

72

1.0
1.0

(A)

0.5

(1)

0.5

(2)

(B)

0.5

tion
Popula

1.0

(3)

0.0

g


0.5
0.0



(4)

0.0

0.5

t (ms

)

1.0

1.5

Figure 5.10: (A) Decay dynamics for ~ = h ⇥ 1.8(1) kHz (at the band
center) for (1) weak coupling with ⌦ = 2⇡ ⇥ 0.4 kHz (g/¯
! = 0.18), (2,3)
intermediate coupling with ⌦ = 2⇡ ⇥ (1.0, 2.3) kHz (g/¯
! = 0.43, 1.0), and (4)strong coupling for a reduced bandwidth (W1 = 0.1Er ) with ⌦ = 2⇡ ⇥ 2.2 kHz(g/¯! = 4.9). The dots are data taken for di↵erent hold times, averaged overat least 3 runs each, with error bars respresenting the standard error of themean (S.E.M). The gray lines represent the predictions of the isolated-emittermodel. (B) Schematic illustrating the competition between coupling g andtunnelling ⇠ !
¯ in the shallow lattice (see text)
trivially performs coherent Rabi oscillations between the emitter and the (nowe↵ectively single-mode) vacuum, in analogy to a simple cavity-QED scenario.
A remnant of this e↵ect persists even in the weak coupling regime for shorttimes t . !
¯ 1 ⇠ 0.2 ms when the associated Heisenberg uncertainty in energy
exceeds the band width such that the band is not resolved and appears as asingle state. This results in a decay dynamics that always starts quadratically,
as a Rabi oscillation.
Generally there is good agreement between the model and the observation before significant population has been emitted (which mostly coincides
with the short-time regime). Deviations, including an o↵set and enhanced
oscillations, are expected to appear when atom reabsorption by neighboring
empty emitters becomes relevant [91]. Proper accounting for these e↵ects
would necessitate a multi-emitter model, possibly including collisional interactions, which is beyond the scope of this paper. That said, the observed decay
dynamics displays oscillations that last to long times, in qualitative agreementwith the single emitter model, and for the spectrum of fig. 5.9, taken at 400 µs,

73the data only di↵er from the model by a constant scaling factor. For g/¯!
1,
the emitters e↵ectively decouple such that the isolated emitter model applies
fully.

5.3.3

Bound States Above and Below the Band

Importantly, the spatial structure of a given bound state qualitatively depends
on whether it is located below or above the band. Using the approach of section
4.4.5, it is straightforward to show that the bound states contain evanescentmatter waves of the form
Z k±
dq ± (q) hz | 1, qi(5.8)
B (z) =
kwith quasimomentum probability amplitudes
±

(q) =

~g
1
±
2k ~!B "(q)(5.9)

For the bound state above the band, where the energetic di↵erence between~!B+ and "1,±q is smallest near q = ±k, this means that | + (q)|2 possesses a
double-peaked structure in quasimomentum space. In contrast, for the boundstate below the band, the quasimomentum is concentrated around q = 0.
To demonstrate these features, we directly detect the spatial features of
two representative bound states symmetrically located on opposite sides ofthe band (where ± (q + k) = ⌥ (q)) with emission energies ~ ± = (1 ± 3)~¯!
and weak coupling (g/ ± )2 ⌧ 1 (such that !B± ⇡ ± ). We prepare thestates by slowly ramping on the coupling g using a sinusoidal ramp. The ramp
duration of 2 ms is long with respect to the bound state frequencies !B± , and no
dynamics are observed for a variable hold time between 0 and 0.5 ms following
the ramp. The resulting quasimomentum distributions are observed in timeof-flight after a band-map of all optical potentials as before, c.f. fig. 5.11. The
observed distributions match qualitatively the predictions for | ± (q)|2 (with asmall higher-band contamination, see methods), with quantitative agreement
if we allow for a blurring of 0.1k due to finite size e↵ects (system size ⇠ 10µm)
and imaging resolution. We note that the predicted below-band bound state,cf. fig. 5.11(A), resembles the state found near a single edge [91] while the
sharp edges at ±k (not resolved in the data) lead to modulations in its spatial
shape with the lattice periodicity, a feature that was suggested earlier in [97].

74num. (arb.)
1.0

(A)

1.0

|+(q)|2 (arb.)|-(q)|2 (arb.)

