Dynamics of Atomic Matter Waves
in Optical Lattices

A Dissertation PresentedbyJeremy Brian Reeves
to
The Graduate Schoolin Partial Fulfillment of the Requirements
for the Degree ofDoctor of Philosophy
inPhysics

Stony Brook University
July 2015

Stony Brook University
The Graduate School

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

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

Thomas Bergeman – Chairperson of Defense
Adjunct Professor, Department of Physics and Astronomy

Mengkun Liu
Assistant Professor, Department of Physics and Astronomy

Robert Konik
Physicist
Brookhaven National Laboratory

This dissertation is accepted by the Graduate School.

Charles Taber
Dean of the Graduate Schoolii

Abstract of the Dissertation

Dynamics of Atomic Matter Waves
in Optical LatticesbyJeremy Brian Reeves
Doctor of Philosophy
inPhysics
Stony Brook University
2015

Quantum gases in optical lattices allow for fundamental studies
in atomic and condensed-matter physics and the exploration ofnovel effects. After a brief introduction to the fundamentals ofquantum gas experiments in optical lattices, we discuss two recent
experiments focusing on driven matter waves in a one-dimensional
optical lattice.
The first experiment uses a tilted bichromatic optical lattice toinvestigate the interplay of disorder and collisional interactions in
the accelerated transport of a Bose-Einstein condensate. Here,a screening effect is observed, in which the interactions effectively
cancel the damping of Bloch oscillations induced by a (quasi-)disordered potential. This effect can be understood through a modification of the underlying band structure by the interactions.
The second experiment studies the dynamics of a weakly trapped
condensate resonantly coupled to the orbitals of a strongly confining state-selective lattice. We observe momentum distributionsiii

that correspond to matter wave diffraction from a periodic structure; however, the diffractive dynamics remain strongly linked tothe internal-state Rabi oscillations. In the regime investigated,which we call the nonadiabatic regime, no diffracting potential canbe defined. We show how only for much stronger coupling, theinternal and external dynamics decouple, transitioning from nonadiabatic diffraction to the well-studied Kapitza-Dirac diffraction.
We further investigate prospects for realizing dissipative spin models in our ultracold atomic gas experiment. To this end, we develop
and test in the laboratory several possible implementations of effective spins with differential coupling to a superfluid background andexamine their viability for a realization of the spin-boson model.

iv

Contents
List of Figuresvii

Acknowledgements

viii

1 Introduction

1

2 Theoretical Background and Experimental Techniques
2.1 Bose-Einstein condensates . . . . . . . . . . . . . . . . . . . .
2.1.1 Mean field description . . . . . . . . . . . . . . . . . .
2.1.2 Experimental apparatus . . . . . . . . . . . . . . . . .
2.2 Atom-light interaction . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Interaction Hamiltonian and Rabi oscillations . . . . .
2.2.2 Dressed states and dipole potentials . . . . . . . . . . .
2.2.3 State-dependent optical potentials . . . . . . . . . . . .
2.2.4 Two-level dynamics in the 87 Rb ground state . . . . . .
2.3 Optical lattices and band structure . . . . . . . . . . . . . . .
2.3.1 Single particle dynamics . . . . . . . . . . . . . . . . .
2.3.2 Effects of weak interactions . . . . . . . . . . . . . . .
2.3.3 Tight-binding and strong interactions . . . . . . . . . .
2.3.4 Lattice loading and characterization . . . . . . . . . . .

5
5
5
7
9
9
10
12
15
17
19
22
22
23

3 Superfluid Bloch Dynamics in a Tilted Incommensurate Lattice26
3.1 Interacting disordered systems . . . . . . . . . . . . . . . . . .
26
3.2 Bloch oscillation basics . . . . . . . . . . . . . . . . . . . . . .
27
3.3 Quasiperiodic potential . . . . . . . . . . . . . . . . . . . . . .
28
3.4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.5 Microscopic model of damping . . . . . . . . . . . . . . . . . .
33
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38v

4 Nonadiabatic Diffraction of Matter Waves
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Kapitza-Dirac diffraction . . . . . . . . . . . . . . . . . . . . .
4.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Dressed states and adiabatic approximation . . . . . . . . . .
4.5 Satisfying the adiabatic condition . . . . . . . . . . . . . . . .
4.6 Application to interferometry . . . . . . . . . . . . . . . . . .
4.7 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . .

39
39
40
42
44
48
49
50

5 Towards the Realization of a Dissipative Quantum Model
5.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Options for implementation . . . . . . . . . . . . . . . . . . .
5.2.1 Occupational spin . . . . . . . . . . . . . . . . . . . . .
5.2.2 Internal states . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Vibrational levels . . . . . . . . . . . . . . . . . . . . .
5.3 Implementation and evaluation . . . . . . . . . . . . . . . . .
5.3.1 Coherence times . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Differential interactions . . . . . . . . . . . . . . . . . .
5.4 Summary and outlook . . . . . . . . . . . . . . . . . . . . . .

52
52
56
56
56
57
59
60
64
65

6 Conclusion

66

Bibliography

68

vi

List of Figures
2.1 The experimental BEC apparatus. . . . . . . . . . . . . . . . .
2.2 Schematic experimental imaging system. . . . . . . . . . . . .
2.3 Rabi oscillations. . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Electronic level structure of rubidium-87. . . . . . . . . . . . .
2.5 State-dependent optical potentials and scattering rates. . . . .
2.6 Hyperfine-ground state control with applied microwave fields. .
2.7 Energy bands of the sinusoidal lattice at varying depth. . . . .
2.8 Bloch wave and Wannier functions of a sinusoidal potential. .
2.9 Band group velocity and Landau-Zener process in a lattice. . .
2.10 Lattice calibration via Kapitza-Dirac. . . . . . . . . . . . . . .

7
8
11
13
15
16
18
20
21
25

3.1
3.2
3.3
3.4
3.5
3.6

Bloch oscillations of an optically trapped atomic cloud. . . . .
28
Band structure of the quasiperiodic lattice. . . . . . . . . . . .
29
Bloch oscillations of weakly interacting atoms in an optical lattice 32
Damping due to the band structure of the bichromatic lattice.
34
Effect of nonlinearities on the structure of the minigaps. . . .
36
BO Damping rates as a function of chemical potential. . . . .
37

4.1
4.2
4.3
4.4
4.5

Experimental scheme for microwave induced diffraction. . . . .
Resonant orbital transfer and diffraction onset. . . . . . . . . .
Calculated dynamics from the nonadiabatic to the adiabatic
regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transition to adiabatic dynamics in large detuning limit. . . .
In-situ probing of matter wave interference. . . . . . . . . . . .

47
49
50

5.1
5.2
5.3
5.4
5.5
5.6
5.7

Atomic occupational spins in a bath. . . . . . . . . . . . . . .
The internal state spin. . . . . . . . . . . . . . . . . . . . . . .
An orbital level spin. . . . . . . . . . . . . . . . . . . . . . . .
Rabi frequencies for shaken lattice. . . . . . . . . . . . . . . .
Properties of the internal state spin. . . . . . . . . . . . . . . .
Coherent behavior in a phase modulated lattice. . . . . . . . .
Coherence decay and the driven steady state. . . . . . . . . .

54
57
58
59
61
62
63

vii

43
45

Acknowledgements
Anyone who has known me these past six years knows I will do my best to keep
this as brief as possible. However, it is impossible to overstate the support I
have had from so many people during the process of my Ph.D. and I hope that
I sufficiently recognize the impact you all have had on me, both personallyand professionally.
I want to first thank my adviser Dominik Schneble, whose guidance andenthusiasm has made this long road as enjoyable and rewarding as it has
been. Dominik has been an outstanding mentor and teacher from my first
introductions to atomic physics and ultracold atoms in my first year as a
graduate student to more recent discussions of the physical motivations andtechnical details of our experiments. In the lab, we can always count on
Dominik for advice and motivation. His motivational powers were probably
enhanced by his near constant presence in the laboratory. Lastly, he should
be recognized as the driving force behind the fact that I began to drink any
coffee at all.
I have had the pleasure of working with an outstanding group of physicists
in the laboratory. Daniel Pertot and especially Bryce Gadway deserve manythanks for initiating me into the world of ultracold atoms and table-top experimental physics. Both of of my senior colleagues set a shining example tostrive toward in their dedication and quality of lab work, that has certainly
influenced my own work and will have a lasting impact on the culture of thelab. The younger generation of physicists in the lab have also been integral
to the success of the lab and my own project. Ludwig Krinner is a constantsource of solid physical intuition, Mike Stewart could always be counted on toarrive early and be ready to get the next set of experiments rolling and ArturoPazmiño has been invaluable with technical support in his early days in thelab. It has been a pleasure to work with three so dedicated graduate students
I want to thank each of them for useful discussions in the lab and at Fridaysocials, as well as time spent on the soccer field and on road trips. I eagerly
look forward to hearing of your successes in the lab in the coming years.
My thesis committee has made the final steps of my Ph.D. official and I

want to thank each of Robert Konik, Mengkun Lui and Thomas Bergeman
for their time in reading my dissertation and attending my defense. Tom
Bergeman has been a constant resource for discussion and ideas across all
aspects of my work the past several years. From working closely with us tomodel our experiments to pushing us with new ideas, Tom has been importantin helping us refine our work in the lab. Martin Cohen has been a valuable
help for the lab and myself, especially concerning optics in the lab, and his
attentive reading of an early draft of my dissertation definitely help to ironout any kinks and inconsistencies.
The broader atomic physics community at Stony Brook has truly provided
a great network for discussion both academic and otherwise. John, Brian,Vlad, Peter, Spencer, Arthur, Oumarou, Bertus, Mehdi, Carl and Vince are agreat group of people and have made my time at Stony Brook a pleasure. I
wish them all the best.
Beyond the atomic physicists, I would be remiss to leave out Rahul, Omer,
Humed, and Josh; my roommates and friends since the first weeks of graduate
school. These guys have made the graduate experience so much fun and it
is been fantastic to see each of our lives and careers develop to where we arenow. Moreover, I would like to thank anyone who has taken to a soccer fieldwith me on Long Island. You all have provided a great outlet, without which
I surely would have gone crazy.
My family has always been there to back me throughout my time at
Stony Brook. My parents, Bob and Karen, have helped guide me throughthe challenges and been there to share in my excitement of successes in thelab. Jonathan, Mark and Ben have always been there for me and each has
been an inspiration in their own way.
Last and far from least, a heartfelt thank you to Megan for being by myside throughout the entire Ph.D. and patiently sharing in my excitement andfrustrations. Your support means all the more as you suffered long commutesand late nights to help make our lives together possible. And as we alreadybegin our lives away from Long Island, I am excited to see what the futureholds for us.

We thank IOP Publishing and the American Physical Society for the right toreproduce, in part, several works that are discussed in this thesis.

We are also thankful for the generous support of the Research Foundation of
the State of New York, the Department of Education and the GAANN
Program, as well as the National Science Foundation.

Chapter 1
Introduction
The 1995 achievement of Bose-Einstein condensation (BEC) in dilute atomic
gases [1–3] provided the physics community with a nearly ideal, laser-like coherent matter wave having very narrow momentum spread. The realizationof degenerate dilute gases was soon followed by the first experiments with
BECs in optical lattice potentials. The early optical lattice experiments focused mostly on the dynamics of matter waves in the lattice. Interference of a
BEC released from an adiabatically loaded optical lattice or after interaction
with a pulsed potential takes advantage of its spectral narrowness to further
refine atom optical techniques [4–8]. BECs in optical lattices readily allow for
the study of single particle dynamics, as well as the effects of weak nonlinear
interactions on a many body lattice system [9, 10]. Optical lattices further
allowed for the study of interacting matter waves in reduced dimensions [11]and of strongly correlated systems [12–14]. These foundational experiments
have kicked off the growing field of quantum simulation using ultracold atoms
in optical lattices [15, 16].
In recent years, experiments in optical lattices have grown from the initial cubic lattices to include more complex lattice geometries, allowing forthe simulation of increasingly complex quantum Hamiltonians [17, 18]. Experiments have also extended to include fermions, allowing for the study ofultracold atomic lattice systems having a more direct comparison with electrons in solid-state systems [19, 20]. Developments in optical lattice quantum
simulation techniques have allowed for the study of orbital physics in superlattices [21, 22], and the study of quantum and classical magnetism [23–25],
while graphene-like systems have allowed for the study of topological physics[26–28], and spin-orbit coupling from synthetic gauge-fields mimics the effectsof large magnetic fields in electronic materials [29–33]. These experiments
largely focus on single-band physics in the lattice, while recent efforts havebegun to take advantage of the simultaneous use of multiple orbital degrees
1

of freedom to engineer the excitation spectra of interacting systems [34–36].
Building on experiments in the early 2000s, the most recent experimental efforts today take advantage of the many accessible degrees of freedom inherent
to optical lattice systems including; internal states, orbital levels, and tunableinteractions. The increased control over the atomic system is showcased by experiments in quantum-gas microscopes with single lattice site control [37–42]
and the extremely precise optical lattice clocks [43–46].
Many of the above experiments use the precise control offered by ultracold atomic experiments to realize ideal versions of model Hamiltonians. Real
condensed matter materials or systems do not always follow the ideal and theeffects of imperfections or disorder can readily be investigated as well. Possible implementations of disorder in ultracold atom experiments include optical
speckle [47–49], quasi-periodic lattices [50–52], and atomic disorder from localized impurities [53, 54]. Ultracold atoms provide an ideal system in whichto study the properties and dynamics of systems beyond single particle disorder physics. The general behavior of disordered, interacting systems can be
broken into two distinct regimes. For strong interactions, in the absence ofdisorder, it is possible to drive a transition to a localized state known as theMott insulator [14]. In the presence of disorder, a gap-less insulating state
known as the Bose glass replaces the gapped Mott insulator [54–56]. The
Bose glass is distinguished by the cooperation of interactions and disorder tospatially localize particles. For weakly or noninteracting particles, disordercan independently drive an insulating transition via Anderson localization, resulting from the destructive interference of quantum mechanical trajectories[47, 51, 57–59]. In this limit, the introduction of weak interactions serves todestroy the Anderson localization, having a delocalizing effect [60, 61]. Recent
work investigating this interplay has focused on the equilibrium properties of
the interacting disordered system [62].
While this sampling of the applications of ultracold atoms optical lattice
is far from comprehensive, it touches on some of the cutting edge of a broad,evolving field. In this thesis, we will focus on applications of weakly interacting Bose gases (matter waves) to the study of disordered systems and a novel
regime of atom optical diffraction. We also touch on applications of these systems to designing open quantum systems to study dissipation and decoherence[63–65].
Weakly interacting matter waves in a disordered potential
Following from the study of interactions in strongly correlated lattices [54],
we investigate the dynamics of a weakly interacting matter wave in a (quasi-)disordered potential. We study the delocalizing effect of interactions in the2dynamics of Bloch oscillations in a disordered potential. It is well known thatdisorder [66] and interactions [67] independently dephase Bloch oscillations,leading to damping of the collective motion. In our experiment, we observedthat the presence of both disorder and weak interactions lead to competition
between the two effects and an interaction induced reduction of the Blochoscillation damping due to disorder. This effect saturates once the interactions
are comparable to the disorder energy scale. The effects studied here can be
related to the physics of granular superconductors [68] and other disorderedmaterials such as superfluid helium in porous media [69].
Matter wave diffraction beyond the Born-Oppenheimer regime
Diffraction techniques have become invaluable to experiments using ultracold
atoms for applications such as lattice characterization using Kapitza-Diracdiffraction [5], Bragg diffraction for spectroscopy [7], and for atom interferometry [70]. These applications and all previous work on atomic matter wave
diffraction [71–73] have relied on the action of a potential. For atoms withinternal structure interacting with optical fields, this potential corresponds
to the optical potential, deriving from the dressed states of the light-atomsystem. Here, we present diffractive dynamics in a little explored regime inwhich the adiabatic potentials are insufficient to describe the time evolutionof the system. The diffraction of a superfluid coupled via microwaves to thedeep wells of a state-dependent optical lattice is studied. We realize strong
coupling between the internal and external state dynamics, in contrast to thefamiliar regimes of Kapitza-Dirac and Bragg diffraction.
Engineering dissipative spin models
Isolating a quantum system from the environment is imperative for the study ofmany sensitive quantum effects. An understanding of the system-environment
coupling, which typically leads to decoherence and dissipation, is critical for
applications in quantum information science. Furthermore, these effects describe the emergence of classical physics in a world governed by quantum
mechanics, and ultracold gases are uniquely poised to study such questions[63, 74]. In this context, it is important to identify a quantum system (in thiscase a spin-1/2) that couples controllably to some environment (in this casea superfluid bath) [75]. We will explore several possibilities for the formationof a pseudospin-1/2 with (potentially) differential spin-bath coupling in ourapparatus and present some first studies on their experimental viability.

