Bose Gases in Tailored
Optical and Atomic Lattices

A Dissertation Presented
byBryce Russell Gadway
to
The Graduate Schoolin Partial Fulfillment of the Requirements
for the Degree ofDoctor of Philosophy
inPhysics

Stony Brook University
August 2012

Stony Brook University
The Graduate School

Bryce Russell Gadway
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

Harold J. Metcalf – Chairperson of Defense
Distinguished Teaching Professor, Department of Physics and Astronomy

Tzu-Chieh Wei
Assistant Professor, C. N. Yang Institute for Theoretical Physics,
Department of Physics and Astronomy

Robert Konik
Associate Physicist
Brookhaven National Laboratory
This dissertation is accepted by the Graduate School.

Charles Taber
Interim Dean of the Graduate
Schoolii

Abstract of the Dissertation

Bose Gases in Tailored
Optical and Atomic Lattices
byBryce Russell Gadway
Doctor of Philosophy
inPhysics
Stony Brook University
2012

Quantum degenerate atomic gases offer a unique platform for the
exploration of a wide variety of interacting many-body systems
in a pristine environment, based on a set of powerful tools for thecoherent control of the atoms’ internal and external degrees of freedom. Here, we present experimental studies of strongly interacting
bosonic mixtures in one-dimensional (1D) systems, in which themobilities of the two species are independently controlled with astate-selective optical lattice. In a first experiment, we freeze out
the tunneling of one species from a binary mixture, and study theformation of “quantum emulsion” states, where immobile atoms
serve as a static, random disorder for a more mobile species. We
investigate the 1D superfluid-to-insulator transition in the presence of this disorder, and make comparisons to the effects of quasidisorder from an incommensurate optical lattice. We observe enhanced localization in the more random potential, highlighting theimportant role of correlations in disordered systems. Through a
combined measurement of transport, localization, and excitationiii

spectra, we are able to obtain strong evidence for observation of adisordered, insulating quantum phase, the 1D Bose glass.
In a second experiment, we introduce a new experimental technique
for the characterization of ultracold gases held in optical lattices.
In analogy to neutron diffraction from solids, we use atomic de
Broglie waves to non-destructively probe the spatial structure of
1D Mott insulators through elastic Bragg diffraction, and to probeinelastic band-structure excitations of more weakly interacting 1D
Bose gases. Furthermore, we use the diffraction of matter waves
to detect the formation of forced-antiferromagnetic ordering in a
crystalline atomic spin mixture.
Lastly, we study the dynamical response of matter waves to a periodically pulsed, incommensurate optical lattice, a situation thatrealizes a system of two coupled kicked quantum rotors. We observethat the coupling induces a suppression of energy growth at quantum resonances, and a localization-to-delocalization transition in
momentum space for off-resonant driving. Our observations confirm a long-standing theoretical prediction for the two-rotor system, and illustrate how classical behavior can emerge from theevolution of a simple quantum system.

iv

Contents
List of Figuresviii

Acknowledgements

xi

1 Introduction

1

2 Weakly interacting
Bose gases I:
theoretical concepts
2.1 Bose–Einstein condensates . . . . . . . . . . . . . . . . . . . .
2.1.1 Non-interacting gases . . . . . . . . . . . . . . . . . . .
2.1.2 Interacting gases . . . . . . . . . . . . . . . . . . . . .
2.2 Atom-light interaction . . . . . . . . . . . . . . . . . . . . . .
2.3 Optical lattices and band-structure . . . . . . . . . . . . . . .

9
9
10
11
14
17

3 Weakly interacting
Bose gases II:
experimental probes
3.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Laser cooling and evaporative cooling . . . . . . . . . .
3.1.2 Optical dipole trap and optical lattices . . . . . . . . .
3.1.3 Imaging of the atoms . . . . . . . . . . . . . . . . . . .
3.2 Experimental characterization methods . . . . . . . . . . . . .
3.2.1 External trapping potential . . . . . . . . . . . . . . .
3.2.2 Optical lattices . . . . . . . . . . . . . . . . . . . . . .

22
22
22
24
27
28
28
30

4 Analysis of Kapitza–Dirac diffraction patterns beyond the Raman–
Nath regime
35
4.1 Introductory discussion . . . . . . . . . . . . . . . . . . . . . .
35
4.2 Raman–Nath regime . . . . . . . . . . . . . . . . . . . . . . .
37
4.3 Experimental procedure . . . . . . . . . . . . . . . . . . . . .
38

v

4.4
4.5
4.6
4.7

Weak-pulse dynamics . . . . . . . . . . . . . . . . . . . . . . .
Strong-pulse dynamics . . . . . . . . . . . . . . . . . . . . . .
Single-shot calibration of optical lattices . . . . . . . . . . . .
Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . .

39
41
42
43

5 Quantum dynamics of matter waves in a pulsed incommensurate lattice44
5.1 Introductory discussion . . . . . . . . . . . . . . . . . . . . . .
44
5.2 Experimental procedure and kicked rotor system . . . . . . . .
45
5.3 A system of coupled kicked rotors . . . . . . . . . . . . . . . .
48
5.4 Off-resonant dynamics . . . . . . . . . . . . . . . . . . . . . .
49
5.5 Suppression of resonant growth . . . . . . . . . . . . . . . . .
53
5.6 2D representation . . . . . . . . . . . . . . . . . . . . . . . . .
54
5.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . .
55
5.8 Supporting considerations . . . . . . . . . . . . . . . . . . . .
56
5.8.1 Classical dynamics . . . . . . . . . . . . . . . . . . . .
56
5.8.2 Floquet eigenstate analysis of Û . . . . . . . . . . . . .
56
5.8.3 Lattice incommensurability . . . . . . . . . . . . . . .
59
6 Strongly interacting
Bose gases I:
theoretical concepts
6.1 Bose–Hubbard model . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Mean-field treatment of the BHM . . . . . . . . . . . .
6.1.2 Exact solutions of the BHM for small systems . . . . .
6.2 Theoretical descriptions of 1D Bose gases . . . . . . . . . . . .
6.2.1 General considerations . . . . . . . . . . . . . . . . . .
6.2.2 Theoretical description of interacting 1D bosonic fields

63
63
67
69
74
74
76

7 Strongly interacting
Bose gases II:
experimental probes
7.1 Visibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Momentum peak width . . . . . . . . . . . . . . . . . . . . . .
7.3 Impulse-response . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Excitation spectra . . . . . . . . . . . . . . . . . . . . . . . . .

80
80
85
87
89

8 Atomic mixtures in state-dependent lattices
8.1 Introductory discussion . . . . . . . . . . . . . . . . . . . . . .
8.2 Hyperfine-state mixtures . . . . . . . . . . . . . . . . . . . . .
8.3 State-dependent optical potentials . . . . . . . . . . . . . . . .

92
92
96
99

vi

8.4

A moving state-dependent optical lattice . . . . . . . . . . . .

103

9 Glassy behavior in a binary atomic mixture
107
9.1 Introductory discussion . . . . . . . . . . . . . . . . . . . . . . 107
9.2 Experimental procedure . . . . . . . . . . . . . . . . . . . . . 109
9.3 Disordered Bose–Hubbard gas . . . . . . . . . . . . . . . . . . 111
9.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 115
9.4.1 Excitation spectra . . . . . . . . . . . . . . . . . . . . 115
9.4.2 Transport and spatial coherence measurements . . . . . 116
9.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . 119
10 Probing an ultracold-atom crystal with matter waves
121
10.1 Introductory discussion . . . . . . . . . . . . . . . . . . . . . . 122
10.2 Matter-wave probing of an atomic crystal . . . . . . . . . . . . 123
10.3 1D collisions and inelastic scattering . . . . . . . . . . . . . . 124
10.4 Elastic Bragg diffraction of matter waves . . . . . . . . . . . . 127
10.5 Bragg diffraction spectra using accelerated probes . . . . . . . 132
10.6 Low energy scattering of free and trapped 1D bosons . . . . . 135
10.7 In-situ investigation of low-velocity impurity transport . . . . 138
10.8 Detecting forced checkerboard ordering . . . . . . . . . . . . . 140
10.9 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . 143
10.10Some experimental details . . . . . . . . . . . . . . . . . . . . 143
10.10.1 Hyperfine state mixtures . . . . . . . . . . . . . . . . . 143
10.10.2 State-selective lattice potential . . . . . . . . . . . . . 144
11 Conclusion and outlook

146

Bibliography

148

A Delocalization of a spinful kicked rotor
174
A.1 Introductory discussion . . . . . . . . . . . . . . . . . . . . . . 174
A.2 System under consideration . . . . . . . . . . . . . . . . . . . 175
A.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . 177
A.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . 179

vii

List of Figures
2.1
2.2
2.3
2.4
2.5
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
4.1

4.2
4.3

5.1

Condensate density profiles with and without atomic interactions. 14
Energy band structure of a 1D lattice for varying lattice depths. 18
Spatial wave functions of the lowest-band Bloch eigenstates. .
19
Spatial wave functions of the localized Wannier functions in thethree lowest energy bands. . . . . . . . . . . . . . . . . . . . .
20
Localized Wannier functions at different sites. . . . . . . . . .
20
Relevant fine and hyperfine level-structure of 87 Rb showing optical transitions used for laser-cooling and imaging. . . . . . .
23
Configuration of laser beam paths in the x − y plane, used for
imaging, optical trapping, and (x and y) optical lattices. . . .
25
Configuration of optical lattice laser beam paths along the z-axis. 26
Absorption images of atoms in time-of-flight and in-trap. . . .
28
Measurement of optical trap frequencies. . . . . . . . . . . . .
29
Pendellösung oscillations of momentum-mode populations in an
optical lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
Lattice depth calibration by Kapitza–Dirac diffraction. . . . .
31
Lattice amplitude modulation spectroscopy of band-structure.
33
Time-of-flight (TOF) absorption images of a condensate diffracted
from a 1064 nm standing-wave optical pulses of constant areaβ = V0 τ /~ = 4.5, and varied durations . . . . . . . . . . . . .
39
Suppression and revival of atomic diffraction from a constantarea standing wave light pulse with increasing pulse length. . .
40(a) Comparison of experimental diffraction spectra and numerical fits. (b) Comparison of numerical lattice depth calibrations,using Eqs. (4.4), from relatively short (8 µs) and long (50 µs)
optical lattice pulses. . . . . . . . . . . . . . . . . . . . . . . .
42
Atomic matter waves in a periodically-pulsed optical lattice potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

46

5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
7.1
7.2

7.3
7.4
7.5
7.6

Scan of the pulse-period T revealing resonant and off-resonant
behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dynamics of a BEC under off-resonant kicking in the presenceof disorder. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dynamical evolution of resonantly kicked matter waves (K1 =1.6) in the presence an additional off-resonant drive (K2 ). . . .
Simulated effective-2D dynamics of a coupled δ-kicked rotor system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Averaged classical dynamics of the energy ε as a function ofkick number N . . . . . . . . . . . . . . . . . . . . . . . . . .
Time-independent analysis of the kick operator Û for off-resonant
kicking (T = 36 µs) by both lattices . . . . . . . . . . . . . . .
Localization-delocalization transition in the K1 -K2 plane. . . .
Influence of lattice commensurability. . . . . . . . . . . . . . .
Bose–Hubbard model parameters t and U as a function of thelattice depth sER . . . . . . . . . . . . . . . . . . . . . . . . . .
Mean-field phase diagram of the Bose–Hubbard model. . . . .
Tuning the ground state from a Mott insulator to a superfluid.
Ground and excited state energies of the Bose–Hubbard Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Momentum-distribution contrast of lattice gases. . . . . . . . .
Decay of off-site correlations as a function across the superfluidto-insulator transition. . . . . . . . . . . . . . . . . . . . . . .
Characterizing the inhomogeneous two-dimensional array of the
1D systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analytical estimates of the transition point to insulating behavior in 1D Bose gases. . . . . . . . . . . . . . . . . . . . . . . .
Loss of matter-wave coherence across the superfluid-to-insulatortransition in 3D. . . . . . . . . . . . . . . . . . . . . . . . . .
Time-of-flight visibility contrast C1D as a function of lattice
depth for an array of 1D Bose gases (γ ≈ 0.6) as a functionof the longitudinal lattice depth sz . . . . . . . . . . . . . . . .
Collapse-and-revival in the matter-wave coherence of 1D Bose
gases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Momentum peak width in time-of-flight across the superfluidto-insulator transition in 1D. . . . . . . . . . . . . . . . . . . .
Impulse response of 1D bosons as a function of lattice depth. .
Excitation spectra across the superfluid-to-insulator transition
in 1D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

47
50
52
54
57
58
60
61
66
68
70
71
73
73
75
78
82

83
85
86
88
91

8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
9.1
9.2
9.3
9.4
9.5

Two-component mixture in a state-dependent optical lattice. .
Hyperfine energy level structure. . . . . . . . . . . . . . . . . .
Transitions between different internal states of the atoms. . . .
Basic scheme for creating state-dependent optical potentials. .
State-dependent lattices and atomic four-wave mixing. . . . .
Energy shifts and scattering rates as a function of optical wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Polarization dependence of state-dependent lattice depths. . .
Moving state-dependent optical lattice. . . . . . . . . . . . . .

94
97
98
100
101

Disordered one-dimensional Bose gases. . . . . . . . . . . . . .
Histograms and correlation properties of disorder distributions.
Disappearance of excitation gap due to disorder. . . . . . . . .
Characterization of the 1D superfluid-to-insulator transition indisordered gases. . . . . . . . . . . . . . . . . . . . . . . . . .
Dependence of transport and coherence properties on the depthof an incommensurate lattice. . . . . . . . . . . . . . . . . . .

110
114
116

104
105
106

117
118

10.1 Free and nearly-free interspecies collisions of one-dimensionalbosons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
10.2 Diffraction from a moving atomic crystal. . . . . . . . . . . . . 128
10.3 Probe scattering from a moving crystalline target. . . . . . . . 129
10.4 Scattering profiles as a function of target confinement. . . . . 131
10.5 Measuring the on-site density profiles of atomic scattering centers.133
10.6 Scattering of an accelerated probe from a stationary crystal . . 134
10.7 Low velocity impurity transport. . . . . . . . . . . . . . . . . . 136
10.8 Low velocity impurity transport as observed in-situ. . . . . . . 138
10.9 Direct in-situ evidence for specular reflection. . . . . . . . . . 139
10.10Detecting forced antiferromagnetic ordering via matter-wave
scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
10.11Scaling of Néel temperature TN with lattice wavelength, λ . . 143
A.1 Periodic driving of a matter-wave field by external (lattice pulse)and internal (spin rotation) state driving. . . . . . . . . . . . .
A.2 Dependence of energy growth rate on the commensurability oflattice kicking and spin rotation frequencies. . . . . . . . . . .
A.3 Dynamics of the mean energy with increasing kick number, fordifferent kicking strengths K and spin-rotation evolution frequencies (µ). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.4 Dependence of the energy growth rate ∆ε/∆N on the spinselective stochasticity parameter K. . . . . . . . . . . . . . . .

x

175
178

179
180

Acknowledgements
First and foremost, I want to thank my advisor Dominik Schneble. Dominik
has been a great teacher and a terrific advisor, and I’ve learned most of what
I know about experimental physics from doing research in his group. He’s
always been around - typically in the lab - to offer advice and motivation, as
well as coffee, Haribo gummy bears, and other fuels for research. It’s been areally great opportunity to work in his group, and I’ll always be thankful formy years spent here at Stony Brook.
Secondly, I want to thank my former labmate Daniel Pertot. Working
alongside Daniel my first few years was a real godsend, as I was able to learn
quite a lot about ultracold atoms and about experimental research in generalfrom Daniel. He is an extremely smart and uniquely talented physicist, andalso just a really nice guy and good friend. Also, I was very fortunate to havejoined a fully functional BEC lab, forged through the hard work of Daniel anda number of other students over the years. In particular, I’d like to thank thestudents with whom I had some overlap, including Rebekah Schiller, Daniel
Greif, René Reimann, and Matthias Vogt, as well as a number of undergraduate researchers, namely Bartosz Bogucki, Gaku Nagashima, Justin Tian andDavid Meltzer.
I’d also like to thank the current members of the BEC lab, Jeremy Reevesand Ludwig Krinner. It’s been a pleasure to work alongside Jeremy for the
past few years, as he’s a really talented and dedicated physicist, and more
recently with Ludwig, who has proven to be an invaluable addition to the
BEC team. I’m looking forward to hearing about all the interesting physics
they will be pursuing in the years to come.
Marty Cohen, a visiting scholar, has been a great help to the lab and tome, and I thank him for all of his advice and feedback over the years. I also
would like to thank Tom Bergeman, who has been a theory collaborator withthe group for several years, and who has helped provide insight on a numberof topics. I’d like to thank Petr Anisimov for some really helpful conversations
about quantum optics and quantum atom optics.
For first pointing me in the direction of Stony Brook many years ago, Ixi

thank my undergraduate advisor, Kiko Galvez. Since arriving at Stony Brook,no one has helped me to maneuver through life within the department morethan Hal Metcalf. Hal has also been a great teacher over the years, and I’d liketo thank him for all his help. I would also like to thank Tom Weinacht, who
has always been there to lend equipment, and to provide insightful questions
and comments at the AMO seminars. I’d also like to thank John Noe of theLaser Teaching Center (LTC), as he and Marty helped to guide me throughthe physics of laser beams carrying orbital angular momentum.
I owe a lot of thanks to my thesis committee: Hal for serving as the chair,as well as the internal and external committee members Tzu-Chieh Wei andRobert Konik. For their help in guiding me towards the end of the graduate
school tunnel, I am indebted to Department Chairs past and present, Peter
Koch and Laszlo Mihaly, as well as to our Graduate Program Director Jac
Verbaarschot. Moreover, I need to thank a number of others for their help
in this regard, including Linda Dixon, Sara Lutterbie, Pat Peiliker, Socoro
Delquaglio, Maria Hofer, Diane Diaferia, Maryann Lodato, and Nathan LeoceSchappin. Thanks also to everybody in the machine shop, including Mark,Paul and Jeff, and to Rich and Bob for keeping the building running, andespecially to Frank Chin, who was nice enough to give my car a jump one
Winter day.
It has been a real pleasure to attend grad school with some really nice folks,
and knowing them has made it a highly enjoyable experience. For lending me
their optics, the occasional beer, and many delightful conversations, I’d like tothank fellow basement dwellers Chris, John, Dan, the original J. Reeves, Xiaoxu, Claire, Brian, Dominik, Marija, Peter, CoCo, Steve, Oumarou, Hyunoo,and Zhedong.
I also need to thank my friends from outside of physics for providing on
many a night’s a diversion from research – thanks Jeremy, Andy and all myother friends from Colgate, as well as Lance, Ryan, Merchant, Garret andother friends from home. More locally, I’d like to thank my good friends andhousemates Peter and Kyle. I am indebted to them for many fond memories,
and to their girlfriends Heather and Devon for many leftover meals.
I owe a great deal of thanks to my family, as over the years my brothers andsister, and all of their spouses and kids, have provided lots of encouragement.
I am especially thankful for the support of my parents Russ and Pam, who
throughout my life have been never-ending sources of love and encouragement.
Thanks to Andrea for being, in so many ways, the wind beneath my wings.
The joys of doing physics research often pale in comparison to the joys of
being with her, and her patience and support these past few years have madeit possible for me to finish my Ph. D.

xii

We thank the Optical Society of America and the American Physical Society
for the right to reproduce, in part, several works that are discussed in thisthesis.

We are also thankful for the generous support of the Research Foundation of
the State of New York, the Department of Education, the Office of Naval
Research, and the National Science Foundation.

xiii

Chapter 1
Introduction
Since the first observations of Bose–Einstein condensation [1–3] in dilute atomic
gases in 1995 [4–7], atomic systems have served as an ideal testing ground fora myriad of many-body quantum phenomena. The subsequent achievementsof quantum degeneracy in atomic Fermi gases [8, 9] and in ultracold moleculargases [10–12] (and near-degeneracy in gases of polar molecules [13, 14]) havemade ultracold atomic physics an even more expansive and diverse arena ofstudy. Ultracold quantum gases allow for the investigation of hydrodynamics, superfluidity, vortices and gases under rotation [15], collective enhancement and suppression [16, 17], transport in optical lattices [18], ultracold collisions, Feshbach resonances [19], few-body physics [20], and the BEC-BCS
crossover [21–24], just to name a few. Atomic physics thus has made connections to a variety of other fields, including but not limited to condensed matterphysics, nuclear physics, mathematical physics, and quantum information science.
Studies of strongly correlated many-body physics [25, 26] in atomic systems
were opened up a decade ago with the experimental realization [27] of a quantum phase transition between superfluid and Mott insulator [28, 29], based on
the controlled competition between tunneling and repulsive on-site interactionenergies of bosons confined to an optical lattice. More recently, explorationsof phase transitions to new quantum states have been undertaken in a numberof strongly-interacting [30–32] and weakly-interacting [33–35] systems. Manyof these have been aimed at studying systems of increased complexity in theform of, for example, long-range interactions (such as with dipolar gases [36]),
disordered potentials [37], synthetic gauge potentials [38], and magnetic interactions [39]. Due to their versatility and dynamical tunability, experimentswith ultracold quantum gases promise to be of continued importance for the
study of strongly correlated physics [25, 26], including non-equilibrium dynamics [40] and quantum critical phenomena [41–43], as well as many other areas1

of study.
In this thesis, I shall discuss several experiments with systems of both
weakly- and strongly-interacting Bose gases, aimed at studying novel quantum phases of matter. In particular, we have studied several effects that canoccur in strongly-interacting mixtures of bosonic gases in optical lattices, inwhich the mobility of each species can be independently controlled [44, 45].
The focus of these studies has been to look at effects going beyond those foundin the single-species Bose–Hubbard model. In the case that the second species
is a superfluid supporting phonon modes, it can mediate effective long-range(off-site) interactions between atoms of the first species. In contrast, if thesecond species consists of randomly positioned atoms pinned in place, it can
act as a static disordered potential. Using such atomic impurities, we havestudied spectral and transport properties of disordered one-dimensional Bose
gases, and observed a disorder-induced shift of the superfluid-to-insulator transition [46]. The resulting state is insulating and has a flat excitation spectrum,
consistent with the formation of a quantum emulsion state [47], whose properties are similar to those of an equilibrium Bose glass [28, 48]. Furthermore,
the study of bosonic mixtures in state-dependent optical lattices has allowed
us to develop a new experimental technique, based on the scattering of atomic
matter-waves [49], which may possibly aid in future endeavors to characterize
novel quantum states. Lastly, we have explored a localization-to-delocalizationtransition occurring in a system of weakly-interacting matter waves that areperiodically “kicked” by two incommensurate optical lattices [50], realizing asystem of coupled kicked quantum rotors.
Atomic mixtures in state-dependent lattices [44]While the single-species Bose gas inan optical lattice still presents a very
active area of study, new classes ofphysical effects can be found in systems of increased complexity. In thisvein, Hubbard models involving morethan one species have been studied
both for Bose–Bose [51–53] and Bose–
Fermi [54–57] mixtures (not to mention Fermi–Fermi mixtures or studies of
SU(N) physics with multi-species alkaline-earth mixtures [58]). Aside fromstudies of quantum magnetism [59–62] that one may hope to explore at very
low temperatures, novel interactions leading to physics beyond the singlespecies Hubbard model, with relevance to condensed matter systems, are expected to appear in two-species mixtures.
2

In studies with atoms of different masses (or effective masses in a lattice),lighter atoms can be used to mediate effective long-range interactions between
the heavier particles [63, 64]. This can be thought of as the heavier particles
coupling to phonons, or density excitations, in the lighter species. Phononcoupling is important to many phenomena in real materials (while it is absent
in optical lattices per se). In atomic systems, phonon-mediated coupling in
multi-component mixtures may, for example, lead to the emergence of supersolidity [65]. In a study of the response of one species of atoms in a lattice toa second superfluid species (“bath”), we observed a reduction of the apparent superfluid coherence of the atoms near the superfluid-to-Mott transition.
These observations are consistent with the formation of lattice polarons [63],composed of an atom on a given site and an induced density dip of the superfluid bath, resulting from repulsive interspecies interactions. In addition to a
reduction of mobility as compared to bare atoms, such a polaronic dressing ofatoms is predicted to lead to the formation of clusters due to induced off-site
attraction [63, 64].
A second effect that can be studied with atomic mixtures is the influenceof disorder. It has been proposed [66, 67] that localized atoms of one state,
randomly placed at sites of an optical lattice, can serve as a nearly ideal
form of random disorder. One advantage as compared to optical methods(such as two-color incommensurate lattices [68] or speckle potentials [69]) isthat the correlations of such an atomic disordered potential decay on a very
short length-scale, making the disorder essentially uncorrelated from site-tosite. For this situation, a metastable out-of-equilibrium state known as aquantum emulsion has been predicted [47], with properties very similar to
those of a Bose glass [47, 70, 71]. In the experiment, we studied such a scenario
in the limit where an added second species is much less mobile than the first,
and observed a sharp decrease of the superfluid coherence due to interactions
with the localized species, consistent with the formation of such a quantum
emulsion.
Glassy behavior in a binary atomic mixture [46]For non- and weakly-interacting parU
ticles placed into a random potential landscape at low energies [33, 34,
72, 73], Anderson localization can oc∆i
cur due to destructive interference between different quantum trajectories.
It has been shown that weak repulsive interactions between particles can screen
the disordered potential and can destroy such localization [74], leading to a3

transition to a superfluid phase. On the other hand, the interplay between
disorder and interactions is less clear-cut in systems of strongly-interacting
particles, where the presence of interaction gaps (Mott-gaps) can result in insulating behavior even in the absence of disorder [28, 48]. When disorder isadded, the presence of several mechanisms that each promote insulating behavior leads to the emergence of a new quantum phase known as the Bose
glass, which is a gapless and compressible insulator, making it distinct fromboth the Mott insulator and the Anderson glass. Despite some preliminary
experiments aimed at studying this Bose glass phase [68, 69, 75], no clear
observations have been made to date.
In this experiment, we have studied one-dimensional (1D) lattice-trapped
gases of bosons subject to disorder, formed by either localized atomic impurities or by an incommensurate optical lattice. In the first case, one species ofa two-species mixture is pinned in place to a state-selective lattice, thereafter
serving as a static disorder potential for the more mobile species. As mentioned
earlier, it is predicted that the resultant disordered quantum emulsion should
behave much like an equilibrium Bose glass state. One motivation for using thefine-grained impurity disorder [66], as opposed to standard optical disorders(pseudorandom incommensurate lattices [33, 68] and speckle patterns [69, 75]),
is to explore the role of spatial correlation properties. For both types of disorder, we study the excitation and transport properties of the one-dimensional
Bose gases, and for deep lattice confinement observe behavior consistent withthe formation of a Bose glass, i.e. insulating behavior with a flat, gaplessexcitation profile. However, we find that near to the expected superfluidto-insulator transition in 1D, the uncorrelated atomic disorder (representing“white noise”) much more strongly drives the system towards an insulating,
localized state. These observations reinforce the notion that the details of adisorder potential, such as its correlation properties, play an integral role indefining the ground state and excited state properties of a system, and that
there is no universal phase diagram of the disordered Bose–Hubbard model
defined solely by the amplitude of disorder.
Probing an ultracold-atom crystal with matter waves [49]In recent years, several approaches have been taken for the characterizationof strongly-interacting atomic gases in optical lattices. High-resolution in-situ
imaging methods based on absorption [76] and fluorescence [77, 78] imaging
of 2D planar systems have been utilized, as well as ones based on electron
microscopy [79]. The observation of new quantum phases of matter, suchas magnetically-ordered spin states and systems with non-trivial topologicalproperties, will likely be enabled by advanced detection techniques.
4Here, motivated by the usefulness of matter-wave scattering techniques invarious scientific disciplines, we explore the scattering of atomic matter waves
from samples of strongly correlated gases confined in optical lattices [80], providing information on spatial ordering and excitations in these systems. Our
method is analogous to the diffraction of electron and neutron de Broglie waves
from crystalline materials, but at energies that are a billion times lower. We“shine” one-dimensional Bose gases (probe) onto a Mott insulator (target),and observe Bragg diffraction peaks that reveal the spatial ordering and localization of atoms on individual lattice sites. For weak lattice confinement, westudy excitations and inelastic scattering processes in the target, connecting
to 1D collisions between distinguishable bosons in the free-atom limit. The
elastic scattering of matter waves provides a non-destructive in-situ interrogation technique, with a large tunability over the de Broglie wavelength, allowing
for the probing of structure and ordering across a wide range of length scales.
We demonstrated the versatility of our technique, as well as its suitability forcharacterizing spin-ordered phases, by using atomic de Broglie waves to detect
forced-antiferromagnetic ordering in a crystalline atomic spin mixture.
Quantum dynamics of matter waves in a pulsed incommensuratelattice [50]
According to the correspondence principle [81] the predictions of quantum mechanics should, in the classical limit, be inagreement with those of classical mechanics.
However, the conditions that specify “theclassical limit” are not uniquely defined.
For non-interacting particles, the classicallimit can relate to the thermal occupationof a large number of quantized energy levels, where effects of quantization becomeless important to the description of the system. It may also be reached in5systems of a macroscopic number of interacting particles, which are faithfully
described by the predictions of classical statistical mechanics. A fundamental, outstanding problem is how exactly classical behavior emerges in closed
quantum systems, i.e. without invoking a coupling to some environment thatis divorced from the system.
Here, we study the dynamics of a system that is based on the so-called δkicked rotor, in which a particle is subject to a time-periodic series of “kicks”.
In the classical case, the rotor dynamics are chaotic, giving rise to a diffusive
increase of the action variable. However, in the quantum case, interferences
lead to a suppression of growth by an effect known as dynamical localization [82], which is analogous to Anderson localization in one dimension [83].
The system that we specifically consider consists of two coupled kicked rotors.
Here, the coupling has been predicted to induce diffusive, classical-like transport behavior [84]. We realized this system, for the first time, by periodicallypulsing two incommensurate optical lattices onto an atomic Bose–Einstein condensate. For off-resonant kicking, we observed a breakdown of the dynamical
localization that is found in the single-rotor case, relating to a transition fromquantum to classical dynamics. Additionally, we observed a suppression ofballistic momentum-space transport in the case of resonant kicking of one of
the rotors. Our findings shed new light on the correspondence between quantum and classical dynamics in multi-dimensional systems, and provide a route
towards further experimental studies of higher-dimensional nonlinear, chaotic
systems.

Outline of the thesis
In Chapter 2, I describe some basic theoretical concepts for weakly-interacting
Bose–Einstein condensates in optical lattices. Chapter 3 provides a brief
overview of the experimental methods used for producing and for probing ourBose–Einstein condensates. In Chapter 4, I discuss in more detail a general
method for lattice depth determination based on Kapitza–Dirac diffraction in
the long-pulse regime. Chapter 5 presents our experiments on a system ofcoupled kicked quantum rotors. Chapters 6 and 7 introduce some important
theoretical and experimental methods used to describe and characterize systems of strongly-interacting bosons. In Chapter 8, I discuss our methods forcreating atomic mixtures, and for creating lattices that address the atoms ina spin-dependent manner. I discuss in brief our study of strongly-interacting
mixtures in such state-dependent lattices. Chapter 9 presents our study ofquantum emulsion states in a one-dimensional bosonic mixture, where one
species (“impurities”) acts as an effectively random disorder potential for thesecond species. In Chapter 10, I describe our experiments aimed at determin6

ing the structural properties and excitations of strongly-interacting lattice gas“targets” via the scattering of weakly-interacting 1D matter-wave “probes”, aswell as detecting the introduction of forced-antiferromagnetic spin-ordering ina crystalline target. Lastly, Chapter 11 provides a conclusion and an outlook
for future endeavors.

7

Publications of this PhD work (described either in part or in full):• Chapter 3
Versatile transporter apparatus for experiments with optically trapped
Bose–Einstein condensates
D. Pertot, D. Greif, S. Albert, B. Gadway, and D. Schneble
J. Phys. B: At. Mol. Opt. Phys. 42, 215305 (2009)• Chapter 4
Analysis of Kapitza–Dirac diffraction patterns beyond the Raman–NathregimeB. Gadway, D. Pertot, R. Reimann, M. G. Cohen, and D. Schneble
Opt. Express 17, 19173 (2009)• Chapter 5
Quantum dynamics of matter waves in a pulsed incommensurate lattice
B. Gadway, J. Reeves, L. Krinner, and D. Schneble
arXiv:1203.3177v1 (2012)• Chapter 8
Collinear Four-Wave Mixing of Two-Component Matter Waves,
D. Pertot, B. Gadway, and D. Schneble
Phys. Rev. Lett. 104, 200402 (2010)
Superfluidity of Interacting Bosonic Mixtures in Optical Lattices,B. Gadway, D. Pertot, R. Reimann, and D. Schneble
Phys. Rev. Lett. 105, 045303 (2010)• Chapter 9
Glassy Behavior in a Binary Atomic Mixture
B. Gadway, D. Pertot, J. Reeves, M. Vogt, and D. Schneble
Phys. Rev. Lett. 107, 145306 (2011)• Chapter 10
Probing an Ultracold-Atom Crystal with Matter Waves
B. Gadway, D. Pertot, J. Reeves, and D. Schneble
Nature Phys. 8, 544-549 (2012)

8

Chapter 2
Weakly interacting
Bose gases I:
theoretical concepts
In this chapter, we discuss some theoretical concepts that are fundamental to
the study of degenerate Bose gases in optical lattices.

2.1

Bose–Einstein condensates

Bose–Einstein condensation was predicted by Einstein in 1924 [1] to occur
for an ideal gas of massive particles obeying Bose statistics, as developed by
Bose for photons [2]. A Bose–Einstein condensate is characterized by the“condensation” of a macroscopic fraction of particles into the single-particle
ground state. In 1938, not long after its prediction in noninteracting gases,
London proposed [85] that the λ-phenomenon (discontinuity in heat capacity)in liquid 4 He associated with the transition to a superfluid state was related
to Bose–Einstein condensation. However, strong interactions between helium
atoms greatly reduce the condensate fraction in this system, and make for acomplicated and somewhat convoluted theoretical description.
In contrast, a nearly ideal system for the study of Bose–Einstein condensation can be found in dilute, ultracold atomic vapors, where typically thespacing between particles greatly exceeds the characteristic length scale ofinteractions. Following advances in the laser-cooling and trapping of neutral
atoms [86] (leading to the 1997 Nobel prize in physics [87–89]), and also of stable confinement [90] and evaporative cooling in magnetic traps [90–92], firstobservations of Bose–Einstein condensation in dilute atomic vapors were madein gases of rubidium at Boulder [4] and sodium at MIT [5] in 1995 (and also

9

reported evidence [6] for BEC in lithium-7 at Rice that same year, confirmed
in 1997 [7]). The realization of BEC in dilute atomic gases (for which the 2001
Nobel prize in physics [93, 94] was awarded) has since led to great advancesin the understanding and application of this novel state of matter (for somereviews, see for example [3, 95, 96]).

2.1.1

Non-interacting gases

The phenomenon of Bose–Einstein condensation, characterized by the emergence of a macroscopic population in the ground state, can be motivated byexamining the Bose distribution function at finite temperature. Here, we consider a gas in the grand canonical ensemble, with temperature T and particle
number N determined by coupling to a thermal and particle reservoir having temperature T and chemical potential µ. The populations Ni in levels ofenergy i are given as
1
.
(2.1)
Ni = (i −µ)/k T
Be−1
where µ is kept less than the ground state energy 0 to avoid negative populations. If the chemical potentialPis made to approach 0 , N0 will diverge, andfor fixed particle number N = i Ni , the population will condense into theground state.
We can also consider a more physical picture for the temperature-driven
transition from a normal (thermal) state to a Bose–Einstein condensate, occurring in a gas of bosonic particles (atoms) at fixed density n. Furthermore, asimple estimate of the transition temperature may be obtained in this manner,
where we relate the atoms’ thermal kinetic energy kB T /2 ∼ p2 /2m (momentum p) to the thermal de Broglie wavelength (coherence length) λdB ∼ h/p ofthe atomic wavefunctions,p(2.2)
λdB = ~ 2π/mkB T .
For decreasing temperature, the spatial extent of the wavefunctions will increase and at some point they will begin to overlap, such that distinguishability of the atoms by spatial coordinate is lost. This defines quantum degeneracy
in the gas, occurring when the de Broglie wavelength exceeds the mean interparticle spacing, i.e. when
λdB & n−1/3 ,(2.3)
or more accurately in free 3D space when nλ3dB > 2.612... (ζ(3/2), where ζ isthe Riemann zeta function) [97]. For dilute gases of alkali atoms with typicaldensities of 1013 − 1014 cm−3 , this corresponds to temperatures in the 100 nK

10

range.
In the experiments that will be described in this thesis, the starting condensate temperatures are on the order of several nK, orders of magnitude below
Tc , such that thermal depletion of the condensate is negligible. In the limitof zero temperature, atoms of a non-interacting Bose gas are all condensed
into the single-particle ground state ψ(x) of the system. For an N -particle
system, the many-particle wavefunction is just given as a product state of Nsuch identical wavefunctions,ΨN (x1 , x2 , . . . xN ) =NYψ(xi ) .

(2.4)

i=1

In the limit of large N , we can describe the condensate by one macroscopic
wave function, or complex order parameter,√(2.5)
φ(x) = N hψ̂(x)i .
Here, we have replaced the bosonic field operator ψ̂ by a classical field withuniform global phase (relating to a spontaneously broken symmetry at the
BECphase transition). The condensate wave function has a normalizationRdx|φ(x)|2 = N , such that the particle density is given as n(x) = |φ(x)|2 .