0.0

0.5
0.0
-2

-1

0
q/k

1

1.0
0.5
0.0
-2

2

(B)

-1

0
q/k

1

2

Figure 5.11: Structure of the bound states at ~ + = 1.0(1)Er and ~=0.5(1)Er below (A) and above (B) the band edges. (A) Observed quasimomentum distribution of the bound state below the band at ~!B = 2~¯! ⇡h ⇥ 1.9(3) kHz, where al = /2 is the lattice spacing, from time-of-flight
following an adiabatic (2 ms long) ramp on of the coupling up to g/¯! = 0.43.
Each data-point is the average of more than 30 individual repetitions. The density plot shows the average time-of-flight picture. The gray curve is the quasimomentum distribution from (A) convolved with a Gaussian blur ( = 0.1k)
to accomodate finite size e↵ects and imaging resolution. (B)Observed distribution for the bound state above the band at ~!B+ = 4~¯! ⇡ h ⇥ 3.9(3) kHz,taken as in (A). The gray curve is blurred with the same Gaussian as in panel
(A).

75

Chapter 6
Theoretical considerations forbound-state induced many-body
physics in optical lattices
Given that our experimental system features many emitters in an array, we
might be tempted to ask whether or not these neighboring emitters play arole in the observable physics. The answer is a resounding yes! Because of
the presence of many emitters is intrinsic to our system, it behooves us toconsider the possibility of observing many-body e↵ects in our system, going
beyond the model of chapter 4. We begin with a brief overview of quantum
master equations as a means to describe the e↵ect of many particles and thenintroduce one of the most well-known many-body models in optical latticephysics, the Bose-Hubbard model. After this (textbook level) overview, we
will demonstrate how a Markovian master equation treatment of our system
is predicted to modify the superfluid to Mott-insulator transition in an arrayof quantum emitters coupled to a free-particle continuum. This calculationis the essential novelty of this chapter, and we round out the discussion withseveral proposals for experiments to investigate the many-body character ofthe quantum emitter array.
As a word of caution before we embark on this journey, we wish to highlight a shift in the usage of the word Markovian. Previously (chapter 4), we
invoked the Markovian approximation when studying emission of matter-waves
into free space in 1D, and found that the emitters should not decay for < 0at this level of approximation. In order for this approximation to be valid,
the typical coupling strengths must be small compared to any energetic divergences in the density of states (specifically, we described (⌦/ )2 ⌧ 1 as being
Markovian). In what follows, we revive the language of systems and baths,
and we take as our definition of Markovian to be that the dynamics of the76excited state in the system S are slow when compared to a typical timescaleof the reservoir. In this case, we will see that a Markovian description applies
even for the case of < 0. It is not strictly necessary to invoke this Markovian
approximation to compute a master equation for a quantum system [140], but
it will suffice for our purposes to work with the Markovian quantum masterequation. We note that some of these ideas were already considered by I. de
Vega, et al. in 2008 [112], and its extension [122]. Specifically, the original
matter-wave emitter proposal [112] did not stop with considering dynamics
of a single quantum emitter in 3D, but rather predicted, using a Markovian
master equation, several many-body e↵ects. In particular, for< 0, novel
Hubbard models with beyond nearest-neighbor coupling were demonstrated
to be realizable in an experiment. This chapter extends this approach for our
1D situation.

6.1

Master equations

In order to treat the physics of a system of many quantum emitters interacting
through a shared bath, we shall turn our attention to the approach based on
quantum master equations first mentioned in section 3.2 [92, 140]. In thissection, we shall introduce the master equation formalism and some of its
general properties. Then we specialize to the case of our free particle modelfrom 4.3 and compute the Markovian master equation for a system of quantum
emitters of matter waves. In the following section, we will use this Markovian
master equation to demonstrate a renormalization of the e↵ective Hubbard
model parameters and discuss the superfluid to Mott-insulator transition in
our emitter system.
Formally, for a time-dependent Hamiltonian, the unitary time-evolution
operator for a quantum state (in the Schrödinger picture) may be written
✓ Z t
◆
1
Û (t, 0) = T exp
Ĥ(⌧ )d⌧ .
(6.1)
~ 0
A quantum system need not start in a single, well-defined quantum state| a i, however. The density operator (equivalently: density matrix) ⇢ˆ candescribe such a mixed state. The density operator satisfies the well-known
von Neumann equation
i
dˆ
⇢
ih
=
Ĥ(t), ⇢ˆ(t)
(6.2)
dt
~
which resembles the Heisenberg picture equation of motion for the evolutionof a time dependent operator, with a di↵erence in sign on the right hand side.

