Quantum dynamics in strongly correlated one-dimensional ...

Quantum dynamics in stronglycorrelated one-dimensional Bose gases

Dissertationby

Florian Meinert

submitted tothe Faculty of Mathematics, Computer Science and Physics

of the University of Innsbruck

Advisor: Univ. Prof. Dr. Hanns-Christoph Nagerl

Innsbruck, February 2016

Abstract

Strongly interacting quantum many-body systems and their inherent complexity giverise to fascinating novel phenomena in physics. In recent years, ultracold atomic gases pre-pared in optical lattice potentials have opened compelling new routes for experimental stud-ies of many-body physics in a highly flexible environment. In particular, the unprecedentedcontrol over system parameters, such as lattice properties, the strength of particle interac-tions, or system dimensionality, allows for detailed studies of paradigmatic model Hamilto-nians known from condensed matter physics, e.g. Hubbard models or Luttinger liquids.

The realization of such models with ultracold atoms not only allows for emulating low-energy phenomena, such as ground-state phases and quantum-phase transitions, but alsoprovides the unique opportunity to investigate many-body systems far from equilibrium,and to study the diversity of phenomena emerging from the coherent evolution. This thesiscovers a series of experiments investigating many-body dynamics in the context of stronglyinteracting bosons confined to a one-dimensional (1D) geometry.

A cesium Bose-Einstein condensate prepared in an optical lattice is employed to realizean array of 1D lattice systems, described by the Bose-Hubbard model. We study many-bodycoherent tunneling dynamics in the Mott insulating phase, after the ensemble is suddenlyexposed to a strong force, which essentially tilts the lattice potential. Strong particle interac-tions in the Mott insulator result in resonant and highly correlated atom tunneling through aprecisely controlled number of lattice sites when the tilt compensates for integer fractions ofthe on-site interaction energy. First, we observe coherent many-body dynamics for resonanttunnel coupling between neighboring lattice sites. In this regime, the system maps onto aneffective Ising spin chain, which we analyze in detail for equilibrium and out-of-equilibriumsituations. The coherent dynamics is further exploited to study a modification of the tunnel-ing rate induced by particle interactions, the so-called density-induced tunneling.

Further, resonant long-range tunneling through multiple barriers of the lattice is inves-tigated. These processes can be assigned to higher-order atom tunneling, where the specificorder increases with the number of penetrated barriers. This allows us to observe many-body dynamics driven by small amplitude terms beyond the scale of the celebrated super-exchange. In another set of experiments, a 1D superfluid in the tilted lattice is explored,which exhibits so-called Bloch oscillations. In the presence of particle interactions we ob-serve regular quasi-periodic evolution marked by quantum phase revivals, and map out thetransition to a quantum chaotic behavior.

Finally, the lattice along the 1D systems is removed. This realizes an array of 1D Bosegases, each described by the Lieb-Liniger model. Employing Bragg-spectroscopy, we mea-sure the excitation spectrum of the gas from the weakly to the strongly interacting regime.The role of the peculiar hole-type excitations in shaping the dynamical response of thestrongly correlated 1D system is revealed.

Zusammenfassung

Stark wechselwirkende Quanten-Vielteilchensysteme zeigen eine Vielzahl faszinieren-der Phanomene auf Grund ihres hohen Grades an Komplexitat. In jungster Zeit haben ul-trakalte Atome in optischen Gitterpotentialen ganz neue experimentelle Wege eroffnet, diePhysik solcher Systeme in besonders flexibler Weise zu untersuchen. Die außergewohnlichgute Kontrolle uber Systemparameter, wie beispielsweise die Gittereigenschaften, die Wech-selwirkungsstarke zwischen den Atomen, oder die Dimensionalitat des Quantensystems,erlaubt es im Speziellen, grundlegende Modelle aus der Festkorperphysik zu studieren. Bei-spiele hierfur sind Hubbard-Modelle oder Luttinger-Flussigkeiten.

Von Interesse ist auf der einen Seite die ”Quantensimulation”der niederenergetischenGrundzustandsphasen und deren Phasenubergange. Daruber hinaus bieten Experimentemit kalten Atomen gleichzeitig die einzigartige Moglichkeit koharente Vielteilchen-Quanten-dynamik, und die damit verbundenen Effekte, zu untersuchen. Die vorliegende Arbeit bein-haltet eine Reihe experimenteller Studien zur Vielteilchendynamik stark wechselwirkenderBosonen in eindimensionalen (1D) Systemen.

Ein Casium Bose-Einstein-Kondensat in einem optischen Gitter dient der Realisierungeindimensionaler Gittersysteme, beschrieben durch das Bose-Hubbard Modell. Wir unter-suchen koharente Vielteilchen-Tunneldynamik im Mott-Isolator-Zustand des Systems, nach-dem wir diesen plotzlich einer starken externen Kraft aussetzten, und damit das Gitterpoten-tial ”kippen”. Die starke Wechselwirkung der Atome im Mott-Isolator fuhrt zu resonantenund stark korrelierten Tunnelprozessen uber eine kontrollierte Zahl von Gitterplatzen, wenndie Kraft die Wechselwirkungsenergie zweier Teilchen auf dem gleichen Gitterplatz kom-pensiert. Zunachst untersuchen wir koharente Vielteilchendynamik fur den Fall resonantenTunnelns zwischen benachbarten Gitterplatzen. In diesem Regime lasst sich das Quanten-system auf ein Ising-Spin-Modell abbilden, das wir im Gleichgewichtsfall und im Nicht-Gleichgewichtsfall analysieren. Uber die Tunneldynamik beobachten wir auch eine Modi-fikation der Tunnelrate auf Grund der Wechselwirkung zwischen den Teilchen, das soge-nannte Dichteinduzierte Tunneln.

Des Weiteren untersuchen wir langreichweitiges Tunneln durch gleich mehrere Barrie-ren des Gitterpotentials. Diese Prozesse konnen Tunneltermen hoherer Ordnung zugeord-net werden, wobei die jeweilige Ordnung gegeben ist durch die Zahl der durchdrunge-nen Barrieren. Im Experiment ermoglicht das die Beobachtung von Vielteilchendynamikdie getrieben ist durch schwache Kopplungsterme, welche uber die Ordnung der wichtigenSuper-Exchange-Wechselwirkung hinausgehen. Eine Reihe weiterer Experimente widmetsich schließlich noch der Quantendynamik eines 1D Superfluids im gekippten Gitterpoten-tial, die durch sogennante Bloch-Oszillationen gekennzeichnet ist. Im Fall wechselwirken-der Teilchen beobachten wir eine quasi-periodische Zeitentwicklung auf Grund sogenannter”quantum phase revivals”, und vermessen den Ubergang zu quanten-chaotischem Verhal-

ten.Abschließend wird das eindimensionale Gas wechselwirkender Bosonen ohne zusatz-

liches Gitterpotential untersucht, beschrieben durch das Lieb-Liniger Modell. Wir vermes-sen das Anregungsspektrum des Vielteilchensystems vom schwach wechselwirkenden bisins stark wechselwirkende Regime mittels Bragg-Spektroskopie. Damit zeigen wir den Ein-fluss sogenannter ”loch-artiger” Anregungen auf die Form des Anregungsspektrums furdas stark korrelierte 1D System.

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CONTENTS

1 Introduction 11.1 Tunable degenerate Bose gases . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.1 Atomic scattering and Feshbach resonances . . . . . . . . . . . . . . . . 81.1.2 Scattering properties of cesium . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Ultracold atoms in an optical lattice and the Hubbard model . . . . . . . . . . 131.2.1 Quantum mechanics and dynamics in a periodic potential . . . . . . . 141.2.2 Strongly correlated many-body system: The Bose-Hubbard model . . 171.2.3 Out-of-equilibrium dynamics in the Bose-Hubbard model . . . . . . . 191.2.4 Extended Hubbard model and density-induced tunneling . . . . . . . 24

1.3 The one-dimensional Bose gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.3.1 Universal low-energy description: Luttinger liquid . . . . . . . . . . . 291.3.2 Microscopic description: Lieb-Liniger gas . . . . . . . . . . . . . . . . . 31

1.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.5 List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2 Publication: Quantum Quench in an Atomic One-Dimensional Ising Chain 392.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.2 Tilted Mott insulator and quantum magnetism . . . . . . . . . . . . . . . . . . 402.3 Driving the Ising quantum phase transition . . . . . . . . . . . . . . . . . . . . 412.4 Quench dynamics near the quantum critical point . . . . . . . . . . . . . . . . 452.5 Interaction-induced modification of the tunneling rate . . . . . . . . . . . . . . 452.6 Collective effects in the quench dynamics . . . . . . . . . . . . . . . . . . . . . 472.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.8 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.8.1 Preparation of the one-atom-per-site Mott insulator . . . . . . . . . . . 482.8.2 Calibration of E and U . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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2.8.3 Estimating the average chain length . . . . . . . . . . . . . . . . . . . . 50

2.8.4 Detection of single and double occupancy . . . . . . . . . . . . . . . . . 51

2.8.5 Numerical simulations of quench dynamics . . . . . . . . . . . . . . . 51

3 Publication: Observation of many-body dynamics in long-range tunneling aftera quantum quench 53

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 Higher-order atom tunneling in a tilted Mott insulator . . . . . . . . . . . . . 54

3.3 Observation of higher-order tunneling resonances . . . . . . . . . . . . . . . . 56

3.4 Many-body dynamics driven by second- and third-order tunneling . . . . . . 57

3.5 Many-body time reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.6 Fourth- and fifth-order long-range tunneling . . . . . . . . . . . . . . . . . . . 59

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.8 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.8.1 Perturbation theory solutions for the E2 = U/2 resonance . . . . . . . 63

3.8.2 Higher-order resonances: E3 = U/3 and beyond . . . . . . . . . . . . . 66

3.8.3 Numerical simulation of the doublon number on resonance . . . . . . 66

3.8.4 Entanglement growth during the quench . . . . . . . . . . . . . . . . . 68

3.8.5 Collective effects in the tunneling rates for higher-order resonances . . 68

4 Publication: Interaction-Induced Quantum Phase Revivals and Evidence for theTransition to the Quantum Chaotic Regime in 1D Atomic Bloch Oscillations 71

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Bloch oscillations in the Hubbard model . . . . . . . . . . . . . . . . . . . . . . 72

4.3 Bloch oscillations of a non-interacting sample . . . . . . . . . . . . . . . . . . . 73

4.4 Interaction-induced quantum phase revivals . . . . . . . . . . . . . . . . . . . 75

4.5 Decoherence in the quantum chaotic regime . . . . . . . . . . . . . . . . . . . . 76

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.7 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.7.1 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.7.2 Numerical simulations of Bloch oscillation dynamics . . . . . . . . . . 82

4.7.3 Modeling the effect of the harmonic trap . . . . . . . . . . . . . . . . . 84

5 Publication: Observation of Density-Induced Tunneling 85

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2 Experimental manifestation of density-induced tunneling . . . . . . . . . . . . 86

5.3 The extended Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4 Numerical simulation of the resonant tunneling dynamics . . . . . . . . . . . 88

5.5 Density dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

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5.6 Density-induced tunneling and higher-order hopping . . . . . . . . . . . . . . 925.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.8 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.8.1 Lattice depth calibration and error bars . . . . . . . . . . . . . . . . . . 945.8.2 Calculation of the occupation-dependent on-site energy shift . . . . . 945.8.3 Experimental preparation of the n = 1 and n = 2 Mott shell . . . . . . 945.8.4 Mode spectrum of the standard Hubbard model . . . . . . . . . . . . . 955.8.5 Finite-size effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.8.6 Fourier analysis of doublon dynamics at occupation n = 2 . . . . . . . 975.8.7 Role of defects for n = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6 Publication: Probing the Excitations of a Lieb-Liniger Gas from Weak to StrongCoupling 1016.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.2 Realization of a one-dimensional Bose gas with tunable interactions . . . . . . 1046.3 Probing the excitation spectrum of the one-dimensional Bose gas via Bragg

spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.4 Finite temperature analysis of the excitation spectrum . . . . . . . . . . . . . . 1076.5 Analysis of the momentum distribution after Bragg excitation . . . . . . . . . 1086.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.7 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.7.1 Lattice depth calibration and error bars . . . . . . . . . . . . . . . . . . 1106.7.2 Atom number distribution across the 1D tubes . . . . . . . . . . . . . . 1116.7.3 Density profile in the tubes . . . . . . . . . . . . . . . . . . . . . . . . . 1126.7.4 Mean γ and kF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.7.5 Sampling of the dynamical structure factor over the array of tubes and

comparison with the experimental data . . . . . . . . . . . . . . . . . . 1136.7.6 The ABACUS algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.7.7 Regime of linear response . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.7.8 Heating effects on the excitation spectra . . . . . . . . . . . . . . . . . . 1156.7.9 Interaction-independent spectral width at small γ . . . . . . . . . . . . 1176.7.10 Momentum distribution of Lieb-I and Lieb-II excitations . . . . . . . . 117

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CHAPTER 1

INTRODUCTION

The development of quantum theory at the beginning of the twentieth century has revolu-tionized our understanding of the physical laws on microscopic length scales in a funda-mentally new way. Despite the great achievement to contract the experimental observationsmade at that time into a theory of beautiful simplicity, at the heart of which is Schrodinger’swave equation, it soon became clear that complex quantum systems, made up of already arelatively small number of interacting particles, are in principle very hard to calculate. Thisis due to the exponentially growing Hilbert space of the many-body quantum system withthe number of its constituents. Specifically, this has profound implications in understand-ing quantum phenomena that emerge in condensed matter systems due to strong particlecorrelations, such as (high-Tc) superconductivity, low-temperature transport, or quantummagnetism. It is the challenge to capture and explain effects and properties of such type byan appropriate low-energy description of the system at hand. Paradigmatic models are, forexample, Heisenberg spin chains, Hubbard-type or Luttinger models.

Rapid experimental advances in the control and manipulation of dilute atomic ensemblesprepared at ultralow temperatures has now reached a point, where some model Hamiltoni-ans know from condensed matter physics can indeed be realized in the laboratory with ahigh degree of parameter control. This research direction follows the idea most prominentlypointed out by Richard Feynman in 1982 [Fey82]: Instead of calculating the properties ofa quantum mechanical system on a classical computer (which in many cases is highly lim-ited), it might be feasible to design a different, yet highly controllable, quantum system thatis described by the same model and allows to simulate the system of interest in an exper-iment. These exciting novel developments have turned the field of atomic physics into ahighly interdisciplinary research area, where ultracold atoms are promising candidates forsuch ”analog quantum simulators” of solid state systems [Blo12].

1

Introduction

From laser cooling to many-body physics

Before discussing pioneering experiments and recent progress in this direction, a short re-view on the development of experimental techniques to prepare and control atomic vapors atultralow temperatures will be given. The research on laser cooling and trapping techniquesin the 1980s has made it possible to prepare dilute vapors of neutral atoms at temperaturesin the microkelvin regime in magneto-optical traps (MOT) [Met99]. In 1995, the temperatureof a laser cooled cloud of rubidium could be further reduced by means of forced evaporativecooling resulting in an increase in phase space density beyond the threshold to quantumdegeneracy [And95]. This was the first experimental demonstration of a Bose-Einstein con-densate (BEC), a new state of matter that is a result of a macroscopic occupation of the ener-getically lowest quantum state of the system [Bos24, Ein25]. Very soon, quantum phenomenaassociated with Bose-Einstein condensation have been experimentally verified, e.g. matter-wave interference [And97], superfluidity and the existence of vortices [Mat99, Mad00], sup-pression of bosonic bunching [Bur97], and Bogoliubov excitations [Sta99].

An important step forward has been the observation of Feshbach resonances, which al-low controlling two-body interactions in the gas [Ino98, Chi10]. Parallel progress in experi-ments with fermionic atoms has led to the observation of a degenerate Fermi gas [DeM99].In two component fermionic mixtures, Feshbach resonances enabled the creation of weaklybound molecules [Reg03, Str03, Cub03, Joc03b]. These molecules are found stable under col-lisions due to the fermionic nature of the atoms they are made of. This stability in com-bination with interaction tuning has opened the door to study the crossover between aBEC of molecules for repulsive interactions [Gre03, Joc03a, Zwi03] and a Bardeen-Cooper-Schrieffer (BCS)-type superfluid for attractive interactions [Reg04, Zwi04, Kin04a, Bar04,Chi04a, Zwi05]. The observation of superfluidity in a BCS-type system, as known in the con-text of superconductivity, together with the spectacular experimental control demonstratedin these studies has been a breakthrough in demonstrating the prospects of cold atoms as asimulator for complex quantum systems.

Along these ideas, optical lattices have been suggested to reach regimes where particlecorrelations become important and mean-field methods break down in describing the sys-tem properties [Jak98]. A groundbreaking experiment in this direction has been the first im-plementation of the Hubbard model with bosonic atoms and the demonstration of the quan-tum phase transition from a superfluid to a Mott insulator [Gre02a]. The versatile potentialof optical lattices has triggered numerous experimental studies in various directions. Theprovided control over dimensionality has led to the first demonstration of the strongly cor-related one-dimensional (1D) Tonks-Girardeau [Kin04b, Par04] and super-Tonks-Girardeaugas [Hal09], or the study of the superfluid-insulator transition in reduced dimensions [Sto04,Spi07, Hal10b]. Various lattice geometries have been implemented to study more sophisti-cated band structures [Tar12], frustrated classical magnetism [Str11], or non-trivial topolog-

2

Introduction

ical properties [Jot14, Duc15]. Recent effort is devoted to the implementation of artificialgauge fields [Dal11] via e.g. specially tailored Raman transitions [Lin09, Aid13, Miy13] ormodulation techniques [Str12] to access the regime of the (fractional) quantum Hall effect.An outstanding goal is the detection of anti-ferromagnetic ordering in the Fermi-Hubbardmodel [Jor08, Sch08]. Pioneering work on super-exchange coupling between pairs of bosonicatoms in neighboring lattice sites of a superlattice potential [Tro08] has now reached a sta-tus where short-range magnetic correlations have been observed in the fermonic Hubbardmodel [Gre13]. Parallel to this spectacular growth of possibilities with neutral atoms in op-tical lattices, lattice models with long-range interactions are being implemented with polarmolecules [Yan13], Rydberg [Sch15] or strongly magnetic atoms [Bai15].

Out-of-equilibrium dynamics in strongly correlated systems

Seen from a condensed matter viewpoint, simplified model Hamiltonians have in thepast been devised to explain low-energy physics, e.g. phase diagrams or low-energy excita-tions. With the implementation of some of these models in ultracold gases and the controlover system parameters, experimentalists are now in a position to study aspects that gobeyond these initial questions. A prominent new frontier is the possibility to study out-of-equilibrium dynamics in a practically closed interacting quantum system [Pol11]. Coherentdynamics can be induced by a rapid change of one or more parameters of the Hamiltonian,thereby inducing a ”quantum quench” into a highly non-equilibrium situation, or via a con-trolled preparation protocol of a highly excited initial state. Remarkable examples are thedemonstrated absence of thermalizing collisions in the dynamics of two highly excited op-posite momentum states in a 1D Bose gas [Kin06], the relaxation dynamics of an initiallydensity-wave ordered state in a 1D Hubbard model [Tro12], and the relaxation towards apre-thermalized state in coherently split 1D Bose gases [Gri12]. Slow changes of system pa-rameters have been shown to allow studies of generalized Landau-Zener transitions in thetunneling dynamics of a many-body system [Che11] or the emergence of coherence whencrossing a quantum phase transition [Bra15]. Analysis of coherent evolution after a quenchcan also be used to extract model parameters as demonstrated via matter-wave dynamicsinduced by multi-body interactions in the Bose-Hubbard model [Wil10]. The spectaculardemonstration of single-site-resolved detection and manipulation of atoms in optical lat-tices in a ”quantum gas microscope” has opened up new possibilities due to the access tolocal observables [Bak09, She10]. This has been demonstrated in studies of light-cone likespreading of correlations [Che12] and the detection of entanglement dynamics [Fuk15] in1D Hubbard chains.

The advances in experiments come together with newly developed theoretical meth-ods for out-of-equilibrium studies of quantum many-body systems, thereby creating in-creasing interest to resolve fundamental question, some of them being longstanding open

3

Introduction

problems [Pol11, Eis15]. Among others, those are studies of equilibration and thermaliza-tion mechanisms in interacting quantum systems that obey unitary evolution [Rig08, Deu91,Sre94] and how they relate to properties such as integrability [Cau11, Rig09]. Universal dy-namics at quantum critical points is investigated, e.g. the formation of defects when pass-ing through a quantum phase transition [Pol08, Cam14]. Another prominent example is(quasi-)particle transport and related correlation and entanglement dynamics after a quench[Ami08, Kol07, Cra08a, Moe08]. 1D systems are of particular interest as numerical tools areavailable, such as time-dependent density-matrix renormalization group (t-DMRG) methods[Vid04, Dal04, Whi04, Ver08], to track the many-body dynamics for experimentally relevantsystem sizes, and thereby allow precise benchmarking for quantum simulations with ultra-cold gases.

Content of this thesis

This thesis is devoted to experimental studies of quantum many-body dynamics in thecontext of strongly interacting bosons confined to a 1D geometry. The experiments presentedfocus on various aspects of such dynamics in two different models. The first setting considersglobal quench dynamics in the 1D Bose-Hubbard Hamiltonian subject to an additional tilt,given by a linear potential. In the Mott insulating regime, the presence of strong interactionsallows us to study highly correlated coherent tunneling dynamics. The main results are theobservation of interaction-induced resonant tunneling processes across multiple sites, drivenby increasingly higher-order terms in the Hubbard chain, and the analysis of the correspond-ing dynamics in the many-body context. In the superfluid regime, the effect of interactionson the matter-wave dynamics is studied in the context of the spectral properties of the Hub-bard model, giving rise to either quantum phase revival dynamics or an irreversible decay. Ina second setting, dynamics of a many-body system is studied from a different perspective.We measure the excitation spectrum of the interacting Bose gas in 1D, which realizes theLieb-Liniger Hamiltonian. This study reveals how collective excitations that have no coun-terpart in higher dimensions shape the spectrum. This serves as an example where ultracoldatoms can help to understand physics that go beyond the typical low-energy description,which in this case is a Luttinger liquid model.

The experiments are presented in a collection of five scientific articles that have beenpublished in the course of this doctoral work. At the heart of these studies is a degeneratequantum gas of cesium (Cs) in an optical lattice with highly tunable interactions [Gus08a].The first chapter is devoted to provide a broad introduction to methods and techniques em-ployed for this work as well as the physics of ultracold atoms in optical lattices, specificallyfocusing on the Bose-Hubbard model and one-dimensional systems. It also includes briefsummaries of the results presented in each publication and finally closes with an outlook.Among others, the latter elaborates on specific future experimental studies that are within

4

Introduction

direct reach in our laboratory given the results of this thesis.Chapter 2 reports the realization of an effective Ising spin chain in a tilted Mott insulator

following its recent demonstration as an example of a quantum simulation [Sim11]. Thequantum simulator is based on a mapping of tunneling degrees of freedom that are subjectto an interaction-induced constraint onto pseudo-spins. As a first step, the quantum phasetransition from the paramagnetic to the anti-ferromagnetic state is probed. In a second stepwe quench the system to the vicinity of the phase transition and study the dynamics in themany-body context. The experiment also reveals that interactions modify the bare tunnelingrate. In chapter 5, we show in a combined experimental and theoretical effort how this canbe explained in an extended Hubbard model including a term known as density-inducedtunneling.

In chapter 3, we report on the observation of long-range tunneling processes that emergein a resonant way due to the presence of other particles and their mutual interaction. Throughan analysis of the many-body tunneling dynamics, it is demonstrated how these processesare driven by higher-order terms in the Hubbard model, going beyond the scale of the super-exchange. The investigation of dynamics in the tilted Hubbard model is complemented inchapter 4 through a study of Bloch oscillations in the superfluid regime. Here, the energyspectrum of the Hamiltonian changes from regular to chaotic level statistics depending onsystem parameters. This shows up in a qualitative change in the dynamics of the Bloch os-cillating matter-wave.

Finally, chapter 6 reports the investigation of the excitation spectrum of a 1D Bose gasthat is probed via Bragg spectroscopy. Tuning the gas from the weakly to the strongly in-teracting regime reveals the role of a new type of excitation that results from the effectivefermionization of the strongly repulsive bosons. Its role in shaping the experimentally mea-sured spectrum is identified via a detailed comparison with numerical calculations, that relyon the integrability of the underlying Lieb-Liniger model.

5

Introduction

1.1 Tunable degenerate Bose gases

On the microscopic level, nature provides two different types of particles with very distinctproperties determined by their quantum statistics: bosons and fermions. While two identicalfermions cannot occupy the same quantum state, in 1925, Albert Einstein predicted that, inan ideal gas of bosons, the lowest single-particle quantum state becomes macroscopicallyoccupied at very low temperature [Bos24, Ein25]. This new phase of matter is now knownas a Bose-Einstein condensate (BEC). A BEC starts to form when the phase space densityexceeds approximately unity. In other words, BEC requires that the de Broglie wavelengthλdB becomes comparable to the mean particle distance, determined by the gas density n.Precisely, condensation starts for λ3

dBn > 2.61.

About a decade later, Fritz London suggested Bose-Einstein condensation as a mecha-nism for the superfluid properties of 4He, although the importance of interactions in theliquid phase makes it very different from the ideal gas BEC. In 1941, Landau found a consis-tent theory for superfluids, suggesting that the spectrum of low-energy excitations is sound-like, and a few years later Bogoliubov developed the first microscopic theory of the interact-ing Bose gas. Yet, despite all theoretical progress, an experimental demonstration of Bose-Einstein condensation in a dilute gas remained elusive.

The first experiments trying to realize BEC in dilute atomic gases started in the late1970s using cryogenic techniques to cool spin polarized hydrogen in helium coated cells.These first attempts, however, failed because of recombination at the cell walls. To avoidany contact with a surface, evaporative cooling in magnetic traps was pursued. Althoughthese experiments got remarkably close to a BEC, condensation was hindered by spin re-laxation. Meanwhile, advances in laser cooling techniques had made it possible to preparecold and dense clouds of alkali metal atoms in magneto-optical traps (MOT) (see [Met99]for a review). It was the combination of these methods with the knowledge of evaporationtechniques gained by the hydrogen community that led to the first achievement of a BECin a dilute atomic vapor in 1995. In a sequential experiment, 87Rb was first loaded from aMOT into a time-orbiting-potential (TOP) magnetic trap, in which it was then evaporativelycooled below the BEC transition [And95]. Astonishingly, this approach allowed for produc-ing almost pure BECs compared to 4He, where the condensed fraction is ∼ 15%.

The approach to use laser cooling and subsequent evaporation in magnetic or off-resonantoptical traps is applicable to many other atomic species. Already in the same year, conden-sates of the alkali metals sodium (23Na) [Dav95] and lithium (7Li) [Bra95] were produced,followed by the demonstration of a BEC of spin-polarized hydrogen [Fri98], metastable he-lium (4He) [Rob01], and potassium (41K) [Mod01]. The success in achieving a BEC heavilydepends on the efficiency of the evaporative cooling stage. The general idea is to allow themost energetic part of the sample to escape from the trap while the remaining atoms ther-malize to a lower temperature. In a trap the reduced temperature in turn leads to a reduced

7

Introduction

occupied volume and thus increased density. The temperature decrease and density increasemakes evaporation so successful in tremendously increasing phase space density.

As a general guideline for evaporation to work, the rate of thermalizing elastic collisionsmust be about two orders of magnitude higher than the rate of inelastic collision processesthat cause heating and trap loss. In a dilute gas, inelastic collisions are either spin-relaxingtwo-body processes (∝ n2) or three-body recombination into bound molecular states (∝ n3).Spin-relaxation can be avoided by preparing the atoms in the energetically lowest spin state,and using far off-resonant optical traps in case this state cannot be magnetically trapped.The remaining challenge is to keep elastic collisions dominant compared to the disturbingthree-body recombination.

A feature unique to systems of ultracold atoms is the tunability of two-body scatteringvia Feshbach resonances [Chi10]. These resonances in the atomic scattering cross section al-low for setting the strength of interactions in the gas from attractive to repulsive values viaadjusting an external magnetic field, or even making the atoms non-interacting. This pow-erful handle also constitutes an important knob to evaporate atomic species with scatteringproperties unsuitable for efficient evaporation. A prominent example is the element used inthis work: cesium. Cesium offers a particularly rich structure of Feshbach resonances andallows for exceptionally precise interaction tuning. Bose-Einstein condensation of Cs wasfirst achieved in Innsbruck in 2002 [Web02], thereby also closing the list of stable alkali metalatoms that were successfully Bose condensed.

Before introducing the main concepts and methods for interaction tuning in ultracoldgases, I cannot close this section without mentioning the more recent demonstration of BECof elements outside the group of hydrogen-like atoms, such as ytterbium [Tak03], the al-kaline earth metals strontium [Ste09] and calcium [Kra09a], as well as the strongly dipolaratoms chromium [Gri05], dysprosium [Lu11], and erbium [Aik12].

1.1.1 Atomic scattering and Feshbach resonances

Not only, as discussed above, does atomic scattering play a major role in the successfulpreparation of ultracold and degenerate samples. Interaction processes are utterly at thevery heart of many exciting phenomena in the field of quantum gases. In weakly interactingsamples, such as a typical BEC in three dimensions, they lead to the formation of a meanfield. Strong interactions, as present in e.g. optical lattice gases or low-dimensional systems,are responsible for interesting particle correlations. The diluteness of ultracold atom samplesallows us to describe basic interaction properties via two-body scattering.

As discussed in many textbooks (see e.g. [Sak94] for reference), the scattering process oftwo quantum particles, say two Cs atoms, with relative momentum ~k and mass m, that in-teract via some potential V (r), is conveniently described by expanding the wave function ofthe relative motion into a series of partial waves each carrying quantized angular momen-

8

Introduction

tum l = 0, 1, 2, ... . The effect of the potential reduces, at distances much larger than its range,to an energy-dependent phase shift δl(k) for each partial wave individually 1. For bosons,only the lowest angular momentum wave, the s-wave, contributes in the regime of ultralowtemperatures. The scattering process can then be characterized by a single parameter, thes-wave scattering length

as = − limk→0

tan δ0(k)k

. (1.1)

In general, the precise form of the scattering potential can be very complicated. The conceptof the scattering length is thus of crucial importance when including the role of interactionsin various contexts. Its definition can be more intuitively understood as the position of thelast node of the interpolated scattering wave function outside the range of V (r). This nodecan be located at negative or positive r, giving a negative or positive scattering length. Avalue as < 0 signifies attractive interaction. Vice versa, a value as > 0 signifies repulsiveinteraction.

It is instructive to imagine a situation, in which V (r) gives rise to a large and negativevalue of as. Suppose, the depth of the potential was increased, thereby causing a strongercurvature of the wave function in the scattering region. Finally, the interpolated node thatdetermines as flips from a large negative to a large positive value. The divergence and signchange of as is related to the emergence of a bound state just below the scattering thresholdfor large and positive as. For as much larger than the range of V (r), the bound state energyuniversally relates to the scattering length via

Eb = − ~2

ma2s

. (1.2)

These bound states just below threshold extend far beyond the classically allowed regionand are thus termed universal halo states [Jen04]. Halo states have been first studied in atomicnuclei. A well-known example in molecular physics is the helium dimer [Luo93]. In recentyears, more sophisticated universal few-body phenomena at large scattering lengths [Bra06]have become experimentally accessible in quantum gases, including the long-predicted Efi-mov trimer [Kra06, Kno09].

Feshbach resonances

With the above reasoning, the scattering length is fixed by the exact shape of V (r). Yet, itbecame clear that bound states close to threshold are of major importance for the strength ofas. This observation is at the heart of a Feshbach resonance, first studied in the context of nu-clear physics [Fes58, Fes62], and most recently employed for interaction control in quantumgases (see [Chi10] for a review). In samples of ultracold atoms, a Feshbach resonance re-

1Note that a spherically symmetric potential V (r) cannot couple partial waves of different l.

9

Introduction

HaL

atomic separation r

ener

gyclosed channel

open channel0

-2 -1 0 1 2

-2

0

2

4

HB-B0LD

ener

gya s

abg

HbL

Eb

Figure 1.1: (a) Sketch of the two-channel model for a Feshbach resonance. The scattering state oftwo atoms colliding at ultralow temperature in the open channel resonantly couples to a molecularbound state of the closed channel. The energy difference, depicted by the black arrows, is tuned viaan external magnetic field. (b) Scattering length (top) and bound state energy (bottom) as a functionof magnetic field near the Feshbach resonance. The gray shaded area indicates the regime of theuniversal bound state, eq. 1.2. Figure adapted from [Chi10].

quires two molecular potentials, which naturally arise from the spin degrees of freedom, e.g.the atomic hyperfine structure. For large internuclear distances, the two potentials asymp-totically connect to a pair of free atoms. Two initially separated atoms approach each other inthe lower lying entrance or open channel. In the ultracold regime, close-to-zero-energy scatter-ing ensures that the energetically higher closed channel is not accessible from the two-atomthreshold. A Feshbach resonance in as occurs, very similar to the situation discussed above,when the energy of a bound molecular state supported by the closed channel approaches thescattering state at threshold in the open channel (Fig. 1.1(a)). A weak coupling between thetwo states leads to mixing and a divergence in as. Tuning of the energy difference betweenthe two channels is typically realized via an external magnetic field, making use of the dif-ference in magnetic moment of the bound and the free atom states and their correspondingZeeman energy shifts.

The dependence of as on the magnetic field B can be parametrized by the simple expres-sion

as(B) = abg

(1− ∆

B −B0

). (1.3)

Here, abg is the background scattering length, associated with the potential of the entrancechannel, B0 denotes the resonance position, and ∆ is the resonance width (Fig. 1.1(b)). Animportant feature of a Feshbach resonance is the zero-crossing of the scattering length atB = B0 + ∆, which allows for the experimental preparation of non-interacting samples.The width of the resonance depends on the coupling strength between the bound and thetwo atom scattering state as well as their difference in magnetic moment, and can vary over

10

Introduction

orders of magnitude typically in the range of 10−6 to hundreds of Gauss.

The coupling bends the molecular state in the vicinity of the resonance and connectsit adiabatically to the scattering state. For large and positive as, the binding energy of themolecule obeys equation 1.2. The extent of this universal regime depends on the character ofthe resonance, which can vary from so-called open- to closed-channel dominated. For an open-channel dominated resonance, the bound state is universal over a large fraction of ∆, whilethis is not the case for a closed-channel dominated resonance, for which the universality ofthe bound state only extends over a small fraction of ∆.

In an experiment, the coupling between the two channels allows for the adiabatic associa-tion of two free atoms into the weakly bound molecular state [Koh06]. For bosons, such Fesh-bach molecules are typically unstable due to collisional quenching into more deeply boundstates. Optical lattice potentials allow for the preparation of long-lived Feshbach molecules,associated from an ensemble of atom pairs that are isolated from each other and located inthe minima of the lattice structure [Tha06].

The first Feshbach resonance in the context of ultracold atomic gases was observed in asodium BEC [Ino98], quickly followed by their studies in other alkalies, such as rubidium-85[Cou98, Rob98] or cesium [Vul99]. Feshbach resonances can be detected in three-dimensionaltrapped samples via the enhanced elastic collision cross section in the vicinity of the reso-nance position, or its reduction at the zero crossing. The strength of elastic collisions canbe inferred from measurements of thermalization rates, evaporation loss, or the mean fieldenergy in a BEC. Apart from elastic scattering, also inelastic processes are modified. For sam-ples prepared in the energetically lowest spin state, three-body recombination is the dom-inant loss channel. In the universal regime, the three-body loss coefficient L3 ∝ a4

s. Thisscaling can be largely modified by few-body quantum effects, such as the Efimov trimer,first found in cesium [Kra06].

1.1.2 Scattering properties of cesium

Precise experimental and theoretical investigations of the unusual Cs collision propertiesand the rich spectrum of weakly bound dimers have been of particular importance for col-lision shifts in cold atom-based atomic clocks [Gib93, San02], the successful realization of aBEC [Web02], or the preparation of ultracold molecular samples [Her03]. Pioneering work,combining experimental and theoretical effort [Chi04b], now sets the stage for experimentalstudies of quantum few-body effects [Kra06, Fer09, Hua14] and its universal aspects [Ber11],scattering and strong particle correlations in reduced dimensions [Hal09, Hal10a], or preci-sion measurements on many-body states in an optical lattice [Mar11a, Mar12].

The starting point for all experiments reported in this work is a BEC of Cs atoms preparedin the internal hyperfine ground state |F = 3,mF = 3〉. The interaction potential betweentwo such atoms consists of the strong electronic Born-Oppenheimer potential (dominated by

11

Introduction

-200 0 200 400 600 800

-1000

0

1000

2000

3000

4000

5000

magnetic field B HGaussL

scat

terin

gle

ngth

a sHa

0L

HaL

5 15 25 35 45 55

-2000

-1000

0

1000

2000


scat

terin

gle

ngth

a sHa

0L

HbL

-200 0 200 400 600 800-1000

-800

-600

-400

-200

0


bind

ing

ener

gyE

bh

HMH

zL

-1H33L6sH6L-2H33L6sH6L

-3H33L6sH6L

-7H

44L8

sH6L

-7H

44L6

sH6L

-6H34

L7sH6L

-6H34

L6sH6L

HcL

5 15 25 35 45 5500-10

00-8

00-6

00-4

00-2

000


bind

ing

ener

gyE

bh

HMH

zL

-1H33L6sH6L

-2H33L6gH6L

-2H

33L4

dH4L

-x2

gH2L

-2H

33L4

gH3L

-2H

33L4

gH4L

6lH3

L

6lH4L 6l

H5L

HdL

Figure 1.2: Scattering length (a,b) and energy levels (c,d) versus magnetic field for the lowest energyspin channel (F = 3,mF = 3) in Cs. The large magnetic field range data in (a) and (c) are calculatedincluding s-wave molecular states only. The data in (b) and (d) show details of the low field region.In (b), the smooth transition of as is due to the s-wave Feshbach resonance located at ∼ −12 G. Tworesonances around 48 G and 53 G arise from coupling to the 4d(4) and 2g(2) states. Additional narrowg-wave resonances at∼ 11, 14.3, 15, and 19.9 G are not shown. Their positions are marked by the bluearrows. In (d), the molecular states corresponding to the resonances at ∼ 11 and 15 G as well as threefurther g-wave states in the field region below 10 G are not plotted. The n(F1F2) quantum numbersfor the l-wave states are −3(33). The x2g(2) state is of mixed character. Relativistic interactions areomitted for the calculation of the bound state energies. Including them results in narrow avoidedcrossings between states of different L or mF . Figure adapted from [Mar07, Ber13].

electron exchange at short distances and by the van-der-Waals interaction at long distances)and a weaker relativistic spin-dependent interaction part (spin dipole-dipole and second-order spin-orbit interaction). The relativistic terms are of particular importance in Cs leadingto an exceptionally rich structure of Feshbach resonances.