3Outline of thesis
Chapter 2 will briefly outline the theory of BEC in weakly interacting dilute atomic gases and introduce relevant concepts from atomic physics used
to trap and manipulate a BEC. With the necessary background established,
Chapter 3 will describe experiments on Bloch oscillations of a weakly interacting BEC in bichromatic optical lattices. Here, we review the problem ofdisordered systems in the strong and weak interaction limit and go on to describe the properties of the bichromatic potential and its use as disorder in our
experiments. The experiment and results are discussed and we present a microscopic model to capture the essential physics. Chapter 4 details experiments
on matter wave diffraction in a dressed state-dependent optical lattice. A review of conventional matter wave diffraction from optical potentials is given.
We then describe our experiments on Rabi oscillations in the presence of astate-dependent potential and highlight qualitative differences in the dynamics before giving an interpretation in terms of nonadiabatic dressed states. We
describe, briefly, the possibility of implementing the diffraction scheme here
to study dissipative systems and superfluid excitations. Building on this in
Chapter 5, and following a brief introduction to the spin-boson model, we examine several mechanisms for realizing effective spins coupled to a superfluid
bath and discuss the prospects for practical implementation of each. Finally,
we make a few concluding remarks in Chapter 6.
Publications of this PhD work (described either in part or in full):• Chapter 3
Superfluid Bloch dynamics in an incommensurate optical lattice
J. Reeves, B. Gadway, T. Bergeman, I. Danshita, and D. Schneble
New Journal of Physics 16, 065011 (2014)• Chapter 4
Nonadiabatic diffraction of matter waves
J. Reeves, L. Krinner, M. Stewart, A. Pazmiño, and D. Schneble
arXiv 1505.06085 Accepted for publication in Physical Review A (2015)

4

Chapter 2Theoretical Background andExperimental Techniques
In this chapter, some of the basic physics behind the creation and manipulationof ultracold atoms will be introduced. There are many excellent reviews on
the physics and creation of Bose-Einstein condensates and of ultracold atoms
in optical lattices [10, 15, 76–78] and several textbooks have comprehensively
covered the subject [16, 79, 80]. This chapter will give a brief review of theory of Bose-Einstein condensates and of important techniques relevant to theexperiments described in this thesis.

2.1

Bose-Einstein condensates

From the conception of condensation of photons in 1924 by Bose and its subsequent generalization to particles by Einstein, to its realization in dilute atomic
gases in 1995, the subject of Bose-Einstein condensation (BEC) has been of
great interest in physics. The concept has found applications in the discussionof superfluid helium [81, 82] and the physics of superconductors, where a condensate of bosonic Cooper pairs occurs [83, 84]. BEC can be extended to thecondensation of quasiparticles such as excitons and polaritons [85] and evento photons interacting with dye molecules [86]. The achievement of BEC in
dilute atomic gases has since given birth to an ever growing field of research,taking advantage of the remarkable properties of quantum-degenerate gases.

2.1.1

Mean field description

Once cooled below degeneracy, the dilute and weakly interacting gas of bosonicatoms in three-dimensions is well described by a macroscopically populated5single-particle wave function Ψ(r, t) subject to the nonlinear Gross-Pitaevskii
equation (GPE) [79]
~2 2∂Ψ(r, t)2−
∇ + V (r) + g|Ψ(r, t)| Ψ(r, t) = i~
(2.1)
2m
∂t
where V (r) describes some external potential. The effects of collisional interactions between atoms in the gas is modeled by the nonlinear term g|ψ(r)|2 ,where g = 4π~2 as /m is parametrized by the s-wave scattering length as . We
typically work in the Thomas-Fermi approximation in which the kinetic energy
term is neglected and we make the ansatz Ψ(r, t) = ψ(r) exp (−iµt/~), withthe chemical potential µ. The equilibrium density is thus given byn(r) = |ψ(r)|2 = (µ − V (r))/g

forµ ≥ V (r).

(2.2)

In this case µ = gn(r) + V (r), such that gn(r) can be considered a local
chemical potential, i.e. the energetic level to which the potential V (r) is “filled”
at position r by the condensate.
In addition to the ground state, we are also concerned with the elementary
excitations of the condensate as well as its modes of collective motion. The
elementary excitations can be found by linearizing the GPE about some change
in the wave function ψ(r, t) + δψ(r, t); the result is the Bogoliubov spectrumwith dispersion relation [79]s2~2 q 2~2 q 2(2.3)+ 2µ(q) =
2m
2m
pwhich is phonon-like (linear) for small q with speed of sound vs = µ/m
and particle-like (quadratic) for large q. The length scale associated with thetransition between phonon and particle excitations is given by the so-calledhealing length
1
ξ=√
,
(2.4)
8πnas
which is the characteristic length scale required to deform the condensate
wave function. Excitations with energy (q)  µ (or 2π/q  ξ), act like freeparticles.
It is also of experimental interest to consider the low lying collective modesof a trapped BEC. For our purposes, we will assume the BEC occupies anisotropic harmonic trap of the form V (r) = 1/2mω 2 r2 . The lowest energy
collective excitations in such a trap correspond to linear center-of-mass motion of the entire condensate, or dipole oscillations. The dipole oscillation is
6

(a)

(b)

(c)

v
v
Figure 2.1: The experimental BEC apparatus. (a) Photograph showing thequadrupole field coils, vacuum chamber and science cell along with nearby
optics. (b) Schematic drawing of the apparatus showing the main componentsof the machine. The red line represents the path followed by the transporter
coils, including a “dog leg” to minimize ballistic collisions with the background
vapor. (c) The science cell with crossed optical dipole trap and Helmholtz coilpairs (used to generate TOP trap in conjunction with the quadrupole coils).
Drawing taken from [89].
characterized by a frequency ω and preserves the density profile throughout
the collective motion. Such modes are easily excited by spatial displacementof the trap minimum and can be exploited to characterize the trapping potential [87]. Dipole oscillations also apply to anisotropic harmonic traps, where
the anisotropic oscillation frequencies can be determined by excitation along
different spatial directions. Higher energy collective motion can be excited bysudden squeezing of the trap to create a breathing mode or a scissors modewith oscillation frequencies dependent on the trap geometry [88].

2.1.2

Experimental apparatus

The experimental apparatus used throughout this thesis has been described indetail in [87, 89–91]. The important features of the apparatus will be briefly
reviewed here; for reference the bulk of the machine is photographed andschematically drawn in Fig. 2.1. The BEC apparatus begins with a room
temperature vapor of natural abundance rubidium (the rubidium source andion pump maintain a background rubidium pressure of a few 10−9 torr). Diodelasers tuned to the 87 Rb D2 line are used to laser cool the atoms in a magneto7

(a) Resonant

(b)

Light

BEC
Lens

CCD

Figure 2.2: Schematic experimental imaging system. (a) Schematic drawingof the absorption imaging process and system. A pair of lenses project the
shadow image of the atomic sample onto a CCD chip. Inset shows raw data
for the imaging on a CCD of a cold atomic cloud. The transparent tubes
represent resonant light and the cold cloud’s shadow. (b) Example time-offlight data following release of a BEC from a periodic optical potential.
optical trap (MOT) [92, 93] formed with a pair of moving quadupole coils.
After loading the MOT and performing brief sub-Doppler cooling in an opticalmolasses [94, 95], the magnetic quadrupole trap is moved by a linear-motor
driven translation stage through a differential pumping tube to a small glass“science cell” (maintained at ∼ 10−12 torr).
In the science cell, the cold atomic cloud is further cooled using forced rfevaporation in a time orbiting potential (TOP) trap [96] before transfer intoa crossed optical dipole trap formed by a pair of 1064 nm laser beams derived
from a 20 Watt Ytterbium fiber laser. After a final stage of evaporation in
the dipole trap, nearly pure BECs of 105 ...106 87 Rb atoms are obtained withtypical temperatures around 100 nK.
The BEC is then manipulated using a combination of optical potentials,
magnetic field gradients, and microwave fields before being imaged in timeof-flight, c.f. Fig. 2.2. The imaging system uses resonant laser light to casta shadow of the BEC on a CCD camera, after release from the optical trap
and subsequent ballistic expansion. For sufficiently long time of flight, thecaptured shadow image reflects the momentum distribution of the trapped
atoms and it contains information about the shape, coherence and collective
motion of the atomic cloud.

8

2.2

Atom-light interaction

Many of the most important tools used throughout ultracold atom experiments can be understood by examining the interaction of a simple two-level
atom with a classical radiation field. In the case of coupling to an optical field,
the interaction consists of dissipative and conservative contributions. The dissipative effects caused by absorption and spontaneous re-emission of photons
by atoms are of central importance for laser cooling and trapping [93]. We
focus here on the conservative effects associated with the dressing of atomic
energy levels by the applied radiation field. The so called light shift (or acStark shift) results from the interaction of the induced dipole moment of theatom with the driving light field. In the presence of spatially inhomogeneous
intensity this shift can be utilized in the creation of potential wells or barriers
for the confinement of atoms in free space [97]. In the last section, we thengeneralize the treatment of the two-level atom to that of multilevel atoms andintroduce effects beyond those described in the two-level picture.

2.2.1

Interaction Hamiltonian and Rabi oscillations

We begin with a simple atom consisting of a ground state |gi and an excitedstate |ei with energy separation ~ω0 = Ee − Eg . Consider the atom’s interaction with a monochromatic field of the form E(r, t) = E0 (x, y)ˆ cos(ωt − kz)
propagating along the ẑ-axis with polarization vector ˆ (we will ignore thecoupling to the magnetic field B for now as its magnitude is reduced by afactor of speed of light c). We can then write the system Hamiltonian as sumof the free atom Ha and the atom-field interaction Hint as
H = Ha + Hint = ~ω0 |eihe| − d · E(0, t)(2.5)

where we have made the dipole approximation, and dropped the spatial dependence of the field, with atomic dipole operator d = −ere for positive fundamental charge e. The ground state energy has been set to 0. Writing the
dipole operator in terms of the atomic states yields d = hg|d|ei(|gihe| + |eihg|)(if |gi and |ei have the same parity we simply have hg|d|ei = 0). Thus we canrewrite the interaction term as ∗E0 iωt E0 −iωt−iω0 t
iω0 t
e +
e
,
(2.6)
Hint = −hg|ed|ei |gihe|e+ |eihg|e22
assuming d = dˆ; for a two level atom the atomic polarization is always along
the field ˆ [93]. For near resonant light ω0 ≈ ω, the cross terms in Eq. 2.6
carrying rapidly oscillating terms exp [±i(ω0 + ω)t] can be neglected (making
9the rotating wave approximation) and Hint can be further simplified to
Hint ≈ |eihg|~ωR iδt~ωR −iδte+ |gihe|
e .
2
2

(2.7)

Here, we have introduced the Rabi frequency ωR = eE0 hg|d|ei/~ and thedetuning δ = ω − ω0 . The terms of the Hamiltonian simply represent theexcitation (de-excitation) of the two-level atom by absorption (emission) ofa photon. In this form we can solve for the dynamics of the atomic system
by applying the time-dependent Schrödinger equation with the ansatz |ψi =cg |gi + ce |ei. The equations of motion for the population amplitudes can thus
be written as
ωR
ce
2
ωR
cg ,
ic˙e = δce −2

ic˙g =(2.8)

The solution of Eq. 2.8 describes Rabi oscillations of the internal state populations

 2

ωR
ΩR
2
2
t
Pe (t) = |ce | =sinΩR2Pg (t) = |cg |2 = 1 − Pe (t)(2.9)
pwith generalized Rabi frequency ΩR = ωR2 + δ 2 . For resonant coupling δ = 0,
Eq. 2.9 predicts full contrast oscillations of the population, while driving offresonance reduces the amplitude and increases the oscillation frequency as seenin Fig. 2.3. Under realistic conditions, long lived oscillations using opticaltransitions are limited by the short lifetime of the excited state leading todamping due to dephasing for an ensemble of atoms.

2.2.2

Dressed states and dipole potentials

While resonant light fields drive transitions between the internal states, offresonant fields still shift the energy levels of the atom. The previous section
focused on dynamics in the basis of atomic eigenstates; here we consider the
eigenstates (dressed states) of the total light-atom system [93]. For this purpose, it is useful to write Eq. 2.8 in matrix form~ −2δ ωR
(2.10)
H=
0
2 ωR

10

1.0
0.8
Pe

0.6
0.4
0.2
0.0

0

1

2
t [2π/ωR]

3

4

Figure 2.3: Rabi oscillations of the excited state population Pe (t) of a two-levelsystem are plotted for δ/ωR = 0 (black), 1 (red), 2 (blue).
and then diagonalize it to find the “dressed” eigenstates and eigenenergies
corresponding to the atom-field system. The dressed states are given by,|χ+ i = cos θ|gi − sin θ|ei

|χ− i = sin θ|gi + cos θ|ei,

and(2.11)

where θ is the mixing angle, with cos(2θ) = −δ/ΩR . The dressed states haveenergies~(2.12)
E± = (−δ ± ΩR ).
2
For an off resonant field δ 2  ωR2 and δ < 0, the mixing angle θ ≈ 0 andthe dressed states are nearly given by the bare states, i.e. |χ− i ≈ |gi and|χ+ i ≈ |ei (and vice versa for δ > 0). In this limit the energy shifts are given
by
~ωR2
~ω 2
Eg =
and
Ee = − R .
(2.13)
4δ
4δ
Utilizing spatially dependent Rabi frequencies, these energy shifts can be used
to produce optical traps and potential barriers. The optical fields are mosttypically derived from laser beams with Gaussian profile such that E0 (x, y) ∝exp [−(x2 + y 2 )/w2 (z)] with beam radius w(z). Since the intensity of the laser
field can be related to the Rabi frequency through I(r) ∝ |E0 |2 ∝ ωR2 , thespatially dependent energy shifts give rise to a force known as the optical
dipole force. Moreover, since the energy shift depends on the sign of thedetuning, optical dipole traps (barriers) for ground-state atoms can be formed
for red δ < 0 (blue δ > 0) detuned optical fields.
It is important to keep in mind that the optical potentials actually correspond to the dressed states of the atom, with generally non-negligible contribu-

11

tions of both ground and excited states. Thus, it is also necessary to consider
the effect of spontaneous scattering. The exact (rotating wave) expressions forthe dipole potentials and the scattering rate are given by [97]3πc2 Γ
I(r),2ω03 δ
 2
3πc2 Γ
Γsc (r) =I(r),2~ω03 δVdip (r) =(2.14)(2.15)

where the excited state decay rate Γ is given byΓ=ω03 |d|2,3π0 ~c3(2.16)

following the work of Wigner and Weisskopf [93].∗ For off-resonant fields,
the scattering rate falls off as δ −2 , while the potential only falls off as δ −1 .
This dependence on the scattering rate calls for the use of larger laser powers
P ∝ I and larger detunings in order to minimize, for a given potential depth,
the unwanted effects of spontaneous photon scattering. The light-induced
dissipative effects of photon scattering typically lead to heating and the decayof any coherent dynamics in ultracold atomic samples.

2.2.3

State-dependent optical potentials

Though the two-level description gives good intuition about the effect of anac electric field’s interaction with an atom, it is not always possible or evendesirable to isolate a single two-level optical transition. The energy shifts
described above are linear in the light intensity and second order in electricfield E0 . Thus, second order perturbation theory for a given state |ii withenergy Ei (in the following we set the bare atom energy Ei = 0 for convenience)in a multilevel system yields the level shifts according to∆Ei =X |hj|Hint |ii|2
Ei − Ej

i6=j,(2.17)

where the sum is taken over all other internal states |ji of the atom. Including
the driving field, the ground state atom plus light system has energy Ei = n~ω
assuming that the field contains n photons. The system with an excited atomthen has energy Ej = ~ω0 + (n − 1)~ω = ~δi,j + n~ω; we thus replace the∗Hence, the Rabi frequency is given by ωR = Γ
λ = 2πc/ω, is the saturation intensity.

12p

I/2Is , where Is = 2π~cΓ/(3λ3 ), with

mF= -3
2

P3/2

2

P1/2

-2

-1

0

+1 +2 +3
F=3
F=2∆'HFS~0.5GHz F=1
F=0∆FS=7THz
F=2
F=1

-

D2=780.24 nmσ
λ=787.59 nm

D1=794.98 nm
F=2
2∆HFS=6.8 GHz

S1/2

F=1

Figure 2.4: Electronic level structure of rubidium-87 [98]. The excited state is
split by ∆FS by spin-orbit coupling of the orbital and spin angular momentum.
Each fine structure manifold is further split, by coupling to the I = 3/2 nuclear
spin, into hyperfine levels with typical energy scales ∆HFS and ∆0HFS . For
specific applied light fields, for instance σ − light at 787.59 nm, the ac-Stark
shift for specific S1/2 levels can be tuned to 0. In the example shown, |2, −2ionly couples to a single excited state (shaded in blue), experiencing a two-levellight shift. On the other hand, |1, −1i couples to several states (shaded in red)in both hyperfine manifolds, and the net ac-Stark energy shift vanishes.
denominator in Eq. 2.17 with the detuning for each state pair δi,j . For a
two-level atom, we can readily show that this reproduces the result given in
Eq. 2.15.
For multilevel atoms, the two-level result generalizes to∆Ei =3πc2 X Γj c2i,j
I
,
3
2
ω
δ
i,j
i,j
j6=i

(2.18)

where we have already ignored the counter-rotating terms† and reduced the†

The counter-rotating term is nonnegligible for very large detunings and should generally
be included. For now, we concern ourselves with light fields near enough to resonance tojustify this exclusion.