77

Furthermore, the expectation value of any observable Ô may be computed if
we know the density matrix via the trace operation:
D E
h iÔ = Tr Ô⇢ˆ
(6.3)

It is at this point that we return our attention to system plus reservoir
models. We again assume that the total Hilbert space consists of a small setof states S of interest, called the system, and a (larger) set of states R, called
the reservoir or bath, with which the system can interact. The Hamiltonian
schematically is assumed to take the form Ĥ(t) = ĤS + ĤR + ĤI (t), withĤS (ĤR ) the Hamiltonian of the system (respectively: bath), and ĤI (t) theinteraction between them. Note that any time-dependence in the Hamiltonian
is contained in the interaction piece. We remember that the time evolution
operator in the full system plus reservoir Hilbert space is unitary, and thus inprinciple, there is no decoherence. However, if we focus our attention on only
the system S and “don’t keep track of” the reservoir R, then we may appear
to have decoherence and dissipation arising in the system S alone.
To be more precise, we assume that the full equations of motion for the system plus bath are too complicated to solve fully. We can, however, restrict ourattention to just the system by tracing over the reservoir degrees of freedom,and thus obtain the reduced density matrix⇢ˆS = TrR [ˆ
⇢]

(6.4)

where the trace in (6.4) is taken over the reservoir states only. The reduced
density operator describes how the system S evolves in the presence of a bath.
By exchanging quantum mechanical probability with the bath, the system may
appear to decay, and we arrive at a possibility of dissipative behavior. Notethat by taking the trace of the von Neumann equation (6.2), we find that the
reduced density operator evolves according to
⇣h
i⌘
dˆ
⇢S
1
= TrR Ĥ(t), ⇢ˆ(t)
dt
i~

(6.5)

Though this equation is in general quite complicated, making some approximations will render it into a well-studied form referred to as the Lindblad
master equation (or just master equation) [92].
The steps to get to the usual form of the quantum master equation from
(6.5) are a textbook exercise and we refer the reader to [92] for the full details.
Borrowing their notations temporarily (and following their convention ~ = 1),78

we find (c.f. [92] eq. 3.136) thatdˆ
⇢S X X
=
dt
! ↵,

↵

(!) A (!)ˆ⇢S A†↵ (!)

A†↵ (!)A (!)ˆ⇢S (t) + H.c.

(6.6)

with A↵ (!) a so-called Lindblad operator. The Lindblad operators are model
dependent, and we shall discuss them when we specialize to our atomic emittermodel. This equation can be simplified by breaking the coefficients ↵ into
its real and imaginary parts↵=

1↵ (!) + iS↵ (!)
2

(6.7)

i [HLS , ⇢ˆS ] + D (ˆ
⇢S )

(6.8)

which reduces (6.6) to the form
dˆ
⇢S
=
dt

We see immediately that (6.8) consists of two parts: a unitary evolution by
HLS and a dissipative piece. Let us discuss everything in turn.
First, we discuss how ↵ is defined. The reservoir is well described bycorrelation functions⌦↵G↵ (!) = B↵† (t)B (0)(6.9)
The complex master equation rates are then obtained from correlations of thereservoir by
Z
↵

1

=

ei!t G↵ (t)(6.10)

0

We have called the reservoir correlations G↵ (t) in analogy to our nomenclaturefrom chapter 4, as they are the many-site extension of the bath-correlation
functions introduced in chapter 4. Now, these complex rates ↵ consist of
two parts. The imaginary part enters (6.8) in the unitary evolution piece. We
define the Lamb shift Hamiltonian HLS byXX
HLS =S↵ (!)A†↵ (!)A (!)(6.11)!

↵,

This name comes from the fact that [HS , HLS ] = 0 and thus the Lamb-shift
Hamiltonian serves to renormalize the energy levels of the system. This contribution will become very important when we consider the modified superfluid
to Mott-insulator transition. The real parts of ↵ , namely, ↵ describe thedecay rates from various system states to the available bath states. They show79

up in (6.8) in the dissipator
✓
XX
D (ˆ
⇢S ) =⇢S A†↵ (!)↵ (!) A (!)ˆ!

↵,

1A†↵ (!)A (!), ⇢ˆS
2◆(6.12)The dissipator (6.12) describes jumps between system states mediated by thetraced-out reservoir with rates coming from the reservoir correlations. We note
finally that the usual Lindblad form of the master equation is obtained from
(6.8) by diagonalization of ↵ (!).

6.2

Modified superfluid to Mott-insulator transition

As an application of quantum master-equations, we return our attention to
the model of section 4.3, namely a single quantum emitter coupled to freely
propagating momentum modes. However, with a suitable generalization, we
will be able to consider an array of emitters, and the master equation treatment will provide us with a way to predict the phase diagram of the superfluid
to Mott-insulator transition in this model. We review very briefly the standard mean-field treatment of the SF-MI phase transition [70, 189, 190] before
discussing the modifications which arise due to the coupling to the freely propagating modes.

