Download Dynamics of cold Fermi atoms in one

Alexey V. Ponomarev
Dynamics of cold Fermi atoms in
one-dimensional optical lattices
Freiburg, März 2008
Dynamics of cold Fermi atoms in
one-dimensional optical lattices
Inaugural-Dissertation
zur
Erlangung des Doktorgrades
der Fakultät für Mathematik und Physik
Albert-Ludwigs-Universität
Freiburg im Breisgau
vorgelegt von
Alexey V. Ponomarev
aus Krasnojarsk
am 31. März 2008
Dekan:
Leiter der Arbeit:
Erstgutachter:
Zweitgutachter:
Prüfer:
Tag der Verkündigung
des Prüfungsergebnisses:
Prof.
Prof.
Prof.
Prof.
Prof.
Prof.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Jörg Flum
Andreas Buchleitner
Andreas Buchleitner
John S. Briggs
Alexander Blumen
Matthias Weidemüller
6. Mai 2008
To my family
My father was writing his doctoral thesis in physics when I was four. Although
the meaning of his work became clear to me much later, I had the impression
that he was doing something very important. I wanted to do something similar,
but I only learned how to read at that time. Somehow the idea of a “new”
language came to my mind. I composed the vocabulary of this language from
ordinary words, by reversing the order of their letters.
At those days I did not know that I was not the first one who invented this
language. After all, few attempts to speak this language give the impression that
it will always remain on a piece of paper or in the best case it may be used to
code a message in cheap spy-movies. However, I still remember the enthusiasm
I had started the process of translation with. When the composition of new words
for my thesis-dictionary became boring, I was suddenly surprised by palindromic
words, like a word “kayak”, which coincide with their translation. The discovery
of this symmetry has significantly inspired me that I even found few sentences
that read the same backward as forward.
Later I realized that one can play with abstract objects and sustain happiness
by revealing new universalities and symmetries in their structure, but as soon
as one has convinced oneself that a formal mathematical language can capture
the behavior of the real world, one cannot stay silent and starts asking physical
questions.
Alexey V. Ponomarev
List of publications covered in this thesis
Chapter 3
• A. V. Ponomarev and A. R. Kolovsky
Dipole and Bloch oscillations of cold atoms in a parabolic lattice
Laser Physics 16 (2), 367-370 (2006)
Chapter 5 and 6
• A. V. Ponomarev, A. R. Kolovsky, and A. Buchleitner
Bloch oscillations of cold interacting Fermi atoms
in preparation
Chapter 7
• A. V. Ponomarev, J. Madroñero, A. R. Kolovsky, and A. Buchleitner
Atomic current across an optical lattice
Phys. Rev. Lett. 96, 050404 (2006)
• A. V. Ponomarev, J. Madroñero, and A. Buchleitner
Directed atomic transport in one-dimensional optical lattices
in preparation
In addition the author has contributed to the following publications
• A. V. Ponomarev, A. R. Kolovsky, and A. Buchleitner
Microscopic dynamics of Bose atoms in a parabolic lattice
in preparation
• J. Madroñero, A. Ponomarev, A. R. R. Carvalho, S. Wimberger, C. Viviescas,
A. Kolovsky, K. Hornberger, P. Schlagheck, A. Krug, and A. Buchleitner
Quantum chaos, transport, and control – in quantum optics
Adv. At. Mol. Opt. Phys 53, 33 (2006)
viii
Zusammenfassung
Die vorliegende Doktorarbeit beschäftigt sich mit der dynamischen Reaktion
el. neutraler, fermionischer, über Stöße miteinander wechselwirkender Atome,
die in ein optisches Gitter geladen sind, auf eine statische Kraft. Sowohl für
ein einzelnes Teilchen als auch für viele nicht wechselwirkende Teilchen sind
zwei fundamentale Ergebnisse bekannt: das Auftreten von Bloch-Oszillationen
des Gesamtimpulses und das Ausbleiben eine Nettostromes, welche von der
Lokalisierung der Teilchen im Konfigurationsraum begleitet werden. In unserer
Arbeit erinnern wir zunächst an die bekannten Einteilchen Ergebnisse (welche
wir auch auf den Fall eines zusätzlichen parabolischen Einschlusses verallgemeinern) um dann das entsprechende Transportproblem für zwei verschiedene
systeme mit nichtverschwindender interatomarer Wechselwirkung zu lösen.