13

matrix elements to transition strengths (Clebsch-Gordan coefficients) ci,j =hF 0 , m0 F |F, 1, mF , qi for initial and final states |ii = |F, mF i and |ji =|F 0 , m0 F i
with a spin-1 photon having polarization q = 0, ±1 corresponding to linear orσ ± polarization with respect to the quantization axis [93, 97].
We are most concerned with the case of alkali atoms, which have a single
unpaired electron in their valence shell and two strong optical transitions in
the form of the D1 and D2 lines, c.f. Fig. 2.4. We can then recast Eq. 2.18 as[91]
3πc2 ΓD1 X c2i,j ΓD2 X c2i,k
∆Ei =
I 3
+ 3
,
(2.19)22
ωD1 j δi,j ωD
δ
i,k
2
k
where the sums over j and k are over the states in the 2 P1/2 and 2 P3/2 manifolds
respectively. For a given polarization q and initial state |ii the participating
excited states will be limited to those allowed by dipole selection rules. The
factor of 2 in the D2 sum accounts for the effective line strength compared to
the D1 line.
For most applications, we will be working with detunings that are large
compared to the splittings within the hyperfine manifolds (few 100 MHz),but small compared to the fine structure splitting (∼ 7 THz). In the case ofeffectively unresolved F levels, we can greatly simplify Eq. 2.19 to [99]
1 − qgF mF
πc2 Γ 2 + qgF mF
I+,(2.20)
∆Ei =
2
∆2,F
∆1,F
3
3
where we have taken ΓD1 /ωD≈ ΓD2 /ωD≈ Γ/ω03 and introduced the Landé
12factor gF of the considered hyperfine ground state |ii = |F, mF i. In this form,
it is easy to see that for a given mF state and q, with ω tuned between theD1 and D2 transitions, a competition can be engineered between the firstand second terms in the brackets, as the detunings from the D2 and D1 lines∆2,F and ∆1,F can take on opposite signs. This allows for the simultaneous
production of an optical potential for one mF state while another experiencesno ac-Stark shift. In our experiment, we typically use the state pair |2, −2i
and |1, −1i for one implementation of this so called “tune-out” scheme, as
shown in Fig. 2.4. Other possible configurations of such a lattice are further
described in [100]. Optical potentials for a few such configurations are plottedin Fig. 2.5; the potentials are calculated using Eq. 2.19.
As the driving optical field is still close to resonance in this configuration,
the admixture of excited states due to the field is relatively high, and spontaneous scattering from these excited state components must be considered.

14

40
σ−

V0 [Er]

20

σ+

π

0
-20
-40
5

Γsc [s-1]

4
3
2
1
0
775

780

785

790

795

800 775

780

785

790

795

800 775

780

785

790

795 800

λ [nm]

Figure 2.5: State-dependent potentials and scattering rates. (a) Potential
depths for an optical standing wave formed by a counter-propagating pair ofGaussian beams with waist w = 1 mm and with power 0.75 W. Red, dashed(blue, solid) lines correspond to |2, −2i (|1, −1i).Plotted from left to rightare q=-1,0,1. (b) Spontaneous scattering rates for an atom trapped in thepotential minima formed by the above configuration.
Generalizing Eq. 2.15 and Eq. 2.17 we can write
Γsc,i =X Γj |hj|Hint |ii|21I.
2
20 c~2 j6=i
δi,j(2.21)

Scattering rates, calculated after applying this to the D1 and D2 transitions
separately, are plotted in Fig. 2.5. For typical intensities applied here, the scattering rates are on the order of a few Hz, which is acceptable for experiments
occurring on timescales up to ∼ 100 ms.

2.2.4

Two-level dynamics in the 87 Rb ground state

The rubidium ground state provides an ideal system for the resonant two-level
physics described above, using magnetic-dipole transitions between hyperfine
levels. As 87 Rb is an alkali atom with a single electron of J = 1/2 andnuclear spin I = 3/2, its ground state is split into manifolds with total angularmomentum F = 1, 2 separated by ∆HFS ≈ h × 6.8 GHz, c.f. Fig. 2.4. In a
weak magnetic field these states are described by the Zeeman-split sublevels|F, mF i with angular momentum projection onto the quantization axis mF =

15

(a)

(b)

1.0

?

(c)

?

?

0.8

Pe

0.6
?

0.4
0.2
0.0

4.20 4.22 4.24 4.26 4.28 4.30
ν-6827 [MHz]

0

20

40

60 80 100 120 0t [µs]

5

10 15 20 25 30 35
ωR [arb.]

Figure 2.6: Hyperfine-ground state control with applied microwave fields. (a)Typical microwave resonance measured using a Rabi π-pulse with ωR ≈ 2π×17
kHz and τ = 28 µs. The peak width is given by the Fourier width of the applied square pulse. (b) Resonant Rabi oscillations between |1, −1i and |2, −2i
hyperfine sublevels, with ωR = 2π × 17.5 kHz. (c) Landau-Zener transitions
between the same sub-levels for varied microwave power. The detuning is
swept from δ = −200 kHz to δ = +200 kHz in 2 ms; near 100% transfer isreliably achieved for the largest Rabi frequencies.
{−F, −F + 1, ..., F − 1, F }. There are no electric-dipole transitions between
these states; however, the magnetic-dipole interaction Hint = −µ · B allows
for Rabi couplings between them.
In the laboratory, this is done by the application of microwave and rf-fields[90, 91]. We can thus control the internal state population of the condensate
by driving Rabi oscillations or performing “Landau-Zener” sweeps of the frequency around resonance.‡ Both processes coherently alter the state of thetrapped atoms, as demonstrated in Fig. 2.6. In principle, arbitrary state superpositions can be prepared, limited only by mean-field effects and systematic
imperfections to lifetimes of a few ms. The splitting between levels of a single
F manifold, determined by the Zeeman effect, are linear to first order in asmall bias field B ∼ 1 G and are on the order of MHz. The spectral widthof the applied fields can be much narrower (on the order of Hz), which allows
for the isolation of transitions between hyperfine sub-levels in different manifolds and selective driving of transitions between them. Transitions within asingle hyperfine manifold can be isolated by taking advantage of the quadratic
Zeeman effect.

‡

The case of off-resonant applied microwave fields, applying an ac-Stark shift (describedabove) has also been used to study spinor gases [101–103].

16

2.3

Optical lattices and band structure

An optical lattice, in its most basic realization, consists of a single laser beamthat is retroreflected back onto itself to form a standing-wave interference pattern with the spatial period d = λ/2. The standing wave interference createsa series of sub-wavelength-sized optical potential wells that remain coupledthrough tunneling. More general lattice configurations in higher dimensions
can be implemented by interfering multiple mutually coherent laser beams(not necessarily retroreflected) [16, 104]. For the work described in this thesis,
we will restrict ourselves to the case of a one-dimensional lattice along the ẑ
direction
V (z) = Vlatt sin2 (klatt z)(2.22)
where Vlatt is the lattice depth and klatt = 2π/λ. As discussed in any basic
solid-state physics textbook [105], the dispersion relation for a free particleof mass m in one dimension E(k) = ~2 k 2 /(2m) is modified by the lattice
into periodic bands because of to the condition that solutions of the timeindependent Schrödinger equation Hψ = Eψ with H = p̂2 /2m + V (z) must
share the periodicity of the lattice potential V (z). These solutions are known
as Bloch waves and have the general formφq (z) = eiqz uq (z)(2.23)

where uq (z) = uq (z + d), and q is known as the quasimomentum. As detailed
further in e.g. [16, 87, 106], inserting φq (Eq. 2.23) into the kinetic energy
term transforms the Schrödinger equation into [(p̂ + q)2 /(2m) + V (z)]uq (z) =Eq uq (z). Writing both uq (z) and V (z) as a Fourier series gives
X (q)X
uq (z) =cl ei(lK)zandV (z) =
Vj eijKz(2.24)
jlwith reciprocal lattice wave numbers K = π/d. Inserting the above into thetime independent Schrödinger equation results in the eigenvalue problem
X X (q,n)X (q,n)X (Kl + q)2 (q,n)
cl eilKz +cl Vj ei(j+l)Kz = Eq(n)cl ei(lK)z
2m
j
l
l
l
(2.25)
where we have included the band index n. This is valid for an any spatiallyperiodic potential and can be written conveniently in matrix form
X
(q,n)
(q,n)
Hl,j · cl= Eq(n) cl(2.26)
l,j

17

35
30
25
E [Er]

20
15
10
5
0
-1.0 -0.5

0.0 0.5
q [klatt]

1.0 -1.0 -0.5

0.0 0.5
q [klatt]

1.0 -1.0 -0.5

0.0 0.5
q [klatt]

1.0 -1.0 -0.5

0.0 0.5
q [klatt]

1.0

Figure 2.7: Energy bands of the sinusoidal lattice at varying depth. Plottedeigenenergies E(q) for the first five bands of the lattice for s = 0, 3, 12, 30 (fromleft to right) in the first Brillouin zone. The dashed red line gives the lattice
depth, indicating the number of bound states in the lattice potential.
where Hl,j is a square matrix with diagonal elements of the form (Kl+q)2 /2m+
Vj=0 and off-diagonal elements V|l−j| . The one-dimensional optical lattice potential (Eq. 2.22) considered above has K = 2klatt and the Fourier decomposition V0 = Vlatt /2 and V1 = −Vlatt /4, leading to a tridiagonal Hl,j . The
eigenvalues and vectors can then be found by taking a finite number of lattice wave numbers K = [−2lmax , ..., −2lmax ] and diagonalizing Hi,j . Plotted in Fig. 2.7 are the eigenenergies of Eq. 2.22 in the first Brillouin zone(q = [−klatt , +klatt ]) for different lattice depths s = Vlatt /Er , where the recoil
2energy Er = ~ωR = ~2 klatt/(2m) is the kinetic energy of an atom of mass mrecoiling from the absorption of a photon with wave number klatt .
For nonzero Vlatt , band gaps (with q and n dependent energy ∆gap ) are
opened in the dispersion relation, leaving bands of allowed eigenstates of thepotential separated by gaps with zero density of states. The energy widthof a band is related to the tunneling amplitude among neighboring latticewells.§ Note that increasing the lattice depth flattens these bands and shifts
the zero-point energy up from the free particle case. The flattening of the
bands corresponds to an increase in a particle’s effective mass and reductionof its kinetic energy. For very deep lattices, the deeply bound energies approach√
those of a harmonic oscillator with trapping frequency given by ωho = 2 sωR .
In this limit, each lattice well acts as an isolated, tightly-confining harmonic
trap.
The wave functions for these bands are generically given by Bloch waves§

In the tight-binding limit, the band width is related to the tunneling parameter by∆band = 4t

18

described in Eq. 2.23. They are identified by a single quasimomentum q andband index n and have infinite spatial extent, thus making the Bloch basis
convenient for describing the coherent states of a shallow lattice potential. For
deeper lattices, it is often convenient to instead consider the Wannier functions
describing a particle localized to a single lattice site [107]. One formulation
for the Wannier functions is given byX
wn (z − zi ) ∝e−iqxi φq(n) (z).
(2.27)q

While the above relation provides a state localized around zi , it is not guaranteed to provide the maximally localized wave function. While wn (z − zi ) converge to the harmonic oscillator eigenstates for Vlatt → ∞, using the harmonic
oscillator wave functions is generally not meaningful, as the most localized
Wannier functions that still span the Bloch wave basis still have significant
structure beyond the typical harmonic oscillator length of a single well. In
fact, even in the case of a deep lattice, the small differences between the harmonic oscillator wave functions and Wannier functions can lead to large discrepancies in calculations of the tunneling rate between wells (see Eq. 2.32).
The necessary accurate calculation of the localized Wannier function can beachieved using well established methods, as described in [26, 108].¶ Several of
the relevant wave functions are plotted in Fig. 2.8.

2.3.1

Single particle dynamics

The band structure of the lattice significantly modifies the dynamics of noninteracting particles in the periodic potential. If we consider a particle in thelowest band, by convention n = 0, of the lattice, we see that the dispersion
relation is flattened and gapped. When subjected to a force F the particle’s
quasimomentum evolves just as in free space~q(t) = F t + ~q(0).

(2.28)

While for a free particle, this corresponds to continuous acceleration (in the
non-relativistic limit), the gapped dispersion relation of the lattice prevents
such continuous acceleration as the particle is confined to move on the lowest
band of the lattice. This leads to so-called Bloch oscillations, as the particle
traverses the Brillouin zone and is Bragg reflected from +klatt to −klatt eachtime it reaches the zone edge. By analogy with optics and the free particle¶

These methods may yield a different set of phase prefactors than in Eq. 2.27 andtypically rely on minimizing the total spread of the wave function hz 2 i − hzi2 .

19

(a)

(c)
2.0

1.0

ψ

1.5
1.0

0.5

0.5
0.0
0.0

(b)

(d)
2.0

1.5
1.0

1.0

0.5

ψ

0.0
- 0.5

- 1.0

- 1.0
- 1.5

-4

-2

0

2

4

z [d]

- 2.0

-2

-1

0

z [d]

1

2

Figure 2.8: Bloch wave and Wannier functions of a sinusoidal potential. Plotted are the lattice potential, the corresponding Bloch wave, Wannier function
and harmonic oscillator wave function (gray, black, orange and dashed blue
respectively).(a) For s=2, n=0. (b) For s=2, n=1. (c) For s=12, n=0.(d) For
s=12, n=1.

20

(a)

(b)
1.0

2
E [arb.]v [vr]

0.5
0.0
-0.5
-1.0
-1.0

1
0
F

-1

ħ∆gap

-2
-0.5

0.0
q [klatt]

0.5

1.0

-4

-2

0
2
q [arb.]4

Figure 2.9: Band group velocity and Landau-Zener process in a lattice. (a)Group velocity of a particle moving in the lowest band of the lattice for s =0.1, 1, 5, 10, s → ∞ (red, gree, orange, blue, black). (b) Schematic LandauZener process at a gap, for a wave packet evolving under some force F .
case, the group velocity of particle moving in the lattice is given by [105]vg (q(t)) =1 dE (n) (q)
.
~ dq
q(t)

(2.29)

Bloch oscillations can thus be seen as an oscillation of the mean momentum
hp̂i = mvg , c.f. Fig. 2.9a.
Bloch oscillations describe the dynamics of a particle as long as it isconfined to a given band of the lattice. This is not always true and whenF d ≈ ∆gap , there is a meaningful probability for the particle to leave the bandin which it originates. This mechanism underlies Zener breakdown of a nonconducting material, and as ∆gap becomes smaller as the band index increases,
this typically leads to rapid population of higher bands, and thus conduction[109]. Near a gap or avoided crossing, the probability to remain in the lowest
band of the lattice is described by the Landau-Zener formalism, c.f. Fig. 2.9b,
which is closely related to that of adiabatic rapid passage. The rate of transfer
into the higher band is given simply by [110]Ω2(2.30)
P = 1 − exp − 2π~(dE/dt)
where ~Ω ≈ ∆gap is the coupling between bands of the lattice (e.g. the matrix
element coupling different bare plane wave states) and dE/dt ∝ F describes
the slew rate through the avoided crossing.

21

2.3.2

Effects of weak interactions

In the noninteracting or ideal gas regime, the dynamics of a condensate ina lattice is described well by the single particle physics described above [10]such that Bloch oscillations and Landau-Zener tunneling can be observed andcompared to the simple band structure theory [9, 111]. Mean-field interactions
tend to alter the properties and dynamics of a lattice-trapped condensate.
For sufficiently shallow lattices and weak interactions, the interactions act asan effective potential term that reduces the lattice depth by a factor (1 +µ/(2Er ))−1 [112]. As demonstrated in [9], dynamical effects such as LandauZener tunneling then behave indeed as if an effectively shallower lattice were
present in this regime.
Some of these dynamical effects can be described microscopically. Taking
the case of Landau-Zener tunneling, the effective reduction in lattice depth
results from a breakdown of adiabaticity at a band gap caused by the nonlinearity [113]. In the presence of strong interactions, the energy spectrum at thegap can be dramatically modified (see Chapter 3 for details). The bands become multivalued near the gap and loops (or swallowtails) form in the energy
spectrum. These loops prevent adiabatic following of the energy levels thatwould be described by Eq. 2.28, resulting in increased tunneling into higher
bands as the gap is traversed.
Interactions also affect the stability of an interacting condensate trappedin a lattice. It can be shown that states near the zone edge are unstable due
to interactions [114]. To understand this, consider the dispersion relation in
the lowest band near the zone edge. For q ≈ π/d, Eq takes on a negative
curvature which corresponds to an effective mass m∗ < 0 [105]. The negative
effective mass implies a change in sign of the interaction parameter g in the
GPE, leading to exponential growth of Bogoliubov perturbations δψ to thecondensate wave function ψ, instead of the usual oscillating sound solutions.
As a result, small perturbations lead to rapid decay of ψ. This so-called
dynamical instability has been observed experimentally as a depletion of thecondensate wave function [115].
For deeper lattices, the interaction energy can become the largest relevantenergy in the system. Consequences for the strongly interacting lattice gas
will be briefly discussed below.

2.3.3

Tight-binding and strong interactions

When the lattice is sufficiently deep that the lowest bands can be considereddispersion-free, it is typically more useful to consider the tight binding limit.