6.2.1

Standard superfluid Mott-insulator transition

This section summarizes standard textbook treatments of the superfluid toMott-insulator transition [191]. We consider the Bose-Hubbard model forbosons in a lattice with nearest neighbor tunneling [192, 193]. The Hamiltonian isX †X
UX † †
ĤBH = Jâi âj +(✏i µ)â†i âi +âi âi âi âi(6.13)2<i,j>iiwhere âi and â†i are bosonic creation and annihilation operators on a site
satisfying the usual commutation relations [âi , â†i ] = ij , J is the tunnelingmatrix element and U the on-site (contact) interaction parameter obtained ina tight-binding manner, ✏i is a site-dependent energy o↵set corresponding to,e.g. an external harmonic trap or on-site disorder, and µ is the usual chemical
potential. Theoretical treatments of the model using Mathieu functions orelse Gaussian approximations to the on-site Wannier functions [194] give the80following useful relationships for J and U as a function of s (the depth of the
optical lattice in recoils):
U
4kas3/4
= p
.
Er
2⇡

p
J
4
= p s3/4 e 2 s ;Er⇡(6.14)

which for U strictly speaking only applies in the case of uniform 3D latticeconfinement. (For completeness, k is the wavevector of the optical lattice
and a is the s-wave scattering length of the condensate atoms. For 87 Rb,a ⇡ 100aBohr .) Note that the notation < i, j > indicates a summation onnearest-neighbor sites only. Introducing the on-site number operator n̂i = â†i âi ,and considering the case of no external potentials or disorder, the Hamiltonianin (6.13) reduces to the form we will consider for computing the Bose-Hubbard
phase-diagram, namely
ĤBH =

J

X

â†i âj

<i,j>

µ

X
i

n̂i +

UX
n̂i (n̂i
2 i

1)(6.15)

as depicted pictorially in fig. 6.1 Experimentally, we may tune J, U , and µ via

JU
Figure 6.1: Schematic depiction of various terms in the Bose-Hubbard model.
Atoms (represented as red circles) in the ground band of a 1-dimensional optical lattice (gray lines) can tunnel between neighboring sites with rate J, andinteract on the same site with interaction energy U .
the lattice depth, atomic Feshbach resonances [37], and the total atom number
respectively.
In order to understand the phases of this model, we consider the groundstates in two limits. First, if U = 0, then it will be energetically favorable forthe atoms in the lattice to delocalize. Therefore
!
Nlatt
1
1 X
1
†
p
| U =0 i = pâj |0i = p â†q=0 |0i(6.16)
Nlatt j=1
NN
We have introduced the notation â†q which creates a coherent state having
quasimomentum q. This state is phase coherent across the whole lattice, and81as such, it represents a superfluid state or a BEC. The other limit we consider
is when J = 0. Because interactions are so strong, the system now favors
localization onto individual sites. Unlike before, however, the specific form of
the ground state depends on the number of atoms present. For the case thatN = Nlatt the ground state will just have one atom per site
| J=0,1 i =and for the case that N/Nlatt =and can be written

N
latt
Y
j=1

â†j |0i(6.17)

2 Z, the ground state has
N
latt
Y

(â†j )
p |0i
| J=0, i =!
j=1

atoms per site,(6.18)

These states, having a fixed number of particles per site, lack the phase coherence of the superfluid state (6.16). Therefore, they represent a di↵erent phase,
which is called the Mott-insulator phase. Furthermore, there is no continuous
way to deform the Mott-insulator ground state to the superfluid state, so weknow that as a function of U and J there must be a transition between thetwo phases. We note that there is no exact expression for the ground state ingeneral, but we can gain insight into the phase diagram properties by using amean-field treatment [189, 195, 196].
Let usthe expectation value of the creation and annihilation opD defineE†erators âj = hâj i = . This quantity, which may be chosen to be a real
number value, serves as an order parameter for the Bose-Hubbard model. We
now make the mean-field approximation. Specifically, we assume that we canreplace the quantum-varying population on each site with a mean-field, andwe will attempt to learn about the Bose-Hubbard model by assuming that theatoms can tunnel and interact with this mean field. The form of the mean-field
decoupling [189] is2â†i âj ⇡ âj + â†i
(6.19)and thus the tunneling term takes the form"
X †ĤJ ⇡ zJ(âj + âj )j2#(6.20)

where we have defined z, the coordination number, as the number of nearest
neighbors of site j. For example, in a simple cubic lattice in 1, 2, and 3
dimensions, z = 2, 4, and 6 respectively.

82

If we now define Ū = U/zJ and µ̄ = µ/zJ we can cast our mean-field
Hamiltonian as a sum of single site Hamiltonia in the following manner:
X
Ĥmf = zJ(h0j + Hj0 )(6.21)
J

with
Hj0 =
and

Ū
n̂j (n̂j 1) µ̄n̂j
2
⇣
⌘
h0j =
â†j + âj2+(6.22)(6.23)We have split (6.21) into two parts: an unperturbed Hamiltonian (6.22) which
is diagonalized in the basis of states |ni with a fixed number n of particles on
every site, and (6.23) which acts as a perturbation in this basis.
By applying standard perturbation theory one finds [189, 191](0µ<0
En(0) = Ū
(6.24)
n(n
1)
µ̄n
U
(n
1)
<
µ
<
U
n
2
(1)

for the unperturbed energies [189, 191], En = 0, andEn(2) =

X hn| (â† + â) |n0 i 2

n6=n0

(0)
En

(0)
E n0

2

=

Ū (n

n
1)