Zuerst untersuchen wir das Verhalten eines zweikomponentigen (d.h. nicht polarisierten) fermionischen Gases unter dem Einfluss einer statischen Kraft. Wir
zeigen, dass, die Wechselwirkung zwischen den Atomen in diesem System keinen
Nettostrom der Fermionen bewirkt und eine Vielzahl neuer Phänomene auftritt.
Diese reichen von der Amplitudenmodulation der Bloch-Oszillationen mit einer
neuen Frequenz, welche proportional zur Wechselwirkungsstärke ist, über die
Verdopplung der Bloch-Frequenz, zu einer Reskalierung der charakteristischen
Zeitskalen bei starker Wechselwirkung.
Als zweites untersuchen wir die Voraussetzungen für das Auftreten von gerichtetem atomaren Transport und schlagen ein Modell zur Beschreibung des normalen Transportverhaltens eines verdünnten Gases, bestehend aus polarisierten
(d.h. nicht wechselwirkenden) Fermionen vor, wobei der Nettostrom durch
die wechselwirkungs-induzierte Kopplung an eins endliches Bad, bestehend
aus bosonischen Atomen welche sich im gleichen optischen Gitter befinden,
entsteht. Das Ergebnis ist eine Strom-Spannungs-Kennlinie, die sowohl Ohmsche als auch negative differentielle Leitfähigkeit aufweist, wobei der maximale Strom genau am Übergang zwischen den beiden Regimes auftritt. Dieser
Übergangspunkt ist eindeutig durch die mikroskopischen Parameter des Modells bestimmt und hängt nur schwach von der Temperatur des Bades ab, während die Amplitude des Stromes mit zunehmender Temperatur asymptotisch
verschwindet.
Abstract
In the present thesis, we discuss the dynamical response to a static forcing of
collisionally interacting, neutral fermionic atoms loaded into a one-dimensional
optical lattice. It is well known that for a single particle, as well as for many noninteracting particles, there are two fundamental results: the Bloch oscillations
of the total momentum and the absence of a net current, which are accompanied
by the localization of the particles in configuration space. In our work, we recall
(and, in the presence of an additional parabolic confinement, generalize) the
single particle results and, then, solve the associated transport problem for two
many-particle systems with non-vanishing atom-atom interactions.
In the first case, we analyze the behavior of a two-component (i.e., non-polarized) fermionic gas under static forcing. We show that, while the atom-atom
interactions do not induce a net current of fermions in this system, a variety of
new phenomena emerge. These range from the amplitude modulation of Bloch
oscillations with a new frequency, which is proportional to the interaction constant, over doubling of the Bloch frequency, to a rescaling of the characteristic
time scales at strong interactions.
In the second case, we discuss the requirements to witness directed atomic
transport, and propose a microscopic model to describe the normal transport
behavior of a dilute gas of polarized (hence non-interacting) fermions, where
the net current emerges due to the collision-induced coupling with the finite
bath, which is made of bosonic atoms loaded in to the same optical lattice. The
outcome is a “current-voltage” characteristics which displays both Ohmic and
negative differential conductivity, and a maximum current right at the transition between both regimes. This transition point is completely determined
by the microscopic model parameters, and weakly depends on the bath temperature, while the current amplitude asymptotically vanishes with increasing
temperature.
Contents
1 Introduction
1.1 Transport with cold atomic gases .
1.1.1 Optical lattices . . . . . . .
1.1.2 What we know and what we
1.1.3 Questions for this work . . .
1.2 Outline of the thesis . . . . . . . .
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1
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2 Description of the system
2.1 Cold atom Hamiltonian . . . . . . . . . . . . . . . . . . .
2.1.1 Static tilt vs. lattice acceleration . . . . . . . . .
2.1.2 Fock representation . . . . . . . . . . . . . . . . .
2.1.3 Boundary conditions . . . . . . . . . . . . . . . .
2.1.4 Character of atom-atom interactions . . . . . . .
2.2 Single band approximation . . . . . . . . . . . . . . . . .
2.2.1 Energy bands in a one dimensional optical lattice
2.2.2 Bloch oscillations and Landau-Zener tunneling . .
2.2.3 Wannier basis and tight-binding model . . . . . .
2.3 Hubbard model . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Effective Hamiltonians at strong interactions . . .