The reason can be seen as follows. In the regime of very weak binding energies it is rea-sonable to label the bare molecular states n(F1F2)fL(mf ). Here, n denotes the vibrational2

and L the rotational state. The spin configuration is conveniently expressed by the atomichyperfine states F1 and F2 to which the molecular state asymptotically connects at largeinternuclear separation. f is the resultant of F1 and F2 and mf = mF1 + mF2 the corre-sponding projection. Note that for s-wave scattering, parity conservation restricts couplingonly to molecular states with L = 0, 2, 4, ..., which are labeled s-, d-, g-wave states, respec-tively. Accordingly, Feshbach resonances associated with these states are called s-, d-, g-waveresonances. For a moment we may neglect the relativistic interactions. Considering s-wave

2The vibrational states are counted from the topmost level downwards starting from n = −1,−2, ... .

12

Introduction

scattering only and recalling that the Cs ground state has zero orbital angular momentum,the isotropic Born-Oppenheimer potential can only couple to molecular levels with L = 0.Fig. 1.2(c) shows the bound state spectrum below threshold calculated only on the basis ofs-wave states, and restricted to states with the total spin projection mf = 6 of our entrancechannel. Fig. 1.2(a) depicts the corresponding calculation of the scattering length, exhibitingthree broad Feshbach resonances that can be traced back to the coupling with the molecu-lar states −7(44)6s(6), −6(34)7s(6), and −6(34)6s(6). Note that the positions are affected byavoided level crossings with the state −1(33)6s(6). This state is so weakly bound, that itsenergy is not resolved on the scale in Fig. 1.2(c).

The state −7(44)6s(6) gives rise to a smooth transition of as in the low magnetic fieldregion from large negative to large positive values. Including now also the relativistic inter-action terms modifies the spectrum and gives rise to many additional Feshbach resonancesdue to coupling to d- and g-states. The scattering length and the energy levels of the weaklybound molecular states in the low magnetic field region are shown in Figs. 1.2(b) and (d).This is essentially the magnetic field range at which our experiment is designed to work[Gus08a], and that is exploited for interaction tuning and Feshbach molecule production.

1.2 Ultracold atoms in an optical lattice and the Hubbard model

The realization of spatially periodic potentials is essential for many approaches to mimiccondensed matter systems with ultracold atoms. Such potentials, termed optical lattices, aregenerated in the laboratory via standing wave light fields, that result from the interferencebetween multiple laser beams [Blo05]. When far off-resonant from any atomic transition, theoscillating electric field component induces a spatially dependent ac-stark shift of the atomicground state, leading to a conservative force pointing towards the minima or maxima of thefield intensity [Gri00]. In the simplest case, a single retro-reflected laser beam, generatesa periodic series of pancake-like 2D systems. Two such standing waves, intersecting at anangle of 90, result in an array of 1D tubes (Fig. 1.3(a)). Adding a further standing wave alongthe third dimension realizes a cubic lattice configuration. Via appropriate adjustment of thepotential depths along the three dimensions, lattice systems of reduced dimensionality arerealized (Fig. 1.3(b)). More complex geometries, such as double periodic structures [Fol07,Seb06], honeycomb [Tar12], triangular [Bec10], or Kagome lattices [Jo12] are accessible viadifferent beam configurations and phase control.

Optical lattices have been of widespread interest in the field of atomic physics long be-fore they were considered a cornerstone for the quantum simulation of strongly interact-ing many-body systems. For example, Kapitza-Dirac [Gou86] and Bragg scattering [Mar88]of atoms from a standing light wave was first demonstrated with mono-energetic atomicbeams. Tight spatial confinement provided by individual lattice sites made Raman sideband

13

Introduction

Figure 1.3: Illustration of optical lattice geometries used for the experiments presented in this thesis.(a) A pair of orthogonal retro-reflected laser beams generates an ensemble of 1D tube-like systems.(b) Three mutually orthogonal standing waves create a cubic lattice geometry. When the lattice depthalong the horizontal plane is sufficiently large, an array of decoupled vertically oriented 1D latticesystems is generated. The waist of the beams and the number and size of the illustrated tubes andlattice sites is not to scale.

cooling techniques [Die89, Mon95] available for neutral atoms [Ham98, Vul98].

Starting from a BEC, the optical lattice is typically populated by a slow adiabatic ramp-up of the light fields that form the potential [Hec02]. Depending on the specific setting, thisallows for a controlled preparation of the atomic ensemble close to its quantum mechanicalground state [Gre02a]. The slow lattice loading procedure, aiming for a minimal increase inentropy, is a prime example for adiabatic transfer from one many-body quantum state toanother. Specifically, it will allow us to reduce the theoretical description of the interactingbosons in the optical lattice to a single-band model, as will be outlined in the followingsections.

1.2.1 Quantum mechanics and dynamics in a periodic potential

Before discussing the description of strongly interacting bosons in an optical lattice via aHubbard model, we will start with basic properties of a single atom confined to a periodicpotential. In the more specific case of the lattice being formed by the standing wave of twocounter-propagating laser beams of wave vector kl = 2π/λ, the single-particle Hamiltonianreads3

H =p2

2m+ V0 cos2 (klz) . (1.4)

The lattice depth V0 is conveniently given in units of the recoil energy ER = ~2k2l /(2m)

associated with the absorption or emission of a photon from the lattice light field. Very muchas for electrons in a crystal [Blo29], the eigenstates of (1.4) are delocalized Bloch waves (seee.g. [Ash76]). They can be written as a product of a plane wave eiqz with quasimomentum

3For the purpose of this introduction, we will restrict the discussion to a single dimension.

14

Introduction

-1 0 10

2

4

6

8

10

12

quasimomentum q Hk lL

ener

gyE

qnHE

RL

HaL

-1 0 10

2

4

6

8

10

12

quasimomentum q Hk lL

ener

gyE

qnHE

RL

HbL

Figure 1.4: Band structure for a cos2 optical lattice potential with lattice depth V0 = 2ER (a) andV0 = 7ER (b). The energy of the Bloch eigenstates is plotted as a function of the quasimomentum.The black wave packets illustrate the process of Landau-Zener tunneling into the next higher Blochband at the edge of Brillouin zone in the shallow potential (a), and Bloch oscillation dynamics withinthe lowest band in the deep potential (b).

q and a spatially periodic function unq (z) = unq (z + d), that reflects the lattice period d =λ/2. Further, the lattice periodicity allows for a discrete Fourier expansion of unq (z) withfrequency components given by integer multiples of 2kl. Specifically, the Bloch waves can bewritten in the form

φ(n)q (z) = eiqzunq (z) =

eiqz√2π

+∞∑j=−∞

cn,qj ei2klzj . (1.5)

A Bloch eigenstate of quasimomentum q therefore consists of a sum of plane waves withwave vectors q + j · 2kl, where j · 2kl are the reciprocal lattice vectors. Periodicity allows re-stricting q to the first Brillouin zone q ∈ [−kl, kl]. The coefficients cn,qj and the correspondingenergies Enq of the Bloch states are found by inserting (1.5) into the stationary Schrodingerequation, and numerically diagonalizing the resulting linear eigenvalue problem in the co-efficients cn,qj . The obtained eigenvalues can be grouped in a set of energy bands labeled bythe index n that are separated by forbidden band gaps (Fig. 1.4).

Already on the single-particle level, the band structure gives rise to interesting and some-times counterintuitive effects that can be beautifully probed in quantum gas experiments. Aprominent paradigm comprises coherent dynamics in the lattice under an external force act-ing on the atoms. Contrary to free space, where the force results in a simple acceleratedmotion, the evolution in the lattice is time-periodic in position and momentum space. SuchBloch oscillations [Blo29, Zen34], can be most intuitively understood by considering an ini-tially localized wave packet in momentum space that is accelerated by the force term andthereby driven through the Brillouin zone. Once the wave packet reaches the zone edge,Bragg scattering by the lattice potential changes its momentum state by one reciprocal lattice

15

Introduction

vector from +kl to −kl. This process, repeated periodically, results in the oscillatory dynam-ics. The observation of Bloch oscillations in cold gases [Dah96] is a wonderful demonstrationof coherence phenomena that are difficult to probe in solid state systems [Fel92, Was93].

While Bloch oscillations as discussed above are a pure single-band effect, inter-band dy-namics can be probed when the force is sufficiently strong compared to the band gap. In thatcase, Landau-Zener tunneling at the edge of the Brillouin zone into the next higher Blochband is favored [And98, Zen09]. The process serves as a sensitive probe for the band gapand has been used more recently to detect closing gaps, as they appear at so-called ”Diracpoints” in graphene-type lattice structures [Tar12]. Finally, resonant tunneling into excitedstates of the periodic lattice is another interesting example for inter-band dynamics of theatomic ensemble subject to a strong force [Sia07].

Experiments focusing on single-particle phenomena in the lattice typically work in pa-rameter regimes where interactions are weak. For example, this is the case when the latticeis generated from a single standing wave, forming a series of pancake like 2D systems. Al-though the BEC loaded into such a potential is a three dimensional object, it is possible toreduce the problem to an effective 1D Gross-Pitaevskii wave equation describing the axialdynamics along the lattice [Sal02]. In this approximation, one neglects particle motion trans-verse to the lattice axis and contracts the transversal density profile into a single parameter.Interactions are incorporated via a mean field, which introduces a nonlinear term. Goingbeyond single-particle dynamics, the BEC in the optical lattice thus serves as an excellentsystem for investigating the effects of the nonlinearity, in particular, when interactions canbe tuned via Feshbach resonances (see [Mor06] for a review).

Among others, experimental studies have demonstrated the role of the mean field forLandau-Zener tunneling dynamics in shallow potentials [Mor01], the interaction-induceddephasing of Bloch oscillations [Gus10] and its absence in non-interacting BECs [Gus08b,Fat08] and non-interacting degenerate Fermi gases [Roa04]. The nonlinear term in the Gross-Pitaevskii equation can, in some cases, cause an exponential growths of a weak deviationfrom the stationary solution. This effect is called dynamical instability, and is predicted forBloch states close to the edge of the Brillouin zone in the absence of a force [Wu01, Wu03].The observed decay of the condensate wave function in a moving optical lattice has beeninterpreted in terms of a dynamical instability arising above a ciritical quasimomentum[Fal04, Sar05].

This short introduction to the statics and dynamics of weakly interacting bosons in anoptical lattice provides a first glimpse how relatively complex behavior can arise already onthe mean-field level. The experiments presented in this thesis leave the realm of mean-fielddynamics and consider regimes of strong interactions. Here, particle correlations becomeutterly important and the ensuing dynamics need to be treated on the microscopic level ofthe individual bosons. For sufficiently low temperature the system is described in terms of

16

Introduction

the Bose-Hubbard Hamiltonian, as will be discussed in the next section.

1.2.2 Strongly correlated many-body system: The Bose-Hubbard model

Experimentally, the strongly interacting regime can be reached with e.g. a cubic optical lat-tice, formed via three retro-reflected and mutually orthogonal laser beams. Typical spacingsbetween the lattice sites (∼ 0.5µm) are comparable to the mean interatomic distance in thetrapped BEC, which leads to an average number of about 1 to . 3 bosons per site, dependingon the trapping parameters. In this system, it is the direct competition between interactionand kinetic energy (here given by the tunnel coupling among neighboring sites) that causesstrong particle correlations and highly suppressed fluctuations of the atom number in eachwell of the potential.

For a theoretical description of the strongly interacting many-body system, it is conve-nient to switch the representation from the single-particle eigenstates, given by the delocalizedBloch waves, to a basis set of localized wave functions. Those are the so-called Wannier func-tions [Koh59], which in 1D can be written as

wn(z − zi) =

√d

2π

∫ +π/d

−π/ddq unq (z)e−iqzi . (1.6)

For each band, the Wannier functions are constructed from a superposition of all Bloch states.They form a complete orthogonal basis set of exponentially localized wave functions cen-tered in the ith potential minimum of the lattice, which makes them useful for describinglocal interactions between particles.

For ultracold bosons that interact via s-wave scattering, the precise form of the inter-atomic potential can be replaced by a contact-interaction potential [Pet02]

U(x) =4π~2asm

δ(x) , (1.7)

where the strength of interaction g = 4π~2as/m, and δ denotes the Dirac delta function. Inthe presence of an external potential V (x), the full second-quantized Hamiltonian reads

H =∫dx Ψ†(x)

(− ~2

2m∇2 + V (x)

)Ψ(x) +

g

2

∫dx Ψ†(x)Ψ†(x)Ψ(x)Ψ(x), (1.8)

where Ψ(x) is the bosonic field operator. The first term quantifies the kinetic energy in thelattice (quantum tunneling), while the second term accounts for particle interactions. In or-der to switch to a Fock state representation, one introduces the bosonic creation (annihila-tion) operators a†i (ai) for a particle on the ith lattice site via expanding the field operatorsin terms of Wannier functions, Ψ(x) =

∑i,nwn(x− xi) an,i. When the external potential is a

3D cubic optical lattice, the Wannier function is a product wn(x) = wnx(x)wny(y)wnz(z), and

17

Introduction

the three-dimensional problem can be separated. The Hamiltonian (1.8) can be drasticallysimplified under the following approximations. First, we restrict the full model to the lowestBloch band n = 0. Second, for the kinetic energy, we only keep terms that include Wannierfunctions located in neighboring lattice sites (tight-binding approximation). Third, for the in-teraction term, we neglect all types of off-site interaction processes and only keep on-siteinteractions. This results in the celebrated Bose-Hubbard Hamiltonian [Fis89, Jak98]

H = −J∑〈i,j〉

a†i aj +∑i

U

2ni (ni − 1) +

∑i

εini . (1.9)

The first term describes tunneling between neighboring lattice sites with a rate J/~, and 〈..〉denotes summation of nearest neighbors only. The second term quantifies the total interac-tion energy, with U representing the on-site interaction energy of a pair of bosons sittingon the same lattice site, and ni = a†i ai counts the number of bosons on the ith lattice site.Finally, εi can account for a site-dependent energy offset due to an external potential. In theexperiment, this could be e.g. a weak harmonic confinement produced by the radial profileof the Gaussian laser beams or an applied external force.

For the purpose of most of the experiments presented in this work, we focus on thesituation where the tunneling rate along one dimension is much larger than along the othertwo dimensions, say Jz Jy, Jx (Fig. 1.3(b)). In that case, the site labels in eq. (1.9) arereduced to a single dimension only, and the parameters of this 1D Bose-Hubbard model arederived via the integrals

J = −∫dz w0(z)

(− ~2

2m∇2 + V0 sin2 (klz)

)w0(z − d) , (1.10)

U = g

∫dx |w0(x)|4 . (1.11)

In the experiment, the approximations used to derive the Hubbard model require thattemperature T and on-site interaction energy U are much less than the gap to the next higherBloch band. Corrections due to strong interactions are discussed in section 1.2.4.

Historically, the Hubbard Hamiltonian was first introduced in its fermionic form for de-scribing correlation phenomena of electrons in solids [Hub63]. Much later, the bosonic ver-sion was studied [Fis89], motivated largely by experiments on 4He absorbed in porous me-dia. The Bose-Hubbard model features a quantum phase transition from a superfluid (SF) toa Mott insulator (MI) with increasing value U/J . While the superfluid is characterized by adelocalized, phase-coherent matter wave, in the insulating phase, the bosons are exponen-tially localized and the excitation spectrum becomes gapped. A mean-field approximationpredicts the phase transition point at a critical value (U/J)c = 5.8z for average site occupa-tion 〈n〉 = 1. Here, z denotes the number of nearest neighbors, which depends on the system

18

Introduction

dimensionality. However, in low-dimensional systems, the deviations from the mean-fieldanalysis are large. Precise studies, using DMRG or Quantum Monte Carlo simulations, lo-cate the phase boundary at (U/J)c ≈ 3.4 in 1D [Kuh00], (U/J)c ≈ 16.7 in 2D [Cap08],and (U/J)c ≈ 29.3 in 3D [Cap07] for 〈n〉 = 1. In 1998, Jaksch et al. realized the connectionbetween a BEC loaded into a cubic optical lattice and the Bose-Hubbard Hamiltonian, asoutlined above [Jak98]. The authors predicted that the strongly correlated insulating regimecan be reached via adiabatically increasing the depth of the optical lattice superimposed onthe trapped atomic cloud.

Following this proposal led to the experimental demonstration of the SF-MI quantumphase transition in the context of ultracold atoms in optical lattices [Gre02a], which has setthe stage for the simulation of condensed matter systems with modern atomic physics tools.In the experiment, the transition is evidenced via a reversible loss of phase coherence, de-tected after time-of-flight in momentum space, together with the opening of an excitationgap for increasing U/J . In contrast to the situation in a widely homogenous solid, the pres-ence of weak harmonic trapping leads to the formation of large concentric Mott insulatingregions, where the number of atoms per site is fixed to integer values with highly suppressedfluctuations. The number and extent of these ”Mott-shells” is controlled via the strength ofthe external trap. The characteristic shell structure has been detected using spatially selec-tive mircowave transitions [Fol06], and more recently via single-site-resolved fluorescenceimaging in a ”quantum gas microscope” [Bak10, She10].

Parallel progress with fermionic samples led to the experimental realization of the Fermi-Hubbard model [Ess10]. Here, fermionic statistics together with spin degrees of freedomenrich the phase diagram at low temperature [Hof02]. Metallic, band insulating, and Mottinsulating phases have so far been observed [Koh05, Sch08, Jor08], and the quest for prepar-ing low enough entropy states necessary to detect magnetic ordering seems to be withinreach [Gre13]. Mixtures of bosonic and fermionic ensembles have also been prepared in theHubbard regime [Gun06, Osp06, Bes09], and extend the field with prospects to study moreexotic quantum phases.

1.2.3 Out-of-equilibrium dynamics in the Bose-Hubbard model

Building on successful demonstrations of strongly correlated quantum phases with ultracoldatoms, recent attention has been devoted to use them for the investigation of phenomena ininteracting many-body systems that take place far from equilibrium [Pol11, Eis15]. This in-terest in out-of-equilibrium studies is based on a few unique properties in cold atom settings[Lan15a]. To begin with, the typical energy scales dictating statics and dynamics in quan-tum gases correspond to timescales in the milli-second range. This provides comparativelyeasy time-resolved experiments. Note the need for ultrafast (sub-)femtosecond pump-probespectroscopy to study transient dynamics in strongly correlated phases of solids [Kra09b].

19

Introduction

Moreover, practically perfect isolation from the environment allows us to track essentiallyunitary time evolution in a closed interacting many-body system. Finally, system parameterscan be precisely controlled and changed in time.

The latter provides an essential handle to take the system out-of-equilibrium, via a so-called ”quantum quench”. In this scenario, the ensemble is first prepared close to the many-body ground state (or alternatively in some excited metastable state) of the underlyingmodel Hamiltonian for a specific set of system parameters. Then one or more parametersare quickly changed in time, such as the lattice depth, the strength of interaction, or the trap-ping potential. This parameter change initiates the dynamics, and the system subsequentlyundergoes unitary time evolution governed by the new Hamiltonian.

One of the central questions in this context addresses the equilibration and possible ther-malization dynamics of closed quantum systems. For open systems, coupling to an externalreservoir provides ways to reach thermal equilibrium, essentially forming the basis for a sta-tistical description. Yet, how a closed quantum system equilibrates, even though the many-body wave function obeys unitary evolution, is still an unresolved question. A first impor-tant insight is, that despite the unitary evolution of the entire system, the expectation valuesof many observables, and specifically local ones, tend to equilibrate [Cra08a, Cra08b, Kol07].Such an equilibration process has been investigated recently in an experiment, starting witha density-wave ordered state prepared in a deep optical lattice. After a sudden quench of thelattice depth, the atoms were allowed to tunnel, and rapid relaxation of the local site occupa-tions was observed [Tro12]. Under which general circumstances equilibrium values can bedescribed in terms of a thermal state, and which generic mechanisms lead to thermalization,is actively debated. For example, one prominent formulation is based on the so-called eigen-state thermalization hypothesis (ETH), which conjectures that thermal properties are alreadypresent in the initial state of the evolution on the level of individual many-body eigenstates[Rig08, Deu91, Sre94].

Not all quantum systems are expected to thermalize. A notable example are (almost) inte-grable models, such as the experimentally realizable one-dimensional Bose gas with contactinteraction (see chapter 1.3). Indeed, the absence of discernible equilibration in the collisiondynamics of two opposite momentum states of 1D bosons has been demonstrated in a hall-mark experiment, known as the ”quantum cradle” [Kin06]. Still, integrable systems mayequilibrate to a state described by a so-called generalized Gibbs ensemble (GGE). For exam-ple, mostly numerical studies have evidenced a breakdown of the ETH in integrable systems(e.g. [Rig08, Rig09]), and dynamical relaxation towards a GGE (e.g. [Rig07, Cra08a, Cas11]).A sufficiently strong breaking of integrability, in turn, appears to be related to the validityof the ETH [Pol11]. Further, a close relation between breaking of integrability and the emer-gence of quantum chaotic behavior in the level statistics of the many-body eigenstates canbe drawn for Hubbard-like models, indicated by a crossover from Poisson to Wigner-Dyson

20

Introduction

spectral statistics [Poi93].

Assuming some equilibrium state after a quench, a direct and obvious question is howthis steady state is essentially reached. The potential complexity of such transients in a many-body quantum system has been experimentally demonstrated with a single 1D Bose gas ina versatile microchip-based magnetic trap. Here, the observed coherence dynamics after asplitting of the gas into two halves revealed a two stage process [Gri12]. An initial rapid re-laxation towards a metastable so-called pre-thermalized state was identified, followed bya further slower evolution. Recently, it was demonstrated that the pre-thermalized stateemerges locally via a light-cone like spreading of correlations across the system [Lan13],and that it can indeed be described in terms of a GGE [Lan15b]. Light-cone like dynam-ics have also been observed in the context of the Bose-Hubbard model [Che12]. Startingthe experiment deep in the Mott insulating phase, the system was taken out-of-equilibriumvia a quench in the lattice depth closer to the phase transition point, and the correlationdynamics was read out utilizing a quantum gas microscope. While for the continuum 1DBose gas, the spreading of correlations can be understood from the propagation of phononicmodes [Gei14], the appropriate quasi-particles that carry information in the strongly corre-lated phase of the Hubbard model are entangled doublon-hole pairs [Che12].

Apart from studying correlation dynamics in view of quasi-particle propagation, ultra-cold atoms prepared in the Hubbard regime also provide means to address questions ofparticle transport in a many-body lattice system [Sch12, Hal10b]. In a recent experiment, theexpansion dynamics of an initially trapped Mott insulating sample has been investigated af-ter quickly removing the external trap [Ron13]. Particularly, the study revealed the influenceof integrability on the transport process. In 1D, the Bose-Hubbard model is integrable in thenon-interacting limit (U = 0) and in the hard-core boson limit (U/J → ∞), showing up inballistic expansion. Away from integrability, diffusive dynamics was observed.

As a closing example, ultracold atoms offer the potential to combine the investigationof quantum phase transitions with out-of-equilibrium dynamics. More specifically, interest-ing dynamics are expected from quenches on or across a phase transition point due to thechange of the ground-state behavior [Pol11]. The latter has been experimentally investigatedin the context of the Hubbard model. A quench from the superfluid into the Mott insulatingregime by a rapid change of the lattice depth was shown to result in coherent quantum phaserevivals of the matter-wave field, caused by its quantized structure in combination with in-teractions [Gre02b, Wil10, Kol07]. In a different setting, a quench through a quantum phasetransition to a ferromagnetic state was realized in a spinor Bose-Einstein condensate. Thislead to the dynamic formation of ferromagnetic domains [Sad06]. A particularly instruc-tive scenario are slow, near-adiabatic ramps through a quantum phase transition point, forwhich universal scaling laws can be derived, describing e.g. the formation of defects [Dzi10].This issue has been addressed in a recent experiment, that investigated the emergence of co-

21

Introduction

herence in the Bose-Hubbard model when ramping the system from the Mott insulating intothe superfluid regime [Bra15]. Interestingly, the observed dynamics appear more elusive andquestion the capability of simple universal scaling laws to capture the full behavior.

The above discussion is intended to provide a brief glimpse on the recent very diverseand active debate on non-equilibrium dynamics in closed many-body quantum systems,particularly seen from an experimental perspective, and with a slight bias on studies in thecontext of the Bose-Hubbard model. By no means is it possible to capture all aspects anddirections of this exciting and rapidly growing field (see [Pol11, Eis15, Lan15a] for recentreviews). The discussion yet demonstrates how quantum gases have brought a topic of pre-viously mostly academic interest into the laboratory.

In chapter 2 to 4 of this thesis, we investigate out-of-equilibrium dynamics in the 1DBose-Hubbard model subject to an additional force in various different contexts. The forceresults in a linear potential gradient, and thereby tilts the lattice. The experiments presentedin chapter 2 address quench dynamics deep in the Mott insulating phase with one-atom-per-site filling. We focus on the situation where the tilt generates a site-to-site energy offset E,that compensates for the on-site interaction energy 4. In that case, resonant tunnel couplingbetween atoms in neighboring sites ensues, provided that the atom located in the target sitehas not itself tunneled (Fig. 1.5(a)). These constrained dynamics map onto an effective Isingspin chain. The spin-spin interaction originates from the conditional tunneling process thatdepends on the occupation of the neighboring lattice site [Sac02, Sim11]. Since the map-ping exploits directly the motional degrees of freedom the dynamical timescale is set by thetunneling rate J . This is quite remarkable, given that standard nearest-neighbor spin-spininteraction in lattice systems with local on-site interaction is mediated by the much weakersecond-order super-exchange scaling as J2/U [Fol07, Tro08]. After investigating the quan-tum phase transition from a paramagnet to an anti-ferromagnet featured by the effectivespin chain (Fig. 1.5(b)), we study a quantum quench to the vicinity of the phase transitionpoint of the Ising model. The quench induces coherent nearest-neighbor tunneling dynam-ics in the many-body system that correspond to dynamics in the orientation of the effectiveIsing spins.

In chapter 3, many-body dynamics driven by long-range tunneling processes extendingover up to five lattice sites is investigated. When introducing the Hubbard model, it has beenpointed out that the matrix elements for direct tunneling between sites separated by morethan one lattice spacing can be safely neglected. Yet, in the tilted Mott insulator resonanttunneling over multiple sites is still possible. Such processes arise from higher-order termsin perturbation theory in J/U . For an example, consider a value of the tilt E ≈ U/2. In thatsituation, tunneling over two lattice sites is resonant, provided the atom on the target site

4Note the close analogy with the presence of an electric field in the context of solid state physics. For typicalsolid state Mott insulators, achievable field strengths well satisfy E U, J . In atomic lattice systems, however,one easily achieves situations where E ∼ U [Sac02].

22

Introduction

(b)

(d)

(a)

U / 2

J J

=

=

=

spin mapping

E≈U (c) EU increasing E paramagnet

an,‐ ferromagnet

condi,onal tunneling

2nd‐order tunneling

higher‐order tunneling

E=U/3 E=U/4

E=U/2

Figure 1.5: Illustration of resonant tunneling processes in the tilted 1D Mott insulator. (a) Tunnelingbetween nearest-neighbor sites is inhibited, unless the energy cost for a particle to tunnel onto theneighboring site vanishes. This happens when the site-to-site energy offset E ≈ U , and the neigh-boring atom has not tunneled itself, realizing strongly constraint dynamics. (b) For E U ,a density-wave of doubly occupied sites forms. The mapping to the Ising spin chain is depicted inthe lower left corner, allowing to realize a quantum phase transition from a paramagnet to an anti-ferromagnet. (c) Second-order tunneling between next-nearest-neighbor sites is resonantly enhancedfor E ≈ U/2. The process involves an intermediate virtual state that is offset in energy. (d) Long-range higher-order tunneling processes emerge for smaller values of the tilt, e.g. E ≈ U/3 (left) andE ≈ U/4 (right).

has not tunneled itself. The latter implies a similar constraint on the tunneling dynamics asfor the effective Ising chain scenario at E ≈ U . The process itself is driven by second-ordertunneling involving intermediate virtual states, e.g. a state where the site in the middle isdoubly occupied (see Fig. 1.5(c) for an illustration in a reduced Mott chain of three sites).Its characteristic rate scales as J2/U . Importantly, second-order tunneling is at the heart ofthe above mentioned super-exchange interaction that builds the basis for quantum magneticordering [And50]. More generally, when the tilt of the Mott insulator is an integer fraction ofthe on-site interaction energy E ≈ U/n, nth-order tunneling over a distance of n lattice sitesis resonantly enhanced (Fig. 1.5(d)). A quantum quench to these values of the tilt therebyallows us to isolate processes that emerge from a specific order in perturbation theory andinvestigate many-body dynamics driven by such terms.

Finally, in chapter 4, we study the response of a 1D superfluid that we suddenly quenchinto a tilted configuration. For a non-interacting gas, we observe simple Bloch oscillations.In the presence of interactions and for a sufficiently strong tilt, the quench induces Blochoscillations that undergo a periodic series of quantum coherent decay and revival. This orig-inates from a combination of atom-atom collisions and the discreteness of the site-occupationin the lattice, and has been predicted in [Kol03a]. Moreover, when we reduce the strength ofthe tilt a transition to an irreversible relaxation of the Bloch dynamics is identified. This qual-itative change comes along with the emergence of quantum chaos in the energy spectrum

23

Introduction

of the interacting many-body system [Buc03]. Our observations thus provide evidence forthe transition to quantum chaotic dynamics, which connects back to one of the central openquestions posed above, on the mechanisms for relaxation and potential thermalization inclosed interacting quantum systems.

1.2.4 Extended Hubbard model and density-induced tunneling

The Hubbard Hamiltonian for Bose and Fermi systems has proven extremely successful indescribing quantum gases in optical lattices as well as electrons in solids. However, as dis-cussed above, deriving its relatively simple form is based on a set of approximations (seesection 1.2.2). This raises the immediate question on the range of validity of the standardHubbard model, and the role of processes that are typically neglected. The need to answerthis question with experiments based on cold atoms can be seen from two perspectives. Onthe one hand, using quantum gases in optical lattices as truly reliable simulators for fun-damental problems in condensed matter physics requires a detailed understanding in whatparameter range a one-to-one mapping is applicable. On the other hand, the ability to probeprocesses that are typically neglected may in turn provide insight for their role in solids. Bothaspects motivate the quest for the study of extended, non-standard versions of the Hubbardmodel that include additional relevant terms (see [Dut15] for a review).

Occupation-dependent on-site interaction

For the purpose of this thesis, let us restrict the further discussion to the Bose-Hubbardmodel, eq. (1.9). The first approximation used to derive the Hamiltonian, the restriction tothe lowest Bloch band, was justified by assuming that the on-site interaction energy U issmall compared to the band gap [Jak98]. When this criterion is not sufficiently well satisfied,the value for U needs to be corrected and becomes occupation dependent due to couplingto higher bands [Bus98, Joh09, Sch09, Buc10]. Note, that off-site scattering processes can stillbe neglected for the calculation of the interaction energy. Intuitively, strong repulsive (at-tractive) interaction between particles on the same lattice site broadens (narrows) the on-sitewave function compared to the lowest band single-particle Wannier function. This modifica-tion can be described by an admixture from higher bands and leads to a multi-orbital on-siteinteraction energy U(n), that depends explicitly on the number of atoms n on the lattice site.Remarkably, it is possible to keep most of the simplicity of the lowest band Hubbard model,but now taking into account the occupation-dependent on-site interaction.

A first sign for higher band admixture has been found in precision microwave spec-troscopy of a rubidium Mott insulator [Cam06], followed by an observation of number de-pendent on-site interactions in the matter-wave dynamics after a quench of the lattice depth[Wil10]. However, the lack of tunable interactions in these experiments did not allow for

24

Introduction

a detailed investigation of the range of validity of the standard Hubbard model. The ener-gies of two- and three-body states in the lattice were measured later over a large range ofthe interaction strength from attractive to repulsive values, by exploiting the precise controlover the scattering length in Cs via a Feshbach resonance [Mar11a, Mar12]. The measuredtwo-body energy U(2) was found in agreement with numerical calculations, taking into ac-count all higher Bloch bands, and modeling explicitly the two-channels of the Feshbach res-onance [Buc10]. Another approach consistent with the data uses the analytic solution for twocontact-interacting bosons in a harmonic trap [Bus98], with an appropriate renormalizationthat accounts for the anharmonicity of the lattice wells [Sch09]. A field-theoretical approachfor the calculation of U(3) [Joh09] could describe the data only for moderate values of thescattering length, but failed for increasing interactions.

Off-site scattering and density-induced tunneling

While the modification of U due to higher bands has been investigated extensively in thepast, the experiments presented in chapter 2 and 5 report on the first direct measurements ofinteraction- and density-induced modifications of the single-particle tunneling rate J . This ismade possible by using the observed nearest-neighbor coherent tunneling dynamics in thetilted 1D Mott chains as a direct measure of the hopping rate, and exploiting the Feshbachtuning of interactions. In chapter 5, we demonstrate that the modified tunneling rate arisesfrom an off-site scattering process, that is neglected in the standard Bose-Hubbard model.This is the so-called density-induced tunneling [Luh12, Bis12]. At first glance, it seems surpris-ing that scattering of particles in neighboring lattice sites causes a correction to the physicaltunneling rate. However, when such a scattering process is accompanied by a hopping eventit can indeed do so.

Contrary to the modifications of U discussed above, density-induced tunneling consti-tutes a relevant extension to the standard Hubbard model within the lowest band. It is de-rived by writing the two-body interactions in eq. (1.8) in the basis of Wannier functions inthe lowest lattice band

Hint =12

∑ijkl

Uijkl a†i a†j akal . (1.12)

The matrix element Uijkl quantifies the corresponding (off-site) interaction terms and reads

Uijkl = g

∫dxw∗(xi)w∗(xj)w(xk)w(xl) . (1.13)

Here, w(xi) is the lowest-band Wannier function, eq. (1.6), localized at the ith lattice site.Taking into account on-site as well as nearest-neighbor scattering processes, the extended

25

Introduction

5 10 15 20 25

10-1

10-3

10-5

10-7

lattice depth HERL

ener

gyHE

RL

U

J

JBC

Jpair

V

HaL

Figure 1.6: (a) Parameters of the extended Bose-Hubbard model including nearest-neighbor scatter-ing processes. The value of the on-site interaction U , single-particle tunneling J , density-inducedtunneling JBC, correlated pair tunneling Jpair, and off-site density-density interaction V is plottedas a function of the lattice depth for an isotropic cubic lattice (Vx = Vy = Vz) at a scattering lengthas/d = 0.014 (as = 140a0 for d = λ/2 = 532 nm). (b) Illustration of the bond-charge interaction ordensity-induced tunneling. The off-site scattering is accompanied by a tunneling process and therebyaffects the total tunneling rate. (c) The nature of the density-induced tunneling can be understoodwithin an effective potential picture. The bare lattice potential V is modified by an interaction- anddensity-dependent term, and the total tunneling rate is derived from the resulting effective potentialVeff . Figure adapted from [Luh12].

Bose-Hubbard Hamiltonian takes the form

H = −J∑〈i,j〉

a†i aj − JBC

∑〈i,j〉

a†i (ni + nj)aj + Jpair

∑〈i,j〉

a†2i a2j +

∑i

U

2ni (ni − 1) + V

∑〈i,j〉

ninj .

(1.14)Here, U = Uiiii quantifies the familiar on-site interaction energy. Nearest-neighbor interac-tions lead to three additional terms compared to the standard Hubbard model. Those are thedensity-induced tunneling or bond-charge interaction with JBC = −Uiiij , the correlated pairtunneling with Jpair = Uiijj/2, and the off-site density-density interaction with V = Uijij .The matrix elements quantifying the strength of the individual terms in eq. (1.14) are plottedin Fig. 1.6(a) as a function of the lattice depth for a particular value of the scattering length.Correlated pair tunneling and off-site density-density interaction can be safely neglectedwhen compared to J and U . Density-induced tunneling, however, is small compared to Ubut can reach ∼ 10% of the single-particle tunneling J . Its effect on the SF-MI phase transi-tion point can become significant, in particular with increasing lattice site occupation num-ber [Luh12].

The interaction terms contributing to the density-induced tunneling (e.g. ∼ a†i a†j aj aj)

correspond to scattering that is accompanied by a hopping event (Fig. 1.6(b)). This allowscombining them with the single-particle tunneling to an overall density-dependent tunnel-

26

Introduction

ing operator.

Jeff = −J∑〈i,j〉

a†i aj − JBC

∑〈i,j〉

a†i (ni + nj)aj (1.15)

= −∑〈i,j〉

[J + JBC(ni + nj − 1)] a†i aj . (1.16)

As JBC ∝ as, the overall physical tunneling rate within the extended Bose-Hubbard modelthereby depends explicitly on the scattering length and the site occupancy. The effect of thebond-charge interaction on the overall tunneling rate can be illustrated via a modificationof the bare lattice potential V (x) [Luh12]. In this picture, the density operator in eq. (1.16) isidentified with an occupation-dependent density ρBC(x) = 〈ni〉|w(xi)|2 + (〈nj〉 − 1)|w(xj)|2

on the neighboring sites i and j excluding the hopping particle. The combined process ofbare single-particle and density-induced tunneling is then viewed as hopping in an effectivepotential V (x) + gρBC(x) with a total rate given by

Jeff = −∫dxw∗(xi)

(− ~2

2m∇2 + V (x) + gρBC(x)

)w(xj) . (1.17)

For repulsive interactions (g > 0), the effective tunneling barrier is reduced, which leads to ahigher overall tunneling rate due to density-induced tunneling, and vice versa for attractiveinteractions (see Fig. 1.6(c)).

The time-resolved study of coherent tunneling dynamics in the tilted 1D Mott chains(chapter 2) and the quantitative experimental and theoretical investigation of its interaction-and density-dependence (chapter 5) is the key to a direct observation of density-inducedtunneling. Exploiting these new techniques provided in experiments with ultracold atoms,we indeed investigate an extension to the Hubbard model that has been discussed theoreti-cally in the context of condensed matter physics long before. Already in his seminal paper,John Hubbard estimated the strength of bond-charge interactions for electrons in transitionmetals [Hub63], with the result that the term was neglected in the standard Hubbard Hamil-tonian. However, its strength may be larger in other materials. For example, for benzenethe bond-charge interaction is estimated to reach a sizable fraction of the on-site interac-tion energy [Par50, Str93]. More importantly, it provides an alternative to phonon-mediatedelectron-electron interaction in the context of high-Tc superconductivity, via its attractive na-ture for holes at the Fermi surface [Hir89, App93]. Further, its role on magnetic propertieshas been also discussed [Str94, Ama96].

27

Introduction

1.3 The one-dimensional Bose gas

In the previous chapter, we have seen how quantum many-body systems are realized in opti-cal lattices and described by the Bose-Hubbard model. Another approach to achieve stronglyinteracting quantum liquids comprises confinement to a one-dimensional geometry [Caz11].This is either established using single elongated magnetic traps [Hof07], or by loading thequantum gas adiabatically into an array of 1D tubes formed by two perpendicular retro-reflected laser beams (see Fig. 1.3(a)) [Gre01, Mor03]. The gas is said to be one-dimensionalwhen its temperature and chemical potential are much smaller than the energy gap to thefirst transversally excited quantum state (kBT, µ ~ω⊥) [Ols98]. In that case, the atomsradially reside in the harmonic oscillator ground state provided by the strong transverseconfinement. Any motion perpendicular to the tubes is effectively frozen out. For the arrayof tubes, a further prerequisite is that tunneling between the tubes is negligible on relevantexperimental timescales.