2.1.2

Interacting gases

In reality, interactions between atoms often play an important role, and arenecessary for describing the equilibrium and dynamical properties of a condensate (and they are a necessary ingredient for reaching degeneracy in thermal
equilibrium through evaporative cooling). The Hamiltonian of a gas of interacting particles in second-quantization of the field can be written as
Z
Ĥ =



~2 ∇2
+ Vpot (x) ψ̂(x) +dxψ̂ (x) −2m
Z Z1dxdx0 ψ̂ † (x)ψ̂ † (x0 )Vat (x − x0 )ψ̂(x)ψ̂(x0 ) , (2.6)
2†where ψ̂ † (x) and ψ̂(x) are the field creation and annihilation operators obeying
bosonic commutation relations. Here, we have included the influence of someexternal potential Vpot (x), typically used to confine the atoms in space, as well
as nonlinear atom-atom interactions characterized by the two-particle interaction potential Vat (x − x0 ). At ultralow temperatures, only the lowest s-wave
term of a partial wave expansion will contribute to the scattering amplitude11between two atoms. With an s-wave scattering length as typically less than
the interparticle spacing in dilute gases (na3s  1, weakly interacting limit), we
can then effectively describe these interactions by a δ-function pseudopotentialof the form
Vat (x − x0 ) ≈ g · δ(x − x0 ) ,(2.7)
where g = 4π~2 as /m is the coupling constant characterizing atom-atom interactions.
By transforming to the Heisenberg picture, we can write down equationsof motion describing the time evolution of the field operators as

 2 2
Z
~∇
0 †
0
0
0
+Vpot (x)+g dx ψ̂ (x , t)δ(x − x )ψ̂(x , t) ψ̂ † (x, t) .
i~∂t ψ̂ (x, t) = −
2m
(2.8)
†

If we again (as in Eq. 2.5) assume a macroscopic wave function of the field givenby the expectation value of the creation (annihilation) operator, a nonlinear
Schrödinger equation (known in this context as the Gross–Pitaevskii [98, 99]
equation) for the matter-wave field can be written as


~2 ∇2
2
i~∂t φ(x, t) = −+ Vpot (x) + g|φ(x, t)| φ(x, t) .
(2.9)
2m
Here, g|φ(x, t)|2 represents the time-dependent mean-field potential of the condensate, where n(x, t) = |φ(x, t)|2 is the condensate density. For attractive
interactions g < 0, this can lead to unstable condensate dynamics [100]. For
positive scattering lengths, as are typically used in experiment, the repulsive
mean-field potential simply acts to modify the ground-state wave functions.
The atomic interactions lead to an effective Hartree–Fock-type potential, whichcounteracts the (typically harmonic) trapping potential Vpot (x), resulting in alarger extent of the wave function.
By assuming a time-independent form of the Hamiltonian, and taking
φ(x, t) = ϕ(x)e−iµt/~ where µ is the condensate’s chemical potential, we can
write


~2 ∇2
2
+ Vpot (x) + g|ϕ(x)| ϕ(x) = µϕ(x) .
(2.10)−
2m
For a sufficiently large numbers of atoms, an approximate form of the groundstate wave function can be found by neglecting the kinetic contribution to theenergy (sufficiently far away from the edges of the cloud, see below), where thechemical potential can then be related to the density of the central or peakcondensate density as µ/g = nc . This so-called Thomas–Fermi approximation12

leads to a density profile of the form
n(x) =µ − Vpot (x)g(2.11)

for regions of x fulfilling µ > Vpot (x), and vanishing density outside theseregions. We now consider a weakly interacting BEC confined to a 3D harmonic
trapping potential


m 2 2
2 2
2 2
(2.12)
Vpot (x) =ωx x + ωy y + ωz z
2with trapping frequencies ωx,y,z , and with mean harmonic trappingpfrequency
ωHO = (ωx ωy ωz )1/3 and mean harmonic oscillator length aHO = ~/mωHO .
The Thomas–Fermi profile of a condensate in such a potential is then givenbyµ Y
x2  
x2 n(x) =
1 − α2 Θ 1 − α2 ,(2.13)gRαRαα∈{x,y,z}with Θ the Heaviside step function. Here, the chemical potential is related tothe total atom number by
2/5~ωHO 15N as
,
µ=
2
aHO

(2.14)

the Thomas–Fermi radii of the condensate along the directions xα are determined by settingµ = mωα2 Rα2 /2 .
(2.15)In Fig. 2.1, we compare the condensate density profiles of harmonicallytrapped atoms for the cases of clouds with no interactions, with interactions,and for an interacting condensate in the Thomas–Fermi approximation. We
see that compared to the non-interacting case, the profile is broader for a repulsively interacting condensate. Also, at the edge of the distribution, theThomas–Fermi approximation leads to discrepancies on the order of the healing length ξ = (8πnas )−1/2 . For simplicity and clarity, and in anticipation of
the discussion in Chapter 6 on one-dimensional (1D) systems, we have plotted
the condensate density profiles for a 1D scenario, assuming particles of mass
m set to that of 87 Rb and assuming a trapping frequency ω/2π = 100 Hz along
the one direction. To account for interactions, we incorporate an effective 1D
coupling constant
g1D /~ = 2 × 105 m/s, and assume a fixed total particleR
number N = n1D (z)dz = 150, where n1D (z) is the 1D density profile. We

13

0.6

non-int
int
TF

n1D [µm-1]

0.5
0.4
0.3
0.2
0.1
0.0
-4

-2

0

z [µm]

2

4

Figure 2.1: Condensate density profiles with and without atomic interactions.
The 1D density profiles n1D (z) of condensates are shown for the case of nointeractions (black, small dashing), with interactions (blue, dashed) as determined using Eq. 2.10, and with interactions in the Thomas–Fermi approximation (red, solid). Details of the assumed trapping potential and interaction
coupling can be found in the text.
note that the Thomas–Fermi approximation gives a slightly different relationof the radius to atom number in one dimension than in three, which can be
determined by normalizing the total atom number asZ1[µ − mω 2 z 2 /2]Θ[µ − mω 2 z 2 /2]dz ,(2.16)
N=
g1D
where Θ[x] is the Heaviside function. This can then be solved as
2g1D N
=
mω 2

Z 

 


3/2
2µ
2µ
4 2µ
2
2
−z Θ
− z dz =
,
mω 2
mω 2
3 mω 2(2.17)

where then using µ = mω 2 R2 /2 as before leads to the 1D Thomas–Fermi
radius R = (3g1D N/2mω 2 )1/3 .

2.2

Atom-light interaction

Neutral atoms interact with electromagnetic light fields in both a dissipativeand conservative manner. The former relates to transitions to excited elec-

14tronic states by photon absorption (followed by spontaneous emission), as
used for laser-cooling and magneto-optical trapping of atoms [86]. The conservative interaction of atoms with light involves a “dressing” or modificationof the energy levels of the atom in the presence of the light field. Physically,
this comes about due to interaction of the light field with the field-induced
dipole moment in the atom, which leads to a shift of the atomic energy levels(ac-Stark shift). For spatially inhomogeneous light fields, such energy shifts
can be used for trapping [101, 102] or anti-trapping ground state atoms in attractive or repulsive potentials (analogous to dispersive trapping of refractive
objects). Here we consider the simplified case of a two-level atom, however wewill treat the topic of atom-light interactions more rigorously in Chapter 8 withrespect to state-dependent optical potentials involving multiple excited-statetransitions.
We consider a two-level atom with ground state |gi and excited state |ei,
separated energetically by ~ω0 = Ee − Eg . We also consider a single-frequency
light field E = E0ˆ cos(ωt − kz) with propagation along the direction z (E0 theelectric field amplitude, ˆ the unit polarization vector), which has an angularfrequency ω = 2πν (wave number k = ω/c and wavelength λ = 2π/k), and isdetuned from the atomic resonance by an amount ∆ = ω − ω0 . This light field
has a local intensity I = 0 cE02 /2 (vacuum permittivity 0 ). Absent the atomlight interaction term, the Hamiltonian describing the atom and the light fieldisĤ0 = ~ω0 |eihe| + ~ω(â† â + 1/2) ,(2.18)
where â† (â) is the light field creation (annihilation) operator. We shall assume
a macroscopic population of the light field hâ† âi = N ' N − 1.
We now consider the interaction term of the Hamiltonian that describes
coupling between the atom and the light field in the dipole approximation(neglecting spatial variations of the light field on the length scale of the atom’s
electronic wavefunctions) of the form Ĥint = −d · E. In terms of the atomic
ground and excited levels and the light field operators, the interaction term
can then be expressed as [103] ∗~Ω (z) † iωt ~Ω(z) −iωtiω0 t−iω0 tâ e +âe. (2.19)Ĥint = |eihg|e+ |gihe|e22Here, Ω(z) = Ωe−ikz (z-dependence of Ω(z) dropped from here on), ~Ω ≡|µ|E0 defines the Rabi frequency, where µ = ehe|r|gi is the dipole moment of
the atom, and he|r|gi is the dipole matrix element describing overlap of theground and excited state orbitals, where r is the electron coordinate. The Rabi
frequency of light-atom coupling will thus be given by Ω = (eE0 /~)|he|r|gi|.

15

We now make the rotating wave approximation (RWA), where we neglect
the cross-terms with rapidly oscillating phase e±i(ω0 +ω)t , which is valid for nearresonant light fulfilling |∆|  ω0 . The interaction Hamiltonian can then be
reduced to~Ωâ† ei∆t~Ωâe−i∆t+ |gihe|,(2.20)
Ĥint ≈ |eihg|22where the first (second) term describes excitation (de-excitation) of the atomaccompanied by absorption (emission) of a photon. Starting in the groundstate, the probability √of occupying the excited state oscillates at the effective0
Rabi frequency Ω = Ω2 + ∆2 , with a maximum value of Pemax = |Ω/Ω0 |2 .
For large detunings |∆|  Ω the population remains within the ground state
to a good approximation.
In the case of such large detunings, coupling to the light field still has theeffect of shifting the energy levels of the ground and excited states. Theseenergy shifts can be determined by second-order perturbation theory, or bydirect solution of the Hamiltonian in a rotating frame [86], to be∆Eg,e = ±3πc2 Γe
~Ω2
=
I ,
4∆
2ω03 ∆(2.21)

when the counter-rotating term can be neglected. At the right hand side wehave related the shifts back to the local intensity of the light field I, and theexcited state decay rate
ω03 µ2.
(2.22)
Γe =
3π0 ~c3
Thus, for red detunings ∆ < 0, corresponding to laser light with a frequency
less than the atomic transition, the ground-state energy is lowered, proportional to the light intensity. For spatially inhomogeneous light fields, theatoms will then be attracted to the energy potential minima, located at positions of maximum light intensity. The atoms we study are typically confined
to an attractive optical dipole potential, formed at the focus of far red-detuned
Gaussian laser beams (Hermite–Gauss HG00 transverse modes) with intensity
distributions
w2
I(ρ, z)22= 2 0 e−2ρ /w (z) ,(2.23)Imaxw (z)with peak intensity Imax = 2P/πw02 , where P is the total laser power, w0
the minimum 1/e2 radius. The beam-radius
expands along the propagationp
direction of the beam as w(z) = w0 1 + (z/zR )2 , where zR = πw02 /λ is thebeam’s Rayleigh range.
In addition to the conservative potential experienced by the ground-state16

atoms, small effects of spontaneous photon (Rayleigh) scattering remain, leading to a residual scattering rate 2
3πc2 ΓeΓsc =I .
(2.24)
2~ω03 ∆
This effect can be understood by the fact that the states with shifted energy
levels are actually “dressed” states, with contributions from the ground andexcited levels (excited-state “contamination”). We note that this scattering
rate can be obtained simply by multiplying the time-averaged excited-state
population with the excited state decay rate Γe . A rigorous derivation of Γsc
in the context of multi-level atoms will be given in Chapter 8.

2.3

Optical lattices and band-structure

In addition to using Gaussian-beam optical dipole potentials for the trappingof neutral atoms, we also use optical lattice potentials to confine the atoms toperiodic “light crystals”. Optical lattices are formed by interfering two or morelaser beams with well-defined relative phase. In the case of a 1D lattice, thisis simply done by full retro-reflection of a Gaussian laser beam from a mirror,
which creates an optical standing wave. The potential along the direction z of
the beam will then be a periodic function, with spatial periodicity d = λ/2,
and modulation depth Vlatt . In the case of unequal local intensities in theforward and back-reflected beams, there will also be an offset potential. We
assume that the transverse dependence of the beam intensity, due to the overall
Gaussian profile, is negligible over the size of the atomic distribution being
addressed, so that the beams forming the periodic potential can be considered
to be plane-waves. The motion along z is transformed from the case of freeparticle dispersion to one described with an energy band structure, which we
now describe in detail, following the descriptions in Refs. [104, 105].
We consider the fate of a particle of mass m moving within some latticepotential V (z) with spatial periodicity d = λ/2 (spatial frequency k = π/d),subject to the Hamiltonian H = p̂2 /2m + V (z). Solutions to this problem
are given by Bloch wavefunctions [106], which are products of plane-waves
exp(iqz/~) and periodic functions
φq(n) (z) = eiqz/~ · uq(n) (z)(2.25)

with the same periodicity as the lattice potential, where the Bloch eigenstates
(n)
(n)
(n)
uq have energies Eq . This leads to a Schrödinger equation for the uq (z)

17

s=0

s=5

s = 10s = 15

E [ER]

10

5

0
-1

0

1 -1

0

1 -1

q [ħπ/d]

0

1 -1

0

1

Figure 2.2: Energy band structure of a 1D lattice for varying lattice depths.
We plot the band structure energies (in units of ER , folded within the first
Brillouin zone) as a function of quasimomentum q (in units of ~π/d) for lattice
depth values of s = Vlatt /ER = {0, 5, 10, 15}.
of the form(n) (n)HB u(n)q (z) = Eq uq (z)

with

HB =

1
(p̂ + q)2 + V (z) . (2.26)2m

For the generic periodic potential V (z), we can write both the lattice po(n)tential and the functions uq (z) as discrete Fourier sums
X n,qX
cl ei2lkz ,Vr ei2rkz
andu(n)(2.27)
V (z) =
q (z) =
rl

with integers r and l. The potential and kinetic energy terms of Eq. 2.26
respectively becomeXX
V (z)u(n)(z)=
Vr ei2(r+l)kz cn,qand
(2.28)qlrl

X (2l~k + q)2 n,q(p̂ + q)2 (n)uq (z) =cl ei2lkz .
2m2m
l(2.29)

For purely sinusoidal potentials of the form
Vlatt (z) =sERcos(2kz) ,2(2.30)

where s is the lattice depth in units of the recoil energy ER = ~2 k 2 /2m, only
two Fourier terms of Eq. 2.27 survive, Vr=±1 = sER /4.

18

(a)

|φ q(0)(z)|2Re [φ (0)
(z)]
q

(b)

-1

0

1

-1

z/d

0

1

z/d

Figure 2.3: Spatial wave functions of the lowest-band Bloch eigenstates, for(n)a lattice depth of s = 5. (a) Real part of the Bloch wave functions φq (z)in the lowest band (n = 0) for quasimomenta of q = 0 (blue solid line) andq = ~k (red dashed line), i.e. at the band center and band-edge, respectively.
(b) Density of the Bloch wave functions for the same two cases as in (a).
The Schrödinger equation for the coefficients cn,q
can then be written inlmatrix form as
X
(n,q)
(n,q)
Hl,l0 · cl
= Eq(n) cl.
(2.31)l

The matrix Hl,l0 has diagonal kinetic energy contributions (2lq/~k)2 ER , aswell as off-diagonal elements for |l − l0 | = 1 of weight sER /4. Here, q is thequasimomentum within the first Brillouin zone of the lattice, running fromq = −~k to ~k. For a lattice of infinite extent, q will be continuous, while
for a finite lattice with N sites it will have spacing 2~k/2N . For a given(n)quasimomentum q, Eq is the eigenenergy in the n-th energy band of thelattice (below we use the convention n = 0 for the ground band, n = 1 forfirst-excited, etc.) and the eigenvector c(n,q) defines the Bloch wave functionof this state.
For illustration, we show a few calculations of the band energy structure
and the Bloch wave functions (limits of l taken out to |l| ≤ 5) for a 1D
lattice. Fig. 2.2 displays band structure energies for varying lattice depth
values, illustrating the transition from a quadratic free-particle dispersion for
low lattice depths to one of flat bands with nearly uniform spacing (as in aharmonic oscillator) for very deep lattices.
Fig. 2.3 illustrates the spatial wave functions of the Bloch eigenstates ofthe lowest band (n = 0), at the center of the Brillouin zone (q = 0) and at the
band-edge (q = ~k). From Fig. 2.3 (a), we see from the real part of the wave
functions that, while the lowest band q = 0 Bloch wave function has a uniform
phase distribution, those at the band-edge have a π phase inversion between
adjacent sites (leading to a shift of the momentum peaks when projected onto

19(b) n = 1| w1 (z) |2| w2 (z) |2(c) n = 2

-1

0

Re [w2 (z)]Im [w1 (z)]Re [ w0 (z)]| w0 (z) |2(a) n = 0

1

-1

z/d

0

1

-1

z/d

0

1

z/d

Figure 2.4: Spatial wave functions of the localized Wannier functions in the 3
lowest energy bands, for a lattice depth of s = 5: (a) Density of the central-site
Wannier function w0 (z) at top, as well as the real part of the wave function
at bottom, for the lowest band (n = 0). (b) The first-excited band (n = 1)
density and imaginary part of the Wannier function w1 (z). (c) The secondexcited band (n = 2) density and real part of the Wannier function w2 (z).
Gaussian

Density

Wannier

V(z)

-1

0

z/d

1

Figure 2.5: Localized Wannier functions at different sites. For a lattice of depths = 5, we show the density distributions of localized lowest band (n = 0)
Wannier functions at the central site (blue solid line) and the neighboringsites (green solid lines). For comparison, we show the central-site atomic wave
function in the harmonic approximation to the lattice site potential (red dashedline). For clarity, we also show the lattice potential V (z) as a black solid line.
plane-wave states, as for time-of-flight imaging). The rather similar density(n)distributions (|φq (z)|2 ) of these same Bloch states are plotted in Fig. 2.3 (b).
So far, we have considered a basis of Bloch states, which are delocalized overthe lattice sites. This is very suitable for the description of atomic wave func20

tions for non- and weakly-interacting Bose gases, where the coherence length(or spatial extent) of the atoms spans the system size. However, in many cases
an equivalent description in terms of localized wavefunctions is more appropriate. This holds especially true for the case of strongly-interacting gases, where
repulsive interactions can force atomic wave functions to become localized onto
individual sites. A basis of localized Wannier functions of a particular energy
band is constructed by superpositions of the different delocalized Bloch wavesof that band. For a finite-size lattice composed of N sites, a summation overthe 2N Bloch states of the n-th band (with q ∈ {−~k, ~k}), of the form
X(2.32)
e−iqzj /~ φ(n)wn (z − zj ) = N −1/2
q (z) ,
qis used to create a localized Wannier function at site position zj , where N
is a normalization constant. In Fig. 2.4, we show the constructed Wannierfunctions at the central lattice site z0 = 0 for the three lowest bands, both
the densities |wn (z)|2 and the relevant (non-zero) real/imaginary parts of the
wave functions. In the limit s → ∞ where the band structure resembles thatof a harmonic potential, these Wannier orbitals map onto the Hermite–Gaussorbitals HG0 , HG1 , and HG2 , where the n-th such wave function has n nodes(relating to phase-inversions).
Finally, in Fig. 2.5 we plot lowest-band Wannier functions on three adjacentsites. As will be discussed in Chapter 6, the nearest-neighbor tunneling energyof the n-th band is determined by the overlap of adjacent Wannier functions
wn (z) and wn (z + d),Z~2 ∂z2∗+ Vlatt (z) wn (z + d)dz .
(2.33)
tn = − wn (z) −2m
For the ground band, a harmonic approximation to the lattice potential is typically made for deep lattices s  1, which yields an effective
wave√√width of thefunctions σ/d = s−1/4 /π and tunneling energy t/ER = (4/ π)s3/4 e−2 s [107].

21

Chapter 3
Weakly interacting
Bose gases II:
experimental probes
In this chapter, we briefly discuss the apparatus used to produce and study systems of ultracold atoms in our experiments. We also discuss some commonlyused techniques used for characterizing these systems.

3.1

Experimental setup

3.1.1

Laser cooling and evaporative cooling

As detailed in Refs. [45, 108–111], our experiments begin with the loading andlaser-cooling of 87 Rb atoms from background vapor into a six-beam magnetooptical trap (MOT). The optical transitions used for the primary cycling (F =2 → F 0 = 3), repumping (F = 1 → F 0 = 2), and depumping (F = 2 →F 0 = 2) to accumulate population in the F = 1 ground state are displayedin Fig. 3.1. Over a course of about 10 s, during which light-induced atomic
desorption (LIAD) [112] is used to enhance the MOT-loading rate, a cloudof ∼ 1 × 1010 atoms is loaded and cooled to a temperature below 1 mK.
The magnetic field of the MOT is then turned off for a brief optical molasses
stage, by which the atoms are cooled to around 25 µK. From the mixture oftrappable hyperfine states (|F, mF i = |1, −1i, |2, 1i, and |2, 2i), the atoms arethen optically pumped into the |F, mF i = |1, −1i hyperfine ground state. The
magnetic quadrupole trap, which uses the coils of the MOT, is then turnedon at a higher magnetic field gradient (40 G/cm) to catch the cooled cloud of
atoms, yielding around 2.2 × 109 atoms at around 150 µK.

22

F=3
266.650(9) MHz2

5 P 3/2

72.9113(32) MHz

F=2

156.947(7) MHz

5 2P 1/2

72.218(4) MHz

D1

D2

F=1
F=0

Cycling

794.978 850 9(8) nm

780.241 209 686(13) nm

377.107 463 5(4) THz
12 578.950 985(13) cm-1
1.559 590 99(6) eV

384.230 484 468 5(62) THz
12 816.549 389 93(21) cm-1
1.589 049 439(58) eV

DepumpRepumpF=22.563 006 GHz2

5 S 1/2
4.271 677 GHz

F=1Figure 3.1: Relevant energy level-structure of 87 Rb showing transitions used
for laser-cooling and imaging. Transition data is taken from Ref. [113].
Because of the relatively high background pressure of Rb atoms in thevapor cell (∼ 10−9 torr), losses due to background gas collisions severely limit
the lifetime of the trapped atoms. To reach quantum degeneracy, at higher
density and lower temperature, we first move the atoms to a lower-pressure
region before performing evaporative cooling. For the purpose of transporting
the atoms, the Helmholtz coil pair forming the quadrupole trap is mounted
onto a two-axis mechanical translation stage, as described in Ref. [111]. This
allows for the atomic cloud to be moved from the vapor cell MOT chamber into
a separate low-pressure chamber (at . 10−11 torr, with differential pumping
between the two chambers), and then into a connected glass cell (hereafterreferred to as the “science cell”) that provides almost unobstructed opticalaccess. As the trap is first moved, it is also stiffened to a field gradient of
Bz0 = 350 G/cm to avoid losses and heating. This compression results inan adiabatic heating of the cloud to a temperature of about 450 µK. Aftera transport time of about 3 s, around 1.6 × 109 atoms remain, serving as astarting point for the evaporative cooling to quantum degeneracy. The firststage of evaporative cooling occurs in the linear quadrupole potential, until
the temperature becomes low enough for spin-flip Majorana losses to become
noticeable. At this point, we convert the magnetic quadrupole trap into a23harmonic (near the trap minimum) time-averaged, orbiting potential (TOP)trap [90], by adding a rotating magnetic bias field in the x − y plane that is
produced by two pairs of bias-field coils in Helmholtz configuration. In the
TOP trap the magnetic field is offset from zero at the trap center, avoiding
Majorana losses.
After reaching temperatures and densities near degeneracy in the TOP
trap, we then transfer a thermal cloud of atoms into a crossed-beam optical
dipole trap (ODT), formed at the intersection of two far red-detuned laser
beams, as discussed in more detail in the following section. As the magnetic
trap is slowly turned off, a magnetic bias field along the z-axis on the orderof a few Gauss is left on, serving to define a quantization axis and preserve
the spin orientation (|F, mF i = |1, −1i). A final stage of evaporative coolingin the ODT is achieved by lowering the intensity of the trapping beams, thus
decreasing the trap depth. Evaporation resulting in essentially pure condensates with no discernable thermal fraction is used to define the starting point
for all of the experiments discussed in this thesis.

3.1.2

Optical dipole trap and optical lattices

As detailed in Refs. [45, 111, 114], we produce an attractive optical dipole trapat the intersection of two far-detuned laser beams of wavelength λ = 1064 nm,with beam-waists (1/e2 radius) of ∼ 135 µm at the BEC position and around
1.5−2 W of power in each beam. The laser light comes from a single-frequency(line-width ∼ 70 kHz) ytterbium fiber laser (IPG YLR LP-SF series). To
avoid interferences between the two trapping beams, they have orthogonallinear polarizations, and also a relative frequency offset of 20 MHz to averageout residual interferences. The beams are separately fiber-coupled (singlemode TEM00 ) to the position of the science cell, with PID-controlled intensity
stabilization. As depicted in Fig. 3.2, in the area of the science cell the two laser
beams, which are referred to as optical dipole trap beams 1 and 2 (ODT1 andODT2), are oriented along the lab axes x and y, respectively (note a different
convention of lab coordinates than in Refs. [45, 109, 114]). In addition to theattractive Gaussian potentials of the two laser beams, in general the atoms
are also subject to the linear gravitational potential along z, such that thetrap center is offset considerably from the center of the two laser beams due
to gravitational sag.
Along with the weak optical trapping of the Bose-condensed atomic clouds,
many of our experiments make use of optical standing waves to further confine
the atoms. Depending on the number of lattices used (in a cubic geometry
with 3 orthogonal axes), this confinement can result in a one-dimensional arrayof two-dimensional “pancake” gases (disc-shaped systems), a two-dimensional24

tocameraimaging
beam 2

y

x

ODT2
+1
-1

science
cell
-1
+1

ODT1

mirrordichroic
mirrortocameraAOM

imaging
beam 1Figure 3.2: Configuration of laser beam paths in the x − y plane, as used
for imaging, optical trapping, and (x and y) optical lattices. The optical
dipole trap (ODT) is formed by two crossed laser beams (ODT1 and ODT2)
of wavelength λ = 1064 nm. Lattices along the directions of these beams,
x and y, are formed through partial retroreflection of the beams ODT1 andODT2, controlled by acousto-optic modulators (AOMs). Near-resonant laser
beams used for absorption imaging of the atoms are combined onto the ODTbeam path by transmission through dichroic mirrors, and removed from thispath for imaging onto a CCD camera by use of dichroic mirrors as well.
array of one-dimensional “tube” gases (rod-shaped systems), or a fully threedimensional array of zero-dimensional sites (restriction to a single spatial mode
per site). This allows for the study of atoms in the Hubbard regime, where
effects of atomic interactions become important. The optical lattices can beused not just for confinement, but also for the diffraction of BECs in order tosplit them up into a number of plane-wave momentum states.
The introduction of optical lattices along the lab axes x and y is effectedby retroreflecting the beams that create the optical dipole trap (ODT). It is
important to have no optical lattice present during evaporation in the ODT, toallow for efficient thermalization of the sample. Thus, we implement a retrore25

mirrordichroic
mirrorpolarization
quarter-wave
platesciencecellzy’
λ = 1064 nm
λ' ~ 785 nm

Figure 3.3: Configuration of optical lattice laser beam paths along the z-axis.
The beam paths for the λ = 1064 nm laser light and λ0 ∼ 785 nm light are
combined by use of a dichroic mirror, which is transmissive for λ0 . Both lattices
are formed by full retroreflection from a mirror above the science cell.
flection scheme as in Refs. [76, 111], by which the amount of light reflectedback can be attenuated over seven orders of magnitude in a controlled way,from essentially no light reflected to ∼ 10 − 20%. As depicted in Fig. 3.2,
each of the optical trapping beams pass through two acousto-optic modulators (AOMs), each driven at 80 MHz, in the +1 and −1 diffraction orders,
respectively. The light is then reflected back by a mirror at normal incidence,passing back through the AOMs upon return. This results in no net frequency
shift or variation in alignment of the retroreflected beam.
In addition to these optical lattices oriented along x and y, we also have
two optical lattices that are both oriented along the z axis. One of the lattice
beams at λ = 1064 nm is derived from the same laser light as for the ODT1 andODT2 beams. The second, co-propagating lattice laser beam has a tunablewavelength operated in the ranged λ0 ∈ {770, 800} nm, typically at λ0 ∼785 nm. This beam is derived from a Titanium-Sapphire (Ti-Sapph) ring laser(Coherent 899) pumped by a 10 W solid-state laser at 532 nm (Coherent Verdi

26V10); more details of its spectral properties can be found in Ref. [114]. The
output of the Ti-Sapph is fiber-coupled and exits the fiber near the positionof the science cell, subject to intensity stabilization after passing through apolarizing element. As with the imaging light, the lattice laser beam at λ0
is made to co-propagate with the 1064 nm z-beam upon passing through adichroic mirror, as depicted in Fig. 3.3. The polarization of the beams is set
by a quarter-wave plate phase retarder below the science cell. As discussed
in Chapter 8, this is important for the relatively near-detuned λ0 light, as thestrength of the induced dipole potential is strongly dependent on polarization.
After passing through the science cell, each of these two laser beams is thentypically retroreflected by a dielectric mirror at normal incidence to form theoptical standing wave lattices along z. As discussed in Chapter 8, additional
beam manipulation above the science cell is used to construct a moving latticealong z, for the experiments described in Chapters 8 and 10.

3.1.3

Imaging of the atoms

To access the atomic column density distribution, we typically perform absorptive imaging in time-of-flight (TOF) after turning off all confining potentials
rapidly (. 1 µs for optical potentials and . 2 ms for the magnetic traps) andallowing the atoms to freely evolve for 10 − 20 ms. For our trapping frequencies, this time scale allows us to image the far-field momentum distributionof the atoms (neglecting the influence of interactions on TOF expansion). As
depicted in Fig. 3.2, TOF imaging paths are available along both the x and yaxes. The beams used for imaging co-propagate with the optical dipole trapping beams, entering and exiting through dichroic mirrors that are transmissive
for the imaging light. As described in Ref. [108, 109], we image the shadow castby the atoms, due to absorption of resonant imaging light, onto a CCD camera (Princeton Instruments PIXIS 1024B). A typical absorption image along
x of atoms released from a 3D optical lattice is shown in Fig. 3.4 (a). The
x direction is the primary TOF imaging path used for characterizing the momentum distributions of the atomic ensembles, while y is predominantly used
for diagnostic purposes. The imaging is performed on the F = 2 → F 0 = 3
cycling transition, concurrent with optical pumping from F = 1 → F 0 = 2,
with imaging timescales between 25 and 100 µs. For experiments with hyperfine state mixtures, as discussed in Chapter 8, a Stern–Gerlach pulse of amagnetic field gradient is first used to separate the different hyperfine statesin time-of-flight.
In addition to time-of-flight absorption imaging of momentum-space distributions, we have also implemented in-situ imaging to provide real-space column density distributions. For this we utilize a different imaging beam-path 27

(a)

(b)

z
y

z
y’

10 µm

100 µmFigure 3.4: Absorption images of atoms in time-of-flight and in-trap. (a) After
18 ms of time-of-flight free expansion, atoms released from a cubic opticallattice are imaged, providing a picture of the momentum-space distribution.
(b) Atoms imaged in-trap with no time-of-flight.
achieving higher magnification and a resolution ∼ 1 µm - that interrogates theatomic cloud at 45◦ with respect√ to the time-of-flight imaging path along x, i.e.
0along the axis x = (x + y)/ 2 (not shown in Fig. 3.2). An aspheric lens withrelatively high numerical aperture NA = 0.49 is used at close distance fromthe science cell for imaging light collection. Real-space absorption images of
the atomic clouds are taken with the same laser light as for the TOF imaging,and the optical trapping and lattice beams are kept on prior to and throughout
the imaging. A typical in-situ absorption image is shown in Fig. 3.4 (b).

3.2

Experimental characterization methods

3.2.1

External trapping potential

In the absence of direct in situ imaging capabilities, details of the atomic density distributions prior to release in TOF may be estimated based on knowledgeof the total atom number and details of the confining potential. While the optical dipole trap is formed at the intersection of two Gaussian beams, often in thepresence of a linear potential gradient due to gravity, we may approximate thepotential as harmonic close to the trap minimum. As discussed in the previous
chapter, a Thomas–Fermi description can be used to describe the density distribution of interacting Bose-condensed atoms in a harmonic trap. To estimate
the BEC’s chemical potential µ, as well as the spatial extent (Thomas–Fermi
2radius) of the gas along the three coordinate axes R{x,y,z} = (2µ/mω{x,y,z})1/2 ,
we measure the harmonic trapping frequencies ω{x,y,z} that characterize the
28

0

(b)

.75

σv [vR]

Velocity [vR]

(a)

.5

-.5

.50

.25
0

10

20

60

Time [ms]70ωmod / 2π [Hz]80

Figure 3.5: Measurement of optical trap frequencies. (a) Center-of-mass dipole
oscillations (in velocity-space) of a trapped cloud along the z direction, after anabrupt displacement of the trap center. The blue line is a sinusoidal fit to thedata yielding a trap frequency ωz /2π = 73 Hz. (b) Resonant excitation (heating) via sinusoidal modulation of the trap position. The rms velocity-width
σv of the clouds is determined after a fixed time of excitation and thermalization. The peak response of the atomic gases to the modulation corresponds
to resonant excitation at the trap frequency. The blue line is a Gaussian fit
to the data. For each case, the velocity is given in units of vR = h/mλ, with
λ = 785 nm.
trapping potential [3] (for a discussion of our atom number determination see
Ref. [109]).
In general, we determine the harmonic trapping frequencies using two
methods: by observing center-of-mass dipole oscillations of the cloud in TOF
and by excitation of the BEC via sinusoidal modulation of the trap position. Data obtained by these two methods are shown in Fig. 3.5. To induce
center-of-mass dipole oscillations, we shift the atoms from the trap center withan added linear potential (magnetic field gradient), which is then quickly (in∼ 100 µs) extinguished. The atoms oscillate back and forth about the trap
center at an angular frequency ω, which we observe through oscillations intheir TOF velocity. The data shown in Fig. 3.5 (a) are for oscillations along
z in an array of 1D gases – such that damping of the dipole oscillations bycoupling to motion along x and y is suppressed – but the method is genericallyapplicable.
For the case of excitation by sinusoidal modulation of the trap position, we
begin with an array of harmonically trapped 1D BECs as in the case above.
We then add a weak linear potential along z, whose amplitude is sinusoidally
modulated with a Gaussian envelope as A sin(ωmod t) exp[−(t − t0 )2 /2∆2t ], fora “wiggling time” of 2∆t ∼ 100 ms. After driving the atoms, we allow for
500 ms of thermalization time before measuring the velocity-width of the distributions along z, to characterize the amount of energy deposited into the gas.

29N / N TOT

1.0

0 ħkL

0.5

±2 ħkL

0.0
0

1

2

Time [ms]

0

0.1

0.2

1

1.1

1.2

2

2.1

2.2

Figure 3.6: Pendellösung oscillations of momentum-mode populations in an
optical lattice. For weak pulses, only the lowest few momentum orders arecoupled by the applied optical lattice field, leading to Rabi-like oscillationsof the momentum-mode populations. The solid lines are fits of sinusoidal
oscillations with a Gaussian decay, observed to occur on the timescale of a few
milliseconds.
Because of parity, we expect resonant excitation for ωmod ≈ ωz . As shown in
Fig. 3.5 (b), the response shows a maximum around ωmod /2π ≈ 75 Hz, inapproximate agreement with the dipole oscillation measurement.

3.2.2

Optical lattices

Precise knowledge of the optical lattice depth is often crucial for proper interpretation of observed physical effects, as well as for faithful modeling of theexperimental situation. Here, we discuss a few experimental methods used todetermine lattice depths for this purpose.
The first method that we use for lattice depth determination is based onmatter-wave dynamics in the long-pulse or channeling regime [115] of Kapitza–
Dirac diffraction [116, 117]. Beginning with a BEC at rest (quasimomentum
q = 0), a stationary optical lattice is turned on abruptly for a variable time T .
The atomic density focuses in towards the lattice potential minima, and thendefocuses back out, undergoing so-called Pendellösung oscillations [115, 118].

30

(a)

(b)

+8 ħkL

Lattice Depth [ER]+6 ħkL+4 ħkL+2 ħkL0 ħkL- 2 ħkL- 4 ħkL

60

40

20

- 6 ħkL- 8 ħkL00

1

2

Intensity [arb. u.]

Intensity

Figure 3.7: Lattice depth calibration by Kapitza–Dirac diffraction. (a) Timeof-flight diffraction patterns of BECs after pulsing by an optical lattice for apulse duration τ = 10 µs. With increasing optical intensity, higher momentum
orders are populated (shown in multiples of 2~kL , with kL = 2π/λL and λL =785 nm). (b) Dependence of the fit-determined optical lattice depth (in unitsof ER = ~2 kL2 /2m) on the optical intensity of the lattice’s laser beams.
A simple way to understand the temporal evolution of the state is to project
the initial state, essentially a q = 0 plane wave, onto eigenstates of the Hamiltonian in the presence of the lattice. By virtue of the symmetry of the situation, the decomposition is only onto q = 0 states of even lattice bands (index
n = 0, 2, 4 . . .). For sufficiently weak lattices, essentially only the lowest twosuch bands contribute, and the initial population can then be written as asuperposition of the states |n = 0, q = 0i and |n = 2, q = 0i, with energies
E0 (q = 0) and E2 (q = 0). As a function of time T , these two states acquirea difference in phase of exp[i[E2 (q = 0) − E0 (q = 0)]T /~]. Upon projecting
the band-states back onto a basis of plane-wave states by release in TOF, thisdifferential phase evolution is converted into population oscillations. We thus
expect that the frequency of the Pendellösung oscillations, for weak lattice
depths, should occur at a frequency [E2 (q = 0) − E0 (q = 0)]/~. From here,
one can make a comparison to band-structure energy calculations to determine
the lattice depth. We show in Fig. 3.6 an example of such Pendellösung oscillations in a lattice along z of wavelength λz = 1064 nm. From the observed
oscillation frequency ω/2π ≈ 11.2 kHz, we determine the lattice depth to besz = 5.68(2) ER . We remark that the oscillations are observed to decay on
the timescale of about 1-2 milliseconds, for which dissipation through atomic
interactions [119] and higher-band tunneling are the most likely causes.
The second, related method investigates the response of atoms to latticepulses in the short-pulse or phase-grating regime [115, 117], also known as

31the Raman–Nath regime. In this short pulse limit, the atom’s motion during
the duration of the pulse can be ignored. Therefore, the dominant effect of
the applied pulse is to imprint a spatially varying phase pattern onto thecondensate, of the form exp[iVL cos2 (kL z)τ /~], where kL = 2π/λL is the wave
number and λL is the wavelength of the lattice light, τ is the duration of
the pulse, and VL is the depth of the lattice. Expanding the wavefunction
in a plane-wave basis of states with momenta pn = 2n~kL [120–122], startingwith a stationary condensate at p = 0, one finds that the coefficients of thedifferent plane-wave orders after application of the pulse can be approximatedas cn = (−i)n e−iVL τ /2~ Jn (VL τ /2~), where the Jn are Bessel functions of thefirst kind. Thus, from populations of the form Pn = |Jn (VL τ /2~)|2 , one candetermine the lattice depth VL . More generally, effects of finite pulse length– covered more in the following Chapter – can be accommodated through asimple solution of the Schrödinger equation for a p = 0 initial state, solving a
set of coupled equations of the formċn = −iVL
4n2 ERcn − i (cn−1 + cn+1 ) ,
~
4~

(3.1)

where ER = ~2 kL2 /2m is the recoil energy. In Fig. 3.7, we show data fora typical calibration of lattice depth (as a function of laser beam intensity)
based on comparisons of experimental momentum distributions and solutionsof Eq. 3.1 for a fixed pulse duration of τ = 10 µs.
Each of the methods described thus far relies on starting with all of theatoms in a single quasimomentum state. For certain systems, in particular those consisting of spin-polarized fermions, these methods fail. The last
method we explore is based on lattice amplitude modulation spectroscopy,
which can be used in systems that lack matter-wave coherence. Here, we startwith atoms loaded into very deep optical lattices, such that the lowest bands ofthe lattice are essentially flat. We then sinusoidally modulate the depth of thelattice, which near to the lattice minima performs an even parity z 2 perturbation. This perturbation can directly couple lattice bands whose indices are
separated by even integers. The excitation can also be viewed as a two-photon
Raman process, in which the modulation effectively adds frequency sidebands
onto the laser beams. When the frequency of modulation matches the thefrequency spacing between such sets of bands (in the atomic limit for deep
lattices one can think of the spacing between lattice orbitals s, p, d, f, g, . . .),resonant (and parity-conserving) excitation of population to higher bands canoccur.
To directly detect the population of atoms in different lattice bands, astandard procedure known as band-mapping [123] can be used. Here, a rela-

32

0

10

ħωmod / ER
20

30

40

(b)
100

Population [%]∆ P0 [%]

(a)
0
-20

P0

P2

P4

50

-40
0

20

40

60

ωmod / 2π [kHz]

0

80

0.0

0.1

0.2

0.3

Tmod [ms]Figure 3.8: Lattice amplitude modulation spectroscopy of band-structure.
(a) Even excitations can be made to higher bands, preserving parity of theatomic orbital wavefunctions (i.e. s → d and d → g allowed, s → p disallowed). By scanning the frequency ω at which we sinusoidally modulate
the lattice amplitude, and monitoring the percentage change of ground-band
atoms (following a band-mapping procedure), we can spectroscopically determine the transition frequencies to higher lattice orbitals. Plotted at the
top is the modulation energy scale in units of ER . By comparison to bandstructure calculations, we determine the lattice depth to be 54 ER . The dip
at ∼ 53 kHz is a direct (two-photon Raman) transition to the second-excited
band, while the feature at ∼ 27 kHz is due to a higher-order processes (with
odd-parity excitation ruled out by an absence of population in the first-excitedband). (b) For stronger amplitude of modulation at resonance, we can leave
the linear-response regime and observe partially coherent dynamics in the band
populations P0 , P2 , P4 (s, d, and g orbitals) as a function of modulation time.
tively fast ramp-down of the lattice is performed, typically on the timescaleτBM ∼ 1 ms, slow enough to prevent redistribution of population between
bands, fulfilling ~/τBM  ∆gap , with ∆gap the bandgap. Once the lattice isramped off and there is a free-particle dispersion, the atoms originally residing
in different bands will have momenta such that different Brillouin zones are
populated in TOF. In Fig. 3.8 (a), we plot as a function of modulation frequency the fraction of atoms remaining in the lowest band (n = 0), following
10 ms of lattice amplitude modulation (at ±5%, after a 0.85 ms bandmap). In
the frequency range explored, the population removed from the lowest band
is observed to populate the n = 2 band. We also note that these measurements are performed with amplitude modulation of the lattice along the y-axis,
starting from a Mott insulator of atoms confined along all three axes (lattice
depths V{x,z} = {40, 30} ER along the other axes) with a lattice wavelength33

of λ = 1064 nm along all axes. From the modulation spectra, we determine an energy spacing between the ground and 2nd-excited band of 26 ER(ωmod /2π ≈ 53 kHz), corresponding to a lattice depth along y of Vy = 54 ER .
While this data is taken for relatively small amplitude of modulation, forlarger amplitude “wiggling” we can drive the system well out of the linear
response regime. In Fig. 3.8 (b), we show data for on-resonance modulation of
the y-lattice amplitude [for slightly different conditions than in Fig. 3.8 (a)],which shows slightly coherent evolution of the population within the lowest andthe second- and fourth-excited bands. More fully coherent dynamics should
result for phase modulation (“shaken” lattice), which couples the lowest andthe first-excited band, where the lower natural linewidth (tunneling rate) of
the excited state provides for a more fully closed transition.