22

Here, the single particle dynamics are determined by the HamiltonianX
H = −t|j + 1ihj| + h.c.
(2.31)jwhere j is the lattice cite index and t is the energy associated with hoppingor tunneling between neighboring sites. The hopping parameter t is given by
the matrix element

 2
Z
p̂
(n)
(n)∗
+ V (z) w(n) (z + d)
(2.32)
t = t = dzw (z)2mand is generically band dependent; for the lowest band t is positive and thus
favors delocalization in the lattice.
In the presence of interactions, the system of ultracold atoms restricted to
the ground band of a lattice can be mapped onto the Bose-Hubbard model[116], a paradigmatic generalization of Eq. 2.31 [55]:H = −tX †X
UX(b̂i b̂j + h.c) +n̂i (n̂i − 1) − µ
n̂i
2 i
i

(2.33)

hi,jiHere the brackets indicate a summation over nearest neighbors, b̂†i is thebosonic creation operator at lattice site i, n̂i = b̂†i b̂i is the number operator
and U represents the on-site interaction energy defined as,
Z
U = g dz|w(z)|4 .
(2.34)
Restriction to the ground band allows us to take t > 0 and include the negative
tunneling energy explicitly. The Bose-Hubbard model contains rich physics including a quantum phase transition between a superfluid and (gapped) Mott
insulator driven by the relative strength of the interactions U . Early realizations of the Bose-Hubbard model have demonstrated ultracold atom experiments to be ideal for controlled studies of the relatively simple physics
it contains [14]; such experiments have more recently paved the way for thesimulation of more complex Hamiltonians [27, 32, 37, 49, 117].

2.3.4

Lattice loading and characterization

In our experiments, the atoms are often loaded into the lattice by beginningwith a condensed atomic cloud and introducing the optical lattice potential at alater time. In doing so, we attempt to remain in the ground state of the system23

by applying an adiabatic lattice ramp, which must be slow compared to any
collective excitations of the superfluid cloud, slow compared to timescales set
by the gaps in the band structure [118], and slow with respect to the tunnelingrate. The latter requirement assures the atomic density is allowed to rearrange
in the trap plus lattice potential. While there is debate over the existence of an
ideally adiabatic ramp [119–122], we can achieve sufficiently adiabatic loadingof the lattice by keeping the rate of change in Vlatt slow compared to the overall
trapping frequencies and also the instantaneous tunneling time.
When considering deeper lattices where tunneling rates can be on the orderof a few Hz, it becomes difficult to satisfy the adiabatic condition. We cancheck the ramp adiabaticity by entering the Mott insulator regime with alattice ramp-up and then returning back to superfluid with a lattice rampdown. For short hold times and filling fractions of order one, we can attribute
any heating seen in the final condensate to nonadiabatic processes in the latticeramp procedure [123].
Once the atoms are loaded into the lattice, several techniques can be used tocharacterize the lattice depth. A simple and fast technique takes advantage ofmatter wave diffraction following the application of a pulsed optical potential.
This form of Kapitza-Dirac diffraction allows us to measure the lattice depth
by looking at the time evolution of diffracted momentum modes during theshort lattice pulse. This procedure was previously characterized in [87, 124].
In Fig. 2.10 we show the time of flight momentum distributions after a shortlattice pulse along with the corresponding lattice depth calculated by fitting
the distributions to calculated diffraction patterns given by the set of equations(2)

iċj =Vlatt
Er j 2cj +(cj−1 + 2cj + cj+1 )~4~(2.35)

describing the evolution of the amplitudes for each diffracted mode cj for(n)integer j, where we define the general n−photon recoil energy Er = n2 Er .
Other methods include lattice modulation spectroscopy, which we shalltouch on in Chapter 5. To briefly introduce the principle, this method relies
on measuring ∆gap by modulating the phase or amplitude of the lattice intime. This allows for the excitation of higher bands in the lattice, inferred
by a reduced superfluid visibility or measured directly via a band-map to
adiabatically transform the Bloch states onto free particle states [12].

24

(b)
30

6
4
2
0
-2
-4
-6

25
Vlatt [Er]p [pr]

(a)

20
15
10
5

0.8

2.9
Vphotodiode [V]

5.1

0

0

1

2

3
4
Vphotodiode [V]5

Figure 2.10: (a) Time of flight Kapitza-Dirac diffraction patterns after shortlattice pulses (τ = 10µs) of variable intensity. (b) Fitted lattice depth as afunction of a pick-off photodiode voltage monitoring the power in the latticebeam. The potential depth is found from the TOF images in (a) by comparison
with solutions to Eq. 2.35. For details see [87, 124].

25

Chapter 3
Superfluid Bloch Dynamics in a
Tilted Incommensurate Lattice
In this chapter, we investigate the interplay of disorder and interactions in theaccelerated transport of a lattice-trapped Bose-Einstein condensate throughan incommensurate optical lattice. While there often is a focus on the cleanliness of and the ease of control in optical lattice systems, the properties of realmaterials critically depend on the presence of “imperfections” such as impurities and lattice defects. Understanding the behavior of interacting disorderedsystems continues to present a challenge to condensed matter theory and experiment alike. By examining the interplay of interactions and disorder in the
dynamics of Bloch oscillations in an optical lattice, we demonstrate that weakinteractions can effectively cancel the damping of Bloch oscillations induced
by disorder due to screening, and we provide a simple model to qualitatively
capture this effect. The competition between disorder and weak interactions
described in this work complements the cooperation between disorder andstrong interactions observed in an earlier experiment on the Bose-glass phase[54].
The following chapter is based on the article Superfluid Bloch Dynamics inan Incommensurate Lattice, New Journal of Physics 16, 065011 (2014) [125].

3.1

Interacting disordered systems

The combined effects of disorder and interactions in condensed-matter systems
can influence their transport properties in profound ways. Beyond merely reducing conductivity, disorder can lead to Anderson localization [57] in the absence of interactions, and repulsive interactions can give rise to localized Mott
phases [126] without the influence of disorder. While disorder and interaction

26

act cooperatively in the limit of strong interactions by promoting disorderedinsulating phases [55, 60], weak interactions in bosonic systems can counteract the localizing effects of disorder by making accessible higher-energy states,
thus screening the disorder potential [61, 62]. The often subtle interplay between disorder and interactions is relevant for condensed-matter systems suchas superfluid helium in porous media and disordered films [69, 127] and granular superconductors [68, 128], but it can also be studied experimentally withultracold atoms in optical lattices. While optical lattices are naturally defectfree [16], it is possible to implement disorder using laser speckle [47, 58], localized impurity atoms [54], and incommensurate standing waves [129]. Recent
experiments with disordered optical potentials have addressed Anderson localization [47, 51, 58], delocalization due to repulsive interactions [50, 62, 130],
as well as the formation of glassy, localized phases in the regime of strong
repulsive interactions [48, 49, 54, 56, 131]. It has also been proposed that disordered many-body systems may experience a novel form of localization known
as many-body localization [132, 133], ultracold atoms have already begun to
study these effects [52, 130].

3.2

Bloch oscillation basics

Disorder and interactions also fundamentally affect the dynamics of particles inlattice potentials. Here, we examine the case of static external forcing resulting
in coherent Bloch oscillations (BOs) [109, 134] that prevent macroscopic transport unless mechanisms for relaxation exist [135]. While BOs are overdamped
in conventional solids due to relaxation times much shorter than an oscillation
period, they can be observed in superlattice semiconductors [136] and alsowith ultracold atoms in optical lattices [4, 10, 137]. Atomic quantum gases
in optical lattices naturally provide an environment free of scatterers such as
defects and lattice-phonons and allow, in principle, for long lived oscillations,c.f. Fig. 3.1a.
In the atomic system, minimizing the collisional interactions is necessary
for the direct observation of extremely long-lived BOs [67, 138, 139]. Typical mean-field interactions in Bose-Einstein condensates already result in significant dephasing, c.f. Fig. 3.1b [67, 139] due to nonlinearities, dynamical
instability, and quantum chaotic dynamics [140–142]. (It has recently been
observed that in certain parameter regimes, such dephasing can be reversible[143, 144].) A qualitatively similar dephasing of BOs in optical lattices was also
observed to occur due to the additional presence of disorder, which scrambles
the regular phase evolution between lattice sites [66]. In addition to dephasing,the damping of BOs by disorder was also accompanied by strong depletion of

27

(a)
p [pR]

0.3

2

0

0

-0.3

(b)

η~0.2

-2

p [pR]

0.3

2

0

0

-0.3

-2

η~0.6
0

1

2

3

4

t [TB]

5

0

6

0.8

1.6

2.4

hold time [ms]Figure 3.1: Bloch oscillations of an optically trapped atomic cloud. (a) Observed oscillations of the atomic momentum distribution and mean momentum
hp̂i for a dilute ultracold atomic gas with mean field µ ∼ 0.1Er in a ∼ 20 Hz
trap. The damping rate is fit to be η ∼ 0.2TB . (b) Same for increased mean
field µ ∼ 0.3Er . The damping rate is increased to η ∼ 0.6TB , and a significant
spreading of the momentum wave packet can be see in time of flight images.
Lines are the result a fit described in the text and points represent the centerof mass of the expanded atomic cloud.
the condensate wave function.
For the case that both interactions and disorder are present, it has been
predicted that the combined effects can modify the BO dynamics, by either
reducing or enhancing the damping due to disorder, depending on their relative
magnitude [145, 146]. In the experiment described here, the interplay between
disorder and interactions is studied in the dynamics of Bloch oscillations in
an incommensurate lattice. The results demonstrate that for a given disorderinduced damping rate, increased interactions can reduce or enhance the overall
damping rate. Furthermore, we identify a characteristic interaction strength
between the regimes of reduction and enhancement, corresponding to a typicaldisorder energy scale. A simple model to capture the basic features of thesystem is proposed and discussed.

3.3

Quasiperiodic potential

For this experiment we choose to implement disorder through the use of abichromatic lattice potential consisting of two collinear superimposed opticalstanding waves of different wavelengths λ1 and λ2 . For rational wavelength
ratios, such lattices describe super-lattices or can be used to generate a latticewith a basis of lattice wells at each site. We will consider here the case that the28(a) 1.8

(c)

(b)

1.7

E [Er]

1.6
1.5
1.4
1.3
1.2

-1.0

-0.5

0.0

q [k1]

0.5

1.0 -1.0

-0.5

0.0

q [k1]

0.5

1.0 -1.0

-0.5

0.0

q [k1]

0.5

1.0

Figure 3.2: Band structure of the quasiperiodic lattice. The red lines represent
the lowest (n = 0) band of the main lattice, while the gray lines represent all
minibands of the total superlattice. The longer periodicity of the bichromatic
lattice results in a shrinking of the first Brillouin zone. (a) The unperturbed
n = 0 band of an s1 = 3 lattice. (b) The lowest set of minibands corresponding
to the original unperturbed band in (a) for a quasiperiodic lattice with β = 1.36
for s1 = 3 and s2 = 0.5/β 2 . (c) Same as (b) for β = 1.15.
wavelengths are incommensurate, i.e. have a ratio that cannot be expressed asm/n, where m and n are integers. Such a lattice forms an optical quasicrystal[147] that lacks crystalline translational symmetry, but still maintains order
on a shorter length scale [148]. For a quasiperiodic potential generated by theoverlap of two incommensurate standing waves, we can approximate the irrational wavelength ratio through a rational number, due to the finite precision
with which the laser wavelengths are known. In this case, the superposition
leads to a superlattice with long-range translational symmetry, rather than atrue quasicrystal. However, as we will see, this is of no practical consequence
as long as the period of the superlattice exceeds the size of the atomic cloud.
In our experiments, we generate quasiperiodic potentials with λ1 = 1064 nm
and λ2 = 785 nm; the commensurate ratio β ≡ λ1 /λ2 ≈ 1.36 can be used todescribe the band structure.
The bichromatic lattice has long range periodicity Λ such that V (z) =V (z + Λ). Assuming a nearly rational β, we find its true period to be givenby the ratio
β=mnsuch that

Λ
=n
λ1

and

Λ
=m
λ2

(3.1)

requiring that m/n be written with the smallest possible integers (in reduced
terms). Taking β = 1.36, we find n = 25 and m = 34. Thus for a potential of29the form
V (z) = s1 Er,1 sin2 k1 z + s2 Er,2 sin2 k2 z

(3.2)

we can use Eq. 2.26 to solve for the band structure of the total quasiperiodic
potential if we select K = 2π/Λ. In our example, Hi,j has diagonal terms (Kl+q)2 /2m + s1 Er,1 /2 + s2 Er,2 /2 and off-diagonal elements H|l−j|=25 = −s1 Er,1 /4
and H|l−j|=34 = −s2 Er,2 /4. The leading effect of a weak secondary lattice on
the band structure is to break up the unperturbed band in Fig. 3.2a into a
series of gapped mini bands in Fig. 3.2b. These gaps are narrow in energy
compared with s2 Er,2 due to the second order nature of the process which
opens the minigaps.
Quasiperiodicity as Disorder
Optical lattice experiments have been able to implement disordered and quasidisordered systems in several ways. Most common among these are the introduction of optical speckle potentials [47, 48, 129, 149], localized atomic
impurities [53, 54, 150], or the bichromatic optical lattices discussed here
[50, 51, 54, 62, 151]. We can get an idea of the physics of the bichromatic
potential by examining the Harper or Aubry-André model for particles hopping on a quasiperiodic lattice with amplitude disorder [152, 153]
XX
H = −t(|jihj + 1| + |j + 1ihj|) + ∆cos(2πβj)|jihj|.
(3.3)
j

j

A system of noninteracting particles in the tight binding limit of a one-dimensional
quasiperiodic potential maps onto this model with [154, 155]√−2.07 s1t ≈ 1.43s0.98
1 e

and

∆≈

s2 β 2.
2(3.4)

The Aubry-André model shows an Anderson-like transition from extended
to exponentially-localized states for ∆/t = 2. Unlike Anderson localization,
which occurs for all states and any nonzero amount of disorder (in one dimension), the Aubry-André model exhibits a finite mobility edge Emob for
sufficiently small ∆. States with energy larger than Emob are delocalized,
while the lowest-lying states experience Anderson-like localization. While thisholds for the infinite system with truly irrational β, for rational β the conceptof the degree of incommensurability (measured by the ratio of the system size
L and the superlattice period Λ) must be considered [155]. For Λ/L . 1,
the true periodicity of the lattice is not sampled by the system and we canconsider the model to be a good approximation for a disordered system. In

30this limit, an effectively disordered energy landscape with no long range orderis seen by the atoms. The relative ease of implementation and the fact thatthe essential physics can be captured in the easily calculated spectrum of thepotential make the incommensurate bichromatic lattice an attractive system
in which to study (quasi-∗ )disorder.

3.4

Experiment

In our experiment we load a BEC of a few 104 87 Rb atoms in the |F, mF i ≡|2, −2i hyperfine state into a crossed optical dipole trap (XODT) [89]. A
magnetic field gradient levitates the BEC in the trap, before a one-dimensional
optical lattice
V (z) = s1 Er cos2 (k1 z) + s2 Er cos2 (k2 z)(3.5)
is adiabatically ramped up along the vertical direction ẑ. Here kj = 2π/λj ,and s1 (s2 ) are the lattice depths in units of the main lattice recoil energy Er =(~k1 )2 /(2m), ~ = h/(2π) is Planck’s reduced constant and λ1 = 1064 nm andλ2 = 785.2 nm are the wavelengths of the lattice. The ratio β = k2 /k1 ≈ 1.3551
is used as a measure of the lattice commensurability; for nearly irrational β,
the lattices will be periodic on length scales exceeding those relevant to theexperiment. Although this true periodicity exceeds the system size, it should
be noted that the bichromatic potential has a quasiperiod d/(β − 1) ∼ 2.8d,with d = λ1 /2 the main lattice spacing. The quasiperiod describes the lengthscale on which the potential approximately repeats itself and is much shorter
than the real period Λ. As our system spans only a few of these quasiperiods
and less than a single Λ, for relevant time and length scales the bichromatic
potential can be considered a good approximation for disorder. The main
lattice has a depth of s1 = 3 for all measurements and the second lattice depth
is varied from s2 = 0 to s2 ≈ 1. We define the disorder strength ∆ as theaverage energy shift of a lattice site due to the secondary lattice, ∆ ≈ s2 /2,
which also corresponds to the energy scale Egap ≈ s2 /2Er of perturbations tothe band structure.
The chemical potential of the condensate is adjusted in a range µ̃ = µ/Er =0.1...0.5 prior to loading the lattice by varying the atom number as well as thetrapping frequency of the XODT from ∼ 17 Hz to ∼ 80 Hz, corresponding
to condensates with radii from ∼ 8d to ∼ 12d. To induce BOs the levitating∗

In previous work [54], we did not observe a shift in the transition to an insulatingstate as a function of U/t, for the case of an incommensurate lattice. On the other hand, ashift in the superfluid-insulator transition is seen for random atomic disorder. The different
behavior in each case can be related back to the fact that disorder correlations do not decay
over short ranges for the incommensurate lattice, unlike in the case of atomic disorder.