The strength of interaction in a homogeneous one-dimensional gas of bosons, inter-acting via a repulsive contact-potential, is described by a single dimensionless parameterγ = mg1D/(~2n1D) [Lie63a]. In this expression, m is the particle mass, g1D the 1D couplingstrength (see chapter 1.3.2 below), and n1D the 1D density. For γ = 0 the sample is non-interacting, and resembles an ideal gas of bosons. With increasing γ the particles start torepel each other. The wave function is more and more decreased at small interparticle dis-tances, and spatial correlations are built up [Gan03, Kin05]. In the limit γ → ∞ one gets agas of impenetrable bosons, known as the Tonks-Girardeau (TG) gas [Gir60].

The TG gas was first observed in 2004 with 87Rb atoms loaded into an optical latticethat forms an array of tubes, as discussed above [Kin04b]. Here, an interaction parameterup to γ ∼ 5.5 was realized. The TG limit has also been studied in the context of a sparselyfilled 1D lattice system [Par04]. In this approach, the dispersion relation of the lowest latticeband increases the effective mass of the atoms, and thereby allows for reaching an effectiveinteraction strength up to γ ∼ 200. In our experimental setup, the 1D Bose gas is realized inan array of 1D tubes (Fig. 1.3(a)) with full control over interactions via Feshbach tuning ofthe scattering length [Hal10a]. This approach made it possible to probe the gas all the wayfrom the weakly to the strongly interacting regime, via the study of collective oscillationmodes [Hal09]. Moreover, by switching interactions quickly from strong repulsive to strongattractive values, a metastable strongly correlated gas-like state was observed, the so-calledsuper-Tonks-Girardeau gas.

In chapter 6, we will go beyond the equilibrium ground-state properties of the 1D Bosegas and probe its excitation spectrum. The following sections are devoted to a more in-depthdiscussion of the peculiar nature of one-dimensional quantum systems and the role of theirelementary excitations.

28

Introduction

1.3.1 Universal low-energy description: Luttinger liquid

The study of one-dimensional systems and associated strong correlation phenomena has along-standing history in condensed matter theory, long before they became available withultracold atoms [Gia04]. Materials where the spin and charge carrying electrons move alongchains [Jer04] or ladders [Dag99] comprise wonderful candidates to investigate 1D physics.This goes along with advances in the fabrication of nanostructures, such as nanotubes [Boc99]or quantum wires [Aus02, Jom09], and the study of their 1D properties. Let me begin dis-cussing the nature of 1D quantum systems from such a condensed matter point of view,having in mind a gas of fermionic electrons.

In higher-dimensional systems, strongly interacting electrons are described within thecelebrated Fermi liquid theory, introduced by Landau in the late 1950s [Lan57, Lan59]. Ina gas of non-interacting electrons, all quantum states up to the Fermi energy are occupiedat zero temperature. Landau realized that, remarkably, the free non-interacting electron gasis essentially very similar to the interacting system. Yet, the individual strongly interact-ing electrons are replaced by new elementary quasiparticles, which are electrons dressed bydensity-fluctuations. These quasiparticles provide new, well defined excitations that obeyagain fermionic statistics. They themselves interact only weakly with each other, describedby the Landau parameter. This residual interaction results in a finite lifetime of the quasipar-ticles. The fact that they are essential free and scatter only weakly makes Fermi liquid theoryand the quasiparticle concept so successful. More specifically, Landau showed that when onelooks at the low-energy excitations closer and closer to the Fermi level, their lifetime actuallydiverges, which makes them always well defined.

In one dimension, however, Landau’s quasiparticle picture fails. This is due to the factthat for interacting quantum systems in 1D, excitations naturally become highly collective.Loosely speaking, one cannot push a single electron without pushing all the others (Fig. 1.7(a)and (b)) [Gia04]. The complementary low-energy description for one-dimensional systemsis known as the Luttinger liquid theory [Hal81, Gia04]. The key to this approach is built onparticle-hole excitations in the Fermi sea as elementary excitations. Different to the situationin higher dimensions, they are well defined at sufficiently low-energy and provide a sound-like linear dispersion relation between momentum and energy (Fig. 1.7(c) and (d)). Theirimportant role can be understood as follows. In higher dimensions, a low-energy particle-hole excitation at the Fermi surface does not have a well defined momentum-energy relationbecause for each momentum smaller than twice the Fermi momentum, one finds a way toexcite a particle near the Fermi energy (Fig. 1.7(e)). In 1D, however, the Fermi ”surface” con-sists of only two points and any excitation with non-zero momentum necessarily becomesgapped. An exception is the process, where a particle is put from one Fermi point to theother, gaining twice the Fermi momentum. This is quite peculiar for 1D systems, and knownas umklapp-scattering (Fig. 1.7(f)).

29

Introduction

2D, 3D Fermi liquid

1D Luttinger liquid

(a)

(b)

(e)

(f)

(c)

(d)

Figure 1.7: Elementary properties of one-dimensional quantum liquids. While higher-dimensionalsystems are described by weakly interacting quasiparticles (a), excitations in 1D systems are nec-essarily highly collective (b). Their description within the Luttinger liquid framework is built onparticle-hole excitations. Characteristic shape of the particle-hole spectrum for 2D and 3D systems(c), compared to 1D systems (d). (e) In 2D and 3D, low-energy particle-hole excitations close to theFermi surface (here denoted by a circle in 2D) exist for all momenta q smaller than twice the Fermimomentum kF . (f) In 1D, the Fermi surface reduces to two points which results in well defined low-energy modes at q ∼ 0 and q ∼ 2kF . Figure adapted from [Gia04].

The collective sound-like excitations in the density (or charge) are bosons, and the fieldtheoretical approach to describe the interacting gas in terms of these modes is termed bo-sonization, accordingly [Gia04, Caz04]. Electrons posses a spin, and next to the density-waves also spin-waves exist, which in 1D propagate with a different velocity. This peculiarsplitting of the electron into two components is called spin-charge separation [Gia04].

Although we have considered fermions so far, the Luttinger liquid theory captures bosonson completely equal footing [Caz04]. In a way, this originates from a less stringent role ofquantum statistics in 1D. In fact, interacting bosons can show fermion-like properties, aprime example being the TG gas, which is in many ways equivalent to a gas of free fermions.To be more specific, the Luttinger Hamiltonian for a gas of identical bosons takes the form

HLL =~v2π

∫dz

[K

(∂

∂zφ(z)

)2

+1K

(∂

∂zθ(z)

)2], (1.18)

with θ and φ related to the density and the phase of the boson field operator [Caz04]. Here,v is the velocity of the sound-like excitations, and K is the so-called Luttinger parameter,which relates to the strength of quantum fluctuations. In particular, K = 1 corresponds tothe TG gas limit (γ →∞), and K →∞ represents a non-interacting gas of bosons (γ = 0).

The Luttinger liquid theory provides a universal low-energy description for 1D quantumsystems, and can be seen truly as important as the Fermi liquid theory for higher dimensions.However, it lacks properties of finite-energy observables, which provide a measure of the ex-citation spectrum beyond the low energy regime [Ima12]. An experimentally accessible ex-

30

Introduction

ample is the dynamical structure factor (DSF), which we probe in chapter 6 via two-photonBragg spectroscopy [Ste99, Sta99]. Loosely speaking, the DSF measures how well the gascan be excited by an external source at a given momentum and energy. It is thus not sur-prising that, within the Luttinger liquid, the DSF is simply a delta function peaked at thelinear dispersion of the infinitely long-lived sound mode. The excitation spectrum shownin Fig. 1.7(d) indicates directly how this picture breaks down with increasing energy, andeven recent extensions of the Luttinger liquid to higher energies cannot capture all features[Ima09, Ima12]. When comparing experiment with theory, it is thus necessary to computethe DSF from the microscopic Hamiltonian. The uniform 1D Bose gas with repulsive contactinteraction provides an ideal candidate to do so for arbitrary strength of interactions as wellas for finite temperature by exploiting the integrability of the model in combination withefficient numerical methods [Cau06, Pan14]. In view of the experiments presented in chap-ter 6, the next section outlines the microscopic description of interacting bosons in 1D andthe role of the elementary excitations in shaping the DSF.

1.3.2 Microscopic description: Lieb-Liniger gas

A theoretical study of repulsively interacting bosons in 1D dates back to the seminal workby Elliott Lieb and Werner Liniger in the early 1960s [Lie63a, Lie63b], motivated by the questfor an exactly solvable quantum many-body problem featuring a simple local two-body po-tential. Specifically, the latter is assumed to be of the familiar contact-type. The Hamiltonianthen reads

H = − ~2

2m

∑i

∂2/∂zi2 + g1D

∑〈i,j〉

δ(zi − zj) , (1.19)

where the first term is the sum over the kinetic energy of the individual particles, and thesecond term quantifies their mutual interaction with strength g1D. In the context of ultracoldatoms, the parameter g1D can be directly derived from two-body scattering as will be out-lined in the following paragraph.

Two-body scattering in one dimension

A quantitative comparison of Lieb’s historical model with ultracold atoms confined to1D geometry is made possible as g1D can be directly connected to the three-dimensionalscattering length as [Ols98]. In fact, although the atomic motion is restricted to one dimen-sion, the two-body scattering process still prevails much of its three-dimensional character.This is because the effective range of the scattering potential is typically much smaller thanthe characteristic length scale of the transverse confining potential. For a harmonic potentialwith trap frequency ω⊥, the latter is given by the quantum oscillator length a⊥. The confine-ment then sets a type of boundary condition on the scattering phase shift [Kim05]. In case

31

Introduction

of a sufficiently small value as a⊥, the 1D coupling strength is directly proportional tothe 3D scattering length via g1D ≈ 2~ω⊥as [Ols98, Pet00, Pet08]. The situation changes whenas ∼ a⊥, which is e.g. the case in the vicinity of a 3D Feshbach resonance. During scatteringvirtual excitations to higher transverse oscillator modes lead to a divergent behavior of g1D,a so-called confinement-induced resonance (CIR). A full expression for the 1D coupling strengthtaking into account the resonant scattering in the vicinity of the CIR has been derived by M.Olshanii and reads [Ols98]

g1D = 2~ω⊥as(

1− 1.0326asa⊥

)−1

. (1.20)

In [Ber03] the CIR has been interpreted as a Feshbach-type resonance, for which the trans-verse states of the confining potential play the role of the open and closed channel. Morespecifically, the CIR is identified with a level crossing between the transversally excitedmolecular bound state and the two atom scattering threshold in the transverse ground state.

Experimentally, the effect of the confinement on the scattering properties has been inves-tigated via a shift of a weakly bound molecular state in a two-component Fermi gas [Mor05],which in 1D exists also for negative values of the scattering length [Ber03]. In our experimen-tal setup, the CIR has been identified via enhanced three-body recombination, and its shiftwith the confinement strength was studied [Hal10a]. More recently, coherent molecule for-mation in the vicinity of a CIR was investigated in a single elongated trap loaded with twolithium atoms [Sal13].

The Lieb-Liniger model and its elementary excitations

Having discussed the two-body scattering properties in a tight atomic waveguide and itsconnection to the 1D coupling strength, let us now return to the many-body Hamiltonian,eq. (1.19) describing the uniform 1D Bose gas. As mentioned already above, the system ischaracterized by one non-trivial and dimensionless parameter γ = mg1D/(~2n1D) [Lie63a].The Lieb-Liniger parameter quantifies the strength of interactions all the way from the non-interacting (γ = 0) to the infinitely repulsive (γ → ∞) TG limit. A first notable difference to2D and 3D systems is that the gas becomes more strongly interacting with decreasing densityn1D.

An intuitive interpretation of γ can be given in terms of the 1D interaction length rg =~2/(mg1D). It quantifies the distance scale on which the repulsion between the particles re-duces the relative wave function [Pet08, Ols98]. The γ-parameter is then the ratio of theaverage particle distance 1/n1D and rg. For weak interactions, the reduction of the wavefunction on the scale of the inverse density is practically absent, namely γ 1. Vice versa,in the strongly repulsive regime, this reduction is large on short distances rg 1/n1D, in-

32

Introduction

dicative for the fermionized TG gas, and correspondingly γ 1. For weak interactions,the γ-parameter can be alternatively seen as the ratio of interaction energy (∼ g1Dn1D) and acharacteristic kinetic energy scale for particles separated by the inverse density (∼ ~2n2

1D/m).This picture, however, breaks down with increasing repulsion due to the built-up of particlecorrelations. In that case, the interaction energy also depends on the reduction of the wavefunction at short distances, or more intuitively speaking on the ”overlap” of the individualatoms [Pet08, Gan03].

A solution of the Lieb-Liniger Hamiltonian in view of its many-body ground state as wellas the spectrum of elementary excitations has been presented in [Lie63a, Lie63b]. The deriva-tion exploits the integrability of the model via Bethe-Ansatz. Historically, this approach hasbeen first applied to the 1D Heisenberg spin chain [Bet31]. In the context of the 1D Bose gas,the Bethe-Ansatz wave function is of the form

ΨB(x1, ..., xN ) =∑P

a(P ) exp(iN∑n=1

kP (n)xn) . (1.21)

The xi denote the ordered positions of the N particles. The sum extends over all permuta-tions P of a set of pseudo-momenta k = k1, ..., kN . The specific form of the Bethe wavefunction can be seen as follows. If the coordinates of all particles were distinct, the contact-interaction term would vanish, i.e. the eigenfunctions could be written as a linear com-bination of products of single-particle plane waves. Collisions happen when the positionof two bosons coincide. In one dimension the collision event has only two possible out-comes that obey energy and momentum conservation. Either the particles leave each otherwith unchanged momenta, or their mutual momenta are interchanged. The permutations ineq. (1.21) account for all sequences of such two-body collisions starting from a specific setk [Caz11].

Technically, the contact-interaction in eq. (1.19) is replaced by a boundary condition forthe many-body wave function at zero interparticle distance. The Hamiltonian reduces to thatof free particles, with the wave function now being subject to this additional constraint. Thatapproach delivers the pre-factors a(P ) and the set k, and thereby the eigenfunctions of theproblem. Importantly, in the thermodynamic limit, many important properties of the groundstate, e.g. the energy, the chemical potential, or the pseudo-momentum distribution can beobtained from a set of integral equations, called the Lieb-Liniger system [Lie63a],

1 + 2λ∫ 1

−1

g(x)dxλ2 + (x− y)2

= 2πg(y) , (1.22)

e(γ) =γ3

λ3

∫ 1

−1g(x)x2dx , (1.23)

33

Introduction

-1 -0.5 0 0.5 10

1

2

pseudo-momentum k HkF L

dist

ribut

ion

fHkL

HaLΓ = 0.034

Γ = 0.121

Γ = 0.405

Γ = 1.233

Γ = 4.525

Figure 1.8: Elementary excitations and the dynamical structure factor of the 1D Bose gas. (a) Pseudo-momentum distribution f(k) for increasing value of the interaction parameter γ (from gray to black).The blue dashed line shows the distribution for the fermionized TG gas. Thin vertical lines indicatethe interaction-dependent pseudo-momentum cutoff. (b) Illustration of Lieb’s particle-like (top) andhole-like (bottom) excitations in pseudo-momentum space. Occupied pseudo-momenta in the groundstate are represented by the orange rectangle, and excitations are indicated by the circles and arrows.(c)-(e) Dynamical structure factor (value shown in gray scale) for γ = 1 (c), γ = 5 (d), and γ = 100 (e).For quantitative comparison, note that here ~ = 1 = 2m. Figures (c)-(e) adapted from [Cau06].

γ

∫ 1

−1g(x)dx = λ . (1.24)

Numerically solving eq. (1.22) for a fixed λ delivers g(x), which encodes the distribution ofpseudo-momenta for a certain γ. The γ-parameter itself is derived from eq. (1.24). Finally,eq. (1.23) delivers e(γ), which directly relates to the total energy of the gas. The distributionof pseudo-momenta f(k) is depicted in Fig. 1.8(a) for different values of the Lieb-Linigerparameter. In the TG limit all pseudo-momenta up to the Fermi wave vector kF = πn1D areequally populated, reminiscent of the non-interacting Fermi gas. It is worth mentioning thedifference between the distribution of pseudo-momenta and the actual momentum distribu-tion of the individual bosons. For the TG gas the latter shows a peak around zero momentumscaling with p−1/2 [Vai79], and a universal p−4 high-momentum tail [Ols03].

Excitations above the ground state of the Lieb-Liniger gas naturally emerge in two types,called the Lieb-I (particle-like) and Lieb-II (hole-like) mode (Fig. 1.8(b)) [Lie63b]. Lieb-I typeexcitations correspond to processes that take a particle from the pseudo-momentum cutoffto a higher momentum state. In the weakly interacting limit, they reduce to Bogoliubov ex-citations with the characteristic phonon dispersion at low momenta. Lieb-II type excitationscan be viewed as taking a particle from a momentum state below the pseudo-momentumcutoff to the edge of the distribution, thereby creating a hole. This type has no equivalent inhigher dimensions, and is absent in Bogoliubov’s theory. Because of interactions, the excitedparticle (hole) affects the remaining pseudo-momenta collectively, which again underlinesthe important role of correlations in 1D. As for the ground state, the dispersion relation for

34

Introduction

the Lieb-I and the Lieb-II mode is derived from a set of integral equations [Lie63b].

The excitations naturally emerging in the Lieb-Liniger model with their peculiar particle-and hole-like character have pronounced effects on the shape of the DSF [Cau06], whichis measured in the experiment. Specifically, the hole-like mode defines a minimal amountof energy needed to excite the gas at a specific momentum, while the particle-like modeprovides a bound of the spectrum from above. In the weakly interacting regime, the spectralweight of the DSF is found predominantly in the vicinity of the Lieb-I mode, approximatelyresembling Bogoliubov’s excitation spectrum (Fig. 1.8(c)). With increasing γ, however, theDSF broadens significantly, with most of the spectral weight lying within the Lieb-I andLieb-II mode (Fig. 1.8(d)). Finally, when the gas is deep in the TG regime, the DSF becomesessentially identical to the spectrum of free fermions in 1D, with a characteristic flat topbounded by Lieb’s particle- and hole-like mode (Fig. 1.8(e)).

In chapter 6, the DSF of the 1D Bose gas is measured all the way from the weakly to thestrongly interacting regime. The data is compared with numerical results of the DSF at arbi-trary interactions and also at finite temperature, based on the Bethe-Ansatz solution of theuniform gas [Pan14]. Inhomogeneities in the density distribution of the 1D systems, arisingfrom weak external trapping potentials, are accounted for in a local density approximation.This analysis allows us to demonstrate the important role of Lieb’s hole-like mode in shapingthe excitation spectrum of the strongly interacting 1D Bose gas.

1.4 Outlook

The experimental investigation of tunneling dynamics in the many-body lattice system pre-sented in this thesis is primarily based on the detection of sample-averaged local observ-ables, e.g. the total number of singly and doubly occupied lattice sites. Although this facil-itates already a powerful handle, the current setup does not provide a direct way to detectspatial correlations. The latter is possible with the recently developed single-site-resolvedfluorescence imaging techniques [Bak09, She10]. A new experimental apparatus is currentlydeveloped in our group [Gro15], pursuing the endeavor of local addressing and read-out inan optical lattice for two atomic species, cesium and potassium.

Such local read-out offers promising prospects for the study of dynamical critical scal-ing in the effective Ising model [Kol12b]. Specifically, measurements of non-local spin cor-relations and defect densities are a prerequisite in this context. For such quantities univer-sal scaling relations with the ramp rate are predicted, when ramping through the quan-tum phase transition [Kol12a, Kol12b]. Furthermore, correlation dynamics after quenchingonto the phase transition point as well as the built-up of entanglement might become di-rectly observable [Dal12]. Going beyond nearest-neighbor tunneling and the correspondingIsing spin chain, local detection techniques would permit similar studies for the observed

35

Introduction

longer-ranged resonant tunneling processes. Specifically, a mapping onto an appropriatespin model and the critical behavior at these resonances remains an open question.

Resonant tunneling in tilted Mott insulators may also be exploited to study correlatedquantum phases in more complex settings. Those involve two-dimensional square latticesof simple cubic type, for which an Ising density-wave along the direction of the tilt is pre-dicted that can co-exists with transverse-superfluid order [Sac02, Pie11, Kol15]. Similar be-havior is found also for more sophisticated lattice geometries, such as the triangular lattice.A qualitatively different situation appears on the Kagome lattice, for which a novel quan-tum liquid state is expected at certain tilt directions [Pie11, Pie12]. An important ingredientto stabilize these phases from decay due to three-body losses is a sufficiently large three-body interaction in addition to the standard Hubbard parameters. Given the exceptionalflexibility for tuning this term in a cesium Mott insulator [Mar11a, Mar12], our experimen-tal setup appears to be well suited to start investigating tilted lattice scenarios beyond theone-dimensional case.

Building on the demonstrated control over correlated atom tunneling, an extension tostudies of periodically driven systems of strongly interacting bosons is within direct reach.Coherent manipulation of single-particle tunneling in e.g. phase-modulated lattices [Lig07]has proven to be a powerful tool for implementing artificial gauge potentials [Str12, Aid13,Miy13]. Our experimental setup is suited for investigations of a lattice gas with periodicallydriven interactions, which features a tunable tunneling rate that additionally depends on thenumber of particles involved [Rap12, Gon09] 5. For such a situation a rich phase diagram isexpected, including regions that exhibit pair-superfluidity [Rap12]. Moreover, combiningmodulated interactions with Raman-assisted tunneling in a spin-dependent optical latticemay open ways to implement artificial gauge fields that are density dependent [Gre14].

For the one-dimensional Bose gas, we have observed the peculiarities of its excitationspectrum when one enters the strongly correlated regime. The specific shape has intriguingconsequences for the motion of an impurity particle embedded in the system [Mat12, Kna14].In a way, the correlated background gas can be seen to provide an effective lattice with aspacing related to the gas density that modifies the dynamics when the impurity is acceler-ated [Gan09]. For sufficiently strong interaction of the impurity with the correlated Bose gas,a damped Bloch-oscillation-type periodic motion for the impurity is predicted that settles toa drift with constant velocity [Kna13]. In the experiment, the impurity particle can be real-ized by means of another hyperfine magnetic sublevel that is accelerated in the gravitationalfield. Pioneering experiments in this direction have observed significant deviations from freefall, yet, the study was limited to comparatively weak interactions [Pal09]. In this context,our realization of a largely tunable one-dimensional Bose gas with cesium appears promis-ing to address the above aspects of quantum transport in the presence of strong interactions.

5Note that we have already carried out first experiments in this context that demonstrate controlledoccupation-dependent tunneling.

36

Introduction

1.5 List of publications

• Quantum Quench in an Atomic One-Dimensional Ising Chain.Florian Meinert, Manfred J. Mark, Emil Kirilov, Katharina Lauber, Philipp Weinmann,Andrew J. Daley, and Hanns-Christoph Nagerl,Phys. Rev. Lett. 111, 053003 (2013).

• Interaction-Induced Quantum Phase Revivals and Evidence for the Transition to the QuantumChaotic Regime in 1D Atomic Bloch Oscillations.Florian Meinert, Manfred J. Mark, Emil Kirilov, Katharina Lauber, Philipp Weinmann,Michael Grobner, and Hanns-Christoph Nagerl,Phys. Rev. Lett. 112, 193003 (2014).

• Observation of many-body dynamics in long-range tunneling after a quantum quench.Florian Meinert, Manfred J. Mark, Emil Kirilov, Katharina Lauber, Philipp Weinmann,Michael Grobner, Andrew J. Daley, and Hanns-Christoph Nagerl,Science 344, 1259-1262 (2014).

• Observation of Density-Induced Tunneling.Ole Jurgensen, Florian Meinert, Manfred J. Mark, Hanns-Christoph Nagerl, and Dirk-Soren Luhmann,Phys. Rev. Lett. 113, 193003 (2014).

• Probing the Excitations of a Lieb-Liniger Gas from Weak to Strong Coupling.Florian Meinert, Milosz Panfil, Manfred J. Mark, Katharina Lauber, Jean-Sebastien Caux,and Hanns-Christoph Nagerl,Phys. Rev. Lett. 115, 085301 (2015).

Additional and forthcoming publications

• A new quantum gas apparatus for ultracold mixtures of K and Cs and KCs ground-statemolecules.Michael Grobner, Philipp Weinmann, Florian Meinert, Katharina Lauber, Emil Kirilov,and Hanns-Christoph Nagerl,arXiv:1511.05044 (2015).

• Floquet engineering of correlated tunneling in the Bose-Hubbard model with ultracold atoms.Florian Meinert, Manfred J. Mark, Katharina Lauber, Andrew J. Daley, and Hanns-Christoph Nagerl,arXiv:1602.02657 (2016).

37

CHAPTER 2

PUBLICATION

Quantum Quench in an Atomic One-Dimensional Ising Chain

Phys. Rev. Lett. 111, 053003 (2013)

F. Meinert1, M. J. Mark1, E. Kirilov1, K. Lauber1, P. Weinmann1,A. J. Daley2, and H.-C. Nagerl1

1Institut fur Experimentalphysik und Zentrum fur Quantenphysik,

Universitat Innsbruck, 6020 Innsbruck, Austria2Department of Physics and Astronomy, University of Pittsburgh,

Pittsburgh, PA 15260, USA

We study non-equilibrium dynamics for an ensemble of tilted one-dimensional atomicBose-Hubbard chains after a sudden quench to the vicinity of the transition point of theIsing paramagnetic to anti-ferromagnetic quantum phase transition. The quench results incoherent oscillations for the orientation of effective Ising spins, detected via oscillations inthe number of doubly-occupied lattice sites. We characterize the quench by varying thesystem parameters. We report significant modification of the tunneling rate induced byinteractions and show clear evidence for collective effects in the oscillatory response.

2.1 Introduction

Ultracold atomic ensembles confined in optical lattice potentials have proven to offer uniqueaccess to the study of strongly correlated quantum phases of matter [Mor06, Blo08]. Un-precedented control over system parameters as well as exceptionally good isolation from theenvironment allow for implementation and quantitative simulation of lattice Hamiltonians

39


[Lew07, Sac08], not only bridging the fields of atomic and condensed matter physics in thestudy of ground-state phases, but also opening fundamentally new opportunities to exploreout-of-equilibrium physics in essentially closed quantum systems [Pol11, Blo12]. For exam-ple, the rapid time-dependent control available over system parameters makes it possibleto observe dynamics arising from a quantum quench, where a parameter such as the latticedepth is changed suddenly in time [Che12, Tro12, Wil10]. Recently it was demonstrated that1D chains of bosonic atoms with a superimposed linear gradient potential exhibit a quan-tum phase transition to a density-wave-ordered state, in which empty sites alternate withdoubly-occupied sites (“doublons”). Beginning in a Mott-insulator phase of a Bose-Hubbard(BH) system [Jak98, Gre02a], where the on-site interactions dominate over tunneling and theatoms are exponentially localized on individual lattice sites, a gradient potential is addeduntil the potential difference between adjacent sites matches the on-site interaction energy,and atoms can again resonantly tunnel. This was monitored for individual 1D chains with alength of about 10 sites with the quantum gas microscopy technique [Sim11], and effectivelymaps onto a 1D Ising model [Sac02], making it possible to simulate the transition from 1Dparamagnetic (PM) spin chains to anti-ferromagnetic (AFM) spin chains in the context ofultracold atoms.

In this letter, we explore the dynamics of a quantum quench for bosonic atoms in sucha tilted optical lattice [Rub11, Kol12a, Kol04]. Specifically, we quench the strength of thetilt to be near the phase transition point between PM and AFM regimes, and hence take thesystem far out of equilibrium, inducing strong oscillations in the number of doublons, whichwe detect through molecule formation. We find clear indications for the collective characterof the ensuing dynamics.

2.2 Tilted Mott insulator and quantum magnetism

We consider a 1D atomic ensemble in a tilted optical lattice potential near zero tempera-ture. For sufficiently weak interaction energy, much smaller than the band gap, the system isdescribed by the 1D single-band BH Hamiltonian [Jak98] augmented by a tilt [Sac02, Kol04]

H = −J∑〈i,j〉

a†i aj +∑i

U

2ni (ni − 1) + E

∑i

ini +∑i

εini . (2.1)

As usual, a†i (ai) are the bosonic creation (annihilation) operators at the ith lattice site, ni =a†i ai are the number operators, J is the tunnel matrix element, and U is the on-site interactionenergy. The linear energy shift from site to site is denoted by E, and εi accounts for a weakexternal confinement. For sufficiently small tilt (E U ) and sufficiently strong interactions(U J) the ground state of the system for one-atom commensurate filling is a Mott insulatorwith exactly one atom per site. When the tilt is ramped from E U across E ≈ U

40


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tilt per lattice site E f HkHzL

sing

leoc

cupa

ncy

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´10

4

doub

leoc

cupa

ncy

Nd

´10

4

Figure 2.1: Adiabatic tilt through the phase transition point. The number of doublons Nd (circles) asdetected through Feshbach molecule formation and the number of singly occupied sitesNs (triangles)are plotted versus the tiltEf at the end of the adiabatic ramp for Vz = 10 ER and as = 225(5) a0, givingU = 0.963(20) kHz for Vy,z = 20 ER. The full ramp takes 40 ms. The solid lines are fits to guide the eyeusing the error function. The dashed line indicates Ec = 1.010(20) kHz. The insets show a schematicof the Ising phase transition for a 1D Mott-insulator chain from the PM (left) to the AFM (right)ground state [Sim11]. All error bars in this figure and in the following figures denote the one-sigmastandard error.

at finite J , the system establishes a regular periodic pattern of dipole states, for which onefinds double occupancy at every second site with empty sites in between [Sac02, Rub11], seeinsets to Fig. 2.1. The formation of such density-wave ordering is due to an effective nearest-neighbor constraint, which results in correlated tunneling of all particles in the 1D chain.When, in the course of the ramp, the tilt compensates for the interaction energy (E ≈ U ), theparticles are allowed to resonantly tunnel onto the neighboring sites, however, only whenthe neighboring particle has not yet tunnelled itself. This system, when |U−E|, J U,E, canbe mapped onto an effective Ising spin model [Sac02, Sim11], where the two distinct groundstates (PM and AFM order, respectively) are connected via a quantum phase transition withthe quantum critical point Ec = U + 1.85J [Sac99].

2.3 Driving the Ising quantum phase transition

Starting with a Cs Bose-Einstein condensate [Web02, Kra04] we prepare an ensemble of 1DBose chains by first creating a 3D Mott insulator in a cubic optical lattice with unity filling(see methods chpt. 2.8). Initially, the lattice depth Vq is 20 ER (q = x, y, z), where ER = 1.325kHz 1 is the photon recoil energy. The residual harmonic confinement along the vertical z-direction is νz = 11.9(0.2) Hz. A tilt E of up to 1.7 kHz along z is controlled by a magneticforce |∇B|. A Feshbach resonance allows us to control U independent of J [Mar11a, Mar12].

1We give all energies in frequency units.

41


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Figure 2.2: (a) Reversibility of the phase transition. The number of doublons Nd is plotted as thesystem is driven from the PM into the AFM ground state (t ≤ 40 ms) and back (t > 40 ms) forVz = 10 ER. The dashed line is a fit using two concatenated error functions to guide the eye. For thismeasurement Vx,y = 25 ER. (b) Phase transition for various values of the lattice depth Vz = 8 (circles),10 (triangles), 12 (squares), and 14 ER (diamonds). The ramp rate is E = (17.3, 17.3, 8.6, 5.8) Hz/msfor Vz = (8, 10, 12, 14) ER. The solid lines are fits using the error function to determine the half widthw. For this measurement Vx,y = 30 ER. We determine U to (1.233, 1.324, 1.401, 1.467) kHz and Ec to(1.309, 1.371, 1.431, 1.487) kHz for Vz = (8, 10, 12, 14) ER with typical error of 20 Hz for U and Ec. Thepositions of Ec are indicated by the dashed lines. (c) Half width w of the phase transition obtainedfrom the fit as a function of Vz . The dashed line shows the calculated 4J . (d) Same data as in (b), butnormalized and plotted as a function of (Ef − U)/J . The dashed line indicates the quantum criticalpoint.

42


We first drive the transition adiabatically from the PM to the AFM state. We do this inparallel for an ensemble of about 2000 1D systems (”tubes” or ”Ising chains”) with an av-erage length of 40 sites (see methods chpt. 2.8). We start with a pre-tilt of E ≈ 0.7U andthen lower Vz to 10 ER (J = 25.4 Hz [Jak98]) within 10 ms to allow for particle motion alongthe vertical direction. Effectively, our 3D one-atom-per-site Mott insulator has now turnedinto an ensemble of 1D tubes with near unity occupation as horizontal tunnel coupling be-tween the tubes can be neglected on the timescale of the experiment. We ramp through thePM-to-AFM transition with a ramp speed of E = 17.3 Hz/ms up to a final value Ef andproject the ensemble of 1D many-body systems onto number states by quickly ramping Vzup to its initial value and instantly reversing the tilt. We detect double occupancy throughFeshbach molecule formation and detection [Win06, Dan10] (see methods chpt. 2.8) withan overall doublon detection efficiency of 80(3)%. Fig. 2.1 plots the number of doublonsNd as a function of Ef. The transition from an ensemble of singly occupied sites to an en-semble with predominant double occupation can be seen clearly. The transition point is invery good agreement with the expectation from the BH model, with U = 0.963(20) kHzand Ec = 1.010(20) kHz. When ramping fully through the transition we create Nd = 3 to4 × 104 doublons, corresponding to ensembles with 79(2)% double occupancy. We believethat defects in the initial one-atom Mott shell due to finite temperature and the superfluidouter ring are the main limitations to a higher conversion efficiency. We estimate the averagelength of AFM chains to ≈ 13 lattice sites (see methods chpt. 2.8). Note that this state withexceptionally high double occupancy is ideal for creating ultracold samples of ground-statemolecules [Dan10].

An unambiguous characteristic of a phase transition is its reversibility. Fig. 2.2(a) showsthe result when the linear ramp (from 0 to 40 ms) is reversed (from 40 to 80 ms). We find thata large fraction of the doublons, more than 85%, is returned to the one-atom-per-site Mottphase.

Adiabatic driving of the quantum phase transition from the PM into the AFM state ispermitted by the finite tunneling rate J between the sites through quantum fluctuations.In Fig. 2.2(b) we drive the phase transition at various values for the lattice depth Vz in or-der to highlight the role of tunneling. To assure adiabaticity, we increase the ramp durationfor larger Vz . Evidently, a reduced tunnel coupling for larger values of Vz gives rise to anarrowing of the transition. Note that the transition point is shifted to a larger tilt as Vz isincreased due to the increase in U . We fit each data set with an error function of the form∝ erf ((E − E0)/w) and plot the extracted half-width w as a function of Vz in Fig. 2.2(c). Weclearly find the width of the phase transition to scale with the tunnel matrix element. Inter-estingly, w fits very well to the value 4J . In Fig. 2.2(d) we plot the data shown in Fig. 2.2(b)as a function of (Ef − U)/J . The data collapses onto a single curve, confirming the role of Jas the natural energy scale [Sac02].

43


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Figure 2.3: Dynamic response for a quench to the resonance pointE ≈ U with U = 1.019(20) kHz. (a)Number of doublons Nd as a function of hold time th for E = 1.038 (squares), 0.973 (triangles), and0.865 kHz (circles). The solid lines are exponentially damped sinusoids fit to the data. (b) Extractedfrequency fδ as a function of δ = E − U . The dashed line is a quadratic fit to the data in the vicinityof the minimum. (c) Extracted damping rate γ as a function of δ. The dashed line is a Lorentzian fitincluding a finite offset. (d) Extracted number of doublons Nmax

d at the first oscillation maximum as afunction of δ. The dashed line is a Lorentzian fit. The vertical dashed lines in (b)-(d) give the quantumcritical point. The shaded areas (red and blue) indicate the one-sigma statistical experimental error(for Ec − U and δ = 0, respectively). For all measurements Vz = 10 ER, Vx,y = 20 ER, and as = 238(5)a0.

44


2.4 Quench dynamics near the quantum critical point

Slow ramping through the critical point allows the system to follow adiabatically the many-body ground state. Non-equilibrium dynamics, however, arises when the initial Mott stateis quenched by a rapidly applied tilt to the vicinity of the transition [Rub11, Kol04]. In ourexperiment we initiate non-equilibrium dynamics by first tilting the lattice and then quicklylowering Vz to 10ER. We let the system evolve for a hold time th and finally record the numberof doublons Nd as before. For the subsequent measurements U = 1.019(20) kHz. Fig. 2.3(a)shows the result for different values of δ = E−U close to the transition. The ensemble of 1Dchains exhibits large amplitude oscillations for Nd. The amplitude is maximal for δ ≈ 0 andsignificantly lower for |δ| > 0. For δ ≈ 0, which we call the resonant case, the first maximumis reached at th ≈ 3 ms. The oscillations damp out to an equilibrium value over a time scaleof about 30 ms. We fit the data using exponentially damped sinusoids. From these we obtainthe frequency fδ, the peak value of the first oscillationNmax

d , and the decay rate γ. Away fromresonance,Nmax

d is lower, however fδ is higher and, interestingly, γ is lower. This can be seenfrom Fig. 2.3(b), (c), and (d), where we plot fδ, γ, and Nmax

d as a function of δ. We fit fδ by aquadratic function, as will be justified below, and obtain the position of the minimum Eres.The data for Nmax

d and γ are fitted by Lorentzians. The damping is consistent with collapseof oscillations in coherent dynamics as observed in our simulations of the constrained Isingmodel (see methods chpt. 2.8), see also Refs. [Sac02, Kol04]. We find that the positions of themaxima for Nmax

d and γ at E = 1035(2) and E = 1030(3) Hz, respectively, agree with theminimum for fδ at Eres = 1028(1) Hz. Within our error bars 2, all values are compatible withthe value for U . We note thatNmax

d is only about 5% lower than the value forNd we obtainedfrom the adiabatic sweep as shown in Fig. 2.1.

The quadratic fit to the data for fδ is justified, in the simplest approximation, by a two-state Rabi model for the case of a single double-well potential [Rub11]. At the point E ≈ U ,the two Fock states |11〉 and |20〉, describing one atom in each site and two atoms in theleft site, respectively, are resonantly coupled by J , leading to Rabi oscillations between thetwo states with frequency

√2J , where the factor

√2 ensues from bosonic enhancement. The

probability to find the system in the dipole state |20〉 thus oscillates in the case of finite de-tuning from resonance with the generalized Rabi frequency fδ =

√8J2 + δ2 [Rub11], which

is quadratic for sufficiently small δ.

2.5 Interaction-induced modification of the tunneling rate

Before we continue this discussion we make use of our capability to tune U by means of aFeshbach resonance. We repeat the measurements shown in Fig. 2.3 for different values of as

2We note that the day-to-day variation of the positions of the extrema for Nmaxd , γ, and fδ is about 10 Hz and

hence considerably larger than the statistical error from the fits.