34

Chapter 4
Analysis of Kapitza–Diracdiffraction patterns beyond theRaman–Nath regime
In this chapter, we report on an experimental study of Kapitza–Dirac diffraction of matter waves from (temporally) short and not so short optical latticepulses. We find that when pulse area is conserved, a longer pulse leads to lessmatter-wave diffraction. We find that for sufficiently weak pulses, the usual
analytical short-pulse prediction for the Raman–Nath regime continues to holdfor longer times, however with a reduction of the apparent modulation depthof the standing wave. We quantitatively relate this effect to the Fourier widthof the pulse, and draw analogies to the Rabi dynamics of a coupled two-statesystem. Our findings, combined with numerical modeling for stronger pulses,
are of practical interest for the calibration of optical lattices in ultracold atomic
systems. This chapter reproduces our publication Analysis of Kapitza–Diracdiffraction patterns beyond the Raman–Nath regime, Opt. Express 17, 19173(2009) [122].

4.1

Introductory discussion

The diffraction of matter-waves from a standing light wave is a fundamental
concept in atom optics [124, 125]. Originally predicted by Kapitza and Dirac[116] for electrons more than 75 years ago (and recently also observed for these[126]), it was first demonstrated in the 1980s with an atomic beam [117], a
decade later with cold atoms [127], and has since become a standard tool inatom interferometry for the coherent mixing of momentum modes [125, 128].
The advent of Bose-Einstein condensates in the 1990s [95] has made it possible35

to directly observe the dynamics of matter-wave diffraction in time-of-flightimages [115, 129, 130]. Moreover, the diffraction of condensate atoms fromstanding light waves finds applications in high-resolution spectroscopy [131]and metrology [132, 133], and plays a fundamental role in superradiance [134,135].
When atoms diffract from a standing light wave, each two-photon (absorption / stimulated emission) scattering event changes the atomic momentum
along the standing wave by either zero or two photon momenta, which formultiple such events results in a series of evenly spaced atomic momenta. Two
scenarios can generally be distinguished, depending on the presence of a resonant coupling. In the Bragg case, the atoms oscillate between two resonantly
coupled momentum states, depending on the strength and duration of theinteraction with the light field [136]. If the Bragg condition is not met (offresonant case), atoms can nevertheless be diffracted into a number of momentum states, provided that the interaction is sufficiently short and strong [117].
In accordance with a common convention in atom optics [137] we refer to thiscase as Kapitza–Dirac diffraction, but we extend it to include times beyond
the Raman–Nath regime. In this context, it is interesting to note a similar
discussion for the diffraction of light from sound waves in acousto-optic devices[120].
In the Raman–Nath regime, i.e. when atomic motion during the interaction
with the light field can be neglected, the populations of the diffracted states
exclusively depend on the product of the strength V0 and the duration τ of theinteraction, i.e. the area of the applied pulse [117]. Outside the Raman–Nathregime, the diffraction dynamics exhibits collapses and revivals for constantinteraction strength [115, 129, 130]. We now specifically analyze the caseof a pulsed interaction of variable duration but constant pulse area. This
allows for a study of the breakdown of the Raman–Nath prediction and, inparticular, for a quantification of deviations when the system is close to, but
not deep into, the Raman–Nath regime. The findings of our study are of directinterest for the elimination of systematic errors in calibration measurements forexperiments with ultracold atoms in optical lattices [18, 26] when the lattice
depth is determined via Kapitza–Dirac diffraction.
This Chapter is organized as follows: Section 4.2 reviews general aspectsof the matter-wave diffraction from a one-dimensional optical lattice, while
Section 4.3 briefly describes our experimental system. In Section 4.4 we analyze diffraction patterns for weak pulses, and give an analytical modification
to the Raman–Nath diffraction formula, which is motivated by a comparisonof the diffraction dynamics with that of a coupled Rabi system, as well as byconsidering the spectral properties of the pulse. Section 4.5 discusses the more36

complicated dynamics of strong pulses, and Section 4.6 deals with calibration
methods for the depth of optical lattices, comparing numerical simulations
with single-shot diffraction patterns not restricted to the Raman–Nath regime.

4.2

Raman–Nath regime

We first briefly review general aspects of the diffraction of a condensate froma standing light wave with wavenumber k = 2π/λ that is switched on fora duration τ . The standing wave gives rise to a sinusoidal optical potential
V0 cos2 kz [18]. The evolution of the condensate in the standing wave can thenbe modeled [121, 138] (neglecting mean-field interactions) as that of a matterwave ψ subject to the HamiltonianĤ = −(~2 /2m)∂z2 + V0 cos2 kz(4.1)

where m is the atomic mass. Expanding the condensate wave function in
the basiswaves populated by diffraction from a standing wave as
Pof plane i2nkz
ψ(t) = n cn (t) e(where n = 0, ±1, ±2, . . . and cn (t = 0) = δn,0 ) andintroducing the dimensionless parameters
α = (Er(2) /~) τ
β = (V0 /~) τ,
(n)

(4.2)
(4.3)
(1)

(where Er = (n~k)2 /2m denotes the n-photon recoil energy, with Er ≡Er ), transforms the time-dependent Schrödinger equation into a set of coupled
differential equations
i

α n2
β
dcn
=
cn +
(cn−1 + 2cn + cn+1 )
dt
τ
4τ

(4.4)

for the amplitudes cn (t) of the diffracted orders n. For a given lattice depth,
the highest momentum order (±2n~k) capable of being populated is given by
the cutoffpn̄ = β/α(4.5)
for which the potential energy is fully converted into kinetic energy (cf. [130]).
Note that in these equations, α is the pulse duration τ in units of the 2-photon(2)(2)
recoil time τr = ~/Er , and β measures the area of the pulse.
The dynamics of the condensate in the standing wave depends on theratio between αn2 and β, i.e. between the kinetic energies acquired during
diffraction and the depth of the potential. The Raman–Nath approximation

37

consists of neglecting the αn2 terms in Eqs. (4.4), which is justified if τ ismuch shorter than the harmonic oscillation period in a potential well [115],(2)such that τ ωho  1, with ωho = [V0 Er ]1/2 /~, or equivalently βα  1. In thiscase, the solution of Eqs. (4.4) is cn (t) = (−i)n e−iβt/2τ Jn (βt/2τ ), such that the
population Pn = |cn |2 of the nth diffracted order after application of the pulse
is given by β2
Pn = Jn,(4.6)
2
where the Jn are Bessel functions of the first kind.

4.3

Experimental procedure

In the experiments described in this Chapter, we subjected an optically trapped87
Rb Bose-Einstein condensate to a vertically-oriented, pulsed standing light(2)
wave at 1064 nm, for which the 2-photon recoil time τr ≈ 20 µs. In brief, ina nearly isotropic crossed-beam optical dipole trap we produced condensates
typically containing 5 × 105 atoms in the |F = 1, mF = −1i hyperfine groundstate without a discernible thermal fraction. The standing light wave, with aGaussian 1/e2 radius of 130 µm at the position of the condensate, was derived
from a single-frequency ytterbium fiber laser with a linewidth of 70 kHz andcould be switched off within 1.5 µs using an acousto-optic modulator. Immediately after application of the pulse, the atoms were released from the opticaltrap and allowed 15 ms time-of-flight. They were then, in 100 µs, simultaneously repumped to F = 2 manifold and imaged on the cycling transition usingnear-resonant absorption imaging.
A series of images taken for pulses of variable duration τ but with constantpulse area β = 4.5 is shown in Fig. 4.1. Based on a naive invocation of Eq. (4.6)
for the Raman–Nath regime one would expect the same diffraction pattern inall the images. Instead, the number of diffracted orders is seen to generally
decrease with the duration of the pulse, with an oscillatory decay of the firstdiffracted orders for long pulse durations.
The populations of the 0~k and ±2~k momentum orders in the series ofFig. 4.1 are shown in Figs. 4.2 (a) and (b), together with corresponding data
for a weaker standing wave with β = 1.5, for which diffracted orders 2n~kwith |n| > 1 are not populated over the full range of pulse durations. For thelarger area pulse with β = 4.5, this is only the case for pulse durations of atleast 90 µs, due to contributions of the ±4~k orders, as shown in inset (c).

38

4 µs 815 30

45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 330 360 µs

+6ħk
+4ħk
+2ħk
0ħk
-2ħk
-4ħk
-6ħk

Figure 4.1: Time-of-flight (TOF) absorption images of a condensate diffracted
from a 1064 nm standing-wave optical pulses of constant area β = V0 τ /~ = 4.5,
(2)
and varied durations, ranging from τ = 4 µs (α = τ /τr ≈ 0.2) to 360 µs(α ≈ 18). For the data with durations τ & 90 µs (for which [β/α]1/2 . 1.0),
the apparent modulation depth of the standing wave undergoes an oscillatory
decay consistent with the form (β/2)sinc(α/2), as described in the text.

4.4

Weak-pulse dynamics

For sufficiently weak pulses with a cutoff n̄ ≈ 1 for which only the lowest orders±2~k are populated, Eqs. (4.4) can be reduced to three coupled equations; thisapproximation is valid for shallow potentials V0 . 4Er . The solutions c0 andc±1 to Eqs. (4.4), which can be obtained in a straightforward way, then leadto!
p
2 /2 + α2
β
β2
sin2
(4.7)
P±1 = 22β + 4α22and P0 = 1−2P±1 . This can be cast into the standard form of a Rabi oscillation 
1  χ 2 2 Ω
P±1 =
sin
τ
(4.8)
2 Ω
2
1/2

with generalized Rabi frequency Ω = [χ2 + ∆2 ] , resonant coupling χ =
√
(2)
(2)
V0 / 2~ and detuning ∆ = Er /~ = ωr between the atomic momentum
states 0~k and ±2~k. Another way to interpret this result is the following:each scattering event that changes the atomic momentum from 0 to ±2~k byenergy conservation leads to a frequency mismatch of the scattered photonby ∆ with respect to the standing wave, requiring a corresponding Fourier
width of the pulse for stimulated scattering to be able to occur. Consequently,
the drop-off of populations in higher orders with increasing α can be seen as
resulting from the decrease in the Fourier width of the square pulse which, in

39

1

  

  
      


 








P0k 0.5













(a)




0.2

(c)














0



0.5









 


         



 







 





0



1





5





2

3

4













5

6

(b)






(d)



  

0





0


0
1

β



0.3

0.15

βeff



P4 k 0.1

 

0

P2k











 




 





















10

15

Figure 4.2: Suppression and revival of atomic diffraction from a constant-areastanding wave light pulse with increasing pulse length. (a and b) Relative
populations of the central condensate 0~k and orders ±2~k vs. normalized(2)
pulse duration α = τ /τr ∼ τ /20 µs. Filled blue dots and open red squares
refer to pulses of area β = 4.5 (cf. Fig. 1) and 1.5, respectively, and the solidlines plot the function Jn2 [(β/2)sinc(α/2)]. The dashed black lines correspondto numerical fits for β = 4.5. (c) Decay of the ±4~k orders in the β = 4.5
data set, with the numerical simulation (dashed line) as well as the function J22 [(β/2)sinc(α/2)]. (d) Behavior of βeff /β, where the atomic diffraction
patterns are fit with the distribution Jn2 (βeff /2). The black solid line is the analytical form sinc(α/2), and the dashed black line corresponds to a numericalsolution of Eqs. (4.4) using β = 4.5.
some analogy to the treatment of sound-wave diffraction in [138], results in amore general modification of Eq. (4.6) to


α
β
2
sinc
,
(4.9)
Pn = Jn22where the sinc function arises from the Fourier transform of the square pulse.
Indeed, the populations P0,±1 predicted by Eq. (4.9) agree with those of
Eq. (4.7) up to O(α2 β 2 ), independent of the pulse duration α or pulse area β.
40For [β/α]1/2 ≤ 1, the agreement of Eq. (4.9) with the experimentally observed
patterns is excellent, as can be seen in Figs. 4.2 (a) and (b).
It is interesting to note the connection of Eq. (4.9) to the resonant caseof nth order Bragg diffraction from a moving standing wave (with a relative
detuning δω = n2 ∆ between the two beams). In this case, the Fourier transform of the applied potential V0 cos2 (kz + δω t/2) with duration τ would leadto a frequency distribution ∝ sinc[(ω − δω)τ /2] (i.e. identical to that of thestanding wave case, but shifted by δω) making available the resonant frequency
component, independent of the pulse duration.
The form of Eq. (4.9) shows that although the populations Pn in a spectrum
may follow a Bessel distribution as predicted by the Raman–Nath Eq. (4.6),
a fit will not necessarily return the correct modulation depth. The apparent
pulse area in Eq. (4.9) is given by βef f /2 = (β/2)sinc(α/2), which means thatfor a pulse of duration τ , the apparent modulation depth of the standing wave
extracted from a diffraction spectrum is reduced from its short-pulse value
V0 = ~β/τ to V0,eff = ~βef f /τ , i.e.
V0,eff = V0 sinc τ /2τr(2) .

(4.10)

The apparent pulse area returned by a fit of Eq. (4.6) to the diffraction
patterns is shown in Fig. 4.2 (d) for both the β = 4.5 and β = 1.5 data sets. For
the latter, the fit results agree with Eq. (4.10) over the full range, consistent
with the fact that orders |n| > 1 are not populated. The first-order revivals canthus be interpreted as arising from a sampling of the Fourier transform of thepulse. For β = 4.5, the observed behavior still shows qualitative agreementwith revivals from sampling of the sinc-shaped frequency distribution withincreasing pulse duration τ , but clearly exhibits deviations when higher-ordermomentum states are present.

4.5

Strong-pulse dynamics

For more intense pulses with n̄ = [β/α]1/2 > 1, the simple generalization of
Eq. (4.9) no longer holds, as is evident by the large discrepancies in Fig. 4.2
for α < 4.5 (τ < 90 µs) in the β = 4.5 data set.
The presence of higher diffraction orders leads to population decays thatare faster than those determined by the two-photon timescale in Eqs. (4.9)
and (4.10). This is shown in inset (c) of Fig. 4.2 for the ±4~k (n = 2)
orders, which require the exchange of four photons in the transition from 0~k.
In general, the presence of many higher momentum orders will complicate
analytic descriptions of the diffraction dynamics.

41

Momentum State Populations









0.4
0.2
0 
• •





• •





τ = 8 µs

(a)






• • • •










• •

•








• • •

0.4



0.2
0 • •
0.8











• • • •





• •





0.6



0.4
0.2



0 • •
0









• •












• •
10

• • •



20



• •



•






 

τ = 25 µs

0.8
0.6










• • •

•







•

• •





τ = 50 µs









•





•

30

40


• • •


•

•






50

(b)

60

Fit Depth [Erec] from 50 µs data

1 
0.8
0.6

50

•

•



60

40





30


20










10
0













0



10

20 30

40 50 60

Expected Lattice Depth [Erec]from 8µs data

Lattice Depth [Erec]

Figure 4.3: (a) Comparison of experimental diffraction spectra and numericalfits. For each data set, numerical simulations [using Eqs. (4.4)] of the momentum state distribution are shown as solid, dashed, and dotted curves for the
0~k, ±2~k, and ±4~k diffraction components, respectively. The red crosses
are data for the 0~k order, and the blue open squares (green filled dots) aredata averaged for the ±2~k (±4~k) orders. (b) Comparison of numerical lattice depth calibrations [using Eqs. (4.4)] from relatively short (8 µs) and long
(50 µs) optical lattice pulses for various intensities of the standing light wave.
The results for numerically-fit lattice depths agree to within 4% (the solid line
has a slope of 1).
Nevertheless, it is possible to accurately describe the observed dynamics
by numerical integration of the coupled differential Eqs. (4.4), truncated for
unpopulated higher orders beyond n̄. This is shown in Fig. 4.2 and Fig. 4.3 (a).

4.6Single-shot calibration of optical lattices

In experiments with ultracold atoms in optical lattices, the tunneling ratedepends exponentially on the lattice depth V0 [18], for which an accurate
knowledge thus is essential. Unlike most other methods (cf. [18]), applying
a “short” Kapitza–Dirac diffraction pulse can conveniently reveal the lattice
depth in a single-shot measurement. However, the application of well-defined
pulses that are short enough to be deeply in the Raman–Nath regime, yetstrong enough to yield significant diffraction, can be technically challenging.
Our study quantified how the results of such a calibration measurement need
to be corrected if pulses of finite length are used.
In Fig. 4.3 we present the comparison of experimentally observed mo42

mentum state distributions to numerical simulations, for three different pulse
lengths (8, 25, and 50 µs), as the lattice depth is varied from 0 to 65 Er . While
the observed momentum state distributions are close to the typical Raman–
Nath form Eq. (4.6) for the 8 µs pulses, this is no longer the case for longer
pulses. For all pulse lengths, numerical simulations agree well, as can be seenin Fig. 4.3 (a). The lattice depth calibrations for the three pulse durations,determined by comparison of experimental data to numerically simulated patterns, agree to within 4%, as can be seen in Fig. 4.3 (b) for the extreme casesof 8 and 50 µs. Despite the complicated form the dynamics take, one canthus obtain a reliable lattice depth calibration even with longer pulse durations. We have independently verified the method described here with anothersingle-shot calibration method for deep lattices (adiabatic lattice rampup, followed by a sudden projection of the ground band population onto the ±2~kplane-wave states [139]), for which we found comparable agreement to within
10%.

4.7

Concluding remarks

We have studied Kapitza–Dirac diffraction of a Bose-Einstein condensate in apulsed standing light wave, considering the case that the pulse area remains
constant as the pulse duration is varied. We find that for sufficiently weakpulses exciting only the ±2~k orders, the usual analytical short-pulse prediction continues to hold for longer times, albeit with a modification of theapparent modulation depth of the standing wave. We relate this effect tothe frequency spread of the square-wave pulse, and also draw analogies to the
Rabi dynamics of a coupled two-state system. Our findings are of practicalinterest for the calibration of optical lattices in ultracold atomic systems, andwe show that for a general length and strength pulse, relatively simple (neglecting mean-field effects) numerical modeling can be used well outside theRaman–Nath regime to accurately determine the lattice depth.

43Chapter 5
Quantum dynamics of matterwaves in a pulsed
incommensurate lattice
In this Chapter, we experimentally study the dynamical response of weaklyinteracting atomic matter waves to a periodically pulsed optical lattice formedby two standing-waves of incommensurate spatial periodicity. While periodic
driving with a single standing wave results in ballistic spreading of momentum
wavepackets at quantum resonances and dynamical localization otherwise, we
observe radically different behavior with the pulsed incommensurate lattice.
Here, the spatial quasidisorder causes a suppression of momentum growth atresonance, and delocalization in momentum space for off-resonant driving.
The observed breakdown of quantum interference effects can be explained bya mapping to a system of coupled kicked rotors, which has been predicted
to display classical-like behavior even for a purely quantum evolution. This
chapter is partially based on our manuscript Quantum dynamics of matterwaves in a pulsed disordered lattice, arXiv:1203.3177v1 (2012) [50].

5.1

Introductory discussion

In isolated ultracold atomic systems, the quantum nature of matter can bemade manifest in striking ways. One example arises in the dynamics of quantum chaotic systems, i.e. systems whose classical counterparts are chaotic,
and in which destructive interference can suppress the onset of chaos [140].
The δ-kicked rotor model [140] has found particular relevance in experimental
studies of driven quantum particles, as realized with cold neutral atoms periodically kicked by trains of optical standing wave pulses [82]. Whereas regimes

44

of fully chaotic behavior with diffusive growth of the momentum variable canbe found in the classical system, the quantum analog generically displays dynamical localization due to quantum interference [140–142], which is analogous to one-dimensional (1D) Anderson localization [143] in real space [83].
Foundational experiments on quantum chaos in driven atomic vapors havedemonstrated such dynamical localization [82, 144]. In recent years, the use ofcoherent rather than thermal sources of atoms has allowed for the observationof ballistic spreading in momentum space [145, 146], relating to a quantum
walk [147, 148], when the frequency of δ-kicking corresponds to a discreteenergy resonance.
Studies of the contrasting dynamics of classical and quantum systems [149],
in particular those addressing how classical behavior arises in the presenceof noise [150–153] and nonlinearities [84, 154–157], are central to the understanding and control of quantum decoherence [158]. Such a transition towards
classical behavior was observed in early studies of δ-kicked rotors under the
influence of random noise and decoherence [159–161]. Remarkably, classicalbehavior has also been predicted to already emerge in a simple driven quantum system consisting of two coupled kicked rotors [84], providing hope thatthe disparate behavior of quantum and classically chaotic systems may bereconciled in the macroscopic limit.
Here, we have explored the influence of controlled spatial quasidisorder on
the dynamics of a quantum kicked rotor, which was realized with ultracold
atomic matter waves in a periodically pulsed optical lattice. We observe thatthe addition of a second, incommensurate optical destroys hallmark behaviorof the kicked rotor system, namely ballistic momentum spreading at quantum
resonances and dynamical localization for off-resonant driving. We show that
our configuration effectively realizes a system of two coupled kicked rotors,
where the coupling suppresses transport at resonance and induces transport
away from it. In addition to a direct connection to the field of quantum chaos,studies of wave propagation in disordered media (as with those in photonic
lattices [162, 163]) may also provide insight into the influence of disorder ontransport in solid materials.

5.2

Experimental procedure and kicked rotor
systemOur system consists of a nearly-pure, optically-trapped Bose–Einstein condensate (BEC) of (1.4 ± 0.4) × 105 87 Rb atoms in the |F, mF i = |2, −2i hyperfine
ground state. The atoms are driven by pulsed optical lattices oriented along

45

(a)
1 T

2

τ

N -1

s1

s1, 2 = { 0 , 100 }

(c)

y

N=1

N=1

z

N = 40s1, 2 = { 50 , 80 }N = 40

10-1

10-1

N = 4010-2|ψ(p)|2|ψ(p)|2

N

s2

(b)

10-3
10-4

λ1
λ2

-10

-5

0

p / ħk1

5

10-3
10-4

10

N = 40

10-2

-10

-5

0

p / ħk1

5

10

Figure 5.1: Atomic matter waves in a periodically-pulsed optical lattice potential. (a) A Bose–Einstein condensate is exposed to a train of N pulses (duration
τ , separation T ) of two incommensurate standing waves with wavelengths λ1,2and lattice depths s1,2 . (b) Time-of-flight diffraction spectra (averaged over 34 images) of atoms released after N = 1 and 40 kicks, for driving with a single
lattice (lattice depths s1,2 = {0, 100}). The momentum distribution along z(integrated along y) after N = 40 kicks is shown below. (c) As in (b), but for
driving with two incommensurate lattices (s1,2 = {50, 80}). The dashed blackline at the bottom of (c), which serves as a guide to the eye, is a Gaussian
profile corresponding to diffusive spreading.
the z-axis, as depicted in Fig. 5.1 (a). We perform simultaneous pulsing by twolattices of wavelengths λ1 = 1064 nm and λ2 = 782 nm, with wave numbers
k1(2) = 2π/λ1(2) . The lattice depths are characterized as V1(2) = s1(2) ER (where
ER = ~2 k12 /2M is the recoil energy of the first lattice and M the atomic mass),
and are calibrated via Kapitza–Dirac diffraction [122]. Subsequently, we use
pulses of duration τ = 2 µs (Raman–Nath regime), spaced at a variable period T . After applying N pulses, we immediately release the atoms and allow
them to freely evolve in time-of-flight for 16 ms before performing absorptive
imaging of momentum distributions, as displayed in Figs. 5.1 (b,c).
The system subject to a train of N lattice pulses is approximately described

46

2

(a)

N = 50s1 = 106

4

N = 30s1 = 10

ε

ε

1

(c)

2

0
35

36

37

0

T [µs]2(b)

N = 50s2 = 10

6

4

115

125

135

115

125

135

T [µs](d)
N = 40s2 = 10

ε

ε

1

105

2

0
35

36

T [µs]37

0105

T [µs]

Figure 5.2: Scan of the pulse-period T revealing resonant and off-resonant
behavior. The average energy per particle, ε (in units of ER ), is measured
for clouds of atoms that are subject to a train of N pulses of a standing-wave
optical lattice, of wavelength λ1 and lattice depth s1 ER . (a,b) Results for apulse period of T ∼ 36 µs, which is off-resonant for both the first and second
lattice (λ1 = 1064 nm, λ2 = 782 nm). (c) For a pulse period of T ∼ 124 µs,
driving with the lattice of wavelength λ1 fulfills the first Talbot resonance,such that κ/4π = 1. (d) For this same value of T , driving with the second
lattice (λ2 ) is off-resonant. A nearby higher-order quantum resonance is seen
for T ∼ 133 µs, with η 2 κ/4π = 2. The solid lines (sinc2 functionals, motivatedby the square-pulse shape) serve only as guides to the eye. The dashed lines
in each plot refer to the pulse periods that were used for off-resonant andresonant driving.
by the 1D Hamiltonian
H = −~2 ∂z2 /2M + S(t)ER /2 ,

47

(5.1)

with
S(t) = [s1 cos(2k1 z) + s2 cos(2k2 z)]NXu(t/τ − jT /τ ) ,

(5.2)

j=1

where u is the normalized boxcar function. The approximation ignores effects of the trapping potential (ωz /2π ≈ 30 Hz along the axis of kicking
and mean frequency ω̄/2π ≈ 20 Hz) and interactions (chemical potentialµ/h ≈ 0.3 kHz), which is justified if the total pulsing duration is kept lessthan both 2π/ωz and h/µ. Also, the actual intensity profiles are slightly
smoother than the assumed rectangular shape, but with an equivalent area.
Further, approximating the pulses
P as δ-functions allows us to write S(t) =[s1 τ cos(2k1 z) + s2 τ cos(2k2 z)] N
j=1 δ(t − jT ), which for pulsing with a single
lattice corresponds to the standard δ-kicked rotor [82]. To make the connection
to this well-studied problem, we reexpress the Hamiltonian in dimensionlessform asNX
02
H = ρ /2 + K(φ)δ(t0 − j) ,
(5.3)
j=1

with
K(φ) = K1 cos(φ) + K2 cos(ηφ)(5.4)

and η defined by the ratio η = k2 /k1 ∼ 1.36, and rescaled quantities t0 = t/T ,H 0 = (8T 2 /~TR )H, φ = 2k1 z, ρ = (2k1 T /M )p (recoil time TR = ~/ER ). Here,the conjugate variables φ and ρ obey the commutation relation [φ, ρ] = iκ,with an effective Planck constant κ = 8T /TR . The so-called stochasticity
parameters K1(2) = 4s1(2) τ T /TR2 serve to delineate regimes of classically regularand chaotic motion. Resonant driving by the first (second) lattice is achieved
when κ/4π (η 2 κ/4π) is equal to a rational number [or in the parlance of Talbot
interference [146], when T matches a rational multiple of the Talbot time
TT = TR π/2 (TR π/2η 2 )]. In Fig. 5.2, we plot experimental results of scans of
the pulse period T , which demonstrate growth on resonance and a lack thereofaway from it.

5.3

A system of coupled kicked rotors

In general, we may expect that adding the incommensurate lattice causes deviations from behavior typical to the δ-kicked rotor. For example, the scenario ofincommensurate and irrational values of η corresponds to a quasirandom realspace lattice [164], where the sets of momentum modes |mi and |ni coupledby each of the two light fields have no intersection (and only sparse overlap for
most incommensurate values). The absence of direct coupling between these48

modes allows us to describe the Hamiltonian as that of an effectively 2D system. Of course, this assumes that the momentum width of the distribution
is negligible compared to the spacing between populated modes, necessitating spatially coherent samples
N as in a BEC. In the discrete plane wave basisof the lattice, |m, ni = |mi |ni with momenta pm,n = 2(m + nη)~k1 , theHamiltonian can then be written as
H = HT + ~φVNX

δ(t − jT ) ,

(5.5)

j=1

where the kinetic and potential terms are, respectively,
X
HT = 4ER(m2 + 2ηmn + η 2 n2 )n̂m,n

and

(5.6)

m,n

φV = (TR /8T )X[K1 (â†m−1,n + â†m+1,n ) + K2 (â†m,n−1 + â†m,n+1 )]âm,n . (5.7)
m,nHere n̂m,n is the number operator and â†m,n (âm,n ) is the creation (annihilation)
operator for the mode |m, ni.
In a stroboscopic description [83], the parameters K1,2 control the strengthof tunneling within each of the two sets of non-overlapping modes, |mi and|ni, whose quasienergies (phase accrual between kicks) are given by the distributions 4ER T m2 mod(2π) and 4ER T η 2 n2 mod(2π), which are pseudorandom
for irrational κ/2π (η 2 κ/2π). While the absence of mode overlap guarantees
a higher effective dimensionality, there is a coupling between the two dimensions in the momentum variables, in the form of the kinetic energy cross term
8ηmnER , which arises from the fact that the system (with quadratic dispersion) is still physically 1D. Systems of coupled kicked rotors have been shownto have a quantum-to-classical transition [84] even for the simplest possible
case of only two rotors. In the following, we experimentally investigate indetail how this coupling influences the kicked rotor dynamics in our system,
both away from and at quantum resonance.

5.4

Off-resonant dynamics

We first investigate the influence of an incommensurate lattice on dynamical
localization [82], using a pulse period of T = 36 µs that is off-resonant for eitherof the two lattices. When kicking with a single lattice only (K1,2 = {0, 4.6}),
the atomic population remains localized in the lowest momentum orders, as canbe seen from the TOF images in Fig. 5.1 (b). For increasing N , we observe in49

(a) K1,2 = {0, 4.6}

6

ε

4
2
0

(b) K1 = K2 = 2.3
0

20

60 0

40

N

K2 = 0 (c)∆ ε / ∆N

.05

20

K1 = 0 (d)

40

N

60

K1 = K2 (e)

0
0

1

K1

0

2

(f )

2

K2

4

0

1

(g)

4 1-b

15

4

1-c

10

ε40 - ε1

3

K2

3

K2

2
1

2
1

expt.
0
0

2

K1,2

1

K1

2

0

sim.
0

0

5

1

K1

2

Figure 5.3: Dynamics of a BEC under off-resonant kicking in the presenceof disorder. (a) Kicking by a single lattice, with stochasticity parameter
K2 = 4.6. Shown is the per-particle energy ε as a function of kick number
N . The blue solid and red dashed lines are simulated quantum and classicaltrajectories, while the black points are experimental data with statistical error
bars (empty circles for individual runs). (b) Off-resonant kicking with two
incommensurate lattices (K1 = K2 = 2.3). (c,d,e) Rate of energy growth,∆ε/∆N , as determined from linear fits to data and simulated points in (a,b).
Filled black points are experimental, open blue disks are numerical, and error
bars represent the standard error of the linear fits. Shown are the cases of (c)only the first lattice, (d) only the second lattice, and (e) both lattices present
with the same strength of kicking K1 = K2 . Red points in (c,d) are classicalgrowth rates. (f,g) Measured and simulated change in the mean energy ε between N = 1 to 40 kicks as a function of K1 and K2 (8-9 sampled points ineach direction). The circles labeled 1-b and 1-c highlight data derived fromthe distributions of Figs. 5.1 (b,c).
Fig. 5.3 (a) that the per-particle energy ε (in units of ER ) shows no net increaseover 60 pulses. This previously observed [157] behavior is unique to the use ofdegenerate gases occupying a single momentum mode, and is distinct from the50case of dynamical localization in laser-cooled samples of atoms in magnetooptical traps (MOTs), where the initial momentum distribution spans all of therelevant phase-space, and where the energy growth saturates after a “quantum
break time” (tB = K 2 /4κ2 , in units of T ), with exponential localization in
momentum-space [82, 159]. In our experiments, the pulse period (κ ≈ 3.7)
and kicking strengths (K1,2 ) are such that tB is less than one pulse period for
all the cases considered, such that localization sets in immediately.
A theoretical comparison based on numerical simulations, without free parameters and accounting for finite pulse length, shown in Fig. 5.3 (a), confirms
the absence of growth. We start with all population in the zeroth momentum
order |ψ0 i = |m = 0, n = 0i, and then describe the total evolution from one
pulse to the next by a unitary “kick” operator Û = R̂−1 ÊL R̂Ê0 , where Ê0accounts for free evolution for a time T − τ . More explicitly,2

hm0 , n0 |Ê0 |m, ni = δm,m0 δn,n0 e−i4(m+nη) (T −τ )/TR .

(5.8)

Likewise, ÊL describes evolution of the lattice eigenstates (as determined bynumerical diagonalization) during the pulse for a time τ . R̂ maps the freeparticle eigenstates onto those of the lattice. After a train of N pulses, theatomic wavefunction is then given by |ψN i = Û N |ψ0 i. From the simulated momentum distributions, we calculate both the mean per-particle energy ε andthe rms momentum width σp (in units of ~k1 ). While typically considering up
to ±15-20 momentum orders for each lattice, we restrict the determination of εand σp to |p|/~k1 ≤ 15, which corresponds to the experimentally observed momenta. We note that the experimental data in Fig. 5.3 (a) differ greatly froma computed classical trajectory (averaged over 105 individual trajectories),
which after 60 pulses has an energy of ε ∼ 200. This rules out the possibility
that classical localization (due to Kolmogorov-Arnol’d-Moser [KAM] barriers
separating classically chaotic regions [165]) causes the observed localization.
We describe briefly at the end of this chapter the determination of the averaged
classical trajectories.
We next address the effects of adding the second, incommensurate lattice.
For the case of off-resonant kicking with two deep lattices (K1,2 = {2.3, 3.7}),
as shown in Fig. 5.1 (c), the observed atomic population clearly delocalizes inmomentum space, and the momentum distribution after 40 kicks has a nearly
Gaussian distribution, which is consistent with diffusive spreading. Fig. 5.3 (b)shows the dependence of ε on N for a similar situation with slightly weakerkicking (K1 = K2 = 2.3). Consistently, both experiment and the simulated
quantum trajectories reveal that the energy increases with N , and that dynamical localization is destroyed.
To gain insight into the disorder-driven crossover from localization to delo51

(d)(a) K2 = 0∆σp / ∆N

σp
σp
σp

3
2
1
0
3
2
1
0
3
2
1
0

.05

K2 = 8 (b)0K2 = 16 (c)
0

5

10

N

15

0

20

5

K2

10

15

Figure 5.4: Dynamical evolution of resonantly kicked matter waves (K1 = 1.6)in the presence an additional off-resonant drive (K2 ). (a,b,c) Momentumwidth σp of the atomic distribution as a function of kick number N , for apulse period T = 124 µs (κ/4π ≈ 1), and K2 = 0, 8, 16. Black points are datafrom individual experimental runs, the blue line is a numerical calculation for
an initial plane-wave, and the dashed red curve takes into account finite-sizecorrections. (d) Growth rate of the momentum width ∆σp /∆N , determinedby a linear fit to the N -dependence, as a function of K2 . The simulated growth
rates (open blue circles) are scaled by a factor of 1/2 to account for effects offinite size.
calization, we characterize the energy-growth per kick ∆ε/∆N using a simple
linear fit to the energy evolution (for 1 ≤ N ≤ 60, allowing an energy offset at N = 1). The rates of growth as a function of K1 and K2 are plottedin Figs. 5.3 (c–e) for the cases K1 = 0, K2 = 0, and K1 = K2 . In particular, Figs. 5.3 (c,d) show the absence of sustained growth for either of thesingle-lattice cases. When simultaneously kicking with both lattices, however,
we observe in Fig. 5.3 (e) a transition to a regime of growth (a region of
weak transient growth near the transition is seen in the simulation but not inexperiment). Signatures of this crossover between dynamical localization anddelocalization are confirmed through a time-independent analysis of the singlekick operator Û , based on a stroboscopic Floquet state analysis [83, 166], as
described in more detail at the end of this Chapter. Along the line K1 = K2 ,such an analysis reveals a transition to delocalized states at a value K1,2 ∼ 2.2,
in approximate agreement with the observations of Fig. 5.3 (e).
To obtain a more complete picture of the dependence of dynamical localization on the two lattice strengths, we now more fully sample the parameter-52space of K1 and K2 . Figs. 5.3 (g,f) show results for the change in ε between
1 and 40 kicks, with good qualitative agreement between data and simulations. As expected, the population remains localized when only a single lattice is used, while significant growth results when strongly kicking with twolattices simultaneously. We note that our observations are in contrast to thecase of a single-lattice driven along 1D at incommensurate temporal frequencies [41, 152, 167, 168], where similar delocalization is expected only for effectively three or more dimensions [152, 169], as opposed to weak localization in2D [170].