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3 Single atom dynamics
3.1 Floquet-Bloch formalism . . . . . . . . . . . . . . . . . . . . .
3.2 Bloch oscillations . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Effects due to an additional parabolic confinement . . . . . . .
3.3.1 Quantum pendulum model . . . . . . . . . . . . . . . .
3.3.2 Dephasing and revivals of Bloch and dipole oscillations
3.3.3 Beyond the single band approximation . . . . . . . . .
4 Interacting fermions in the absence of static forcing
4.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Invariance under spatial transformations . . .
4.1.2 Spin symmetry . . . . . . . . . . . . . . . . .
4.1.3 Conditional symmetries . . . . . . . . . . . .
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xii
CONTENTS
4.2
4.3
4.4
Integrability and spectral properties . . . . . . . . . . . . . . . . . . . . . .
Spectrum under twisted boundary conditions . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Spectral properties of interacting fermions under linear static forcing
5.1 The Floquet-Bloch operator . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Limit of vanishing interaction strength (U → 0) . . . . . . . . . . .
5.1.2 Limit of strong static forcing (F d ≫ J) . . . . . . . . . . . . . . . .
5.2 Spectrum of the Floquet-Bloch operator: general case . . . . . . . . . . . .
5.2.1 Symmetries of the Floquet-Bloch operator . . . . . . . . . . . . . .
5.2.2 Parametric dynamics of the quasienergies . . . . . . . . . . . . . . .
5.2.3 Spectral statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Bloch oscillations of interacting fermions
6.1 Dynamics at moderate interactions (U . J) . . . . . . . . . . . . .
6.1.1 Amplitude modulations at strong static forcing (F ≫ J) . .
6.2 Dynamics at strong interactions (|U| ≫ J) . . . . . . . . . . . . . .
6.2.1 Retrieved coherence of oscillations (U > 0) . . . . . . . . . .
6.2.2 Fermionic pairing: dynamics of bosonic compounds (U < 0)
6.3 Instantaneous spectrum in (non-)adiabatic evolution . . . . . . . . .
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7 Directed atomic current across an optical lattice
7.1 Model ingredients . . . . . . . . . . . . . . . . . . .
7.1.1 Bosonic bath . . . . . . . . . . . . . . . . .
7.1.2 Initial conditions . . . . . . . . . . . . . . .
7.2 Numerical experiment . . . . . . . . . . . . . . . .
7.3 Master equation approach . . . . . . . . . . . . . .
7.3.1 Bosonic density-density correlation functions
7.3.2 Interaction-induced decoherence . . . . . . .
7.4 Beyond the Markov approximation . . . . . . . . .
7.4.1 Thermalization of the environment . . . . .
7.4.2 Current-voltage characteristics . . . . . . . .
7.5 Summary . . . . . . . . . . . . . . . . . . . . . . .
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Conclusions, remarks and new open problems
99
A Realization of optical lattices
A.1 One-dimensional elongated geometry . . . . . . . . . . . . . . . . . . . . .
A.2 Ring-shaped geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3 Static forcing of cold atoms in an optical lattice . . . . . . . . . . . . . . .
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B Diagonalization procedure
109
CONTENTS
xiii
C The Floquet-Bloch operator at strong static forcing
111
Bibliography
113
Acknowledgements
121
xiv
CONTENTS
List of Figures
1.1
1.2
Elongated 1d optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . .
Bright and dark optical lattices within ring-shaped geometry . . . . . . . .
4
4
2.1
2.2
2.3
Energy band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The ground energy band at different potential depths . . . . . . . . . . . .
Spectrum of the effective Hamiltonian at strong attractive interaction . . .
17
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3.1
3.2
3.3
3.4
3.5
Energy eigenstates of the quantum pendulum in coordinate representation
Frequency of the classical pendulum vs. the classical action . . . . . . . . .