31

(a)

(b)

0

(d)

<p> [ħk1]0.33

e-(ηt)

1.2

0

2.4 t [ms]

η [T B -1] μ/E R = 0.42
0.07
0.75

2

0.50

0

-0.33

0
2

-0.33

(c)

-2

p [ħk 1]

0.33

0.25
0
0

1

2
t [T B ]

0

3

0.2

s2

0.4

0.6

Figure 3.3: Bloch oscillations of weakly interacting atoms in an optical lattice. (a) Absorption images for different evolution times in a single-wavelength
lattice. (b,c) Observed average momentum for a weakly interacting cloud(µ̃ = 0.07) for s2 = 0 and 0.46 respectively. The red lines are fits to the data
while the dashed gray lines trace the Gaussian envelope with decay rate η. The
extracted damping rates in b) and c) are 0.02 TB −1 and 0.55 TB −1 respectively.
(d) Damping rate as a function of secondary lattice depth for µ̃ = 0.07 (blue,
dashed, circles) and for µ̃ = 0.42 (red, solid, triangles). The lines are linear
fits to guide the eye. In (b-d), each data point represents three repetitions of
the experiment, error bars represent standard deviations.
gradient is rapidly switched off in < 200 µs, and the atoms are allowed to
evolve in the lattice plus XODT potentials for a variable hold time. During
this time, the atoms undergo BOs with period TB = h/F d ≈ 0.9 ms whereF = mg is the gravitational force; for our system F d = 0.56 Er . The atoms
are then suddenly released and the atomic momentum distribution is imaged
on a CCD camera after 18 ms time of flight, as shown in Fig. 3.3a.
After repeating the experiment for different hold times, the oscillations arerecorded as seen in Fig. 3.3b,c. A damping rate η is extracted from the databy fitting the mean of the linear momentum with2

hpz (t)i = e−(ηt)

dE(q)
.
dq

(3.6)

Here E(q) is the energy dispersion of the lowest energy band of the lattice as afunction of quasimomentum q which evolves as ~q = F t. The Gaussian envelope is chosen because at short times the pz (t) ≈ (1−t2 ) behavior best matches32dephasing due to two beating frequencies [146]. This also matches previously
observed and simulated BO damping behavior due to both interactions andincommensurate lattices [144, 146], while a pure exponential envelope has beensuggested in the disorder-free case [142].
In the weakly interacting (µ̃ = 0.07) and disorder-free case (Fig. 3.3b) weobserve only weakly damped oscillations, with damping due to the nonzero
interaction energy and the harmonic confinement of the XODT. When adding
the second lattice with s2 = 0.46 the damping time is markedly reduced tojust a few TB , as seen in Fig. 3.3c. As illustrated in Fig. 3.3d, increasing thesecondary lattice depth increases the damping rate for a weakly interacting
cloud; however, as shown for two interactions µ̃ = 0.07 and 0.42, the effect ofthe disorder is weaker when stronger interactions are present. Furthermore,
the effect of interactions depends on the depth of the disorder potential: forsmall s2 < 0.35, the damping rate is higher for low µ̃ while the opposite is true
for s2 > 0.35.

3.5

Microscopic model of damping

To better understand the impact of interactions in the disordered lattice, weinvestigate the damping in the noninteracting and interacting regimes separately. In the noninteracting case, damping can be described by considering
the single particle band structure of the lattice potential. Starting with all
the atoms at q = 0 in the lowest band, for s2 = 0 the atoms experience a
continuous, periodic dispersion relation E(q) giving rise to long-lived Blochoscillations described by Eq. 3.6. The chief effect of the second lattice, seen in
Fig. 3.4a, is to open minigaps of energy ∆gap in the band structure at k2 and|k1 − k2 | [151]. The minigaps serve to disrupt the evolution of the momentum
distribution by splitting the condensate among the minibands of the latticeas described by Landau-Zener tunneling [109], as illustrated in Fig. 3.4b. The
minigaps appear at values of q that are irrational fractions of the original Brillouin zone width 2~k1 . As a result of the incommensurability, the estimatedrephasing time for a single particle is much longer than the oscillation lifetimein our experiment.
The fraction of atoms which tunnel to those that pass the minigap adiabatically is given approximately, in the Landau-Zener formalism, by−2π∆2gap i.
P = exp
F (dE/dq|gap )
h

(3.7)

where dE/dq|gap is the slope of the unperturbed band at the gap. In absorption

33

(a)

0.4

-2ħk2

0.3

(c)

(d)

E [ER]

-2ħk1

0.2
Egap

(b)

p

0.1
0
-1.0

-0.5

0.0

0.5

1.0

0.40

2ħ|k1-k2|
2ħk1

0.36
0.32

0

2ħk2
0.56

0.60

0.64 0.68
q [ħk1]

0.72

0.76

Figure 3.4: Damping due to the band structure of the bichromatic lattice. (a)Band structure of the lowest energy band of the two-color lattice calculated
in the tight-binding limit for s1 = 3 and s2 = 0.1. Our irrational β = λ1 /λ2 isapproximated by β = 1.36, which is still nonperiodic on experimental lengthscales. The bold red curve outlines the perturbed lowest band of the s2 = 0
lattice while the gray lines represent minibands of the bichromatic potential.
The primary effect of the second lattice is to open minigaps in the lowest
band. (b) Schematic illustration of a Landau-Zener tunneling event at a mini
gap. (c,d) Momentum distributions of atoms undergoing oscillations in a band
structure described in a) after 0.65 TB and 2.4 TB respectively. The red lines
represent a relative momentum of 2~|k1 − k2 |.
images of the momentum distribution, the splitting among mini-bands of thefirst Brillouin zone is clearly seen after evolving for just TB /4 (i.e. passing
through just one half of the Brillouin zone), Fig. 3.4c. Unlike in Fig. 3.3a,
where a single color lattice gives peaks separated by 2~k1 , we observe an extra
peak at a momentum separation of 2~|k1 − k2 |. For longer evolution times
as in Fig. 3.4d, the condensate breaks up into a series of peaks with spacing2~|k1 − k2 |, spanning a range of momenta between ±2~k1 consisting of thefirst Brillouin zone. This spreading of the momentum wave packet leads toa breakdown of collective oscillation and accelerated damping as described inFig. 3.3c.
To demonstrate the BO damping effect of the bichromatic lattice, we numerically propagate a wave packet through the bichromatic dispersion relation.
For simplicity, the calculation does not include coherent effects involved when
the wave packet crosses a minigap, resulting in a purely probabilistic treatmentof the Landau-Zener process. At times when a wave packet reaches a minigap,
34

it is instantaneously split according to the adiabatic crossing probability given
by Eq. 3.7. The resulting calculation reproduces the damped BO behavior
seen in the experiment and demonstrates spreading of the momentum wave
packet in discrete peaks.
Mean-field interactions in the BEC modify the damping by changing theminiband shape near the minigap, causing the minibands to develop a swallowtail structure [114]. The multivalued (multivalued) energy spectrum can becalculated from a simple model [113]
!
∆gap
∆E(t)
cEr
2
2
+
(|u|
−
|v|
)
2
2
2
,
(3.8)
HNLZ =
∆gap
∆E(t)
cEr
−
−
(|u|2 − |v|2 )222
where ∆E(t) gives the energy splitting between upper and lower branches of
the avoided crossing for an atom moving in the band structure as a functionof time, c = µ/4Er parameterizes an interaction induced detuning, and |u|2
and |v|2 are the populations in the upper and lower branches respectively. The
eigenvalues of this system are shown in Fig. 3.5a for cEr ∼ ∆gap . The shapeof the avoided crossing is effectively changed, with the lower band narrowing
near the avoided crossing and becoming multivalued. This change leads to thebreakdown of adiabaticity in the strong-interaction limit µ & ∆gap , resulting
in an expected increase in the fraction P of atoms that tunnel through the gap
and remain on the perturbed original band.
The probability to tunnel from one band to the next can be calculated fromthe time evolution of HNLZ . Indeed, the nonadiabatic transition rate increases
with increased µ. Including these modified tunneling rates in our numericalcalculation allows us to estimate the contribution of the nonlinear LandauZener process to the damping rates. The tunneling results for the modified
gap, shown for s2 = 0.4 in Fig. 3.5b, predict a reduction of the damping rate as
the interactions are increased. Our numerical calculation effectively treats thenonlinearities as rescaling the gap size in Eq. 3.7, according to the interactionenergy. The effective ∆gap is as much as 50% smaller for µ ≈ ∆gap , predicting
as much as a 1/3 reduction of the damping rate.
We note that this simple calculation does not separately address the damping induced by the interactions themselves, which perturb the evenly-spaced
Wannier-Stark energy levels of the tilted(forced)-lattice, and give rise to irreversible damping for our lattice parameters in which the energy offset between
lattice sites is comparable to the tunneling matrix element [142, 144].
The predicted decrease of the minigap-induced damping due to interactionsin Fig. 3.5b agrees with our experimental data to within a factor of 3. The discrepancy could result from a number of factors. For example, our calculation

35

(b)

1.0

1.5

0.5

η[TB-1 ]E [∆gap]

(a)

0.0

0.5

-0.5
-2

1.0

-1

0
1
q [arb.]

0.0

2

0.0

0.5
~
μ

1.0

Figure 3.5: Effect of nonlinearities on the structure of the minigaps. (a)Dashed red lines describe the unperturbed energy bands in the vicinity ofa minigap, while the solid blue lines describe the band shape for the caseof µ ∼ ∆gap . Note the “sharpening” or narrowing of the lower band andthe appearance of a loop at the quasimomentum corresponding to the exact
avoided crossing. The damping rate due to the minigaps for s2 =0.4 is plottedin (b) for a range of µ̃ as calculated using a nonlinear Landau-Zener formalism(see text)
does not take into account the three dimensional nature of the experimental setup nor does it include coherent effects involved in the (non)adiabatic
crossing of a minigap, which were excluded to minimize computational timeand complexity for propagation in the quasiperiodic potential. The assumption that the tunneling through a minigap happens exactly at the gap is also
unrealistic as the momentum width of the condensate is comparable to thequasimomentum spacing between the gaps and one would expect coherent dynamics beyond the probabilistic, isolated gap treatment used here to alter thedetails of the dynamics, but not the qualitative effect observed.
Experimentally, we have selected our parameters to best probe the relationship between interaction-induced damping and Landau-Zener damping
due to the potential’s quasi-disorder. Fig. 3.6a plots the measured damping
rate with varied chemical potential for different disorder depths, as described inFig. 3.3b,c. In the s2 = 0 case we observe, as expected from earlier related work[67], a steady increase in the interaction-induced damping. For s2 > 0, theinitial damping in the weakly interacting case is significantly enhanced. However, for increasing interactions, within an experimentally accessible range,
we subsequently observe a decrease in the damping rate, only to be followed
later by a small increase. A heuristic parabolic fit yields a chemical potentialµ̃co = 0.29 for minimum damping.
The observed behavior is consistent with a simple, qualitative picturein which the interactions initially screen out the disorder, thus allowing for
longer lived oscillations. Above µ̃co , the damping driven by interactions out36

(a)

(b)

0.75

0.75

s 2= 0.56
0.00

0.50

0.25

0.25

0

0

η [T B -1]

0.50

0

0.1

0.2
~
μ

0.3

0.4

0.1

0.2

~
μ

0.3

0.4

Figure 3.6: Damping rates of BOs as a function of chemical potential. (a)Measured damping rate as a function of µ̃ for different s2 . For the s2 = 0
case (red, solid, triangles), the damping rate increases roughly linearly withµ̃. For a larger disorder s2 = 0.56 (blue, dashed, circles) a clear decrease in
the damping rate is observed as the µ̃ is increased. Lines are fits to guide theeye. From a parabolic fit to the s2 = 0.56 data a minimum at µ̃c = 0.29 is
obtained. In b) we plot the same data obtained from a 3DGPE simulation. A
fit finds a minimum damping rate at µ̃c = 0.26.
weighs the screening and the damping rate begins to increase. We note thatµ̃co ≈ ∆ = 0.28, indicating that the crossover from disorder-dominated tomean-field-dominated behavior occurs when the interacton strength becomescomparable to the disorder. This is consistent with findings for the equilibrium case [62], where a weakly-interacting condensate with µ̃ < ∆ tends tobecome fragmented, corresponding to a breakdown of global coherence acrossthe cloud, and where the fragmentation becomes less severe leading to increased phase coherence as the potential is “filled” by the interaction energy.
Above µ̃co , the amount of fragmentation is limited with only weak dependenceµ̃ and mean-field effects take over as the disorder is of minimal significance tothe cloud’s coherence.
To confirm our experimental finding that µco ' ∆, we perform a three
dimensional Gross-Pitaevskii equation (3DGPE) simulation with experimental parameters as inputs (see Fig. 3.6b) [156]. Data from the simulation isanalyzed in a manner identical to the experimental data and a very similar
behavior of the damping rate is observed, with a minimum at µ̃co = 0.26
in good agreement with our experimental findings. We note that the factorof ∼ 1.3 absolute discrepancy between the experimental and simulated rates
37

could result from the fact that the full 3DGPE does not account for depletionof the condensate.

3.6

Conclusions

In summary, we have demonstrated that interactions and disorder can compete
to determine the transport properties of a lattice-trapped superfluid. The competition results in the decrease of the disorder-induced damping rate of BOs.
Our findings for Bloch oscillations are also consistent with other recent transport measurements in disordered potentials, including dipole oscillations in aquasiperiodic potential [50] and in a 2D speckle potential [49]. The observed
effect saturates at a minimum damping rate when the interaction energy becomes comparable to the characteristic energy scale of the disorder.
The interaction-induced delocalization effects observed here are also relevant for the description of systems ranging from superfluid helium in porous
materials to superconductivity in granular or disordered materials [68, 69, 127,128]. Here the competition between disorder and interactions ultimately contrasts with cooperative behavior in the regime of stronger interactions, which
can give rise to strongly localized, disordered phases such as the Bose glass[55, 60]. Our findings in the crossover between the two regimes give an exampleof how subtle the parameter dependence can be in a given system.

38

Chapter 4
Nonadiabatic Diffraction of
Matter Waves
While diffraction phenomena usually can be formulated in terms of a potentialthat induces the redistribution of a wave’s momentum, we demonstrate in thischapter a novel regime of matter wave diffraction from an optical lattice inwhich the diffractive dynamics cannot be captured by the action of an optical potential, but in which the adiabatic dressed states are strongly mixed.
We show how, in the adiabatic limit (characterized by strong coupling), theobserved coupling between internal and external dynamics gives way to standard Kapitza-Dirac diffraction. We demonstrate the utility of our scheme foratom interferometry and discuss prospects for studies of dissipative superfluid
phenomena.
The contents of this chapter are based on the preprint Nonadiabatic Diffraction of Matter Waves, arXiv 1505.06085, accepted for publication in PhysicalReview A (2015) [157].

4.1

Introduction

Diffraction, the bending of waves around obstacles, is one of the most fundamental and ubiquitous phenomena in optics, with a centuries-old history
going back to the works of Grimaldi, Huygens, and Young on the wave nature of light. In the modern era, it has led to the understanding of x-rays[158, 159], has provided direct proof for the wave nature of particles [160],
and today finds many applications in physics and materials science, ranging
from electron, x-ray and neutron diffraction, to applications in atom optics[70, 161].
Quite generally, diffraction is caused by a position-dependent potential

39

with absorptive (imaginary) and/or dispersive (real) character. While the former includes material structures and diffraction gratings, an example for thelatter is the ponderomotive potential exerted on electrons by an optical standing wave, as originally suggested by Kapitza and Dirac [162]. Kapitza-Diracdiffraction (KDD) of electrons was first demonstrated only fairly recently [163],
long after the first observation of an analogous effect on neutral atoms basedon the ac Stark shift near an atomic resonance [71]. It has also been applied
to Bose-Einstein condensates (BEC) [5, 7] and has become an important andoften-used tool in atom interferometry and metrology [70].
Here, we demonstrate a regime of diffraction for which the potential description does not describe the dynamics of the system. In our experiment,
we diffract an atomic matter wave from a microwave-dressed optical lattice,with a diffractive dynamics that is qualitatively different from Kapitza-Diracdiffraction of dressed matter waves from a periodic optical potential. We notethat the coherent mixing of states interacting with an external field is often
used for the engineering of dressed potentials [96, 164–171]. Deviations fromadiabaticity have previously been found to have deleterious effects on dressedstate lifetimes in rf-dressed lattices [172, 173]. In our experiment, we enter
the strongly nonadiabatic regime, in which coherent Landau-Zener transitionsof the atomic wave function between the adiabatic dressed potentials (driven
by zero-point motion) induce a strong coupling between internal and external
degrees-of-freedom, leading to a breakdown of the usual Born-Oppenheimer(adiabatic) approximation [30].

4.2

Kapitza-Dirac diffraction

For reference, we first describe the usual case of ”adiabatic” diffraction. We
focus on the atom optical implementation of diffraction in which an atomic
sample (a trapped cold gas, atomic beam, or degenerate gas) interacts for ashort time τ with a standing wave optical potential. Diffraction of atomic
matter waves can be described in terms of the two-level Hamiltonian Eq. 2.10
with spatially varying Rabi couplings corresponding to the spatially periodic
intensity pattern [161]


~
p2
0
ωR (t) cos(klatt z)
+
.
(4.1)
H=
−2δ
2m 2 ωR (t) cos(klatt z)
To limit the effects of spontaneous emission, atomic diffraction is typically
performed with light far detuned from resonance. This makes it possible tocalculate an adiabatic potential felt by the atomic ground state. We consider

40

only the action of this adiabatic potential on the ground state atoms and write
the Hamiltonianp2~ωR (t)2
H=+
cos2 (klatt z)
(4.2)
2m
4δ
where the potential depth is given by Eq. 2.13. Working in the plane-wave
basis by taking the ansatz
X
22Ψ(z, t) =cj e−i(2j)klatt z × e−i4j ~klatt /(2m)t ,(4.3)
jwith cj is the amplitude for the plane wave state with hpi = 2j~klatt , we canrecover Eq. 2.35
(2)iċj =Er j 2
Vlatt (t)cj +(cj−1 + 2cj + cj+1 )
~
4~

(4.4)

where we have replaced the potential depth Vlatt = ~ωR (t)2 /(4δ) and defined(n)
the n−photon recoil energy Er = n2 Er . To allow for easy solution of Eq. 4.4,
we assume the time dependence of ωR (t) to follow a simple rectangular pulseof duration τ . The solutions of this equation are typically split into the shortpulse (Raman-Nath) regime and the long pulse (Bragg) regime [124]. These
regimes differ in the relative importance of the detuning introduced by the kinetic energy of a diffracted (recoiling) atom. In the long pulse regime, the small
uncertainty of the pulse energy prevents resonant excitation of the higher lying states. This leads to Pendellösung behavior in which atoms are exchanged
between the j = 0 and the j = ±1 diffraction orders. This population oscillation never fully depletes initial j = 0 state without populating at least j ≥ 2orders.
In the short pulse limit, the kinetic detuning is washed out by the spectral
width of the pulse. This allows for higher order scattering events to take place
and the population of many diffracted orders when the potential depth Vlattis large. In this limit, we can ignore the kinetic term in Eq. 4.4 and find thesolutions for the population in each diffraction order Pj in the form [5]22 Vlatt τ,(4.5)
Pj = |cj | = Jj
2~
where Jj (x) are Bessel functions of the first kind. This describes a spreadingof the momentum wave packet due to the short interaction with the opticalstanding wave. For practical purposes, this picture and thus the wave packet
spreading, is limited by the potential strength. For finite lattices and nonzero
τ , the wave packet will undergo revivals.
41

We will explore below just how this picture breaks down in the limit of
weak coupling, even while we are able to satisfy a resonant condition for all
diffraction orders.