2

µ̄

+

(n + 1)µ̄ Ū n(6.25)The ground state energy to this order in perturbation theory is then (as afunction of the mean-field order parameter )
✓
◆
n
n+1
(0)
En ( ) = En + zJ+
+ 1 2 + ...
(6.26)Ū (n 1) µ̄ µ̄ Ū n
Now, in the ground state, the energy will be a minimum, so we minimize (6.26)
with respect to the order parameter . The first term is independent of , andtherefore the minimization only depends on the term proportional to 2 . The
term in parentheses determines the behavior of in the ground state. If theterm in parentheses is positive, (6.26) is minimized by taking = 0, which
we interpret to be the Mott-insulating phase since there will not be phasecoherence across the system in this state. If however the coefficient of 2 in
(6.26) is negative, then we can minimize the ground state energy by taking
2> 0, which is to say the mean-field order parameter becomes non-zero andthe system will develop phase coherence, so we may identify this phase with83

the superfluid state. The boundary between these two phases is found when
the coefficient of 2 is equal to zero, so we can solve for the phase boundary.
The result is [189, 191]
zJ
(µ/U
=
U

n) (µ/U n + 1)µ/U + 1

(6.27)

This classic result has informed much theoretical and experimental e↵ort on
the Bose-Hubbard model [41, 197]. The phase diagram, c.f. fig. 6.2(A), takes
the form of lobes of Mott-insulating phase having fixed number of particles
surrounded by lobes of superfluid for which n is not a good quantum number,
but which has a coherent phase.
The preceding description is modified in the presence of an external trapping potential. As in an experiment, consider a roughly harmonic trapping
potential V (x) / x2 . Then the term ✏i in (6.13) plays a role, and we replace µby µi = µ ✏i a site dependent chemical potential in (6.15). The result due toa spatially varying chemical potential is that the model now supports regionsof fixed n Mott-insulating phase surrounded by rings of superfluid phase in thecelebrated wedding cake structure [198, 199]. This structure was observed ina spatially resolved way starting in 2006 [199] in the group of Wolfgang Ketterle using spectroscopic methods, and later in 2009 [200]. The realization ofquantum gas microscopy [61, 63] provided an even more transparent method
for viewing the wedding cake structure experimentally. We note that even if
we do not have spatial resolution of the wedding cake structure, we can still
access the di↵erent Mott-insulating lobes by careful spectroscopic measurements following the technique of the Ketterle group [199], c.f. the thesis of L.
Krinner [172].

6.2.2

The SF-MI transition with long-range tunneling

Here, we follow the same steps [112, 122] to derive a Markovian quantum master equation and e↵ective tunneling model at negative detunings for our 1D
emitter system. Then as an interesting extension, we show how the previously
considered superfluid to Mott-insulator phase transition is modified in thise↵ective model. We will find that the previously discussed mean-field treatment applies with one key modification: the e↵ective coordination z ⇤ will be
modified by the coupling to the bath of propagating states.
Consider the model of section 4.3, in which a single red emitter in the
excited state is initially coupled to a contiuum of blue freely-propagating momentum modes. If we now consider multiple red emitters located on the sites

84

(A)
SF

zJ/U

n=1
n=2

MI
0

n=3

1

2

/U

n=4
3

4

Atoms per site

(B) 3
MI
2
MI

MI

1

SF
MI

MI

0
-1

0

r/Rtf

1

Figure 6.2: Various features of the Superfluid to Mott-Insulator phase transition. (A) Phase diagram in the mean field treatment. The shaded aread represents parameter combinations that yield a superfluid (coherent) phase acrossthe condensate in an optical lattice, while the white lobes represent Mottinsulating phases of fixed atom number per site n, with the first 4 Mott-lobes
fully shown. The phase boundary is determined by (6.27). (B) Illustrationof the wedding-cake structure of a harmonically trapped condensate having
N = 150, 000 atoms in a Harmonic trap of trap frequency ! = 2⇡ ⇥ 60 Hz.
The calculation assumes a 3D square lattice (z = 6) at = 1064 nm and a
depth of 25 Er . The chemical potential is computed as a function of distancefrom the trap center in the Thomas-Fermi approximation, and it corresponds
to the red dashed line in (A).
j of an optical lattice, we must modify the Hamiltonian (4.9) as follows:ĤI =X X ~⌦
j

k

2

ke

i

kt

85

ikrj †b̂k |0j i h1j | + H.c.