Oscillations of the mean particle momentum: dephasing and revivals . . . .
Eigenstates of a single particle in a 1d parabolic optical lattice . . . . . . .
Single particle dynamics in a parabolic optical lattice . . . . . . . . . . . .
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4.1
4.2
4.3
Parametric energy level dynamics of the Fermi-Hubbard model . . . . . . .
Statistical analysis of the Fermi-Hubbard model spectrum . . . . . . . . .
Same as in Fig. 4.2, but with twisted boundary conditions . . . . . . . . .
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5.1
5.2
Parametric dynamics of the Floquet-Bloch operator quasienergies . . . . .
Statistical analysis of the Floquet-Bloch operator spectrum . . . . . . . . .
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6.1
6.2
6.3
6.4
6.5
6.6
6.7
Decay and revivals of Bloch oscillations . . . . . . . . . . . . . . . . . . .
Interaction-induced modulation of Bloch oscillations . . . . . . . . . . . .
Bloch oscillations in the regime of strong repulsive interactions . . . . . .
Frequency doubling of Bloch oscillations. Decay and revivals . . . . . . .
Bloch oscillations of the mean energy and velocity in the adiabatic regime
Adiabatic vs. non-adiabatic evolution for the system below half-filling . .
Same as in Fig. 6.6, but for the system exactly at half-filling . . . . . . .
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7.2
7.3
7.4
7.5
7.6
Parametric energy level dynamics of the Bose-Hubbard model . . . .
Spectrum and density of states of the Bose-Hubbard model . . . . . .
Initial mean energy of the bosonic bath, as a function of temperature
Mean velocity of a single fermion vs. mean energy of the bosonic bath
Bosonic correlation functions and their decorrelation time scales . . .
Irreversible decay of fermionic mean velocity . . . . . . . . . . . . . .
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xvi
7.7
7.8
7.9
LIST OF FIGURES
Mean energy of the bosonic bath, as a function of temperature . . . . . . . 93
Decrement of energy difference (EB (t) − EB∞ ), as a function of the static tilt 94
Dependence of the drift fermionic velocity on the static tilt . . . . . . . . . 96
A.1 Ring-shaped optical potential . . . . . . . . . . . . . . . . . . . . . . . . . 106
A.2 Generation of bright and dark optical lattices with ring-shaped geometry. . 107
Chapter 1
Introduction
Curious physicists and small kids have surprisingly much in common – they all tend to
ask fundamental questions. Unfortunately, most of scientists use specialized terms, whose
meaning has been defined at some stage in the process of answering fundamental questions.
In that way the ability to ask and trying to answer fundamental questions in a simple form
(including the question – “What are you doing as a physicist?”) decreases in the course of
time.
In order to describe the elementary constituents of matter, their wave-particle nature,
and the character of their interactions, physical society has developed a very technical
language. Currently there is a very small number of open physical problems, such as the
great unification of all fundamental interactions, or the origin/long-time behavior of our
universe, which are both fundamental and meaningful to a nonspecialist.
This kind of problems is extremely hard to deal with: it may take decades until a solution will be found, and what is much more dramatic: to check any solution experimentally
seems to be at least challenging or even impossible. The natural question to ask is what
else except fundamental physics might be important and interesting to do?
It turns out a lot! The list of understood collective properties of the elementary constituents of real or artificially composed materials is still incomplete. In solid state physics,
the rational quantum Hall effect and high-temperature superconductivity may serve as well
known examples of such collective effects, the appearance of which cannot be explained in
the framework of current theories.