45


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Figure 2.4: Dependence of the resonance position Eres (a) and of the on-resonance oscillation fre-quency f0 (b) on as as deduced from the dynamical response of the system. The solid line in (a) isthe linear prediction of the BH model. In (b) the solid line is a linear fit to the data. For these mea-surements, Vz = 10 ER and Vx,y = 20 ER. (c) On-resonance oscillation frequency f0, interpolated toas = 0, as a function of Vz (circles). The dashed and dotted lines show the expected result for an Isingchain of three sites and the double well in the non-interacting limit, respectively. (d) On-resonancequench dynamics for the number of doublons in the 1D Ising chain (circles) and the number of singlyoccupied sites in nearly isolated double wells (triangles) for U = 1.060(20) kHz.

to which we tune after producing the 3D one-atom Mott-insulator state and before initializ-ing the quench. We determine the resonance position Eres and the on-resonance oscillationfrequency f0 = fδ=0. The result is shown in Fig. 2.4(a) and (b). For repulsive interactions wefind good agreement forEres with U calculated from lowest-band Wannier functions [Jak98].For attractive interactions there is a significant shift to lower absolute values ofEres as correc-tions taking into account higher bands and the properly regularized pseudopotential wouldhave to be included [Mar12]. We attribute the considerable dependence of the data for f0 onas to an effectively altered tunnel barrier as U and hence Eres are changed [Bis12, Luh12].

46


2.6 Collective effects in the quench dynamics

We now quantitatively compare our findings for f0 with the result from the simplified double-well system. Firstly, to remove the effects of the modified tunneling rate due to the tilt andinteractions, we interpolate our data to find f0 for as = 0 with the linear fit as shown inFig. 2.4(b). The resulting values are shown in Fig. 2.4(c) for data sets taken at three differentvalues of Vz , and are compared to the prediction for the double well on resonance, f0 = 2

√2J

(dotted line). Evidently, the tunneling in the 1D chains is significantly faster when comparedto the double-well system. This arises due to the nearest-neighbor constraint present in the1D chains, which prohibits having two adjacent dipoles and which leads to an increase ofthe oscillation frequency even for three sites by a further factor of

√2, to 4J (dashed line)

[Rub11]. Through numerical simulations with up to 60 lattice sites we have verified that theoscillation frequency shows only a slight further increase for longer chains beyond 3 sites(see methods chpt. 2.8). The weak harmonic trapping potential causes an additional smallincrease of the computed oscillation frequency. However, both effects are presently withinour experimental uncertainty.

Finally, we compare the on-resonance oscillatory dynamics of the ensemble of 1D Isingchains with that of an ensemble of nearly isolated effective double-well systems to checkthe role of the constraint. The double wells are engineered by highly non-adiabatic loadingof the 3D lattice, thereby creating a comparatively dilute sample of doublons. We clean outthe singly occupied lattice sites by temporarily hiding the doublons in a molecular state andremoving singular atoms by resonant light. We subsequently initiate the quench dynamicsas before and detect the number of singly occupied sites (see methods chpt. 2.8) as a functionof th as shown in Fig. 2.4(d). The oscillation frequencies in the two systems, determined tof0 = 128(2) Hz and f0 = 93(2) Hz, respectively, differ by a factor 1.38(3), in good agreementwith the collective effect discussed above.

2.7 Conclusion

In conclusion, we have investigated quench dynamics by tuning a tilt suddenly onto thetransition from PM to AFM Ising order in a large ensemble of 1D atomic Mott-insulatorchains. We observe significant shifts of the tunneling parameter arising from interactions,and collective effects in the oscillations arising from the effective constraint in the model.This study of quench dynamics opens the opportunity to explore many aspects of the dy-namics in these systems that up to now were only addressed theoretically. This includesscaling relations for smaller quenches across the phase transition [Kol12a, Kol12b, DeG10],as well as possibly quenches at other resonance points (e.g., near E = U/2).

47


We are indebted to R. Grimm for generous support, and thank Johannes Schachenmayerfor discussions and contributions to numerical code development. We gratefully acknowl-edge funding by the European Research Council (ERC) under Project No. 278417, and sup-port in Pittsburgh from NSF Grant PHY-1148957.

2.8 Materials and Methods

2.8.1 Preparation of the one-atom-per-site Mott insulator

To prepare the ensemble of 1D Bose chains we start with a 3D Bose-Einstein condensate(BEC) without detectable non-condensed fraction of typically 8.5 × 104 Cs atoms in the en-ergetically lowest hyperfine ground state |F = 3,mF = 3〉 confined to a crossed dipole trap.Trapping and cooling procedures are described in Refs. [Web02, Kra04]. The sample is levi-tated against gravity by a vertical magnetic field gradient of |∇B| ≈ 31.1 G/cm. The BEC isadiabatically loaded from the trap into a cubic 3D optical lattice generated by three mutu-ally perpendicular retro-reflected laser beams at a wavelength of λl = 1064.5 nm to inducethe phase transition to a 3D Mott insulator. The final value for the lattice depth Vq in eachdirection (q = x, y, z) is typically 20 ER. For some of the experiments the depth is set to ahigher value. Here, ER = 1.325 kHz denotes the photon recoil energy for Cs atoms at λl. Wetake care to prepare the system with a clean singly occupied Mott shell with less than 4% ofresidual double occupancy. When the dipole trap is extinguished the residual harmonic con-finement in the direction of gravity as a result of the transversal profile of the lattice beamsis νz = 11.9(0.2) Hz. A tilt up to E = 1.7 kHz along the vertical z-direction can be intro-duced by lowering the strength of the levitation field. A broad Feshbach resonance allowsus to precisely tune the atomic scattering length as and thus to set U [Mar11a, Mar12]. Theday-to-day variations in the number of particles in the BEC lead to variations in the numberof doublons Nd, as can be seen in e.g. Fig. 2.2(b). There, the different data sets have beenrecorded on different days.

2.8.2 Calibration of E and U

The tilt E is set by reducing the vertical magnetic field gradient to a certain fraction. Wecalibrate E by lattice modulation spectroscopy. For the calibration measurements we set Eto a value far detuned from the resonance pointEres such that tunneling remains suppressedby the on-site interaction. Modulation of Vz with a frequency ν± = E±U bridges the energygap and results in driven tunneling along and against the applied potential gradient ontoneighboring sites [Ma11].

We modulate the lattice sinusoidally for 100 ms with an amplitude of typically 5% aroundits mean value Vz = 10ER. Three typical excitation spectra in the number of doublons created

48


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Figure 2.5: (a) Typical lattice modulation spectra showing tunneling resonances at modulation fre-quencies E ±U as detected by an increase in the number of doublons Nd. We have set as = 50 a0 andE = 1.73 kHz (squares), as = 100 a0 and E = 1.73 kHz (triangles), and as = 50 a0 and E = 0.69 kHz(circles). The solid lines are Lorentzian fits to determine the center position of the resonances. (b) Tiltper lattice site E as a function of the fraction of the magnetic levitation field. The dashed line is alinear fit to the data. The data is taken at as = 50 a0. (c) On-site interaction energy U as a function ofas. The prediction from the BH model is depicted by the dashed line. In (b) and (c) all error bars aresmaller than the data points.

49


0 10 20 30 40 50 60

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oms

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Figure 2.6: (a) Calculated number of atoms as a function of tube length using the Gutzwiller meanfield method. (b) Calculated number of atoms as a function of chain length after introducing 5 %randomly distributed defects. The resulting average chain length is calculated to ≈ 13 lattice sites.

are shown in Fig. 2.5(a). In each spectrum we identify two strong and narrow tunnelingresonances at ν± = E ± U . The resonance positions provide E and U . Keeping as constantthey allow for precise calibration ofE as a function of the magnetic levitation field, as shownin Fig. 2.5(b). Analogous calibration of U as a function of as for a fixed tilt is depicted inFig. 2.5(c). The data are in very good agreement with the prediction from the BH model[Jak98] for as > 0, whereas for as < 0, U is considerably shifted to smaller absolute valuesdue to multi-body interaction effects [Mar12].

2.8.3 Estimating the average chain length

To get a rough estimate for the average chain length for our ensemble of tubes we first nu-merically determine the ground state of the 3D Mott insulator using the Gutzwiller mean

50


field method [Gut63], feeding in our typical values for the particle number, the trappingstrength, and the interaction strength. Fig. 2.6(a) shows the calculated atom number distri-bution. The average atom number per tube is ≈ 40. Defects are introduced by removingatoms from lattice sites at random positions. The remaining uninterrupted chains with acertain length can then be counted. Fig. 2.6(b) shows the resulting distribution when intro-ducing 5 % defects. The average chain length decreases to ≈ 13 lattice sites, but evidently areasonable number of longer chains survives.

2.8.4 Detection of single and double occupancy

We detect double occupancy by associating atom pairs on a lattice site to weakly boundmolecules crossing a narrow g-wave Feshbach resonance with a pole at 19.8 G [Mar07]. Thesystem is cleaned from remaining singly occupied sites by combining rapid adiabatic pas-sage from |F = 3,mF = 3〉 to |F = 4,mF = 4〉 using a microwave field with a resonant lightpulse [Dan10]. After dissociating the molecules again we detect the number of atoms withstandard absorption imaging. Single occupancy is measured by associating doubly occupiedsites into Feshbach molecules, thereby hiding these during the imaging process.

Our measurement noise is mainly affected by atom number fluctuations in the BEC.When detecting double occupancy, additional noise arises from small shot-to-shot variationsin the molecule association, cleaning, and dissociation efficiency. Especially for experimentswith comparatively low absolute atom numbers, detection of single occupancy is favored asnoise from the cleaning and dissociation process is absent.

2.8.5 Numerical simulations of quench dynamics

We performed numerical simulations of the quench dynamics both within the effective con-trained Ising model, and directly using a single-band Bose-Hubbard model. Using combina-tions of exact diagonalization techniques and time-dependent density matrix renormaliza-tion group (t-DMRG) methods [Vid04, Dal04, Whi04, Ver08], we calculated the propagationin time, beginning with an initial state with one particle per lattice site. Example calculationsfor the number of doubly-occupied sites as a function of time with different numbers of sitesin a Bose-Hubbard model with open boundary conditions are shown in Fig. 2.7.

There are several features that we recognize from these calculations. First, we note thatalthough collective effects for chains longer thanN = 3 lead to higher frequency componentsappearing in the oscillations, these do not substantially affect the period of oscillations overthe first few cycles - as used in the manuscript to determine the frequencies in Fig. 2.4 (notethat in the experiment, we implicitly average these values over chains of different length).These high-frequency components that enter due to collective effects instead lead to dephas-ing of the oscillations, as can be seen in the curves for systems larger than three sites. Forlarge chain lengths N & 30, this dephasing, combined with some averaging over chains of

51


0 1 2 3 4 50

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tota

ldou

ble

occu

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y

Figure 2.7: Number of doubly occupied sites as a function of time tJ (~ = 1), as computed from theBose-Hubbard model beginning from an initial state with one particle per lattice site. Here we chooseE = U = 12J , and vary the number of lattice sitesN = 3 (black), 10 (blue), 20 (red), 30 (green), and 40(purple). These calculations are performed with t-DMRG methods, which are converged for matrixsizes D = 100–200.

different length, is sufficient to account for the rate of damping seen in the experiment. Inpractice, other sources of dephasing, e.g., due to the harmonic confinement or noise, will alsocontribute to damping in the experiment. However, we estimate in our experiments that thedominant contribution arises from these coherent collective effects.

52

CHAPTER 3

PUBLICATION

Observation of many-body dynamics in long-range tunneling aftera quantum quench

Science 344, 1259 (2014)

F. Meinert1, M. J. Mark1, E. Kirilov1, K. Lauber1, P. Weinmann1,M. Grobner1, A. J. Daley2,3, and H.-C. Nagerl1


Universitat Innsbruck, 6020 Innsbruck, Austria2Department of Physics and Astronomy, University of Pittsburgh,

Pittsburgh, PA 15260, USA3Department of Physics and the Scottish Universities Physics Alliance, University of Strathclyde,

Glasgow G4 0NG, UK

Quantum tunneling is at the heart of many low-temperature phenomena. In strongly cor-related lattice systems, tunneling is responsible for inducing effective interactions, andlong-range tunneling substantially alters many-body properties in and out of equilibrium.We observe resonantly enhanced long-range quantum tunneling in one-dimensional Mott-insulating Hubbard chains that are suddenly quenched into a tilted configuration. Higher-order tunneling processes over up to five lattice sites are observed as resonances in thenumber of doubly occupied sites when the tilt per site is tuned to integer fractions of theMott gap. This forms a basis for a controlled study of many-body dynamics driven byhigher-order tunneling and demonstrates that when some degrees of freedom are frozenout, phenomena that are driven by small-amplitude tunneling terms can still be observed.

53

Observation of many-body dynamics in long-range tunneling after a quantum quench

3.1 Introduction

Quantum tunneling is ubiquitous in physics and forms the basis for a multitude of funda-mental effects [Ank10] related to electronic transport, nuclear motion, light propagation, andsuperfluidity in lattice systems [Mor06]. Whereas for weakly interacting particles tunnelingat a rate J will occur as an individual process for each particle, in strongly interacting sys-tems the behaviour of each particle is correlated with the behaviour of other particles. Suchcorrelated processes are believed to play an important role, for example, in superconduc-tivity of the cuprate systems [Mac88, And04, Nor11]. Second-order tunneling has been ob-served in cold atom experiments as driven resonances [Ma11] or directly as a dynamical pro-cess for pairs of strongly interacting particles in arrays of double well potentials [Fol07]. Thatprocess results in an effective nearest-neighbor super-exchange interaction [Tro08, Gre13],which forms the basis of important forms of quantum magnetism [Dag94], and provides astarting point for the formation of quantum many-body phases. Such tunneling processeshave also recently been observed for electrons in systems of quantum dots [Bra13].

Higher-order processes involving correlated tunneling across multiple lattice sites cangive rise to longer-range effective interaction terms, and more complex many-body criticalphenomena [Sac08], as well as marked changes in out-of equilibrium dynamics. Parallelscan been drawn between long-range tunneling processes in tilted lattices and multi-photonelectron-positron creation in strong electric fields, with connections to relativistic phenom-ena such as the Sauter-Schwinger effect in tilted Mott insulators [Que12], and also to long-distance electron transport in molecular systems, for example [Sed11, Win14]. However, al-though single-particle tunneling loss via higher band resonances [Sia07] has been demon-strated, it has been difficult to observe coherent quantum dynamics due to higher-order tun-neling processes, because the small amplitude driving these terms places challenging upperlimits on the energy scales for required temperatures and allowed disorder.

3.2 Higher-order atom tunneling in a tilted Mott insulator

Our experiment is based on an array of one-dimensional (1D) Mott-insulating ”Ising” chainsof bosons in an optical lattice near zero temperature [Sim11, Bak11, Sac02, Mei13, Kol12a].We model the system by a single-band Bose-Hubbard (BH) Hamiltonian [Jak98] (see meth-ods chpt. 3.8). For large on-site interaction energy U J , the many-body ground state is aMott insulator with unit occupation at commensurate filling (Fig. 3.1 A). This phase is char-acterized by exponentially localized atoms and highly suppressed tunneling. In addition, wesuperimpose a linear gradient potential, which introduces a site-to-site constant energy shiftE. We perform a quantum quench to a highly nonequilibrium situation by rapidly tiltingthe initial Mott state to an integer fraction of the Mott gap E ≈ U/n. The quench initiatesresonantly enhanced long-range tunneling to the nth neighbor for all sites simultaneously

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Figure 3.1: Tunneling resonances in a tilted 1D Mott insulator. (A) Schematic view of the long-range correlations across n sites for a tilt of E = U/n after the quench from the initial 1D one-atomMott insulator (top) to the tilted configuration (bottom). Here, n = 3. (B) Number of doublons Nd

as a function of E at th = 200 ms after the quench. Here, Vz = 10ER and as = 252(5) a0, givingU = 1.077(20) kHz for Vx,y = 20ER. The solid lines are Lorentzian (for E = U ) and Gaussian (forE = U/2 and E = U/3) fits to the data to determine the center positions and widths. The insets showmatter-wave interference patterns obtained in TOF at E1 = U , E2 = U/2, and E3 = U/3 taken afterth = 1 ms, 9 ms, and 28 ms, respectively. (C to E) Integrated line densities of the TOF images shown inthe insets in (B). The solid lines are fits according to double-slit interference patterns with Gaussianenvelopes of the form n(p) = N0e

−(p−p0)2/w2(1 + V cos[k(p− p0) + φ]) with the fringe visibility V ,

the wave vector k, and a phase φ. Error bars in all figures reflect ±1 standard deviation.

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(lower part of Fig. 3.1 A). For n = 1, one couples to nearest-neighbor dipole states and ob-serves strong coherent oscillations in the number of doubly occupied sites (doublons) witha characteristic frequency 4J [Mei13]. For n > 1, resonant tunnel coupling occurs acrossn − 1 intermediate lattice sites. The process involves up to n other particles, giving rise tooccupation-dependent nth-order tunneling, with a characteristic rate of atom-pair formationset by αn × Jn/(U/n)n−1 in nth order perturbation theory. Here, αn is a proportionality fac-tor that includes the effect of Bose enhancement (see methods chpt. 3.8). Because all particlesparticipate in a tunnel process across n sites, one expects the build-up of massive correla-tions in the interacting many-body system. As we discuss in chpt. 3.8, the quench onto thecritical point in the many-body system results in many-body dephasing of oscillations in theatom-pair number, corresponding to a characteristic growth in many-body entanglement[Ami08, Tro12, Kol07] in our numerical simulations.

3.3 Observation of higher-order tunneling resonances

We prepare an ensemble of 1D Mott insulators [Mei13] starting from a 3D Bose-Einstein con-densate (BEC) of typically 8.5× 104 Cs atoms without detectable uncondensed fraction. TheBEC is levitated against gravity by a magnetic field gradient of |∇B| ≈ 31.1 G/cm and ini-tially held in a crossed optical dipole trap [Mei13, Web02]. We load the sample adiabaticallyinto a cubic 3D optical lattice generated by laser beams at a wavelength of λl = 1064.5 nm,thereby creating a singly occupied 3D Mott insulator for a lattice depth of Vq = 20ER

1 in eachdirection (q = x, y, z) with less than 4% residual double occupancy. Here, ER = ~2k2

l /(2m) isthe photon-recoil energy, with kl = 2π/λl and m the mass of the Cs atom. The optical latticeresults in a residual harmonic confinement of νz = 11.9(2) Hz in the z direction of gravity.A broad Feshbach resonance allows us to set the atomic scattering length as, and thus Uindependently of J , by means of an offset magnetic field B [Mei13].

Tunneling resonances are observed by quickly tilting the lattice in the z direction througha reduction of |∇B| and then lowering Vz to 10 ER within 1 ms, giving J ≈ 25 Hz 2 [Jak98].All dynamics are now restricted along 1D Mott chains with an average length of 40 sites[Mei13]. The chains, in total ≈ 2000, are decoupled from each other on the relevant experi-mental time scales. We let the systems evolve for a hold time th of up to 200 ms in the tiltedconfiguration and then quickly ramp back Vz to its original value and remove the tilt. Theensemble is characterized by measuring the number of doubly occupied sites Nd throughFeshbach molecule formation with an overall efficiency of 80(3)% [Mei13]. Alternatively, wedetect the emergence of momentum-space coherence in time-of-flight (TOF) by quickly turn-ing off all trapping potentials and allowing for 20 ms of free levitated expansion at as = 0

1The lattice depth Vq is calibrated via Kapitza-Dirac diffraction. The statistical error for Vq is 1%, although thesystematic error can reach up to 5%.


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[Web02] before taking an absorption image.

The experimental result for a specific choice of U = 1077(20) Hz is shown in Fig. 3.1 B.For a hold time of th = 200 ms, the transient response as discussed below has settled toa steady-state value. Besides a broad resonance at E = 1095(2) Hz with full width at halfmaximum (FWHM) = 172(9) Hz, two narrower resonances at E = 532(1) and 351(1) Hzwith FWHM = 44(2) and 27(2) Hz can be seen. Whereas the broad resonance is the result ofresonant tunnel coupling to nearest-neighbor dipole states at E1 = U [Mei13], the positionsof the narrower resonances are consistent with E2 = U/2 and E3 = U/3, and we hence inter-pret them to emerge from tunnel processes extending over a distance of two and three latticesites, respectively. The reduced widths reflect the smaller amplitude of the higher-order tun-nel processes. We believe that the resonances are slightly broadened inhomogeneously bythe external harmonic confinement. The assignment of the resonance features to tunnelingprocesses over multiple lattice sites is supported by TOF images (insets to Fig. 3.1 B) takenfor each resonance En in the course of the transient response. The images clearly exhibitmatter-wave interference patterns, indicating delocalization of the atoms during the tunnelprocesses. The integrated line densities are presented in Fig. 3.1, C to E. The periodicity of thesinusoidal density modulation, found to be 2~kl/n, is in agreement with spatial coherence ofthe atomic wave function over a distance of n sites.

3.4 Many-body dynamics driven by second- and third-order tun-neling

We now investigate the transient dynamics after the quantum quench. Figure 3.2, A and C (Band D), shows the on-resonance response ofNd and the fringe visibility V in the TOF imagesforE1 (E2). The quench toE1 results in large amplitude oscillations forNd; calculations showthat the decay is due to many-body dephasing, which plays an increased role for largerchain lengths [Mei13]. The oscillatory response at E1 is clearly reflected in the dynamics forV , as each local minimum coincides with an extremum for Nd. The dynamics for E2 are, incontrast, highly overdamped and fit to a saturated growth function of the form∝ (1−e−th/τ ),with a characteristic rate 1/τ ; a simple three-site BH model predicts oscillations at frequencyν2 = 4(2

√2+√

2)J2/U (see methods chpt. 3.8). In the experiment, we find a single maximumfor V before it decays. The dephasing here results from more complicated dynamics in BHchains longer than three sites (see methods chpt. 3.8).

We now focus on the scaling of the resonant doublon growth rate 1/τ with J and U

for the resonance E2. Example data sets (Fig. 3.3 A) clearly demonstrate that 1/τ dependsnot only on Vz but also on U when Vz and thereby J are kept constant. In Fig. 3.3 B, weplot the same data with the time axis rescaled by the energy scale J2/(U/2) for a second-order tunneling process. The data collapse onto a single curve, demonstrating that indeed

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second-order tunneling dominates the transient dynamics after the quench. The numericaldata for 10 to 30 site BH chains (see methods chpt. 3.8) show similar rise characteristics andreveal the same scaling collapse (Fig. 3.3 C). The values for 1/τ from measurements takenat different combinations of Vz and U have a linear dependence on J2/(U/2) (Fig. 3.3 D)with a surprisingly large prefactor α2 = 38(2), which we analyze in two ways. First, wecompare to the frequency of coherent doublon oscillations in the simple three-site model.The role of many-body dephasing faster than a full second-order tunneling cycle is estimatedby assuming τ as a quarter of the full tunneling period. The value 1/τ ≈ 4×ν2 is indicated bythe solid line in Fig. 3.3 D. Second, we extract a characteristic growth rate from the numericaldata, indicated by the dashed regions in Fig. 3.3, C and D, revealing quantitative agreementwith the experiment.

A similar behavior in the dynamical scaling of the resonant response at the resonanceE3 = U/3 is seen in Fig. 3.3, E to H. Scaling collapse is observed when rescaling time byJ3/(U/3)2, indicating a third-order tunneling process. Residual oscillations after the initialgrowth period in the numerical data (Fig. 3.3 G) relate to the finite system size and the lackof averaging over positions in the trap (see methods chpt. 3.8). From the linear fit to thegrowth rate 1/τ in Fig. 3.3 H, we obtain a slope of α3 = 34(2), in good agreement with acharacteristic growth rate determined from the numerical data, which we indicate by thedashed region as before. We note that the signature of the third-order process is not maskedby the presence of second-order energy shifts (see methods chpt. 3.8).

3.5 Many-body time reversal

To what extent can one reverse this many-body dephasing dynamics? In Fig. 3.4 A, we showthe result of a many-body echo experiment for which we switch the sign of U and E at theE2 = U/2 resonance in the course of the transient response. A clear, although only partial, re-versal in the time evolution forNd can be seen beforeNd reaches the same steady-state valueas before. It would be interesting to test whether the revival could be improved by switchingthe sign of J as well. Naively, the second-order process scaling with J2 should not dependon the sign of J . Switching J by means of modulation techniques [Lig07] may allow a de-tailed benchmarking of many-body damping versus the presence of mere inhomogeneousbroadening in our system.

3.6 Fourth- and fifth-order long-range tunneling

Finally, in Fig. 3.4 B, we show resonances corresponding to many-body tunneling acrossfour and five lattice sites. For these data, the lattice depth was reduced to Vz = 7ER to speedup the processes; the system was initially in the Mott-insulating regime. With decreasing

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Figure 3.3: Second- and third-order tunneling dynamics. (A) Double occupancy at E2 for Vz/ER,as/a0 = 10, 253(5) (squares), 12, 253(5) (triangles), and 10, 400(5) (circles). (E) Double occupancy atE3 for Vz/ER, as/a0 = 7, 253(5) (squares), 9, 253(5) (triangles), and 9, 175(5) (circles). The solid linesare fits to the data with saturated growth functions. (B and F) Collapse of the data shown in (A) and(E) for rescaled time axes. (C and G) Result of a numerical simulation of the resonant response atE2 and E3, respectively. (D and H) Growth rates 1/τ for E2 and E3, respectively. In (D), the datafor Vz = (8, 9, 10, 12, 14)ER with fixed as = 253 a0 (squares) and as = (175, 253, 325, 400) a0 withfixed Vz = 10ER (circles) are plotted as a function of J2/(U/2). The solid line gives the predictionfrom a three-site BH model. In (H), the data for Vz = (7, 8, 9, 10)ER with as = 253 a0 (squares) andfor as = 175 a0 at Vz = 9ER and as = 300 a0 at Vz = 7ER (circles) are plotted as a function ofJ3/(U/3)2. The dashed line is a linear fit to the experimental data. The shaded areas in (C), (D), (G),and (H) indicate the spread in the growth rate extracted from the numerical data with fixed steady-state values from the experiment (see methods chpt. 3.8).

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´10

4 A

E ® -E

U ® -U

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U5U6

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0.0 0.2 0.4 0.6

0.5

1.0

1.5

2.0

tilt per lattice site E HkHzL

doub

leoc

cupa

ncy

Nd

´10

4

U2

U3

U4

U5

2U3

B

Figure 3.4: Many-body echo and higher-order tunneling resonances. (A) Double-occupancy Nd asa function of hold time th at E = U/2 for Vz = 8ER and as = −250(5)a0, giving U = −994(20) Hz(squares). Partial time reversal of the many-body dynamics (circles) occurs after switching as to+250(5)a0 and simultaneously reversing E to −E at th = 6 ms within 1 ms (gray bar). For the echodata (circles), a typical error bar is given for the data point at th = 16 ms. (B) Nd as a function of Eafter th = 200 ms at Vz = 7ER with U = 959(20) Hz, for as = 252(5) a0. The arrows indicate the ex-pected positions of the tunneling resonances at En = U/n. An additional resonance at 2U/3 appears.The inset gives a fine scan of the U/4 and U/5 resonances. The solid line is a fit based on the sum ofmultiple Gaussians to guide the eye.

Vz , the resonances at U/2 and U/3 slightly broaden, which we attribute to the increase ofthe second- and third-order tunneling rates. The new resonances at U/4 and U/5 are clearlydetectable. We note that these fourth- and fifth-order tunneling processes greatly benefitfrom substantial Bose enhancement (see methods chpt. 3.8) and speculate that even higher-order processes should become accessible when one eliminates residual parabolic energyshifts due to the trapping laser beams.

3.7 Conclusion

Our results underline the utility of cold atoms in optical lattices for the investigation of fun-damental physical processes driven by small-amplitude terms and specifically higher-ordertunneling. By partly freezing the motion in the deep lattice, these sensitive processes canbe observed here despite finite initial temperatures (which here are converted into defectsand missing atoms in an ensemble of initial states). This will motivate further investiga-tion of quantum phases and critical properties near these higher-order resonances, whichare presently unknown, including systems with tilts along multiple axes [Sac02, Pie11]. Ourinitial studies of parameter reversals also open the door to the study of many-body dephas-ing and echo-type experiments on a quantum many-body system, as well as investigationsinto the nature of the many-body dephasing and (apparent) thermalization [Rig08]. Parallelscan be drawn with arrays of quantum dots, opening further possibilities to model electron

61


tunneling over multiple sites [Bra13] by using fermionic atoms.

We are indebted to R. Grimm for generous support, and thank J. Schachenmayer fordiscussions and contributions to numerical code development. We gratefully acknowledgefunding by the European Research Council (ERC) under Project No. 278417, and support inPittsburgh from NSF Grant PHY-1148957.

62



In this supplementary material, we provide more details of the theoretical treatment of theresonance behavior. We begin by outlining the perturbation-theory models for the behaviornear E = U/n, starting with a few-site case, and then address the generalization to themany-particle case and the resulting many-body effects. We then specifically discuss thecomparison between experimental results and data from numerical simulations.

3.8.1 Perturbation theory solutions for the E2 = U/2 resonance

In Ref. [Sac02], a simplified spin model is derived describing the resonant dynamics near theE = U resonance for our type of system. The dynamics seen here at theEn = U/n resonancescan be considered in a similar fashion, by determining the corresponding resonant states andapplying perturbation theory to determine the tunneling amplitudes and energy shifts.

Here we discuss this in more detail. As a simple introduction, we first discuss resonantoscillations in a three-site Hubbard model for the E2 = U/2 resonance, showing how thecorresponding oscillation frequency is obtained. We then give an outline of the perturbation-theory description for multiple sites, and discuss the more complicated dynamics that arisein longer chains. In each case, we begin by identifying the manifold of atom configurationsthat have an equal energy (in the tilted system) to an initial state with a single atom on everylattice site, and that are connected by small numbers of processes in the lowest relevant orderin perturbation theory. This is analogous to the starting point for the analysis of dynamicsnear the E1 = U resonance in Ref. [Sac02], and comprises the manifold of states that will berelevant in the dynamics at moderate times in a parameter regime where J U,E.

Simple case: Three lattice sites

To the simplest approximation, we can discuss the dynamics atE2 on the basis of a three-site Hubbard model. Starting with one atom per site, |i〉 = |1, 1, 1〉, we let the system evolvewith E = E2 = U/2 for which it couples resonantly to the dipole state |f〉 = |2, 1, 0〉.For U J , the system is found to oscillate between |i〉 and |f〉 at a single frequencyν2 = 2(2

√2 +√

2)J2/(U/2). This result is most intuitively understood from second-ordertunneling, which constitutes the dominant process due to energetically suppressed nearest-neighbor tunneling. The effective coupling between |i〉 and |f〉 arises in second-order per-turbation theory via contributions from two virtual intermediate states

ν(1) = 〈f |Ja†1a2|1, 2, 0〉〈1, 2, 0|Ja†2a3|i〉/(U/2) (3.1)

ν(2) = 〈f |Ja†2a1|2, 0, 1〉〈2, 0, 1|Ja†1a2|i〉/(U/2) , (3.2)

63


0 0.05 0.1 0.150.0

0.5

1.0


104do

uble

occu

panc

yN

d10

4

UJ = 22.5

UJ = 42.5

UJ = 70.3

Figure 3.5: Double occupancy Nd as a function of hold time th for E = U/2 in a three-site Bose-Hubbard model at different values U/J . The dashed line shows the prediction in second-order per-turbation theory for large U/J giving a characteristic frequency ν2.

giving ν2 = 2(ν(1) + ν(2)). Here, a†i (ai) are the bosonic creation (annihilation) operators atthe i-th lattice site. The additional factor of 2 arises as ν2 gives the oscillation frequencies inthe probabilities of finding the system in |i〉 or |f〉. As discussed below, there is also a smallenergy shift in second-order perturbation theory for three sites.

General case: Occupation-dependent second-order tunneling

For longer chains at the E2 = U/2 resonance, the configurations that are resonantly cou-pled involve situations where particles from the initial state have tunneled over two lat-tice sites, e.g., a configuration |1, 1, 1, 1, 1〉 → |0, 1, 2, 1, 1〉, or |1, 1, 1, 1, 1〉 → |1, 0, 1, 2, 1〉, or|1, 0, 1, 2, 1〉 → |0, 0, 2, 2, 1〉, and so forth. There is also a constraint that particles separatedby two sites should not both tunnel simultaneously. In dynamics with chain lengths longerthan 3 atoms, this constraint will produce a variety of frequency scales in the dynamics,and lead to a rapid dephasing of oscillations, as was observed for the E1 = U resonancein Ref. [Mei13]. The dephasing observed in the experiments at the E2 and E3 resonances is,however, even stronger than that observed at the E1 resonance. This occurs because for thehigher-order resonances, interactions arise from additional sources.

The first additional source is that the sign and magnitude of the tunneling amplitude forany two-site tunneling process will depend on whether neighboring atoms (on either side)have tunneled. Let us consider the tunneling of the second atom in a chain, and look at thedifferent amplitudes that can arise dependent on the position of the remaining atoms. Takinga general configuration of three atoms and considering the relevant amplitudes if we transfera particle from the second site to the fourth site, we find the following possible amplitudesdepending on the positions of the remaining atoms:

64


• |1, 1, 1, 1, 1〉 → |1, 0, 1, 2, 1〉: There are two ways of obtaining this process in secondorder, |1, 1, 1, 1, 1〉 → |1, 0, 2, 1, 1〉 → |1, 0, 1, 2, 1〉 or |1, 1, 1, 1, 1〉 → |1, 1, 0, 2, 1〉 →|1, 0, 1, 2, 1〉 so that we obtain an amplitude (2

√2J2 +

√2J2)/(−U/2), as discussed

above.

• |1, 1, 0, 1, 2〉 → |1, 0, 0, 2, 2〉: There is one way of obtaining this term, with amplitude(√

2J2)/(U/2).

• |0, 1, 2, 1, 1〉 → |0, 0, 2, 2, 1〉: There are two ways of obtaining this, |0, 1, 2, 1, 1〉 →

|0, 0, 3, 1, 1〉 → |0, 0, 2, 2, 1〉 or |0, 1, 2, 1, 1〉 → |0, 1, 1, 2, 1〉 → |0, 0, 2, 2, 1〉, and we obtainan amplitude (3

√2J2)(−3U/2) + (2

√2J2)/(U/2).

Energy offset terms

In addition to this, there are significant terms of order J2/U that arise in second orderin perturbation theory and are diagonal in the occupation configuration basis. For example,the energy of the state with occupation numbers |1, 1, 1, 1, 1〉 is shifted compared with theunperturbed energy (at second order in the tunneling) by−64J2/(3U). When we investigatethe energy values at second order in J/U of states that have unperturbed energies (in thelimit J → 0) equal to that of the state with occupations |1, 1, 1, 1, 1〉, we find that the spreadis comparable to the magnitude of the effective tunneling terms. Because of this, these termslead to an exceptionally rapid dephasing of any oscillations in the dynamics of doubly oc-cupied sites, which is seen in the calculations and measurements of the number of doublyoccupied sites as a function of time.

Comments on the resulting effective model

Up to now, we have written the perturbation theory in terms of occupation number con-figurations, and have looked at second-order perturbation theory within the context of theBose-Hubbard model. In Ref. [Sac02], a spin model was derived for the E = U resonance,where spin up and down denoted whether particles beginning at particular sites had tun-nelled or not. Here, beginning in a configuration with singly occupied sites, the dynamicsof a short chain could be modelled by a similar model, where spin up and down wouldnow denote atoms having tunneled or not tunneled over two sites, respectively. In that case,defining the standard Pauli matrices for a spin at position l as σxl , σyl and σzl , we would notonly obtain terms proportional to σxl terms that flip the spins, but in addition terms involv-ing σzl±1σ

xl that give an interaction-dependent spin flip. For short chains up to five sites, such

a model can be derived, however multiple tunneling events in longer chains can lead to par-ticle configurations that are outside the list of configurations that can be represented in thisway.

65


Up to now, there have been limited theoretical studies of the resulting effective modelsand their critical behaviour, in part because of the difficulty of realising this behaviour inexperiments. The present work should open new questions and avenues for theoretical andexperimental investigation of this and related models and their critical behaviour, as well asdynamical studies involving, e.g., finite-speed quenches both to and across the critical point.

3.8.2 Higher-order resonances: E3 = U/3 and beyond

Naturally, similar things will arise at E3 = U/3 and E4 = U/4, etc. In each case, the order ofperturbation theory for the coupling terms will increase (J3/U2 for E3 ≈ U/3 and J4/U3 forE4 ≈ U/4), and the corresponding spin models will become more complicated. In each case,the diagonal shifts in second-order (i.e., J2/U ) terms will remain, and other diagonal correc-tions at third and higher order can enter, again leading to strong dephasing of oscillationsand some suppression in the creation of doubly occupied sites, because the coupling termsare smaller than the energy shifts. We note that both in numerical simulations and in theexperimental data, the rate of doublon growth for E3 = U/3 is still seen to be approximatelyproportional to J3/U2 despite the presence of the second-order energy shifts, because thetunneling that gives rise to growth in the doublon number is generated by the third-orderterms.

3.8.3 Numerical simulation of the doublon number on resonance

For theE2 andE3 resonances, we present numerical data for the number of doubly occupiedsites as a function of time calculated within the Bose-Hubbard model augmented by a tilt[Sac02]

H = −J∑〈i,j〉

a†i aj +∑i

U

2ni (ni − 1) + E

∑i

ini + ε∑i

i2ni , (3.3)

where ni = a†i ai are the number operators and ε accounts for a weak harmonic confine-ment. These are computed using exact diagonalization (ED) for small system sizes and longtimes, or using time-dependent density matrix renormalization group (t-DMRG) methods[Vid04, Dal04, Whi04, Ver08] for longer system sizes but shorter times. In each case, the com-putations presented were converged in the numerical parameters including time steps andt-DMRG truncation error. We calculated the propagation of the system in time, beginningwith an initial state with one particle on each lattice site.

For the initial growth region, we found that beyond a very small number of particlesand lattice sites (ca. 6 sites), the initial growth of doublons for E2 = U/2 and E3 = U/3 col-lapses essentially on a single line when scaled with J2/(U/2) for the E2 resonance and withJ3/(U/3)2 for the E3 resonance. In the main article this was only calculated for compara-tively small system sizes. In Fig. 3.6 we show the same results for short times, now including

66


0 0.01 0.02 0.03 0.040.0

0.1

0.2


Ndno

rm.d

oubl

eoc

cupa

ncy0

4E=U2A

UJ = 42.5 HEDLUJ = 67.2 HEDL

UJ = 40 Ht-DMRGL

0 0.01 0.02 0.03 0.040.0

0.1

0.2


Ndno

rm.d

oubl

eoc

cupa

ncy0

4

E=U3B

UJ = 18.4 HEDLUJ = 22.5 HEDL

UJ = 20 Ht-DMRGL

Figure 3.6: Initial growth in the number of doublons normalized to the total number of particles forE = U/2 (A) and E = U/3 (B) when increasing the system size. Numerical data for a 10 site Bose-Hubbard chain computed via exact diagonalization (blue and green) is compared to a system with 30particles on 30 sites obtained by t-DMRG methods (black). The simulations include a weak harmonicconfinement ε/J = 0.01 for E = U/2 and ε/J = 0.005 for E = U/3.

t-DMRG simulations for 30 particles on 30 lattice sites, showing that this does not change inlonger chains.