(6.28)

which now contains a sum over red sites j and where rj = d0 ⇥ j is the positionof red emitter j in terms of the lattice spacing d0 .
The quantities of interest in a master-equation treatment are the complex
Markovian rates jl , defined by (6.10). For our system, we may compute thesequantities as follows:
D
E
1 X 2
†
Gjl (⌧ ) = 2
g exp ( i k ⌧ ikrj l ) b̂k b̂k
~ k k
1 X 2
⇡ lim 2
gk exp ( (" + i k )⌧ ikrj l )(6.29)"!0 ~k
DE
where we have approximated b̂†k b̂k ⇡ 1, valid only in the Born-Markov
regime, and we have introduced an epsilon-regulator to ensure the convergence of the integral over ⌧ which we must compute in order to determine jl .
SpecificallyZ 1
X g 2 eikrj lkGjl (⌧ )d⌧ = lim+
(6.30)jl ="!0~2 " + i k
0k

which is very reminiscent of our Markovian limit treatment of this emitter
system in section 4.3.3. Taking a continuum limit for the k-summation in(6.30), and utilizing the definitions of gk = ~⌦ k /2 and k = ~k 2 /2m
, we
find
Z
2 2
(⌦/2)2 aho 1 e k aho e ikrj lpdk(6.31)
jl = lim
"!0+
"+i k
⇡
1
or, upon defining a change of variables  = kaho(⌦/2)2
jl = lim p
"!0
⇡

Z 1

2

e  e id0 (j l)/ahod
2i
1 " + i!0  /2

(6.32)

This integral is intractable in general, but we note that for the limit of strong
confinement, !0⌦, and any other energy scale, we may approximate thegaussian term in the numerator of (6.32) as 1, exp( 2 ) ⇡ 1, whereupon(⌦/2)2jl ⇡ lim p
"!0
⇡

Z 1

e id0 (j l)/ahod
2i
1 " + i!0  /2

(6.33)

which may be integrated to yield
jl =

i

L

2

r

⇡!0
exp
2

86

d0
aho

r

2
|j
!0

l|

!

(6.34)

By inspection, this result depends upon the sign of the detuning,define the following auxiliary functions [112, 122]:
(
1
<0
⌫=
i
>0
which keeps track of the complex analytic character of the rates
tion of the sign of the detuning,r⇡!0L0( ) =
2 2| |, so we(6.35)

jl as a func-(6.36)

which determines the strength of the couplings ( L = ⌦2 /!0 as before), andr!0⇠( ) = aho(6.37)
2| |which determines the e↵ective range (in lattice sites) of the e↵ective tunnelingterm. With these definitions, the Markov master equation rates become
✓
◆
i
d0
⌫
|j l|
(6.38)
jl ( ) =0 ( ) exp⌫⇠( )We note that importantly, for< 0, (6.38) is a purely imaginary quantity.
This means that for negative detunings, the master equation for the reduced
density matrix of red emitters consists only of the coherent, Lamb-shift termwith dissipator identically equal to zero! (Recall equations (6.7), (6.8) and(6.11).)We are thus led to the following model: we are considering a lattice of
red emitters in deeply confining wells of an optical lattice (harmonic oscillator
frequency !0 much greater than any other energy scale) in one dimension. Dueto this strong confinement, the atoms will not tunnel directly, so e↵ectively,
their Hamiltonian consists only of the on-site and chemical potential terms in
the Bose-Hubbard model. After we turn on the coupling, the atoms acquire,through the traced out blue momentum states, a tunneling term of the form
X
Ĥtunnel =Jjl â†j âl(6.39)
j,lwith
Jjl = ~ 0 exp

✓

87

d0
|j
⇠

l|

◆

(6.40)

Therefore, the new e↵ective Hamiltonian for this model becomes
ĤLRT =~ 0X

exp d0 |i j|/⇠ â†i âj

µ

i,j

X

n̂i +

i

UX
n̂i (n̂i
2 i

1)(6.41)

We now proceed as before and make a mean-field decoupling of the tunneling term, i.e. we replace hâj i = hâ†j i = . For a fixed site index j, we consider
the following change of notation: let r(j) = r = j l, andJr = ~ 0 exp( d0 |r|/⇠)(6.42)The mean-field decoupling of the tunneling term now takes the form for a fixedsite iX
Ĥtunnel ⇡ ~ 0
exp( d0 |r|/⇠)(âj + â†j )(6.43)
rThus the mean field Hamiltonian of the model again
decouples as a sum upon
P M
MFsingle site Hamiltonians of the form ĤLRT = j Ĥi M F (here MMF stands
for Modified Mean Field) withĤjM M F =µn̂j +

Un̂j (n̂j
2

1)