The main difficulties in dealing with such problems are due to the complexity of real
objects such as solid materials – natural systems of interest, with a huge number of constituents. First-principle calculations do not allow one to find an analytical solution even to
the general three-body Coulomb problem, therefore many assumptions and phenomenology
are introduced to get some insight into interaction-induced properties of a many-particle
system. For instance, if one substitutes electrons in a crystal lattice by a gas of noninteracting particles such that only the spectral properties of a quantum particle in a
periodic potential together with Fermi statistics are taken into account, then an analysis of the electronic transport reveals periodic oscillations of the electronic current (Bloch
oscillations), rather than directed electronic transport. To explain the appearance of the
2
1. Introduction
normal conductivity in real crystals at finite temperatures [1], it is necessary to include
a relaxation mechanism into the model. This can be done phenomenologically, or in the
framework of a microscopic approach directly by taking into account the electron-electron
interactions and the electron-environment coupling. Both approaches have weak points:
the first, phenomenological, model contains free parameters – the relevant relaxation constants, which may only be extracted from the experimental data; the second, microscopic,
approach gives a complete result only for a very small number of particles or at special
boundary conditions, but for a real solid with ∼ 1023 interacting particles only the ground
state with low-energy excitations, and, thus, only linear response to external perturbations
can be calculated [2].
The dynamical behavior of a quantum many-particle system taken out of equilibrium
by an external force cannot be always predicted by using only the properties of low-energy
excitations of the unperturbed system. The general solution to a dynamical problem
requires the knowledge of the complete spectrum, of the initial state and of all eigenstates
of the system in the perturbed configuration.
When analytical methods do not work any more, one turns to numerics. Application of
the latter to an eigenvalue problem becomes time-consuming even for several interacting
particles in a finite potential, since the size of the corresponding Hilbert space grows
exponentially with increasing particle number. However, for the same reason of a rapidly
growing Hilbert space, one may expect to find a universal dynamical behavior, which does
not depend on the system size, even with a relatively small number of particles, if there is
a scale-free universality in the system at all.
1.1
Transport with cold atomic gases
Recent progress in trapping and manipulating cold atomic gases in magneto-optical potentials made it experimentally possible to change the shape of a trapping potential dynamically [3], to tune the strength of inter-particle interactions [4], and to test the robustness
of the system properties to a variation of the particle number [5, 6]. With the help of theoretical models such experimental flexibility makes possible to address different dynamical
processes independently, and identify their contributions to the complex behavior of the
system. That is in the highest degree relevant to the theory of quantum transport, where
the central problem is to characterize the flow of the globally conserved quantities such as
heat (energy), spin, charge or mass across the system.
In this thesis, we will focus on transport properties of cold atomic gases in spatially
periodic trapping potentials referred to as optical lattices in analogy to crystal potentials.
Before we come to the context of the central physical questions addressed in this work, we
shortly introduce optical lattices with various one-dimensional geometries which have been
realized recently.
1.1 Transport with cold atomic gases
1.1.1
3
Optical lattices
The whole variety of the dipole traps used to confine and operate with cold atoms relies on the same principle based on interaction between a laser-induced dipole moment
with the same laser field (for a detailed discussion on the physics of the confining optical
potentials and well known methods to realize optical lattices of different geometries, see
supplementary information in Appendix A). An important result of this interaction is that
the form of the induced optical potential felt by an atom is given by the light intensity
profile. Moreover, such potential can be inverted by changing the sign of the detuning
δ of the laser frequency from the atomic resonance. Then one distinguishes the case of
negative detuning, δ < 0 (“red detuning”), with the potential minima at the positions
of maximal intensity (“bright” trapping potential), from that of positive detuning, δ > 0
(“blue detuning”), when the potential and the intensity minima coincide (“dark” trapping
potential).
Nowadays, to construct very different trap geometries by superimposing laser beams
with different propagation directions, frequencies and radial power profiles, as well as their
combination with an additional magnetic and/or gravitational confinement, has become
experimental routine. One to three dimensional optical lattices [7], 3D box-like potentials
with light sheets, which represent hard walls for the trapped atoms [8], are some of the
many experimentally realized implementations [3].
In the present work, we restrict ourselves to one-dimensional optical lattices. These
fall into two general classes distinguished by their topological properties. The first class
is formed by finite elongated periodic potentials, while the second is represented by those
with ring-shaped geometries.
Originally, the idea to create a one-dimensional periodic potential was proposed by
Letokhov in 1977 [9]. He pointed out that atoms might be periodically confined in the
local minima or maxima of a standing wave formed by counterpropagating laser beams.