The key noticeable discrepancy between the data presented in Fig. 3.3 of the main articleand the numerical simulations is the substantial oscillations in the numerical data after theinitial growth region. While the dynamics of these oscillations is not the main subject ofthis article, let us comment briefly on these based on our numerical calculations and therelationship to the experimental data. These oscillations are clearly much more pronouncedfor smaller system sizes, and we find that they become smaller (though not non-existent)for longer chains in our calculations. In the experiment, where many chains are measured inparallel there are three types of averages taking place automatically that are not accountedfor in the simulations, each of which lead to a suppression of the oscillations under averagingin our calculations.

The first is an average over chain length, because the trapping potential confining thesystem leads to different chain lengths in different parts of the system. The second is aver-ages over position within the harmonic trap, and the third is an average over positions ofmissing atoms in the initial state. We have simulated each of these contributions for smallsystem sizes around 10 lattice sites, and find that the exact form of the oscillations - includingtheir position and amplitude - depends significantly on the choice of trap strength, positionwithin the harmonic trap, and position of missing atoms in the chain. To give an exampleof the effects of averaging, in Fig. 3.7, we plot the same results as in Fig. 3.3 of the mainarticle, but now averaging the numerical data over trap positions. Here we simply shift thetrap either one or two sites up or one or two sites down the gradient potential, and averagethe results from the resulting five trap positions. It is clear from the figure that for the U/3

67


0 0.05 0.1 0.150.0

0.1

0.2


Ndno

rm.d

oubl

eoc

cupa

ncy0

4E=U2A

UJ = 42.5

UJ = 67.2

UJ = 70.3

0 0.1 0.20.0

0.1

0.2


Ndno

rm.d

oubl

eoc

cupa

ncy0

4

E=U3B

UJ = 18.4

UJ = 22.5

UJ = 32.6

Figure 3.7: Number of doublons normalized to the total number of particles as a function of hold timeth for E = U/2 (A) and E = U/3 (B) when averaging over trap positions. The harmonic confinementis set to ε/J = 0.01 for both E = U/2 and E = U/3. The grey shaded areas indicate results of the fitmethod to the numerical data used to extract estimates of the characteristic growth rate 1/τ .

resonance, this almost completely suppresses the oscillations (compare with the numericaldata shown in panels C and G of Fig. 3.3 of the main text). Nevertheless, the numerical datastill shows the characteristic scaling behavior.

3.8.4 Entanglement growth during the quench

One of the signatures of the many-body nature of the quench dynamics, and a means to seeclearly that more many-body states are involved than the simplified picture of single (es-sentially uncorrelated) tunneling, is the entanglement growth across the center of the chainduring the dynamics. We can compute this with t-DRMG methods, in terms of the von Neu-mann Entropy S = −Trρ log2 ρ of a reduced part of the system (containing half of thelattice sites) for which the reduced density operator is ρ. Quench dynamics in a many-bodysystem can produce rapid growth of entanglement, and we can see this clearly in Fig. 3.8,which shows the time dependence of S shortly after the quench to E = U and to E = U/2for half of a 30-site chain. We see similar behaviour here to that observed in other studies ofdynamics with quenches to transition points in a many-body system [Ami08, Tro12, Kol07].While we do not have the access to local measurements necessary to measure this directly,our experimental results, combined with a recent proposal showing how Renyi entropiescould be measured with a quantum gas microscope [Dal12] could form a basis for such mea-surements in the future.

3.8.5 Collective effects in the tunneling rates for higher-order resonances

In addition to the many-body dephasing and entanglement growth demonstrated in the pre-vious section, the other collective effect we observe here is an enhancement in the tunneling

68


0 0.5 1.0 1.50

1

2

3

4

5

6

hold time th ´ J

von

Neu

man

nE

ntro

pyS

Figure 3.8: Growth of the von Neumann entropy S as a function of time in the middle of a 30-site sys-tem described by the Bose-Hubbard model, shown for a quench to E = U (blue lines) and E = U/2(red lines) with U = 24.4J and beginning with 30 particles each on a separate lattice site. The calcu-lations are performed with t-DMRG methods, which intrinsically limit the growth of entanglement.For comparison we plot results with matrix size D = 100 (dashed lines) and D = 200 (solid lines) toseparate the behaviour of the system from limitations of the numerical method.

rate due to the presence of additional particles. In this section, we briefly discuss this en-hancement.

In the first paragraph of this supplementary material, the Rabi oscillation frequency forsecond-order tunneling in a three-site Hubbard model at E2 is given. The presence of anatom in the central site increases the frequency by a factor of 3 compared to a chain |1, 0, 1〉 (inthe limit U/J 1) due to Bose enhancement. Here, we discuss the corresponding collectiveeffects for the higher-order resonances at En with n ≥ 3. We numerically compare the timeevolution in a chain of n + 1 sites with initially unit occupancy (|1, 1, ..., 1, 1〉) tilted to En

to the dynamics in a system with two atoms on the outmost sites and n − 1 empty sites inbetween (|1, 0, ..., 0, 1〉). An example case with n = 3 is depicted in Fig. 3.9 A. Evidently,the coherent tunneling dynamics is strongly Bose enhanced for the unity filled chain. Fittingthe numerical data with a squared sinusoidal, sin2(ωt), yields a Bose enhancement factor inthe oscillation frequency between the two cases GBose = ω|1,1,1,1〉/ω|1,0,0,1〉 ≈ 10. We analyzethe numerical results beyond the third-order tunneling resonance analogously and plot theBose enhancement factorG as a function of the order of the tunneling resonance in Fig. 3.9 B.Remarkably, the data suggests an exponential scaling of G with n. The large collective effecton the tunneling rates arising from Bose enhancement is thus essential for the observationof the higher-order tunneling processes on realistic experimental timescales.

69


0 0.2 0.4 0.6-0.0

-0.5

-1.0


doub

leoc

cupa

ncy

Nd

A

æ

æ

æ

æ

æ

2 3 4 5 6

5

10

50

100

500

J2resonance order nJ2

Bos

een

hanc

emen

tfac

tor

G B

Figure 3.9: (A) Double occupancy Nd as a function of hold time th for E = U/3 in a four-site sys-tem either initially in state |1, 1, 1, 1〉 (red) or in state |1, 0, 0, 1〉 (black). Here, U/J = 40. (B) Boseenhancement factor G as a function of the order of the tunneling resonance n at En. The vertical axisis logarithmic.

Increasing the system size beyond n+1 sites atEn collectively enhances the effective tun-neling rate. For example at E = U the oscillation frequency in the number of created doublyoccupied sites increases by an additional factor of

√2 already for 3 sites [Mei13]. On the

second- and higher-order tunneling resonances the effect is comparatively small when com-pared to the Bose enhancement discussed above and within the experimental uncertainty.

Extracting estimates for the growth rate 1/τ from the numerical data

In the main article, we extract growth rates 1/τ from the numerical simulations shownin Figs. 3.3 (C) and (G) and compare them to our experimental data in Figs. 3.3 (D) and (H).Specifically, we fit the initial increase of doubly occupied sites in the numerical data with asaturated growth functionNmax

d (1−e−th/τ ) while fixing the steady-state valueNmaxd to what

we obtain in our experimental data. The uncertainty in the obtained values for 1/τ mainlyreflects experimental fluctuations in Nmax

d and is indicated by the grey shaded regions.

70

CHAPTER 4

PUBLICATION

Interaction-Induced Quantum Phase Revivals and Evidence for theTransition to the Quantum Chaotic Regime in 1D Atomic Bloch

Oscillations

Phys. Rev. Lett. 112, 193003 (2014)

F. Meinert1, M. J. Mark1, E. Kirilov1, K. Lauber1, P. Weinmann1,M. Grobner1, and H.-C. Nagerl1


Universitat Innsbruck, 6020 Innsbruck, Austria

We study atomic Bloch oscillations in an ensemble of one-dimensional tilted superfluidsin the Bose-Hubbard regime. For large values of the tilt, we observe interaction-inducedcoherent decay and matter-wave quantum phase revivals of the Bloch oscillating ensem-ble. We analyze the revival period dependence on interactions by means of a Feshbachresonance. When reducing the value of the tilt, we observe the disappearance of the quasi-periodic phase revival signature towards an irreversible decay of Bloch oscillations, indi-cating the transition from regular to quantum chaotic dynamics.

4.1 Introduction

The response of a single particle in an ideal periodic potential when subject to an externalforce constitutes a paradigm in quantum mechanics. As first pointed out by Bloch and Zenerthe evolution of the wave function is oscillatory in time rather than linear, due to Braggscattering of the matter wave on the lattice structure [Blo29, Zen34]. Yet, the observation

71

Interaction-Induced Quantum Phase Revivals and Evidence for the Transition to the QuantumChaotic Regime in 1D Atomic Bloch Oscillations

of such Bloch oscillations in condensed-matter lattice systems is hindered by scattering ofelectrons with crystal defects [Fel92], which causes rapid damping of the coherent dynamics,eventually allowing for electric conductivity [Ash76].

Ensembles of ultracold atoms prepared in essentially dissipation-free optical lattices guar-antee long enough coherence times [Mor06] to serve as ideal systems for the observation ofBloch oscillations [Dah96, And98]. Furthermore, unprecedented precise control over atom-atom interactions in a Bose-Einstein condensate (BEC), via e.g. Feshbach resonances [Chi10],allows to engineer controlled coherent dephasing of the atomic matter wave and even tocancel interactions entirely. Bloch oscillations have been studied with BECs in quasi one-dimensional ”tilted” lattice configurations created by a single retro-reflected laser beam withtypically thousands of atoms per lattice site [Mor01]. Here, comparatively weak interactionsresult in strong dephasing observed in a rapid broadening of the atomic wave packet in mo-mentum space [Gus08b, Fat08], and are understood by a modification of the wave functionin terms of a mean field [Wit05, Gus10]. In contrast, for sufficiently tight confinement whenonly single spatial orbitals are relevant, the microscopic details determine the quantum co-herent time evolution, covered within the Bose-Hubbard (BH) model [Jak98].

In this letter, we study atomic Bloch oscillations in the one-particle-per-site regime real-ized with a large ensemble of 1D superfluid Bose gases trapped in an array of ”quantumtubes” and subject to a tilted optical lattice. In particular, we observe regular interaction-induced dephasing and revivals of the quantum matter-wave field for strongly tilted lattices[Kol03a]. In contrast, for sufficiently small values of the tilt, an irreversible decoherence ofthe wave function in the presence of interactions is seen, giving a strong indication for theonset of quantum chaos [Buc03].

4.2 Bloch oscillations in the Hubbard model

At ultralow temperatures, much below the lattice band gap, our system is well described bythe one-dimensional BH Hamiltonian augmented by a tilt [Jak98, Kol03a]

H = −J∑〈i,j〉

a†i aj +∑i

U

2ni (ni − 1) + E

∑i

ini . (4.1)

As usual, a†i (ai) are the bosonic creation (annihilation) operators at the ith lattice site, ni =a†i ai are the number operators, J is the tunnel matrix element, and U is the on-site interactionenergy. The linear energy shift from site to site is denoted by E. Let us first discuss the caseof a strong force E J . For U = 0 the energy spectrum is the famous equidistant Wannier-Stark ladder, which gives rise to a single frequency, fB = E 1, contributing to the dynamicsof any arbitrary initial wave packet [Glu02]. This is the origin of Bloch oscillations for a


72


wave packet in momentum space that is periodically driven across the first Brillouin zoneand Bragg reflected at the zone edge with the Bloch frequency fB [Kol03b]. An intermediateinteraction energy U ≈ J causes splitting of the degenerate energy levels of the Wannier-Stark ladder into a regular pattern of energy bands and results in quasi-periodic coherentdecay and revival of the Bloch oscillations with a new fundamental frequency frev = U

[Kol03a, Buc03]. This can be understood with the previous assumption of a strong tilt (E J) for which the eigenstates of a single atom in the lattice coincide with the localized Wannierstates. The Stark localization of the wave function together with the discreteness of the siteoccupation number leads to an evolution of the mean atomic momentum

〈p〉(t) ∝ exp (−2n(1− cos(2πfrevt))) sin (2πfBt) , (4.2)

where n denotes the mean occupation number per lattice site 2 [Kol03a]. Note the close anal-ogy of the dynamical evolution to the experiment of Ref. [Gre02b]. For the sake of complete-ness, we stress that for sufficiently strong interaction energy (U J) and sufficiently smalltilt (E U ) the ground state is a Mott insulator for one-atom commensurate filling, whichshows resonant tunneling dynamics when subject to a sudden tilt [Mei13, Mei14].

4.3 Bloch oscillations of a non-interacting sample

Our experiment starts with a BEC of typically 8 × 104 Cs atoms prepared in the internalhyperfine ground state |F = 3,mF = 3〉 and trapped in a crossed-beam optical dipoletrap [Web02, Kra04]. The sample is levitated against gravity by a magnetic field gradientof |∇B| ≈ 31.1 G/cm. The BEC is loaded adiabatically into an optical lattice of three mu-tually orthogonal retro-reflected laser beams at a wavelength of λl = 2π/k = 1064.5 nmwithin 500 ms. During the lattice loading the scattering length is as = 115 a0. At the end ofthe ramp, the final lattice depth is Vx,y = 30ER in the horizontal and Vz = 7ER (J = 52.3Hz) in the vertical direction, where ER = 1.325 kHz is the photon recoil energy. This createsan array of about 2000 vertically oriented 1D Bose-Hubbard systems (”tubes”) at near unityfilling that are decoupled over the timescale of the experiment. The residual harmonic con-finement along the vertical z direction is measured to νz = 16.0(0.1) Hz. We now ramp as

slowly (with≈ 1.5 a0/ms) to values of typically 0 to 90 a0 by means of a Feshbach resonance[Mar11a], and thereby prepare the 1D systems near the many-body ground state for an on-site interaction energy U of typically 0 to 400 Hz. This constitutes the initial condition for theobservation of Bloch oscillations. Bloch oscillations are then initiated by quickly applyinga gravity induced tilt E = 1740(4) Hz through a reduction of the magnetic levitation field,giving a Bloch period TB ≡ 1/fB = 575(1)µs. After a variable hold time th we switch off

2The analytic expression is derived approximating the initial state with a product of Bloch waves with quasimomentum k = 0.

73


HaL2Ñk HbL HcL

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æ

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æ

æ

ææ

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0 5 10 15 20

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Figure 4.1: Interaction-induced collapse and revival of atomic Bloch oscillations in a large ensembleof 1D tubes. The lattice depth along the tubes is Vz = 7ER (J = 52.3 Hz). Time-of-flight absorptionimages of one Bloch cycle are shown after th = 0, 7, 14TB for zero interaction (a-c) and for as =21.4(1.5) a0 (e-g), respectively. The vertical bar in (a) indicates the extent of the first Brillouin zone.Full time evolution of the mean atomic momentum is shown for zero interaction (d) and for as =21.4(1.5) a0 (h). Solid lines are fits to the data using the analytic model function (see text). The shadedareas indicate the data points shown in the time-of-flight pictures.

74


the lattice beams within 300µs, remove the tilt, and allow the sample for a free levitated ex-pansion of 30 ms to measure the atomic momentum distribution by standard time-of-flightabsorption imaging. During expansion, as is set to zero to avoid any additional broadeningdue to interactions.

First, we study a non-interacting sample by setting as = 0 a0 to quantify the effect of theresidual harmonic trapping potential. Time-of-flight absorption images spanning one Blochoscillation cycle are shown in Figs. 4.1(a)-(c) after th = 0, 7, and 14 TB . Note that the aspect ra-tio has been adapted for better visualization to compensate the faster expansion transversalto the orientation of the tubes. The typical linear motion of the atomic wave packet throughthe first Brillouin zone together with Bragg reflections at the zone edge shows a slow de-phasing due to the harmonic confinement. The mean atomic momentum 〈p〉 extracted fromtime-of-flight images is depicted in Fig. 4.1(d) as a function of th. We model the dephasingeffect of the harmonic trapping potential by a local variation of the Bloch frequency over theextent of the initial wave packet (see methods chpt. 4.7).

4.4 Interaction-induced quantum phase revivals

To demonstrate the effect of atom-atom interactions on the dynamics, we repeat the abovemeasurement now with the sample prepared at as = 21.4(1.5) a0, corresponding to U =102(8) Hz. Analogously to the previous measurement, time-of-flight images of full Bloch os-cillation cycles after th = 0, 7, and 14 TB are shown in Figs. 4.1(e)-(g). In contrast to thenon-interacting sample, we observe a rapid initial decay of the Bloch dynamics, resulting ina spreading of the atomic cloud over the entire Brillouin zone, see Fig. 4.1(f). Following thedynamical evolution we find a near perfect recovering of the Bloch oscillations as evidentfrom Fig. 4.1(g). This arises from the strong coherent dephasing of the initial atomic wavepacket in the presence of atom-atom interactions discussed above, leading to a subsequenthigh contrast matter-wave phase revival. The mean atomic momentum 〈p〉 as a function ofth is plotted in Fig. 4.1(h) and fit by the model function Eq. (4.2) including the overall de-caying envelope discussed above to account for the residual harmonic trap (see methodschpt. 4.7). We do such measurements at different values for as. The values for frev and fB

extracted from the two-mode fit function are depicted in Fig. 4.2(a). While fB is not affectedby interactions, frev increases linearly with as and is in good agreement with the predic-tion for U calculated from lowest-band Wannier functions [Jak98]. Two additional datasetsshowing the evolution of the atomic sample in the tilted lattice are shown in Fig. 4.2(b) and(c) for intermediate and comparatively strong interaction energy, respectively, in order todemonstrate the experimental robustness of the quantum phase revival. While we find clearseparation between the timescales given by fB and frev and observe four distinct decay andrevival periods of the matter-wave packet in Fig. 4.2(b), Bloch and revival period start to

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Figure 4.2: Dependence of the revival frequency frev on the on-site interaction energy U . (a) Extractedfrev (circles) and Bloch frequency fB (triangles) from model fits of Eq. (4.2) to time traces 〈p〉(th) asshown in Fig. 4.1 as a function of as. Error bars are smaller than the data points. The shaded gray areadepicts the calculated U taking into account a 5% error on the lattice depth. The solid line denotes aconstant fit to the data for fB . Mean atomic momentum as a function of hold time for as = 40.6(1.5) a0

(b), giving U = 194(10) Hz, and as = 111.5(1.5) a0 (c), giving a comparatively large U = 533(15) Hz.Solid lines are fits to the data based on Eq. (4.2).

mix when U is increased to ≈ E/3 in Fig. 4.2(c). Interestingly, the model function, Eq. (4.2),still provides a surprisingly precise fit to the data. Note that our technique allows the directmeasurement of the Bose-Hubbard interaction parameter U in the superfluid regime downto very small values, limited only by the residual external confinement.

4.5 Decoherence in the quantum chaotic regime

So far, we have restricted the discussion to large values of the tilt. In the remainder of this pa-per, we study Bloch oscillations in the regime when all energy scales in the BH Hamiltonianare of comparable magnitude, E ≈ J ≈ U . Consequently, it is impractical to assign a mean-ingful set of quantum numbers to the energy levels in a perturbative approach. Instead, theenergy spectrum emerges densely packed and requires a statistical analysis, revealing theonset of quantum chaos in a Wigner-Dyson distribution of the energy level spacings for suf-ficiently small E [Buc03]. The transition from the regular to chaotic regime is predicted tobecome manifest in a rapid irreversible decoherence of Bloch oscillations within a few oscil-lation cycles. To probe this regime, we prepare the sample in a more shallow lattice Vz = 4ER(J = 114.2 Hz) at a fixed value of U = 106(8) Hz, and now vary E. Two example datasets areshown in Fig. 4.3(a) taken at E = 855(15) Hz and 266(5) Hz, respectively, which show verydifferent qualitative behavior. For the strongly forced lattice we clearly identify the regulardecay and revival dynamics described above. In contrast, for smaller E the revival, expectedto appear at th = 1/U , is missing. Instead, we observe a single, rapid irreversible decay of

76


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Bloch oscillations. This observation is quantified twofold. First, we fit the initial decay in ourdata to a decaying sinusoid with an envelope ∝ exp(−t/τ). The extracted Bloch frequencyfB and the exponential decay time τ as a function of E are plotted in Fig. 4.3(b) and (c) (cir-cles). Second, we extract the amplitude of the revival δp in momentum space 3 and show itin Fig. 4.3(c) (triangles). As expected, fB is directly given by E. Further, τ is found constantfor E & 600 Hz, quickly increases, and finally saturates for E . 400 Hz. The revival signal

3We evaluate δp =qP

i(〈p〉i − 〈p〉)2/(N − 1) where the summand is over the sample of N datapoints in the

range 8.8ms ≤ t ≤ 10.8ms, and 〈p〉 denotes the mean over the sample.

77


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Figure 4.4: Dependence of the irreversible decay of Bloch oscillations on atom-atom interaction in thechaotic regime. (a) Mean atomic momentum as a function of hold time for as = 4.5(1.5) a0 (squares),as = 20.8(1.5) a0 (triangles), and as = 83.4(1.5) a0 (circles) at E = 346(10) Hz. Here, Vz = 4ER(J = 114.2 Hz). Solid lines are exponentially damped sinusoidal fits to the data. (b) Exponential decayrate 1/τ as a function of as extracted from datasets as shown in (a). The dashed line shows a predictionfrom numerical simulations for a homogeneous system without a trap.

δp exhibits an opposite behavior and decreases with decreasing tilt.

We interpret our data as a strong indication for the transition from the regular to thequantum chaotic regime. For large E/J & 6, we find a coherent dephasing and revival ofBloch oscillations with a characteristic dephasing timescale that is solely given by U and notaffected by the strength of the tilt, as expected from Eq. (4.2). With decreasing E, the revivalcontinuously disappears accompanied by a significant change of τ , indicating the onset ofa regime where the system dynamics decay irreversibly for E/J . 3. A spectral analysis ofthe tilted BH model for our system parameters (n & 1) reveals the onset of Wigner-Dysonstatistics in agreement with the observed crossover in τ (see methods chpt. 4.7). In additionwe compare our data in Fig. 4.3(c) to results obtained from numerical simulations of Blochoscillations within the BH model, indicated by the shaded areas. We emphasize again thatthe irreversible decay is due to interaction-induced decoherence in the system arising fromthe multitude of avoided crossings in the chaotic level structure of the many-body energyspectrum and ought to be contrasted from a coherent dephasing as observed for E J

[Buc03]. In this sense, the atomic ensemble itself acts as the bath responsible for decoherenceof the quantum many-body system [Rig08, Pol11].

Finally, we report on a last set of measurements to emphasize the role of interactionson the irreversible decay discussed above. We study Bloch oscillations for a fixed tilt E =346(10) Hz at Vz = 4ER (J = 114.2 Hz) and vary U . Three example datasets are shownin Fig. 4.4(a). We identify an overall irreversible decay indicative for the chaotic regime.

78


Moreover, the decay rate strongly depends on the interaction strength and decreases withdecreasing U . This is expected from the non-interacting limit U → 0, for which the systemturns regular again. Note that the observation of decay rates . 70 Hz is currently limited bythe overall dephasing due to the presence of the harmonic trap discussed above. We extractthe decay rate 1/τ as before and plot it as a function of U in Fig. 4.4(b). Our data reveal asaturating monotonic increase of 1/τ with U . Scaling of the decoherence rate with U is ex-pected to change from a quadratic (regular regime) to a square-root (chaotic regime) depen-dence [Ven11], indicated by the dashed line in Fig. 4.4(b) (see methods chpt. 4.7). However,a precise experimental confirmation requires compensation of the harmonic trap to allowfor longer observation times of the Bloch oscillations and will be the issue of a forthcomingexperiment.

4.6 Conclusion

In conclusion, we have studied Bloch oscillations in the context of a strongly interactingmany-body system properly described within the Bose-Hubbard model. In this regime the”granularity” of matter in combination with strong atom-atom repulsion causes coherentquasi-periodic decay followed by a high contrast quantum phase revival of the Bloch oscillat-ing matter-wave field. The revival period is entirely determined by the interaction strengthand thus provides a direct, precise measure for the on-site interaction energy in the super-fluid regime, in contrast to a related technique to measure U in a Mott insulator [Wil10]. Forpractical applications we point out that the phase revivals effectively extend the observationtime of Bloch oscillations even in the presence of interactions with potential prospects toe.g. precision force measurements [Car05, Mah14]. Further, we have investigated the Blochdynamics of the interacting atomic ensemble as a function of the applied tilt and foundclear evidence for the transition from regular to quantum chaotic dynamics. Our resultsmay open the experimental study of quantum chaos in such systems including its implica-tion on the decoherence and thermalization of interacting 1D quantum many-body systems[Rig08, Pol11, Kin06]. Moreover, quantitative studies on the parameter dependence of thetransition from the regular to the quantum chaotic regime are of interest [Eck11].

We are indebted to R. Grimm for generous support, and thank M. Hiller and A. Buchleit-ner for fruitful discussions. We gratefully acknowledge funding by the European ResearchCouncil (ERC) under Project No. 278417.

79


0.5 0.7 0.9 1.1 1.3 1.5

-0

-1

-2

-3

-4

-5

-6

tilt E

ener

gyHaL

4.5 4.7 4.9 5.1 5.3 5.5

-5

0

5

10

15

tilt E

ener

gy

HbL

Figure 4.5: Central part of the energy spectrum of the tilted BH Hamiltonian as a function of E withJ = U = 1 and N = L = 5. The regime of small tilt (a) is contrasted to the situation of large tilt (b).


Here, we provide theoretical background to the observed transition from a regular decayand revival to an irreversible decay of Bloch oscillations in the experiment. First, we discussthe statistical analysis of the energy spectrum of the tilted BH Hamiltonian [Buc03] in theparameter regime of Fig. 4.3 of the main article. Second, we provide numerical simulationsof Bloch oscillations within the BH model that can be directly compared to the experiment.

4.7.1 Spectral analysis

Following the approach presented in Ref. [Buc03], we evaluate the energy spectrum of thetilted BH Hamiltonian, Eq. 4.1 of the main article, as a function of the applied tilt E. We setJ = U = 1 in the calculation, which approximates the experimental situation of Fig. 4.3. InFig. 4.5(a) and (b) the central part of the energy spectrum is plotted as a function of E (givenin units of J) in the regime of small and large tilt for a system of N = 5 atoms in L = 5lattice sites. One clearly identifies the strong non-perturbative mixing of energy levels whenall energy scales J , U , and E are of comparable magnitude. For a fixed value of E we now

80


0 1 2 3 40.0

0.5

1.0

s

IHsL

HaL

1 2 3 40.0

0.5

1.0

tilt E

ΣP

,GO

E

HbL

1 2 3 40.0

0.5

1.0

tilt E

ΣP

,GO

EHcL

1 2 3 40.0

0.5

1.0

tilt E

ΣP

,GO

E

HdL

Figure 4.6: Statistical analysis of the energy spectrum. (a) Cummulative nearest neighbor level spac-ing distribution I(s) for E = 1 (black), 3 (red), and 5 (blue) with U = J = 1 for a system of N = 7atoms on L = 7 lattice sites. The dashed and dash-dotted lines show Poisson and GOE cummulativedistributions, respectively. (b-d) Crossover from regular to chaotic level statistics when decreasingthe tilt E for fixed U = J = 1. The deviation σP,GOE of I(s) from Poissonian (black) and GOE (red)cummulative level spacing distribution is plotted as a function of E for a system with N = 6, L = 8(b), N = L = 7 (c), and N = 8, L = 6 (d).

compute the cumulative nearest neighbor energy level spacing

I(s) =∫ s

0P (s)ds (4.3)

where P (s) is the probability distribution of the normalized nearest neighbor energy levelintervals ∆E/∆E. The result for I(s) for three different values ofE, calculated in the centralpart of the spectrum, is shown in Fig. 4.6(a) for a system of N = L = 7. The numerical datais compared to Poissonian and Wigner-Dyson statistics for a Gaussian orthogonal ensemble(GOE), which are given by PP(s) = exp(−s) and PGOE(s) = π

2 s exp(−πs2/4), respectively.We identify the change from Poissonian to GOE level statistics with decreasing E, an indica-tion for the onset of quantum chaos [Buc03, Gia91].

Next, we characterize the crossover from regular to chaotic level statistics in more de-tail by evaluating the deviation of the numerically computed I(s) from Poissonian andGOE statistics σP,GOE = 1/N

∫∞0 (I(s)− IP,GOE(s))2 ds. The normalization constant is N =∫∞

0 (IP − IGOE)2 ds. The result for three different mean site occupation numbers n = 0.75, 1,and 1.33 is shown in Figs. 4.6(b)-(d), respectively. The numerical data for σ clearly reflect the

81


crossover from Poissonian to GOE level statistics with decreasing E. We notice a systematicshift of the crossover to larger values of E with increasing n. The transition to irreversiblydecaying Bloch oscillations, as observed in the experiment, is found in good agreement withthe onset of chaotic spectral statistics for n & 1.

4.7.2 Numerical simulations of Bloch oscillation dynamics

We simulate Bloch oscillation dynamics via exact diagonalization of the finite size tilted BHsystem for our experimental parameters in analogy to Ref. [Ven11]. ForE J, U tunneling issuppressed and thus the time evolution of the initial many-body state is essentially driven byon-site dynamics. This changes when the tilt is decreased toE ≈ J, U . Tunneling then causesboundary effects in the simulation arising from the finite size of the system. Therefore, wefirst calculate the ground state for a non-tilted lattice with N atoms on L sites for a certaincombination J and U , which we place into a larger lattice with M > L sites. This constitutesthe initial condition at t = 0. We then compute the time evolution with a superimposed tiltE. From the many-body wavefunction |Ψ(t)〉 the mean atomic momentum

〈p〉(t)~k

=m

N~k4Jdi~〈Ψ(t)|

M−1∑j=1

a†j+1aj + h.c.|Ψ(t)〉 (4.4)

is derived. In Fig. 4.7 we plot the computed 〈p〉 as a function of time t for different E andcompare it to the experimental datasets that are analyzed to produce the data shown inFig. 4.3 of the main article. For E U, J the simulation data exhibit Bloch oscillations withpronounced decay and revival dynamics. With decreasing E the amplitude of the revivaldecreases and finally vanishes, resulting in an irreversible decay of the Bloch oscillations.Additional small oscillations can be attributed to the finite size of the sample. Note that thecrossover regime in the Bloch dynamics is accompanied by the onset of chaotic level statisticsin Fig. 4.6(c).

The generic behavior in the experimental data is closely reproduced by our numericalsimulation. The main difference is the lack of the overall damping due to the weak harmonicconfinement. Including this effect in the calculation extends the simulation to experimen-tal system sizes, which requires much more elaborate numerical methods. For the sake ofa quantitative comparison we model the trap by multiplying the numerical data with thesingle-particle damping exp(−t2/(2τ2

0 )) discussed in the main text and in the following sec-tion. From that we proceed in analogy to the experimental data analysis and extract a char-acteristic exponential dacay time τ of the initial dacay in 〈p〉(t) and the revival amplitude δp.The shaded areas in Fig. 4.3(c) of the main text reflect numerical results with varying meansite occupation 1 ≤ n ≤ 1.4. Numerical data for E . 300 Hz are excluded from this analysisas the assignment of an exponential damping rate becomes delicate when the Bloch period

82


200

400

600

800

tilt E HHzL

05

1015time t HmsL

-0.30.00.3

Xp\HÑkL

HaL

200

400

600

800

tilt E HHzL

05

1015

20hold time th HmsL

-0.30.00.3Xp\

HÑkL

HbL

Figure 4.7: Transition from a regular decay and revival to an irreversible decay of Bloch oscillation inthe experiment (b) and numerical simulation (a). The experimental datasets in (b) are the ones usedfor Fig. 4.3 of the main article. The simulations in (a) show results for a system of initially N = 7atoms in L = 7 lattice sites that is propagated in time in a lattice with M = 10 sites. Here, U/J = 0.93,which corresponds to the experimental parameters.

exceeds τ .

In Fig. 4.4(b) of the main article, we provide a numerical prediction of γ = 1/τ that isbased on simulations of Bloch oscillation dynamics now with varying U at fixed E/J = 3.03in a system with N = L = 7 and M = 9 4. Fitting a proper decay rate to the simulateddynamics is hindered by residual small oscillations due to the finite system size especially forlarge values of U for which strong damping ensues. Alternatively, we fit the first minimumin p(t) at tm and evaluate γm = − ln (p(tm)/pU=0(tm)) 1/tm [Ven11].

4Note that for this value of E it appears sufficient to enlarge the initial lattice by one site in both directions tosuppress boundary effects.

83


4.7.3 Modeling the effect of the harmonic trap

We model the dephasing effect of the harmonic trapping potential by a local variation of theBloch frequency over the extent of the initial wave packet, giving

〈p〉(t) ∝∑i

exp(−i2/(2γ2)) sin ((2πfB + νi/~)t) .

Here, ν = m(2πνz)2d2, with m the mass of the Cs atom, and d = λl/2 the lattice spacing. Thewave packet is approximated by a Gaussian envelope of half width γ. Replacing the sum byan integral yields 〈p〉(t) ∝ exp(−t2/(2τ2

0 )) sin(2πfBt), with τ0 = ~/γν [Pon06, Mar11b]. A fitof the model to our data presented in Fig. 4.1(d) of the main article delivers τ0 = 11.0(0.2)ms, from which we deduce a mean extension of the matter wave over ≈ 30 lattice sites, inagreement with what we expect from our loading procedure.

84

CHAPTER 5

PUBLICATION

Observation of Density-Induced Tunneling

Phys. Rev. Lett. 113, 193003 (2014)

O. Jurgensen1, F. Meinert2, M. J. Mark2,H.-C. Nagerl2, and D.-S. Luhmann1

1Institut fur Laser-Physik, Universitat Hamburg, Luruper Chaussee 149,

22761 Hamburg, Germany2Institut fur Experimentalphysik und Zentrum fur Quantenphysik, Universitat Innsbruck,

6020 Innsbruck, Austria

We study the dynamics of bosonic atoms in a tilted one-dimensional optical lattice andreport on the first direct observation of density-induced tunneling. We show that the inter-action affects the time evolution of the doublon oscillation via density-induced tunnelingand pinpoint its density- and interaction-dependence. The experimental data for differentlattice depths are in good agreement with our theoretical model. Furthermore, resonancescaused by second-order tunneling processes are studied, where the density-induced tun-neling breaks the symmetric behavior for attractive and repulsive interactions predicted bythe Hubbard model.

5.1 Introduction

The Hubbard model is the primary description for strongly correlated electrons in solids.It takes into account the interaction between the particles at a lattice site and the tunnel-ing between the sites, whereas other interaction processes are neglected. It was pointed outthat these additional interactions may have crucial influence in strongly correlated materials

85


such as superconductors or ferromagnets [Hir89, Str93, Hir94, Ama96]. Of particular impor-tance is the so-called bond-charge interaction that represents a density-induced tunneling ofan electron. In solids, this interaction-driven process cannot be studied systematically due tothe lack of direct control over the electron density and the interaction strength. Furthermore,the complexity of the investigated materials hinders a direct observation of this interactioneffect. Hence, the role of interaction-induced tunneling has remained an open question incondensed matter physics.

Ultracold atoms in optical lattices allow the realization of extremely pure lattice systemswithout defects and phononic excitations. Furthermore, the unique control of both the latticepotential and the interaction strength permits a systematic study of static and dynamic prop-erties. In optical lattices, density-induced tunneling [Maz06, Mer11, Luh12, Jur12, Dut15] iseven more pronounced due to the characteristic shape of the Wannier functions [Luh12].Several indications for density-induced tunneling have been found: It has a strong influenceon the superfluid to Mott-insulator transition in bosonic [Luh12, Pil12, Mar11a] and mul-ticomponent systems such as Bose-Fermi mixtures of atoms [Osp06, Gun06, Bes09, Mer11,Jur12, Luh08]. As a tunneling process, it also modifies the effective band structure, which hasalso been observed in a Bose-Fermi mixture [Hei11]. A direct observation of density-inducedtunneling processes was hindered mainly by the fact that the Mott insulator transition de-pends only on the ratio of on-site interaction and total tunneling and by the averaging overdifferent on-site occupancies in experimental systems.

5.2 Experimental manifestation of density-induced tunneling

Here, we report on the direct observation of density-induced tunneling with ultracold atomsin a tilted one dimensional optical lattice. We study the dynamics of a 1D Mott insulator afterquenching the tilt energy E between neighboring sites into resonance with the on-site inter-action energy U . We show that the resulting resonant particle oscillation between neighbor-ing sites (see inset of Fig. 5.1) is driven by interaction-induced tunneling on top of conven-tional tunneling. The experimental control over the on-site occupancy and the interactionstrength via a Feshbach resonance [Mar11a] allows us to isolate the effect of interaction-induced tunneling and to study it systematically.

The observed oscillation frequency f0 is expected to be directly proportional to the tun-neling matrix element J and is plotted in Fig. 5.1 as a function of the atomic scattering lengthas. The plot shows that the tunneling rate is modified by the interaction strength and thedensity of the sample. The Hubbard model predicts a constant value for J (dashed lines inFig. 5.1). In contrast, density-induced tunneling ∆J changes linearly with as and the on-siteoccupancy n, contributing to the total tunneling energy via

Jtot = J + (ni + nj − 1)as∆J, (5.1)

86


−400 −200 0 200 40040

60

80

100

120

140

160

180

200

220

−400 −200 0 200 4000

50

100

150

200

250

300

350a) b)

Figure 5.1: Oscillation frequency f0 of the doublon number in a tilted lattice as a function of as forVz = 8ER (diamonds), 10ER (squares), and 12ER (circles) with initial on-site occupancy (a) n = 1 and(b) n = 2 (see insets). The dashed lines show the prediction of the standard Hubbard model. The solidlines depict the interaction dependence due to density-induced tunneling (see Eq. (5.1)). Theoreticalpredictions correspond to a 4% lower lattice depth (see methods chpt. 5.8). The experimental data of(a) is taken from Ref. [Mei13].

for a tunneling process involving neighboring sites i and j. The measured oscillation fre-quency agrees very well with this modified tunneling rate (solid lines in Fig. 5.1). Increasingthe on-site occupation from n = 1 to n = 2 enhances the slope of the experimental data inaccordance with Eq. (5.1). This allows us to identify the density-induced tunneling processunambiguously and shows that its amplitude can even be as strong as the one of conven-tional tunneling.