4.3

Experiment

Our experimental procedure, as sketched in Fig. 4.1a, uses microwave radiation
to coherently couple an optically-trapped condensate of 87 Rb atoms (chemical
potential µ = h × 1.0(1) kHz in a 60Hz trap at 1064nm [89]) in internal state|bi = |F = 1, mF = −1i to another internal state |ri = |2, −2i (separated from|bi by 6.8 GHz) exposed to a deep (30Er ) state-selective 1D optical lattice(wavelength λ = 2π/k = 792 nm, σ + polarization) not seen by the condensate[99, 174]. We study the transfer of population into the lattice upon applying apulse of microwave radiation, as a function of the applied Rabi frequency Ω up
to h × 17 kHz, the pulse duration τ , and the detuning δ from the bare (i.e. no
lattice) resonance between |bi and |ri. In our experiments, residual magneticfield instabilities limit the shot-to-shot stability of the |bi ↔ |ri transition
to δZ /2π ∼ 0.6 kHz; the typical uncertainty in the determination the Rabi
frequency is ∼ 0.2 kHz.
The presence of the lattice potential Vr (z) = V0 sin2 (kz) modifies the microwave resonance condition, since the atoms in |ri are now confined to latticeorbitals |ni (band index n = 0, 1, 2, ...). In our effectively blue-detuned lattice, the orbital resonances are shifted upward by the zero-quasimomentum (q)band energies ~ωn . We characterize the lattice depth V0 = sEr with s = 30(2)in terms of the recoil energy Er = ~ωr = (~k)2 /2m = h × 3.7 kHz, where m(n)(n)
is the atomic mass (we also define the n-photon recoil energy Er = ~ωr =1(n~k)2 /2m√ used below). In the harmonic limit, ~ωn ≈ (n + 2 )~ωho , with~ωho = 2 sEr = h × 40(1) kHz.
To study the orbital transfer, we first apply a rectangular microwave pulseof fixed duration τ = 60µs and Rabi frequency ω/2π = 8.1kHz (corresponding
to a π-pulse in the bare case), but variable detuning δ. We determine theorbital populations Pn after applying the microwave pulse followed by a bandmap sequence, consisting of a fast (1 ms) ramp down of the lattice depth,
subsequent free expansion (18 ms), Stern-Gerlach separation of |ri and |bi,
and absorption imaging.∗ The spectroscopic location of the transitions, cf.
Fig. 4.1b, is in excellent agreement with the expectation from the lattice calibration, with line shapes that follow theory for a Rabi pulse with a spectrally∗

The band-map sequence prevents residual four-wave mixing during the time-of-flightevolution [174, 175]. We correct for differential detection efficiencies for |ri and |bi by
removing correlations between Nb,r and Nr+b .

42

∆

r
6.8 GHzb

Nr/Nr+b(c) 1.0
0.5
0

0



(b)
140
120
100
80
60
40
20
0

0

δ/2π [kHz](a)0.4 0.8 1.2 1.6
τ [2π/Ω]

3

1

4

2

4

2
0

0

0.5
Nr/Nr+b

1.0

Figure 4.1: Experimental scheme for microwave induced diffraction. (a) Experimental scheme, for resonant coupling (see text). (b) Fraction of atoms
in the 30Er deep lattice after applying a rectangular pulse (coupling strengthΩ/2π = 8.1kHz, duration τ = 60µs) for variable detuning δ. The red solidline is the incoherent sum of predicted line shapes for resonances centered atdetunings given by the q = 0 level structure of the lattice; the pulse areas are
πγn , where γn = 0.72, 0, 0.61, 0, 0.32 for n = 0, 1, 2... are the Franck-Condon
overlaps between BEC and the orbitals |ni. The light gray points and linerefer to a reference measurement without lattice (pulse area π). The inset
shows time-of-flight absorption images after band-mapping, revealing the orbital populations. (c) Population dynamics in orbital |0i for resonant drivingwith Ω/2π = 8.8kHz and δ/2π = 19.2kHz. The curve is a resonant Rabi
oscillation with frequency γ0 Ω. All error bars represent statistical standard
deviations of at least three experimental iterations.
limited FWHM of 14 kHz. However, while the free-space Rabi frequency isgiven by Ω = hr|µ · B|bi/~, it now is modified by the Franck-Condon overlaps
γn = hn|ψ0 i between the orbital Wannier functions |ni and the locally flat,
single-particle BEC wave function |ψ0 i. In particular, parity conservation imposes an orbital selection rule disallowing transfer to the odd-n orbitals. For
the lowest orbital, |0i, on which we will concentrate in the following, Ω iseffectively reduced by γ0 = 0.72.
We next study the coherent dynamics on resonance with the transition to
the |0i orbital. For parameters comparable to the spectroscopic measurement,
Rabi-type oscillations between BEC and orbital population can be observed
for times up to (δZ )−1 ∼ 0.3 ms. The first cycle for resonant (within a systematic uncertainty of 1.1 kHz) coupling is shown in Fig. 4.1c. Comparing
the fitted frequency to Ω yields a reduction by 0.70(1), consistent with thecalculated Franck-Condon factor and the spectroscopic data. We note that43

the oscillation amplitude is slightly reduced below unity, corresponding to a
finite detuning of 1.5(1) kHz in the Rabi model, which is within the systematic
lattice calibration error (and comparable to differential mean-field shifts fromthe density compression in the lattice [176]).
However, the simple Rabi picture breaks down for stronger couplings. We
find that, while the measured oscillation frequency increases with Ω, the amplitude of the orbital population oscillation is strongly suppressed (by up to
30% for the maximum Ω), cf. Fig. 4.2a. At the same time, the population in|bi undergoes high-contrast, Pendellösung-type (albeit not number-conserving)
oscillations between the initial momentum state |p = 0~ki and |p = ±2~ki, cf.
Fig. 4.2b, in phase with the population oscillation. As discussed further below, the coupled internal-external dynamics is the characteristic feature for the
breakdown of the Born-Oppenheimer approximation. To further characterize
the coupling, the fraction of atoms remaining in |0~ki and the fraction of atoms
transferred to |±2~ki, for an effective π pulse with γ0 Ωτ = π, are plotted together in Fig. 4.2c. We find that, as a function of Ω, the diffracted fraction(2)
increases rapidly once the coupling strength becomes comparable to Er , while
the population in |0~ki gets suppressed nearly completely, cf. Fig. 4.2c. We
note that for the smallest couplings Ω . 5kHz a significant fraction of |bi
atoms remain in |0~ki. However, this behavior is explained by the Rabi model
if one allows for a small effective detuning of 1.1 kHz, which is similar to ourexperimental uncertainty for the microwave transition frequency.

4.4

Dressed states and adiabatic approximation

To capture the appearance of additional momentum orders, one might naively
describe the effect of extracting atoms from the condensate as that of an imaginary potential formed by the q = 0 Bloch wave (i.e. the array of “absorbing”
Wannier functions). However, as for the case of a real periodic potential, theobserved complete depletion of the initial state |0~ki in this case would entail significantly populating at least one higher order |±2j~ki (j = 2, 3, ...),
while the observed Pendellösung dynamics involving |0, ±2~ki exists only inthe perturbative limit of a weak optical potential [124]. Our observations arethus qualitatively inconsistent with such a description.
In order to properly describe our system, we first write down the full Hamiltonian in the internal state basis {|ri, |bi} after making a rotating wave ap-

44

Nr/Nr+bγ0Ωτ=π0.8

Nb,0/Nr+b

Fraction(a)0.6

Nb,±2/2Nb

Nb,±2/Nr+b

(c)

0.4

0.5

0.2
Fraction

0
(b) 4
2
0
-2
-4
p [ħk]

0.25

0

1

2
τ [2π/Ω]030.25 0.5 0.75 1.0ħΩ [ Er(2)]Figure 4.2: (a) Resonant orbital transfer to |0i and diffraction for stronger coupling Ω/2π = 16.9kHz. Shown are the transferred fraction (red filled points),
and the diffracted fraction (blue open circles) in the momentum state |±2~ki.
The red and blue shaded regions are the results of a numerical simulation
including experimental uncertainties in δ and Ω. The population in higher
orders (not shown) is less than 2% of the total number. (b) Observed diffraction patterns for |bi atoms. Shown is a sequence of time-of-flight images forvarying pulse duration τ . Each line is a single image summed over the direction perpendicular to the lattice. (c) Total diffraction signal as a function ofΩ for a Rabi pulse with γ0 Ωτ = π. The shaded blue (gray) area shows the
band-structure prediction for atoms in the first diffracted order Nb,±2 /Nr+b(undiffracted fraction Nb,0 /Nr+b ) allowing for effective detunings up to 1.1
kHz (see text).

45

proximation,
p̂2
~
H=
I+
2m
2



2Vr (z)/~ − δ Ω≡ HT + HV ,
Ω
δ

(4.6)

where p̂ = −i~∂z is the canonical momentum operator. Usually, one then diagonalizes HV in order to obtain the dressed states |χi i = αi |bi + βi |ri (i = 1, 2)
and corresponding dressed optical potentials Ṽi , cf. Fig. 4.3a, and the stateof the system evolving under H is then expressed as Ψ = Σj cj |χj i. However,this approach only works if ∂z |χj i  ∂z cj (Born-Oppenheimer approximation)[30]. This is typically satisfied in optical potentials where both Ω and δ canbe very large compared to the energy associated with the spatial variation of
the coupling fields. In this limit, ∂z |χj i ∼ 0 and thus the resulting dressed potentials describe the eigenstates of the system. In our case (cf. Fig. 4.3a), thebare-state energies cross at two positions within each lattice site and for weakcoupling (in our experiment, ~Ω ≤ 0.17V0 ) the dressed-state mixing angle experiences a rapid spatial variation around these positions. Hence, the adiabatic
condition is not fulfilled, and off-diagonal terms of the form Aij ∝ hχi |p̂|χj i
become significant, leading to momentum-dependent mixing of the adiabatic
states. With inseparable kinetic and potential terms in H, the notion of an
optical potential is not meaningful (i.e. the eigenstates associated with the
adiabatic potentials do not describe the system dynamics), and we therefore
diagonalize the full Hamiltonian H, with numerical results shown in Fig. 4.3b.
(2)
We first discuss the case of very weak microwave couplings ~Ω/Er  1(2)(where Er /V0 < 1). Here, the energy levels are split by the two-level ac
Stark shift corresponding to the leading-order matrix element in the coupling. For the energetically lowest states |e0 i and |e1 i, the shifts are given by±0.72~Ω/2, consistent with the Franck-Condon overlap γ0 of the states |b, 0~ki
and |r, n = 0i between which the Rabi oscillation is observed. The next-higher|e± i states experience a shift due to off-resonant coupling of the symmetric√(antisymmetric) momentum-state combinations (|b, 2~ki ± |b, −2~ki)/ 2 to|r, n = 0i (|r, n = 1i). For all Ω of experimental interest, these are the primary couplings describing the dressed levels (note that parity conservation
forbids population of the state |e− i).
(2)
For intermediate couplings ~Ω ∼ Er , the admixture of the symmetricmomentum-state combination to |e0 i and |e1 i induces a downward curvingof their energies and gives them the periodicity of the lattice, cf. Fig. 4.3c.
While at t = 0 the two states add to produce |b, 0~ki, at√ Ωt ≈ π/(γ0 ) theyacquire a π phase shift and add to (|b, 2~ki + |b, −2~ki)/ 2 , which produces
the observed diffraction pattern (with a small contamination by |b, 0~ki dueto the admixture of |e+ i).
46

0

1

2

0

1

2

0.5

0.5
1

1
2

2
5

5
10

10

20

20
50

50

100

100

200
2

200

3

0

(b)

(i)

-5

-5

r

45

(iii)

e0

-1

-1 0 1z [d/2]

0
0

1

2

(c)

1 2 3
ħΩ [Er(2)]

e0

e1

1.0
0.5
0
0.5
0

10 0
10 0
τ [2π/Ω](i)0z [d/2]10
8
Nb=1 6
4
2
0
Nr=1
-2
-4
0
1

5

0

1.5
1 -1
0.5

10

0

5

0
10 15

Figure 4.3: Calculated dynamics from the nonadiabatic regime to the adiabatic
regime. (a) (i-iii) Adiabatic dressed potentials for coupling strengths ~Ω =(2)
{0, 1, 15}Er and s=30.
The
position dependent internal state composition
1
0
is encoded in color. (i) shows the bare potentials, while (iii) shows potentials
in the adiabatic limit characterized by a near-constant dressed-state mixing
angle. For weak coupling (ii), gaps are small and the mixing angle varies
rapidly. (b) Nonadiabatic q = 0 band energies for the four lowest states as afunction of Ω with δ = ~ω0 . The dashed line is a state of opposite parity that isdecoupled from the other three states. (c) Spatial structure of the ground and(2)
first excited states in the bare-state basis, for ~Ω = 1.25Er . The shaded red
and blue areas denote the square of the wave function, and the solid lines thewave function itself. (d) Diffractive dynamics in three regimes characterizedby Ω/V0 = 0.01, 0.05, 15 with V0 = 100Er , and (e) internal state dynamics inthe case Ω/V0 = 15. The red and blue lines trace total population in |ri and |bi
while the dashed lines represent the expected envelopes for the m = 0, 1, 2, 3
(from top to bottom) diffraction orders for standard Kapitza-Dirac diffraction.

47

(iii)

(d,i)

Relative Number

(e)

10 0

40
20
0
20
10
0
-1

0

0

3

-1 0 1z [d/2]

r

4
0
-4

2

(iii)

b

4
0
-4

p [ħk]

e1

0

-30

(d)

e-

1

E [Er(2)]E [Er]

(ii)

e+

2

b

30

(ii) (c)ψr,b(z)(a) 30

1

E [ER]

1

ψr,b

0

0

1 -1
z [d/2]

0

3

(2)

Strong couplings ~Ω  Er lead to the admixture of higher-order momentum(2)state combinations, and since Er ∼ ~ωho (for s ∼ 10) the dynamics also involves higher even orbitals. Taken together, the internal and external dynamics
eventually decouple: the internal dynamics proceeds as a fast oscillation between |bi and |ri (with Rabi frequency √Ω), while in the rotating frame the adiabatic dressed states |χ1,2 i = (|bi±|ri)/ 2 undergo diffraction in the half-depth
optical potentials V1,2 = V (z)/2 ± ~Ω/2, with populations Pn (τ ) = Jn2 [V0 τ /4~]
spread over many orders n as expected for standard Kapitza-Dirac diffraction
Fig. 4.3d and 4.3e. Indeed, for a π-pulse with τ Ω ∼ π, the condition Ω  ωhoreproduces the usual Raman-Nath criterion τ ωho  1 [5]. The predicted transition from weak to strong coupling is illustrated in Fig. 4.3d. We cautionthat using our scheme in this adiabatic limit would require increasing the microwave power by three orders of magnitude, which is outside our technicalcapabilities.
(2)
Our model accurately reproduces all the observed dynamics for ~Ω ≈ 4Er
shown in Fig. 4.2. We note that the narrowing of the diffraction signal in
Fig. 4.2b (as compared to the orbital transfer) is easily explained by consideringthat the signal gets more pronounced with the depletion of the background in|b, 0~ki.

4.5

Satisfying the adiabatic condition

In the previous section, it is noted that we can reproduce typical adiabatic
diffraction patterns in the limit of very strong microwave coupling. The huge
microwave powers needed prevent this from being easily achieved; however,
it is possible to examine the transition to adiabaticity along another pathin parameter space. The adiabatic potentials describe the dynamics of the
Hamiltonian, equation (4.6), as long as the mixing angle does not vary strongly
in space.
p The mixing angle θ(z) for a two-level system is given by cos 2θ(z) =−δ(z)/ Ω + δ(z)2 , where δ(z) includes the spatial detuning introduced by
the state dependent potential. It can clearly be seen that for large Ω (hδ(z)iz ),θ(z) → π/4 (→ 0). In either case, the spatial dependence of θ becomes
negligible.
Though we do not access the large Ω limit, we can observe a transition toadiabaticity by increasing the overall detuning such that δ becomes comparable to V0 . The dynamics of the system, c.f. Fig. 4.4, transition from being
described by the nonadiabatic spectrum described in Fig. 4.3, to being described by the generalized
Rabi model with population dynamics described by√2
2
frequency Ω̃ = Ω + δ . In an intermediate regime, with δ ∼ 4Er , the twotime scales can readily be seen in the dynamics of the system, c.f. Fig. 4.4a.

