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Complex dynamics of ultracold atoms [Elektronische Ressource] / vorgelegt von Patrick Plötz

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Complex Dynamics of Ultracold AtomsInaugural–Dissertationzur Erlangung der Doktorw¨urdeder Naturwissenschaftlich–Mathematischen Gesamtfakult¨atder Ruprecht–Karls–Universit¨at Heidelbergvorgelegt vonDipl.-Phys. Patrick Plo¨tzaus GreifswaldTag der m¨undlichen Pr¨ufung: 13. Oktober 2010Gutachter: Dr. Sandro WimbergerProf. Dr. Manfred SalmhoferZusammenfassungAktuelle Experimente mit ultrakalten Atomen bilden die Motivation fu¨r dasStudium komplexer Quantendynamik von Bose–Einstein-Kondensaten in opti-schen Gittern, das den Gegenstand dieser Arbeit bildet. Ausgehend von diesenExperimenten wird ein Gittermodell wechselwirkender Bosonen unter dem Ein-fluss einer externen Kraft abgeleitet und im Weiteren der Arbeit in verschiede-nen dynamischen Regimen untersucht. Im ersten Teil entwickeln wir dabei einneues Mass zur Detektion vermiedener Kreuzungen in komplexen Energiespek-tren und wenden es auf das quantenchaotische Regime des Ein-Band Systemsan. Im zweiten und la¨ngeren Teil der Arbeit wird die Kopplung zweier En-ergieba¨nder anhand eines Zwei-Band-Modells untersucht. Die komplexe Zeit-entwicklung zeigt sich hierbei schon im wechselwirkungsfreien Problem in derhorizontalen und vertikalen Populationsdynamik, unter anderem in der Existenzvon Resonanzen im Interband-Transport.
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Complex Dynamics of Ultracold Atoms
Inaugural–Dissertation
zur Erlangung der Doktorw¨urde
der Naturwissenschaftlich–Mathematischen Gesamtfakult¨at
der Ruprecht–Karls–Universit¨at Heidelberg
vorgelegt von
Dipl.-Phys. Patrick Plo¨tz
aus Greifswald
Tag der m¨undlichen Pr¨ufung: 13. Oktober 2010Gutachter: Dr. Sandro Wimberger
Prof. Dr. Manfred SalmhoferZusammenfassung
Aktuelle Experimente mit ultrakalten Atomen bilden die Motivation fu¨r das
Studium komplexer Quantendynamik von Bose–Einstein-Kondensaten in opti-
schen Gittern, das den Gegenstand dieser Arbeit bildet. Ausgehend von diesen
Experimenten wird ein Gittermodell wechselwirkender Bosonen unter dem Ein-
fluss einer externen Kraft abgeleitet und im Weiteren der Arbeit in verschiede-
nen dynamischen Regimen untersucht. Im ersten Teil entwickeln wir dabei ein
neues Mass zur Detektion vermiedener Kreuzungen in komplexen Energiespek-
tren und wenden es auf das quantenchaotische Regime des Ein-Band Systems
an. Im zweiten und la¨ngeren Teil der Arbeit wird die Kopplung zweier En-
ergieba¨nder anhand eines Zwei-Band-Modells untersucht. Die komplexe Zeit-
entwicklung zeigt sich hierbei schon im wechselwirkungsfreien Problem in der
horizontalen und vertikalen Populationsdynamik, unter anderem in der Existenz
von Resonanzen im Interband-Transport. Des Weiteren k¨onnen wirzeigen, dass
die interatomare Wechselwirkung zu Kollaps und Wiederkehr dieser resonanten
Interband-Oszillationen fu¨hrt und sind in der Lage alle Zeitskalen dieser kom-
plexen Interband-Dynamik vorherzusagen. Letzteres erfolgt durch ein exakt
lo¨sbares effektives Modell, das durch eine Vielzahl numerischer Ergebnisse mo-
tiviert und gestu¨tzt wird.
Abstract
Motivated by current experiments with ultracold atoms, the study of complex
dynamics of Bose–Einstein condensates in optical lattices forms the central
subject of this work. A lattice model of interacting bosons under the influence
of an external force is motivated and derived from the experimental setup.
Several dynamical regimes of this model are discussed in this thesis. In a first
part we will develop a new measure for detecting avoided crossings in complex
energy spectra and apply it to the quantum chaotic regime of the single-band
system. The second and main part of this work is dedicated to the coupling
between energy bands described in terms of a two-band model. The complex
time evolution is already apparent in the horizontal and vertical population
dynamics of the non-interacting problem. We find resonances in the interband
transport, and, inasecond step, studytheeffectofinter-particleinteractionson
these resonantoscillations. We are ableto predict alltimescalesof the complex
interband dynamics even in the presence of interactions. This is possible via
the introduction of an effective model that is motivated and supported by a
multitude of numerical results and proves exactly solvable.ivPreface
Wissenschaft wird vom Menschen gemacht.
Werner Heisenberg
The last three years passed very quickly and the work on this thesis took many, some-
times unexpected turns. Whilst working on the problems that cumulated in the present
text, I experienced help and support from many sides and I would like to acknowledge the
most important ones here.