In agreement with Letokhov’s proposal, in the minimalistic setup employed in current
experiments [7, 10], two counter-propagating beams with equal characteristics produce an
optical lattice (standing wave) along the z-axis, with period d given by half the wave length
of the laser beams.1 The red-detuned laser automatically provides central confinement in
the radial direction, while the blue-detuned one, in contrast, repels atoms from the beam
center and thus requires some trapping potentials in addition.
Typical two-beam configurations with red-detuned lasers result in a pancake-like shape
of the surfaces of constant potential (Fig. 1.1, on the left). Thus, a weak radial confinement
comparable to that of the local site might cause the transverse modes to compete with
those along the lattice direction in a nontrivial manner. Therefore, a one-dimensional
description of this system is questionable, and it is preferable to refer to the minimalistic
two-beam setup as a quasi one-dimensional optical lattice, if no additional confinements
are mentioned.
One of the experimentally successful realizations of truly one-dimensional optical lattices
1
The lattice constant d can be additionally adjusted by changing the angle θ between the counterpropagating beams, d(θ) = d/ sin(θ/2).
4
1. Introduction
Figure 1.1: Constant potential surfaces of the central part of an elongated quasi one-dimensional
optical lattice produced in the minimalistic two-beams setup (left), and of an array of onedimensional optical tubes (right) [11] produced in the four-beam setup. In the second setup, an
additional, weaker modulation along the tubes by a third pair of laser beams might be used to
achieve an array of truly one-dimensional optical lattices [12]. The picture on the right is taken
from [11].
HaL
HbL
HcL
HdL
Figure 1.2: Different ring shaped geometries in experiment (a,b) and theory (c,d), to implement
bright (left) and dark (right) optical lattices (see our discussion on page 5). The picture is taken
from [13].
1.1 Transport with cold atomic gases
5
has required three pairs of counterpropagating laser beams [11]. The first two pairs with
equally strong intensities, Vx = Vy , confine the atoms in the xy-plane by dividing the space
into arrays of elongated tubes along the z-axis (Fig. 1.1, on the right), while the third pair
with a weak intensity as compared to that of the others, Vz ≪ Vx , Vy , is responsible for
periodic modulations of the potential along the tubes.
While the elongated trapping geometry has been routinely implemented already for
decades, one way to realize ring-shaped periodic potentials was only recently proposed
in [14], and an another was partially2 accomplished in the lab [13] (for more details, see
Appendix 1.1.1). In short, the radial intensity dependence of a single laser beam can be
modified arbitrarily by spatial light modulators (SLMs) acting as reconfigurable diffractive optical components, i.e., holograms [15]. In particular, two Laguerre-Gaussian (LG)
laser modes with discrete rotational symmetry around the propagation direction can be
produced using SLMs, and then superimposed to form a ring-shaped optical lattice (see
Fig. 1.2). In contrast to elongated one-dimensional optical lattices, the latter possesses periodic boundary conditions, thus making possible to minimize effects related to the finite
size of the system.
1.1.2
What we know and what we would like to know
Given the above, it is legitimate to conclude that optical lattices represent the cleanest
periodic confinements for quantum particles ever produced or seen in nature. In contrast to
crystal lattices, where electrons scatter on defects and phonons (eigenmodes of the lattice),
there is only one dissipation mechanism for cold atoms in an optical potential – spontaneous
emission of photons by cold atoms. However, the latter can be brought to negligible values
by increasing the detuning of the laser frequency from the atomic resonance (for more
technical details, see Appendix A). Such versatility of optical lattices enables to isolate
the effect of particle-particle interactions and particles statistics on the transport behavior
of quantum particles in periodic potentials.
One dimensional (1d) systems, which are the central objects of this thesis, represent the
simplest spatially periodic structure. They were extensively studied during the last several
decades in solid state physics, where a number of analytical approaches was developed.
Due to that, not only the ground state of cold atoms in a 1d optical lattice, but also the
low energy properties described by Bogolyubov excitations [16, 17, 18] for bosons in the
superfluid regime (i.e., at week interactions), and by the Luttinger [19] liquid for fermions
are well known. Remarkably, locally interacting fermions in a deep optical lattice even have
a complete analytical solution in the framework of the Hubbard model [20, 21, 22], though
this solution is written in a nonexplicit way, through a number of coupled nonlinear equations. Similar to solid state systems, cold atoms in optical lattices exhibit quantum phase
transitions, but, in contrast to the former, a phase transition in optical lattices can be also
tuned dynamically, by an adiabatic change of one of the system parameters. For instance,
2
The desired interference pattern was created, but loading of cold atoms into the trap has not been
performed yet.