~ 0

X

exp d0 |r|/⇠ (âj + â†j )(6.44)r

We may simply carry out the r summation in (6.44), and if we identify
Jef f = ~ 0 , then we may identify the r dependent parts above as an e↵ective coordination number z ⇤ . What we find is that
✓
◆
✓ ◆
X
d0 |r|d0⇤z =exp= coth
(6.45)⇠
2⇠r
such that the mean field Hamiltonian reduces to the same form as the mean
field Hubbard model, provided we replace zJ by z ⇤ Jef f . Then all of the preceding computations from section 6.2 carry over unchanged, so we obtain thesame phase diagram (when plotted as a function of µ/U vs. z ⇤ Jef f /U ). We
note, however, that because in 1D, z = 2, if ⇠1, z ⇤
2.16, so the real
physical values at which the lobes occur do shift with respect to the nearest
neighbor model. This can potentially have observable experimental consequences, c.f. fig. 6.3. We note also that the e↵ective coordination number
can be less than 2, i.e. when ⇠ ⌧ d0 , the excitations in the quantum emitters
cannot hop to their neighbors, and the model is e↵ectively less able to tunnel
than a standard 1D lattice.
A simple experiment to check for this shift in the Mott-insulator transition

88

(A)
SF

z*J/U

n=1
n=2

MI
0

1

(B)

n=3
2

n=4
3

/U

4

(C) 1.0

(D)10
z*

(ii)

Je/0

(i)

0.5

0.0

0

2

4

6

8 10

|i-j|/d0

5

0

0

2

4

/d0

Figure 6.3: Modifications to the superfluid to Mott-insulator transition in thee↵ective Hamiltonian model of (6.41). (A) Phase diagram in the mean field
treatment. The only di↵erent from fig. 6.2(A) is that the vertical axis isscaled by z ⇤ instead of z. (B) Illustrations of cases where (i) z ⇤ = 2 for atypical optical lattice experiment and (ii) z ⇤ = 4.08 for the e↵ective model
having ⇠ = 2d0 . (C) E↵ective tunneling rate for sites i and j separated byinteger multiples of the lattice spacing d0 , assuming ⇠ = 2d0 . (D) E↵ective
coordination number z ⇤ as a function of ⇠/d0 in the 1D system. The tunabilityof the system allows for a 1D system having z ⇤1, i.e. we can tune thee↵ective dimensionality of the system while remaining fully 1D.
would start with a deep n = 1 Mott insulating state in a state selective opticalpotential. (For concreteness, we consider as the trapped state 87 Rb’s F =1 hyperfine ground state |ri = |1, 1i in an sErec lattice along z and theuntrapped state |bi = |2, 0i as in chapter 5, both subject to 40Erec tubes in89

x and y.) Applying a coupling microwave field at negative detuning createsbound states which leads to the e↵ective tunneling Hamiltonian considered
here, and whose decay length ⇠ is given as a function of . Applying thisradiation as a function of s should show the transition from superfluid to a
Mott insulator at a di↵erent value than for the uncoupled case. Using a peakwidth measurement or contrast measurement, one would look for the point
at which the transition occurs as a function of s for several values of ⇠ andthen compare to the predictions of (6.45) (see [201] for a treatment with truly
long-range tunneling.)
A further experiment of interest might be described as “purification of a
Mott-Insulator via bound state formation”. The set-up we consider is now
a Mott-Insulating phase in a harmonic trap having distinct lobes of di↵ering
atom number. The goal is to create a uniform n = 1 Mott-Insulator across thewhole sample. Because the interaction shift on each site is number-dependentas E(n) = U n(n 1)/2, the resonance condition will be shifted for sites ofdi↵erent atom number. If we denote the splitting between the n = 1 redemitter and the zero energy blue propagating state by= 1 , then fora fixed microwave detuning of 1 , the other sites will have detuning n =1 + E(n) >1 , c.f. fig. 6.4. We now propose the following method to bringthe Mott-Insulator to a uniform filling of n = 1.
(A)

(B)

Figure 6.4: Purification scheme for a Mott-insulator having shell structure.
(A) Purification scheme based on on-site energy shifts for sites having morethan one atom. For a properly chosen detuning ramp, highly occupied sites
can be sequentially emptied until they reach unity occupation, (B) resulting
in a Mott-insulator with hni i  1. See text for details.
First, we will create a Mott-Insulator with a large number of atoms, and90