5.5

Suppression of resonant growth

We now turn to study the influence of the incommensurate lattice on a resonantly kicked quantum rotor, where atoms undergo ballistic transport inmomentum space (σp ∝ N ). The use of a BEC precludes trivial decoherence
from a rapid loss of spatial overlap between the momentum wavepackets. We
set the pulse period to T = 124 µs, which realizes a resonance condition for
the first lattice (κ/4π ≈ 1), while the driving with the second lattice remains
off-resonant. In Fig. 5.4 (a), we observe a (nearly) linear increase in the momentum width for up to N = 20 for weak resonant kicking (K1,2 = {1.6, 0}).
The measured growth rate is a factor of two off from what is expectedbased on plane-wave simulations. This deviation can be explained throughthe initial rms momentum spread ∆p/~k1 ≈ 0.03 of the trapped atomic distribution, which is limited by its finite size (Thomas–Fermi radius ∼ 9 µm
along z). Fig. 5.4 (a) also plots the expected on-resonance momentum growth,
averaged over the initial momentum distribution of the sample. More explicitly, we assume a Gaussian distribution with momentum width σ = 0.03~k1 ,and employ the analytical formula describing the mean energy per particle (inunits of ER ) after N lattice pulses for particles with initial quasimomentum
β = p/2~k1
s21 ER2 τ 2 sin2 (2πβN ),(5.9)ε(β, N ) =2~2
sin2 (2πβ)
which is used in and appears as Eq. (2) in Ref. [146], taken from Eq. (13) ofRef. [171].
With the addition of a deep incommensurate lattice [with K2 = 8 and 16,
Figs. 5.4 (b,c)] the momentum growth is markedly reduced with respect to thesingle-lattice case, despite a large effect of the initial kick. We again characterize the growth rate ∆σp /∆N by a simple linear fit in the range 1 ≤ N ≤ 20,
which reveals that this rate decreases with the strength of the incommensu-

53

20

20

(a)
15

a1
a2

5

5

b1

0

20

40

N

60

80

5
0

100

1

a1

0

20

b1

n 0

40

N

60

80

0

10

b2

10

-5
5

b2

10

10
0

(b)

σp

σp

15

n0

.5

a2

-10

n 0
-5
-10

0

m

10

0

Pm,n

-5 0 5m-10m

Figure 5.5: Simulated effective-2D dynamics of a coupled δ-kicked rotor system. (a) Dependence of the momentum width σp on kick number N , forthe case of resonant kicking (κ/4π = 1 ; τ = 10 ns) with a single lattice(K1,2 = {1.6, 0}, black solid line a1), and with an added deep incommensurate lattice (K1,2 = {1.6, 12.8}, red dashed line a2). (b) Dependence of σpon N for off-resonant kicking (T = 36 µs ; τ = 10 ns) with a single lattice(K1,2 = {0, 11.7}, black solid line b1), and for off-resonant kicking with twolattices (K1,2 = {5.8, 5.8}, red dashed line b2). The lower panels show the 2D
momentum distributions (Pm,n = |hm, n|ψN i|2 ) at N = 100 for all trajectories,
decomposed with respect to the two standing waves.
rate lattice K2 , as shown in Fig. 5.4 (d). A simulation for a wider range ofkicks (and for more ideal δ-kicking exactly at resonance) suggests that the
inhibition of ballistic transport is due not to a crossover to classical diffusion
as in the off-resonant case, but rather to the onset of dynamical localizationin the strong incommensurate lattice. This becomes even more apparent upon
examining the distributions in the 2D space of modes with indices m and n of
the two lattices, c.f. Fig. 5.5 (a).

5.6

2D representation

Naturally, we expect that the mechanism underlying the suppression of resonant growth should again be due to the coupling term 8ηmnER . It is clearthat when the atoms have non-zero momenta from the second lattice (n 6= 0),
this additional term destroys coherent phase revival of all modes of the firstlattice [integer 4ER T m2 /2π]. Surprisingly, however, a complete suppression54

of resonant growth is found, even though the atomic population spends a significant amount of time in the n = 0 subspace. The suppression of coherent
growth due to strong coupling into and out of n = 0 is somewhat reminiscentof the physics of the Kapitza pendulum [172] and of ponderomotive potentials
for charged particles, where strong off-resonant driving dynamically suppresses
transport.
Depiction of the momentum distributions in the 2D space spanned by thetwo lattices also helps us to visualize the breakdown of dynamical localization
for the case of off-resonant driving. Fig. 5.5 (b) shows simulated dynamics similar to the experimental situation in Fig. 5.3, however for couplings K1,2 exceeding those attainable in experiment. The dynamics of σp show that dynamic
localization occurs for kicking with a single lattice only (K1,2 = {0, 11.7}),
while kicking off-resonantly with two lattices (K1,2 = {5.8, 5.8}, same total
strength) gives rise to a dramatic delocalization in the 2D m-n space.

5.7

Concluding remarks

In conclusion, we have studied the dynamics of a BEC subject to a periodicallykicked quasidisordered optical lattice that is formed by two incommensurate
standing waves. We find that the addition of the second, incommensuratelattice leads to radically different behavior from that of the standard kicked
rotor (occuring for a single lattice), characterized by a suppression of resonant
momentum spreading at quantum resonance and diffusive transport instead ofdynamical localization for off-resonant kicking. We relate these observations
to the emergence of classical behavior in a system of two coupled kicked rotors.
A possible extension of our study would be the realization the kicked Harper
model [173, 174], for kicked harmonic oscillators, with a strong nonlinear coupling term. This could be studied, for example, by examining the single-siteorbital (band population) dynamics of a periodically-tilted Mott insulator,
where the nonlinear term may be controlled through either a tunable on-sitefilling or via the interaction strength with a Feshbach resonance. Additionally,a natural extension in our system would be to study a “spinful” kicked rotor, i.e. by using atoms in a superposition of different spin states and driving
them with a state-dependent lattice beam. It has recently been proposed [175]that such a system might allow for a study of the anomalous quantum Hall
effect with driven cold atoms. A simple scenario of spinful kicking, leading toclassical diffusion in momentum-space, is discussed in Appendix A.

55

5.8

Supporting considerations

5.8.1

Classical dynamics

It is well known that classical localization can occur in kicked-rotor systems forsmall values of the stochasticity parameter, where the classical system is notglobally chaotic but supports bounded orbits due to KAM barriers [83]. To
demonstrate that this effect does not contribute to the observed localizationin momentum space in Fig. 2 (a,c,d) of the main text (i.e. for atoms kickedby only a single optical lattice), we compare the experimental data and thesimulated quantum trajectories of the per-particle energy ε as a function ofkick number N to simulated classical dynamics, averaged over 105 classicaltrajectories. To briefly detail the determination of the averaged classical dynamics, each individual classical trajectory is determined by probabilisticallyprojecting out the particle’s momentum space wavefunction after each application of the kick operator Û . We show in FIG. 5.6 (i-iv) the classical dynamics
of ε, as averaged over n = 10, 102 , 103 , and 105 individual trajectories, for thesame system parameters as used in Fig. 2 (a) of the main text [T = 36 µs ;
τ = 2 µs ; K2 = 4.6 (s2 = 100) ; K1 = 0]. For large sample sizes, the classical growth rate is found to be in fair agreement with the expected classicaldiffusion constant [166],∆ε/∆N ∼ η 2 V22 τ 2 /2~2 = 2η 2 K22 /κ2 ,(5.10)

for the second lattice (2K12 /κ2 for the case of only using lattice 1). It isobserved that for large kick numbers, the simulated classical trajectories growmore slowly than one would expect from the classical diffusion constant. Thisdeviation is due to the finite pulse length (τ = 2 µs) of the applied kicks [122],and in FIG. 5.6 (iv) we also plot the simulated trajectories for the same pulse
area but for τ = 1 µs and 0.1 µs, showing that better agreement is reached in
the limit of δ-like pulses.

5.8.2

Floquet eigenstate analysis of Û

To gain a deeper insight into the expected response of the matter waves to agiven kick operator Û (i.e. for given pulse lengths, lattice depths, and pulse
period), we analyze the properties of the matrix Û in a time-independent
manner. While monitoring the response of the matter waves as a functionof the kick number N makes a direct connection to experiment and can discern between regimes of localization and delocalization in momentum space,at short times it may be subject to transient behavior and small-scale fluctu-

56

200

(ii)

n = 10n = 102

(iii)

(iv)

n = 103n = 105

ε

150

(i)

100
50
0
200

ε

150
100
50
0

0

20

N

40

60 0

20

N

40

60

Figure 5.6: Averaged classical dynamics of the energy ε as a function of kicknumber N for different sample sizes consisting of n trajectories, for off-resonant
kicking with the second lattice [K2 = 4.6 (s2 = 100) ; K1 = 0 ; τ = 2 µs].
(i-iv) Averaged trajectories are shown for the cases of n = 10, 102 , 103 , and
105 as solid red lines. In all plots, we also show as a dashed black line theexpected classical diffusion for delta-function kicks. In (iv), we also showclassical trajectories for the cases in which the pulse area is the same, butwith smaller pulse durations of τ = 1 µs (black dashed-dotted line) and 0.1 µs(blue solid line).
ations. We are experimentally restricted to a modest number of kicks in orderto maintain a near-field treatment (i.e. assuming identical, overlapped spatial
wavefunctions for all the matter-wave fields) and to minimize contributions
from nonlinear atom-atom interactions. Motivated by pioneering work thatmade a stroboscopic connection between the quantum δ-kicked rotor modeland the 1D Anderson model on a lattice [83], we study the properties of the
Floquet (or Bloch–Floquet) quasi-energy eigenstates of the kick operator Û .
For regimes in which the growth dynamics result in dynamical localization,these Floquet eigenstates should all be localized in momentum space, while
they are delocalized in the case of diffusive growth.
We first investigate the case of kicking off-resonantly with two lattices of
equal strength (stochasticity parameter K1 = K2 ≡ K1,2 ), using the same
parameter values as in Fig. 5.3 (e). In Fig. 5.7 (i,ii), we plot the momentum
space distribution of the lowest energy Floquet eigenstates of Û , |ψ0U i, forK1,2 = 2.15 and 2.34. The plotted distributions are shifted by the mean
momentum p̄ = hψ0U |p̂|ψ0U i to p0 = p − p̄. While the distribution is localized

57

0.5

(i)

K1,2 = 2.15 (ii)K1,2 = 2.34

Pp’
0.0

-40 -200

20p’ / ħk1p’ / ħk1-40

40-40 -200

20p’ / ħk1

40

1

(iii)

-20

Pp’

0
20
40
2.0

σp’

30

2.1

2.2

K1,2

2.3

0

2.4

(iv)

20

10

0

2.0

2.1

2.2

K1,2

2.3

2.4

Figure 5.7: Time-independent analysis of the kick operator Û for off-resonant
kicking (T = 36 µs) by both lattices, with equal kicking strengths K1,2 foreach. (i,ii) Momentum distribution of the lowest energy Floquet eigenstate ofÛ , |ψ0U i, in the plane-wave basis. The momenta p are shifted by the meanvalue p̄ = hψ0U |p̂|ψ0U i to p0 = p − p̄. For low values of the kicking strength, asin (i), the eigenstates are localized in momentum space, while for larger values
they became delocalized as in (ii). (iii) Momentum distributions as in (i,ii) as
a function of the stochasticity parameter K1,2 . Beyond a value of K1,2 ∼ 2.2
a bifurcation and delocalization of the momentum distributions is observed.
(iv) For the same range of stochasticity parameter values as in (iii), we plot
the rms momentum width σp0 of the lowest energy eigenstates of Û , which
exhibits a sharp increase across a value K1,2 ∼ 2.2.
for the weaker kicking strength, it is delocalized into two regions for the larger
strength. In fact, in Fig. 5.7 (iii) we observe a bifurcation of the eigenstate
distributions as the stochasticity parameter K1,2 is increased. We can readily58

calculate the rms momentum width σp0 of these distributions, and as shownin Fig. 5.7 (iv) there is a sharp rise in the width across a value of K1,2 ∼ 2.2.
Not surprisingly, this value is in general agreement with the observed onset of
non-zero energy growth versus kick number in Fig. 2 (e) of the main text.
There exist certain superficial analogies between our experiment involving
atoms driven with two spatial frequencies and experiments performed withmultiple temporal frequencies, which result in an effectively higher-dimensional
localization when the frequencies are incommensurate. In order to explore thisconnection, we perform a Floquet eigenstate analysis in the full parameter
space of K1 and K2 . In Fig. 5.8 (i), we plot as white the regions in which
the Floquet eigenstates are localized in momentum space and as dark blue theregions in which they are delocalized, with the criterion for delocalization being that more than 10% of the population resides further than 5.5 momentum
units (~k1 ) away from the most populated mode. This plot shows localization
along either axis, where dynamical localization is expected for any value of the
kicking strength when only a single lattice is used, while delocalization generally occurs beyond some line in the parameter space spanned by K1 and K2 .
Much as in the case of driving with multiple temporal frequencies, delocalization only occurs when the strengths of each of the incommensurate frequency
components are significant. We also show in Fig. 5.8 (ii) the simulated changein ε between one and N = 40 kicks, as in Fig. 3 (b) of the main text but overa larger range of K1 values. This plot demonstrates that qualitatively similar regions of localization and delocalization are also observed in the energy
growth dynamics.

5.8.3

Lattice incommensurability

In a theoretical treatment of the quantum kicked-rotor model with two spatial
frequencies [153], the incommensurability of the two frequencies is found to
be a necessary condition for inducing a quantum-to-classical transition in the
dynamical behavior. Moreover, this incommensurability is a well-known condition for realizing higher dimensional localization in multi-frequency driving in
the temporal domain [41, 152, 166, 168], as well as for achieving quasi-random
lattices in configuration space with multiple standing-waves [33, 46]. While experimentally we work with a fixed commensurability parameter η ∼ 1.36, here
we investigate the role of this condition through simulations of the off-resonant
and resonant dynamics.
In Fig. 5.9 (a), for T = 36 µs and τ = 2 µs, we show the expected energy
growth rate ∆ε/∆N (determined for N = 1 to 60 kicks) for atoms kicked by
two lattices of equal depth V1 = V2 = 100 ER as a function of the parameter η.
A minimum in the induced growth can be seen for η = 1, as expected. However,
59

(ii)

4

4

3

3

15

10

ε40 - ε1

(i)

K2

K2
2

2

1

1

0
0

1

2

K1

3

4

0
0

5

0

1

2

K1

3

4

Figure 5.8: Localization-delocalization transition in the K1 -K2 plane. (i) As
a function of the stochasticity parameters K1 and K2 , we plot as white theregions in which the Floquet eigenstates of Û are localized in momentum space
and as blue the regions in which they are delocalized, as detailed in the text.
A general trend of delocalization for strong kicking with both lattices, andlocalization when kicking with only a single lattice, is observed. (ii) Here we
replot the simulation data from Fig. 5.3 (g), showing the change in energy
ε from the first to N = 40 kicks, but over a larger range of K1 values. Behavior qualitatively similar to the localization-delocalization plot in (i) can beobserved in this dynamical response data.
the dependence of the growth rate on the commensurability condition becomessomewhat intertwined with other η-dependent effects. Most importantly, while
the wavelength of the first lattice is kept fixed at 1064 nm, the second lattice’s
wavelength varies with η, such that full or fractional quantum resonances [146]are encountered whenever 4η 2 T ER /h equals a rational number. To disentangle
these two contributions, we show in Fig. 5.9 (b) the growth rate dependence on
η, averaged over a large sample of pulse periods to smooth out effects of quantum resonances. Specifically, we average over the randomly chosen set of values T = {36, 32, 59, 67, 98, 68, 62, 86, 84, 82, 76, 85, 48, 88, 41, 87, 99, 90, 55} µs.
Here, we observe a much clearer minimum at η = 1 for the case of full commensurability.
Now we investigate the dependence of resonant dynamics on the commensurability of the two lattices. It was observed in Fig. 5.4 of the main text thatthe addition of an incommensurate lattice reduces the growth of the rms momentum width as a function of kick number, ∆σp /∆N . In Fig. 5.9 (c) we show
the dependence of the growth rate on η for the case V1 = V2 = 10 ER with
60

0.8

Δ ε / ΔN

0.6
0.4
0.2
0.0

(i)

0.4

Δ ε / ΔN

0.3
0.2
0.1
0.0

(ii)

∆σp / ∆N

0.25

0.20

0.15

0.10

(iii)
0.6

0.8

1.0

η

1.2

1.4

Figure 5.9: (a,b) Dependence of induced energy growth for off-resonant driving by two lattices on the commensurability parameter η. In (a) the growthrate ∆ε/∆N is shown as a function of η = λ1 /λ2 , for kicking with two latticesof equal strength (V1 = V2 = 100 ER for τ = 2 µs) and a pulse period of
T = 36 µs. The wavelength of the first lattice is kept fixed at λ1 = 1064 nm,
while the second is varied with η. A local minimum at η = 1 can be observed,
while other features may be due to full or fractional quantum resonances ofthe second lattice. (b) Same dependence of ∆ε/∆N on η, but averaged over anumber of pulse periods T to smooth out features due to quantum resonancesof the second lattice. (c) Effect of commensurability parameter η on the breakdown of momentum growth for on-resonant kicking. The pulse period is set
to T = 123.3 µs (κ/4π = 1) to be resonant with the first lattice, with pulse
widths of τ = 2 µs and lattice depths of V1 = V2 = 10 ER . For values of η 6= 1
there is a strong suppression of the growth rate ∆σp /∆N , while local maxima
appear at η 2 ∼ 0.5 and 1.5, i.e. when η 2 κ/4π is near a rational number.

61

timings T = 123.3 µs and τ = 2 µs. A clear maximum is observed for η = 1,
as both lattices contribute to the resonant growth, while a general suppression
is observed otherwise. Smaller resonances can be observed for η 2 ∼ 0.5 and
1.5, when the second lattice fulfills a fractional quantum resonance.

62

Chapter 6
Strongly interacting
Bose gases I:
theoretical concepts
In this chapter, we discuss some fundamental concepts for the study of stronglyinteracting, lattice-confined atomic gases, where effects of atom-atom interactions are of central importance. Theoretical descriptions are given both for thecase of deep lattice confinement, where a basis of localized atomic wavefunctions is appropriate, and the case of strongly-interacting 1D gases in a weak(or vanishing) longitudinal optical lattice.

6.1

Bose–Hubbard model

A few years after the achievement of BEC [4, 5], it was realized [29] thatultracold bosons confined to an optical lattice potential could be an ideal
experimental implementation of the Bose–Hubbard model (BHM) [28]. As
the simplest description of interacting bosons in a period potential that fully
maintains quantum correlations, the BHM is important for describing a number of experimental systems, including that of superfluid 4 He in porous media,
Josephson junction arrays, and strongly-coupling BCS pairs. The experimental realization of the BHM several years later [123, 176], and the observationof a quantum phase transition between superfluid and an insulator of bosons,has thus been of great importance, not only with respect to study of bosonicsystems, but in paving the way for quantum emulation of non-trivial systems
with highly-controllable ensembles of ultracold atoms [27]. Here, we discuss
concepts relevant to implementation of the Bose–Hubbard model with ultracold neutral atoms.

63Let us first recall the second quantized Hamiltonian describing a system
of interacting (via elastic two-body collisions) scalar bosonic particles in a
potential Vpot (x), as introduced in Chapter 2,
Z
Ĥ =



~2 ∇2
+ Vpot (x) ψ̂(x) +dxψ̂ (x) −2m
Z Z1dxdx0 ψ̂ † (x)ψ̂ † (x0 )Vat (x − x0 )ψ̂(x)ψ̂(x0 ) , (6.1)
2†where ψ̂ † (x) and ψ̂(x) are the bosonic field creation and annihilation operators. The cold atom-atom collisions are described by the interatomic potential
Vat (x) = gδ(x), where the coupling constant g is given by g = 4π~2 as /m, withas the s-wave scattering length.
To faithfully describe the correlations in a system of strongly interacting bosons, it is necessary to forgo a mean-field description – as adopted in
Chapter 2 in introducing the Gross–Pitaevskii equation – and maintain field
quantization. For the case of an optical lattice potential, and allowing for a
slowly-varying external potential, Vpot (x) = Vlatt (x) + Vext (x), a more approximate description of the Hamiltonian for strongly-coupled lattice bosons canbe found. We first expand the field operator ψ̂ † (x) in the basis of localized
Wannier states as
X (n)
(6.2)
ψ̂(x) =
âi wn (x − xi ) ,i,n(n)

where âi is the annihilation operator for a boson in the n-band Wannierorbital located on site i.
Under typical experimental conditions, a great simplification of the Hamiltonian is achieved through the tight-binding approximation, by considering
only ground band Wannier orbitals (n = 0) and by neglecting off-site interactions. Furthermore, direct tunneling beyond the nearest neighbor can usually
be neglected. These conditions require that the atomic samples be of sufficiently low temperature not to populate higher bands, and that the lattices beof sufficient depth such that the energy gap to the first-excited band is muchlarger than either the tunneling or on-site energies.
With these conditions fulfilled, the Hamiltonian can then be recast as
ĤBH = −tX †XUXn̂i (n̂i − 1) +n̂i (εi − µ) .
(âi âi+1 + â†i+1 âi ) +
2 i
i
i

(6.3)

Here, â†i and âi are the creation and annihilation operators for atoms at sitei, obeying the standard bosonic commutation relations [âi , â†j ] = δij , and64n̂i = â†i âi is the number operator. The chemical potential µ serves as a Lagrangian multiplier to fix the mean atom number of the system in the grandcanonical ensemble. The nearest-neighbor tunneling matrix element t, theon-site interaction energy U , and the site-dependent energy offset εi , are respectively defined in relation to the Wannier orbital wavefunctions and theapplied potentials as


Z
~2 ∇2
∗
+ Vlatt (x) w0 (x − xi+1 ) ,(6.4)
t = ti = − dxw0 (x − xi ) −2m
Z4πas ~2U=
dx|wo (x)|4 , and
(6.5)
m
Z


(6.6)
εi = dxw0∗ (x − xi ) Vext (x) w0 (x − xi ) ≈ Vext (xi ) .
For fixed chemical potential, a quantum phase transition between superfluid to Mott insulator is encountered as a function of t/U , due to a competition between kinetic and interaction energy. Here, superfluid refers to a stateof atoms with non-zero number variance, a continuous spectrum of excitedstates, and supporting mass currents, i.e. density response to variations of
the chemical potential. Mott insulator refers to a state that has zero number variance, has a gapped excitation spectrum, which makes it insensitive tosmall changes in the chemical potential, suppressing mass currents. In practice, AMO systems allow for a dynamical tuning of these parameters over awide range, and provide access to a large range of the phase diagram. The
on-site interaction U may be directly tuned via a Feshbach resonance [19],where the magnitude of as can be controlled over orders of magnitude, andthe sign may even be inverted [177]. In our system, we control both thetunneling t and the on-site energy U through change of the optical lattice
depths sER (recoil energy ER = ~2 k 2 /2m, lattice wavevector k = 2π/λ). We
plot in Fig. 6.1 the dependence of t and U on the depth s, based on bandstructure calculations, for the case of an isotropic 3D lattice of wavelength
λ = 1064 nm and a s-wave scattering length of 100 a0 . For comparison with
the numerically determined parameter values, we also provide analytical forms
for t and U applicable for deep lattices s  1, based on analytical solutionsof the Mathieu equation and on the approximation of Gaussian wavefunctionsin harmonic potentials, respectively [107]. For deep
the tunneling√√ lattices,
3/4 −2 s, and the onparameter t is approximately given as t/ER = (4/ π)s esite interaction between two√atoms for the case of isotropic
latticedepths is√
3/4
given as U/ER = 4kas s / 2π, or as U√ = 2~ω(as /σ)/ 2π in terms of theharmonic oscillation frequency ω = 2ER s/~ and harmonic oscillator length

65

0.8

t [ER]U [ER]

10-2
10-3
10-4

(b)

0.6

100
50

0.4

10
5

U/t

(a)

10-1

1
0.5

0.2

10

20

s

30

40

(c)

10

20

s

30

40

5

10

s

15

20

Figure 6.1: Bose–Hubbard model parameters t and U as a function of the lattice depth sER . (a) Magnitude of the tunneling matrix element t/ER obtained
from the width of the lowest lattice band t = ∆E/4. (b) On-site interactionenergy U/ER for a 3D geometry (red), with an assumed s-wave scattering
length as = 100 a0 and wavelength λ = 1064 nm. Also shown in blue is U/ER
for a 1D geometry, with lattice depths of s⊥ = 40 along the transverse axes.
(c) Ratio of t/U as a function of lattice depth sER , for the 3D and 1D geometries, in red and blue. For all cases, the dashed lines represent approximate
analytical forms as discussed in the text.
p
σ = ~/mω. In the case of unequal lattice depths or√ spacings along different
axes, the on-site energy is given as U = 2~ω̄(as /σ̄)/ 2π, where the bars referto geometric means of the different axes. The distribution of site-dependent
energy offsets εi is usually determined by the slowly-varying Gaussian envelope of the optical dipole trapping beams. However, as will be discussed insubsequent chapters, a more active control over the εi distribution – which
can be implemented, e.g., with additional optical potentials [46, 68, 69, 75] orwith atomic potentials in the two-component BHM [44, 46, 47, 66, 67, 71] –can be used to study phases of the disordered BHM.
A qualitative understanding of the BHM phase diagram can be obtainedby examining the extreme limits of the competition between kinetic (t) andinteraction (U ) energies, which promote delocalization and localization the
single-particle wavefunctions, respectively. In the limit of vanishing interactions U/t → 0, the problem reduces to that of non-interacting bosons in a
lattice band-structure. The lowest energy eigenstate in the ground-band is at
q = 0, and all the particles will Bose condense into this lowest energy Blochstate. We can express the state of N particles condensed into this Bloch state,delocalized across a system of M sites with uniform phase, in the Wannier

66

orbital basis as

N

M
X
†
−1/2
âj |0i .
|ψSF i ∝ M

(6.7)

j=1

This pure superfluid (SF) state is a coherent number state, and it can be
written as a product state of uncorrelated single-atom wavefunctions. We
note that, although we use the standard terminology of a “superfluid”, inthe complete absence of interactions the typical hallmarks of superfluidity –the possibility of creating vortices, a linear phonon branch of the dispersion
relation at low quasimomentum, etc. – are not present in the system.
In the opposite limit of negligible tunneling and/or strong interactions(t/U → 0), repulsive interactions between atoms cause the position of a given
atom to depend on the positions of the other atoms. The total energy of the
Hamiltonian is minimized by reducing the sum of non-linear interaction energies on each site, which is achieved by having uniform site occupancies acrossthe sample. This strongly-correlated state may be written in the occupationbasis of Wannier states as
MY|ψM I i ∝(â†j )N/M |0i ,

(6.8)

j=1

and is referred to as the Mott insulator (MI) state (assuming an integer valueof N/M relating to a commensurable density). The MI state has a minimized variance in the number distribution (number-squeezed), and a maximized variance in the relative phase distribution, characterized by a loss ofphase coherence across the sample. For more general ratios of t/U , the groundstate and excitations may be determined either by solving the system exactly,or through approximate theoretical techniques, as described below. In theseintermediate regimes, even when the superfluid density is non-zero and thestate may be characterized as a superfluid, it can display strong many-bodycorrelations [178].

6.1.1

Mean-field treatment of the BHM

One approach used to approximately determine the state of a system for agiven set of parameter values is to make a mean-field approximation [29, 179–
181] that neglects the correlations between the individual particles. Similarly to the approach used to describe mean-field interactions in the Gross–
Pitaevskii
equation, we introduce a superfluid order parameter defined aspψ = hn̂i i = hâ†i i = hâi i, which reflects the mean value of a spatially homogeneous condensate field (with ψ chosen to be a real number). The creation and
67

(a)
|ψ|

3

n=3
µ/U

2

(b)

0.5
0.4
0.3
0.2
0.1
0.0
0

n=2
1.0

1

0.8

|ψ|

n=1
0
.05

.1

.2

.3

.4

zt / U

.5

(c)

0.6
0.4
0.2

n = 0 (vacuum)
0

.1

.15

zt / U

.25

.2

0.0
-1

-.5

0

Log10 [z t / U]

.5

1

Figure 6.2: Mean-field phase diagram of the Bose–Hubbard model (BHM).
(a) Areas of zero (blue) and non-zero (white) superfluid order parameter are
shown, delineating the superfluid and insulating regions of the BHM. The
phase diagram is plotted as a function of the chemical potential scaled by theinteraction energy, µ/U (where the mean occupancy is hni = µ/U + 1/2), andthe scaled tunneling energy zt/U (with z the number of nearest neighbors).
The insulating regions are characterized by a fixed atom number per site.
(b) Superfluid order parameter |ψ| (normalized), across the phase-transition
boundary for variable zt/U and fixed average filling of unity (µ/U = 0.5).
(c) Over a larger range of zt/U values (in log-scale), a fuller growth of thesuperfluid order parameter can be seen.
annihilation operators are then given by their mean values and a fluctuation
term about the mean, as âi = ψ + ∆âi and â†i = ψ + ∆â†i . We may then insert
the expressionâ†i âj = (ψ + ∆â†i )(ψ + ∆âj ) ≈ ψ 2 + ψ(â†i − ψ) + ψ(âj − ψ) = −ψ 2 + ψ(â†i + âj ) ,(6.9)
where we have neglected second-order fluctuations, into the Bose–Hubbard
Hamiltonian. The Hamiltonian then reduces to a set of decoupled single-site
Hamiltonians of the form [181]Ĥi0 = −

µ
1
zt
zt †
(âi + âi )ψ − n̂i + n̂i (n̂i − 1) + ψ 2 ,U
U2U(6.10)

where we have rescaled all terms by the on-site energy U . Here, z = 2d is thenumber of nearest neighbors for a lattice of dimensionality d.
To find the ground state for a given set of parameters z, t, U , and µ, we
variationally minimize the total energy as a function of ψ. Fig. 6.2 (a) plots68the variationally determined phase-diagram depicting regions in which the
expectation value of Ĥi is minimized by either a zero or a non-zero superfluidorder parameter, relating to insulating and superfluid regions. The value of thesuperfluid order parameter ψ becomes non-zero outside of the Mott insulator
lobes, as shown in Fig. 6.2 (b,c) for fixed chemical potential µ/U = 0.5, andapproaches its maximum value in the limit zt  U . In nearly all experimental
realizations, the system has a fixed total atom number but a spatially varying
chemical potential µ, as defined by the inhomogeneous trapping potential. Fora uniform value of zt/U about a sample in the atomic limit, this results in socalled “wedding cake” or “ziggurat” density structures [29, 76–78, 182, 183]. in
which domains of different quantum phases - here SF and MI - coexist within
the same inhomogeneous sample, as determined by the local density.

6.1.2

Exact solutions of the BHM for small systems

For systems of relatively small size, one can also perform exact numerical
diagonalization of the BHM (Eq. 6.3) for fully soft-core bosons (i.e. no restrictions on site occupancy). For a fixed atom number N and fixed numberof lattice sites M , the number of possible
PMstates of the system (of the form
|n1 , n2 , · · · , nM −1 , nM i with a total of
j=1 nj = N particles) is given by
Ns = (N + M − 1)!/N !(M − 1)!, growing exponentially with the system size.
Here, we examine the case of commensurate filling N/M = 1 for N = 6 particles, assuming a uniform site-dependent offset energy εi = 0 and working withperiodic boundary conditions i ≡ M + i.
To start, we shall consider only the ground state of the system |φg i, whichshould change character as a function of t/U . To study the transition between
the superfluid and the Mott insulator regimes, we examine the overlap of
the exact ground state with the superfluid and Mott insulator ground states,|ψSF i and |ψM I i, which are exact only in the respective limits t/U = ∞and t/U = 0. The overlap between two pure quantum states |φA i and |φB i
can be characterized by the fidelity F = |hφA |φB i|. In Fig. 6.3 (a), we plot
the square of the fidelity of the ground state with the SF and MI ground22
2
states, i.e. FSF= |hφg |ψSF i|2 and FM
I = |hφg |ψM I i| , as a function of t/U .
Another way to characterize the overlap between two pure quantum states|φA i and |φB i is through the statistical distance [184] between them, given as
D2 = 4(1 − |hφA |φB i|2 ). In Fig. 6.3 (b), we plot the distance D between theexact ground state and the SF and MI ground states, as a function of t/U .
While studying the fidelity and statistical distance of the ground statewith respect to some prototypical many-body states can be instructive, it is a
bit detached from experiment, where one does not typically have information
about the full quantum state of the system (as through quantum state tomog69

(b) 1.0

FMI ; FSF

DMI / 2 ; DSF / 2

2(a) 1.0

2

0.5

0.0
0

.25

.5

.75

1

t/U

0.5

0.0
-2

-1

0

1

Log10 [t / U]

Figure 6.3: Tuning the ground state from a Mott insulator to a superfluid.
(a) As a function of the ratio of tunneling to interaction energies t/U , weplot the square of the fidelity F of the ground state with respect to a perfect
Mott insulator (MI, red) and perfect coherent state or superfluid (SF, blue),
characterizing the degree of overlap with these two delineating distributions.
(b) Over a larger range of t/U and in log-scale, we plot the statistical distanceof the ground state from the MI and SF quantum states, with the same color
code as in (d).
raphy). Thus, we now focus more on some properties of the system that havea connection to experimental measures, and which can provide signatures ofa transition between different quantum phases. One common experimental
technique is to study the excitation spectrum of a system [27, 185]. Here,one perturbs the system in some way at a controlled frequency or energy, andexamines an observable of the system – relating to susceptibility, for example – as a function of the perturbation frequency or energy. This can yield
information about the excited state spectrum of the system [186, 187].
Going back to the extreme limits of t  U and t  U , excitations of
the system can be easily visualized. In the atomic limit (t/U = 0), the excitations are those of particle-hole pairs, in which there is a holon (or 1 atom
less than for uniform density) on one site and a charge-on (or 1 atom morethan for uniform density) on another. For commensurate filling, this leads
to doublon-holon pairs, with 2 particles on a given site and 0 particles onanother, and other variations of vacancies and multiple occupancies. In the
delocalized SF limit (t/U = ∞), the energy spectrum should be exactly asin the case for non-interacting atoms as studied in Chapter 2. As the BHM
restricts particles to the lowest band, one would thus expect a band of energies with width 4t, with quasimomenta ranging from −π/d to π/d in steps of
π/2N d, continuous in the limit of large M . In Fig. 6.4, we plot the numerically determined eigenergy spectra as a function of t/U . In Fig. 6.4 (a), we see

70

(a) 15

(b)

3

E/U

E/U

10

4

5
0

2
1

-5

0

-10
0

(c)

.25

.5

t/U

.75

1

0

(d)

1.20

0.05

0.10

t/U

0.15

0.20

E / N × (U / t)

2

E/U

1.15
1.10

1
0
-1
-2

1.05
0.13

0.14

t/U

0.15

0

5

t/U

10

Figure 6.4: Ground and excited state energies of the Bose–Hubbard Hamiltonian. (a) We plot the eigen-energy distribution (scaled by U ) of a six-site commensurate BHM, as a function of t/U . In the atomic limit (t  U ), the energy
spectrum is gapped with a typical energy spacing ∼ U , and with particle-holeexcitations (doublon-holon, 2 doublon-holons, triplon-biholon, etc.). In thelimit of delocalized particles (t  U ), a nearly continuous distribution of excited state energies can be seen. (b) As in (a), but plotted over a small range tohighlight the lowest excitations. (c) Zooming in on some avoided many-bodycrossings as t/U is increased. (d) Distribution of eigen-energies scaled by thetunneling t and particle number N , relating to a single-particle band-width of
4t in the limit of weak interactions (t  U ).
that for vanishing t/U the spectra are gapped, relating to discrete particle-holeexcitations (doublon-holon, triplon-biholon, etc). The low-energy features forsmall t/U are highlighted in Fig. 6.4 (b). As the tunneling energy is increased,
the eigenenergies spread out and form essentially continuous spectra. Avoided
many-body energy crossings can be observed in the transition between the two
regimes, as highlighted in Fig. 6.4 (c). Finally, in the limit t  U , we see in
Fig. 6.4 (d) that the single-particle energy spectra indeed have the expected
width 4t (eigenergies scaled by the total particle number N).
We now discuss some other experimentally relevant measures, which are
better aimed at detecting (non-local) correlations in the system. The contrast71

of time-of-flight interference patterns [27, 188, 189] is a standard observableused to distinguish a coherent superfluid state from a number-squeezed, phaseincoherent Mott insulator state. This is in analogy to observing the far-fieldinterference pattern from a laser-illuminated grating, such that when the phaseacross the slit-pattern is uniform, sharp interference peaks are seen, and thepattern is washed out if the phase is randomized. Ignoring effects of interactions, the far-field time-of-flight single-particle interference pattern reveals thein-trap momentum distribution of the atoms [188],
S(k) ∝MX0ei(j−j )kd hφg |â†j âj 0 |φg i .

(6.11)

j,j 0 =1

Here, we have neglected an additional term, the Fraunhofer or Wannier envelope function W (k), given by the Fourier transform of the on-site wavefunctions. For completely delocalized wavefunctions, the distribution of kvectors is in complete
analogy to the laser-illuminated slit, with S(k) ∝2
2−2
N sin N kd/2 / sin kd/2 . The on-site repulsion serves to reduce thespatial coherence length of the single-particle wavefunctions, and leads toa decreasing contrast of the interference pattern. For a 1D distribution of
atoms, with only N = M = 4 atoms and sites, we plot in Fig. 6.5 the visibility contrast of the momentum distribution, defined as C = S− /S+ with
S± = S(2π/d) ± S(π/d). The visibility contrast serves to characterize thefirst-order, single-particle coherence of the sample. Higher-order correlations
can be measured through higher moments of the matter-wave field, such as
through noise-correlation measurements [190, 191] or through elastic diffraction experiments [49, 192].
Under certain conditions, such as at finite temperature or in reduced dimensions, the contrast of the interference pattern is not a particularly good
observable to characterize correlations in a system [188, 193, 194], while themomentum peak-width serves as a better indicator [188]. The width of theinterference peak is inversely proportional to the in-trap correlation lengthof the single-particle wavefunctions, i.e. the typical length scale over which
the off-diagonal coherences of the single-particle density matrix decay. As afunction of site distance r, the off-diagonal coherence from a site j is givenby [188]p(6.12)
Dj (r) ∝ hφg |â†j âj+r |φg i/ n̂j n̂j+r ,which in the homogeneous system is independent of j. In the superfluid regime,these coherences decay algebraically as D(r) ∝ A|r|−K/2 , while in the Mott
insulator regime the decay is exponential as D(r) ∝ Be−|r|/ξ , with ξ the correlation length. Across the superfluid-to-insulator transition, the peak-width72

(b) 1.0
1

0.8
0.6

Contrast - C

Contrast - C(a)1.0t/U = 10t/U = 0.1
t/U = 0

S(k)

0.4
0

0.2
0.0

0

0.2

0

1

2

k [π/d]

0.4

0.6

t/U

0.8

0.8
0.6
0.4
0.2
0.0
-3

1

-2.5

-2

-1.5

-1

Log10 [t / U]

-.5

0

.5

Figure 6.5: Momentum-distribution contrast of lattice gases. (a) As shown in
the inset, the momentum wavevector distribution S(k) of a lattice-modulated
gas displays interference peaks at multiples of k = 2π/d, akin to the farfield interference pattern from an illuminated grating. For perfectly coherent
samples [as shown for t/U = 10 (red)], the momentum distribution has a large
contrast, defined as C = S− /S+ with S± = S(2π/d) ± S(π/d). For morestrongly interacting gases (t/U = 0.1 in blue and t/U = 0 as dashed grey) thecontrast of the pattern decreases. In the greater graph, we plot the contrast
C as a function of t/U . (b) The contrast C plotted as a function of t/U , overa larger range and in log-scale.
1.0

-0.5

0.6

-1.0

0.4

r=1

0.2
0.0

(b)

Log10 [D ( r )]

0.8

D(r)

0.0

(a)

r=2
0

0.2

0.4

0.6

t/U

0.8

1

-1.5

r=1

-2.0

r=2

-2.5
-2

-1.5

-1

-0.5

Log10 [t / U]00.5

Figure 6.6: Decay of off-site correlations as a function acrosssuperfluid-toP the†insulator transition. (a) The off-site correlator D(r) ∝ j hâj âj+r i is plotted
as a function of t/U for both off-site distance r = 1 (solid blue) and r = 2(dashed red). (b) Log-Log dependence of D(r) on t/U .
increases abruptly from w ∝ L−1 , where L is the system size for an inhomogeneous system, to w ∝ ξ −1 [188]. In Fig. 6.6, with plot the off-site correlator
D(r) for nearest-neighboring and next-nearest neighboring sites (for a small
system of N = M = 4).
Finally, we recall the limits of validity for the BHM model, which consid-

73

ers only a single-band of the lattice, local on-site interactions, and nearestneighbor tunneling. Of particular relevance to the remaining chapters are thelimitations of this model at weak lattice depths in systems of strongly interacting one-dimensional (1D) Bose gases. In the following, we discuss sometheoretical descriptions of interacting bosons in one dimension.