The experiments are performed as reported in detail in Ref. [Mei13]. Starting from a CsBose-Einstein condensate, we prepare a bosonic Mott insulator in a 3D cubic optical latticewith a lattice depth Vq = 20ER (q = x, y, z), where ER = h × 1.325 kHz denotes the photonrecoil energy, and h is Planck’s constant (see methods chpt. 5.8). Adjusting initial density,interactions, and external confinement during lattice loading allows us to prepare eithera clean one-atom- or a two-atom-per-site Mott shell (see methods chpt. 5.8). We set as tothe desired value (−400 a0 ≤ as ≤ +400 a0) by means of a Feshbach resonance. Tunnelingdynamics along 1D chains is initiated by first setting the tilt E along the vertical z-directionvia a magnetic force |∇B| and then quickly lowering Vz along the direction of the tilt. Wemeasure the number of doubly occupied sites (doublons) after a variable evolution time ththrough Feshbach molecule formation and detection [Mei13]. On resonance (E ≈ U ) thedoublon number exhibits large-amplitude oscillations. For n = 1 the oscillation frequencyf0 is deduced from a damped sinusoidal fit to the data [Mei13]. In Fig. 5.1(a) we give f0

as a function of as for three different Vz from data sets taken for Ref. [Mei13]. Time tracesfor n = 2 need a more refined spectral analysis (see below). Fig. 5.1(b) plots the (mean)

87


frequency deduced from measurements with n = 2 as a function of as for Vz = 10ER andVz = 12ER. The observed frequencies clearly depend on both the interaction strength as andthe on-site occupancy n in the lattice. The Hubbard model (dashed lines) predicts constantJ , which solely depends on the lattice depth Vz [Jak98], and cannot reproduce this behavior.

5.3 The extended Hubbard model

For the theoretical description of the experiment we make use of the generalized HubbardHamiltonian for the one-dimensional lattice including density-induced tunneling, which isgiven by

H = −J∑i

b†i bi+1 + c.c.+U

2

∑i

ni(ni − 1) + E∑i

nii

−as ∆J∑i

b†i (ni + ni+1)bi+1 + c.c. (5.2)

with the tunneling matrix element J , the on-site interaction U , a tiltE per lattice site, bosonicannihilation (creation) operators b(†)i on site i, and ni = b†i bi. The first line is the standardBose-Hubbard model, whereas the second line represents the density-induced tunneling op-erator. It originates from the two-body interaction operator and represents the dominantoff-site contribution for neutral atoms in optical lattices [Luh12, Jur12, Dut15]. Its amplitude

∆J = −4π~2

m

∫d3r w∗(r− d)w∗(r)w2(r), (5.3)

is determined by the Wannier functions w(r) of the lowest band of the lattice with the lat-tice spacing d using a δ-shaped interaction potential. This tunneling operator is explicitlyoccupation-dependent due to the factor (ni + ni+1). Assuming that the time-evolution onneighboring sites is predominantly given by a constant total occupation ni + ni+1 = 2n, wecan define an effective total tunneling operator as

Jtot = −Jtot∑i

b†i bi+1 + c.c. (5.4)

using Eq. (5.1), which allows us to retrieve the standard Bose-Hubbard model with a modi-fied tunneling rate Jtot.

5.4 Numerical simulation of the resonant tunneling dynamics

It is a priori not clear that conventional and density-induced tunneling can be combined toone total hopping process. To verify this simplification, we perform exact numerical simu-

88


0

0 100 2000

1

1

20 40 60 80 1000

100

200

300

400

500

−400 −200 0 200 400

40

60

80

100

120

a)

b) c)

0 300

3

0 300

3

Figure 5.2: (a) Fourier spectrum of the simulated doublon number dynamics for E = U without (up-per panel) and with (lower panel) inclusion of density-induced tunneling. Here, n = 1, Vz = 10ER,and as = 400 a0. The dashed (solid) line shows the predominant frequency component predictedwithout (with) density induced tunneling. The shaded areas incorporate a Gaussian broadening witha width w = 4 Hz for visualization purposes. The insets show the time trace for the first 30 ms. (b)Mode spectrum as a function of the single-particle tunneling rate J/h tuned via the lattice depth Vzfor as = 400a0 (w = 4 Hz). (c) Mode spectrum as a function of as at fixed Vz = 10ER (w = 1 Hz). In(b) and (c) the dashed line shows 4J/h, while the solid line shows 4Jtot/h, including density-inducedtunneling.

89


lations of the time evolution of the initial state by diagonalizing the generalized Hubbardmodel (5.2) for a finite lattice withN = 8 sites. Using the exact solution, we are not restrictedto short time traces, allowing us to resolve the full Fourier spectrum. We first discuss thecase of an initial on-site occupation n = 1. As an example, the insets in Fig. 5.2(a) show timetraces of the number of doublons (ni = 2) for Vz = 10ER and as = 400a0 at the resonanceE = U . Here, the number of triply occupied sites (triplons) is negligible. The Fourier spec-trum (Fig. 5.2(a)) contains a broad range of frequencies that are peaked around f0 = νJtot/h

(bottom) or around f0 = νJ/h (top) for the standard Hubbard model (∆J = 0) with theprefactor ν ≈ 4, as indicated by the vertical lines. By means of time-dependent DMRG of upto 40 sites, it has been shown in Ref. [Mei13] that the system size does not significantly affectthe characteristic oscillation frequency f0 for Mott chains beyond 3 sites but causes increasedmany-body damping with increasing system size (see also methods chpt. 5.8).

In Fig. 5.2(b), the frequency spectrum is plotted against the bare tunneling rate J/h forthe generalized Hubbard model (∆J 6= 0). The centroid of the two strongest modes matcheswith the total tunneling rate 4Jtot/h, while the dashed line corresponds to the standard Hub-bard model with 4J/h. Plotted as a function of the scattering length as (Fig. 5.2(c)), the differ-ence between standard and generalized Hubbard model becomes even more obvious. As theabsolute value of the on-site interaction is compensated by the resonance condition E = U ,the doublon dynamics is independent of as within the standard Hubbard model (dashedline). In contrast, the interaction-dependence of density-induced tunneling imprints a lineardependence on the observed frequency modes f0 ∝ J + as ∆J .

Although the theoretical spectrum of the time evolution contains several distinct fea-tures, we can conclude that the main frequency can be attributed to an oscillation with4Jtot/h. Fitting the experimental time traces for n = 1 with a damped oscillation [Mei13]allows us to extract this central frequency plotted in Fig. 5.1(a). The experimental data pointspinpoint the dependence on the interaction strength as discussed above and agree wellwith the generalized model including density-induced tunneling. Note that the interaction-induced admixture of higher-bands will lead to a slightly modified rate for the total tunnel-ing [Luh12, Jur12, Bis12, Dut15].

5.5 Density dependence

Let us now turn to the direct verification of the density dependence of the interaction-driventunneling process by preparing an initial state with on-site occupancy n = 2. Due to Jtot =(2n−1)as ∆J , the impact of the density-induced tunneling is expected to increase by a factorof three. In Fig. 5.3(a) we plot the number of doubly and singly occupied sites as a functionofE measured after th = 50 ms for two different values of U . Close to the expected resonanceposition at E = U , we observe two minima in the doublon number (Fig. 5.3(a)) that can be

90


0 200 400 0

100

200

300

400

500

0

1

100-100 300

b) c)

0 100 200 300 400 500 600 −100

100

400

0 200 400 600 800 1000 1200 1400 1600

0.5

1.0

1.5

2.0

0.4

0.8

1.2

1.6

a)

Figure 5.3: Doublon dynamics for on-site occupancy n = 2. (a) Number of doubly (upper lines) andsingly (lower lines) occupied sites as a function of E after th = 50 ms for as = 245(5) a0 (circles)and as = 354(5) a0 (diamonds) at Vz = 12ER, giving resonances for the |2, 2〉 ↔ |3, 1〉 oscillations atE22 = h× 940(20) Hz and E22 = h× 1283(20) Hz, respectively (see methods chpt. 5.8 for details). Thedashed lines indicate the calculated U and U/2. (b) Measured Fourier spectra extracted from timetraces of the doublon number at Vz = 10ER and E = E22 for different as. The solid blue lines resultfrom multiple Gaussian fits to the data (see methods chpt. 5.8). The shaded area projected into the axisplane depicts the simulated Fourier spectra shown in (c). The lines show the expected frequencies forthe standard (dashed) and extended (solid) Hubbard model (Eq. (5.1)). (c) Simulated Fourier spectraas a function of as incorporating a Gaussian broadening with a width w = 1 Hz. The markers denotepeak positions deduced from the experimental data shown in (b). The marker size scales linearly withthe area under the respective fits. Open circles indicate broad Gaussian fits with a large ratio of widthand amplitude w/A > 0.2.

attributed to the processes |2, 2〉 ↔ |3, 1〉 and |2, 0〉 ↔ |1, 1〉. The latter arises from tunnelingat residual defects (empty sites) in the n = 2 shell (see methods chpt. 5.8). The splittingof the resonance arises from corrections to the on-site energy U due to multi-orbital effects[Buc10, Wil10, Mar11a, Mar12, Luh12] causing an intrinsically occupation dependent on-site energy Un. While for the defect process |2, 0〉 ↔ |1, 1〉 the resonance is at E11 = U2,the process |2, 2〉 ↔ |3, 1〉 is resonant at E22 = 3U3 − 2U2 < U2. The resonances at E22/2correspond to the second-order tunneling process |2, 2, 2〉 ↔ |3, 2, 1〉. Second-order tunnelingto hole defects |2, 2, 0〉 ↔ |1, 2, 1〉 at E11/2 does not contribute, since the intermediate state|1, 3, 0〉 is strongly off resonant. For the measurement of the time traces we determine E22 fora fixed lattice depth and scattering length. Since the two resonances at E11 and E22 are notfully separated, we expect (off-resonant) contributions from defect sites to contribute withup-shifted frequencies [Mei13].

In Fig. 5.3(b) Fourier spectra extracted from the experimental time traces (see methodschpt. 5.8) are shown for three different values of as in the range −100 a0 ≤ as ≤ +400 a0.We obtain the main frequency modes using Gaussian fits (blue areas). While the lowest ob-servable frequency is caused by decoherence and particle loss, we can identify the dominantmode at the expected position f0 = ν(J + 3as ∆J) (red solid line), with ν = 4

√3 for the

|2, 2〉 ↔ |3, 1〉 process. For the two positive values of as, where the frequency range leads tobetter resolved peaks, a splitting of this mode can be observed. This splitting is in generalalso visible in the theoretical spectrum in Fig. 5.3(c), where the circles denote the experimen-tally extracted modes. Moreover, we can identify a mode with lower frequencies that could

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−400 −200 0 200 400

d)c)

500 25500 250

1

2

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5b)a)

120

60

0

Figure 5.4: (a) Time-averaged number of the sum of doublons and triplons (upper lines) and triplonsonly (lower lines) as a function of E for Vz = 8ER at as = 250a0 (blue) and as = −250a0 (red). Thesolid lines show numerical simulations including density-induced tunneling, while for the dashedlines ∆J = 0. The gray area indicates sampling over E in the experiment with an estimated widthof ≈ 50 Hz. (b) Numerical Fourier spectrum of the doublon dynamics as a function of as revealingthe breaking of symmetry between attractive and repulsive scattering for E = U/2. The dashed lineindicates f0 = ν(J + as∆J)(J + 2as∆J)/U with a prefactor ν = 19 in accordance with [Mei14].Experimental time traces at the U/2 resonance for (c) as = +250a0 and (d) as = −250a0 at Vz = 8ERshowing an asymmetric behavior between repulsive and attractive interaction caused by density-induced tunneling. The dashed lines are guides to the eye that indicate the steady-state doublonnumber.

probably be assigned to the mode around ν ≈ 3√

3/2 in the numerical spectrum. However,defect sites will effectively lead to decoupled chains with different lengths surrounded byunoccupied sites affecting the prefactor ν.

The extracted dominant frequencies are plotted for Vz = 10ER and Vz = 12ER (see meth-ods chpt. 5.8) in Fig. 5.1(b), where we use the centroid for split resonances. For both latticedepths, we see a good agreement with the theoretical expectation f0 = 4

√3(J + 3as ∆J). In

combination with results for n = 1, this serves as direct confirmation of the density depen-dence of the tunneling.

5.6 Density-induced tunneling and higher-order hopping

As has been demonstrated recently [Mei14], the dynamics in tilted lattices also allows thestudy of higher-order tunneling processes. In Fig. 5.4(a), the numerically determined dou-blon number, averaged over the time evolution, is plotted as a function of E, depicting sev-

92


eral distinct resonances at fractional values of U in accordance with the experimental ob-servation in Ref. [Mei14]. The resonance at E = U/2 is caused by second-order tunnelingprocesses via an intermediate site. In this case, the hopping to next-nearest neighbor sitesrestores the resonant tunneling condition (2E = U ), whereas direct nearest-neighbor tunnel-ing is suppressed. Higher-order resonances at E = U/n are caused by long-range tunnelingprocesses proportional to J(J/U)n−1. Density-induced tunneling causes a broadening of theresonances for repulsive interactions (blue line) and a narrowing for attractive interactions(red line).

The numerical Fourier spectrum for the resonant second-order tunneling dynamics isplotted in Fig. 5.4(b). A clear evidence for the impact of density-induced tunneling is thebreaking of the symmetry between attractive and repulsive interaction [Sak10] (see methodschpt. 5.8), which holds for the Hubbard model, i.e., ∆J = 0. This frequency shift can also beobserved in the experimental time traces in Figs. 5.4(c) and (d), where the faster initial in-crease for repulsive interactions indicates a higher frequency. In addition, we find a decreasein the average doublon number for attractive interactions, which we attribute to the reducedwidth of the second-order tunneling resonance (see Fig. 5.4(a)), as the variation of E acrossthe sample due to the small residual harmonic confinement and a variation in as due to themagnetic field gradient (gray area) is comparable with the resonance width.

5.7 Conclusion

We have presented the first direct measurement of density-induced tunneling of ultracoldatoms in optical lattices. We observe resonant doublon dynamics when compensating the in-teraction energy U by an applied tilt. The measured frequency exhibits a linear dependenceon the on-site occupancy and on the scattering length. Our numerical simulations show thatan extended Hubbard model incorporating the density-induced tunneling accurately de-scribes the experiment. For approximately constant densities both tunneling processes canbe described with a single effective amplitude J + (2n − 1)as ∆J that can differ stronglyfrom the conventional tunneling J . Furthermore, we have studied second-order tunnelingprocesses and observe an asymmetry between repulsive and attractive interactions causedby density-induced tunneling. This underlines its importance for exchange interactions thatare, e.g., responsible for antiferromagnetic properties in solids [And50]. Our results grantsfuture perspectives for detailed investigations of complex interaction effects caused e.g. byhigher orbitals and off-site interactions [Luh12, Bis12].

We are indebted to R. Grimm for generous support, and thank A. Daley for fruitfuldiscussions. We gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft(grants SFB 925 and GRK 1355) and the European Research Council (ERC) under Project No.

93


278417.


5.8.1 Lattice depth calibration and error bars

The lattice depth Vq is calibrated via Kapitza-Dirac diffraction. The statistical error for Vq is1%, though the systematic error can reach up to 5%.

The scattering length as is calculated via its dependence on the magnetic field [Mar11a]with an estimated uncertainty of±5a0 arising from systematics in the magnetic field calibra-tion and conversion accuracy. Additionally, the magnetic field gradient leads to a variationof less then ±3a0 across the sample.

5.8.2 Calculation of the occupation-dependent on-site energy shift

Occupation-dependent corrections to the on-site interaction energy Un arise from multior-bital effects and the regularization of the pseudopotential. It has been shown in Ref. [Mar11a]that the correct two-body interaction energy U2 and therefore the resonance condition E11

can be approximated by the analytic result of two atoms in a harmonic trap including arescaling due to the anharmonicity of the lattice potential. For E22 = 3U3 − 2U2 the analyticexpression E22 = U2/(1 + 1.34U2/(~ω)), with ~ω the band gap, has been found. In orderto calculate E22 for an anisotropic lattice well we use the geometric mean of the bandgapalong the three axis as value for ~ω and find good quantitative agreement to the measuredresonance positions as shown in Fig. 5.3(a) of the main article.

5.8.3 Experimental preparation of the n = 1 and n = 2 Mott shell

Adjusting the chemical potential of the harmonically trapped Bose-Einstein condensate viainitial density, interaction strength, and external confinement during the lattice loading al-lows us to control the relative occupation of the Mott insulating shells with occupation n = 1and n = 2. For the experimental data shown in Fig. 5.1(a) and Fig. 5.4 of the main article, weprepare a one-atom-per-site Mott insulator with less than 4% of residual double occupancy.For the data plotted in Fig. 5.1(b) and Fig. 5.3 of the main article we prepare a Mott insulatorwith a central two-atom-per-site Mott shell containing ≈ 4.4 × 104 atoms. We remove thesurrounding singly-occupied shell by combining microwave rapid adiabatic passage with aresonant light pulse after association of doubly occupied lattice sites to weakly bound Fesh-bach molecules. Subsequent to the cleaning, we dissociate the molecules again to free atoms.In total the cleaning procedure has an efficiency of 80(4)% [Mei13].

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−400 −200 0 200 400

40

60

80

100

120

−400 −200 0 200 400

a) b)

Figure 5.5: Comparison of mode spectra as a function of the scattering length between (a) the stan-dard Hubbard model and (b) the generalized Hubbard model with density-induced tunneling. Re-sults are shown for a chain of 8 sites at occupation n = 1 and lattice depth Vz = 10ER. The dashedlines depict the Hubbard model prediction f0 = 4J/h, whereas the solid lines are modified bydensity-induced tunneling f0 = 4(J + as ∆J)/h.

The finite cleaning efficiency results in defects in the n = 2 shell that are predominantlyempty sites or remaining Feshbach molecules, with an estimated defect density of ≈ 20%.In combination with tunneling at the chain boundaries this explains the relatively strongresonance corresponding to the process |2, 0〉 ↔ |1, 1〉 observed in Fig. 5.3(a) of the mainarticle.

5.8.4 Mode spectrum of the standard Hubbard model

In the experiment at resonant tilt E = U the on-site interaction U for a tunneling event in aMott insulator is exactly compensated for by the tilt energyE. Consequently, no dependenceof the dynamics on the scattering length is expected within the standard Hubbard model(dashed lines in Fig. 5.5). This is confirmed by our numerical simulations without density-induced tunneling, shown in Fig. 5.5(a). In contrast the generalized Hubbard model withdensity induced tunneling clearly shows the expected dependence of the resonant frequencyon the scattering length (solid lines).

5.8.5 Finite-size effects

The numerical simulations are performed using an exact diagonalization algorithm, which islimited to small systems of up to N ≈ 10 lattice sites. In Fig. 5.6 we illustrate the dependenceof the spectra on the system size for the extended Hubbard model with an initial occupancyof n = 1. For chains of length up toN = 4 finite-size effects play a very important role, whilelarger systems approximate the large-N limit. The common feature is a broad resonance

95


0 200100

Figure 5.6: Frequency spectra of the doublon dynamics for various system sizes N . The areas aresubject to a Gaussian broadening with a width of w = 4 Hz. The broad resonance centered aroundf0 = νJtot (indicated by the red vertical line) is characteristic for all system sizes N ≥ 5. The latticedepth is Vz = 10ER and the scattering length is as = 400 a0.

96


centered around the frequency f0 = νJtot, where the empirical factor ν ≈ 4 does not changesignificantly with the system size. For the majority of our numerical simulations for theoccupancy n = 1 we use N = 8 sites, whereas for n = 2 shorter chains are brought into playby hole defects (see below).

5.8.6 Fourier analysis of doublon dynamics at occupation n = 2

The dynamics of the n = 2 Mott insulator involves several distinct frequencies that we iden-tify via a Fourier analysis. The experimental data at the resonant tilt E = U , Vz = 10ER andas = 100 a0 is shown in Fig. 5.7(a). We enhance the resolution by mirroring the data at th = 0.We associate the first data point with a holding time of ∆th = 1 ms allowing for equidistantdata. The consecutive Fourier transformation is shown in Fig. 5.7(b) together with a fit ofmultiple (in this case seven) Gaussians. This procedure allows us to extract the resonantfrequencies from the doublon dynamics, which can be compared with the theoretical pre-dictions (see Fig. 5.3 of the main text). We attribute the lowest frequency resonance to losses.Especially broad resonances, such as the third and last one in Fig. 5.7(b), probably stem fromseveral resonances that cannot be resolved. From the fit in the Fourier space we reconstructthe time trace (solid line in Fig. 5.7(a)) and compare it with the original data, which gives usan estimate of the quality of the fit.

5.8.7 Role of defects for n = 2

After the applied cleaning procedure, we estimate an amount of p = 20% randomly dis-tributed empty defect sites in the initial n = 2 Mott insulator shell. Consequently, the one-dimensional lattice is divided into chains of the form |0, 2, ..., 2, 0〉, i.e., doubly occupied sitesbounded by hole defects, where the tilt energy increases from left to right. Within the doublyoccupied domain the process |2, 2〉 ↔ |3, 1〉 is dominant and on the right side of the chainthe process |2, 0〉 ↔ |1, 1〉 takes place. Tunneling processes to the leftmost site are generallyoff resonant, since they are associated with a gain in tilt energy that cannot be compensatedby an increase in interaction energy. Therefore, the leftmost hole site does not contribute tothe dynamics within this subsystem. As a consequence, the hole site on the left side sep-arates the one-dimensional lattice into chains of the form |2, ..., 2, 0〉, which can be treatedseparately. The only exception are chains with only one occupied site, where the transition|...2, 0, 2, 0〉 → |...1, 1, 1, 1〉 can occur via two resonant processes.

Neglecting this coupling and higher-order processes, we simulate the effect of hole de-fects by statistically summing up separated chains |2, ..., 2, 0〉 of a total length of N sites,where we assume a relative probability PN = p (1− p)N−1. The dynamic behavior of chainswith length N > 9 is approximated by that of the N = 9 chain, which is the upper limit forour numerical study.

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−20 −15 −10 −5 0 5 10 15 20

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200

400

600

data

fit

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0.6

0.8

1

1.2

1.4

1.6 data

fit

Figure 5.7: (a) Mirrored time trace of the doublon dynamics at Vz = 10ER and as = 100 a0 forn = 2. The solid line is the Fourier transform of the fit in frequency space shown in (b). (b) Frequencyspectrum calculated via Fourier transformation of the data shown in (a). The solid line is a fit usingmultiple Gaussians.

0 2 4 6 8 10 12 14 16 18 200.6

0.8

1

1.2

1.4

1.6

Figure 5.8: Comparison of experimental data with the corresponding numerically simulated timetrace for Vz = 10ER and as = −100 a0, where only the scaling factor and the offset are adjusted.

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−400 −200 0 200 4000

100

200

300

400

500

−400 −200 0 200 400

a) b)

Figure 5.9: Frequency spectra of the n = 2 Mott insulator as a function of the scattering length forVz = 9.6ER and E = U . (a) Statistical average of chains |2, ..., 2, 0〉. The solid lines mark the rangeν(J + 3as ∆J)/h with 3

√3 < ν < 5

√3. The dashed line in (a) shows the resonance at 2

√2(J +

as ∆J)/h. The markers are the resonances extracted from the experimental data (see Fig. 5.3(c) inthe main text). The spectra are obtained from the first 2.5 s of the time evolution with a Gaussianbroadening of w = 1 Hz. (b) Results for a defect free chain with 8 sites (compare Fig. 5.3(c) in themain text).

In Fig. 5.8, we show a direct comparison between statistically averaged theoretical andexperimental time traces. Deviations at long evolution times are attributed to the uncertaintyof the distribution of defects in the lattice, particle loss, as well as the contribution of off-resonant |2, 0〉 ↔ |1, 1〉 processes.

Figure 5.9 shows a comparison between the Fourier spectrum obtained for the statisti-cally averaged chains and for a defect free chain with N = 8 sites as discussed in the maintext. In both cases we observe a broad band of frequencies at ν(J + 3as ∆J)/h with ν in therange 3

√3 < ν < 5

√3 (red solid lines) and a number of weaker resonances with smaller

values of ν. The factor 3 in front of the density-induced tunneling term as ∆J allows to as-sociate the resonances with the main process |2, 2〉 ↔ |1, 3〉. In addition, a frequency modewith 2

√2(J + as ∆J) appears for the defect averaged spectrum (dashed red line), which

stems from the |2, 0〉 ↔ |1, 1〉 process in N = 2 chains. In the experiment, it is expected to besuppressed and shifted to higher frequencies for increasing |as| due to multiorbital effects.

We conclude that the dynamics of the n = 2 Mott insulator is not affected qualitativelyby defects and thus we restrict our discussion in the main text to defect free simulations.

99

CHAPTER 6

PUBLICATION

Probing the Excitations of a Lieb-Liniger Gas from Weak to StrongCoupling

Phys. Rev. Lett. 115, 085301 (2015)

F. Meinert1, M. Panfil2, M. J. Mark1,3, K. Lauber1,J.-S. Caux4, and H.-C. Nagerl1

1Institut fur Experimentalphysik und Zentrum fur Quantenphysik, Universitat Innsbruck,

6020 Innsbruck, Austria2SISSA-International School for Advanced Studies and INFN, Sezione di Trieste,

34136 Trieste, Italy3Institut fur Quantenoptik und Quanteninformation, Osterreichische Akademie der Wissenschaften,

6020 Innsbruck, Austria4Institute for Theoretical Physics, University of Amsterdam,

1090 GL Amsterdam, The Netherlands

We probe the excitation spectrum of an ultracold one-dimensional Bose gas of Cesiumatoms with repulsive contact interaction that we tune from the weakly to the strongly in-teracting regime via a magnetic Feshbach resonance. The dynamical structure factor, ex-perimentally obtained using Bragg spectroscopy, is compared to integrability-based cal-culations valid at arbitrary interactions and finite temperatures. Our results unequivocallyunderly the fact that hole-like excitations, which have no counterpart in higher dimensions,actively shape the dynamical response of the gas.

101

Probing the Excitations of a Lieb-Liniger Gas from Weak to Strong Coupling

6.1 Introduction

Interacting quantum systems confined to a one-dimensional (1D) geometry display quali-tatively different behavior compared to their higher-dimensional counterparts [Gia04]. Sys-tems of strongly interacting electrons recently realized in electronic nanostructure devices[Aus02, Bar10] have evidenced the breakdown of Landau’s Fermi liquid theory of quasi-particles in 1D, a world in which new types of excitations emerge out of the inevitablycollective nature of the dynamics. The understanding of these requires approaches goingbeyond Landau’s paradigm. The best-known, valid for sufficiently small temperatures andenergies, is the Luttinger Liquid (LL) formalism [Hal81]. When probing dynamical corre-lation functions, one however typically leaves this low-energy and large-wavelength limitand enters a regime where even recent extensions of the LL formalism to higher energies[Ima09, Ima12] cannot capture all features. Instead, one must rely on nonperturbative calcu-lations to understand the correct basis of excitations and quantitatively explain experiments,a recent example being spinon dynamics in quantum spin chains [Mou13, Lak13]. Thesesystems however lack the tunability required to track the whole transformation occurringbetween the limits of weak and strong coupling.

Very recently, systems of ultracold bosons have opened up new routes to study strongcorrelation effects in 1D [Caz11, Kin04b]. In particular, tuning the interaction strength [Hal09],as characterized by the dimensionless Lieb-Liniger parameter γ [Caz11], gives access to thefull range from weak (γ . 1) to strong (γ 1) interactions and the fermionized Tonks-Girardeau (TG) regime [Gir60, Kin04b, Hal09, Hal10a, Hal11]. Moreover, the 1D Bose gaswith contact interactions is one of the few integrable many-body problems that allows forcombining experiment with numerically exact studies of the excitation spectrum, making itan ideal setting for observing interaction effects on dynamical correlation functions [Fab15].In their seminal work [Lie63a, Lie63b], Lieb and Liniger have shown that next to a particle-like mode (Lieb-I mode), which resembles Bogoliubov excitations in the limit of weak in-teractions, a second mode naturally emerges (Lieb-II mode) that stems from hole-like exci-tations in the effective Fermi sea in 1D. The coexistence of these two types of elementaryexcitations leads to a significant broadening of the dynamical response functions, clearlyvisible in the strongly interacting regime [Cau06, Che06] (see Fig. 6.1).

In this Letter, we measure the dynamical structure factor (DSF) of the Lieb-Liniger Bosegas realized with ultracold atoms confined to 1D quantum tubes. Previous work [Fab15]has addressed a fixed intermediate interaction regime (γ ' 1). Here, tunability of inter-actions allows us to enter deeply into the strongly interacting TG regime. Our analysis isbased on careful disentangling of the experimental traits and allows us to identify the role ofthe Lieb-Liniger dynamics in shaping the response of the system. Comparison of the mea-sured spectra with state-of-the-art numerical calculations [Cau06, Pan14] ranging from theweakly to the strongly interacting regime allows for a clear distinction between interaction

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Figure 6.1: Sketch of the experimental setup. (a) A pair of retro-reflected laser beams creates an en-semble of ≈ 4000 independent one-dimensional Bose gases. (b) The excitation spectrum is probed byilluminating the gas with a pair of Bragg laser beams. (c),(d) Zero-temperature dynamical structurefactor S(k, ω) (value shown in gray scale) for a moderately (c) (γ = 3.3) and strongly (d) (γ = 45) in-teracting homogeneous gas. The solid (dashed) line in (d) shows the dispersion of the Lieb-I (Lieb-II)mode. Insets indicate averaging over an ensemble of trapped systems. The thin lines show fixed mo-mentum cuts at representative densities. The corresponding values for k/kF are indicated as verticallines in the k-ω-plane. The thick line shows the averaged response S(ω) in a local density approxima-tion.

103


and temperature effects and demonstrate the contribution of the Lieb-II type excitations tothe response of the system.

6.2 Realization of a one-dimensional Bose gas with tunable in-teractions

Our experiment starts with a Cesium Bose-Einstein condensate (BEC) of typically 1.1 × 105

atoms confined in a crossed dipole trap [Web02, Kra04]. The BEC is adiabatically loaded intoan array of ≈ 4000 quantum wires created via two mutually perpendicular retro-reflectedlaser beams at a wavelength λ = 1064.5 nm. At the end of the ramp the lattice depth along thehorizontal direction is Vx,y = 30ER, creating an ensemble of independent one-dimensional”tubes” with a transversal trap frequency ω⊥ = 2π × 14.5 kHz oriented along the verticalz-direction (Fig. 6.1(a)). Here, ER = h2/(2mλ2) is the photon recoil energy with the massm of the Cs atom. During lattice loading the scattering length as is set to as = 173(5)a0 viaa broad Feshbach resonance (see methods chpt. 6.7). In the deep lattice we then ramp as

within 50 ms to the desired value in the range 10a0 . as . 900a0 to prepare the tubes closeto the adiabatic ground state. The ramp of as is carefully adapted to avoid any excitation ofbreathing modes.

The gas in each tube is described by the Lieb-Liniger Hamiltonian [Lie63a]

H = − ~2

2m

∑i

∂2/∂zi2 + g1D

∑〈i,j〉

δ(zi − zj) , (6.1)

with g1D = 2~ω⊥as (1− 1.0326 as/a⊥)−1 the coupling strength in 1D [Hal09, Ols98, Hal10a]and a⊥ =

√~/(mω⊥) the transverse harmonic oscillator length. The Lieb-Liniger param-

eter is then defined as γ = mg1D/(~2n1D), where n1D denotes the one-dimensional linedensity [Caz11]. The density sets the characteristic Fermi wave-vector kF = πn1D of thesystem. In our experimental setup we have to consider two sources of inhomogeneity. First,the tubes are harmonically confined along the longitudinal direction with a trap frequencyωz = 2π × 15.8(0.1) Hz. This gives rise to an inhomogeneous density distribution in eachquantum wire. Second, the loading procedure leads to a distribution of the number of atomsacross the ensemble of 1D systems (see methods chpt. 6.7). For comparing measurementswith theoretical predictions both effects can be accounted for by averaging over homoge-neous subsystems in a local density approximation (LDA) (see insets to Fig. 6.1(c) and (d)).

104


6.3 Probing the excitation spectrum of the one-dimensional Bosegas via Bragg spectroscopy

We probe the spectrum of elementary excitations via two-photon Bragg spectroscopy [Ste99].In brief, the sample is illuminated for 5 ms with a pair of phase coherent laser beams at awavelength λB ≈ 852 nm and detuned by ≈ 200 GHz from the Cs D2 line. The beamsintersect at an angle φ at the position of the atoms and are aligned such that the wave-vector difference points along the direction of the tubes (Fig. 6.1(b)). Its magnitude k =4π/λB sin(φ/2) sets the momentum transfer, while a small frequency detuning ω betweenthe laser beams defines the energy transfer to the system. In linear response, the energyabsorbed from the Bragg lasers for a fixed pulse area ∆E(k, ω) directly relates to the dy-namical structure factor S(k, ω) =

∫dx∫dteiωt−ikx〈ρ(x, t)ρ(0, 0)〉 at finite temperature T via

∆E(k, ω) ∝ ~ω(1− e−~ω/(kBT ))S(k, ω) with Boltzmann’s constant kB [Bru01].

In our experiment, we probe the ensemble of 1D tubes at a fixed k = 3.24(3)µm−1, whichis comparable to typical mean values for kF averaged over the sample 1. The absorbed en-ergy as a function of ω is measured in momentum space. For this, we switch off the latticepotential quickly (within 300µs) and allow for 50 ms time-of-flight at a small positive scatter-ing length of ≈ 15a0. From the integrated line density along the z-direction of the tubes weextract 〈p2〉 and plot it as a function of ω. The result for five different values of γ is depictedin Fig. 6.2(a)-(e).

The datasets cover the regime from weak to strong interactions 0.1 . γ . 50 probed at0.3 . k/kF . 1. The variation in k/kF ensues from the change of the density distributionin the tubes with increasing as, evolving from a Thomas-Fermi profile at weak interactionstowards the TG profile at strong interactions. The values for γ and kF given in Fig. 6.2 denotethe average over the ensemble of 1D systems using the mean n1D in each wire calculatedfor T = 0 from the solution of the Lieb-Liniger integral equations in LDA (see methodschpt. 6.7). Error bars reflect mainly a ±10% uncertainty in the total atom number. A clearinteraction-induced broadening and shift of the spectra with increasing γ is observed inaccordance with the calculated position of the Lieb-I and Lieb-II modes (vertical dashedlines) [Lie63b]. We compare the dataset in the strongly interacting regime (Fig. 6.2(e)) to thecalculated response for an ensemble of trapped TG gases at zero temperature averaged overthe ensemble of tubes (dashed line) [Gol09] (see methods chpt. 6.7). The agreement with thedata underlines the contribution of Lieb’s hole-like excitation to the dynamical response.

Prior to a detailed discussion on the exact lineshape for finite γ and finite T , we attempt asimplified zero-temperature analysis of our spectroscopy signal. A function∝ ω G(ωc) is fit tothe data (solid lines), where the DSF averaged over the ensemble of tubes is approximated bya simple Gaussian function G centered at ωc. The extracted ωc as a function of as is shown in

1We calibrate k from the measured Bragg excitation spectrum of a weakly interacting BEC in 3D.

105


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Figure 6.2: Bragg-excitation spectra for different values of the 1D interaction strength γ. (a)-(e) Trans-ferred energy ∼ 〈p2〉 (normalized to unit area) as a function of the Bragg detuning ω. The scatteringlength is set to 15a0 (a), 173a0 (b), 399a0 (c), 592a0 (d), and 819a0 (e), giving an average γ (kF /µm−1)of 0.12(5) (9(1)), 3.3(1) (4.4(2)), 11.0(4) (3.7(1)), 21.7(7) (3.5(1)), and 45(1) (3.3(1)), respectively. Solidlines show fits to the data using a Gaussian multiplied by ω. The vertical dashed lines indicate the po-sition of the Lieb-I (L1) and Lieb-II (L2) mode calculated with the averaged values for γ (see methodschpt. 6.7). The dashed line in (e) shows the calculated response for ensemble-averaged trapped TGgases. (f) Central excitation energy ωc extracted from the Gaussian fit model as a function of as (cir-cles). Solid (dashed) lines denote the calculated position of the Lieb-I/II modes (Bogoliubov mode).The dotted lines indicate the energy of particle and hole excitations in the TG limit. (g) ωc extractedfrom the data shown in (a) (triangle), (b) (square), (c) (diamond), and (e) (inverted triangle) plottedin the dimensionless energy-momentum plane. Solid (dashed) lines denote the Lieb-I (Lieb-II) mode.The shaded area shows the continuum of excitations in the TG limit. Vertical lines in (f) and (g) givethe fitted FWHM.

106


Fig. 6.2(f) (circles). The vertical lines denote the fitted full width at half maximum (FWHM).We find the spectral weight of the data within the Lieb-I and Lieb-II modes (solid lines)calculated using the ensemble averaged values for γ. Bogoliubov’s quasi particle energyalone (dashed line) does not explain the observed ωc with increasing γ & 3. We summarizethe measured spectral position and width for four selected values of γ in the dimensionlessenergy-momentum plane in Fig. 6.2(g), with ωF = εF /~ = ~k2

F /(2m). Comparing to thedispersion relation of Lieb’s particle and hole-like excitation, we observe a clear signaturefor the contribution of both branches with increasing interactions, finally approaching thelimit of the excitation spectrum of the fully fermionized Bose gas (shaded area).

6.4 Finite temperature analysis of the excitation spectrum

We now turn to a detailed analysis of the spectral lineshape as a function of interactionstrength and temperature. The effect of temperature on the measurement of the DSF arisesfrom two distinct contributions. First, the DSF itself is temperature dependent. This becomesmost evident in the TG limit of fermionized bosons. For kBT εF the effective Fermi seain quasi-momentum space has a sharp edge giving rise to a homogeneous continuum ofexcitations lying between Lieb’s hole- and particle-like modes. When kBT ∼ εF the Fermiedge washes out, resulting in a smoothening of the DSF with its spectral weight being shiftedto higher energies [Pan14]. Second, finite temperature affects the density distribution in our1D systems leading to a decreasing mean n1D with increasing T for fixed as. This changesthe average k/kF at which the tubes are probed.

For our theoretical analysis we take both effects into account. First, we calculate the den-sity distribution in each of the tubes at a temperature T by numerically solving the Yang-Yang integral equations for the 1D Bose gas [Yan69]. The DSF is evaluated at finite T viathe ABACUS algorithm, a Bethe Ansatz-based method to compute correlation functions ofintegrable models [Cau09]. The effect of the trapping potential is incorporated by making aLDA for each tube. The response of the array of tubes is finally calculated by weighting thecontribution of each subsystem by the number of atoms (see methods chpt. 6.7).

The result of our theoretical analysis for four different values of γ is presented in Fig. 6.3and compared to the corresponding experimental data (taken from Fig. 6.2 (b-e)). Althoughfinite temperature leads to small shifts and broadening of the excitation spectrum, the mostrelevant contribution to the spectral shape stems from the broadening of the dynamicalstructure factor with increasing interactions. The analysis underlines the contribution ofhole-like excitations to the overall response when entering deep into the strongly correlatedregime. Further, a reduced χ2 analysis of our data with the computed spectra serves as athermometry tool in the tubes and points to gas temperatures in the range of 5 to 10 nK. Amoderate increase in temperature is seen for increasing values of γ (see methods chpt. 6.7).

107


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6.5 Analysis of the momentum distribution after Bragg excita-tion

So far, we have characterized the excitations in the gas by measuring 〈p2〉. Now, we analyzethe full momentum distribution n(p) of the excited 1D Bose gases. In Fig. 6.4(a)-(c) we plotthe atomic line density after time-of-flight, integrated transversally to the direction of thetubes, which reflects the in-trap momentum distribution of the atomic ensemble. The mea-surements are taken at three different values of γ, ranging from the weakly to the stronglyinteracting regime, and at Bragg excitation frequencies ω slightly below (left column), just at(central column), and slightly above (right column) the peak of the resonance.