thus, a large maximal site occupancy of, e.g. nmax = 7. Two-photon spectroscopy of this Mott-Insulator will reveal the relative populations at eachfilling fraction [199]. Then, we will choose a detuning such that the maximal
site-occupation is nearly resonant with the propagating states, i.e. initial ⇡nmax = 0. The sites with n = nmax will lose one atom into a propagating state
which can escape the system. Next, the microwave detuning will be decreasedso that approximately nmax 1 = 0, removing another atom from all sites having n = nmax 1. This process, which is akin to evaporative cooling, is thenrepeated until we are left with n = 1 everywhere. Specifically, for all detunings
chosen to remove atoms from highly occuppied states, 1 < 0, and thus the
singly occuppied sites form bound states only. These atoms will not escape thesystem, and by an adiabatic turn o↵ of the coupling pulse, will be returned totheir initial emitter states. A 2-photon spectroscopic measurement after the
detuning-ramp procedure should reveal a lower average occupation number nin the lattice, which serves to cool or remove entropy from the system.
We note two practical experimental concerns with regard to this proposed
experiment. First: the emitted blue atoms need to be able to escape thesystem. The current experimental set-up utilizing vertical lattice tubes withharmonic confinement along the tubes does not truly allow the blue atoms
to escape. However, by utilizing horizontal tubes formed from blue-detuned
light at, e.g. 532 nm, we could create “anti-confining” tubes in the sense thatthe trapped atoms would tend to seek low intensity regions, and thus, couldroll out of the tubes and be lost from the system. We do not yet have thecapacity to produce such horizontal tubes, though we note experimental e↵orts
are underway to construct such a lattice using an accordion lattice geometry[202, 203]. Second: the state-selective light scatters many photons from thetrapped atoms, with rate ⇡ 4 s 1 (for |ri = |1, 1i atoms at = 790.0nm), which leads to heating and an overall loss of signal. For our previously
discussed experiments, this is not a problem, as the timescales involved are all
shorter than the scattering time from the state selective light. However, theproposed 2-photon spectroscopy is slow, requiring times on the order of 100
ms for 1 Hz resolution, which now runs the risk of being on a similar time scale
to 1/ SDOL . This problem in principle is harder to solve, though we note thatin-situ microscopy of the trapped cloud could lead to direct imaging of the
Mott-Insulator shells (up to parity) [61, 63]. Overcoming these challenges is
an area of active e↵ort in our lab, and the rich physics of our emitter-reservoir
systems in a many-body context remains an exciting area of study.

91

Chapter 7
Conclusion
Having reached the nominal end to this work, we hope to have convinced thereader of both the theoretical richness in and the experimental excitement enabled by the emission of matter waves from quantum emitters into engineered
reservoirs. First, we showed how to understand the spontaneous emission ofphotons from an excited atom in free space as a toy open quantum system, andthen we considered extensions of the model in two ways: replacing photons
with matter waves and changing the properties of the emitted matter-wave
states. A full theoretical characterization of the emission of freely propagating
matter waves in one-dimension showed a strong Markovian to non-Markovian
evolution of the emitter population, and we also characterized the momentum space profiles of the emitted radiation, with an eye towards an ultracold
atomic realization in the laboratory. Excitingly, the system was also shown to
exhibit the long predicted “atom-photon bound state”, whose spatial profileand decay length we compute. Then we considered the case of quantum emitters coupled to a band structure, in which the modifications to the dynamics
are predicted to be stronger than for a single energetic edge. The dynamics
were shown to be tunable between exponential decay and Rabi-oscillations,allowing us to study the novel intermediate regime. A second bound state waspredicted, living above the band, whose spatial shape is strongly modified by
the lattice.
We then described in detail how to implement a system of matter-wave
emitters in the lab using state-dependent optical potentials. We reported on
experiments on emission into 1D free-space, obtaining good agreement withthe theoretical models previously considered, and accounted for discrepancies
between the observed data and the models by appealing to a breakdown of theisolated emitter assumption. Furthermore, we observed directly the shape ofthe emitter and matter-wave bound state for the first time. We then modified
the emission platform to add a lattice for the emitted radiation and demon92

strated how the relative strength of coupling to the bath of modes allowed
us to tune between fully Markovian (open) decays and fully coherent (closed)
Rabi oscillations, all while also being able to access the intermediate regime.
We furthermore demonstrated the existence of two bound states and showedthat their momentum composition is strongly dependent upon which bandedge they couple to. Furthermore, the observed oscillations are shown to beconsistent with a beating between the bound states in the system!
Going forward, we have shown how our emitter platform may be used to
study many-body e↵ects. In particular, via the machinery of quantum masterequations [112], we showed how to derive an e↵ective Hamiltonian with beyond
nearest neighbor tunneling, and we showed how this can modify, for example,
the superfluid to Mott-insulator transition in our system. We then gave an
example of how this novel system might allow for experiments on cooling ofa Mott-insulating phase. We also can use the quantum emitter system tostudy the formation dynamics of bound states, as well as attempt to study
their transport properties through tilted optical lattices. Though we did notconsider it in this thesis, the emitter platform is also predicted to exhibit
superradiance [112, 204], which we seek to study in future work. Finally, we
can study higher dimensional analogues of our platform [115–118], in which
exotic behaviors beyond the scope of our current work are predicted.
Given the exciting directions in quantum simulation that the Schneble labcan explore, I am excited to see the future directions of this work.

93

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