6
1. Introduction
in recent experiments an adiabatic change of the amplitude of the periodic potential was
used to force the bosonic condensate to undergo a transition from the superfluid (SF) to
the Mott-insulator (MI) state [23].
Nevertheless, not that much is known about the dynamical response of interacting
particles to a non-adiabatic change of the confinement potential, even in one dimensional
systems. Less than a decade ago a new wave of theoretical activity in the field of quantum
many-particle dynamics has been triggered by the successful experimental preparation of
the atomic Bose gases trapped in optical lattices and their subsequent cooling to the ground
state [23, 24], where the aforementioned SF-MI transition occurs. Since that time most
theoretical papers with a focus on dynamical problems are dealing with bosonic systems of
cold atoms driven out of equilibrium, while the same problems for interacting non-polarized
fermions still remain open.
The following selection of topics represents the current activity in the field. Resonant
system response to a dynamical quench [25] by using the time dependent density matrix renormalization group approach (tDMRG), the rescaling of the value of the critical
point of the SF-MI transition under periodic modulation [26] of the lattice depth, and the
interaction-induced decoherence of Bloch oscillations in a tilted optical lattice [27] treated
within an exact diagonalization of the Hamiltonian were recently studied for a finite number of bosonic atoms. Also the case of cold bosonic atoms on a lattice at strong repulsive
interactions (hard-core boson limit) has attracted much attention due to the well known
Fermi-Bose correspondence established by the Jordan-Wigner transformation, which essentially simplifies calculations by mapping the bosons to an integrable model of free,
non-interacting fermions. Relaxation of the latter from the ground state to a new equilibrium in the case of a sudden lattice extension by shifting its rigid wall boundaries was
analyzed theoretically [28], and in the case of a shift of an additional parabolic confinement
with respect to the position of the atomic cloud was subjected to both theoretical [29] and
experimental [30, 10] analyses. In addition to the dynamical problems listed above one
can also mention stability analyses of a superfluid bosonic current. The magnitude of the
critical velocity calculated in the frame of a mean-field approach [31] was measured directly
in recent experiment with one-dimensional optical lattice [32].
Despite the integrability of the standard one-dimensional fermionic Hubbard model introduced to describe interacting, non-polarized fermions in a deep periodic potential, it
is much harder to calculate the response to a dynamical change of the periodic confinement for fermions than for bosons. Standard mean field (Gross-Pitaevsky) [33, 34] or more
general Gutzwiller approximations [35, 36], which assume macroscopic or local coherent
state, respectively, can not be used to describe properly the behavior of fermions. The
more appropriate, BCS-type [37] mean field approximation is shown to be valid only for
fermions on a lattice with attractive interactions and at zero temperature [38]. This approach captures the formation of Cooper-pairs and tightly bounded singlets (molecules),
which can be condensed as composite bosons at week and at strong interactions, but it
is hardly extendable to treat high-energy excitations to strong perturbations. Since the
tDMRG approach mentioned above is based on a local basis truncation algorithm [39], it
is limited to particular systems, which do not develop intrinsic correlations under external
1.1 Transport with cold atomic gases
7
perturbations, and it cannot be always applied. For instance, the tDMRG fails to describe
the evolution of bosons in a deep 1d optical lattice amended by a constant tilt [40], due
to quantum chaotic properties found in the problem spectral structure [41]. In the case of
fermionic atoms, the tDMRG may be useful only at strong interactions, where, depending
on whether the interactions between fermions are attractive or repulsive, the system behaves effectively as hardcore bosons, with week repulsive nearest neighbor interactions, or
as spinless fermions, respectively3 .
From the above discussion, it is obvious that there is an impressive number of open
problems in the dynamics of interacting fermions, even in 1d periodic potentials, which
became important with their realization in tunable optical lattices. Especially, the controllable quantum transport of fermions as a dynamical response to the superposition of
an additional external potential turns out to be very relevant for practical purposes, and
challenging from theoretical point of view.