6.2

Theoretical descriptions of 1D Bose gases

6.2.1

General considerations

A system of bosonic atoms with motion restricted to one dimension (1D) canbe realized by tight radial confinement in an array of “tubes” that are formed
using deep lattices along the two transverse directions x and y. We may
describe this system with an effective 1D Hamiltonian. To describe such asystem, one can start from the description introduced in Chapter 2 based on
the non-linear Schrödinger equation (Gross–Pitaevskii equation) describing
bosonic fields with interactions, which are characterized by a 3D coupling
constant g = g3D = 4π~2 as /m (as the s-wave scattering length). We canthen introduce an effective 1D coupling constant g1D to describe boson-boson
interactions in the 1D version of this equation. For this purpose, we assume
that the wavefunctions of all the atoms are energetically restricted to thelowest radial mode φ0 . We can then factorize the wavefunctions in terms ofradial and longitudinal coordinates (neglecting
any azimuthal pvariation), as√2−ρ2 /2σ⊥, with ρ = x2 + y 2 andΨ(r) =pφ0 (ρ)ψ(z), where φ0 (ρ) = 1/ 2πσ⊥ eσ⊥ = ~/mω⊥ , where ω⊥ is the radial harmonic frequency [195, 196]. The
transverse degrees of freedom may then be integrated out, so that one arrives
at an effective 1D description of the system, where the typical 3D atom-atominteraction potential Vat (r−r0 ) ≈ g3D ·δ(r−r0 ) is replaced by the 1D interaction
potential Vat,1D (z − z 0 ) ≈ g1D δ(z − z 0 ). The 1D coupling strength g1D may thenbe described in terms of that for 3D by the relation [195]


g3D
as
2
g1D = g3D |φ0 (0)| 1 − C √≈ g3D |φ0 (0)|2 =,(6.13)22πσ⊥2σ⊥where C is a constant with value ∼√ 1.46, and where we neglect the second
term in the parentheses as Cas  2σ⊥ for typical experimental conditions.
The 1D coupling strength may also be reexpressed in terms of the transverse
harmonic oscillator frequency as g1D ≈ 2as ~ω⊥ .
For non-interacting and weakly-interacting trapped gases of bosons in 1D,
much is the same as in 3 dimensions. Absent interactions (and at T = 0) all

74

5

10

0

0
-10

i

10

(b)

8
6
4
2
0

-10

0

j

Percentage [%]

0

10

10

n1D [µm

-1 ]

(a)

0

1

2

γ

3

4

Figure 6.7: Characterizing the inhomogeneous two-dimensional array of the
1D systems. (a) Distribution of central densities of individual 1D systems (ortubes), n1D (z = 0), as a function of site indices i and j. (b) Binned percentagesof tubes that are characterized by a given Lieb–Liniger parameter of value γ.
The characteristic value, taken as a weighted (by tube population) averageover the entire distribution, is γeff ≈ 0.6.
the atoms will occupy the single-particle ground state. For weak interactions, amean-field description can be retained, with an approximately Thomas–Fermi
profile [197] as in 3D, albeit with increased density due to the transverse confinement [198]. For very strong repulsive interactions, however, even in theabsence of a lattice potential, correlations between 1D bosons can have a large
influence on a system’s spectral properties and spatial distribution. To minimize interaction energy, repulsive bosons seek to minimize the spatial overlap of their wavefunctions. In the limit of infinite interactions, an effective
fermionization of bosons takes place in 1D [199], where interacting bosons canbe described as non-interacting fermions, with Pauli-blocking mimicked by
the infinite repulsion (similar as to the case of non-identical fermions [200]).
In this limit, the atoms form a Tonks–Girardeau gas [197, 201] where thesingle-particle coherence length decays on the length scale of the typical interparticle spacing, and higher-order correlations vanish at short length scales(anti-bunching). An effective Fermi wavevector can also be defined as for noninteracting fermions, with kF = πn1D , where n1D is the 1D density (locallydefined in a trapped system).
For the most general case, the state of a system of interacting 1D bosons canbe described by the Lieb–Liniger parameter, which compares their interaction
and kinetic energies, as γ ∼ Eint /Ekin [202]. For a lattice-free gas of bosons
described by an effective 1D coupling strength g1D and a 1D density n1D , thisn1D
. In terms of the Lieb–Liniger parameter, a generalis given as γ = ~g21D
n2 /m
1D

75

rule of thumb is that a mean-field description is applicable for γ  1, whilefor γ  1 the Tonks–Girardeau regime is reached [197]. In the intermediate
regime γ ∼ 1, which characterizes most of our 1D gases in the absence ofa z-lattice, correlations are important but coherent and superfluid behavior
can still be observed. In our experiments, we typically populate ∼ 1000 onedimensional tubes of bosons from an initially three-dimensional BEC, witharound 100 atoms per tube on average. Due to the external confinement, the1D density n1D varies amongst the 1D tubes and also within each 1D tube, suchthat it is a function of z as n1D (z). To characterize our collection of 1D gases,
we use 2/3 of the central density n1D (z = 0) as representative value for n1D
in each tube (this value (2/3)n1D (z = 0) is equal to N/2RT F in the Thomas–
Fermi approximation). We then characterize the entire system by a mean-γ
parameter, that is weighted by the population in each tube. Fig. 6.7 (a) plots atypical distribution of central densities in our array of 1D tubes. The resulting
distribution of γ values is plotted in Fig. 6.7 (b), for a typical sample that ischaracterized by a mean value γeff = 0.6.

6.2.2

Theoretical description of interacting 1D bosonicfields

So far we have attempted to give some qualitative insight as to how 1D bosons
behave for weak and strong interactions, and how a system of these particles
can be parameterized in experiment. We now discuss some commonly usedtheoretical descriptions of 1D Bose gases at low temperatures, with shortrange δ-like interaction potentials, in weak or completely absent longitudinallattice potentials, following the descriptions in Refs. [203, 204]. While fordeep lattices a Bose–Hubbard model description can be used, as described inearlier sections, localized on-site wavefunctions are not a priori an appropriate
description in the absence of deeply modulated lattice potentials. For a lowenergy description of 1D quantum fluids, we again begin with a quantized
Hamiltonian as in Eq. 6.1,

 2 2
Z
Z
1
~∇
†
−µ ψ̂(z)+dzdz 0 ψ̂ † (z)ψ̂ † (z 0 )g1D δ(z−z 0 )ψ̂(z)ψ̂(z 0 ) ,Ĥ = dz ψ̂ (z) −2m
2(6.14)
here neglecting any additional external potentials, either slowly-varying as in
Vext (z) or with short modulation length as for an added lattice. The density ρ(equal to n1D ) is controlled through the Lagrangian multiplier µ. In the caseof trapped gases the local density ρ will vary within the 1D system due to theexternal trapping potential.
We now follow the bosonization approach for a collective field description76

of Tomonaga–Luttinger liquids, taking the bosonic field operator as [203–205]ψ̂ † (z) = [ρ̂(z)]1/2 e−θ̂(z) ,(6.15)

where conjugate density and phase operators, ρ̂ and θ̂, obey the commutation
relation [ρ̂(z), θ̂(z 0 )] = iδ(z − z 0 ). Again, we assume a homogeneous systemwith average density hρ̂(z)i = ρ̄ = µ/g1D , which is also given by N/L for
N particles in a uniform system of length L. The density operator can thenbe described to lowest order as ρ̂(z) ' ρ̄ − ∂z φ̂(z)/π, where φ̂(z) is a slowly
varying quantum field.
The Hamiltonian containing only kinetic and interaction contributions,Eq. 6.14, may then be expressed in the form of the Tomonaga–Luttinger (TL)
Hamiltonian [206, 207]


Z
~υ
2
−1
2
dz K(∂z θ̂(z)) + K (∂z φ̂(z)) .
(6.16)
ĤT L =
2π
Here, the system has a linear dispersion relation ω = υ|k| at low energies,
where υ is the speed of sound in the gas. The Luttinger parameter K servesto describe the correlation properties of the system, and is related to theearlier-used Lieb–Liniger parameter γ. In the limit of vanishing interactions(γ = 0) the Luttinger parameter takes a value K = +∞, while in the limit of
infinite zero-range interactions (γ = +∞) it takes a value K = 1. Approximate√
forms can be found near these limits, with K(γ) = π/ γ − 1/2 for γ  1
and K(γ) = 1 + 4/γ 2 + O(γ −3 ) for γ  1, and a sound velocity given by
υ(γ) = ~π ρ̄/mK(γ) [204].
Finally, the effect of weak periodic perturbations, as due to a longitudinallattice, may be taken into account. We assume a lattice potential Vlatt (z) =V0 cos2 (kz), with small depth V0  µ such that it only weakly modulates thedensity distribution. The total Hamiltonian may then be written as Ĥef f =ĤT L + ĤV , where the contribution due to the lattice potential is [204]
Z


g̃u
dz cos 2pφ̂(z) + zδ .
(6.17)ĤV =πHere, p is a parameterizing integer describing the density of atoms with respect
to the lattice, where for p = 1 there is one particle per site, and for p = 2 there
is one particle for every 2 sites, and so on. The second term relates to the incommensurability of the lattice potential with respect to the bosonic density,with δ = nk −2pπρ0 , where n is an integer chosen such that δ takes its smallest
value. The coupling strength is then given by g̃u ≈ π ρ̄V0 (V0 /µ)n−1 [204]. The
added lattice potential can lead to a number of interesting ordered quantum
77

8
BH

scrit = 6.2

s

6

SG

scrit = 4.7

4

0.4

0.5

0.6

γ

0.7

0.8

0.9

Figure 6.8: Analytical estimates of the transition point to insulating behavior
in 1D Bose gases. As a function of the Lieb–Liniger parameter γ, the criticallattice depth that induces insulating behavior in 1D Bose gases, based on thesine-Gordon model and Bose–Hubbard models, are plotted in blue and red,respectively. At a typical value for our 1D gases of γ ∼ 0.6, we expect toobserve insulating behavior beyond critical depths of 4.7 and 6.2, based on thesine-Gordon and Bose–Hubbard models, respectively.
phases for a scalar bosonic field, including Mott-insulating phases for commensurate potentials and charge-density wave states for incommensurate ones.
For a fully commensurate lattice potential (δ = 0), the addition of the lattice term ĤV leads to the so-called sine-Gordon model. This model describes
a superfluid to Mott insulator transition, with n/p bosons per site on average, as a function of the Luttinger parameter K. The value of the Luttinger
parameter at the transition point for fixed filling is given by K ∗ = 2/p2 , andthus K = 2 for unit filling.
Based on the sine-Gordon model, analytical estimates for the superfluid toMott-insulating transition point as a function of lattice depth s = V0 /ER havebeen determined for relatively strong interactions and weak potentials [31].
Forpvalues γ . 10, the Luttinger parameter may be approximated as K(γ) ≈π/ γ − γ 3/2 /(2π) [31], and the critical value for insulating behavior as Kc =2 + |s|K/2 [107], reexpressed as a critical lattice depth in terms of γ ashiπscrit = 2 p−2 .
(6.18)
γ − γ 3/2 /2πThis can be compared to analytical estimates based on the Bose–Hubbard
model, which apply in the limit of deep lattices and for weaker interactions.

78

In Ref. [107], a transcendental equation defining the s − γ phase boundary to
the insulating state was derived as √
√
1 2 2 2π(U/t)crit s ,s = ln(6.19)
4γ
with a critical ratio of the interaction to hopping parameter (U/t)crit ≈ 3.85.
The critical depths for insulating behavior, based on these two analytical forms,
are plotted in Fig. 6.8 as a function of the Lieb–Liniger parameter γ.

79Chapter 7
Strongly interacting
Bose gases II:
experimental probes
In this chapter, we review some common experimental techniques used tocharacterize strongly interacting quantum gases.

7.1

Visibility

For weakly interacting, degenerate lattice gases, in which all the atoms may
approximately be described as having the same wavefunction, the time-offlight (TOF) far-field momentum distribution contains the full information
about the spatial wavefunction of the atoms [18]. As a sum of many identical
single-particle interference patterns after release from the lattice, a Fouriertransformed image of the BEC’s spatial distribution builds up atom by atom.
For strongly-interacting quantum gases, in which the atoms are no longer all
Bose–condensed into the same quantum state, but rather have wavefunctions
that are spatially anti-correlated, the loss of single-particle phase coherence
across the lattice sites leads to a washed-out TOF interference pattern. Here,
we shall first show how TOF spectra can provide evidence for a loss of global
phase coherence in the superfluid-to-Mott insulator transition [27]. After this,
we discuss limitations to using this technique as the sole diagnostic tool, andfailures as a useful observable in reduced dimensions.
We typically operate with a system of ∼ 7 × 104 atoms of 87 Rb in the|1, −1i hyperfine state that are slowly loaded into a 3D optical lattice. The
lattice is symmetric with respect to both the lattice wavelength (λ{x,y,z} ≡λ = 1064 nm) and the lattice depth (sER , with ER = h2 /2mλ2 ). Shortly after80

ramping up the lattice to its final depth s, we suddenly turn off all confining
potentials and release the atoms in time-of-flight. After 18 ms, we absorptively
image the atomic distribution along the x axis. We then measure the visibility
contrast C of the time-of-flight momentum distribution P (k) [27, 189], as
defined in Fig. 7.1. Unlike in the simple 1D example of the previous Chapter,here we measure the number of atoms in different regions about a ring of
constant ky2 +kz2 values (integrated over all kx values by imaging along this axis).
Specifically, we measure at and directly in between the first (and minus first)
reciprocal lattice vectors along these two axes [189]. For a completely coherent
cloud of atoms, we expect a visibility contrast of C = 1, while for a completely
incoherent sample – e.g. due to either thermal or quantum depletion – we
expect to measure C = 0. Fig. 7.1 (a) plots the contrast C as a function of
the final lattice depth s, and reveals a steep decline of C beyond values ofs ∼ 14. In Fig. 7.1 (b) we plot the numerical derivative of the visibility, i.e.
the difference between successive s-values, which has a sharp feature centeredaround s ∼ 15 (with negative values first appearing around s ∼ 14). We notethat upon ramping back down to low s-values from a deep lattice, a highvisibility interference pattern is recovered, with little indication of heating.
This reversible loss of phase coherence with increasing lattice depth s (andincreasing U/t), is an indicator of the well-studied superfluid-to-Mott insulatortransition in 3D.
In the 3D case, visibility contrast appears to be a reasonably accurate observable for detecting the SF-MI transition; the observed transition at a valueof U/t ∼ 36 for s = 14 is realistic for our distribution of local densities. However, in the more general situation, visibility contrast is not the best indicator
for the loss of superfluidity. It has been shown that at zero temperature,measurement of the visibility contrast is not the most direct method for observing the superfluid-to-insulator transition in a system of Hubbard-regime
bosons [188] (alternative measures discussed later). Moreover, visibility contrast can even be a false indicator if used as a proxy for superfluidity. It has
been shown that thermal gases of atoms in the normal phase [193, 194] can
exhibit sharp peaks in their time-of-flight momentum distribution.
In experiments with quantum-degenerate gases, the unsuitability of visibility contrast as an indicator of the SF-MI transition is most severe in lowdimensions. To emphasize this point, we show data for the visibility contrast
as a function of lattice depth for an array of 1D Bose gases. Instead of using asymmetric 3D lattice as above, to create the 1D gases we first load the atoms
into very deep transverse lattices along the axes x and y, with wavelength
λ = 1064 nm and depth s⊥ = 40 ER . This results in a collection of isolated 1D gases, which can be characterized by a mean Lieb–Liniger parameter

81

C=

Nb - Nrz

Nb + Nr

y

0.0

s=9

∆C / ∆sContrast - C

1.0

-0.1

0.5

-0.2

0.0

(a)

-0.3
10

15

20

(b)
10

Lattice Depth - s × ER

15

20

Lattice Depth - s × ER

Figure 7.1: Loss of matter-wave coherence across the superfluid-to-insulatortransition in 3D. (a) Visibility contrast C as a function of the lattice depth
V = sER , symmetric along all axes. The inset illustrates the determinationof C (shown for s = 9). (b) The numerical derivative (difference between
successive points) of the contrast shows a distinct minimum near the expected
position of the SF-MI transition in 3D.
γeff = 0.6. We then slowly ramp up a lattice of wavelength λ = 1064 nm andvariable depth sz ER along z. Due to the stronger interactions in 1D (fromthe enhanced density) and the smaller coordination number z = 2, we expectthat the transition to a Mott-insulating state should occur at a smaller valueof the lattice depth as compared to in 3D. For the characteristic density of
the 1D systems, estimates based on the sine-Gordon [SG; Eq. 6.18] and Bose–
Hubbard [BH; Eq. 6.19] models predict that the transition should occur atcritical lattice depths of sc = 4.7 and 6.2, respectively.
After releasing an array of interacting 1D Bose gases, no distinguishing
momentum features are found along the axes of tight confinement (here x andy), due to the lack of phase coherence between the independent 1D systems.
Thus, we use a slightly modified visibility contrast measure for the case of
1D gases. First, to ensure that the atoms always have the same projection
onto Bloch states prior to release, we quickly (in 50 µs) load the lattice tothe same depth sz = 20 [185]. Then we perform a π phase-shift between
adjacent lattice sites (half a Bloch oscillation, by turning off the ODT for∼ 450 µs while leaving the z-lattice on), which shifts the velocity envelopeof the diffraction pattern by 1 vR , resulting in symmetric peaks along z [27].
Finally, we measure the visibility contrast of the time-of-flight interferences as82

n (v) [arb. u.]

1

C1D

0.6
0.4

0

0.2

-2

0

v / vR

2

0.0
0

5

10

sz

15

20

Figure 7.2: Time-of-flight visibility contrast C1D as a function of lattice depth
for an array of 1D Bose gases (γ ≈ 0.6) as a function of the longitudinal lattice
depth sz . Inset: Time-of-flight velocity distribution (along z).
C1D = η− /η+ [44], with η± = nv=+1 + nv=−1 ± 2nv=0 ; nv=0,±1 apertures shownin the inset of Fig. 7.2. We note that unlike the symmetric measure in Fig. 7.1,
here the overall shape of the Wannier envelope (having greatest intensity at
nv=0 ) results in a visibility contrast not bounded by 1 and 0, but rather 1 andsome slightly negative value (typically ∼ −.2) that depends on the width ofthe envelope.
We now measure this 1D visibility contrast as a function of sz . In contrast
to the 3D case, we observe a continuous, slow decrease of C1D , with no abruptchanges that might signal a transition between a coherent and phase-incoherent
state. In the proximity of lattice depths for which we expect the transition to
an insulating state (sc ∼ 4.7 − 6.2, shown in green), no significant changes inthe visibility measure can be discerned. The lack of any sharp change in thevisibility measure for 1D gases across the expected SF-IN transition is consistent with theoretical predictions [208] and previous observations [68, 185](based on the measured coherent fraction or peak fraction in time-of-flight),
which indicated that the measurement of sharp visibility fringes is not an appropriate stand-in for the detection of superfluidity in a generic system. This
limitation notwithstanding, visibility contrast may still be used as a complementary observable to characterize the state of a system. As is seen in Fig. 7.2,
the contrast still changes monotonically with the lattice depth sz (and thusU/t), and the same may be said for the temperature dependence [209].

83

As a brief aside, we illustrate how visibility measurements in a system of
1D Bose gases can be used to estimate important parameters of the Bose–
Hubbard Hamiltonian, based on collapse-and-revivals of the interference of amatter-wave field [178, 210, 211]. Starting from a lattice depth of sz,i = 6 ER
along z, near the SF-MI transition (with s⊥ = 40 ER ), we quickly (in 50 µs)
linearly ramp the lattice depth to a value deep within the 1D Mott-insulating
regime, sz,f = 18. The lattice ramp speed is adiabatic with respect to theexcitation of higher lattice bands, but fast enough to essentially freeze out
tunneling in the system on relevant timescales. We then vary the amount ofhold time τ in this deep lattice prior to release in time-of-flight, and monitor
the visibility contrast of the released 1D Bose gas array as a function of τ .
To paint a simplified picture of how we expect the visibility contrast to
evolve in time, we imagine the fate of a single particle interacting with a sea ofother bosons (in reality all are indistinguishable, so it makes no sense to pick
one out from the others). Since the freeze-out of the tunneling happened while
the atoms were still in the 1D superfluid regime, it is fair to assume that theparticle’s spatial wavefunction is delocalized over several lattice sites. Withtunneling frozen out, the particle’s site-dependent temporal phase evolution
will depend on the local site-dependent energy εi , as well as the on-site interaction energy U ni (ni − 1)/2. If only interaction terms contributed, we would
expect phase evolution at multiples of U/h (1, 3, 6, etc. for double, triple, andquadruple occupancies). Somewhat similar to Bloch oscillations in a linear
gradient, we thus expect that the particle’s wavefunction on different sites willrephase with a characteristic times h/U (∼ 0.87 ms for the sz,f = 18). Taking
the inhomogeneous trapping potential into account, however, we also expectthat the different sites will phase evolve at an additional rate reflecting the
local potential εi , of which there is a broad distribution, leading to an overall
dephasing across the sample.
In Fig. 7.3, we plot the measured visibility contrast C1D as a functionof τ . We indeed see revivals of the interference pattern’s contrast at times
relating to the expected on-site interaction energy U/ER = 0.56, however due
to the external trapping potential these oscillations die out rather quickly. In
more homogeneous external potentials, which allow for the observation of morethan a few cycles, such collapse-and-revival measurements have been used asa spectroscopic tool for measuring site occupation statistics [210], observing
number-squeezed superfluids upon approaching the MI regime [178, 210], andobserving deviations from the single-band Hubbard Hamiltonian due to atomatom interactions [210].
We shall now discuss some additional observables that may be used tocharacterize an ultracold lattice-gas system, in particular for the signaling of

84

C1D

0.5

0.0
0

1

2

Hold Time τ [ms]3

Figure 7.3: Collapse-and-revival in the matter-wave coherence of 1D Bose
gases. For a cloud of 1D Bose gases (s⊥ = 40), the lattice confinement alongz is quickly ramped up in 50 µs from an initial depth of sz,i = 6 to sz,f = 18.
Oscillations in the contrast of time-of-flight interference patterns (quantified
by the 1D visibility measure C1D ) at a frequency of 1.13(1) kHz are in good
agreement with the expected on-site interaction energy U/h = 1.15 kHz.
a transition between superfluid and Mott-insulating states.

7.2

Momentum peak width

We recall the earlier discussion regarding what information about a system canbe deduced from typical TOF momentum-space images. The number of momentum peaks, as well as their relative weights, is simply given by the Fraunhofer envelope function (or Wannier envelope, analogous to the Debye–Waller
function for material crystals) that reflects the size of the atomic wavefunctions
on individual lattice sites. The width of these peaks, on the other hand, reflects
the number of sites that coherently contribute to the interference pattern. Fora fully degenerate sample of bosons in a lattice, the width σ of the peaks justscales inversely with the system size L, as σ ∝ L−1 . As soon as repulsive
interactions cause the onset of quantum depletion, i.e. when atoms cease to
exclusively occupy the single-particle ground state, the single-particle coherence length (decay length of Green’s function, single-particle density matrix,etc.) becomes smaller than the system size (the same holds for thermal depletion [212]). Deep in the Mott-insulating regime, single-particle coherence does
85

0.5

n (v) [arb. u.]

Peak-width σv [vR]

U / tz 1.0 2.5 5 10 20

50

100

200

1

2σv
0

-2

0

v / vR

2

0.3
0

5

10

sz

15

20

Figure 7.4: Momentum peak width in time-of-flight across the superfluid-toinsulator transition
in 1D. As illustrated in the inset, the momentum-peak√width σ (1/ e half-width) is determined by a symmetric multi-Gaussian fit
to time-of-flight interference patterns. The dependence of σ on sz is plotted,
along with the fit of a linear increase beyond a critical depth sc .
not extend off-site. The width of the TOF momentum peaks increases when
the coherence length in the system becomes smaller. Moreover, it has been
demonstrated that the momentum peak-width is a good indicator of the SF-MI
phase transition, showing a sudden change about the critical point [185, 188].
Here, using the same set of time-of-flight images that produced the 1D
visibility measurements shown in Fig. 7.2, we instead determine the width ofthe momentum peaks. We do this by performing a symmetric multi-Gaussian
fit to the integrated density profiles as shown in the inset of Fig. 7.4, consistingof symmetric diffraction peaks (in blue) on top of a broad background(dashed√red). From such fits, we determine the peak-width σv (1/ e half-width) as
a function of the lattice depth sz along z, as plotted in Fig. 7.4. To thedependence of peak-width on lattice depth, we fit a function that is constant
below a critical value sc , and then increases linearly beyond this point (fitting
to the range of data with sz ≤ 13). The fit (plotted along with the 1 standard
deviation confidence regions in Fig. 7.4) returns a value of sc = 5.7±0.5, whichis in good agreement with the expected region for the transition to insulating
behavior, as shown in green and discussed in the previous section.

86

7.3

Impulse-response

Another way to observe the superfluid-to-insulator transition in lattice-confined
atomic gases is through transport measurements [213], such as by monitoring
the response of the atoms to an applied force. For repulsive 1D Bose gases inparticular, it has been observed that collective motion in response to applied
potential gradients is inhibited in weak optical lattice potentials [185, 214].
In the strongly-correlated regime, the redistribution of density between latticesites costs a discrete energy ∼ U , and for weak gradients (weak with respect
to U ) that do not bring adjacent sites into resonance, response to impulse
is suppressed [27, 186]. In recent years, the response to weak impulses has
been used as an experimental diagnostic to signal the superfluid-to-insulatortransition in 1D [31, 215] and in 3D [75, 215].
To make things explicit, the measurement of impulse-response in its idealized form does not probe physical transport in the lattice (i.e. mass currents ordensity redistribution); rather it probes the coherence of the atomic wavefunctions. To explain how this works, we consider the extreme case of a superfluid
BEC, with quasimomentum q = 0, in a deep lattice with essentially zerotunneling (as can be encountered in non-interacting gases), such that physical transport is a priori disallowed. Just as we wrote the localized Wannier
states as superpositions of delocalized Bloch functions in Chapter 2, we canalso represent the Bloch states as superpositions of localized Wannier orbitals.
(n)
The delocalized Bloch wave function φq (z), of the nth lattice band and withquasimomentum q, can be written as
X
−1/2
φ(n)
eiqzj /~ wn (z − zj ) ,(7.1)
q (z) = Njwhere we recall that wn (z−zj ) is the Wannier orbital of the nth band, localizedat position zj , and that N is a normalization constant. Our q = 0 Bloch statethus represents an equal-phase superposition of all localized Wannier states.
If we apply a weak potential gradient U (z) = F z onto the sample for sometime Timp , a site-dependent phase is accrued of the form ϕj = F zj Timp /~,which means that the Bloch state will be transformed from q = 0 to q =F Timp . This evolution in quasimomentum space (and in momentum space after
band-mapping) comes about solely due to interference between contributionsof the wavefunction on different sites (with Bloch oscillations occurring atlong times, or Josephson oscillations in the two-site case). In contrast to thislinear evolution of quasimomentum for the delocalized Bloch state, a single
particle localized to one site of the lattice will be unresponsive to applied
gradients. We thus expect that if we transform the ground state of the system,87I / Imax

(a)

v/vR

-1 0

0

s=2

v/vR

1

-1 0

1

s=6

slope α1

U / tz 2.5 5(b) 1.0

10

20

6

8 10

α

0.5
0.0
0

2

4

sz

Figure 7.5: Impulse response of 1D bosons as a function of lattice depth.
(a) The response to impulse is determined by a straight-line fit (with slope α,
normalized to the case of free atoms) to the dependence of velocity on applied
impulse I (recoil velocity vR = h/mλ). (b) The dependence of α on the lattice
depth sz . The lines and surrounding shaded regions are fits to the data withconfidence regions (1 s.d.) of a linearly decaying response, with no response(α = 0) beyond a critical lattice depth sc .
by control over t and U , from a superfluid (with delocalized, algebraicallydecaying single particle wavefunctions) to a Mott insulator (with exponentiallylocalized single particle wavefunctions), that the response to applied impulse
will go to zero across the quantum phase transition, serving as a good indicator
for the superfluid-to-insulator transition [31, 75].
Here, we study impulse-response in a system of 1D Bose gases as in theabove examples, again characterized by a tube-averaged value of γeff = 0.6(s⊥ = 40 as above). We apply weak magnetic-field gradients along z, briefly
pulsed on for a duration T = 1.2 ms, which applies forces of variable strength F
ranging from 0 to Fmax /m = 1.2 m/s2 (the atoms are in the |F, mF i = |2, −2istate). A corresponding impulse I = F × T is applied to the atoms. The
88

magnitude of the strongest impulses are such that the final quasimomentum
reached by nearly-free atoms is far from the band-edge, avoiding excitations
that can occur for velocities in excess of some critical value vcrit (with low
values of vcrit in 1D [213, 215]). After impulse, we monitor the response of
the atomic cloud in time-of-flight, by measuring the center-of-mass velocity ofthe distribution along z, after a brief (∼1 ms) ramp-off of the z-lattice (bandmapping). As illustrated in Fig. 7.5 (a), we characterize the response as afunction of the impulse strength I (with slope α). As plotted in Fig. 7.5 (b),we observe a nearly linear decrease of the atomic response to impulse as afunction of the optical lattice depth sz . We determine the critical point at
which the atoms become unresponsive to impulse by fitting a linear decay tothe response α as a function of lattice depth, quenched beyond a critical depth
sc . The fit-determined critical depth of sc = 6.2±0.5 is in good agreement withthe peak-width measurement of Fig. 7.4, and with the expected position of the
SF-MI transition in our 1D bosonic gases (shown in green in the plot). We notethat the observed suppression of transport is due to many-body correlations,as over this range of weak lattice depths the modification of the atoms’ effective
mass alone does not explain the observed suppression.

7.4

Excitation spectra

Lastly, we discuss lattice amplitude modulation spectroscopy [185], which is
a commonly used technique for characterizing the energy spectrum of excitedstates in systems of lattice-confined ultracold atoms, based on a sinusoidal
modulation of the optical lattice depth. Similar to the methodology used in
Chapter 3 for exciting higher lattice bands, in weakly-interacting samples orat large energies this method can provide access to the single-particle excitation spectrum, revealing band-gap energies and band-widths, etc. To study
excitations relevant to gases in the Hubbard regime, however, we now concentrate on excitations with energy below the single-particle band-gap. For
non-interacting gases, the spectrum simply reflects the structure of the zerothenergy band. In the atomic limit (t  U ), there is a charge gap with energy
U relating to the creation of a particle-hole pair (doublon-holon). Additional
interaction bands, all with typical bandwidth on the order of t, appear aroundenergies of 2U (2 doublon-holons), 3U (either 3 doublon-holons or a triplonbiholon), and so on, as plotted in Fig. 6.4 (a). The emergence of an energy gap
has been used as an indicator of the transition between superfluid and Mott
insulator in ultracold atom systems [27, 31, 185].
In lattice amplitude modulation spectroscopy, the lattice depth is variedsinusoidally in time as s(T ) = s0 + δs sin(ωT ). Thus, the tunneling energy
89t and the on-site interaction strength U , which both depend on the lattice
depth s, are also modulated in time. The time-varying perturbation to thesystem allows coupling to excited states at a spectroscopically defined energy~ω (to lowest order in the linear response regime [31, 187]). The excitation
mechanism itself [185, 187] can be related to a symmetric two-photon Bragg
process [131] involving the lattice laser beams, in which the modulation effectively introduces frequency sidebands. Typically, the state of the system ischaracterized by observing how much energy is deposited at a particular modulation frequency (with fixed modulation time Tmod and modulation amplitudeδs). We note that there is a wealth of additional techniques used to characterize excitations, including phase modulation spectroscopy [205], direct Bragg
spectroscopy [216, 217], coupling to excited states via applied gradients [27],and photon-assisted tunneling in a gradient [218].
In the experiment, we investigate the excitation spectrum of a sample of
1D Bose gases, having the same properties as in the above examples (s⊥ = 40,
γeff = 0.6), for a variable depth sz of an applied lattice (wavelength λ =1064 nm) along z. After slowly ramping up the lattice to a depth sz , wesinusoidally modulate its amplitude by ±15% for Tmod = 80 ms. We thenramp down the depth to sz = 4 in 5 ms, and allow for 15 ms of thermalization.
We do not directly measure the energy deposition rate, or the energy increaseof the system for fixed Tmod , but rather we release the atoms from this latticein time-of-flight (following a π phase-shift as before) and measure the visibility
contrast as a proxy for the amount of thermal energy in the system. We plot
the excitation spectra for lattice depths of sz = 4, 9, and 14 in Fig. 7.6,
with energies normalized to the value of U for sz = 14 (U is only slowly
varying with lattice depth). For the lowest depth of sz = 4, which is expected
to be in the superfluid regime (same transition region as in the preceding
sections), we observe an essentially continuous dependence of the visibility on
the excitation frequency ω. For sz = 9 however, a non-monotonic dependenceof the visibility on ω is observed, and for a depth of sz = 14 well into the Mottinsulating regime (U/t ≈ 66), resonant peaks at energies of ∼ U and ∼ 2U are
observed. The emergence of these excitation features with increasing sz thus
signal a transition to gapped excitations of particle-hole pairs. We note thatthe production of a doublon excitation relies on nearest-neighbor tunneling,
and thus in an inhomogeneous system also depends on the difference between
the site-dependent energy offsets ∆i = εi − εi+1 , with excitation at energies atroughly U ± ∆ [218]. Thus, neglecting the small spread in the values of U andt due to the slowly varying envelope of the lattice beams, the large widths ofthe observed excitation peaks in Fig. 7.6 (b,c) are due to the inhomogeneous
external trapping potential, and the resultant spread of the local resonance90

1D Visibility C1D(b)sz = 9(a)sz = 4

0.5

0.3

(c)
sz = 14

0.1
1

2

1

ħω / U2

1

2

Figure 7.6: Excitation spectra across the superfluid-to-insulator transition in
1D. (a) Visibility as a function of amplitude modulation frequency (normalized
to U/ER = 0.53 for sz = 14), for a lattice depth of sz = 4. (b) Similar data,
but for a lattice depth along z of sz = 9. Fit lines are two Gaussians on alinear slope. (c) Similar data, but for sz = 14. The emergence of peak-likestructure in (b) and (c) reflect the appearance of a charge gap in the systemin the Mott-insulating regime.
condition for nearest neighbor tunneling.
Here, we have used measurements of the excitation spectrum to highlightthe transition from a superfluid state to a Mott-insulating state. Furthermore,
excitation spectra characteristic of different quantum phases – such as gapped
and ungapped – are also important for the detection of different states ofquantum matter in disordered bosonic systems [28, 46, 48, 68].

91

Chapter 8
Atomic mixtures instate-dependent lattices
More Is Different.
– P. W. Anderson, Science 177, 393–396 (1972)

In this chapter, we discuss the generation of hyperfine state mixtures in our
experiment and briefly review our method for implementing state-dependent
optical potentials. We include a discussion of areas of experimental study
that are made possible with these techniques, and briefly highlight two suchstudies that we have performed. These deal with nonlinear atom-optical fourwave mixing and implementation and study of the two-component bosonic
Hubbard model, as based on our publications Collinear Four-Wave Mixing of
Two-Component Matter Waves, Phys. Rev. Lett. 104, 200402 (2010) [219]and Superfluidity of interacting bosonic mixtures in optical lattices, Phys. Rev.
Lett. 105, 045303 (2010) [44], which have been discussed in detail in Ref. [45].

8.1

Introductory discussion

Nearly a decade ago, the experimental study of scalar bosons in ordinary(single-frequency and stationary) three-dimensional (3D) optical lattices [27]allowed for realization of the 3D Bose–Hubbard model with ultracold atoms [29],with later studies in 2D [220] and 1D [31, 185]). While the study of scalar particles in optical lattices is a wide-ranging field in and of itself (especially with
the inclusion of dynamical effects, superlattice structures, and higher-orbital
physics), the introduction of a (pseudo)spin degree-of-freedom (not restricted92

to 1/2) in the particles’ internal state greatly enlarges the number of physicalphenomena that can be studied.
For the situation of having two types of bosons in an optical lattice, which
are both in the Hubbard regime, the system can generally be described by
the two-component Bose–Hubbard model [60–62] (2BHM). One interest is instudying exotic quantum ground states particular to this Hamiltonian. Interspecies pairing phenomena – in some analogy to the BCS mechanism – can bestudied in the emergence of so-called pair and counterflow (anti-pair) superfluidity [60, 221], in systems in which the single-particle charge excitations are
gapped. Furthermore, the 2BHM can be used to study quantum magnetismand spin Hamiltonians [59], with state-dependent lattices allowing for the control of spin-exchange coupling constants. Spin dynamics in these systems canbe used to directly engineer entangled-state resources for quantum computing [222]. Additionally, more direct methods with spin-dependent potentials
may be used for the generation of many-particle entanglement and squeezing in lattices [223, 224] and in bulk systems [225]. As a matter of practicalimportance, studies of the 2BHM may also provide insight into the mechanism underlying the observed loss of decoherence in mixture experiments withpseudospin-asymmetric tunneling, both for boson-boson [51, 52] and bosonfermion [54–57] mixtures.
In a recent work of ours [44, 45], we have implemented the 2BHM with ahomonuclear mixture of bosonic particles moving in a 3D lattice with statedependence along one lattice axis. Compared to studies of heteronuclear mixtures in far-detuned lattices, study of the 2BHM with homonuclear particles
allows for a species-specific tunneling that is not fixed by the mass ratio andrespective polarizabilities of the two species, but rather is tunable through thewavelength or polarization of the state-dependent lattice potentials. This has
allowed us to span our system across different regimes of the 2BHM. One likely
contribution to the observed coherence loss in atomic mixture experiments occurs when the tunneling of one of the species is much weaker than that of theother species, and in fact is nearly frozen out: loading mixtures into a latticewith these tunneling characteristics can result in the dynamical formation ofdisorder, so-called quantum emulsions [47, 70]. These quantum emulsions effectively result from the probabilistic projection of the slower-tunneling atoms’
density onto sites of the lattice. While this comes about somewhat naturally
in multi-species mixtures with species-specific tunneling, there have also been
a number of proposals [66, 67, 71] for the controlled generation of disorderusing localized impurity atoms. One key motivation for implementing disorder in this way is the small length scale over which atoms can be localized,
leading to a rather small correlation length of the disordered landscape. As

93

(a) z

2

2σ

(b)

N+

pz (ħk)

1

N0

-2

N–

0

-1

Visa

σa (ħk)(c)

N + N − 2N

Vis = N+ + N− + 2 N0
+

−

0

0.4
0.3
0.2
0.6
0.4
0.2

Vb = V a0

1020

Vb (E R )30

40

Figure 8.1: Two-component mixture in a state-dependent optical lattice [44,45]. (a) Single-axis momentum-peak width measurement σ. The 1/e halfwidth σ of the diffraction peaks is determined by a symmetric multi-Gaussian
fit to the integrated (within the dashed region) 1D momentum profiles, following a π phase-shift of the diffraction envelope. (b) Single-axis visibility
measurement Vis, determined by the contrast of time-of-flight interference patterns. (c) Visibility Vis and peak width σa of the |ai component (filled circles),with fixed lattice depths of Va = 12 ER and V⊥ = 12 ER⊥ , in the presence of |bi
atoms [fb ≈ 3/4, Na + Nb = (3.7 ± 0.2) × 105 ] as Vb is increased. Here, the recoil energies of the transverse and z lattices, with wavelengths λ⊥ = 1064 nm
and λz = 785 nm, respectively, are ER⊥ = h2 /2mλ2⊥ and ER = h2 /2mλ2z .
In reference data taken without |bi atoms and with Na = (1.0 ± 0.1) × 105(open circles), the visibility and peak width are roughly constant at 0.52(3)and 0.26(2), shown as the dashed lines. Data points are averaged over 3-5
runs, with statistical error bars shown. Cartoons at bottom demonstrate theregimes in which polaron and quantum emulsion effects can be expected. For
an extended discussion of this experiment, see Refs. [44, 45].