First of all, I am grateful to my supervisor, Dr. Sandro Wimberger, for the possibility
to work in his group and for introducing me to the fascinating topic of ultracold atoms.
He provided many contacts with researchers working in this and related fields leading to
stimulating discussions. I would also like to thank him for many suggestions during this
whole project and his proof-reading and comments concerning this thesis.
Furthermore I would like to thank Prof. Manfred Salmhofer for being the second referee of
my thesis.
Special gratitude is devoted to Andrea Tomadin for providing his code, for his help in
how to use it and for several good discussions. I am also grateful to Dr. Javier Madron˜ero
for his help in improving the speed of the numerical routines. I am indebted to Prof. Peter
Schlagheck for his constant support and interest. He came up with the idea for an effective
spin model in a personal discussion in Krasnoyarsk.
I profited by financial and organisational support from various sides. I would like to
thank the Klaus Tschira Foundation for generously granting me a scholarship and offering
several interesting events. I also wish to express my gratitude to the Graduate School of
Fundamental Physics in Heidelberg and in particular to Gesine Heinzelmann and Prof. San-
dra Klevansky. Both were always friendly, encouraging, and supportive. I am furthermore
grateful to the Graduate Academy of the University of Heidelberg for two travel grants
allowing me to learn and discuss in Krasnoyarsk and Utrecht.
Friends and colleagues, in particular Ghazal Tayebirad, Christoph Karrasch, Benedikt
Probst, Tobias Paul, and Michael Henke, contributed with many chats, discussions, and
support. Thank you all.
Besondersdankenmo¨chteichmeinenEltern, RitaundGerhardPlo¨tz,diemichstetsund
in allem unterstu¨tzt haben und damit die vorliegende Arbeit erst erm¨oglicht haben. Meiner
Mutter Rita Pl¨otz danke ich zus¨atzlich fu¨r Ihre Offenheit, Toleranz und ihren wunderbaren
Humor gerade in der letzten Zeit.
Die allergro¨ßte und unscha¨tzbare Unterstu¨tzung erhalte ich von Kerstin Klett. Ich kann
ihren Beitrag sowie meine Dankbarkeit und Freude daru¨ber nicht beschreiben.
Heidelberg, July 2010.
vviContents
Introduction 1
Prerequisites: Interacting Bosons in Optical Lattices 7
1 Cold Quantum Gases and the Bose–Hubbard Model 7
1.1 Ultracold Atoms in External Fields . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Wannier–Stark Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Derivation of the Bose–Hubbard Model . . . . . . . . . . . . . . . . . . . 13
1.4 Some Background on Bose–Hubbard Models . . . . . . . . . . . . . . . . . 23
Single-band Bose–Hubbard Model: Quantum Chaos 35
2 Fidelity and Avoided Crossings in the Bose–Hubbard Model 35
2.1 Fidelity and Avoided Crossings . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.1 Introduction: The Quantum Fidelity Measure . . . . . . . . . . . . 36
2.1.2 Detection and Characterization of Avoided Crossings . . . . . . . . 39
2.1.3 Application to Random Matrix Model . . . . . . . . . . . . . . . . 47
2.2 Application to the Bose–Hubbard Model . . . . . . . . . . . . . . . . . . . 50
2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2.2 Density of Avoided Crossings . . . . . . . . . . . . . . . . . . . . . 54
2.2.3 Characterising the System by Avoided Crossings . . . . . . . . . . . 57
2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Two-band Bose–Hubbard model: Complex Dynamics 67
3 The Non-interacting Two-band Model 67
3.1 Introduction and Review of Single-Band Case . . . . . . . . . . . . . . . . 67
3.1.1 Motivation and Hamiltonian . . . . . . . . . . . . . . . . . . . . . 68
3.1.2 Single Band Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 Momentum Space Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2.1 Hamiltonian and Absence of Explicit Solution . . . . . . . . . . . . 72
3.2.2 Perturbative Solution and Existence of Resonances . . . . . . . . . 75
3.2.3 Magnus Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 79
viiCONTENTS
3.3 Real Space Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.3.1 Transformation of Hamiltonian . . . . . . . . . . . . . . . . . . . . 86
3.3.2 Non-resonant Regime . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.4 System in Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.4.1 Effective Model and Analysis of Resonances . . . . . . . . . . . . . 91
3.4.2 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4 The Interacting Two-band Model: Collapse and Revival 101
4.1 Collapse and Revival of the Interband Oscillations . . . . . . . . . . . . . . 101
4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.1.3 Collapse and Revival of the Interband Oscillations . . . . . . . . . . 107
4.1.4 Stability of the Collapse and Revival Effect . . . . . . . . . . . . . 110
4.1.5 Eigenbasis Expansions . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.2 Theoretical Understanding . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.2.2 Coherent State Model . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.2.3 Details of the Coherent State Approximations . . . . . . . . . . . . 135
4.2.4 Exact Diagonalisation: Eigenfrequencies . . . . . . . . . . . . . . . 138
4.3 Effective Spin Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.3.1 The Effective Model and First Results . . . . . . . . . . . . . . . . 141
4.3.2 Exact Solution of the Effective Spin Model . . . . . . . . . . . . . 145
4.3.3 Results for the Revival Time . . . . . . . . . . . . . . . . . . . . . 148
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Summary and Outlook 157
Appendices 161
A Remarks on Numerical Implementation 161
B The Rabi Problem & Collapse and Revival in Quantum Optics 169
C Bessel Functions of the First Kind 173
D Degenerate Perturbation Theory 175
List of Figures 178
Bibliography 179
viiiIntroduction
Ultracold Atoms
The experimental realisation of Bose–Einstein condensation with dilute gases and the un-
precedented control of these systems has created a new and fastdeveloping field of research
inspiring many scientist [Blo08b]. By combining atomic physics, condensed matter and
quantum optics, various areas of physics are connected through this link and fascinating
experiments have become possible to test existing theories and to develop new models. In
theseexperiments, cloudsofbosonicor fermionicatomsarecooled toverylow temperatures
and can be manipulated by various external fields to create, for instance, traps and lattice
potentials or to address different internal degrees of freedom [Blo08b]. They offer the re-
alisation of many different models of quantum physics ranging as far as condensed matter
theory [Blo08b], quantum information [Blo08a], quantum chromodynamics [Rap07, Wil07],
and cosmology [Bra08].