48

Nj/Nb

0.0

(c)

(b)

0.8
0.6
0.4
0.2
0.0
0.5

30

E [Er]

Nr/Nr+b(a) 1.0

j=0
0

50

j=1

100 150 200t [µs]-5
0

50

100 150 200t [µs]-10

z [d/2]

1

Figure 4.4: Transition to adiabatic dynamics in large detuning limit. (a)(2)
Dynamics for Ω/2π = 17 kHz, δ ∼ Er , two separate timescales can easily
be discerned. These correspond to the dressed potential splitting and the(2)timescale set by the two-photon recoil. (b) For larger detuning δ ∼ 3Er ,Rabi oscillations at the generalized Rabi frequency anticipated by the twolevel Rabi model are recovered. In (a) and (b) the upper portions show the
population oscillation of |ri in red, the lower sections show Nb,0~k in blue andNb,±2~k in green. (c) The dressed states in this limit are split by δ and featurevery weak mixing of the bare states. As seen in the color code, the mixing
angle is largely uniform within one lattice site.
For larger detunings, e.g. δ ∼ 13Er in Fig. 4.4b, we recapture the typicalRabi dynamics with oscillation frequency consistent with the generalized Rabi
frequency.
Because the population transfer in the limit is very weak, we observe nosignificant diffraction of the bath. This is expected if the adiabatic potentials Fig. 4.4c are considered. Here, the dressed potentials do not exhibit
strong mixing of |ri and |bi and the dressed potential for the mostly |bi stateremains flat. Previous experiments have adiabatically prepared the dressedstates studied here in the strong coupling limit and noted the formation of
periodic structures in the atomic species not confined to the lattice [170, 173].
On the other hand, we study these states in the time domain and demonstratecoherent dynamics in the dressed state picture.

4.6

Application to interferometry

As a simple application of our orbital transfer scheme going beyond merediffraction, we implement an in-situ detection scheme for matter wave interferometry, in which the initially unpopulated lattice sites act as an array ofprobes for the condensate, cf. Fig. 4.5a. The orbital transfer can be viewed
as (microwave-induced) local tunneling of atoms into individual lattice sites.

49

(b)

(a)
p

Ωp

r

Ωr
b

Tdelay0.5

0.25

Nr/Nr+b

0.5

0

0
0

0.2 0.5
0.9
Tdelay [ms]1.7

Figure 4.5: In-situ probing of matter wave interference. (a) A pump pulse(γ0 Ωτ = 2π/3, τ = 40µs)) creates excitations in the BEC by removing atoms
into a second trapped state |pi ≡ |2, −1i, and is followed by free evolution of
the BEC density profile for time Tdelay before a probe pulse (γ0 Ωτ = 2π/5,τ = 14µs) transferring atoms into the trapped state |ri. (b) Probe pulse yield
as a function of delay time, displaying oscillations at the recoil frequency. The
solid line is an oscillatory fit with frequency ω/2π = 15.0 kHz, and damping
time τ̃ = 0.66ms, corresponding to the beat note and characteristic time forthe wave packets separating after the probe pulse. The insets illustrate theevolving wave-packet interference after applicationΩ of the pump pulse.
rWhen the transferred atoms are detected, the operation is thus reminiscent ofTdelaythat of a tunneling electron microscope. We first imprint a modulation
into
the BEC (“pump”), again using orbital transfer. To keep this step separate
from the subsequent detection (“probe”), a third internal state |pi ≡ |2, −1iwith a lattice potential Vp = (2/3)Vr is used, precluding any coherence effects in detection. Figure 4.5b shows the resulting probe signal Nr as a function of the time delay Tdelay between the two pulses. We observe an oscilla(2)tion in Nr , consistent with an on-site beat note at ωr , generated by matterwave interference between recoiling |b, ±2~ki and stationary |b, 0~ki packets.
The decaying envelope reflects a gradual loss of spatial overlap between the
wave packets. Compared to Kapitza-Dirac-based matter wave interferometry
schemes [177, 178], our 2-pulse sequence allows for a direct, in-situ detectionof the matter wave interference contrast, without the need for an additional
probe laser.

4.7

Conclusion and outlook

In the present chapter, we have studied the dynamics of a BEC coupled tothe orbitals of an array of lattice sites. We point out that we do not observe effects of the collisional interaction µ (such as a modification of the
50

0

0.2 0.4 0.6 0.8

ground-state wave functions [179]), as it is on the order of fluctuations of theorbital-resonance frequency. In the interacting regime ~Ω < µ, the healing
length ξ = ~/(2mµ)−1/2 is compatible with phonon excitation in the BEC,
at a wavelength λs /(2π) = (µ/m)1/2 /Ω = (µ/~Ω)ξ > ξ. The coupling of
the single-site dynamics to phonons should allow for the study of dissipative
phenomena [63], including dissipative quantum phase transitions driven byphonon-mediated coupling between sites [74]. The physics of these systems
and methods for engineering them will be discussed in Chapter 5. Entering
this regime in our setup will require a reduction of the ambient-magnetic field
fluctuations down to . 100µG, which is reachable with present-day technology.
In summary, we have demonstrated a novel regime for the diffraction of
weakly-dressed matter-waves from an optical lattice. In this regime, nonadiabatic transitions induce a strong coupling between the internal and external
degrees-of-freedom, such that a description of the diffracting object througha potential is not possible. In the limit of strong microwave coupling, thedecoupling of the internal and external dynamics results in the well known
Kapitza-Dirac diffraction from an optical potential. While the coherent internal dynamics give rise to particle-like excitations in the regime explored inthis work, an extension into the regime of even weaker microwave coupling, inwhich the superfluid excitations become sound-like, should prove useful in the
study of dissipative phenomena.

51

Chapter 5
Towards the Realization of a
Dissipative Quantum ModelIn this chapter, we consider a proposed implementation of the so called spinboson model using an ultracold atomic system. The spin-boson model describes the controlled coupling of a two-level system to a dissipative bath andallows for the study of decoherence and dissipation in open quantum systems. While a bath can in principle be easily realized using an ultracold or
quantum degenerate atomic gas, the implementation of an effective two-levelsystem coupling to it is more challenging. Since differential interactions between hyperfine states for ultracold atoms can be weak,∗ this endeavor may
involve realizing effective spins, such as mapping the occupation number ofa lattice site or the use of motional states of a lattice potential (similar to
ion trap qubits) [34, 180]. We further note that the manipulation of the motional degrees of freedom of a lattice-trapped gas has led to a number of othereffective spin models and has been useful in engineering novel Hamiltonians[25, 27, 29, 31, 34–36, 181, 182]. This chapter will focus on experimental
techniques that may be used to implement localized, driven spin impurities
coupled to a superfluid. We highlight three possibilities for implementing suchan effective spin in our experiment, and preliminary experimental results for,and the prospects of, each option are presented.

5.1

The model

Macroscopic objects rarely exhibit quantum effects due to their many degrees
of freedom and coupling to the environment. In defining a system interacting∗

Use of a Feshbach resonance or species with large differential scattering lengths canovercome this limitation.

52

with an environment, one starts by selecting the degrees of freedom of importance to the system and integrating out the remaining degrees of freedom
pertaining to the environment. We consider here the two-level spin system andits interaction with a superfluid, c.f. Fig. 5.1. While dynamics of the spin in
the superfluid will be of concern, the dynamics of the superfluid itself will beignored. We start by considering a bath of bosonic harmonic oscillator modes.
This so-called spin-boson model can be written down as [75, 183]~
1 X p2α~∆2 2σx − σz ++ mα ωα xα − σz cα xα .
(5.1)
HSB = −
22
2 α mαHere, the first two terms describe the two-level system (much like Eq. 2.10)
with tunneling (coupling) ∆ and bias (detuning) ; σx,y,z are the Pauli matrices. The sum is over bosonic modes, with [xα , pα ] = i~, indexed by α.
Within the sum, the bosons each have kinetic and potential energies, andthe third term describes the interaction of the spin with each bosonic mode.
This interaction
P can be thought of as a fluctuating polarization energy or bias
E(t) = 1/2 α cα xα (t) with coupling strength cα (for details see [75, 183]).
We define the familiar bosonic annihilation and creation operators; b̂α , b̂†α ,with [b̂α , b̂†α0 ] = δα,α0 , usingrr~~mα ωα †(b̂α + b̂†α )andpα = i(b̂α − b̂α )
(5.2)
xα =
2mα ωα2This leads to a reformulation of Eq. 5.1 as
HSB = −

X
~∆
~
1 X
σx − σz − σz~λα (b̂α + b̂†α ) +~ωα b̂†α b̂α
2
2
2
α
α

where bosonic mode α is coupled to the spin with strength
s
c2α
.
λα =
2~mα ωα

(5.3)

(5.4)

The third term encapsulates coupling between bosons and the spin state, while
the fourth term gives the total energy of the bosonic field (zero-point energy
disregarded).
The details of the coupling are given by the bath’s spectral density
X
J(ω) =λα δ(ω − ωα ).
(5.5)
α

53

(a)

(b)

Va

Uaa
ψb

∆

Figure 5.1: Atomic occupational spins in a bath. (a) A strongly localizing
state dependent optical trap Va is coupled to a bath, ψb , via microwave or
Raman transitions with Rabi frequency ∆.(b) Interactions between localized
atoms Uaa prevent multiple occupations of the site.
Let us consider how this system can be engineered in our ultracold atomic
system and look at the implications of a bath of phonon modes in a superfluid.
To this end, we consider detailed proposals from [63, 184]. A schematic of
the proposed system is sketched in Fig. 5.1. A strongly confined atom in astate-selective potential well is immersed in a degenerate Bose gas. External
coupling (derived from Raman lasers or microwaves as in Chapter 4) drives
internal state transitions (with Rabi frequency ωr ) between the trapped statelabeled |ai and untrapped states, i.e. the Bose gas labeled with |bi. The
localized atoms interact strongly with Hubbard Uaa , thus multiple occupationof the site is energetically forbidden due to collisional blockade. The interaction
between the localized atom |ai and the bath |bi can then be described by the
Hamiltonian [63, 184]
Z
2
Ha + Hab = − ~δ0 + gab dx|ψa (x)| ρ̂b (x) â† âZUaa † †â â ââ + dx~ωr (Ψ̂b (x)ψa (x)â† + h.c.),
(5.6)
+
2
where we have the annihilation (creation) operators for the localized atomâ(†) and ρ̂b (x) is the density operator for the bath atoms with bosonic field
operators Ψ̂b (x). The first term includes the detuning δ0 and collisional interactions with interspecies gab . The second term describes a collisional blockade
for atoms in ground state wave function ψa (x). The last term describes theinduced coupling between a and b atoms.
The Bose gas is assumed to be degenerate, described hydrodynamically by54

the Hamiltonian
1
Hb =
2

Z

 2

~
2
dx
ρb |∇φ̂| (x) + gbb Π̂(x) ,
m

(5.7)

with density ρb and intraspecies interaction gbb . Here, we have introduced thephonon operators in terms of the canonical pair of phase φ̂(x) and fluctuation
Π̂(x) operatorsφ̂(x) = iXq

Π̂(x) =

Xq

mvs
2~qV ρb~qρb
2vs V m1/2

eiq·x (b̂q − b̂†−q )(5.8)

eiq·x (b̂q + b̂†−q )

(5.9)

1/2

with phonon wave vector q, speed of sound vs , and volume of the condensate
V.
Working in the perturbative limit such that ρb (x) ≈ ρb (x) + Π̂(x) andassuming long wavelength excitations (small q such that Π̂(x) ≈ Π̂(0)) under
the condition Uaa /~  δ0 , ωR it is possible to map Eqs. 5.6, and 5.7 onto thespin-boson Hamiltonian Eq. 5.1 [63, 184], by setting ωr = ∆ and  = δ0 . The
sum is now over phonon modes indexed by wave vector q. Given the contact
interaction between the localized spin and the superfluid, we find the phonon
coupling parameter

1/2 
gab
gbb ωq−1 .
(5.10)
λq =2~Vgbb
With phonon coupling of this form, we find the spectral density is given byJ(ω) ∝ (gab /gbb − 1)2 ω D for a D-dimensional bath. Here we immediately see
the importance of differential interactions in the prefactor to ω D .
When D > 1, the system realizes “super-Ohmic” dissipation, which leads todamped oscillations [63]; such dissipation is analogous to spontaneous emissionof an excited atom in free space.
For D = 1, corresponding to “Ohmic” dissipation, the physics exhibits
nontrivial dynamics. In one dimension, the system undergoes a dissipative
quantum phase transition for α ≈ (gab /gbb − 1)2 = 1. For α < 1, the spin
exhibits damped Rabi oscillations, but for α > 1, the spin can become frozen
in the “spin-up” (occupied trap) state, as is discussed in detail in [75].
The spin-boson model in atomic gases has been generalized to extended
spin arrays coupled to a shared bath [74]. This model exhibits similar orderdisorder quantum phase transitions and is naturally applicable to our statedependent lattice.
55

5.2

Options for implementation

For the model above, we discuss possible generalizations of the two-level spinin order to identify an optimal system which allows for tunable interactions
and coupling strengths. We focus here on using the internal state degrees of
freedom and lattice degrees of freedom to create pseudospin models.

5.2.1

Occupational spin

Let us first consider the spin described in the previous section. Chapter 4
demonstrates the relevant theory and experimental technique behind driving
transitions between the bath |bi and localized states |ai. To successfully realize this model, we must work in the collisional blockade where Uaa is thelargest energy scale in the system. This will require a very strongly confining
state dependent trap (such as a lattice site in our state dependent optical lattice). For Uaa /~  δ0 , ωR , the Rabi driving leads to the transfer of maximallya single atom per site, due to the large detuning associated with the multiply
occupied potential well [185–187]. This collisional blockade allows us to associate the Fock states na = 1 with |↑i and na = 0 with |↓i and thus create an“occupational spin.”
We also note that the phonon coupling strength scales with α ∼ (gab /gbb −1)2 ; to maximize the coupling effects, an interspecies Feshbach resonance isrequired. For any pair of 87 Rb hyperfine ground states, typical differential
interactions are gab = (1 ± 0.03)gbb [188], limiting the coupling strength. Fortunately, there exist low field (∼ 9 G) Feshbach resonances in 87 Rb thatshould allow us to adjust the elastic interspecies scattering length up to 30%[188], allowing for stronger coupling to the phonon bath. Associating |↑i ≡|F = 2, mF = 0i and |↓i ≡ |F = 1, mF = 1i would allow for occupational spins
to be realized using the microwave coupling described above and the scheme
realized here.

5.2.2

Internal states

A second natural candidate for a spin in any atomic system utilizes the internal states of the atoms themselves. For localized spins, this setup would
require the use of atomic clock states in the 87 Rb ground state which experience nearly identical state-dependent lattice potentials. The spins are Rabi
coupled by two-photon transitions, c.f. Fig. 5.2. The localized spins wouldinteract collisionally with the superfluid consisting of a third hyperfine groundstate.

56

(a)

(b)

mF

− 2 −1 0 +1 + 2

F=2

Va
ψb

ωµw
ω
∆2γ rf
F=1

Figure 5.2: The internal state spin. A strongly localizing state dependent
optical trap Va traps an atom driven on a clock transition between hyperfine
states. The transitions is driven by two field with radio and microwave frequencies ωrf and ωµw decoupled from the intermediate state by two photon
detuning ∆2γ . The atom |ai is immersed in a superfluid bath of |bi atoms
with wave function ψb , with which the trapped atomic states have differential
interactions with the superfluid atoms.
This scheme for spins can be realized by defining the spin to consist of|↓i ≡ |1, 1i and |↑i ≡ |2, −1i while the background environment would consistof weakly trapped gas of |2, −2i ≡ |bgi. To achieve differential interaction
strengths with the background, we can again turn to a Feshbach resonance[188]. A Feshbach resonance between |bi and |↓i establishes a tunable differential interaction strength that, in principle, allows even for a decoupling of
the spin from the bath.