First, we recognize a dramatic qualitative change in n(p) with increasing γ for all detun-ings presented. In the weakly interacting regime (Fig. 6.4(a)), we observe a clear particle-likeexcitation located at p = ~k as expected from the non-interacting limit. Yet, with increasinginteractions this feature smears out and evolves towards an overall broadening of n(p), in-dicative of a strong collective response of the system to the Bragg pulse. This observationdemonstrates one of the key features of 1D systems: any excitation to the system is neces-sarily collective, and therefore leads to energy-broadened response functions, in contrast tosharp coherent single-particle modes. This broadening however only becomes clearly visiblefor strong enough interactions, where the hole-like modes become dynamically relevant.

In a further measurement, we attempt to quantify the response in momentum space inmore detail. Note that our previous measurement of 〈p2〉 captures both a broadening of themomentum distribution as well as an increase in the mean momentum 〈p〉. In order to sep-arate both contributions spectroscopically, we plot the relative change δw of the width w ofthe central part of n(p) around p = 0 and 〈p〉 as a function of ω for different values of γ in

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Figure 6.4: Momentum distribution n(p) for resonant and off-resonant Bragg excitation measured atweak and strong interactions. (a)-(c) Integrated line density after 50 ms time-of-flight when excitingthe 1D systems at γ = 0.12(5) (a), 3.3(1) (b), 45(1) (c) for red-detuned (left), resonant (middle) andblue-detuned (right) excitation with respect to the Bragg resonance. Datapoints show the average offour measurements. The frequencies given denote the Bragg detuning ω/(2π). The red line shows afit to the central part of the line density around p = 0 using a linear combination of a Lorentzian andGaussian profile to extract the width w. The red shaded area denotes the fraction of atoms asymmet-rically scattered to higher momentum states. (d)-(f) Imparted mean momentum 〈p〉 extracted fromn(p) (squares) and relative change in the fitted width δw of the central peak around p = 0 (trian-gles) as a function of the detuning ω/(2π) for γ = 0.12(5) (d), 3.3(1) (e), and 45(1) (f). Solid lines areinterpolating functions to guide the eye.

109


Fig. 6.4(d)-(f). The data indicate how collective excitations in the gas, expressed via energydeposition in w rather than in 〈p〉, become dominant with increasing γ. Interestingly, the twocurves peak at different values for ω. This observation is confirmed by the momentum-spacecharacter of elementary excitations in the interacting 1D systems, which changes from a col-lective broadening for the Lieb-II mode to a particle-like feature for the Lieb-I mode withincreasing ω (see methods chpt. 6.7).

6.6 Conclusion

In conclusion, we have measured the excitation spectrum of a strongly correlated 1D systemfor a wide range of interaction parameters. Comparison with integrability-based calcula-tions at finite temperature allows for a direct observation of the contribution of the collectiveLieb-II mode to the DSF. Our results demonstrate the successful application of an integrablemodel to analyze dynamics of the 1D Bose gas. Furthermore, the collective nature of elemen-tary excitations in 1D with increasing interaction strength has been demonstrated throughan analysis of the momentum distribution. Our results pose questions on the time evolutionand potential equilibration of these collective excitations. This could be seen as an alternativequantum cradle setting [Kin06] in which, instead of colliding two highly energetic clouds ofatoms, relatively low energy excitations propagate through the system, whose individualfeatures can then be more easily studied.

We are indebted to R. Grimm for generous support, and thank M. Buchhold and S.Diehl for fruitful discussions. We gratefully acknowledge funding by the European ResearchCouncil (ERC) under Project No. 278417 and under the Starting Grant No. 279391 EDEQS,by the Austrian Science Foundation (FWF) under Project No. P1789-N20, and from the FOMand NWO foundations of the Netherlands.


6.7.1 Lattice depth calibration and error bars

The lattice depth Vx,y is calibrated via Kapitza-Dirac diffraction. The statistical error for Vx,yis 1%, though the systematic error can reach up to 5%.

The scattering length as is calculated via its dependence on the magnetic field [Mar11a]with an estimated uncertainty of±5a0 arising from systematics in the magnetic field calibra-tion and conversion accuracy. Additionally, the magnetic field gradient for sample levitation[Web02, Kra04] leads to a variation of less than ±3a0 across the sample.

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Figure 6.5: (a) Atom number distribution over the 1D tubes for a total atom number N = 1.1× 105 inthe BEC and a 3D scattering length during loading a3D = 173a0. (b) Number of tubes that are filledwith Ni,j = N ′ atoms. (c) Total number of atoms in tubes filled with Ni,j = N ′ atoms.

6.7.2 Atom number distribution across the 1D tubes

In order to calculate a mean value of the interaction parameter γ and the Fermi wave vectorkF we need to know the atom number distribution across the ensemble of tubes, as shownin Fig. 6.5. We follow the calculation presented in [Hal11].

The BEC is adiabatically loaded from a crossed dipole trap into the optical lattice. Whenthe lattice is fully ramped up to its final depth Vx,y = 30ER, the laser beams forming the arrayof tubes give rise to an additional background harmonic confinement. The total backgroundharmonic confinement of lattice and dipole trap laser beams is measured to ωx = 2π ×15.4(0.1) Hz, ωy = 2π×20.1(0.1) Hz, and ωz = 2π×15.8(0.1) Hz. We deduce the atom numberNi,j for the tube (i, j) from the global chemical potential µ. Assuming that interactions aresufficiently small during the loading process so that all tubes are in the 1D Thomas-Fermi(TF) regime, the local chemical potential in each tube reads

µi,j = µ− 12m(λ/2)2

(ω2xi

2 + ω2yj

2),

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Figure 6.6: 1D line-density distribution of a tube withN = 30 atoms at as = 150a0 (a) and as = 800a0

(b). The blue line shows the result from the numerical solution of the Lieb-Liniger system. The dashed(dotted) gray lines depict the analytic results for the profile in the Thomas-Fermi (Tonks-Girardeau)limit [Dun01]. Here, ω⊥ = 2π × 15 kHz and ωz = 15 Hz. Finite temperature profiles using the Yang-Yang thermodynamic equations are shown for T = 10nK (orange line) and T = 30nK (red line).

with m the mass of the Cs atom and λ = 1064.5 nm the wavelength of the lattice light. Fromµi,j the atom number in each tube can be derived via

µi,j =(

3Ni,j

4√

2g1Dωz

√m

)2/3

,

with g1D the 1D coupling strength

g1D = 2~ω⊥as(

1− 1.0326asa⊥

)−1

.

Here, a⊥ =√

~/(mω⊥) is the radial oscillator length. The global chemical potential is calcu-lated iteratively from the condition N =

∑i,j Ni,j(µ).

6.7.3 Density profile in the tubes

For T = 0 we model the 1D density distribution n(z) in each tube individually by numer-ically solving the Lieb-Liniger system and making a local density approximation [Lie63a,Dun01]. For T > 0 we solve the Yang-Yang thermodynamic equations of the 1D Bose gasmaking a local density approximation to calculate n(z, T ) [Yan69]. Representative examplesfor two different values of as at zero and finite temperature are shown in Fig. 6.6.

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6.7.4 Mean γ and kF

From the atom number distribution, we calculate the density profile for each tube individu-ally as described above. The mean 1D density in each tube n1D

i,j then delivers local and meanvalues for γ and kF

γi,j =mg1D

~2n1Di,j

, γ =1N

∑i,j

Ni,jγi,j

andki,jF = πn1D

i,j , kF =1N

∑i,j

Ni,jki,jF .

As an example, the distribution of atoms over tubes with local γi,j and k/ki,jF is depicted inFig. 6.7(a) and (b) for a specific value of as. In Fig. 6.7(c) and (d) we show the mean value ofγ and kF as a function of as.

6.7.5 Sampling of the dynamical structure factor over the array of tubes andcomparison with the experimental data

In linear response, the energy ∆E(k, ω, T ) dumped into a single 1D gas after the Bragg pulserelates to the dynamical structure factor S(k, ω, T ) via [Pit03]

∆E(k, ω, T ) ∝ ~ω(1− e−~ω/(kBT ))S(k, ω, T ) .

Provided that the system is probed at sufficiently large momentum k the DSF of the trappedsystem can be computed in a local density approximation

S(k, ω, T ) =1

2L

L∫−L

Shom(k, ω, T ;n(z))dz ,

with L the system length and Shom the DSF calculated for a homogeneous system with uni-form density. In practice, we find that dividing the profile of each tube in≈ 10 homogeneoussubsystems approximates the DSF of the trapped gas sufficiently well. The DSF in each tube

with Ni,j atoms obeys the f -sum rule [Pit03],+∞∫−∞

ωS(k, ω, T ) dω ∝ Ni,j . In combination with

detailed balance S(k, ω) = e~ω/(kBT )S(k,−ω) we find

+∞∫0

ωS(k, ω, T )(1− e−~ω/(kBT )) dω ∝ Ni,j .

This allows us to calculate the dynamical response of the entire ensemble by first normaliz-ing the DSF of each individual tube (i, j) by its f -sum, respectively, and then weighting its

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114


contribution to the total signal by the number of atoms Ni,j . For a direct comparison withthe experimental signal, we finally normalize the ensemble averaged response to unit area.

In the experiment, we extract 〈p2〉 as a function of ω from the momentum space distribu-tion obtained after time-of-flight. For a direct comparison, each spectrum is normalized tounit area as the overall signal depends on the intensity of the Bragg lasers, which we slightlyincrease for data taken at stronger interactions. A changing offset due to an overall broaden-ing of the momentum distribution with increasing γ is subtracted from the data. This offsetdoes not affect the shape of the excitation spectrum and stems from the broadening of theunperturbed momentum distribution with increasing γ.

6.7.6 The ABACUS algorithm

The DSF for a homogeneous system is computed using the ABACUS algorithm. The com-putations are performed for a finite system of length L with a finite particle number N andwith periodic boundary conditions. This means that the experimental situation is recoveredonly in the thermodynamic limit N,L → ∞ with a density n = N/L fixed through the lo-cal density approximation. We have confirmed that the particle numbers N chosen for thecomputations (up toN = 128) are large enough for the results to faithfully represent the ther-modynamic limit. The only exceptions are the highest temperature curves in Fig. 6.3 (b)-(d)where some oscillations due to the finite size are still visible at high energies. The correla-tion function is evaluated numerically by summing contributions according to the Lehmannspectral representation. The spectral sum is infinite and needs to be truncated. The ABACUSalgorithm performs this operation in an efficient way capturing the most relevant contribu-tions. The error caused by the truncation is easily tractable by evaluating the f -sum rule and,for the presented data, does not exceed 5% of the total spectral weight.

6.7.7 Regime of linear response

Measuring the DSF via Bragg spectroscopy requires to probe the system in the regime oflinear response. This we tested experimentally by validating that the excitations created de-pend linearly on the pulse length and quadratically on the intensity of the Bragg lasers forthe range of parameters used in the experiment, see Fig. 6.8 [Bru01].

6.7.8 Heating effects on the excitation spectra

Besides the detailed theoretical analysis of finite temperature effects on the measured DSFreported in the main article, we have checked in an experiment that heating during the rampof as to large values only marginally influences the shape of the excitation spectrum whencompared to the effect of interactions. In Fig. 6.9(a) we show Bragg excitation spectra taken

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Figure 6.8: (a)-(c) Bragg excitation as a function of the laser pulse length measured in a weakly in-teracting 3D BEC (a), in 1D at as = 15a0 (b), and in 1D at as = 819a0 (c). The laser power in bothBragg beams is set to 0.5mW (circles) and 0.2mW (squares) in (a) and (b), and 0.5mW (circles) and0.3mW (squares) in (c). (d),(e) Bragg excitation as a function of the laser power in both Bragg beamsmeasured in a weakly interacting BEC (circles), in 1D at as = 15a0 (squares), and in 1D at as = 819a0

(diamonds). Here, the laser pulse length is fixed to 5ms. The solid lines are exponentially dampedsinusoids fit to the data. The dashed lines are linear fits in (a)-(c) and quadratic fits in (d) and (e) tothe initial increase denoting the regime of linear response.

116


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Figure 6.9: (a) Heating effects during the ramp of the scattering length as on the Bragg spectroscopydata. Bragg spectroscopy data taken directly after the ramp of the scattering length to 173a0 within50 ms (squares) is compared to data taken after ramping to 819a0 and back within 100 ms (circles).(b) Full width at half maximum (FWHM) extracted via the Gaussian fit (see main article) from theBragg spectroscopy data as a function of γ. The dashed (solid) arrow denotes an estimate for linebroadening due to the finite pulse length (quantum finite size effect) in the non-interacting limit.

at moderate interaction strength, corresponding to the data shown in Fig. 6.2(b) of the mainarticle. For the two datasets shown, we either ramp directly to 173a0 (squares) or rampdeep into the regime of strong interactions, as=810a0, and back (circles) prior to applyingthe Bragg pulse.

6.7.9 Interaction-independent spectral width at small γ

In the limit of weak interactions we observe a width of the excitation spectra that levels offat a constant value as shown in Fig. 6.9(b). We attribute this to the onset of two effects thatprevent the observation of a δ-like peak as expected in a non-trapped homogeneous systemfor γ → 0. First, the finite length of the Bragg excitation pulse causes Fourier broadening,which results in an estimated residual width of ≈ 120Hz in the non-interacting limit [Pit03].Second, the finite size of the sample due to the presence of the harmonic trap leads to anuncertainty-limited energy width (not accounted for in the LDA) that can be estimated toδω = 2~k/(maz) for a sample of non-interacting particles [Gol09]. Here, az =

√~/(mωz)

denotes the quantum length scale of the trap.

6.7.10 Momentum distribution of Lieb-I and Lieb-II excitations

In order to illustrate the effects of additional particle-hole excitations on the momentum dis-tribution, we consider two situations in which the total momentum of an excited state (ob-tained by adding a single particle-hole excitation with total momentum equal to the Fermimomentum on the ground state) is carried either by the hole (Lieb type II) mode (Fig. 6.10(a))

117


-2 -1 0 1 20.0

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0.2

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momentum k kF

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LHa

rb.u

nits

LHaL

-2 -1 0 1 20.0

0.1

0.2

0.3

0.4

0.5

momentum k kF

nHk

LHa

rb.u

nits

L

HbL

Figure 6.10: Momentum distribution of two states excited with a single (particle-hole) excitationover the ground state for γ = 64. In the first case (a) the momentum is carried by a hole-like LiebII excitation, in the second (b) by a particle-like Lieb I excitation. Both states have the same totalmomentum equal to k/kF = 1. The calculation is done for a system of 32 particles. The shaded areasindicate the difference to the ground-state momentum distribution.

or by the particle (Lieb type I) mode (Fig. 6.10(b)). The computations are performed againusing the ABACUS method [Cau07]. The results show that while in both cases the momen-tum distribution function is augmented by broadened peaks, the hole excitation leads to anoverall broadening of n(k) while the particle excitation appears as a more distinguishablepeak. This highlights the collective nature of the Lieb-II hole-like excitations and qualita-tively reproduces the features seen in Fig. 6.4 of the main text.

118

BIBLIOGRAPHY

[Aid13] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Realiza-tion of the Hofstadter Hamiltonian with Ultracold Atoms in Optical Lattices, Phys. Rev.Lett. 111, 185301 (2013).

[Aik12] K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler, R. Grimm, and F. Ferlaino, Bose-Einstein Condensation of Erbium, Phys. Rev. Lett. 108, 210401 (2012).

[Ama96] J. C. Amadon and J. E. Hirsch, Metallic ferromagnetism in a single-band model: Effectof band filling and Coulomb interactions, Phys. Rev. B 54, 6364 (1996).

[Ami08] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entanglement in many-body systems,Rev. Mod. Phys. 80, 517 (2008).

[And50] P. Anderson, Antiferromagnetism. Theory of Superexchange Interaction, Phys. Rev. 79,350 (1950).

[And95] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell,Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor, Science 269, 198(1995).

[And98] B. P. Anderson and M. A. Kasevich, Macroscopic Quantum Interference from AtomicTunnel Arrays, Science 282, 1686 (1998).

[And04] P. W. Anderson, P. A. Lee, M. Randeria, T. M. Rice, N. Trivedi, and F. C. Zhang,The physics behind high-temperature superconducting cuprates: the ’plain vanilla’ versionof RVB, J. Phys. Condens. Matter 16, R755 (2004).

[And97] M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, andW. Ketterle, Observation of Interference Between Two Bose Condensates, Science 275, 637(1997).

119

Bibliography

[Ank10] J. Ankerhold, Quantum Tunneling in Complex Systems: The Semiclassical Approach(Springer, Berlin, 2010).

[App93] J. Appel, M. Grodzicki, and F. Paulsen, Bond-charge repulsion and hole superconduc-tivity in the atomic representation of the CuO2 plane, Phys. Rev. B 47, 2812 (1993).

[Ash76] N.W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, Philadel-phia, 1976).

[Aus02] O. M. Auslaender, A. Yacoby, R. de Picciotto, K. W. Baldwin, L. N. Pfeiffer, andK. W. West, Tunneling Spectroscopy of the Elementary Excitations in a One-DimensionalWire, Science 295, 825 (2002).

[Bai15] S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz, Z. Cai, M. Baranov,P. Zoller, and F. Ferlaino, Extended Bose-Hubbard Models with Ultracold MagneticAtoms, arXiv:1507.03500 (2015).

[Bak09] W. S. Bakr, J. I. Gillen, A. Peng, S. Folling, and M. Greiner, A Quantum Gas Microscopefor detecting single atoms in a Hubbard regime optical lattice, Nature 462, 74 (2009).

[Bak10] W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Folling, L. Pollet, andM. Greiner, Probing the Superfluid-to-Mott Insulator Transition at the Single-Atom Level,Science 329, 547 (2010).

[Bak11] W. S. Bakr, P. M. Preiss, M. E. Tai, R. Ma, J. Simon, and M. Greiner, Orbital excitationblockade and algorithmic cooling in quantum gases, Nature 480, 500 (2011).

[Bar10] G. Barak, H. Steinberg, L. N. Pfeiffer, K. W. West, L. Glazman, F. von Oppen, andA. Yacoby, Interacting electrons in one dimension beyond the Luttinger-liquid limit, Nat.Phys. 6, 489 (2010).

[Bar04] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. Hecker Denschlag, andR. Grimm, Collective Excitations of a Degenerate Gas at the BEC-BCS Crossover, Phys.Rev. Lett. 92, 203201 (2004).

[Bec10] C. Becker, P. Soltan-Panahi, J. Kronjager, S. Dorscher, K. Bongs, and K. Sengstock,Ultracold quantum gases in triangular optical lattices, New J. Phys. 12, 065025 (2010).

[Ber03] T. Bergeman, M. G. Moore, and M. Olshanii, Atom-Atom Scattering under Cylindri-cal Harmonic Confinement: Numerical and Analytic Studies of the Confinement InducedResonance, Phys. Rev. Lett. 91, 163201 (2003).

[Ber11] M. Berninger, A. Zenesini, B. Huang, W. Harm, H.-C. Nagerl, F. Ferlaino, R. Grimm,P. S. Julienne, and J. M. Hutson, Universality of the Three-Body Parameter for EfimovStates in Ultracold Cesium, Phys. Rev. Lett. 107, 120401 (2011).

120

Bibliography

[Ber13] M. Berninger, A. Zenesini, B. Huang, W. Harm, H.-C. Nagerl, F. Ferlaino, R. Grimm,P. S. Julienne, and J. M. Hutson, Feshbach resonances, weakly bound molecular states,and coupled-channel potentials for cesium at high magnetic fields, Phys. Rev. A 87, 032517(2013).

[Bes09] T. Best, S. Will, U. Schneider, L. Hackermuller, D. van Oosten, I. Bloch, and D.-S. Luhmann, Role of Interactions in 87Rb-40K Bose-Fermi Mixtures in a 3D Optical Lat-tice, Phys. Rev. Lett. 102, 030408 (2009).

[Bet31] H. Bethe, Zur Theorie der Metalle, I. Eigenwerte und Eigenfunktionen der linearen Atom-kette, Z. Phys. 71, 205 (1931).

[Bis12] U. Bissbort, F. Deuretzbacher, and W. Hofstetter, Effective multibody-induced tunnelingand interactions in the Bose-Hubbard model of the lowest dressed band of an optical lattice,Phys. Rev. A 86, 023617 (2012).

[Blo29] F. Bloch, Uber die Quantenmechanik der Elektronen in Kristallgittern, Z. Phys. 52, 555(1929).

[Blo05] I. Bloch, Ultracold quantum gases in optical lattices, Nat. Phys. 1, 23 (2005).

[Blo08] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev.Mod. Phys. 80, 885 (2008).

[Blo12] I. Bloch, J. Dalibard and S. Nascimbene, Quantum simulations with ultracold quantumgases, Nat. Phys. 8, 267 (2012).

[Boc99] M. Bockrath, D. H. Cobden, J. Lu, A. G. Rinzler, R. E. Smalley, L. Balents, andP. L. McEuen, Luttinger-liquid behaviour in carbon nanotubes, Nature 397, 598 (1999).

[Bos24] S. Bose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift fur Physik A 26, 178(1924).

[Bra13] F. R. Braakman, P. Barthelemy, C. Reichl, W. Wegscheider, and L. M. K. Vander-sypen, Long-distance coherent coupling in a quantum dot array, Nat. Nanotechnol. 8,432 (2013).

[Bra06] E. Braaten and H.-W. Hammer, Universality in few-body systems with large scatteringlength, Phys. Rep. 428, 259 (2006).

[Bra95] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Evidence of Bose-EinsteinCondensation in an Atomic Gas with Attractive Interactions, Phys. Rev. Lett. 75, 1687(1995).

121

Bibliography

[Bra15] S. Braun, M. Friesdorf, S. S. Hodgman, M. Schreiber, J. P. Ronzheimer, A. Riera,M. del Rey, I. Bloch, J. Eisert, and U. Schneider, Emergence of coherence and the dynam-ics of quantum phase transitions, PNAS 112, 3641 (2015).

[Bru01] A. Brunello, F. Dalfovo, L. Pitaevskii, S. Stringari, and F. Zambelli, Momentum trans-ferred to a trapped Bose-Einstein condensate by stimulated light scattering, Phys. Rev. A64, 063614 (2001).

[Buc03] A. Buchleitner and A. R. Kolovsky, Interaction-Induced Decoherence of Atomic BlochOscillations, Phys. Rev. Lett. 91, 253002 (2003).

[Buc10] H. P. Buchler, Microscopic Derivation of Hubbard Parameters for Cold Atomic Gases,Phys. Rev. Lett. 104, 090402 (2010).

[Bur97] E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and C. E. Wieman,Coherence, Correlations, and Collisions: What One Learns about Bose-Einstein Condensatesfrom Their Decay, Phys. Rev. Lett. 79, 337 (1997).

[Bus98] T. Busch, B.-G. Englert, K. Rzazewski, and M. Wilkens, Two Cold Atoms in a HarmonicTrap, Found. Phys. 28, 549 (1998).

[Cam06] G. K. Campbell, J. Mun, M. Boyd, P. Medley, A. E. Leanhardt, L. G. Marcassa,D. E. Pritchard, and W. Ketterle, Imaging the Mott Insulator Shells by Using AtomicClock Shifts, Science 313, 649 (2006).

[Cam14] A. del Campo and W. H. Zurek, Universality of phase transition dynamics: Topologicaldefects from symmetry breaking, Int. J. Mod. Phys. A 29, 1430018 (2014).

[Cap07] B. Capogrosso-Sansone, N. V. Prokof’ev, and B. V. Svistunov, Phase diagram andthermodynamics of the three-dimensional Bose-Hubbard model, Phys. Rev. B 75, 134302(2007).

[Cap08] B. Capogrosso-Sansone, S. G. Soyler, N. Prokof’ev, and B. Svistunov, Monte Carlostudy of the two-dimensional Bose-Hubbard model, Phys. Rev. A 77, 015602 (2008).

[Car05] I. Carusotto, L. Pitaevskii, S. Stringari, G. Modugno, and M. Inguscio, Sensitive Mea-surement of Forces at the Micron Scale Using Bloch Oscillations of Ultracold Atoms, Phys.Rev. Lett. 95, 093202 (2005).

[Cas11] A. C. Cassidy, C. W. Clark, and M. Rigol, Generalized Thermalization in an IntegrableLattice System, Phys. Rev. Lett. 106, 140405 (2011).

[Cau06] J.-S. Caux and P. Calabrese, Dynamical density-density correlations in the one-dimensional Bose gas, Phys. Rev. A 74, 031605(R) (2006).

122

Bibliography

[Cau07] J.-S. Caux, P. Calabrese, and N. A. Slavnov, One-particle dynamical correlations in theone-dimensional Bose gas, J. Stat. Mech. P01008 (2007).

[Cau09] J.-S. Caux, Correlation functions of integrable models: A description of the ABACUS algo-rithm, J. Math. Phys. 50, 095214 (2009).

[Cau11] J.-S. Caux and J. Mossel, Remarks on the notion of quantum integrability, J. Stat. Mech.2011, P02023 (2011).

[Caz04] M. A. Cazalilla, Bosonizing one-dimensional cold atomic gases, J. Phys. B: At. Mol. Opt.Phys. 37, S1 (2004).

[Caz11] M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol, One dimensionalbosons: From condensed matter systems to ultracold gases, Rev. Mod. Phys. 83, 1405(2011).

[Che11] Y.-A. Chen, S. D. Huber, S. Trotzky, I. Bloch, and E. Altman, Many-body Landau-Zenerdynamics in coupled one-dimensional Bose liquids, Nat. Phys. 7, 61 (2011).

[Che12] M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauß, T. Fukuhara, C. Gross,I. Bloch, C. Kollath, and S. Kuhr, Light-cone-like spreading of correlations in a quantummany-body system, Nature 481, 484 (2012).

[Che06] A. Y. Cherny and J. Brand, Polarizability and dynamic structure factor of the one-dimensional Bose gas near the Tonks-Girardeau limit at finite temperatures, Phys. Rev.A 73, 023612 (2006).

[Chi04a] C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. Hecker Denschlag, andR. Grimm, Observation of the Pairing Gap in a Strongly Interacting Fermi Gas, Science305, 1128 (2004).

[Chi04b] C. Chin, V. Vuletic, A. J. Kerman, S. Chu, E. Tiesinga, P. J. Leo, and C. J. Williams,Precision Feshbach spectroscopy of ultracold Cs2, Phys. Rev. A 70, 032701 (2004).

[Chi10] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases,Rev. Mod. Phys. 82, 1225 (2010).

[Cou98] P. Courteille, R. S. Freeland, D. J. Heinzen, F. A. van Abeelen, and B. J. Verhaar,Observation of a Feshbach Resonance in Cold Atom Scattering, Phys. Rev. Lett. 81, 69(1998).

[Cra08a] M. Cramer, C. M. Dawson, J. Eisert, and T. J. Osborne, Exact Relaxation in a Class ofNonequilibrium Quantum Lattice Systems, Phys. Rev. Lett. 100, 030602 (2008).

123

Bibliography

[Cra08b] M. Cramer, A. Flesch, I. P. McCulloch, U. Schollwock, and J. Eisert, Exploring LocalQuantum Many-Body Relaxation by Atoms in Optical Superlattices, Phys. Rev. Lett. 101,063001 (2008).

[Cub03] J. Cubizolles, T. Bourdel, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov, and C. Sa-lomon, Production of Long-Lived Ultracold Li2 Molecules from a Fermi Gas, Phys. Rev.Lett. 91, 240401 (2003).

[Dag94] E. Dagotto, Correlated electrons in high-temperature superconductors, Rev. Mod. Phys.66, 763 (1994).

[Dag99] E. Dagotto, Experiments on ladders reveal a complex interplay between a spin-gappednormal state and superconductivity, Rep. Prog. Phys. 62, 1525 (1999).

[Dah96] M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Bloch Oscillations ofAtoms in an Optical Potential, Phys. Rev. Lett. 76, 4508 (1996).

[Dal04] A. J. Daley, C. Kollath, U. Schollwock, G. Vidal, Time-dependent density-matrixrenormalization-group using adaptive effective Hilbert spaces, J. Stat. Mech.: Theor. Exp.P04005 (2004).

[Dal12] A. J. Daley, H. Pichler, J. Schachenmayer, and P. Zoller, Measuring EntanglementGrowth in Quench Dynamics of Bosons in an Optical Lattice, Phys. Rev. Lett. 109, 020505(2012).

[Dal11] J. Dalibard, F. Gerbier, G. Juzeliunas, and P. Ohberg, Colloquium: Artificial gauge po-tentials for neutral atoms, Rev. Mod. Phys. 83, 1523 (2011).

[Dan10] J. G. Danzl, M. J. Mark, E. Haller, M. Gustavsson, R. Hart, J. Aldegunde, J. M. Hut-son, and H.-C. Nagerl, An ultracold high-density sample of rovibronic ground-statemolecules in an optical lattice, Nature Physics 6, 265 (2010).

[Dav95] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee,D. M. Kurn, and W. Ketterle, Bose-Einstein Condensation in a Gas of Sodium Atoms,Phys. Rev. Lett. 75, 3969 (1995).

[DeG10] C. De Grandi, V. Gritsev, and A. Polkovnikov, Quench dynamics near a quantum crit-ical point, Phys. Rev. B 81, 012303 (2010).

[DeM99] B. DeMarco and D. S. Jin, Onset of Fermi Degeneracy in a Trapped Atomic Gas, Science285, 1703 (1999).

[Deu91] J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046(1991).

124

Bibliography

[Die89] F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Laser Cooling to theZero-Point Energy of Motion, Phys. Rev. Lett. 62, 403 (1989).

[Duc15] L. Duca, T. Li, M. Reitter, I. Bloch, M. Schleier-Smith, and U. Schneider, An Aharonov-Bohm interferometer for determining Bloch band topology, Science 347, 288 (2015).

[Dun01] V. Dunjko, V. Lorent, and M. Olshanii, Bosons in Cigar-Shaped Traps: Thomas-FermiRegime, Tonks-Girardeau Regime, and In Between, Phys. Rev. Lett. 86, 5413 (2001).

[Dut15] O. Dutta, M. Gajda, P. Hauke, M. Lewenstein, D.-S. Luhmann, B. A. Malomed,T. Sowinski, and J. Zakrzewski, Non-standard Hubbard models in optical lattices: a re-view, Rep. Prog. Phys. 78, 066001 (2015).

[Dzi10] J. Dziarmaga, Dynamics of a quantum phase transition and relaxation to a steady state,Adv. Phys. 59, 1063 (2010).

[Eck11] M. Eckstein and P. Werner, Damping of Bloch Oscillations in the Hubbard Model, Phys.Rev. Lett. 107, 186406 (2011).

[Ein25] A. Einstein, Quantentheorie des einatomigen idealen Gases. Zweite Abhandlung.,Sitzungber. Preuss. Akad. Wiss. 3 (1925).

[Eis15] J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many-body systems out of equilibrium,Nat. Phys. 11, 124 (2015).

[Ess10] T. Esslinger, Fermi-Hubbard Physics with Atoms in an Optical Lattice, Annu. Rev. Con-dens. Matter Phys. 1, 129 (2010).

[Fab15] N. Fabbri, M. Panfil, D. Clement, L. Fallani, M. Inguscio, C. Fort, and J.-S. Caux, Dy-namical structure factor of one-dimensional Bose gases: Experimental signatures of beyond-Luttinger-liquid physics, Phys. Rev. A 91, 043617 (2015).

[Fal04] L. Fallani, L. De Sarlo, J. E. Lye, M. Modugno, R. Saers, C. Fort, and M. Inguscio, Ob-servation of Dynamical Instability for a Bose-Einstein Condensate in a Moving 1D OpticalLattice, Phys. Rev. Lett. 93, 140406 (2004).

[Fat08] M. Fattori, C. D‘Errico, G. Roati, M. Zaccanti, M. Jona-Lasinio, M. Modugno, M. In-guscio, and G. Modugno, Atom Interferometry with a Weakly Interacting Bose-EinsteinCondensate, Phys. Rev. Lett. 100, 080405 (2008).

[Fel92] J. Feldmann, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier,G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, Optical investigationof Bloch oscillations in a semiconductor superlattice, Phys. Rev. B 46, 7252 (1992).

125

Bibliography

[Fer09] F. Ferlaino, S. Knoop, M. Berninger, W. Harm, J. P. D’Incao, H.-C. Nagerl, andR. Grimm, Evidence for Universal Four-Body States Tied to an Efimov Trimer, Phys. Rev.Lett. 102, 140401 (2009).

[Fes58] H. Feshbach, Unified theory of nuclear reactions, Ann. Phys. (N.Y.) 5, 357 (1958).

[Fes62] H. Feshbach, Unified theory of nuclear reactions II, Ann. Phys. (N.Y.) 19, 287 (1962).

[Fey82] R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982).

[Fis89] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Boson localization andthe superfluid-insulator transition, Phys. Rev. B 40, 546 (1989).

[Fol06] S. Folling, A. Widera, T. Muller, F. Gerbier, and I. Bloch, Formation of Spatial ShellStructure in the Superfluid to Mott Insulator Transition, Phys. Rev. Lett. 97, 060403(2006).

[Fol07] S. Folling, S. Trotzky, P. Cheinet, M. Feld, R. Saers, A. Widera, T. Muller, and I. Bloch,Direct observation of second-order atom tunnelling, Nature 448, 1029 (2007).

[Fri98] D. G. Fried, T. C. Killian, L. Willmann, D. Landhuis, S. C. Moss, D. Kleppner, andT. J. Greytak, Bose-Einstein Condensation of Atomic Hydrogen, Phys. Rev. Lett. 81, 3811(1998).

[Fuk15] T. Fukuhara, S. Hild, J. Zeiher, P. Schauß, I. Bloch, M. Endres, and C. Gross, SpatiallyResolved Detection of a Spin-Entanglement Wave in a Bose-Hubbard Chain, Phys. Rev.Lett. 115, 035302 (2015).

[Gan03] D. M. Gangardt and G. V. Shlyapnikov, Stability and Phase Coherence of Trapped 1DBose Gases, Phys. Rev. Lett. 90, 010401 (2003).

[Gan09] D. M. Gangardt and A. Kamenev, Bloch Oscillations in a One-Dimensional Spinor Gas,Phys. Rev. Lett. 102, 070402 (2009).

[Gei14] R. Geiger, T. Langen, I. E. Mazets, and J. Schmiedmayer, Local relaxation and light-cone-like propagation of correlations in a trapped one-dimensional Bose gas, New J. Phys.16, 053034 (2014).

[Gia04] T. Giamarchi, Quantum Physics in One Dimension (Oxford Univ. Press, New York,2004).

[Gia91] M.-J. Giannoni, A. Voros, and J. Zinn-Justin, Chaos and quantum physics (North-Holland, Amsterdam, 1991).

[Gib93] K. Gibble and S. Chu, Laser-cooled Cs frequency standard and a measurement of the fre-quency shift due to ultracold collisions, Phys. Rev. Lett. 70, 1771 (1993).

126

Bibliography

[Gir60] M. Girardeau, Relationship between Systems of Impenetrable Bosons and Fermions in OneDimension, J. Math. Phys. 1, 516 (1960).

[Glu02] M. Gluck, A. R. Kolovsky, and H. J. Korsch, Wannier-Stark resonances in optical andsemiconductor superlattices, Phys. Rep. 366, 103 (2002).

[Gol09] V. N. Golovach, A. Minguzzi, and L. I. Glazman, Dynamic response of one-dimensionalbosons in a trap, Phys. Rev. A 80, 043611 (2009).

[Gon09] J. Gong, L. Morales-Molina, and P. Hanggi, Many-Body Coherent Destruction of Tun-neling, Phys. Rev. Lett. 103, 133002 (2009).

[Gou86] P. L. Gould, G. A. Ruff, and D. E. Pritchard, Diffraction of atoms by light: The near-resonant Kapitza-Dirac effect, Phys. Rev. Lett. 56, 827 (1986).

[Gre13] D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, and T. Esslinger, Short-Range QuantumMagnetism of Ultracold Fermions in an Optical Lattice, Science 340, 1307 (2013).

[Gre01] M. Greiner, I. Bloch, O. Mandel, T. W. Hansch, and T. Esslinger, Exploring Phase Co-herence in a 2D Lattice of Bose-Einstein Condensates, Phys. Rev. Lett. 87, 160405 (2001).

[Gre02a] M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, and I. Bloch, Quantum phasetransition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature 415, 39(2002).

[Gre02b] M. Greiner, O. Mandel, T. W. Hansch, and I. Bloch, Collapse and revival of the matterwave field of a Bose-Einstein condensate, Nature 419, 51 (2002).

[Gre03] M. Greiner, C. A. Regal, and D. S. Jin, Emergence of a molecular Bose-Einstein condensatefrom a Fermi gas, Nature 426, 537 (2003).

[Gre14] S. Greschner, G. Sun, D. Poletti, and L. Santos, Density-Dependent Synthetic GaugeFields Using Periodically Modulated Interactions, Phys. Rev. Lett. 113, 215303 (2014).

[Gri05] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, Bose-Einstein Condensa-tion of Chromium, Phys. Rev. Lett. 94, 160401 (2005).

[Gri00] R. Grimm, M. Weidemuller, and Y. B. Ovchinnikov, Optical Dipole Traps for NeutralAtoms, Adv. At. Mol. Opt. Phy. 42, 95 (2000).

[Gri12] M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets,D. A. Smith, E. Demler, and J. Schmiedmayer, Relaxation and Prethermalization inan Isolated Quantum System, Science 337, 6100 (2012).

127

Bibliography

[Gro15] M. Grobner, P. Weinmann, F. Meinert, K. Lauber, E. Kirilov, and H.-C. Nagerl, A newquantum gas apparatus for ultracold mixtures of K and Cs and KCs ground-state molecules,arXiv:1511.05044 (2015).

[Gun06] K. Gunter, T. Stoferle, H. Moritz, M. Kohl, and T. Esslinger, Bose-Fermi Mixtures in aThree-Dimensional Optical Lattice, Phys. Rev. Lett. 96, 180402 (2006).

[Gus08a] M. Gustavsson, A quantum gas with tunable interactions in an optical lattice, Ph.D.thesis, University of Innsbruck (2008).

[Gus08b] M. Gustavsson, E. Haller, M. J. Mark, J. G. Danzl, G. Rojas-Kopeinig, and H.-C. Nagerl, Control of Interaction-Induced Dephasing of Bloch Oscillations, Phys. Rev.Lett. 100, 080404 (2008).

[Gus10] M. Gustavsson, E. Haller, M. J. Mark, J. G. Danzl, R. Hart, A. J. Daley, and H.-C. Nagerl, Interference of interacting matter waves, New J. Phys. 12, 065029 (2010).

[Gut63] M. C. Gutzwiller, Effect of Correlation on the Ferromagnetism of Transition Metals, Phys.Rev. Lett. 10, 159 (1963).

[Hal81] F. D. M. Haldane, Effective Harmonic-Fluid Approach to Low-Energy Properties of One-Dimensional Quantum Fluids, Phys. Rev. Lett. 47, 1840 (1981).