94

shown in Fig. 8.7 we have measured the coherence properties [visibility andmomentum-peak width as detailed in Fig. 8.7 (a,b)] of one species (called |ai)in contact with a second species (|bi) whose tunneling was tunable through its
lattice depth Vb . The properties of the |ai species were kept constant throughout, and by itself the |ai species was a strongly interacting superfluid near theMott insulator transition. For large lattice depths for the |bi atoms Vb , i.e. ina regime where one would expect the formation of quantum emulsion states,
we indeed observe in Fig. 8.7 (c) greatly reduced coherence of the |ai species.
A second possible contribution to observed coherence loss in mixture experiments occurs in the opposite limit, when one of the species is very mobile.
Indeed, a reduction of the |ai component’s coherence can also be observed inthis regime, as seen in Fig. 8.7 (c) for very small values of Vb . The mechanismthat can lead to reduced spatial coherence in this regime is that of a polaronic
coupling between the |ai atoms in the lattice and the highly superfluid background of mobile |bi atoms. In this scenario, the slower-tunneling atoms couple
to density deformations of the faster-tunneling species, represented as phonon
excitations. This coupling between lattice-based atoms and phonon modes of asuperfluid background leads to quasiparticles known as polarons [63] consistingof atoms surrounded by a density deformation (a dip for repulsive coupling anda bulge for attractive coupling), described by a coherent cloud of phonons. The
effects of the background on the lattice-confined atoms can be approximately
described through a renormalization of tunneling (effective mass [226]) andon-site interactions, and the coupling can in fact induce a premature (in termsof lattice depth) transition to an insulating state as compared to the singlespecies case. In dynamical studies of multi-species mixtures [227, 228], thecoupling of the motional state of one species to density excitations of anothermight be used to study the phenomenon of Bloch-phonon coupling [229–231],
which is relevant to condensed matter systems.
Polaronic coupling can also induce long-range off-site interactions between
the lattice-confined atoms [63, 64]. For atoms that are mobile, enough to beable to reconfigure their density distribution through hopping, induced nearestneighbor attraction can lead to the formation of polaron clusters, which minimize the kinetic energy of density deformations in the bath. In the regime inwhich the tunneling of the lattice-confined atoms is completely frozen out, thecoupling of the localized atoms to the delocalized bath can be used to study the
spin-boson model [232], which is fundamental to the study of decoherence andthe dissipative dynamics of qubits coupled to an environment. The qubit may
be represented by a number of possible states of the localized atom - such as
its internal state [233], site-occupancy [234, 235], or orbital wavefunction [119]- with coupling governed by collisional interaction with the atoms of the bath.

95

In recent years, a number of proposals have been presented that suggest using
bath-induced dissipation as a mechanism for cooling atoms in a bath [236],
and more generally for quantum state engineering through dissipation [237].
As a final example, we mention that in the extreme case of one species confined to a lattice and the other unconfined and behaving as an ideal gas, the useof contact thermometry has been proposed for the characterization of temperature in strongly-correlated lattice gases [238] [however the equilibration time
between atoms with large ratios of (effective) mass may be prohibitively longin practice due to a mismatch in their dispersion relations [49, 219]]. Alsoin this limit, the scattering of free massive particles [49, 80] may be used formeasurement of the dynamic and static structure factors of strongly-correlated
lattice gases, in analogy to photon scattering [131, 192, 239–241].

8.2

Hyperfine-state mixtures

We now discuss the specific internal state structure that allows for atoms to
be in different (pseudo)spin states in our experiments. We consider atoms in
the electronic ground state of 87 Rb, the state 2 S1/2 [86]. With a spin angularmomentum of S = 1/2 and no fine structure splitting of the L = 0 groundstate, there is a total electron angular momentum J = 1/2. Inclusion of thenuclear spin I (I = 3/2 for 87 Rb) then leads to the total angular momentum
F = I+J, with allowed values within |J −I| ≤ F ≤ I +J. Thus, interaction of
the nuclear spin and the electron total angular momentum leads to hyperfine
splitting of the ground-state into the F = 1 and F = 2 manifolds, by a frequency ∆hfs ∼ 6.8 GHz. Each manifold is split into different Zeeman sublevels,
defined by their projection along the magnetic field quantization axis mF , withallowed values of |mF | ≤ F . The Zeeman effect shifts the internal state energies with applied magnetic field B, and as shown in Fig. 8.2 (a) at low fields,
the separation between adjacent Zeeman sublevels |F, mF i is roughly proportional to field, as ∆E = gµB mF B (here µB = e~/2me c is the Bohr magnetonand g is the Landé g-factor, with gF ={1,2} = {−1/2, +1/2} for the ground-state
hyperfine manifolds). A more general dependence of the internal state energy
levels on magnetic field B is plotted in Fig. 8.2 (b), based on the so-called
Breit–Rabi formula [242]. This is based on a combination of the Zeeman shifts
for a given magnetic moment of the electron µJ = gJ µB J/~ and of the nucleus µI = gI µB I/~, described by the Hamiltonian HZeeman = −(µJ + µI ) · B,
and the coupling of these magnetic moments via the hyperfine interaction
Hhf = Ahf I · J.
For magnetic fields that are spatially inhomogeneous, conservative potentials can be constructed for atoms in a given internal state, as used in magnetic

96

mF =

-2 -1 0 +1 +2

(a)

15

F=2

(b)

2,+2

~ 6.8 GHz

2.65

F,mF = 2,+2

2.55
2.45
0

2,-2

E / h [GHz]E / h [GHz]F=1

25 50 75 100

B [G]

-4.2

E / h [GHz]10

F,mF = 1,-1

01,-1

-5

2,-2

-10

-4.3
-4.4
0

2,-1

5

1,+1

-15

1,+1

0

25 50 75 100

0.5

1

B [T]

B [G]

Figure 8.2: Hyperfine energy level structure. (a) Hyperfine state structure of87
Rb electronic ground state, with energy dependence of the Zeeman sublevels(|F, mF i states) at low magnetic fields B. (b) Ground-state energies as afunction of magnetic field B, at high fields (in Tesla, T = 104 G).
trapping potentials (quadrupole trap and time-orbiting potential (TOP) trap
used for the BEC production, c.f. Chapter 3). The introduction of forces via
spatially-varying magnetic fields is also used for Stern–Gerlach separation of
the different internal states in our experiments. Shortly after releasing theatoms for time-of-flight expansion, we apply a magnetic field gradient pulse,which “kicks” the atoms in a state-dependent way. After a suitable duration,
the different “spin”-components fully separate in time-of-flight, and can beseparately imaged onto different regions of the camera’s field of view.
To create mixtures of atoms in different hyperfine states, we need to address the atoms with an oscillatory field with frequency matching the hyperfine splitting. For this, we use microwave radiation resonant with the hyperfine splitting, at a tunable frequency of ∼ 6.8 GHz (with the microwavefrequency source and antenna used to address the atoms are described in detailin Ref. [45]). In Fig. 8.3, we display two methods used for populating different mixtures of internal states. The first method, as shown in Fig. 8.3 (a), is
simply to address the atoms with microwave radiation of constant intensity(∝ V ) and with a fixed frequency that is resonant only between two states,
shown here for |ri = |1, −1i and |bi = |2, −2i. For the two-state systemwith resonant driving, the ground and excited state populations will display
an oscillatory dependence with 100% contrast, so-called Rabi-flopping. Thismethod of internal state control is ideally suited to rapidly transfer population, and for creating coherent superpositions of different internal states. The
97ω (t) - ω rb(b)

N / Ntotb = 2,-2

τ

V (t)

(c)

1.00

1.00

0.75

0.75

0.50

V0

V (t)

0.50

0.25

r = 1,-1-δN / Ntot(a)+δω (t) - ω rb

0

0.25

0.00

0.00
0

0.1

τ [ms]

0.2

-35

-25

-15

µ - wave Power [dBm]

Figure 8.3: Transitions between different internal states of the atoms. (a) We
drive coherent Rabi oscillations in the internal state populations, using microwave radiation that is on resonance with a transition between two internal
states, shown here as red (|ri = |1, −1i) and blue (|bi = |2, −2i). (b) We perform Landau–Zener sweeps of the microwave frequency across the resonance,
enabling a well-controlled probability of transition between the two states.
The transition probability can be tuned through the microwave power.
second method is displayed in Fig. 8.3 (b), and is based on the Landau–Zener
protocol [243]. Here, population initially resides solely in the ground-statelevel |ri, and the frequency of the microwave field is ramped linearly acrossthe transition frequency ωrb . Either the speed of the ramp or the intensity ofthe microwave field may be adjusted to control the probability that the population “tunnels” to the excited state |bi. In Fig. 8.3 (b), we plot final state
populations as a function of the microwave power. This method is especiallywell suited to provide highly controllable mixture ratios, with little sensitivity
to variations in the magnetic fields.
In addition to these microwave transitions, we can also make intramanifold
transitions to different states via radiofrequency radiation [109]. Note that, at
low fields the near degeneracy in the energy-splittings between different pairsof states prohibits a two-level description. However, as seen in Fig. 8.2 (b), at
higher fields this restriction is lifted as the linear description of the Zeeman
shift breaks down. At low magnetic fields, the degeneracy can also be lifted by“dressing” the energy levels - i.e. shifting the levels in a non-uniform, statespecific way - by applying microwave fields that are off-resonant with respect tothe hyperfine structure splitting [244, 245]. For homogeneous microwave fields,
98this technique has become an important tool for manipulating internal state
dynamics [246, 247]. In the following section, we shall discuss an analogous
technique of applying off-resonant light fields to shift energy levels in a statespecific way. However, the applied light fields will in general be spatiallyinhomogeneous, and thus can be used to exert forces, including trapping.

8.3

State-dependent optical potentials

We now describe the general methodology used to create state-dependent optical potentials [248–250] in our experiments, using laser beams with tunablewavelength λ and polarization ˆ
. Firstly, when trying to create completely
state-selective potentials, we need to achieve a cancellation of trapping andanti-trapping potentials. As described in Chapter 2, depending on the signof the detuning of a laser frequency ω from an excited-state transition at
frequency ω0 , a far-detuned optical potential can either create an attractive
trap (red-detuned, ω < ω0 ) or a repulsive anti-trap (blue-detuned, ω > ω0 ).
To create such a cancellation with a single-frequency laser beam, we need totune between two different excited-state transitions, such that the laser frequency is red-detuned with respect to one and blue-detuned with respect tothe other. For alkali atoms there are two strong optical transitions from theelectronic ground state, the D1 and D2 lines. For 87 Rb these transitions occur
at wavelengths 795 nm and 780 nm, respectively, and the wavelength of ourstate-dependent laser light is tunable within this range. In Fig. 8.4 (a), weshow how light with wavelength between the D1 and D2 lines leads to red- andblue-detuned contributions from different excited-state transitions.
Secondly, we need to create an optical potential that is particular to agiven internal state of the atom. This is based on the conservation of angularmomentum, which for a given polarization state of light only allows transitions
between certain sets of |mF i sublevels. Here, certain ground states may only
experience optical coupling via either the D1 or D2 transitions. In Fig. 8.4 (a,b)
we illustrate how this takes place for the specific case of σ − circularly polarized
light, with a wavelength of λ = 787.56 nm. The |1, −1i state couples to excitedstates via the D1 and D2 lines, while the |2, −2i state is only coupled to the P3/2
level by the D2 line. Thus, the |1, −1i experiences a light-shift cancellation(with zero net potential for the given λ and ˆ), while the |2, −2i does not andexperiences an attractive potential.
We now demonstrate how such a state-selective potential can be used inpractice. In Fig. 8.5 (a,b) we show diffraction patterns for two different hyperfine states of atoms that are (separately) subject to a state-dependent opticallattice. One of the species exhibits diffraction peaks, due to coupling to the99(a) m F = -3 -2 -1 0 +1 +2 +3(b) m F = -3 -2 -1 0 +1 +2 +3

}

P3/2

F=3
F=2
F=1
F=0

P1/2

F=2
F=1

}

P1/2

F=2
F=1

D1 = 794.98 nmσ−
λ = 787.56 nm

S1/2

F=2
F=1

D2 = 780.24 nm

P3/2

F=3
F=2
F=1
F=0

}

σ−
λ = 787.56 nm

S1/2

F=2
F=1

Figure 8.4: Basic scheme for creating state-dependent optical potentials.
(a) For certain internal states (shown for |F, mF i = |1, −1i), σ − polarized
light with a wavelength halfway between the D1 and D2 optical transitions
leads to zero net light-shift, i.e. no optical dipole potential. This effect is due
to a complete cancellation of negative and positive light-shifts, from the lightbeing red-detuned with respect to the D2 line and blue-detuned with respect
to the D1 line. (b) For a different internal state, |2, −2i, only coupling to excited states via the D2 transition is allowed for σ − polarized light, such that nolight-shift cancellation occurs and this state experiences an attractive optical
dipole potential.
light-field, while the other does not. Interesting effects can take place whenboth species are present, leading to diffraction peaks being observed for eachof the states, as in Fig. 8.5 (c). As studied in detail in Ref. [219], pair-wise
momentum-exchanging collisions between the two species lead to the appearance of diffraction peaks for the species that experienced no optical potential[cf. Fig. 8.5 (d)], due to nonlinear atomic four-wave mixing.
After having introduced the general idea behind the method we employ
for producing state-dependent optical potentials, and after discussing some
practical examples, we explain in a bit more detail how one can calculate
such state-dependent light-shifts [45, 114, 248, 250, 251]. We again consider
the dispersive effect of off-resonant laser light acting on neutral single-electron
atoms [102], but now a bit more rigorously than in Chapter 3. For light

100

(a)

only a(b) only b20−2
pz /ħkL

z
sz=6

x

sz= 0

a and b
(c)

(d)

a

b
2
0
−2

Figure 8.5: State-dependent lattices and atomic four-wave mixing [45, 219].
(a) Diffraction pattern of |ai atoms from an optical lattice of wavelength λ =785 nm and σ + polarization, pulsed on briefly for τ = 25 µs. Calibration of thelattice reveals a depth Va = 6 ER , with recoil energy ER = h2 /2mλ2 . (b) Samefor |bi atoms only, with no observable diffracted population. The lattice isfully state-selective, with Vb = 0 ER . (c) When both species are present andsubject to the state-selective lattice, atoms of the |bi species appear in thediffracted momentum modes. (d) The mechanism for this apparent diffraction
is a pair-wise momentum redistribution between the two hyperfine states, i.e.
nonlinear four-wave mixing of matter-wave fields. For an extended discussionof this experiment, see Refs. [45, 219].
fields far from resonance, for which direct transitions to excited states can beignored (for a detuning |∆/Γ|  1), the effect of the atom-light interaction
Hint = −d · E (with d the dipole operator and E the local electric field) on
the ground state energy can be treated in second-order perturbation theory.
For some state |ii, coupling to excited states |ji by E induces an energy-levelshift
X |hj|d · E|ii|2
,
(8.1)
∆Ei =
E
i − Ejj6=iwith the assumption of a defined quantization axis. For a light field withintensity I(r, t) possibly varying both spatially and temporally (with I =0 cE02 /2 and E0 the electric field amplitude), with frequency ω, and of uniform101

polarization ˆ, this can be reexpressed (now including both the rotating andcounter-rotating terms) as
1 X |hj|d · ˆ
|ii|2
|ii|2 |hj|d · ˆ
Ui (r, t) =−I(r, t) ,(8.2)
20 c j6=i ~(ω − ωji )~(ω + ωji )where ~ωαβ = Eα − Eβ . Due to dipole selection rules stemming from conservation of angular momentum and symmetry considerations [86], only transitions
satisfying ∆L = ±1, |∆J| ≤ 1, |∆F | ≤ 1, and ∆mF ≤ 1 are allowed, suchthat typically only a few excited states will contribute to the level shift for agiven polarization of the light field.
We now focus more narrowly on the system at hand, 87 Rb atoms within the2
S1/2 ground state, either in the F = 1 or 2 manifold, coupled to the 2 P1/2 and2
P3/2 excited states by the D1 and D2 transitions, respectively. Here we can,ignoring counter-rotating terms, approximate the induced dipole potential asΓD2 X |cji |2
3πc2 ΓD1 X |cji |2+2 3
I(r, t)(8.3)
Ui (r, t) =32 ωD
ω − ωjiωD2 2
ω − ωji2
1
j∈ P1/2

j∈ P3/2

for a given polarization of the light field q = 0, ±1 (q = 0 corresponding tolinear π polarization and q = ±1 to circular σ ± ), with the factor of two comingfrom the different line strength factors of the D1 and D2 lines. Here ΓDα andωDα are the natural line-width and transition frequency of the Dα transition(α ∈ {1, 2}), respectively. The term cji is the Clebsch–Gordon coefficient for agiven polarization q and set of states |ii = |F, mF i and |ji = |F 0 , m0F i, defining
the angular part of the dipole matrix element as cji = hF 0 m0F |F 1 mF qi [113].
For detunings ∆1,2 = ω − ωD1,2 that are small compared to the finestructure splitting ∆fs (∆fs /2π ∼ 7 THz for 87 Rb) but large compared tothe excited-state hyperfine structure splittings ∆0hfs (with the largest being∼ 2π × 800 MHz), small differences in the transition frequencies ωji can be33
neglected. After also setting the nearly equal quantities ΓD1 /ωD≈ ΓD2 /ωD
1
2
3
to Γ/ω0 , we then arrive at a much simplified form of the dipole potential experienced by the state |ii ≡ |F, mF i for a given polarization q of the lightfield [250, 251]
πc2 Γ 1 − gF mF q 2 + gF mF q+
I(r, t) .
(8.4)
U (r, t) =
2ω03
∆1
∆2
One serious consideration to keep in mind when using state-dependent optical potentials, given their relatively small detuning from excited-state transitions, is the effect of spontaneous photon scattering or Rayleigh scatter102

ing. By ignoring the counter-rotating term in Eq. 8.2 and incorporating theearlier used form of the scattering rate for far off-resonant dipole potentials
Γscatt = ΓUdipole /~∆, we can describe the total scattering rate for a state |ii
from coupling to excited states |ji as
|ii|2
1 X Γj |hj|d · ˆI(r, t) .
(8.5)
Γi (r, t) =20 c j6=i ~2 (ω − ωji )2
Alternatively, we can apply the expression for Γscatt separately to the D1 andD2 line contributions of Eq. 8.4, yielding the approximate form

1 − gF mF q
πc2 Γ
2 + gF mF q
Γ1
I(r, t) .
(8.6)
Γi (r, t) =
+ Γ2
2~ω03
∆21
∆22
For the typical laser intensities used in our studies with state-dependent
optical lattices, heating due to Rayleigh scattering becomes the main limitationon the lifetime of the experiments, on the order of a few 100 ms. While thesetimescales allow for the study of many interesting effects as discussed in thefollowing chapters, processes occurring at rates of a few Hz or requiring very
low temperatures, such as those based on engineered spin-exchange [59], will
likely be hard to observe due to this incoherent photon scattering.
Having introduced the essentials for calculating light shifts and scattering
rates for the most general of situations, we now plot in Fig. 8.6 calculated
dipole potentials and scattering rates (using Eqs. 8.2 and 8.6) as a function of
wavelength in the vicinity of the D1 and D2 transitions (using 87 Rb transitionline data from Ref. [113]). Our experiments incorporate in some capacity
optical potentials with wavelengths in this range. We note that laser light fields
in general will not be restricted to either linear or circular polarizations, but
may be represented as a coherent superposition of σ + and σ − contributions.
Fig. 8.7 shows the experimentally measured polarization dependence of thestate-dependent optical lattice depth, in a fully retro-reflected geometry using
light with wavelength λ = 785 nm. The induced lattice depth due to a statedependent coupling to excited states is measured with respect to the states|ai = |1, −1i and |bi = |2, −2i.

8.4

A moving state-dependent optical lattice

Here, we briefly also detail the implementation of a moving state-selective
lattice potential. A standard method for the generation of a moving opticallattice is through the interference of two laser beams of the same wavelength103∆E / I [ER / (W/cm2)]

(a)

σ−

0.5

π

σ+

0

-0.5

0.3

Γ / I [s-1 / (W/cm2)]

(b)

780

785

790

795 780

σ−

785

790

λ [nm]
π

795 780

785

790

795

σ+

0.2
0.1
0

780

785

790

795 780785

790

λ [nm]

795 780

785

790

795

Figure 8.6: Energy shifts and scattering rates as a function of optical wavelength. (a) Ratio of energy level shift ∆E to light intensity I, shown for σ − ,π, and σ + polarizations and for the hyperfine states |ai = |1, −1i (red dashedline) and |bi = |2, −2i (blue solid line). (b) Ratio of incoherent scattering rate
Γ to light intensity I, for polarizations and internal states as in (a).
λ, but with a slight frequency difference δf [215, 252, 253]. For fixed frequency
detuning δf , the optical interference pattern moves with a constant velocity
vL = δf (λ/2). This frequency detuning may be experimentally achieved inmany ways, such as by changing the optical path length in a fully retroreflected geometry - either by a moving mirror, a rotating dispersive element,
or with a modulated electro-optical element - or by frequency-shifting withacousto-optic modulators (AOMs). Due to a greater stability and control of
the frequency-shifts, we use radiofrequency-driven AOMs. We use a phasestable setup shown in Fig. 8.8 (a). Some portion of the incident laser beam isreflected by a polarizing beam-splitter (with the transmitted pathway blockedoff). The reflected light passes through two AOMs, driven by two frequencies different by δf , and is diffracted into the first (+1) and minus-first (−1)
orders respectively. This beam, frequency-shifted by δf , is then rotated in polarization and transmitted back through the polarizing beam splitter, nearly
antiparallel to the forward-traveling beam. Finally, the light’s polarization is104

Vb / Va2

[arb. u.]

3

0

Vb

Va

0

 / 90º 1

σ+



1
0

0.0

0.2

0.4

σ0.6

 / 90º

0.8 1.0

Figure 8.7: The polarization dependence of a fully retro-reflected optical lattice acting on atoms in two internal states, |ai = |1, −1i and |bi = |2, −2i.
The main plot shows the polarization dependence of the ratio of optical lattice depths experienced by the two states Vb /Va , with the inset showing (inarbitrary units) the individual dependence of each of the two states on polarization. The polarization axis is parameterized by the rotation angle ofa polarization λ/4 waveplate, which is used to change the polarization fromlinear π to circularly polarized, either σ + or σ − . Data are determined from
Kapitza–Dirac diffraction spectra.
again changed through use of quarter- and half-waveplates, so that at the position of the atoms the forward-going and backwards-going pathways have thesame polarization. In Fig. 8.8 (b), we show time-of-flight images of expanded
clouds of atoms from a moving lattice (originally loaded in a stationary lattice
to form a deep Mott insulator state) for several velocities.
In Chapter 10, we shall use this moving state-selective lattice to study thescattering of free neutral atoms from crystals of localized atoms. In regard
to characterizing strongly interacting systems, it can be also used as a toolfor inelastic Bragg spectroscopy [131], phase-modulation spectroscopy [205],
and, in the state-dependent case, for detecting the dynamic spin-responseof bosonic mixtures [254]. Furthermore, arbitrary control over the phasemodulation [255] of a state-dependent lattice may also allow for the engineering of spin-dependent gauge potentials for ultracold mixtures, i.e. to create
spin-orbit coupling [256] in lattice-dressed matter-wave mixtures.

105

(a)

0

+1

-1

0

rf2

[80 MHz][80 MHz + δf ]

atoms

rf1

polarization
waveplates

acousto-optic
modulator (AOM)

mirrorpolarizing
beam-splitterbeam-stop

3
2
1
0

3
2
1
0

vL / vR = 1

vL / vR = 2

v / vRv / vR(b)vL / vR = 3

Figure 8.8: (a) Moving optical lattice setup. The interference of two laser
beams, with equal polarization and wavelength λ, up to a slight frequency
difference δf , creates optical lattices moving with a velocity vL = δf λ/2.
(b) Time-of-flight images of expanded clouds of atoms released from a moving
optical lattice. The change in center-of-mass positions along z corresponds tothe change in vL , shown for frequency detunings of δf = {15, 30, 45} kHz andcorresponding lattice velocities vL /vR = {1, 2, 3}, with vR = h/mλ the recoil
velocity and λ = 785 nm.

106

Chapter 9
Glassy behavior in a binary
atomic mixture
In this chapter, we report on the experimental study of disordered one-dimensional(1D) Bose gases, where the disorder is generated with an incommensurate optical standing wave consisting of two spatial frequencies. The standing wave
plays the role of a disordering agent, either directly through an associated optical lattice potential or by selectively pinning impurity atoms to the intensity
maxima of one of the two constituent standing waves. We use such randomly
distributed localized impurity atoms to study the effect of uncorrelated disorder on 1D lattice-modulated Bose gases. We then make comparisons to thecase of correlated quasi-disorder, which is formed by an incommensurate lattice potential that acts directly on the 1D gas. While the effects of the twodisorder realizations are comparable deeply in the strongly interacting regime,
both showing signatures of Bose glass formation, we find a dramatic difference near the superfluid-to-insulator transition. In this transition region, we
observe that random, uncorrelated disorder leads to a shift of the critical lattice depth for the breakdown of transport as opposed to the case of correlatedquasi-disorder, where no such shift is seen. Our findings, which are consistent
with recent predictions for interacting bosons in one dimension, illustrate theimportant role of correlations in disordered atomic systems. This chapter isbased on our publication Glassy behavior in a binary atomic mixture, Phys.
Rev. Lett. 107, 145306 (2011) [46].

9.1

Introductory discussion

The presence of disorder is inherent to solid state systems, and it has profoundeffects on transport in a variety of contexts, ranging from electron conductivity107

in metals to dirty superconductors [257]. Quantum gases in optical lattices [26]can, with a high degree of experimental control, elucidate the role played bydisorder in a number of physical phenomena. Recently, Anderson localizationof matter-waves in disordered potentials has been observed for non-interacting
gases [33, 34], and additionally, the reemergence of superfluidity due to repulsive interactions [74]. Discerning the (at times) competing roles of disorder andinteractions is key to the understanding of Bose-glass behavior [28, 48, 258] instrongly interacting disordered systems. Ultracold atomic systems [68, 69, 75]
should provide a versatile testbed to aid in this endeavor [37].
Previous studies of ultracold atoms in disordered potential landscapes, generated by optical fields, have generally suffered from strong correlations of thedisorder that decay over length scales greater than either the healing lengthof the superfluid or the lattice spacing. This is true for both speckle potentials [34, 69, 75] that are diffraction-limited to structures on the order ofthe generating laser field’s wavelength, and quasi-disordered bichromatic lattices [33, 68, 74], which over large distances exhibit perfect correlations that
may make them rather unsuitable for the realization of true disorder. To circumvent these limitations, recently it has been proposed [47, 66, 67, 71] touse atomic impurities, which can be confined to regions much smaller than alattice spacing, to act as point-like defects.
The sudden quench of an atomic impurity field acting on mobile particles
has been proposed [67, 71] for the study of dynamical, out-of-equilibrium response to disorder. Alternatively, theoretical studies [47, 70] have shown thateven a slow “freeze-out” of the tunneling of one species from an initially homogenous mixture can lead to metastable “quantum emulsion” states, typified
by local separation between frozen and mobile atoms (for repulsive interactions) and displaying properties similar to an equilibrium Bose glass. The
study of quantum emulsions may help shed light on the coherence-loss mechanism in a number of experiments involving mass-imbalanced atomic mixtures,
both for the boson-boson [51] and boson-fermion [54–56] cases.
In this chapter, we report on experimental studies of interacting 1D Bose
gases in the presence of disorder. We have studied the effects of uncorrelateddisorder formed by atoms of an auxiliary spin state “frozen” to sites of an
incommensurate lattice, and compare to the case of correlated quasi-disorderfrom an incommensurate bichromatic optical lattice. While both disordertypes drive strongly interacting samples into an apparent Bose glass state,a large difference is seen for intermediate interactions, where we find that
uncorrelated disorder has a dramatic effect in driving the system towards aninsulating state. Our observation of enhanced localization for a more random
disorder is consistent with recent theoretical predictions for interacting bosons

108in 1D [259].

9.2

Experimental procedure

To briefly describe our experimental procedure, we begin with an opticallytrapped Bose–Einstein condensate of 87 Rb atoms. In 200 ms we load an arrayof isolated, one-dimensional tubes formed by the intersection of two opticallattices. These lattices are of period d = λ/2 and depth 40 ER (with λ =1064 nm, ER = (h/λ)2 /2m, Planck’s constant h, m the atomic mass). The
atoms are trapped along the tube axis z by a nearly harmonic potential oftrapping frequency ωz /2π = 80 Hz. A lattice along z, also of period d andwith variable depth s (in units of the recoil energy ER ), is smoothly ramped
up within 100 ms. This primary lattice will serve to define the sites (withindex i) and parameter values (tunneling t, and on-site interaction U ) of thethe (disordered) Bose–Hubbard Hamiltonian we shall use to characterize thesystem.
Initially, the tubes contain only atoms in the |F, mF i ≡ |2, −2i hyperfine
ground state. To create localized atomic impurities that act as uncorrelateddisorder, a fraction (fimp ) of the total population of 8×104 atoms is transferredto the |1, −1i state via a microwave Landau–Zener sweep. The impurity atoms
are loaded into a completely state-selective lattice [44] along z in 20 ms. Thislattice has spacing d0 = λ0 /2, with λ0 = 785 nm, and the impurity atoms
are deeply localized at 20 ER0 (recoil energy ER0 = (h/λ0 )2 /2m ∼ 1.8ER ). As
we begin with an initially homogeneous spin-mixture (i.e. with no spatial
dependence of the spin expectation value), and the lattice loading is fast withrespect to global mass redistribution in our 1D system (but slow with respect tointra-site band dynamics), this situation has been shown in several theoreticalstudies to lead to a so-called “quantum emulsion” state [47, 70, 71], in which
the localized or slow-tunneling atoms can serve as static disorder for the fastertunneling atoms. When considering the case of impurity creation starting froma homogeneous spin mixture, the loading should lead to a superposition ofmany different random distributions or impurity atoms, i.e. to many different
realizations of disorder. In fact, the use of atomic quantum fields in such amanner has been suggested because of this quantum parallelism [67, 71]. In
the end, the measurement will show the response of the system to one suchprobabilistic spin projection (for each of the independent tubes). In what
follows, we refer to the typically equal mixture used (fimp = 0.5, equal partsfrozen and mobile atoms) as “the mixture”.
The cartoon in Fig. 9.1 (a) qualitatively depicts such a one-dimensional
system of lattice-trapped bosons with embedded impurities. It shows that, af-

109

(a)

(b)

∆i = εi - εi+1

Figure 9.1: Disordered one-dimensional Bose gases. (a) To create a disordered
impurity-field, half the atoms of a lattice-trapped Bose gas are converted toan auxiliary spin state (red) and localized to a state-selective incommensurate lattice (dashed green). (b) Alternatively, a weak incommensurate lattice(green) is superimposed onto a lattice-trapped Bose gas. In both cases, thesecondary potential (atomic or optical) causes site-dependent energy shifts εiand site-to-site energy differences ∆i . Here we show an average filling of 1
atom of either spin state per site, while in the experiment we have an inhomogeneous system with a typical central filling factor of n ∼ 3 (total) atoms
per site.
ter creating a homogeneous spin mixture, we slowly freeze the impurity atoms(red) to a deep state-selective lattice (dashed green) of incommensurate spacing. For comparison to a well-studied case of correlated disorder, we also
study bosons in an incommensurate bichromatic lattice system [33, 68, 74],
as depicted in Fig. 9.1 (b). For this alternate disorder implementation, we
begin with a sample of 4 × 104 atoms in the state |2, −2i (i.e. same numberof mobile atoms as for the case of uncorrelated disorder). We then ramp up asecondary lattice in 20 ms, of spacing d0 and variable depth s0 × ER0 , onto the|2, −2i atoms. In general, due to the external trapping potential, our system
is not homogeneous but can be characterized by a typical central filling factorof n ∼ 3 (total) atoms per site. The depictions in Fig. 9.1 are intended toshow the key aspects of the two schemes for disorder implementation, while
they are not necessarily representative of our actual densities.
A brief note on the timescales of adiabaticity in our system, in relation
to our experimental sequence: the timescale over which we “turn on” thedisorder ensures that we are adiabatic with respect to local excitations (i.e.
on-site band excitations for either of the species), however it strongly deviatesfrom adiabaticity with respect to the global ground state of the system. In
fact, the dynamical creation of disorder in a quantum emulsion relies on thisnon-adiabaticity, as the true ground-state in such a system will typically becharacterized by global phase separation between the slow-tunneling and fasttunneling species (assuming, as in our experiments, that interactions between
slow-tunneling atoms are stronger than those between fast-tunneling atoms

110and interspecies interactions, i.e. Uss > Usf > Uf f ), with the more mobile
atoms moving to the outer regions of the trap. This is a rather boring state,at least with regard to studies of disorder. The quantum emulsion state [47],on the other hand, is a metastable state characterized by a disordered arrangement of the densities of the two species. The pathways by which thismetastable state may relax, such as by global mass redistribution of the two
species, are effectively cut off by the insulating nature of the state.

9.3

Disordered Bose–Hubbard gas

In both cases, the dynamics of the mobile atoms can approximately be described by the Bose–Hubbard Hamiltonian (BHH) [28, 29]
Ĥ = −tXi(â†i âi+1 + â†i+1 âi ) +XUXn̂i (n̂i − 1) +n̂i εi ,
2 i
i

(9.1)

where t and U are the tunneling and interaction energies of the mobile atoms,
and âi , â†i , and n̂i = â†i âi are the annihilation, creation, and number operators
for particles at lattice site i. While in a typical system the site-dependent
energy shifts εi will take the slowly varying form εi = mωz2 d2 i2 /2 due to harmonic trapping, here they will be largely determined by the added disordering
potentials. Neglecting harmonic confinement, we consider the effect of disorder in generating random site-to-site energy differences ∆i = εi − εi+1 , which
define the resonance conditions for single-particle intersite tunneling. The disordering potentials, inhomogeneous over individual sites, will in general havea second (likely more minor) effect of modifying the local on-site wavefunctions of the atoms. For strong perturbations this will necessitate a multi-band
treatment, and will lead to a spread of site-dependent values for both t and U ,with the possibility of asymmetric tunneling. In the following and in Eq. 9.1,
we consider only modifications of the ∆ distribution and neglect the effects on
t and U .
The introduction of random or pseudorandom ε (∆) distributions leads tothe disordered Bose–Hubbard model (dBHM), which has been important forthe description of experimentally relevant systems [28, 37, 204], such as cold
4
He atoms in porous media and bosonic Cooper pairs in Josephson junction
arrays. Aside from understanding better the physics of these analogous systems, experimental studies of the dBHM with ultracold atoms are importantfor several reasons. Currently, there is still active theoretical research aimed
at detailing the phase diagram of the dBHM and understanding the nature of
the superfluid-to-insulator transition [28, 260], and experimental efforts may

111

play a complementary role in this regard. These studies are also interesting as
the dBHM displays a non-trivial many-body phase – the Bose glass – whichto date has not been unambiguously observed in any cold atom experiment.
The Bose glass phase is insulating, as in the Mott insulating state, however
it has no discrete excitation gap and displays non-zero number fluctuations,i.e. non-zero compressibility. Study of disordered scalar bosons should also
help pave the way for the study of disordered spin Hamiltonians and spin glass
phases [37]. Lastly, the study of the dBHM in 1D is interesting as it examines the intersection of two fundamentally different localization phenomena
in low-dimensional bosonic systems. In the absence of interactions, random
disorder leads to localization of the one-dimensional bosonic wavefunctions forarbitrarily weak values of the disorder strength. At the opposing case, for fixeddensity but in the absence of any modulating potential - disordered or otherwise - strong repulsive interactions can mimic Pauli blocking and induce thelocalization of single-particle wavefunctions, through the so-called fermionization of bosons into a Tonks–Girardeau gas [199]. The physics of the Bose glass
phase, which results from a competition between disorder and strong repulsive
interactions, thus interpolates between these two extremes.
In general, the many-body ground state and excitations of the dBHM
will be defined by the filling factor n̄ = µ/U + 1/2, as well as by a come where ∆e characterizes the scale of the ranpetition between t, U , and ∆,dom energy shifts. There will also be a dependence on the details of the∆-distribution [28], e.g. quasirandom [259, 261] vs. random (including unbounded Gaussian noise [28], box-top distributions [28], random binary disorder [262], etc.). Here, we contrast the disordered ∆-distributions characteristicof randomly located impurity atoms and an incommensurate lattice potential,
respectively, by showing histograms of site-to-site energy shifts (for 1000 sites).
To calculate each distribution, we begin by assuming ground-state Wannier orbitals of the |2i ≡ |2, −2i atoms in a three-dimensional lattice with isotropic
lattice spacing d and with depths sx,y,z = {40, 40, 6}ER , ωi (r) (with i the site
index). For a generic disordering potential Vdis (r) that is Rrandom across thelattice, the site-dependent energy shift is given by εi = d3 r|ωi (r)|2 Vdis (r)with the integral taken over site i (±d/2 along each axis). For the incommensurate lattice potential, we homogeneity along x and y (as the lattice beam
has a 230 µm waist) and a form Vinc (z) = s0 ER0 sin2 (k 0 z), where k 0 = 2π/λ0
and s0 = 1. The histogram of ∆ values for this quasiperiodic lattice is plotted in Fig. 9.2 (b). For the random impurity potential, Vimp (r), we begin byassuming a fifty-percent mixture of impurity atoms (state |1i ≡ |1, −1i) thatare localized to sites of the very deep state-selective lattice along z, of spacing
d0 and depth sz = 20ER0 , with a corresponding Wannier orbital ω̃j (r). The

112

site indices that play host to impurity atoms, jk with index k ranging from
1 to 500, are determined by projecting out impurities from a distribution ofhard-core bosons on the primary lattice, i.e. with no multiple occupancies
per site and with positions forced to the nearest site of the secondary
P lattice.
We then represent the total impurity potential as Vimp (r) = g12 k |ω̃jk (r)|2 ,where g12 is the interspecies scattering length between |1i and |2i atoms, nominally taken as 100 a0 . A histogram of ∆ values for the impurity potential isshown in Fig. 9.2 (a). We note that given our actual densities and the inhomogeneity of the system as discussed earlier, a more realistic representation ofour ∆ distributions would vary over the system and would support multiple
impurity occupancies per site.
While both of these distributions are contiguous and extend beyond ±∆/U ,the details differ considerably. The impurity distribution is peaked about∆/U = 0 due to adjacent impurity-free sites, while the bichromatic lattice
distribution is peaked at the outer-bounds of the distribution. In Fig. 9.2 (c,d)
we plot the normalized autocorrelation function
χj = h∆i ∆i+j ii /h∆i ∆i ii .