5.2.3

Vibrational levels

A second set of discrete levels that can be selectively addressed consists of thequantized motional levels of the lattice, see Fig. 5.3. An effective spin can becreated by applying a drive, such as lattice amplitude or phase modulation,that couples vibrational degrees of freedom of the potential. Alternatively,
Bragg or Raman lasers can be used to drive transitions between internal statesand vibrational levels of the lattice [182, 189].
Bands of opposite parity can be addressed by “shaking” the lattice via
phase modulation. The time-dependent potential we consider here is given by
V (z, t) = V0 cos2 [kl (z + z0 (t))](5.11)

where z0 (t) describes the displacement of the lattice. The simple case is given
by z0 (t) = L sin (ωt + φ), with L the amplitude of the lattice displacement inspace with frequency ω and arbitrary phase φ. In the frame co-moving with

57

Va
ψb

Figure 5.3: An orbital level spin. A strongly-localizing state dependent opticaltrap Va traps an atom immersed in a superfluid bath ψb . Raman lasers (orparametric modulation of the trap) drive transitions between orbital levels of
the localizing trap Va , giving rise to differential interactions with the superfluid.
the lattice, a coordinate transformation recasts Eq. 5.11 as
V (z, t) = Vlatt cos2 (kl z) + mLω 2 z cos (ωt)(5.12)

thus introducing a time-dependent term describing an inertial force [190, 191].†
Using this transformed potential and for modulation near resonance with theband splitting ∆gap we write an effective two-level model in analogy with
Eq. 2.5
H = Ha + Hint = ~∆gap |nihn| + Vlatt cos2 (kl z) + mLω 2 z cos (ωt)(5.13)

where the assumed basis states are the relevant Bloch waves (or equivalently
Wannier functions). Paralleling the derivation in Chapter 2, we can write aRabi frequency for the transition between lattice bands |ni and |n + 1i.
Ωn =2mLω 2hn + 1, q|z|n, qi~(5.14)

which has general behavior illustrated in Fig. 5.4. Note that the z perturbation
only couples bands with ∆n = 1‡ . We consider an effective spin formed by thes (n = 0) and p-bands (n = 1) of a strongly-confining lattice (denoted again
as |↓iand |↑i respectively), see Fig. 5.6a.
Using vibrational levels as spins may naturally provide a differential colli†

It should be noted that the transformation in Eq. 5.12 requires the atoms to adiabatically follow the lattice, i.e. ω  ωho . While this is not true when ~ω approaches the bandgap, we can still get an intuitive picture of the physics involved here‡
A similar Rabi frequency can be found for amplitude modulation Ωn,n+2j =αV02i~ hn| cos kl z|n + 2ji, which connects bands separated by ∆n = 2j for j = 1, 2, 3, ...

58

(b)

0.30

|Ωn| [4 m L ω2/ħaho]|Ωn| [4 m L ω2/ħaho]

(a)

0.25
0.20
0.15
0.10
0.05
0.00

0

1

2

3

4

n

5

6

7

8

9

0.30
0.25
0.20
0.15
0.10
0.05
0.00

0

10

20

s

30

40

50

Figure 5.4: Rabi frequencies for shaken lattice. (a) Amplitude of matrix elements |hn + 1|z|ni| for s = 1, 10, 100 (in red, blue, green respectively). The
dashed line represents the harmonic oscillator limit for s = 100. (b) Amplitudeof the matrix element |h1|z|0i| as a function of s is plotted in blue. The redline represents the harmonic oscillator limit for a lattice of depth Vlatt = sEr .
sional interaction strength. Consider the energy U of the interaction with thesuperfluidZ
U = gabdx|wa(n) (x)|2 ρ̂b (x),(5.15)(n)with the Wannier state wa (x), for the localized atom in band n. In the longwavelength limit as considered in Eq. 5.3, U = gab ρ̂b (0) ∗ V does not dependon the band index n. Therefore, applications of this spin model will require
more careful consideration of the b atoms’ local response to the impurity wave
function, as the total energy will depend on the curvature of ψb (x). One may
intuit, however, that the larger spatial extent of a vibrationally excited state
will require more “gentle” curvature of ψb (x) and lead to an effectively reduced
spin-bath interaction energy compared to the n = 0 state.

5.3

Implementation and evaluation

The above methods have been experimentally explored in our apparatus, eachhaving its own advantages and limitations. Below, we consider each method’s
practical prospects and discuss results associated with each measurement.

59

5.3.1

Coherence times

To access the long-wavelength phonon excitations the driving energy ∆ mustbe small compared to the local mean field µ. Practical considerations in our
experiment, set µ/~ ≈ 2π × 1 kHz. Thus, to avoid creation of particle-like
excitations, ∆ . 2π ×500 Hz would be ideal. This requires long spin coherence
times; for this purpose any damping and or decoherence must be understoodand eliminated.
Hyperfine-state degrees of freedom
In the case of internal state based pseudospins, the main damping/dephasing
mechanisms seem to be due to instability of the bias magnetic field. Based on
measurements of the ambient field using a fluxgate magnetometer (Bartington Instruments, Mag-03MS1000), we estimate the ac-line-driven field fluctuations to have a magnitude of ∼ 5 mG and offset drifts on the order of ∼ 1
mG/hour. This is particularly concerning for the occupational spin as realized in our apparatus, since the microwave transition resonance center shiftsby ∼ 2.1 kHz/mG. Synchronizing the experiment timing to the 60 Hz acpower-line signal can mitigate these issues. As discussed in Chapter 4, for the|1, −1i ↔ |2, −2i transition, we experience only ∼ 300 µs of coherent oscillation,§ corresponding to shot-to-shot fluctuations on the order of 0.1 mG. In
order to observe Rabi oscillations at 500 Hz, this number would need to be
improved by at least an order of magnitude.
Using a clock-state transition requires a two-photon transition, this secondorder coupling naturally provides a narrow resonance feature, plotted in Fig. 5.5a.
Furthermore, the clock states allow for long lived Rabi oscillations due to insensitivity to background field fluctuations, where the bias field-induced detuning
is on the order of ∼ 10−2 Hz/µG. The 20 ms of high contrast oscillation plottedin Fig. 5.5b compares very favorably with < 500µs of stability observed formagnetically sensitive transitions described above.
Orbital degrees of freedom
To engineer the coupling model described in Eq. 5.13, we realize a shaken
lattice by passing the retroreflected lattice beam through a pair of AOMs
arranged to give a time-dependent frequency difference ∆ν(t) between theinterfering lattice beams, for details see [87]. This frequency difference isrelated to the lattice shake amplitude by L = π∆νmax /(klatt ω). After applying§

Using the |1, 1i ↔ |2, 0i as proposed above would result in a factor of 3 improvementin these numbers.

60

1.0

(b)

(c)

Pe

0.8
0.6
0.4
0.2
0.0
2600 2800 3000 3200 3400
n - ??? [Hz]

0

5

10
15
t [ms]

20

20

130

15

100

10

79

asc [a0] [red]Natom (104) [blue](a)9.00 9.05 9.10 9.15
|B| [G]

Figure 5.5: Properties of the internal state spin. (a) Two photon resonance,
the width of the resonance is just ∼ 100 Hz. (b) Two photon Rabi oscillations
with Rabi frequency of ∼ 80 Hz. Here, we see coherent oscillations over 20 ms;
much longer than observed for a typical single photon transition (see text).
(c) The measured Feshbach loss feature for |bgi and |↓i as well as the blueline shows a fit to a three-body loss process. The red curve shows the elasticchange in asc due to the Feshbach resonance.
shaking, we read out the band populations using the band-map technique. The
experimental signature of transfer out of the s-band into the p- and d-bands
is shown in Fig. 5.6b.
Using the shaken lattice we can measure the excitation spectrum for as = 35Er lattice as in Fig. 5.6c. We see here a clear two-photon excitation
into the second excited band along with strong single photon transfer into thefirst excited band. The second order process can be suppressed by reducing thelattice depth (creating a more anharmonic spectrum) or the shaking amplitude
L. Indeed we note, while a harmonic potential has a ladder of evenly spaced
states, the transformation to an isolated two-level system is justified by theanharmonic nature of the lattice band spectrum. A single pair of bands canbe isolated for Ωn < ∆gap (n = 0 → 1) − ∆gap (n = 1 → 2)¶ . Driving for longer
times we observe coherent dynamics over several ms. Coherent lattice band
dynamics is plotted in Fig. 5.6d; Rabi oscillations between s- and p-bands areobserved and some contamination in further excited bands is also seen.
While orbital spins are not sensitive to the magnetic field instabilities experienced here, their coherence lifetime still seems to be limited to ∼ 2 ms.
To elucidate the short coherence times of these oscillations, we consider the
lifetime of atoms located in the first excited band of the lattice. To do so
we quickly pulse the shaken lattice for τ = 0.3 ms and allow the atoms torelax in the lattice potential. Typical observed relaxation data can be seen in¶

The detuning ∆gap (n = 0 → 1) − ∆gap (n = 1 → 2) here becomes worse for deeper
lattices as the harmonic approximation for a single lattice site is improved for deeply boundlevels.

61

(a) 100(b)

0.8

Fraction

E [Er]

80

(c)

1.0

60
40

0.6
0.4

20

0.2

0
-1.0 -0.5 0.0 0.5 1.0

0.030

q [klatt]

35

40

45

ω/2π [kHz]

50

0

2

4

6

t [ms]

8

10

Figure 5.6: Coherent behavior in a phase modulated lattice. (a) Transitionof interest illustrated in a s = 30 lattice. The two photon transition to thesecond excited band is off resonant even for modest s. (b) Spectroscopy of the
bands. Population in the bands n = 0, 1, 2 are plotted in red, blue and green
respectively. For the 35Er lattice, the two photon transfer into the second
band of the lattice represents a strong feature in the spectrum. The resonant
features are consistent with band structure calculations. (c) Resonant drivingof the n = 0 to n = 1 transition. We see several oscillations at a Rabi frequencyof ∼kHz as well as damping over a 1.5 ms time scale. Solid lines represent
fits to a decaying cosine with some offset, colors are the same as in (b). Inset
shows raw TOF band map images for the first half period of oscillation. Cleartransfer of most of the atoms into the first excited lattice band can be seen,
with faint signal in higher bands. The maximal transfer is limited to ∼ 70%of the total atom number.
Fig. 5.7. The number of atoms in the excited band experiences a rapid decay
to a nonzero final value over ∼ 10 ms.
The decay from the excited band is driven by collisional processes in the
excited band like those illustrated in Fig. 5.7a [192, 193]. The most significant
collisional decay channel consists of a collision between atoms in n = 1 bandwith the atoms relaxing into the n = 0 and n = 2 bands of the lattice. For
sufficiently deep lattices, this band-changing collision is energetically allowed
due to the near harmonic nature of the spectrum. However, collisions can only
occur in lattices shallow enough to exhibit significant tunneling in the n = 1
band. We expect to observe a rapid decay of sites with na > 1, followed by aslow decay of singly-occupied sites. The initial decay to a steady state (steady
up to at least 1 s) in a time slower than typical tunneling rates, suggestingthat we eliminate multiple occupations before decay is eventually frozen out.
Decay from an excited band can also occur for collisions between bands and

62

(a)

(b)

0.30

Fraction

0.25

Fraction

0.20
0.15
0.10

1.0
0.8
0.6
0.4
0.2
0.0
0.0

0.8
0.6

0.5

1.0

1.5

2.0

0.4
0.2

0.05
0.00
0

1.0

20

40

60

t [ms]

80

100

0.0

0

20

40

60

t [ms]

80

100

Figure 5.7: Coherence decay and the driven steady state. (a) Decay of then = 1 excited state population. Line represents a fit to a decaying exponential
with 1/e decay time of 8 ms. (Inset: collisional mechanism of decay fromfirst excited band.) (b) Coherent dynamics for weakly driven, off-resonanttransitions. Decay of coherence is observed similarly to that seen in Fig. 5.6c,
while relaxation to a steady state is observed to occur on a timescale consistent
with the band decay rate. Steady state populations in bands with n = 2 andn = 3 are significant. (Inset shows coherent behavior at short times.)due to a superfluid bath [194].
To suppress transfer to a higher bands, we look at the coherent dynamics
for blue detuned driving. We see for short times a similar coherence decay rateof ∼ 2 ms. On a longer timescale, consistent with the band decay rate we see arelaxation to a steady state, including a significant population transfer into then = 2 band. This observation is consistent with the collisional decay processdescribed above. While the decay data do not demonstrate accumulation in
the n = 2 band, this is to be expected as collisional decay for atoms in n = 2
can be very rapid in dense atomic clouds [192].
While the measured decay and decoherence rates are slow compared to thetunneling rates as predicted by [195]; we do not observe lifetimes consistent
with those measured by early realizations of p-band lattice gases [193] (mostlikely due to an overly dense atomic sample increasing the collisional decay
rates). Furthermore, we do not measure a significant change of the coherence
decay rate over a wide range of parameters varying the shake amplitude andthe total atom number. The persistent 400 Hz timescale could be due toinhomogeneous lattice potential or perhaps due to unstable differential lightshifts in the lattice. The vibrational level pseudospin may be very practical if
the lattice depth can be further stabilized and the density of spins reduced.kkAmplitude modulation may be used to similar effect. This excitation couples only
states separated by ∆n = 2, and is thus further sensitive to tunneling effects and lattice63

5.3.2Differential interactions

Using the internal state spins requires using one of the three low-field Feshbachresonances described in [188]. These resonances allow, in principle, for themodulation of gab of up to 30%. Given that α ∼ (gab /gbb − 1)2 , this precludes
observation of the dissipative quantum phase transition, but it should allow formeasurement of enhanced damping due to the superfluid bath. The calculated
damping rate is given approximately by [63]


πα
∆.
(5.16)Γ ≈ sin2(1 − α)Away from the Feshbach resonance, α ∼ (1.02 − 1)2 = 4 × 10−4 corresponds
to a damping rate of 6 × 10−4 ∆, which is prohibitively slow. However, forgab = 1.3gaa , α ∼ 0.1 and the damping rate is ∼ 0.17∆. For reasonable∆ ∼ 2π × 500 Hz, this would require driving the spin for about 10 ms to see
damping to a steady state of Pe = 1/2, a very realistic figure for our experimentconsidering the coherence times above.
Practical application of these resonances will be limited by the resonancewidths ∆B. Theoretical elastic widths are on the order of ∆B ≈ 1 mG andinelastic widths γB = 3 mG. Comparing to a representative three-body loss
measurement in Fig. 5.5c, we see that the measured elastic width is ∼ 10 mG.
Preliminary measurements suggest that this broadening is due to averaging
over several ac-power line cycles during the loss measurement. We have recently succeeded in reducing these fluctuations to the ∼ 100µG level [196],
making use of the Feshbach resonance a realistic prospect.
In terms of driving, these reduced fluctuations would amount to a timevarying detuning with amplitude 200 Hz for the occupational spin and 1 Hz
for the clock transition. Together with the tuning of gab , this should indeed
allow for the observation of phonon-induced damping of Rabi oscillations.
Compared to hyperfine state spins, orbital levels will not allow for tuningoftherelative interaction√strength. If we argue based on the relative size
p
2
(n=0)
hx i
/hx2 i(n=1) = 1/ 3, we might suggest α ∼ 0.17 and a damping rateof ∼ 0.3∆. Because of the rapid damping already observed for “bare” spins,
it does not appear that this implementation could lead to immediate success.

depth instabilities. At present we are not yet able to observe even a single period of coherent
oscillation using an amplitude modulated lattice.

64

5.4

Summary and outlook

The methods explored here are promising for the implementation of quantum
dissipative models in our experimental setup. Coherence times and differential
interaction strengths are of the correct order-of-magnitude to observe dissipative effects in the spin-boson model. However, to implement this model werequire increased stability of the bias magnetic field, particularly against 60
Hz noise leaking into the setup from the wall power. Recent work has characterized this noise and the use of some compensating fields has allowed forsignificant coherence time improvements of a factor of three or more. Thesemethods may be further improved by the use of active stabilization.
Recent work, in other groups, has focused on the interaction of a bath witha spin superposition state using a few of the pseudospins described above. Efforts have used the vibrational-level effective spin to investigate the decoherence and decay of a spin superposition due to interactions with a bosonic bath[64, 189]. The effects of a Fermi sea on mobile bosonicspins has also been√studied and a loss of coherence of the (|↑i + |↓i)/ 2 superpostion state due
to quasiparticle scattering from the fermionic background has been observed[65]. However, these systems do not fully realize the spin-boson model as there
is no active tunneling between |↑i and |↓i and in each case, scattering of freeparticle-like excitations of the reservoir dominate the dissipative effects. The
experimental methods described here should allow for the extension of thisexperimental work into the regime of a driven spin coupled to a phonon bath.

65

Chapter 6
Conclusion
In this thesis, I have described two experiments investigating the dynamics ofatomic matter waves in optical lattices. Further, I have detailed the experimental requirements and prospects for the implementation of effective spin-1/2
systems coupled to a superfluid bath. The first experiment reveals an interplay between disorder and interactions in the dynamics of a matter wave ina tilted optical lattice [125]. A reduction of disorder induced damping is observed due to the addition of mean field energy to the system, an effect that is
maximized for comparable disorder and interaction strengths, and which canbe explained through a screening argument. The second experiment investigates the dynamics of microwave induced diffraction of a superfluid from astate-dependent optical lattice [157]. The dynamics are interpreted in terms ofthe nonadiabatic dressed states, revealing diffractive behavior in an unfamiliar
regime compared to typical atom diffraction experiments. This experiment establishes a technique for simple interferometric measurements. Furthermore,
we propose the use of this diffraction scheme in a limit which probes the mean
field excitations of the superfluid. In this limit, we suggest to use this scheme
to realize an effective spin-1/2, along with other options in orbital and internal state spin models. The possibility of realizing the spin-boson model in an
atomic system is discussed.
Current directions in the laboratory are working to improve the feasibilityof experimentally realizing the spin-boson model and other dissipative phenomena. Much of the work has initially been in improving the field stabilityof the apparatus, with the dual purpose of improving the usefulness of rubidium’s low-field Feshbach resonances and extending the coherent lifetime ofdriven microwave transitions.
These improvements, will allow for the investigation of dissipative models
in optical lattices such as [63, 74, 194, 197]. The techniques established here
could be used to investigate the concept of quantum parallelism in disordered66potentials [150]. Our system, capitalizing on state-dependent optical lattices,
may be greatly enhanced by the stabilization of a Feshbach resonance, since
this would potentially open up the possibilities to study in detail bosonic latticepolarons [123, 198, 199], perhaps extending the dynamical spin-boson model
into the equilibrium setting. More general studies of impurity physics may
also provide fruitful research directions [200–202].

67

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