[Hal09] E. Haller, M. Gustavsson, M. J. Mark, J. G. Danzl, R. Hart, G. Pupillo, and H.-C. Nagerl, Realization of an Excited, Strongly Correlated Quantum Gas Phase, Science325, 1224 (2009).

[Hal10a] E. Haller, M. J. Mark, R. Hart, J. G. Danzl, L. Reichsollner, V. Melezhik,P. Schmelcher, and H.-C. Nagerl, Confinement-Induced Resonances in Low-DimensionalQuantum Systems, Phys. Rev. Lett. 104, 153203 (2010).

[Hal10b] E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsollner, M. Gustavsson, M. Dal-monte, G. Pupillo, and H.-C. Nagerl, Pinning quantum phase transition for a Luttingerliquid of strongly interacting bosons, Nature 466, 597 (2010).

[Hal11] E. Haller, M. Rabie, M. J. Mark, J. G. Danzl, R. Hart, K. Lauber, G. Pupillo, and H.-C. Nagerl, Three-Body Correlation Functions and Recombination Rates for Bosons in ThreeDimensions and One Dimension, Phys. Rev. Lett. 107, 230404 (2011).

[Ham98] S. E. Hamann, D. L. Haycock, G. Klose, P. H. Pax, I. H. Deutsch, and P. S. Jessen,Resolved-Sideband Raman Cooling to the Ground State of an Optical Lattice, Phys. Rev.Lett. 80, 4149 (1998).

128

Bibliography

[Hec02] J. Hecker Denschlag, J. E. Simsarian, H. Haffner, C. McKenzie, A. Browaeys, D. Cho,K. Helmerson, S. L. Rolston, and W. D. Phillips, A Bose-Einstein condensate in anoptical lattice, J. Phys. B: At. Mol. Opt. Phys. 35, 3095 (2002).

[Hei11] J. Heinze, S. Gotze, J. S. Krauser, B. Hundt, N. Flaschner, D.-S. Luhmann, C. Becker,and K. Sengstock, Multiband Spectroscopy of Ultracold Fermions: Observation of ReducedTunneling in Attractive Bose-Fermi Mixtures, Phys. Rev. Lett. 107, 135303 (2011).

[Her03] J. Herbig, T. Kraemer, M. Mark, T. Weber, C. Chin, H.-C. Nagerl, and R. Grimm,Preparation of a Pure Molecular Quantum Gas, Science 301, 1510 (2003).

[Hir89] J. E. Hirsch, Bond-charge repulsion and hole superconductivity, Physica (Amsterdam)158C, 326 (1989).

[Hir94] J. E. Hirsch, Inapplicability of the Hubbard model for the description of real strongly corre-lated electrons, Physica (Amsterdam) 199-200B, 366 (1994).

[Hof07] S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, Non-equilibrium coherence dynamics in one-dimensional Bose gases, Nature 449, 324 (2007).

[Hof02] W. Hofstetter, J. I. Cirac, P. Zoller, E. Demler, and M. D. Lukin, High-TemperatureSuperfluidity of Fermionic Atoms in Optical Lattices, Phys. Rev. Lett. 89, 220407 (2002).

[Hua14] B. Huang, L. A. Sidorenkov, R. Grimm, and J. M. Hutson, Observation of the SecondTriatomic Resonance in Efimov’s Scenario, Phys. Rev. Lett. 112, 190401 (2014).

[Hub63] J. Hubbard, Electron Correlations in Narrow Energy Bands, Proc. R. Soc. Lond. A 276,238 (1963).

[Ima09] A. Imambekov and L. I. Glazman, Universal Theory of Nonlinear Luttinger Liquids,Science 323, 228 (2009).

[Ima12] A. Imambekov, T. L. Schmidt, and L. I. Glazman, One-dimensional quantum liquids:Beyond the Luttinger liquid paradigm, Rev. Mod. Phys. 84, 1253 (2012).

[Ino98] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn, andW. Ketterle, Observation of Feshbach resonances in a Bose-Einstein condensate, Nature392, 151 (1998).

[Jak98] D. Jaksch, C. Bruder, J.I. Cirac, C.W. Gardiner, and P. Zoller, Cold Bosonic Atoms inOptical Lattices, Phys. Rev. Lett. 81, 3108 (1998).

[Jen04] A. S. Jensen, K. Riisager, D. V. Fedorov, and E. Garrido, Structure and reactions ofquantum halos, Rev. Mod. Phys. 76, 215 (2004).

129

Bibliography

[Jer04] D. Jerome, Organic Conductors: From Charge Density Wave TTF-TCNQ to Supercon-ducting (TMTSF)2PF6, Chem. Rev. 104, 5565 (2004).

[Jo12] G.-B. Jo, J. Guzman, C. K. Thomas, P. Hosur, A. Vishwanath, and D. M. Stamper-Kurn, Ultracold Atoms in a Tunable Optical Kagome Lattice, Phys. Rev. Lett. 108, 045305(2012).

[Joc03a] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Hecker-Denschlag, and R. Grimm, Bose-Einstein Condensation of Molecules, Science 302, 2101(2003).

[Joc03b] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, C. Chin, J. Hecker Denschlag,and R. Grimm, Pure Gas of Optically Trapped Molecules Created from Fermionic Atoms,Phys. Rev. Lett. 91, 240402 (2003).

[Jor08] R. Jordens, N. Strohmaier, K. Gunter, H. Moritz, and T. Esslinger, A Mott insulator offermionic atoms in an optical lattice, Nature 455, 204 (2008).

[Joh09] P. R. Johnson, E. Tiesinga, J. V. Porto, and C. J. Williams, Effective three-body interac-tions of neutral bosons in optical lattices, New J. Phys. 11, 093022 (2009).

[Jom09] Y. Jompol, C. J. B. Ford, J. P. Griffiths, I. Farrer, G. A. C. Jones, D. Anderson,D. A. Ritchie, T. W. Silk, and A. J. Schofield, Probing Spin-Charge Separation in aTomonaga-Luttinger Liquid, Science 325, 597 (2009).

[Jot14] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, andT. Esslinger, Experimental realization of the topological Haldane model with ultracoldfermions, Nature 515, 237 (2014).

[Jur12] O. Jurgensen, K. Sengstock, and D.-S. Luhmann, Density-induced processes in quan-tum gas mixtures in optical lattices, Phys. Rev. A 86, 043623 (2012).

[Kim05] J. I. Kim, J. Schmiedmayer, and P. Schmelcher, Quantum scattering in quasi-one-dimensional cylindrical confinement, Phys. Rev. A 72, 042711 (2005).

[Kin04a] J. Kinast, S. L. Hemmer, M. E. Gehm, A. Turlapov, and J. E. Thomas, Evidence forSuperfluidity in a Resonantly Interacting Fermi Gas, Phys. Rev. Lett. 92, 150402 (2004).

[Kin04b] T. Kinoshita, T. Wenger, D. S. Weiss, Observation of a One-Dimensional Tonks-Girardeau Gas, Science 305, 1125 (2004).

[Kin05] T. Kinoshita, T. Wenger, and D. S. Weiss, Local Pair Correlations in One-DimensionalBose Gases, Phys. Rev. Lett. 95, 190406 (2005).

130

Bibliography

[Kin06] T. Kinoshita, T. Wenger, and D. S. Weiss, A quantum Newton’s cradle, Nature 440, 900(2006).

[Kna13] M. Knap, C. J. M. Mathy, M. B. Zvonarev, and E. Demler, Quantum flutter versus Blochoscillations in one-dimensional quantum liquids out of equilibrium, arXiv:1303.3583v1(2013).

[Kna14] M. Knap, C. J. M. Mathy, M. Ganahl, M. B. Zvonarev, and E. Demler, Quantum Flut-ter: Signatures and Robustness, Phys. Rev. Lett. 112, 015302 (2014).

[Kno09] S. Knoop, F. Ferlaino, M. Mark, M. Berninger, H. Schobel, H.-C. Nagerl, andR. Grimm, Observation of an Efimov-like trimer resonance in ultracold atom-dimer scat-tering, Nature Phys. 5, 227 (2009).

[Koh05] M. Kohl, H. Moritz, T. Stoferle, K. Gunter, and T. Esslinger, Fermionic Atoms in aThree Dimensional Optical Lattice: Observing Fermi Surfaces, Dynamics, and Interactions,Phys. Rev. Lett. 94, 080403 (2005).

[Koh06] T. Kohler, K. Goral, and P. S. Julienne, Production of cold molecules via magneticallytunable Feshbach resonances, Rev. Mod. Phys. 78, 1311 (2006).

[Koh59] W. Kohn, Analytic Properties of Bloch Waves and Wannier Functions, Phys. Rev. 115,809 (1959).

[Kol07] C. Kollath, A. M. Lauchli, and E. Altman, Quench Dynamics and Nonequilibrium PhaseDiagram of the Bose-Hubbard Model, Phys. Rev. Lett. 98, 180601 (2007).

[Kol12a] M. Kolodrubetz, D. Pekker, B. K. Clark, and K. Sengupta, Nonequilibrium dynamicsof bosonic Mott insulators in an electric field, Phys. Rev. B 85, 100505 (2012).

[Kol12b] M. Kolodrubetz, B. K. Clark, and D. A. Huse, Nonequilibrium Dynamic Critical Scal-ing of the Quantum Ising Chain, Phys. Rev. Lett. 109, 015701 (2012).

[Kol03a] A. R. Kolovsky, New Bloch Period for Interacting Cold Atoms in 1D Optical Lattices,Phys. Rev. Lett. 90, 213002 (2003).

[Kol03b] A. R. Kolovsky and A. Buchleitner, Floquet-Bloch operator for the Bose-Hubbard modelwith static field, Phys. Rev. E 68, 056213 (2003).

[Kol04] A. R. Kolovsky, Bloch oscillations in the Mott-insulator regime, Phys. Rev. A 70, 015604(2004).

[Kol15] A. R. Kolovsky, On quantum phase transitions in tilted 2D lattices, arXiv:1510.02671(2015).

131

Bibliography

[Kra04] T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H.-C. Nagerl, and R. Grimm, Op-timized production of a cesium Bose-Einstein condensate, Appl. Phys. B 79, 1013 (2004).

[Kra06] T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange,K. Pilch, A. Jaakkola, H.-C. Nagerl, and R. Grimm, Evidence for Efimov quantum statesin an ultracold gas of caesium atoms, Nature 440, 315 (2006).

[Kra09a] S. Kraft, F. Vogt, O. Appel, F. Riehle, and U. Sterr, Bose-Einstein Condensation ofAlkaline Earth Atoms: 40Ca, Phys. Rev. Lett. 103, 130401 (2009).

[Kra09b] F. Krausz and M. Ivanov, Attosecond physics, Rev. Mod. Phys. 81, 163 (2009).

[Kuh00] T. D. Kuhner, S. R. White, and H. Monien, One-dimensional Bose-Hubbard model withnearest-neighbor interaction, Phys. Rev. B 61, 12474 (2000).

[Lak13] B. Lake, D. A. Tennant, J.-S. Caux, T. Barthel, U. Schollwock, S. E. Nagler, andC. D. Frost, Multispinon Continua at Zero and Finite Temperature in a Near-Ideal Heisen-berg Chain, Phys. Rev. Lett. 111, 137205 (2013).

[Lan57] L. D. Landau, Theory of Fermi-liquids, Sov. Phys. JETP 3, 920 (1957).

[Lan59] L. D. Landau, On the theory of the Fermi-liquid, Sov. Phys. JETP 8, 70 (1959).

[Lan13] T. Langen, R. Geiger, M. Kuhnert, B. Rauer, and J. Schmiedmayer, Local emergenceof thermal correlations in an isolated quantum many-body system, Nature Phys. 9, 640(2013).

[Lan15a] T. Langen, R. Geiger, and J. Schmiedmayer, Ultracold Atoms Out of Equilibrium,Annu. Rev. Cond. Mat. Phys. 6, 201 (2015).

[Lan15b] T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuhnert, W. Rohringer,I. E. Mazets, T. Gasenzer, and J. Schmiedmayer, Experimental observation of a general-ized Gibbs ensemble, Science 348, 207 (2015).

[Lew07] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen De, and U. Sen, Ultra-cold atomic gases in optical lattices: mimicking condensed matter physics and beyond, Adv.Phys. 56, 243 (2007).

[Lie63a] E. H. Lieb and W. Liniger, Exact Analysis of an Interacting Bose Gas. I. The GeneralSolution and the Ground State, Phys. Rev. 130, 1605 (1963).

[Lie63b] E. H. Lieb, Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum, Phys.Rev. 130, 1616 (1963).

132

Bibliography

[Lig07] H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo,Dynamical Control of Matter-Wave Tunneling in Periodic Potentials, Phys. Rev. Lett. 99,220403 (2007).

[Lin09] Y.-J. Lin, R. L. Compton, K. Jimenez-Garcıa, J. V. Porto, and I. B. Spielman, Syntheticmagnetic fields for ultracold neutral atoms, Nature 462, 628 (2009).

[Lu11] M. Lu, N. Q. Burdick, S. H. Youn, and B. L. Lev, Strongly Dipolar Bose-Einstein Con-densate of Dysprosium, Phys. Rev. Lett. 107, 190401 (2011).

[Luh08] D.-S. Luhmann, K. Bongs, K. Sengstock, and D. Pfannkuche, Self-Trapping of Bosonsand Fermions in Optical Lattices, Phys. Rev. Lett. 101, 050402 (2008).

[Luh12] D. S. Luhmann, O. Jurgensen, and K. Sengstock, Multi-orbital and density-inducedtunneling of bosons in optical lattices, New J. Phys. 14, 033021 (2012).

[Luo93] F. Luo, G. C. McBane, G. Kim, C. F. Giese, and W. R. Gentry, The weakest bond: Exper-imental observation of helium dimer, J. Chem. Phys. 98, 3564 (1993).

[Ma11] R. Ma, M. E. Tai, P. M. Preiss, W. S. Bakr, J. Simon, and M. Greiner, Photon-AssistedTunneling in a Biased Strongly Correlated Bose Gas, Phys. Rev. Lett. 107, 095301 (2011).

[Mac88] A. H. MacDonald, S. M. Girvin, and D. Yoshioka, t/U expansion for the Hubbard model,Phys. Rev. B 37, 9753 (1988).

[Mad00] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Vortex Formation in aStirred Bose-Einstein Condensate, Phys. Rev. Lett. 84, 806 (2000).

[Mah14] K.W. Mahmud, L. Jiang, E. Tiesinga, and P. R. Johnson, Bloch oscillations and quenchdynamics of interacting bosons in an optical lattice, Phys. Rev. A 89, 023606 (2014).

[Mar07] M. Mark, F. Ferlaino, S. Knoop, J. G. Danzl, T. Kraemer, C. Chin, H.-C. Nagerl, andR. Grimm, Spectroscopy of ultracold trapped cesium Feshbach molecules, Phys. Rev. A 76,042514 (2007).

[Mar11a] M. J. Mark, E. Haller, K. Lauber, J. G. Danzl, A. J. Daley, H.-C. Nagerl, PrecisionMeasurements on a Tunable Mott Insulator of Ultracold Atoms, Phys. Rev. Lett. 107,175301 (2011).

[Mar11b] M. J. Mark, E. Haller, J. G. Danzl, K. Lauber, M. Gustavsson, and H.-C. Nagerl,Demonstration of the temporal matter-wave Talbot effect for trapped matter waves, New J.Phys. 13, 085008 (2011).

[Mar12] M. J. Mark, E. Haller, K. Lauber, J. G. Danzl, A. Janisch, H. P. Buchler, A. J. Daley,and H.-C. Nagerl, Preparation and Spectroscopy of a Metastable Mott-Insulator State withAttractive Interactions, Phys. Rev. Lett. 108, 215302 (2012).

133

Bibliography

[Mar88] P. J. Martin, B. G. Oldaker, A. H. Miklich, and D. E. Pritchard, Bragg scattering ofatoms from a standing light wave, Phys. Rev. Lett. 60, 515 (1988).

[Mat12] C. J. M. Mathy, M. B. Zvonarev, and E. Demler, Quantum flutter of supersonic particlesin one-dimensional quantum liquids, Nature Phys. 8, 881 (2012).

[Mat99] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, andE. A. Cornell, Vortices in a Bose-Einstein Condensate, Phys. Rev. Lett. 83, 2498 (1999).

[Maz06] G. Mazzarella, S. M. Giampaolo, and F. Illuminati, Extended Bose Hubbard model ofinteracting bosonic atoms in optical lattices: From superfluidity to density waves, Phys.Rev. A 73, 013625 (2006).

[Mei13] F. Meinert, M. J. Mark, E. Kirilov, K. Lauber, P. Weinmann, A. J. Daley, and H.-C. Nagerl, Quantum Quench in an Atomic One-Dimensional Ising Chain, Phys. Rev.Lett. 111, 053003 (2013).

[Mei14] F. Meinert, M. J. Mark, E. Kirilov, K. Lauber, P. Weinmann, M. Grobner, A. J. Daley,and H.-C. Nagerl, Observation of many-body dynamics in long-range tunneling after aquantum quench, Science 344, 1259 (2014).

[Mer11] A. Mering and M. Fleischhauer, Multiband and nonlinear hopping corrections to thethree-dimensional Bose-Fermi-Hubbard model, Phys. Rev. A 83, 063630 (2011).

[Met99] H. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer-Verlag, NewYork, 1999).

[Miy13] H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, Realizingthe Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices, Phys. Rev.Lett. 111, 185302 (2013).

[Mod01] G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni, and M. Inguscio, Bose-Einstein Condensation of Potassium Atoms by Sympathetic Cooling, Science 294, 1320(2001).

[Moe08] M. Moeckel and S. Kehrein, Interaction Quench in the Hubbard Model, Phys. Rev. Lett.100, 175702 (2008).

[Mon95] C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, D. J. Wineland,and P. Gould, Resolved-Sideband Raman Cooling of a Bound Atom to the 3D Zero-PointEnergy, Phys. Rev. Lett. 75, 4011 (1995).

[Mor03] H. Moritz, T. Stoferle, M. Kohl, and T. Esslinger, Exciting Collective Oscillations in aTrapped 1D Gas, Phys. Rev. Lett. 91, 250402 (2003).

134

Bibliography

[Mor05] H. Moritz, T. Stoferle, K. Gunter, M. Kohl, and T. Esslinger, Confinement InducedMolecules in a 1D Fermi Gas, Phys. Rev. Lett. 94, 210401 (2005).

[Mor01] O. Morsch, J. H. Muller, M. Cristiani, D. Ciampini, and E. Arimondo, Bloch Oscilla-tions and Mean-Field Effects of Bose-Einstein Condensates in 1D Optical Lattices, Phys.Rev. Lett. 87, 140402 (2001).

[Mor06] O. Morsch and M. Oberthaler, Dynamics of Bose-Einstein condensates in optical lattices,Rev. Mod. Phys. 78, 179 (2006).

[Mou13] M. Mourigal, M. Enderle, A. Klopperpieper, J.-S. Caux, A. Stunault, andH. M. Rønnow, Fractional spinon excitations in the quantum Heisenberg antiferromag-netic chain, Nat. Phys. 9, 435 (2013).

[Nor11] M. R. Norman, The Challenge of Unconventional Superconductivity, Science 332, 196(2011).

[Ols98] M. Olshanii, Atomic Scattering in the Presence of an External Confinement and a Gas ofImpenetrable Bosons, Phys. Rev. Lett. 81, 938 (1998).

[Ols03] M. Olshanii and V. Dunjko, Short-Distance Correlation Properties of the Lieb-LinigerSystem and Momentum Distributions of Trapped One-Dimensional Atomic Gases, Phys.Rev. Lett. 91, 090401 (2003).

[Osp06] S. Ospelkaus, C. Ospelkaus, O. Wille, M. Succo, P. Ernst, K. Sengstock, and K. Bongs,Localization of Bosonic Atoms by Fermionic Impurities in a Three-Dimensional OpticalLattice, Phys. Rev. Lett. 96, 180403 (2006).

[Pal09] S. Palzer, C. Zipkes, C. Sias, and M. Kohl, Quantum Transport through a Tonks-Girardeau Gas, Phys. Rev. Lett. 103, 150601 (2009).

[Pan14] M. Panfil and J.-S. Caux, Finite-temperature correlations in the Lieb-Liniger one-dimensional Bose gas, Phys. Rev. A 89, 033605 (2014).

[Par04] B. Paredes, A. Widera, V. Murg, O. Mandel, S. Folling, I. Cirac, G. V. Shlyapnikov,T. W. Hansch, and I. Bloch, Tonks-Girardeau gas of ultracold atoms in an optical lattice,Nature 429, 277 (2004).

[Par50] R. G. Parr, D. P. Craig, and I. G. Ross, Molecular Orbital Calculations of the LowerExcited Electronic Levels of Benzene, Configuration Interaction Included, J. Chem. Phys.18, 1561 (1950).

[Pet02] C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (CambridgeUniversity Press, Cambridge, 2002).

135

Bibliography

[Pet00] D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven, Regimes of Quantum Degener-acy in Trapped 1D Gases, Phys. Rev. Lett. 85, 3745 (2000).

[Pet08] D. S. Petrov, D. M. Gangardt, and G. V. Shlyapnikov, Low-dimensional trapped gases,J. Phys. IV France 116, 5 (2004).

[Pie11] S. Pielawa, T. Kitagawa, E. Berg, and S. Sachdev, Correlated phases of bosons in tiltedfrustrated lattices, Phys. Rev. B 83, 205135 (2011).

[Pie12] S. Pielawa, E. Berg, and S. Sachdev, Frustrated quantum Ising spins simulated by spin-less bosons in a tilted lattice: From a quantum liquid to antiferromagnetic order, Phys. Rev.B 86, 184435 (2012).

[Pil12] S. Pilati and M. Troyer, Bosonic Superfluid-Insulator Transition in Continuous Space,Phys. Rev. Lett. 108, 155301 (2012).

[Pit03] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford Univ. Press, NewYork, 2003).

[Poi93] D. Poilblanc, T. Ziman, J. Bellissard, F. Mila, and G. Montambaux, Poisson vs. GOEStatistics in Integrable and Non-Integrable Quantum Hamiltonians, Europhys. Lett. 22,537 (1993).

[Pol08] A. Polkovnikov and V. Gritsev, Breakdown of the adiabatic limit in low-dimensional gap-less systems, Nat. Phys. 4, 477 (2008).

[Pol11] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Colloquium: Nonequilib-rium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83, 863 (2011).

[Pon06] A. V. Ponomarev and A. R. Kolovsky, Dipole and Bloch oscillations of cold atoms in aparabolic lattice, Laser Physics 16, 367 (2006).

[Que12] F. Queisser, P. Navez, and R. Schutzhold, Sauter-Schwinger-like tunneling in tiltedBose-Hubbard lattices in the Mott phase, Phys. Rev. A 85, 033625 (2012).

[Rap12] A. Rapp, X. Deng, and L. Santos, Ultracold Lattice Gases with Periodically ModulatedInteractions, Phys. Rev. Lett. 109, 203005 (2012).

[Reg03] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Creation of ultracold molecules from aFermi gas of atoms, Nature 424, 47 (2003).

[Reg04] C. A. Regal, M. Greiner, and D. S. Jin, Observation of Resonance Condensation ofFermionic Atom Pairs, Phys. Rev. Lett. 92, 040403 (2004).

136

Bibliography

[Rig07] M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Relaxation in a Completely Inte-grable Many-Body Quantum System: An Ab Initio Study of the Dynamics of the HighlyExcited States of 1D Lattice Hard-Core Bosons, Phys. Rev. Lett. 98, 050405 (2007).

[Rig08] M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for genericisolated quantum systems, Nature 452, 854 (2008).

[Rig09] M. Rigol, Breakdown of Thermalization in Finite One-Dimensional Systems, Phys. Rev.Lett. 103, 100403 (2009).

[Roa04] G. Roati, E. de Mirandes, F. Ferlaino, H. Ott, G. Modugno, and M. Inguscio, AtomInterferometry with Trapped Fermi Gases, Phys. Rev. Lett. 92, 230402 (2004).

[Rob01] A. Robert, O. Sirjean, A. Browaeys, J. Poupard, S. Nowak, D. Boiron, C. I. West-brook, and A. Aspect, A Bose-Einstein Condensate of Metastable Atoms, Science 292,461 (2001).

[Rob98] J. L. Roberts, N. R. Claussen, J. P. Burke, Jr., C. H. Greene, E. A. Cornell, andC. E. Wieman, Resonant Magnetic Field Control of Elastic Scattering in Cold 85Rb, Phys.Rev. Lett. 81, 5109 (1998).

[Ron13] J. P. Ronzheimer, M. Schreiber, S. Braun, S. S. Hodgman, S. Langer, I. P. McCul-loch, F. Heidrich-Meisner, I. Bloch, and U. Schneider, Expansion Dynamics of Interact-ing Bosons in Homogeneous Lattices in One and Two Dimensions, Phys. Rev. Lett. 110,205301 (2013).

[Rub11] C. P. Rubbo, S. R. Manmana, B. M. Peden, M. J. Holland, and A. M. Rey, Resonantlyenhanced tunneling and transport of ultracold atoms on tilted optical lattices, Phys. Rev.A 84, 033638 (2011).

[Sac99] S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge,England, 1999).

[Sac02] S. Sachdev, K. Sengupta, and S. M. Girvin, Mott insulators in strong electric fields,Phys. Rev. B 66, 075128 (2002).

[Sac08] S. Sachdev, Quantum magnetism and criticality, Nature Phys. 4, 173 (2008).

[Sad06] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn,Spontaneous symmetry breaking in a quenched ferromagnetic spinor Bose-Einstein conden-sate, Nature 443, 312 (2006).

[Sak10] K. Sakmann, A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Quantum dynam-ics of attractive versus repulsive bosonic Josephson junctions: Bose-Hubbard and full-Hamiltonian results, Phys. Rev. A 82, 013620 (2010).

137

Bibliography

[Sak94] J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley Publishing, 1994).

[Sal13] S. Sala, G. Zurn, T. Lompe, A. N. Wenz, S. Murmann, F. Serwane, S. Jochim, andA. Saenz, Coherent Molecule Formation in Anharmonic Potentials Near Confinement-Induced Resonances, Phys. Rev. Lett. 110, 203202 (2013).

[Sal02] L. Salasnich, A. Parola, and L. Reatto, Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates, Phys. Rev. A 65, 043614 (2002).

[San02] F. P. D. Santos, H. Marion, S. Bize, Y. Sortais, A. Clairon, and C. Salomon, Controllingthe Cold Collision Shift in High Precision Atomic Interferometry, Phys. Rev. Lett. 89,233004 (2002).

[Sar05] L. De Sarlo, L. Fallani, J. E. Lye, M. Modugno, R. Saers, C. Fort, and M. Inguscio,Unstable regimes for a Bose-Einstein condensate in an optical lattice, Phys. Rev. A 72,013603 (2005).

[Sch15] P. Schauß, J. Zeiher, T. Fukuhara, S. Hild, M. Cheneau, T. Macrı, T. Pohl, I. Bloch,and C. Gross, Crystallization in Ising quantum magnets, Science 347, 1455 (2015).

[Sch08] U. Schneider, L. Hackermuller, S. Will, T. Best, I. Bloch, T. A. Costi, R. W. Helmes,D. Rasch, and A. Rosch, Metallic and Insulating Phases of Repulsively InteractingFermions in a 3D Optical Lattice, Science 322, 1520 (2008).

[Sch09] P.-I. Schneider, S. Grishkevich, and A. Saenz, Ab initio determination of Bose-Hubbardparameters for two ultracold atoms in an optical lattice using a three-well potential, Phys.Rev. A 80, 013404 (2009).

[Sch12] U. Schneider, L. Hackermuller, J. P. Ronzheimer, S. Will, S. Braun, T. Best, I. Bloch,E. Demler, S. Mandt, D. Rasch, and A. Rosch, Fermionic transport and out-of-equilibrium dynamics in a homogeneous Hubbard model with ultracold atoms, NaturePhys. 8, 213 (2012).

[Seb06] J. Sebby-Strabley, M. Anderlini, P. S. Jessen, and J. V. Porto, Lattice of double wells formanipulating pairs of cold atoms, Phys. Rev. A 73, 033605 (2006).

[Sed11] G. Sedghi et al., Long-range electron tunnelling in oligo-porphyrin molecular wires, Nat.Nanotechnol. 6, 517 (2011).

[She10] J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Single-atom-resolved fluorescence imaging of an atomic Mott insulator, Nature 467, 68 (2010).

[Sia07] C. Sias, A. Zenesini, H. Lignier, S. Wimberger, D. Ciampini, O. Morsch, and E. Ari-mondo, Resonantly Enhanced Tunneling of Bose-Einstein Condensates in Periodic Poten-tials, Phys. Rev. Lett. 98, 120403 (2007).

138

Bibliography

[Sim11] J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss, and M. Greiner, Quantum simula-tion of antiferromagnetic spin chains in an optical lattice, Nature 472, 307 (2011).

[Spi07] I. B. Spielman, W. D. Phillips, and J. V. Porto, Mott-Insulator Transition in a Two-Dimensional Atomic Bose Gas, Phys. Rev. Lett. 98, 080404 (2007).

[Sre94] M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).

[Sta99] D. M. Stamper-Kurn, A. P. Chikkatur, A. Gorlitz, S. Inouye, S. Gupta, D. E. Pritchard,and W. Ketterle, Excitation of Phonons in a Bose-Einstein Condensate by Light Scattering,Phys. Rev. Lett. 83, 2876 (1999).

[Ste09] S. Stellmer, M. K. Tey, B. Huang, R. Grimm, and F. Schreck, Bose-Einstein Condensa-tion of Strontium, Phys. Rev. Lett. 103, 200401 (2009).

[Ste99] J. Stenger, S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, D. E. Pritchard, andW. Ketterle, Bragg Spectroscopy of a Bose-Einstein Condensate, Phys. Rev. Lett. 82, 4569(1999).

[Sto04] T. Stoferle, H. Moritz, C. Schori, M. Kohl, and T. Esslinger, Transition from a StronglyInteracting 1D Superfluid to a Mott Insulator, Phys. Rev. Lett. 92, 130403 (2004).

[Str93] R. Strack and D. Vollhardt, Hubbard model with nearest-neighbor and bond-charge in-teraction: Exact ground-state solution in a wide range of parameters, Phys. Rev. Lett. 70,2637 (1993).

[Str94] R. Strack and D. Vollhardt, Rigorous criteria for ferromagnetism in itinerant electronsystems, Phys. Rev. Lett. 72, 3425 (1994).

[Str03] K. E. Strecker, G. B. Partridge, and R. G. Hulet, Conversion of an Atomic Fermi Gas toa Long-Lived Molecular Bose Gas, Phys. Rev. Lett. 91, 080406 (2003).

[Str11] J. Struck, C. Olschlager, R. Le Targat, P. Soltan-Panahi, A. Eckardt, M. Lewenstein,P. Windpassinger, and K. Sengstock, Quantum Simulation of Frustrated Classical Mag-netism in Triangular Optical Lattices, Science 333, 996 (2011).

[Str12] J. Struck, C. Olschlager, M. Weinberg, P. Hauke, J. Simonet, A. Eckardt, M. Lewen-stein, K. Sengstock, and P. Windpassinger, Tunable Gauge Potential for Neutral andSpinless Particles in Driven Optical Lattices, Phys. Rev. Lett. 108, 225304 (2012).

[Tak03] Y. Takasu, K. Maki, K. Komori, T. Takano, K. Honda, M. Kumakura, T. Yabuzaki, andY. Takahashi, Spin-Singlet Bose-Einstein Condensation of Two-Electron Atoms, Phys.Rev. Lett. 91, 040404 (2003).

139

Bibliography

[Tar12] L. Tarruell, D. Greif, T. Uehlinger, G. Jotzu, and T. Esslinger, Creating, moving andmerging Dirac points with a Fermi gas in a tunable honeycomb lattice, Nature 483, 302(2012).

[Tha06] G. Thalhammer, K. Winkler, F. Lang, S. Schmid, R. Grimm, and J. Hecker Denschlag,Long-Lived Feshbach Molecules in a Three-Dimensional Optical Lattice, Phys. Rev. Lett.96, 050402 (2006).

[Tro08] S. Trotzky, P. Cheinet, S. Folling, M. Feld, U. Schnorrberger, A. M. Rey,A. Polkovnikov, E. A. Demler, M. D. Lukin, and I. Bloch, Time-Resolved Observa-tion and Control of Superexchange Interactions with Ultracold Atoms in Optical Lattices,Science 319, 295 (2008).

[Tro12] S. Trotzky, Y-A. Chen, A. Flesch, I. P. McCulloch, U. Schollwock, J. Eisert, andI. Bloch, Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas, Nature Phys. 8, 325 (2012).

[Vai79] H. G. Vaidya and C. A. Tracy, One-Particle Reduced Density Matrix of ImpenetrableBosons in One Dimension at Zero Temperature, Phys. Rev. Lett. 43, 1540 (1979).

[Ven11] H. Venzl, Ultracold bosons in tilted optical lattices - impact of spectral statisticson simulability, stability, and dynamics, PhD thesis, University of Freiburg, 2011;http://www.freidok.uni-freiburg.de/volltexte/8126/

[Ver08] F. Verstraete, V. Murg, and J. I. Cirac, Matrix product states, projected entangled pairstates, and variational renormalization group methods for quantum spin systems, Adv.Phys. 57, 143 (2008).

[Vid04] G. Vidal, Efficient Simulation of One-Dimensional Quantum Many-Body Systems, Phys.Rev. Lett. 93, 040502 (2004).

[Vul98] V. Vuletic, C. Chin, A. J. Kerman, and S. Chu, Degenerate Raman Sideband Cooling ofTrapped Cesium Atoms at Very High Atomic Densities, Phys. Rev. Lett. 81, 5768 (1998).

[Vul99] V. Vuletic, A. J. Kerman, C. Chin, and S. Chu, Observation of Low-Field Feshbach Reso-nances in Collisions of Cesium Atoms, Phys. Rev. Lett. 82, 1406 (1999).

[Was93] C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Kohler, Coherentsubmillimeter-wave emission from Bloch oscillations in a semiconductor superlattice, Phys.Rev. Lett. 70, 3319 (1993).

[Web02] T. Weber, J. Herbig, M. Mark, H.-C. Nagerl, and R. Grimm, Bose-Einstein Condensa-tion of Cesium, Science 299, 232 (2002).

140

Bibliography

[Whi04] S. R. White and A. E. Feiguin, Real-Time Evolution Using the Density Matrix Renor-malization Group, Phys. Rev. Lett. 93, 076401 (2004).

[Wil10] S. Will, T. Best, U. Schneider, L. Hackermuller, D.-S. Luhmann, and I. Bloch, Time-resolved observation of coherent multi-body interactions in quantum phase revivals, Nature465, 197 (2010).

[Win06] K. Winkler, G. Thalhammer, F. Lang, R. Grimm, J. Hecker Denschlag, A. J. Daley,A. Kantian, H. P. Buchler, and P. Zoller, Repulsively bound atom pairs in an opticallattice, Nature 441, 853 (2006).

[Win14] J. R. Winkler and H. B. Gray, Long-Range Electron Tunneling, J. Am. Chem. Soc. 136,2930 (2014).

[Wit05] D. Witthaut, M. Werder, S. Mossmann, and H. J. Korsch, Bloch oscillations of Bose-Einstein condensates: Breakdown and revival, Phys. Rev. E 71, 036625 (2005).

[Wu01] B. Wu and Q. Niu, Landau and dynamical instabilities of the superflow of Bose-Einsteincondensates in optical lattices, Phys. Rev. A 64, 061603(R) (2001).

[Wu03] B. Wu and Q. Niu, Superfluidity of Bose-Einstein condensate in an optical lattice: Landau-Zener tunnelling and dynamical instability, New J. Phys. 5, 104 (2003).

[Yan13] B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. A. Hazzard, A. M. Rey, D. S. Jin,and J. Ye, Observation of dipolar spin-exchange interactions with lattice-confined polarmolecules, Nature 501, 521 (2013).

[Yan69] C. N. Yang and C. P. Yang, Thermodynamics of a One-Dimensional System of Bosonswith Repulsive Delta-Function Interaction, J. Math. Phys. 10, 1115 (1969).

[Zen34] C. Zener, A Theory of the Electrical Breakdown of Solid Dielectrics, Proc. R. Soc. LondonA 145, 523 (1934).

[Zen09] A. Zenesini, H. Lignier, G. Tayebirad, J. Radogostowicz, D. Ciampini, R. Mannella,S. Wimberger, O. Morsch, and E. Arimondo, Time-Resolved Measurement of Landau-Zener Tunneling in Periodic Potentials, Phys. Rev. Lett. 103, 090403 (2009).

[Zwi03] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadz-ibabic, and W. Ketterle, Observation of Bose-Einstein Condensation of Molecules, Phys.Rev. Lett. 91, 250401 (2003).

[Zwi04] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman, andW. Ketterle, Condensation of Pairs of Fermionic Atoms near a Feshbach Resonance, Phys.Rev. Lett. 92, 120403 (2004).

141

Bibliography

[Zwi05] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck, and W. Ketterle,Vortices and superfluidity in a strongly interacting Fermi gas, Nature 435, 1047 (2005).

142

ACKNOWLEDGMENTS

Owing to the growing experimental complexity, working in the field of ultracold quantumgases always requires a collaborative effort. From the very beginning of my work in Inns-bruck, I found myself in a team of great people, providing an excellent atmosphere for anexciting scientific voyage. I would like to thank all of you for help and contributions thatmade this PhD work possible. Specifically, I thank

Hanns-Christoph Nagerl, my supervisor, for giving me the opportunity to be part of hisgroup and for all support and assistance during the last years. It has been a pleasure to workon a great and flexible experiment, to be given the freedom to pursue and develop ownideas, and to get numerous possibilities for scientific exchange on conference, workshops, orjoint discussion sessions.

Rudi Grimm for his commitment to provide an excellent environment for experimentalresearch on ultracold gases in Innsbruck. Thanks to this effort, we profit from a valuableinfrastructure and the possibility to exchange knowledge between a number of differentlaboratories.

Manfred Mark, our Postdoc, for a perfect introduction to the ”nuts and bolts” of the Cs-IIIexperiment, specifically in the beginning of my PhD, and for many hours of joint measure-ments. Thank’s a lot for numerous exciting discussions that have not just once been inspiringin view of new ideas.

the whole Cs-III team, specifically Emil Kirilov for sharing his impressive pool of experi-ence and for many stimulating discussions and Katharina Lauber for the joint effort to alwayskeep the experiment on a high performance level and searching for improvements. Thankyou Michael Grobner and Philipp Weinmann for the great teamwork and the constant attitudeto exchange knowledge and experience.

Andrew Daley for all the valuable contributions and for providing support from the theoryside many times during this work. Thank you Ole Jurgensen, Dirk-Soren Luhmann, MiloszPanfil, and Jean-Sebastien Caux for the pleasant collaborations.

Christine Gotsch-Obmascher and Karin Kohle for all support concerning administrative is-sues, and the staff in the mechanical and electronic workshop for the professional help whenneeded.

A very special ”Thank you” goes to my parents and my brother and finally to Elisabeth!

Quantum dynamics in strongly correlated one-dimensional ...

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