(9.2)

The perfectly regular correlations in Fig. 9.2 (d) are a known attribute of quasidisordered incommensurate lattices [259, 261]. In contrast, for the atomic impurity field – which combines the irregular spacing of bichromatic lattices andthe irregular, probabilistic filling of binary disorder – off-site correlations arestrongly suppressed. This difference can have profound effects, as the localization properties of a system will generally depend on both the strength andcorrelation length of the disorder potentials [263, 264]. An extreme case canbe found for non-interacting particles in 1D, where random disorder leads toAnderson localization for any finite disorder strength ∆ 6= 0, while incommensurate lattices induce localization only beyond a critical lattice depth [33, 204],
i.e. in the Aubré–Andre transition [164]. One final, fairly natural way to characterize the two distributions is by their respective spatial frequency components. As is clear, the potential landscape of the quasi-disordered bichromatic
lattice should have only two spectral components at spatial frequencies k1 andk2 . This, however is not so for the case of impurity atoms that are randomly
located at the sites of the incommensurate lattice. Here, random occupation(and thus vacancies) can lead to many more spatial frequency components. In
Fig. 9.2 (e,f), we plot the spatial frequency power spectra (zoomed-in over asmall scale) of the two distributions (random and quasi-disordered). Here, thespectra are for the continuous potential landscapes, and not discrete in termsof site indices. Both spectra contain dominant peaks at spatial frequencies
k/k1 = 1 and k/k1 = d1 /d2 ∼ 1.35, due to the primary and secondary lattices,113

Figure 9.2: (a,b) Histograms of calculated ∆-distributions for the cases ofthe mixture (assuming either 0 or 1 impurities per site) and an incommensurate lattice of depth s0 = 1. Both are for a primary lattice depth s = 6(U/ER = 0.4). (c,d) Autocorrelation function χj = h∆i ∆i+j ii /h∆i ∆i ii of the∆-distributions in (a,b), as a function of the site-to-site distance j. The averaging is over 1000 sites. (e,f) Spatial frequency power spectra for the same
distributions as above. Both spectra contain dominant peaks at spatial frequencies k/k1 = 1 and k/k1 = d1 /d2 ∼ 1.35, due to the primary and secondary
lattices, respectively. In the case of impurity atoms, however, the probabilistic filling of |1, −1i atoms within the incommensurate lattice leads to higher
harmonics at k/k1 ∼ 2.7 and 4, as well as to a broad continuum at low spatial
frequencies, reminiscent of a “white noise” distribution.

114

respectively. In the case of impurity atoms, however, the probabilistic fillingof |1, −1i atoms within the incommensurate lattice leads to higher harmonics
at k/k1 ∼ 2.7 and 4, as well as a broad continuum at low spatial frequencies,
reminiscent of a “white noise” distribution.

9.4

Results and discussion

9.4.1

Excitation spectra

We begin our investigation into the effects of disorder by measuring excitation
spectra, which relate most directly to the distribution of site-to-site energy
shifts. Along with a finite compressibility [265, 266], a gapless excitation spectrum is a characteristic feature distinguishing a disordered Bose glass statefrom a (homogenous) Mott insulator. We measure the excitation spectra by
performing amplitude-modulation spectroscopy [31, 68, 185] of the primary
z-lattice at driving frequencies ωmod /2π. As described in Chapter 8, we sinusoidally modulate s by ±15% for 80 ms, ramp down to s = 4 in 5 ms,
and allow 15 ms of thermalization. When used, the incommensurate lattice isturned on in 20 ms prior to modulation and off in 5 ms concurrent with primarylattice ramp-down. From time-of-flight interference we measure the visibility
η− /η+ [44], with η± = N+1 +N−1 ±2N0 ; N0(±1) apertures in Fig. 9.4 (d). For thedisorder-free case (∆i ≈ 0) in Fig. 9.3 (a), with the sample chosen to be deep
into the 1D Mott regime (s = 14 , U/t ≈ 66), the excitation spectrum exhibits
resonant structure. The resonance positions are consistent with the excitationof particle-hole pairs at U/h [27, 185] (and 2U/h due to either higher-orderprocesses or excitation at the edge of Mott domains [27, 185]). In contrast,
for both the atomic impurity mixture (Fig. 9.3 (b)) and for an incommensurate lattice of depth s0 = 1 (Fig. 9.3 (c)) having comparable ∆-distribution
bounds, we observe flat excitation spectra (cf. [68]). These observations are
expected [28] for broadly-filled ∆-distributions with bounds ∆max > U , andare consistent with the system being in a Bose glass state (future compressibility measurements [266] should allow for the disambiguation between a true
Bose glass [28, 48, 258] and a disordered Mott state [267]).
While the observed spectral properties are consistent with Bose glass formation, transport measurements are necessary to confirm insulating behavior.
Here, we study the effects of disorder in the transition region between superfluid and insulator, determining the critical lattice depth at which the systems
become insulating through the study of localization and transport. In regard
to the former, the momentum-peak width of a released sample (related to theinverse correlation length ξ −1 of the sample in-situ [188]), exhibits a sudden

115

Figure 9.3: Disappearance of excitation gap due to disorder. (a) Visibility as
a function of amplitude modulation frequency (normalized to U/ER = 0.53
for s = 14), in the absence of disorder for s = 9 and s = 14 (open and filled
black circles). Fit lines are two Gaussians on a linear slope. (b) For s = 14,
with atomic impurities, open purple squares and filled red squares represent
fimp = 0.1 and 0.5, respectively. (c) For s = 14, with an incommensuratelattice of depth s0 = 1 (orange triangles) and no impurities. Solid error bars are
statistical over several runs, while dashed are estimated errors for individual
runs (120% of maximum statistical error).
increase accompanying an abating superfluid fraction and loss of off-diagonal
long-range order [212]. As for the latter, it has been shown [185, 214] that the
response to an applied impulse dies away upon entering the strongly correlated
regime, and can serve as a signature of insulating behavior [31, 75].

9.4.2

Transport and spatial coherence measurements

To study transport, we look at how the system evolves after an applied impulse.
A magnetic-field gradient along z is pulsed on for a duration of T = 1.2 ms
and applies a variable force F ranging from 0 to Fmax /m = 1.2 m/s2 , resulting
in an impulse I = F × T . As illustrated in Fig. 9.4 (a), we characterize theresponse as a function of I (with slope α) by monitoring the center-of-mass
velocity along z in time-of-flight absorption images following a brief (∼1 ms)
ramp-off of the z-lattice. As explained in Chapter 8, we access the momentumpeak width as in [68, 185] by releasing the atoms in time-of-flight following
a 50 µs lattice ramp up to s = 20 and a gravitational phase-shift along z(without impulse or lattice ramp-off). We then determine the peak width σ
by a fit to the profile of symmetric diffraction peaks on top of an incoherent
background, as shown in Fig. 9.4 (d).
116

Figure 9.4: (a) The response to impulse is determined by a straight-line fit(with slope α, normalized to the case of free atoms) to the dependence ofvelocity on applied impulse I (profiles shown for disorder-free and impurity
mixture cases; recoil velocity vR = h/2md). (b) α versus lattice depth s forbosons without disorder (black circles) and with atomic impurities (fimp = 0.5,
red squares). Lines and surrounding shaded regions are fits to the data withconfidence regions (1 s.d.) of a linearly decaying response, with no response(α = 0) beyond a depth sc . (c) Similarly, but for an incommensurate lattice(s0 = 3, green diamonds), with disorder-freedata reproduced for comparison.
√(d) The momentum-peak width σ (1/ e half-width) is determined by a symmetric multi-Gaussian fit to time-of-flight interference patterns. The shaded
regions about v/vR = 0(±1) are used to count N0(±1) for visibility measurements of Fig. 9.3. (e) σ versus s for 1D bosons without disorder and for the
mixture (colors/symbols as in (b)). The straight line for the disorder-free data
is a fit of a linear increase beyond a depth sc . The first few data points forthe case of atomic impurities are connected as a guide to the eye. (f) Similarly, but for an incommensurate lattice with s0 = 2 (blue triangles) and withdisorder-free data reproduced.

117

7 (a)
sc 6
5
0

1

s’

2

7
sc 6
5 (b)
0 1

1.0

α0
s’

2

3

0.5 (c)
0
1

s’

2

3

Figure 9.5: (a) Critical intercept sc as a function of the incommensurate lattice
depth s0 from linear fits to momentum-peak-width data. (b) Critical intercept
sc as a function of s0 from the impulse-response data. (c) The fit values of α0(y-axis intercept of linear fit to impulse-response data) as a function of theincommensurate lattice depth s0 .
We find that for the impurity mixture, the mobile atoms are more easily
driven towards insulating behavior. Similar to [31], we determine the criticalpoint at which the atoms become unresponsive to impulse by fitting a linear
decay to the response α as a function of lattice depth. As shown in Fig. 9.4 (b),this fit to the transport measurements yields a critical depth sc = 3.0 ± 0.5.
While we do not fit the peak width data, a kink near sc ≈ 2 can be observed in Fig. 9.4 (e). These values are roughly half of those measured for thedisorder-free case, with values of sc = 6.2 ± 0.5 and 5.7 ± 0.5 based on impulse
response and peak width, respectively. The value of sc ≈ 6 is close to the expectation for our given densities and interaction strengths. For our sample ofone-dimensional systems, which have a tube-averaged Lieb–Liniger parametervalue of γ = 0.6, estimates (see Eqs. 6.19 & 6.18) based on the Bose–Hubbard
and sine-Gordon models predict sc = 6.2 and sc = 4.7, respectively [31]. The
Lieb–Liniger parameter is determined as γ = mg1D /~2 n1D [204], with a 1D density n1D , an effective 1D interaction strength g1D ≈ 2~ω⊥ a [195], s-wave scattering length a ≈ 5.3 nm, and transverse trapping frequency ω⊥ = 2π×26 kHz.
For the incommensurate lattice, transport data for s0 = 3 is shown in
Fig. 9.4 (c). In this case no shift of the critical depth is seen. The momentumpeak width data mirrors this lack of a shift of the transition point, shown for avalue of s0 = 2 in Fig. 9.4 (f). Indeed, for the range of incommensurate lattice
depths considered (we restrict to s0 ER0 /sER . 1 in the transition region to keep
the depth of the secondary lattice less than the first), no shift of the criticalpoint can be seen for either of the two measures. We show in Fig. 9.5 (a) and(b) the fit-determined values of sc as a function of s0 from the peak-width andimpulse-response measures, respectively. Although no shift of the critical pointis seen, we do find that the incommensurate lattice has a detrimental effect ontransport. The response to impulse is generally suppressed, characterized by118

a reduced y-axis intercept α0 (i.e. α for s = 0), as is plotted in Fig. 9.5 (c) as
a function of s0 .
While both incarnations of disorder resulted in an apparent Bose glass statefor very deep lattices, a clear difference was seen in their effect on more weaklyinteracting samples. In attempting to account for the observed difference, anatural consideration is the disparity in their correlations [χj , cf. Fig. 9.2 (c,d)].
In general, one expects that the less correlated the disorder, the more enhancedis the localization [263, 264]. For interacting bosons in 1D, it has been shown
theoretically [259] that the localization transition occurs for a more weakly interacting gas (larger values of the Luttinger exponent K or lower values of the
Lieb–Liniger parameter γ [204]) in uncorrelated disorder than for correlateddisorder. Our observations of a sizeable shift of the transition point for impurities and a negligible shift for an incommensurate lattice are thus in qualitative
agreement with expectations based on their dissimilar correlation properties.
Also relevant to our observations is the reduction of phase-space density
in the presence of localized impurities, as well as of the atomic density due
to the dynamical formation of impurities from the mobile species. The first
effect has been shown [268, 269] to be responsible for adiabatic heating andloss of coherence in recent Bose–Fermi mixture experiments [56], due to reduced entropy following a reduction in effectively occupiable sites, both for
attractive and repulsive interactions. The second effect is more particular to“quantum emulsion” [44, 47, 71] experiments. Here, reduced density leads tomore strongly correlated many-body states in 1D and thus favors increased
localization and insulating behavior.

9.5

Concluding remarks

In conclusion, we have observed signatures of Bose glass formation in 1D Bose
gases with superimposed disorder, both for atomic impurities and for quasidisordered bichromatic lattices. The two disorder types have dramaticallydifferent effects in the transition region between superfluid and insulator, withatomic impurity disorder inducing localization in much more weakly interacting gases. Our observation that a more weakly correlated disorder leads
to enhanced localization is in qualitative agreement with recent theoreticalpredictions for interacting 1D boson systems. Future experiments aimed atmeasuring a non-zero compressibility in insulating regions of our system may
provide the “smoking gun” for Bose glass formation. Direct observation ofcompressibility through local number fluctuations [76] would be extremely difficult in our system, even provided an available in-situ imaging capability withresolution on the order of ∼ 1 µm, due to a line-of-sight interrogation through119many planes of our overall 3D system. Some information may be gained however through measuring the overall response to trap compression [266].

120

Chapter 10
Probing an ultracold-atom
crystal with matter waves
In this chapter, we report on the experimental study of the Bragg scattering ofdelocalized matter-waves from an ultracold atomic crystal, in which scattering
centers are formed by lattice-trapped neutral atoms. In recent years, atomic
quantum gases in optical lattices have served as a versatile testbed for important concepts of modern condensed-matter physics. The availability of methods to characterize strongly correlated phases is crucial for the study of thesesystems. Diffraction techniques to reveal long-range spatial structure, which
may complement in situ detection methods, have been largely unexplored.
Here we experimentally demonstrate that Bragg diffraction of neutral atoms
can be used for this purpose. Using a one-dimensional Bose gas as a source ofmatter waves, we are able to infer the spatial ordering and on-site localizationof atoms confined to an optical lattice. We also study the suppression of inelastic scattering between incident matter waves and the lattice-trapped atoms,
occurring for increased lattice depth. Furthermore, we use atomic de Broglie
waves to detect forced antiferromagnetic ordering in an atomic spin mixture,demonstrating the suitability of our method for the non-destructive detectionof spin-ordered phases in strongly correlated atomic gases. In implementing
the matter-wave crystallography of lattice-trapped gases, we make use of two
techniques that are compared and contrasted in this chapter. This chapter
is partially based on our publication Probing an Ultracold-Atom Crystal withMatter Waves, Nature Phys. 8, 544–549 (2012) [49].

121

10.1

Introductory discussion

The diffraction of electron and neutron matter waves from crystalline structures is a standard tool in crystallography, complementing x-ray techniques [106].
The advent of quantum gases in optical lattices has introduced a new class ofhighly controllable systems that mimic the physics of solids at lattice constants
that are three orders of magnitude larger [26], and it appears natural to ask
about a possible role for atomic matter-wave diffraction in the characterizationof these systems [80, 270].
Several approaches for detecting the spatial structure of strongly correlated phases of ultracold atoms in optical lattices have so far been taken.
These include the analysis of noise correlations in momentum space [190, 191],
accessible after release of the atoms from the lattice, as well as dedicatedin situ detection methods based on optical imaging [76–78] and electron microscopy [79]. In this context, diffraction experiments have the potential toreveal important information on the existence of long-range order, such as
spin-ordered phases in atomic mixtures [235, 271, 272], in a non-destructive
manner and with substantially lower experimental requirements. Here, opticaland atomic matter-wave diffraction are equivalent in the sense that scatteringof photons and atoms can be sensitive to both the external and internal state of
atomic scatterers. However, there are certain advantages to using matter-wave
probes. The de Broglie wavelength of an atomic probe can be tuned freely bycontrolling its velocity, thus precluding limits on spatial resolution and also
providing access to Bragg resonances without the need of varying the angle
of incidence. As in the optical case [192, 273], matter-wave probes of highspectral brightness are readily available by using atoms from a Bose–Einsteincondensate.
In this work we study the scattering of a probe species, consisting of onedimensional Bose gases, from target atoms confined to an optical lattice. For
weak confinement, we observe free-particle-like, one-dimensional (1D) collisions [195, 274] between the two species, corresponding to inelastic bandstructure excitations of the target by the incident probe field. The inelastic
scattering is suppressed for more deeply confined target atoms, giving way
to elastic scattering and Bragg diffraction, from which the underlying crystalline order can be inferred. We use this matter-wave scattering technique tocharacterize the introduction of forced antiferromagnetic ordering [52] in thesystem.

122

10.2

Matter-wave probing of an atomic crystal

Our experiments begin with a virtually pure Bose–Einstein condensate of 87 Rb
atoms in the |F, mF i = |1, −1i hyperfine ground state, prepared in an optical
dipole trap (ODT) of nearly isotropic harmonic trapping frequency 2π × 50 Hz
in the transverse (x and y) and vertical (z) directions. Along the transverse
axes, far-detuned attractive optical lattices (λ⊥ = 1064 nm) are smoothly
ramped up in 200 ms via partial retroreflection of the ODT’s laser beams [111],
resulting in an array of one-dimensional Bose gases, with trapping frequency
ωz /2π ≈ 70 Hz along z. The final depths of these transverse lattices, s⊥ = 40
(measured in units of the transverse recoil energy (h/λ⊥ )2 /2m, where m is theatomic mass), are sufficiently deep so as to suppress tunneling between the 1D
tubes on our experimental timescales. Moreover, given a transverse oscillation
frequency in each tube of ω⊥ /2π = 26 kHz (in the harmonic approximation),and a corresponding energy spacing to the first allowed transverse excitedmode of 2~ω⊥ ≈ 14 ER , all dynamics are energetically restricted to the lowest
mode. For all the studies that follow, in the absence of additional latticepotentials along z, the 1D bosons are not deeply within the Tonks–Girardeau
regime [199], and can be characterized by a Lieb–Liniger parameter valueγ . 1 [197, 202].
After forming the array of 1D tubes, we next create a mixture of probe(|pi ≡ |2, −2i) and target (|ti ≡ |1, 0i or |1, −1i, see below) species throughmanipulation of the atoms’ internal hyperfine state via microwave Rabi pulsesand Landau–Zener sweeps at a magnetic bias field of 1.7 G. By adiabatically
ramping up a magnetic field gradient, we then fully separate the probe andtarget atoms along the longitudinal (z) axis. We thereafter smoothly ramp up
a state-selective lattice along z, formed by light of wavelength λz = 785 nm,
between the D1 and D2 lines of 87 Rb. This allows ([44, 219, 248]; for moredetails see the end of the chapter) for a cancellation of light-shifts (and zerolattice depth) for the probe atoms, while the target atoms experience an attractive lattice along z, with period d = λz /2 and variable depth sz (measured
in units of the longitudinal recoil energy ER = (h/λz )2 /2m). This allows forthe target to be driven to a 1D Mott insulator state [185], while the probe
species remains a superfluid 1D Bose gas. Finally, as detailed below, we examine scattering processes between probe and target at a well-defined relative
velocity, vrel , relating to a probe de Broglie wavelength of ΛdB = h/mvrel .

123

10.3

1D collisions and inelastic scattering

We begin by studying collisions between the two species, as the target atoms
become localized to a state-selective lattice of increasing depth. In the limit ofzero lattice confinement along z, the collisions occurring between the |pi and|ti atoms are essentially free-particle collisions, with all scattering restricted
to the 1D tubes. Such binary collisions have previously been studied with asingle species as a quantum analog of a “Newton’s cradle” [274], and with a
two-species mixture in the context of damped spin impurity transport [275].
Here, we realize such collisions by accelerating the probe atoms into an initiallyseparated sample of target atoms, which is itself at rest.
As illustrated in Fig. 10.1 (a), we set the relative velocity of the collisions
by controlling the magnetic field gradient along z that initially separates thetwo species. The probe atoms experience a longitudinal trapping potential
shifted by a distance ∆z, while the target atoms (|F, mF i = |1, 0i, insensitive
to the applied gradient) remain at the trap center. The magnetic field gradientis quickly extinguished, and after a quarter oscillation in the trap the probe
atoms have accelerated to a nominal velocity of vp0 ≡ ωz ∆z upon interactingwith the stationary (vt0 = 0) target atoms. At this point, we access their
momentum distributions by turning off all confining potentials and separatingthem by applying a pulsed magnetic field gradient during time of flight (TOF),
followed by absorption imaging.
Typical TOF absorption images are displayed in Fig. 10.1 (b), for the caseof free (along z) target atoms and probe atoms incident at vrel = vp0 −vt0 = 2vR(z-lattice recoil velocity vR = h/2md ' 5.8 mm/s). Here, and also for weakz-lattice depths, two distinct velocity components at 0 and vrel can be seen
for both species. As the target atoms are initially at rest and the accelerated
probe atoms are initially at vrel before colliding with each other, the scattering
spectra show energy- and momentum exchanging binary collisions between
probe and target atoms, i.e. reflections in the center-of-mass frame of atomic
pairs. Whereas reflection and transmission events are indistinguishable forcolliding atoms of the same spin [274], a spin-mixture gives experimental accessto the reflection probability [195]. The atoms collide with a high kinetic energythat far exceeds the mean-field energy of either species, such that the collisions
have free-particle character. Using a slightly uneven mixture of the two species(3:2 target to probe atoms), we find that 11 ± 1% of the target atoms and
14 ± 3% of the probe atoms are reflected, in fair agreement with the calculated
reflection probability of R = 10% [195] for our system parameters. Here, wehave taken
R ≈ 1 − |1 − 1/(1 + ikzCOM a1D )|2 ,(10.1)

124

∆z

(a)

mωz2z22pt

vp0= ωz∆ z(b) sz=0
t

p

y

z

vp,t/ = -2vR(c) s =8
δE

4

10

3
2
1
0
-1

0

1

q [�/d]2

3

0.5
scaled 1D
velocity profile

z

t
t
t
(N+2
- N-2
)/NTOT

Energy [ER]

5

(d) 0

5

0

0

5

0

δE/ER

1

2

1.5

t

1 sz=4
t
N-2

0

-2

sz

t

N+2

0

vt / vR

2

10

Figure 10.1: Interspecies collisions of one-dimensional bosons. (a) To study
collisions between two bosonic species in one-dimensional systems, probe (|pi)
atoms of one species are displaced (by distance ∆z) and then accelerated
within a trap potential (trap frequency ωz ) to a final velocity vp0 = ωz ∆z.
The |pi atoms then interact with the stationary target (|ti) atoms at the trap
center. (b) Absorption images showing Stern–Gerlach separated TOF spectra
for the target (top) and probe (bottom), with velocity components at vp,t = 0
and 2vR , due to momentum-exchanging “Newton’s cradle” (NC) collisions.
(c) Illustration of momentum and energy exchange in the band structure of
the optical lattice, shown for a lattice depth sz = 8 (in units of the recoil energy
ER ). (d) Percentage of target atoms that participate in NC-type collisions,as a function of the energy mismatch δE. The solid curve is an exponential
fit with a decay constant of 0.35 ER . The percentage is determined from thepeak asymmetry in the TOF velocity distribution (after summation over the y
direction) of the target species, as shown for vp0 = 2vR in the inset, with atomttnumbers N±2(peaks) and NTOT
(total).
where the center-of-mass wavevector of the colliding particles is given by
kzCOM = (vrel /2vR )π/d = π/d, and the effective 1D scattering scattering length
is
ai
a2 h
,
(10.2)
a1D = − ⊥ 1 − C
2aa⊥with constant C ≈ 1.4603 and harmonic oscillator length, sometimes also

125

−1/4

written as σ⊥ , a⊥ = λ⊥ s⊥ /2π ≈ 67 nm in the harmonic approximation fors⊥ = 40. As an aside, we mention that such probabilistic free-particle collisions
between 1D bosons may find future use in generating atomic distributions
with nonclassical correlations [276], much as in atomic four-wave mixing [219].
Indeed, population transferred to the unseeded modes via pairwise momentumexchange should display properties akin to two-mode squeezed vacuum [277],
and moreover these collisions can create atomic beams with non-separable
spin-correlations [278].
As mentioned above, much effort has been devoted to studying collisionsof lattice-free 1D Bose gases, both for the spin-polarized [274] and spin-mixed
cases [275]. In our system, we now investigate what happens when the dispersion relations are qualitatively different, with one of the species subject toa longitudinal optical lattice. To discuss in simple terms our expectations forthe case that the target atoms reside in a lattice of finite depth, we describe
the momentum and energy exchange in the periodic zone scheme associatedwith the lattice. For zero lattice depth, the dispersion relations of target andprobe atoms coincide, such that the momentum exchange is resonant. However, as illustrated in Fig. 10.1 (c), for non-zero lattice depth sz 6= 0, eachcollision demotes an incident probe atom from the first band to the groundband, and vice versa for the target atom involved in the collision. With anenergy mismatch δE that increases with sz , the momentum-exchange becomes
off-resonant, amounting to an inelastic band-structure excitation of the target. This process can occur as long as the energetic uncertainty ∆E ∼ h/τ
associated with the finite interaction time τ is larger than δE.
The observed changes of the target’s TOF spectra with increasing lattice
depth are analyzed in Fig. 10.1 (d). In the absence of collisions, sz -dependent,
symmetric peaks due to optical diffraction are observed at vt = ±2vR as expected (for up to sz ∼ 20). Collisions at the incident velocity vp0 = 2vR giverise to an asymmetry between the two peaks, depending on the depth of thelattice. The observed asymmetry decays with the mismatch δE, in qualitative agreement with our expectation. We estimate the time for probe atoms
to traverse half of the target (at which point we switch off the lattice) to beτ ∼ 10µm/2vR ' 0.9ms, giving an associated energy uncertainty ∆E ∼ 0.3ER .
This value is in good qualitative agreement with the observed 1/e exponential decay constant of 0.35ER . Our observation of the suppression of elastic
two-body collisions due to band-structure mismatch should be of direct consequence for recently proposed thermometry schemes in strongly correlated
systems that are based upon the use of a lattice-free spectator species in contact with lattice-modulated atoms [238].

126

10.4

Elastic Bragg diffraction of matter waves

While inelastic scattering events are seen to die off with increasing lattice confinement, we instead can expect to observe elastic scattering of probe matterwaves from the crystal of target atoms. The distribution of elastically scattered probe atoms is determined by the static structure factor S(q) of thetarget (where ~q is the probe momentum transfer). As S(q) is given by thesquare of the Fourier transform of the target’s density, probe scattering canthus reveal information about the underlying spatial distribution of the target
atoms. As a caveat, we briefly mention that the study of the dynamic structure factor through low-energy inelastic scattering, which could provide insight
into correlations of the target atom array [80], is largely precluded in our system due to dominant reflection at low energies (described below), as well as
our use of a superfluid probe gas, which itself supports low-energy collective
excitations. Such a study may thus be more well-suited to energy-deposition
measurements based on optical Bragg spectroscopy [216, 217].
The static structure
for a deeply modulated target, can be approxP factor,
iqzj 2
imated as S(q) = | j fj (q)e | , where the fj (q) are the amplitudes of isolated scatterers at positions zj = jd, with j an integer. If one assumes them tobe identical (fj (q) ≡ f (q)), as in a Mott phase with uniform filling, the strucP
2ture factor is a product of two terms, | j eiqzj |2 and |f (q)|2 = |hϕ0 |eiqz | ϕ0 i| ,where ϕ0 (z) describes the on-site density distribution of each scatterer. The
first term determines the positions of Bragg diffraction resonances, and reveals the periodicity of the array of scatterers. The second term, the atomic
form factor |f (q)|2 (or equivalently the single-scatterer or Fraunhofer envelope
function) reveals information of the on-site density distribution.
In order to perform crystallographic measurements, we begin by introducing a controlled method to vary the relative velocity (de Broglie wavelength) ofmatter waves incident on a deeply-confined crystal. Such a wavelength scan isnecessary to identify Bragg diffraction resonances, which are expected when amultiple of the probe’s de Broglie wavelength coincides with the target’s latticespacing (i.e. 2d = nΛdB , or vrel = nvR , with n the order of diffraction), andmore generally it allows study of the dependence of scattering on the probe’sde Broglie wavelength. As illustrated in Fig. 10.2 (a), the target atoms (herein the |1, −1i state) are loaded into a very deep state-selective lattice initiallyat rest in the lab frame. This lattice is then made to move at a well-definedvelocity vL = δω(λz /4π) by detuning the relative frequency of the two interfering laser beams that create the lattice by an amount δω. The pinned
target atoms, which are initially far-separated from the probe atoms, follow
the moving lattice minima at a variable velocity vrel for a total time Tmove , in-

127

(a)

ω1

p

δω = ω1 - ω2

ω2vL = δω × ( λz / 4π )

t

(b)

sz=50

p

t y

vp/ = -4vR

vt/ = -4
vR

-2

0

z

-2

0

Figure 10.2: Diffraction from a moving atomic crystal. (a) Target (|ti) atoms,
strongly confined to a state-selective lattice, move towards a stationary cloudof probe (|pi) atoms at a velocity vL = δω × (λ/4π), by frequency detuning(by δω) the two lattice beams. (b) Absorption images show the |ti and |pi
species after interaction and expansion in TOF.
cluding a set delay time before the probe and target make contact. The large
initial separation helps to avoid the formation of a quantum emulsion [47],resulting when homogeneous mixtures are loaded into state-dependent latticepotentials, which can affect interspecies transport properties [44, 46].
In Fig. 10.2 (b), we show TOF absorption images of both species followinginteraction, taken for the case of deep confinement (sz = 50) and a velocity
vrel = 2vR . The target’s velocity distribution is very broad, as expected for a
deep 1D Mott state, and is centered around the velocity of the moving lattice(−2vR ). The velocity distribution of the probe, initially centered around vp =0, displays a peak of atoms transferred to −4vR . A series of such scattering
spectra (integrated along the transverse y axis) is shown in Fig. 10.3 (a),for a range of relative velocities vrel . For each velocity, the time Tmove ischosen so that the target crystal enters the probe and then moves for 1.4 ms(kept much less than the trap period Tz = 2π/ωz ∼ 14 ms to avoid thedispersion of out-coupled atoms to different velocities due to propagation in
the trapping potential). From the scattered probe spectra as in Fig. 10.2 (b)we observe that, in addition to the line of “transmitted” atoms at vp = 0,
some probe atoms are out-coupled (i.e. reflected) due to the target crystal.
The reflection of probe atoms comes from two apparent elastic mechanisms specular reflection and resonant Bragg diffraction. We note that both of thesemechanisms are to be distinguished from observations of free-space four-wave
mixing in two-component mixtures [219, 238], which does not persist when
one species loses matter-wave coherence.
The specular contribution is due to the reflection of probe atoms from thepotential “step” of the target crystal (energy mismatch outside and inside thecrystal). This reflection is dominant at low velocities and gives way to transmission at larger probe velocities (kinetic energies), where the band structure

128

(a)

1

0

0

1

3

2

vrel / vR

2

1

Ndiff [a.u.]

vrel / vR

0.5

Ndiff [a.u.]

1

1

(b)

3
0

2

- vp / vR

4

6

0

(c)

2

3

4

Tmove [ms]5

Figure 10.3: Probe scattering from a moving crystalline target. (a) Probe TOF
spectra after interaction with a target at sz = 50, at varied relative velocity
vrel . Each horizontal line is obtained from a yz spectrum as in Fig. 10.2 (b) byintegrating along y and normalizing to the total |pi atom number. The solid
white lines near vrel /vR ∼ 2 illustrate the slope of the clearly visible secondorder Bragg resonance, i.e. extrapolations of a linear fit to the positions ofmaximum out-coupling for 1.8 . vrel /vR . 2.2. (b) Dependence of reflection
from the crystal on vrel . We count the number of atoms scattered to vp ∼−2vrel , i.e. within dashed lines of Fig. 10.3 (a). Bragg resonances of the |ti
crystal are observed for vrel /vR = 1, 2, 3, on top of a background contribution
due to specular reflection, which decays with increasing vrel . (c) Reflectedprobe population as a function of the time during which the crystal is moved(Tmove ), for fixed velocities vrel /vR = 1.5 (black data) and vrel /vR = 2 (reddata). The data fall off at long times due to the evolution in the trapping
potential.
of the target “atomic crystal” is free-particle-like. The specular contribution
leads to the transfer of probe atoms to vp = −2vrel and shows no resonant
structure. In contrast, Bragg reflection from scattering centers of the target
crystal occurs at values vrel /vR = n and results in the transfer of probe atoms
to velocities vp = −vrel − nvR . Such a “locking” to a Bragg resonance isobservable through a change in the slope of the out-coupled branch of probe
atoms. Results of the velocity scan, which displays these features, are shownin Fig. 10.3 (a).
In Fig. 10.3 (b) we count the number of probe atoms out-coupled to vp ≈2vrel , and plot it as a function of vrel /vR . Three resonant peaks in the numberof reflected atoms are observed for vrel /vR = 1, 2, 3, corresponding to first-,
129

second-, and third-order Bragg processes. In addition to these resonances, asignificant non-resonant contribution due to specular reflection is observed,
which as expected decays with increasing velocity vrel . We fit the secondorder resonance peak with a Gaussian
on top of a linear slope [red line in√
Fig. 10.3 (b)], and extract a 1/ e-width of 2σv = 0.3vR , reflecting the intrap velocity width of the probe species. The width is a factor of two smaller
than the directly observed TOF velocity width of the probe atoms [279], likely
owing to effects of interaction-induced expansion during TOF [131].
Using our ability to vary the total time Tmove that the target crystal
moves at constant velocity, we can also study the temporal buildup of theout-coupled probe population. In contrast to optical scattering, this may
in the future be used to provide “line-of-sight” information in matter-wave
scattering experiments. In Fig. 10.3 (c), we plot the temporal buildup ofprobe population to vp ∼ −2vrel , for the cases vrel = 1.5vR and 2vR . The
difference between incoherent (specular reflection) and coherent (Bragg) processes should lead to differences in the growth behavior. Indeed, there aresome indications of a more nonlinear initial growth for vrel = 2vR than forvrel = 1.5vR case. However, since the relatively short coherence length of theprobe [L ∼ ~/(mσv ) = 0.8 µm] precludes a fully coherent temporal evolution,
the observed growth will be mostly determined by the time-dependent overlapof the inhomogeneous profiles of the probe and target.
Finally, we recall the dependence of the static structure factor S(q) on theatomic form factor |f (q)|2 , in relation to the three Bragg resonances observedin Fig. 10.3 (b). In the harmonic approximation, the form factor is proportional to exp(−q 2 σz2 /2), where σz is the harmonic-oscillator length characterizing the extent of the atomic distribution on each site. While the third-orderpeak (q = 3) is smaller than the second-order peak (q = 2), the expected
momentum dependence is partially masked, owing to a fixed probe-target interaction time that results in the probe atoms of lower velocity (such as for the
first-order peak) interacting with fewer scattering centers of the target crystal. However, we can directly probe the contribution from the atomic form
factor, which can be seen to be formally identical to the Debye–Waller factor
W = exp(−q 2 hu2 i/2) describing the reduction of elastic scattering due to position fluctuations hu2 i of scatterers in an ionic crystal [106]. The larger σz ,the smaller |f (q)|2 , in analogy to the decrease of the Debye–Waller factor withtemperature. We point out, however, that in our case these fluctuations are
not thermal but arise from zero-point motion, and may be tuned through σz
by varying sz .
To extract the sz dependent amount of elastic Bragg diffraction, we recall
the experimental situation of Fig. 10.1, where the earlier described method

130

(a) 3
2
1

vp
vR 0

1

-1
-2
-3

0.5

(b) 3
2

vt
vR

1
0

0

-1
-2
-3

0

5

10

15

sz

20

25

30

Figure 10.4: Scattering profiles as a function of target confinement. (a,b) Integrated TOF spectra of the |pi and |ti species, following collisions of incident
probe atoms (at velocity vrel /vR = 2) from a stationary target of atoms confined to a lattice of varied depth sz . With increasing depth, we observe thedecay of pairwise momentum-exchanging collisions as studied in Fig. 10.1,
which lead to the initial asymmetry in target population of the ±2vR diffraction peaks, as well as stationary probe atoms at vp = 0. Accompanying thisdecay of energy-changing collisions is the appearance of an elastic Bragg reflection peak of probe atoms at vp /vR = −2.
of probe acceleration in the longitudinal trapping potential (with |1, 0i target
atoms) was employed. For a complete picture of probe-target scattering eventsin the crossover from localized-to-delocalized target atoms, we plot in Fig. 10.4
the integrated 1D scattering profiles of both species as a function of sz . Atvery low depths, profiles consistent with Fig. 10.1 (b) are observed, owing
to (nearly-)free-particle momentum- and energy-exchanging collisions between|pi and |ti atoms. As these collisional events cease to occur, as seen in boththe peak of probe population at vp = 0 and the asymmetry of the vt /vR = ±2
target peaks, a probe feature at vp = −vrel = −2vR becomes visible. This peakcan be attributed to elastic second-order Bragg diffraction, and in Fig. 10.5 (a)we plot the percentage of diffracted atoms as a function of sz . The percentage
of second-order Bragg-reflected probe atoms Ndiff is determined from a fit of131the TOF spectrum with Gaussian peaks for (i) the transmitted probe aroundvp ≈ vp0 ; (ii) a broad incoherent background centered at vp = 0 ; (iii) Braggreflected atoms at vp = −2vR . Ndiff is determined as the amount of Braggreflected atoms normalized with respect to the total probe population.
The data in Fig. 10.5 (a) reveal an increase of diffraction efficiency as thetarget atoms become more localized, and we find the data to be in good qualitative agreement with the calculated dependence of |f (q)|2 on sz (using Bloch
wavefunctions, taken over a single site, for ϕ0 ). We show cartoon depictionsin Fig. 10.5 (b,c) of the atomic wavefunctions and the Fraunhofer diffraction
envelope. While there is good general agreement, the observed signal saturates
at large sz while the expected curve does not, and a better agreement withthe data can be obtained by taking into account sz -dependent heating due
to Rayleigh scattering from the laser beams forming the longitudinal lattice.
We attribute this mostly to heating of the probe species, leading to reduced
coherence and an increased spectral width, and thus less population at thevelocity class that matches the Bragg resonance condition. Indeed, if for a
fixed lattice depth of sz = 25.5 we hold the species for a variable time prior
to probe acceleration to vp0 = 2vR , we observe such heating through a linear
increase in the probe’s TOF velocity-width, as seen in Fig. 10.5 (d). The ensuing Bragg diffraction signal exhibits a roughly exponential decay with holdtime (1/e–time theat ∼ 150 − 200 ms), slightly faster than what we should
expect due to probe heating alone - i.e. expecting a 1/σv (t) decay, with timedependent spectral width σv (t). Assuming that all contributions from heating
due to Rayleigh scattering (of either |pi or |ti atoms) should scale as the time
integral over the lattice depth, we expect a correction factor of exp(−0.01sz )to modify the form factor. A fit to the data with |f (q)|2 × exp(βsz ) yields
β = −0.015 [value used for the dashed curve in Fig. 10.5 (a)], consistent withthe expected role of such heating.

10.5

Bragg diffraction spectra using accelerated probes

We now make a qualitative comparison between the spectrally well-resolved
method of moving a crystal through a stationary probe, and the method ofaccelerating a displaced probe cloud into a stationary target crystal, at a
relatively well-defined final velocity vp throughout the time of probe-target
interaction. In Fig. 10.6 (a), we show a relative velocity scan similar to that ofFig. 10.3 (a), however here the velocity is controlled by the initial displacement,∆z, of the probe species in the external trapping potential. We have here used132

Ndiff [%]

6 (a)
4
2

|φ0| [arb. u.]

0
1

0

(b)

0
-1

1

2 σz

0

1

z/d

0

-5

0

qpd/� + 2

5(e)σv / vR

Ndiff [%]

30

(c)

1.0

(d)

5

0

sz 20

10

0.5

0

.2

.4

THold [s]

0.0

.6

0

.2

.4

THold [s].6

Figure 10.5: Measuring the on-site density profiles of atomic scattering centers.
(a) Percentage of (second-order) Bragg-diffracted probe atoms as a function ofsz . The dashed curve is proportional to the form factor |f (2 × 2π/d)|2 calculated as a function of sz ; the solid curve accounts for additional heating effects
as described in the text. The black square is a reference measurement without|ti atoms. (b) Plot of the modulated wavefunction of an array of target atoms
in a lattice, with the extent on each site characterized by a harmonic-oscillator
length σz . (c) Cartoon depiction of the Bragg reflection efficiency, as governedby the atomic form factor |f (q)|2 (i.e. the Fraunhofer envelope of individual target scatterers, or quantum-limited Debye–Waller factor W ). With ~qp
referring to the momentum change of probe atoms after scattering from thetarget crystal, the Fraunhofer envelope of allowed probe momentum change is
2 2
approximated by e−qp σz /2 , determining the percentage of probe atoms that arereflected. (d) Dependence of the percentage of probe atoms, Ndif f undergoing second-order Bragg diffraction, on the hold time prior to acceleration andtarget interrogation, THold . The red line is a fit exponential decay, with a 1/etime constant of 185 ms. (e) Spectral width σv of the probe component, basedon a fit to the transmitted peak, as a function of the THold .