After a focus on weakly interacting systems in the first years, the high degree of control
of almost all system parameters has allowed a study of strongly correlated quantum systems
by now [Blo08b]. Prime examples of quantum effects and of the perfect control of systems
of ultracold gases are the interference of matter waves [And95a, Gat07], where a large
regime of parameters is accessible in a single experimental setup, and the direct study of
a quantum phase transition in a cold gas realisation of the Bose–Hubbard model [Gre02a].
An extension of earlier experiments on the coupling of energy bands in weakly interacting
gases of cold atoms to the regime of stronger inter-particle interactions is now possible.
The required phase coherence of the time evolution in many-body systems for the complex
interference effects has again been demonstrated in a recent experiment with ultracold
bosons in optical lattices [Wil10].
Complex Dynamics
We take these opportunities as a motivation to study complex dynamics in quantum sys-
tems. The time evolution of a quantum state in a many-level quantum system, very often
quantum many-body models, is a central topic in many fields of physics [Aku06]. The com-
plexity arises from a large number of levels contributing to the time evolution and allowing
multiple phase interference. It can also be the result of an explicit time-dependence of the
Hamiltonian which can be translated into a many-level problem by Floquet analysis. An-
other typical feature of such complex systems is that a small perturbation can have a large
effect and that the system shows completely different behaviour at different time scales.
Direct signatures of an intriguing phase interference in these systems are present in the
1CONTENTS
population dynamics. We will often discuss this simple observable since it bears witness of
the complex time evolution of the wave function.
There are several ways to approach a complex many-level quantum system. These
approaches could be classified as statistical, analytical, and numerical. To begin with, it
is natural to use a statistical description for a system of many contributing levels. The
difficulty lies in finding the right degree of abstraction, i.e., to remove almost all details
without removing the characteristic effects, for instance by replacing matrix elements by
random numbers and keeping only the symmetries of the original problem. Historically,
this proved successful in the description of complex nuclei and led to the development of a
systematic theory of the spectral structure of random matrices [Meh04]. It found a large
revival with the conjecture that quantum chaotic systems have the same spectral properties
as random matrices [Boh84].
Secondly, complex quantum systems do usually not allow a solution in a simple analyt-
ical way, involving only known analytical functions or a small number of integrals. One is
then forced to apply approximations in order to obtain an analytical understanding of the
system’s behaviour. The application of higher order perturbation theory and of different
bases are means to capture the effect of even small perturbations. For the dynamics they
might furthermore allow a separation of time-scales to find a useful approximate description
which can be compared to numerical simulations for further improvement.
But even in seemingly simple cases, where an exact solution is possible, this solution may
not be helpful in a better understanding of the physical system. To give a simple example,
the eigenvalues and eigenvectors of any 2,3, or 4 dimensional matrix can always be given
exactly (the characteristic polynomial is maximally of fourth order and has an explicit solu-
tion in terms of radicals). They can be found for instance with a computer algebra system
and the number of bytes necessary to store such a solution can be taken as a measure for
its length. This length grows very quickly with the system size and where the eigensystem
of an arbitrary two-level system is only two lines long, the full four-level system takes al-
ready 18 pages (with 45 lines per page). Such a solution is thus possible, but clearly not
useful and the need to find good physical approximations is apparent already for the case
of solvable few-level systems.
A third approach is to immerge oneself into a complete numerical study of the system.
If one is able to obtain the required matrix elements (which can be an obstacle right from
the beginning on [Aku06]) and to simulate systems large enough, one faces a different
problem: The vast amount of data for a many-body (or many-level) wave function is often
too detailed and even veiling the relevant physical mechanisms at work which are, in the
end, what we are interested in and not the individual set of data points. This means one
has to distill the physical characteristics out of an abundance of numbers. However, if one
managestogetaglimpseonthebasiceffects, onecantrytobuild—orevenguess—effective
models and compare their predictions to further numerical data.
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