133

(a)

-3

0
1

0.5

-2

vp / vR

-1

0

1

2

3

sz=25.50(b) vp / vR
-2

0

2

vp1vR

vrel / vR

1vp2vR

2vp3vR

3

Figure 10.6: Scattering of an accelerated probe from a stationary crystal.
(a) Probe TOF spectra from a crystal of atoms localized to a lattice of depthsz = 25.5. The inset shows the second-order Bragg resonance (within thedashed line) with color enhanced. (b) Plots of integrated probe spectra, atvrel /vR ' 1, 2, 3. The arrow indicates the expected position of the first-order
Bragg peak; Gaussian fits to the second- and third-order Bragg peaks arehighlighted in red.
a crystal of target atoms that are slowly loaded into a state-selective opticallattice of depth sz = 25.5. Broadly speaking, the two methods produce quite
similar results, with similar observations of first-, second-, and third-order
Bragg reflection peaks as shown in Fig. 10.6 (b). We note that the second-order
Bragg reflection peak in Fig. 10.6 (a) is fixed at vp /vR = −2, as the Bragg
condition is defined in the reference frame of the stationary target crystal.
This difference in slope from the specular contribution is exactly related tothe observed Bragg peak’s slope in Fig. 10.3 (a), although in the case of amoving crystal the reference frame in fact changes with vrel .
However, the method of using a moving target crystal, at an extremely welldefined velocity throughout the time of interaction, allows for a less ambiguous
observation many of the features, especially at low impact velocities. This isowing to the methodology for achieving the relative probe-target velocity. At
low vrel , the method of probe acceleration begins with small displacements ∆z.
In fact, the separation between the center-of-mass positions of the two species
can in general be less than their combined extents, i.e. the two species should
overlap at the outer edges of their distributions up to a value of vrel /vR ∼ 1.2,
in the central-most 1D “tubes”. As the two clouds are strongly-interacting, weshould expect in general a significant deviation from the expected distributions
based on the single-component scenario, with the likelihood of full phaseseparation at low temperatures, even for very weak applied gradients [280].

134

Even without such deviations from the non-interacting case, the fact that theprobe species is not at a fixed velocity, but is rather accelerating towards thetrap minimum, will ensure that during the time of probe-target interaction a
range of vrel will be experienced. This issue is most severe at small vrel , as boththe absolute and relative (as a fraction of vrel ) range of velocities occurring
throughout probe-target overlap decrease as a function of vrel . We expect thatthese such effects, as well as possible interspecies instabilities [281] and effects
of dispersion in the trap, may contribute to the obscuration and ambiguity of
the first-order Bragg reflection peak in Fig. 10.6 (b). By dispersion effects, wemean the fact that specularly reflected atoms will experience a loss of energy
and decrease in velocity they travel back up the dipole trap “hill”, in the shorttime between reflection and release for TOF imaging. As discussed briefly
earlier, the fact that the probe species is weakly-trapped not only leads toan increased spectral width, but limits the useful timescale of probe-target
interaction [cf. Fig. 10.3 (c)].
Finally, there is one last effect that can occur when the two species areoverlapped prior to loading of the state-selective lattice. This is the possibility that a quantum emulsion, as studied in the previous chapter and in
Refs. [44, 46, 47, 70]), can be formed during the loading process. In the following section, we study low-energy scattering between probe and target atoms,
with an emphasis on the transport of 1D probe gases through a crystal oftarget atoms, or in the reference frame of the gas, the dragging of impurities
through 1D Bose gases.

10.6

Low energy scattering of free and trapped
1D bosons

We now turn our attention directly to the case of small values of vrel /vR (for accelerated probe atoms), depicted again in the top row of Fig. 10.7 (a). Naively,
apart from the transmitted probe, one would expect to simply see a first-order
Bragg peak around vrel /vR = 1. As mentioned earlier, a broad distribution ofoutgoing probe velocities with |vp | ≤ vrel is seen instead, obscuring the peak(see also Fig. 10.6 (b), top row). In this context, it is important to recall
that vrel = ωz ∆z is controlled by varying the initial probe displacement ∆z.
For small displacements with vrel /vR ≤ 1.2, the target and probe never fully
separate, encompassing most of the range of the expected Bragg resonance. It
seems likely that the overlap contributes to the blurring of the distribution.
In the following, we analyze these effects more closely.
In the range vrel /vR . 0.5, most of the |pi atoms remain near zero velocity,

135

0.8

sz=25.5

1
0

sz=6

1
0

(b)

0.6

α

vrel / vR(a)target

probe

0

sz=3

0.4

1
0

sz=0

1

0.2
-2

-1

0

1

vp / vR

2 -2

-1

0

1

vt / vR

2

0

2

sz

4

6

Figure 10.7: Low velocity impurity transport. (a) Probe |pi and target |ti velocity distributions for four values of sz ; for comparison the data of Fig. 10.6 (a)is partially reproduced. (b) Inhibition of probe transport due to interactions
with the target. For low velocities (vrel < 0.5) we fit a straight line [solid blacklines in (a)] with slope α to the probe distribution’s peak locations (the peak
for a given vrel is defined as the region around the maximum that comprises
all points down to 50%; its location is then determined from a Gaussian fit).
The error bars indicate the linear regression standard error of the fit.
despite the force exerted on them by the harmonic potential after switching off
the gradient. In order to study this effect further, we take displacement scansat different target lattice depths sz as shown in Fig. 10.7 (a). For vrel /vR . 0.5,
we observe that the peak of the |pi distribution is at velocities that linearly
scale as vpmax /vrel = α < 1. In the range considered (sz ≤ 6), the coefficient α
decreases smoothly with sz , as seen in Fig. 10.7 (b) (a reference run without|ti atoms at sz = 25.5 yields α = 1.0 ± 0.1 as expected). Without attempting
to quantitatively model this behavior, we note that in this range the depthof the effective atomic lattice increases nearly linearly with sz , surpassing theprobe’s chemical potential µ ' 0.4 ER at sz = 3. By atomic lattice we refer tothe modulated potential landscape acting on the probe atoms due to repulsive
interactions with the localized target atoms, with modulation depth given by
Vtp = hnigtp |ϕ⊥ (0)|4 (|ϕz (0)|2 − |ϕz (d/2)|2 ) [238]. Here, gtp = 4π~2 atp /m is the
3D probe-target coupling strength with probe-target s-wave scattering length
atp , and ϕz (z), ϕ⊥ (x, y) are the target’s Bloch wave functions (lowest bandat q = 0), and hni the mean site occupancy. This modulation is appreciablealready at sz = 3, Vtp /hni ' 0.5 ER , growing only modestly to Vtp /hni '136

1.4 ER at sz = 25.5. For our experimental parameters, we expect hni = 1.7
for the site occupancy in the center of the target.
We expect that the observed inhibition of transport for increasing sz islikely the result of the increasing atomic lattice, with the possibility of added
localization due to the formation of a quantum emulsion in the overlap region [44, 46, 47, 70]. Similar observations of greatly inhibited transport inone-dimensional Bose gases have also been made in purely corrugated opticalpotentials [31, 214]. At sz = 0, for which the target is a superfluid 1D Bose gas
just as the probe, the maximum of the probe distribution suffers appreciable
retardation (α ' 0.75), with momentum transfer to the |ti atoms (as seen in acorresponding shift of the center of mass). The slowdown of |pi atoms entering
the target volume can be viewed as resulting from an increase in mean-fieldenergy (i.e. with the target atoms acting as a repulsive potential “hill”), which
can be expressed as a change of the refractive index for matter waves similar
to the case of photons [133]. We note that this mean-field “slow-down” effectcan also be observed (to a small extent) in the position of the transmitted
probe peak at vp /vR ∼ 2 in Fig. 10.4 (a). As the lattice depth sz is increased,
the induced “slow-down” in Fig. 10.4 (a) is seen to disappear.
Over the larger range of velocities considered in Fig. 10.7 (a), we observea broad distribution of |pi atoms that is centered at negative velocities andbounded by retroreflection vp /vrel = −1. The observed broadening of the |pi
spectra is consistent with the expected
presence of dissipation for velocitiespnear the speed of sound c̃s = ( µ/m)/vR ' 0.4 [282]. The presence ofbackscattered atoms (i.e. atoms with vp /vR < 0) does not depend muchon the degree to which the |ti atoms are localized, but persists to low sz .
Remarkably, the velocity spectrum of the target does not contain atoms at thecorresponding momentum change (which would violate energy conservation).
This means that the momentum of the backscattered |pi atoms is collectivelytransferred to the optical lattice rather than to single |ti atoms, similar to theMössbauer effect. An intuitive explanation for this surprising effect is thatthe backscattering originates from the target’s density modulation, which is
not free to move around but is spatially locked to the optical lattice potential.
Indeed, when no lattice is present (sz = 0), the probe spectrum does notcontain atoms at negative velocities. In this case, it is bounded at vrel while
the target spectrum contains atoms with velocities up to vt = vrel , indicating
the presence of energy- and momentum exchanging binary collisions between|pi and |ti atoms, as studied earlier.

137

(b) 50-5impurity
concentration

10 µm∆z [µm]

(a)

4
0
-4
2
0
-2

z
y’

0

10

20

30

“Slosh” time [ms]Figure 10.8: Low velocity impurity transport as observed in-situ. (a) Typicalin-situ absorption image of an array of 1D Bose gases, oriented along z. (b) By
taking images at different times after an abrupt displacement of the trapcenter, dipole oscillations or “sloshing” of the center-of-mass displacement ∆z
can be observed for a gas of |pi atoms. With the addition of impurity |ti
atoms, which are confined to a very deep state-selective lattice, we find astrong damping of the |pi species’ dipole oscillations.

10.7

In-situ investigation of low-velocity impurity transport

In addition to investigating the influence of localized impurities on transport
in momentum-space, we can also directly monitor the temporal evolution of
the probe in-situ density profiles as the 1D Bose gases come in contact with an
impurity-field. For this, we utilize the in-situ imaging beam path discussed in
Chapter 3, having higher magnification than the TOF imaging path and witha resolution ∼√1 µm. A typical in-situ absorption image taken along the axis
x0 = (x + y)/ 2 is shown in Fig. 10.8 (a).
We again use a mixture of hyperfine states, here |ti = |1, −1i and |pi =|2, −2i, which are confined to an array of 1D tubes (s⊥ = 40), with on average∼ 80 total atoms per tube. By controlling the fractional amount of the two
species, we can control the concentration of “impurity ” |ti atoms within eachtube. The impurity atoms experience a deep, state-selective lattice potential
along z, while the |pi experience only the longitudinal trapping potential, withharmonic frequency ωz /2π ≈ 75−80 Hz. To perform transport measurements,
we begin by applying a relatively strong magnetic field gradient to the atomic
mixture, displacing each of the species from the initial equilibrium trap position along z in opposite directions. Next, we first quickly (in 200 µs) ramp
up a state-selective lattice to a depth of sz ∼ 40, and then quickly turn off
the magnetic field gradient. When there are no impurity atoms present, we

138

10

(b)

55

z [µm]
 m
∆z

00

COM

COM

z [µm]
 m
∆z

(a)

-55
10

10

55
00
-55

00

5

10
10

15

20
20

time ms

25

30
30

35

“Slosh” time [ms]

10

00

5

10
10

15

20
20

time ms

25

30
30

35

“Slosh” time [ms]Figure 10.9: Direct in-situ evidence for specular reflection. (a) In the complete
absence of impurities, undamped dipole oscillations of the center-of-mass position of the |pi species are observed following a trap displacement. After a holdtime of about 6 − 7 ms, the atoms have reached the outer turning point of theoscillation. (b) With added impurity atoms we observe a strong damping ofthese oscillations. We also observe evidence of specular reflection, in that thecenter-of-mass position reverses course at the impurity-field interface (meanextent of |ti atom distributions shown in blue) instead of reaching the outerturning point, as highlighted by the arrow.
observe that the |pi atomic distribution undergoes center-of-mass dipole oscillations following the gradient turn-off, as can be seen in the upper panel ofFig. 10.8 (b). Separately, we confirm that the |ti distribution remains fixed asa function of hold time, due to the deep lattice confinement.
Finally, we can investigate how the presence of an inhomogeneous impurityfield influences the dipole oscillation of the |pi atoms. For this, we image only
atoms that are in the F = 2 hyperfine state. As shown in Fig. 10.8 (b), for
an increasing amount of impurity atoms (for a few percent and for roughly 25
percent), a strong damping of the center-of-mass oscillations is observed. Dueto the large extent of the atomic distributions as compared to the interspecies
separation, as well as the optical interrogation through a number of 1D systems
along x0 , the mechanism for the observed damping is not completely transparent. We expect that a combination of effects can contribute to the observed
damping, including excitation of the 1D gases and possible localization within
the impurity field [282], as well as an apparent damping of the COM position
due to specular reflection at the interface of the impurity-field distribution. If
the incoming atomic wave-packets split into transmitted and reflected components, the center-of-mass position may die off even if no energy is dissipated.
We note that we only consider contributions of specular reflection, and notcoherent Bragg diffraction, as the probe’s maximum center-of-mass velocity is
only ∼ 3mm/s (∼ 0.5 vR ).
Indeed, we observe some evidence for specular reflection in the dynamical
evolution of the in-situ probe density profiles. In Fig. 10.9 (a), we plot the139

center-of-mass displacement from the trap center of a cloud of |pi atoms, inthe absence of impurities. In Fig. 10.9 (b), we plot again, as the dashedline, the expected path of the COM position if there were no impurities. We
also illustrate the position and extent of the impurity-atom distribution, inlight blue. The center-of-mass trajectory of the |pi atoms – in addition to
exhibiting a strong damping and slow “creep” into the impurity field at longtimes – actually appears to double-back on itself upon reaching the outer edgeof the impurity field, corresponding to specular reflection at the boundary.
These in-situ observations complement the earlier observations of a decaying(with vrel ) specular component to the elastic diffraction signal in Fig. 10.3 (b).
In the future, the engineering of inhomogeneous atomic (or optical [283] forthat matter, e.g. using projected 2D light patterns [77]) potential landscapes
may be used for creating bandgap materials and tunnel junctions [284, 285] orresonators for propagating matter-waves.

10.8

Detecting forced checkerboard ordering

We now return to the spectrally well-resolved case of a moving target crystal, and we seek to further demonstrate that matter-wave diffraction may be
used to probe non-trivial structure. Here, we show that it gives access to aclear signature of forced antiferromagnetic, or checkerboard, ordering [52] ina mixed-spin crystal. As before, we start with a mixture of atoms in onedimensional tubes, with a probe that sees no longitudinal lattice along z (|pi,here |1, 1i atoms, with λz ∼ 788 nm) and “crystal” atoms that are loaded into
and then follow an optical lattice potential. However, instead of using only one
additional species that experiences an attractive lattice along z, our target now
consists of two species. The first one |ri ≡ |1, −1i again experiences an attractive lattice potential, whereas for the second species |bi ≡ |2, −2i the potential
is repulsive. Thus, the |ri atoms will be drawn to the intensity maxima, whilethe |bi atoms will be forced to the minima, as illustrated in Fig. 10.10 (a).
While we keep the total population of these two species fixed, we vary theirrelative population, which can be quantified by the net “polarization” of thecrystal P = ∆N/N , with ∆N = Nr − Nb and N = Nr + Nb .
When the target is “spin-polarized” consisting of either all |ri or all |bi
atoms, the situation is as before, with a lattice constant of d = λz /2 ∼ 400 nm.
However, when both species are present, the new dominant length scale, between atoms of the two species sitting on distinct sets of sites, will be given by
d0 = λz /4 ∼ 200 nm. This may be viewed as a crystal with the original periodicity and a two-atom basis, or as a new crystal structure having half the lattice
period, given that the interactions of the probe with the |ri and |bi atoms are

140

(a)

0
-2
-4

784

(c)
1

(b)

P ≈ -.9786

788

λz [nm]790

vrel = vR792

d = λz /2d’ = λz /4P≈0

P ≈ .85

3

N+2 [a.u.]

VLatt [arb. u.]

2

(d)

2

0

0 1 2 30 1 2 3

vp / vR

0 1 2 3

-1

0

P = ∆N / N

1

Figure 10.10: Detecting forced antiferromagnetic ordering via matter-wave
scattering. (a) Calculated dependence of state-dependent optical lattice depth(in arbitrary units of energy, for σ + polarization), of the hyperfine states |1, −1i(|ri, red line), |2, −2i (|bi, blue line), and |1, 1i (probe atoms, green line). Using a lattice of wavelength ∼ 788 nm, and for slightly elliptical light, the |1, 1i
atoms experience a vanishing potential. (b) A stationary cloud of |1, 1i atoms
is used as a matter-wave probe for a moving crystal of |ri and |bi atoms, attracted to and repelled from intensity maxima of a state-dependent lattice,respectively. (c) Velocity distributions of scattered probe atoms following interaction with a crystal of atoms in a tunable spin-mixture of the |ri and |bistates, moving at a velocity vrel = −vR . Shown are the cases of spin-polarized
crystals of nearly all |ri (P ∼ −1) and |bi (P ∼ 1) atoms, as well as for an evenmixture of the two states (P ∼ 0). (d) The number of reflected probe atoms[shaded regions in (c)], as a function of the spin imbalance of the target crystal, P = ∆N/N . A distinct minimum is observed for P = 0, relating to thecase of a spin-mixed crystal with maximal forced antiferromagnetic ordering
at spacing d0 = λz /4.
approximately the same. This change results in a different diffraction spectrum for the matter wave probes. With the addition of a second species, as
scattering centers separated by d give way to those of a smaller spacing d0 , theoriginal first-order diffraction peak at vrel /vR = −1 will become diminished,and entirely disappear if an equal mixture of the two species uniformly fills the141lattice, with a new first-order peak now occurring at half the momentum-spacefrequency, vrel /vR = −2.
To probe the mixed-spin crystal, we move it at a constant velocity ofvrel /vR = −1 with respect to the probe atoms, for a fixed interaction timeof 1 ms. In Fig. 10.10 (a), we show probe velocity distributions in TOF forthe three cases P ∼ −1, 0, and 1. As can be seen, the number of probe atoms
out-coupled to 2vR is much lower for the spin mixture (P = 0) than for eitherof the nearly spin-polarized cases. We note that an appreciable number ofout-coupled atoms appears even for the balanced spin-mixture, which is most
likely due to specular reflection as in the previously studied case, while thepresence of unoccupied sites of either the attractive or repulsive lattices may
also cause some Bragg diffraction consistent with the original lattice spacing.
To characterize how the crystal structure changes as the population imbalance is continuously tuned, we count the number of probe atoms transferred
to a velocity region around vp ∼ 2vR . As a function of the crystal “polarization”, the transferred population shows a distinct minimum near P = 0
for a balanced mixture, as shown in Fig. 10.10 (b). This example readilydemonstrates how matter-wave scattering can be used to detect changes tothe crystal structure of an ultracold lattice gas, and through species-selective
scattering may eventually be used to detect quantum-magnetic spin-orderedstates. Finally, we point out that the full tunability of the wavelength foratomic matter waves should provide a distinct advantage over optical scattering in certain circumstances, namely in studying density structures not formed
in optical lattice potentials, such as through self-organization or with magnetic trapping potentials. By using probes of small de Broglie wavelength,
this could also include the study of novel quantum states realized at moreeasily attainable temperatures in systems with smaller characteristic lengthscales. For example, at fixed lattice depth s and fixed tunneling-to-interaction
ratio (t/U )c (e.g., by control of the scattering length as ), the Néel temperature TN , marking the onset of antiferromagnetic ordering, scales with lattice wavelength λ as TN ∝ λ−2 [286]. For fixed as and with variable lattice
depth s, an even more favorable scalingis achieved for realistic system pa√
3/2 −1−32πas (t/U )c ] ∼ λ−2.7 for as = 100 a0 andrameters, with TN ∝ λ ln [λ(t/U )c = 1/32. For a more complete determination of the optimal conditions
for stabilizing a desired quantum phase, entropy considerations and effects oflattice-cooling [287] should also be taken into account.

142

100

constant asconstant s

TN [nK]

10

1

0.1
0.1

0.2

0.3

0.5

λ [μm]

0.7

1.0

1.5 2.0

Figure 10.11: Scaling of Néel temperature TN with lattice wavelength, λ, forfixed interaction-to-tunneling ratio U/t = 32. Shown are the cases of (blueline) fixed scattering length as = 100 a0 but variable lattice depth s and (reddashed line) fixed lattice depth s = 13 (in units of ER = h2 /2mλ2 but variable
scattering length as ). Scalings of TN ∝ λ−2 and (roughly, see text) TN ∝ λ−2.7
are found for the two methods.

10.9

Concluding remarks

In this work, we have demonstrated that matter-wave diffraction can be used
to characterize the crystalline structure of strongly correlated atoms in an
optical lattice. In the future, these techniques may be extended to the characterization of various novel states of ultracold matter, such as charge- andspin-density waves, magnetically-ordered ground states of quantum gas mixtures, and even self-assembled structures such as Tonks–Girardeau gases offermionized bosons, Abrikosov vortex lattices [288, 289], and dipolar crystals [290].

10.10

Some experimental details

10.10.1

Hyperfine state mixtures

The initial two-species mixture of probe |pi ≡ |2, −2i and target |ti ≡ |1, 0i
atoms, used to study in Figs. 10.1, 10.4, 10.5, 10.6 & 10.7 the scattering dependence on sz , contains a total of (1.6±0.2)×105 atoms, with 60% of atoms in thetarget state. For this mixture, a final magnetic bias field of 7.4 G is employed143to suppress |1, 0i ↔ |1, ±1i spin-changing collisions. A second binary mixtureof probe |pi ≡ |2, −2i and target |ti ≡ |1, −1i atoms is used in conjunction
with the moving optical lattice [Figs. 10.2 & 10.3]. This mixture contains
a total of (2.8 ± 0.5) × 105 atoms, with 33% target atoms. A three-species
mixture of one probe species (|pi ≡ |1, 1i) and two target species (|2, −2i and|1, −1i) is used to study forced antiferromagnetic ordering [Fig. 10.10]. This
mixture contains a total of (1.5 ± 0.2) × 105 atoms, with 50% of the atoms in
the target. The target, consisting of two different species, has a fully tunablespin composition [c.f. Fig. 10.10 (b)]. The intra- and interspecies scattering
lengths for all the states used are approximately equal to 5.3 nm, i.e. thebackground scattering length for 87 Rb atoms.

10.10.2

State-selective lattice potential

The state-selective lattices are formed by interfering two laser beams (1/e2radius ∼ 230 µm , the same polarization) with tunable wavelength between the87
Rb D1 and D2 lines to effect a light-shift cancellation for the probe atoms.
In the case of a stationary target of |1, 0i atoms, the lattice is loaded to avariable depth sz with an s-shaped curve in 75 ms and held for an additional
5 ms prior to acceleration of the probe atoms. This lattice is made from fully
retroreflected laser light of wavelength λ = 785 nm of σ + polarization. For
the case of a moving target of |1, −1i atoms (wavelength and polarization as
in the stationary case), the lattice is first smoothly loaded in 45 ms to a depthof sz = 10, exceeding the critical depth of the 1D Mott insulator transition,and then loaded in 5 ms to a depth of sz = 50 to freeze the atoms to the sitesof the lattice. The lattice is then moved, by introducing a relative frequency
detuning of δω between the forward and retroreflected laser beams comprising
the z-lattice, via two acousto-optic modulators driven by phase-locked function
generators.
For the case of a spin-mixed crystal with forced antiferromagnetic ordering,
the two species are quickly loaded into the state-dependent lattice in 2 ms
following creation of the target mixture to avoid spatial separation in a small
magnetic field gradient used to separate the probe. The lattice has modulation
depths sz = 5 and sz = 12, for the |bi = |2, −2i (repulsive lattice) and |ri =|1, −1i (attractive lattice) atoms, respectively. This lattice is formed with lightof wavelength λ ∼ 788 nm, of slightly elliptical polarization. After loading,this lattice is moved at a fixed velocity as above, with a restriction to relatively
low velocities to ensure that both target species faithfully follow the opticalpotential at these modest depths, as observed in the velocity distributions of
the target species.
All the lattice depths are calibrated using Kapitza–Dirac atom diffrac144

tion [122], with a systematic uncertainty of about 3%. For probe atoms, thelattice potential is sufficiently “zeroed” even for greatest available optical potentials such that no diffraction is observed for a pulsed-on lattice or for application of a linear potential gradient while the lattice is present (i.e. no Blochoscillations). To further “zero” the optical lattice, the probe atoms are loaded
into 1D tubes, and we either minimize excitations of probe atoms undergoing
dipole oscillations, or in the case of a moving lattice we move with velocity ofvL = vR , with no observable transfer of probe atoms to non-zero velocities.

145

Chapter 11
Conclusion and outlook
In this thesis, I have detailed several experiments studying the equilibrium anddynamical properties of ultracold Bose gases in optical and atomic potentials.
We have explored [44] phases of the two-component Bose–Hubbard model,using a binary mixture in a state-dependent optical lattice that allows forindependent control of the species’ tunneling and on-site interactions. Here,an apparent loss of spatial coherence of a given species was observed, mostpronounced when the other species was either a delocalized superfluid or was
made up of immobile impurity atoms, consistent with expected formation ofpolarons due to superfluid immersion [63] in one case, and of a quantum emulsions [47] in the other case. We further studied [46] such quantum emulsions ina fully one-dimensional geometry, and compared the effects of random (“white
noise”) impurity disorder to the quasi-disorder of a pseudorandom incommensurate lattice. In studying the transition from superfluid to an insulating Boseglass-like state [28, 48], we found that the loss of superfluid behavior occurred
for a much larger mobility (hopping) in the case of random impurity disorder.
Moreover, we have introduced [49] a new technique for the characterization of
atomic gases in optical lattices, based on the scattering of atomic de Broglie
waves. Through inelastic and elastic scattering, we were able to study excitations and the spatial structure of lattice-modulated one-dimensional Bose
gases, as well as to detect the introduction of forced antiferromagnetic orderingin an atomic crystal. Lastly, we observed [50] an unusual effect, namely thatdisorder can induce transport, when a Bose–Einstein condensate was pulsed by
two incommensurate standing waves, corresponding to a quantum-to-classicaltransition in a system of two coupled, kicked quantum rotors [84].
In addition to the topics I have presented in this thesis, several lines ofresearch are currently being undertaken in the laboratory. These include
the dynamical screening of disordered potentials due to mean-field interactions (observed through the lifetime of coherent Bloch oscillations in applied146

gradients), the use of low-field Feshbach resonances [291, 292] to tune theinterspecies scattering length in one-dimensional bosonic mixtures, and theimplementation of state-selective dynamical control over the particles’ timeaveraged band-structure [293] for modification of the tunneling [294–296] andcreation of tunable gauge potentials [255].
Moreover, our experimental ability to independently control the localization of different species in an optical lattice lends itself to a number of directions of future research. In an earlier study [44], we observed some indicationsof possible polaronic coupling when atoms were immersed in a superfluid bath.
More focused investigations along these lines, such as to measure directly the
induced density “dips” (by modification of Franck-Condon overlaps) throughrf spectroscopy [52] or to observe the clustering of polarons through Bragg
spectroscopy [63], could be made to study polarons in this system. In a similar system, one can explore the physics of the spin-boson model [232, 297],
where coupling of localized atomic “qubits” (with coherent control over e.g.,some internal pseudospin or lattice orbital occupation) to a bosonic bath canlead to local dissipation and also long-range interactions between the “qubits”.
I certainly look forward to seeing the interesting physics that will come out of
this lab for years to come.

147

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173

Appendix A
Delocalization of a spinful
kicked rotor
In this chapter, we consider (in simulations) a problem similar to that addressed experimentally in Chapter 5, namely the dynamical response of amatter-wave field to time-dependent “kicking” with an optical lattice. However, instead of studying the influence of spatial quasidisorder on kicked rotor
behavior, here we consider “spinful” δ-kicking of a quantum rotor system, forwhich the atoms are driven in a pseudorandom, aperiodic way by both latticepulses and spin rotations. As we will show, this analogously leads to classicaldiffusion in momentum space.

A.1

Introductory discussion

Coherent spin-dependent transport of atomic matter-waves has been considered recently in several contexts, including the generation of entanglement [223] and for quantum simulation of charged particles [298]. Some recent
proposals [175, 299, 300] have considered using spin-dependent “kicking” ofcold atoms (in a kicked-rotor type experiment) to study the influence of spinorbit coupling, i.e. broken time-reversal symmetry, on localization phenomenaof charged particles. In the spirit of studying the effects of spin-orbit couplingon Anderson localization, here we consider a very simple case of the spinful
kicked rotor, where non-interacting atoms are periodically kicked by a latticethat is completely state-selective (as described in Chapter 8), and couples different plane-wave momentum orders. Between each lattice kick, a variablespin-rotation is applied to the atoms, causing coherent transitions between
two internal states. We find that for certain spin rotations applied, a breakdown of dynamical localization is achieved, leading to classical-like diffusion

174

T/µ

θ
K

T

Figure A.1: Periodic driving of a matter-wave field by external (lattice pulse)and internal (spin rotation) state driving. Lattice pulses (at bottom) withkicking strength K occur with a frequency 2π/T , and periodic spin rotations(at top, occurring with the same frequency) by an amount θ, where θ (modulo2π) varies at a frequency 2πµ/T .
in momentum space, similar to the case of two incommensurate lattices in
Chapter 5.

A.2

System under consideration

We consider a collection of non-interacting atoms beginning in the plane-wave
state with zero momentum, approximately realized by a weakly-trapped Bose–
Einstein condensate. We consider the case of two internal states of the atoms,|ai and |bi (e.g. as with hyperfine states |1, −1i and |2, −2i), where the possibility of coherent internal state manipulation is assumed, for example byapplying resonant microwave radiation. Furthermore, we consider an opticallattice oriented along one axis, with wavelength λ and wave number k = 2π/λ.
This lattice is assumed to be state-selective, addressing only atoms in the internal state |ai, with a depth measured as sER (ER = ~2 k 2 /2M the recoil
energy and M the atomic mass). A train of N pulses of this state-selective
lattice act on the atoms, having duration τ and periodic spacing T , and thestrength of the lattice is the same for each of the pulses (labeled j ∈ {1, N }).
For internal state manipulation we assume that some short fixed time after the j-th lattice-pulse, a δ-like (in time) microwave spin-rotation by anangle θj is performed (where a π rotation relates to a spin flip from |ai to |bi,|bi to −|ai), where the subscript j denotes that, unlike for the lattice-kicking,the rotation amount is not the same for each pulse. As shown in Fig. A.1, to
realize the case of two-frequency driving (at frequencies 2π/T and 2πµ/T ) we

175

define the j-dependent spin-rotation angles asθj = θ0 + 2πµj .

(A.1)

The rotation angle θj thus increases linearly with the pulse-number j, andmodulo 2π it varies periodically (in the saw-tooth pattern shown in Fig A.1
with frequency 2πµ/T . The case of µ = 1 corresponds to commensurate frequencies, where the spin is flipped by the same amount every period. We
note that the results that follow are found to be generalizable to other periodic forms, such as θj = θ0 + 2πα cos2 (µπj). Also, while we have considereddiscrete-time spin-rotations, these results should likely be generalizable to having a transverse field always on, with its strength varying in a time-periodic
way (neglecting effects of the state-selective light-shifts on the microwave transitions and effects of microwave dressing on the state-selective lattice).
In what follows, we write the Hamiltonian in the form of a δ-kicked rotor,
while in the simulations that follow we fully take into account the finite pulse
length (assumed to be τ = 10 ns). Additionally, we restrict ourselves toa discrete basis of plane wave states as in Chapter 5, here with states |mihaving momenta pm = 2m~k. We then write the Hamiltonian as
Ĥ = Ĥ0 + ~φ̂L

N
X
j=1

δ(t − jT ) + ~NX(θj /2)σ̂y δ(t − jT − ) ,

(A.2)

j=1

P
where Ĥ0 = (~κ/2T ) m m2 n̂m is the time-independent kinetic energy part of
the Hamiltonian, the termφ̂L =K (σ̂0 + σ̂z ) O X †(âm+1 âm + â†m−1 âm ) ,
κ
2
m

(A.3)

accounts for state-selective lattice pulsing, and the final term accounts for the
spin rotations (we have neglected spin-dependent phase accrual in the formof scalar σ̂z terms). Here, the term κ = 8ER T /~ defines the resonances of
the rotor (occurring for rational values of κ/4π) and K = κsER τ /2~ is thestochasticity parameter, which delineates regimes of classically regular andchaotic motion. The Pauli matrices used are given by σ̂0 = |aiha| + |bihb|,σ̂y = −i(|aihb| − |biha|), and σ̂z = |aiha| − |bihb|. The terms n̂m and â†m
(âm ) are the number operator and creation (annihilation) operator for the
momentum mode |mi.
In the following section, we shall present some numerical results for the
dynamical response of a spinful quantum field to spin-dependent lattice kicking
and spin-rotations, as a function of the kick number N , as well as of the tunable176

parameters K (relating to lattice depth) and µ (relating to the ratio of the
kicking frequency to the frequency at which the microwave spin-rotation rate
evolves).

A.3

Numerical simulations

We consider the case of off-resonant kicking with a lattice of wavelength λ =785.5 nm, where the fixed pulse period T = 29 µs relates to κ/4π ≈ 0.43,
such that kicking the |ai atoms alone leads to dynamical localization (and afixed pulse duration τ = 10 ns). If no spin-rotations are applied, neither is
any increase in energy achieved for mixtures of the two species (as the atoms
are assumed non-interacting). Additionally, if the atoms are kicked by onlya single state-independent lattice (addressing both states equally), dynamical
localization is observed regardless of the form of the spin-rotation applied.
We first examine the dependence of growth on the incommensurability of
the kicking frequency and the frequency at which the spin-rotation evolves,characterized by the ratio µ, where for µ = 0 or 1 the same spin rotation is
performed each time step. Prior to the first lattice kick, we begin with an equalsuperposition of the two spin states. In the definition of the j-dependent spinrotations of Eq. A.1, we set θ0 = 0, without loss of generality. To characterize
the dynamical response of the system, we fit a linear dependence of the energy
per atom ε (in units ER ) on the pulse number N , i.e. a rate of growth ∆ε/∆N ,fit in the range N = 1 to N = 200 pulses. We assume that a linear dependence
will be a reasonable approximation to the dynamical response, for either thecase of dynamical localization (where ∆ε/∆N will be zero) or for induced
classical diffusion where we expect ε ∝ N .
In Fig. A.2, we plot the value of this growth rate ∆ε/∆N as a function of
the commensurability parameter µ (in a limited range), for a kicking strengthof K = 3.17. We observe that when µ is equal to a rational number - such as
for 1/2, 4/7, 3/5, 5/8, 2/3, etc. - there is very little growth (with essentiallyno growth for µ = 0, 0.5, 1), while otherwise non-zero growth rates can befound for irrational values. This is consistent with the expectation that theaddition of an incommensurate second frequency should lead to a disruptionof the quantum interference that causes dynamical localization. In Fig. A.2,
we also point out three values of µ in particular, which we will examine inmore detail as follows. These values are µ = 0.6 (rational number with small
amount of continued fraction elements), 0.502 (rational number√ near to 1/2
and with slightly more continued fraction elements) and 2/(1 + 5) = 0.618...
(irrational number, the golden ratio).
We now look directly at the temporal (N ) dependence for each of these

177

0.5

∆ε/∆N

0.4
0.3
0.2
0.1
0.0
0.50

0.55

µ

0.60

0.65

Figure A.2: Dependence of energy growth rate on the commensurability oflattice kicking and spin rotation frequencies. (a) Growth rate of the averageenergy per kick N , ∆ε/∆N , as a function of the incommensurability parameterµ. The growth rate is fit between N = 1 and 200 kicks, for a stochasticity
parameter of K = 3.17. The values of µ = 0.502, 0.6, and 0.618034... (thegolden ratio) are highlighted, as they will be examined further.
values of µ, for three different kicking strengths K = 1.585, 3.170, and 4.755.
What we observe is that for the most rational value, µ = 0.6, an oscillatory
behavior is generally observed, with destructive interference causing dynamical
localization. However, for very large kicking strength, some growth (possiblytransient over N = 200 kicks) is observed. We note that by looking at themean-energy per spin-state (as in the inset for µ = 0.6 and K = 4.755),
oscillations of a greater magnitude can often be seen than in the spin-averagedenergy. The suppression of growth for rational values of µ seems to rely on afine cancellation of this behavior. We observe for values of µ slightly away fromratios of small integers, such as for 0.502, net growth in energy with N canbe achieved. For completely irrational values such as for the golden ratio, we
observe that the spin-dependent kicking always leads to an essentially linear
increase of ε with N , consistent with classical-like diffusion.
Lastly, we determine the dependence of the growth rate ∆ε/∆N on thestochasticity parameter K. We find that the rational value of µ = 0.6 has a
growth rate of essentially zero for most values of K (and the non-zero value
at large K may be only a transient effect), and the value of µ = 0.502 has aslightly irregular dependence on K, albeit with non-zero values. For the irrational ratio of the spin-rotation evolution to the kicking frequency, µ = 0.618...,
we find the dependence on K is in good agreement with the classical diffusion rate D = K 2 /2κ2 (with κ ≈ 5.42), i.e. also suggesting that classical-like
diffusion in momentum-space is occurring due to the addition of the incommensurate frequency component.

178

20

µ = 0.6

20 µ = 0.502

20 µ = 0.618...

15

K = 1.585

15 K = 1.585

15 K = 1.585

ε 10

ε 10

ε 10

5

5

5

0

0

0

20

µ = 0.6

20 µ = 0.502

15

K = 3.170

15 K = 3.170

40 µ = 0.618...

ε 10

ε 10

ε 20

5

5

10

0

0

0
40
30
20
10
0

40
30

ε 20

40 µ = 0.502
100

150

µ = 0.6
K = 4.755

100 150 200N

0100 K = 3.170
50

10
50

µ = 0.618...

ε

ε 20

10
0
0

K = 4.755

30

K = 3.170

30

0

50

100 150 200

N

0

0

50

100 150 200

N

Figure A.3: Dynamics of the mean energy with increasing kick number, for different kicking strengths K and spin-rotation evolution frequencies (µ). Threestacks of plots are shown for different values of µ, with the most commensurate values on the left (µ = 0.6), moving to the most incommensurate at right(µ = 0.502 and µ = 0.618...). For each value of µ, we show the dependence ofthe particle energy ε on the kick number N , for kicking strengths K = 1.585,
3.170, and 4.755. As an inset for the case µ = 0.6 and K = 4.755, we show the
dynamics of the total energy (black) as well as that of the |ai state (red) and|bi state (blue). Note the different scales of the energy axis for the different
cases.

A.4

Concluding remarks

In conclusion, we have performed numerical simulations of a simple situationof spinful kicking of a quantum rotor, leading to classical-like diffusion when
multi-frequency driving at incommensurate frequencies is present. This example highlights how added spectral noise can cause a breakdown of quantum179∆ε/∆N0.15µ = 0.618...

0.10µ = 0.502
0.05µ = 0.6

0.00
0

1

2

K

3

4

5

Figure A.4: Dependence of the energy growth rate ∆ε/∆N on the spinselective stochasticity parameter K, as determined by a linear fit from N = 1
to 200 kicks. We plot the dependence√ on K for the cases of µ = 0.6 (blue line),µ = 0.502 (red line), µ = 2/(1 + 5) = 0.618... (green line). We also show the
expected classical diffusion constant D = K 2 /2κ2 (dashed black line).
interference phenomena, and also how non-separable coupling between internaland external degrees of freedom can affect the localization of spinful quantum
fields.

180