1.1.3
Questions for this work
In this thesis, we explore the situation where interacting fermionic atoms trapped in a deep
1d optical lattice are affected by a sudden tilt of the latter.
Although the reaction of a single particle to a tilted periodic potential is well known
from the electronic theory of solids, to predict the interaction effects on the fermionic
dynamics quantitatively remains a formidable problem. Starting from the single particle
model, we will come to a many-particle generalization, where we will obtain results for the
two following cases: non-polarized fermionic atoms (two interacting spinless species), and
polarized, hence non-interacting, fermions coupled to an interacting bosonic component,
loaded into the same optical lattice.
To formulate the explicit problems for this thesis more specifically, let us recall a single
quantum particle behavior in a linearly tilted periodic potential: This is based on the
interplay between two qualitatively different dynamical regimes: the acceleration towards
the global potential minimum, and oscillations in momentum and configuration space.
The first regime has an obvious classical counterpart, while the second is a pure quantum
mechanical phenomenon, due to periodicity of the particle energy as a function of its
quasimomentum, in periodic potentials.
In more specific terms of solid state physics,4 a particle being initially in the ground
Bloch band increases its quasimomentum linearly in time under a static force, up to the
edge of the Brillouin zone, where there is a certain probability to populate the first excited
band due to Landau-Zener tunneling [44, 45], and to be Bragg reflected to the opposite
3
Also, recently, the adaptive tDMRG approach has been applied to study the response of integrable
and nonintegrable fermionic Hubbard models to a dynamical quench [42] similar to that what has been
done for the bosonic Hubbard model in [25]. In both the bosonic and fermionic cases, the parameter
regime was chosen in such a way that only a finite number of excited states should have been populated
due to a dynamical quench. Therefore, the consequent system relaxation toward a quasistationary state
was successfully captured by the adaptive tDMRG approach.
4
For terminology, see one of the standard text books on solid state physics, for example, [43].
8
1. Introduction
edge of the Brillouin zone inducing periodic oscillations, referred to as Bloch oscillations
of the kinetic energy [46] (see also the example at the very beginning of the introduction
in the present chapter).
In this thesis we assume Landau-Zener tunneling to excited bands can be neglected,
and address the two following, one-dimensional problems:
• Dephasing of Bloch oscillations of non-polarized, ultracold fermionic atoms, due to
on-site interactions between atoms of different polarization;
• Decoherence of Bloch oscillations and directed current across an optical lattice of
polarized, ultracold, noninteracting, fermionic atoms, due to coupling to a finite
bath of ultracold, interacting, bosonic atoms loaded into the same optical lattice.
1.2
Outline of the thesis
In the next chapter 2, we specify the form of the atom-atom interaction, define boundary
conditions and furnish our system with a suitable mathematical description. We recall the
derivation of the well known Hubbard model, with emphasis on the dynamical response to
static forcing when going through the standard approximations.
To get insight into dynamical effects induced by a static forcing, in chapter 3, we
present the analysis of the single particle problem. For the first time in the thesis, we use
the Floquet-Bloch formalism employed later for the original many-particle problem as well.
Within the single-particle description, we also address the effect of an additional parabolic
confinement as a typical attribute of real experiments with cold atoms in the elongated
lattice geometry.
Properties of the fermionic Hubbard model without static forcing are brought to the
separate chapter 4, where we list all symmetries of the model and perform a statistical
analysis of its energy spectrum. The universal statistical behavior of the latter for the periodic and twisted boundary conditions are discussed in the general context of integrability
in quantum and classical systems.
In chapter 5, we calculate the many-particle Floquet-Bloch operator with its spectral characteristics in various regimes of the model parameters. Also we discuss how the
symmetries of the system predefine some properties of the spectrum.
The dynamical behavior of the mean fermionic energy and mean momentum under
static forcing is obtained and than classified in the next chapter 6. In the limit of strong
static forcing (static tilt dominates among other model parameters), we use the exact
form of the Floquet-Bloch operator found in the previous chapter, and identify dynamical
manifestations of the spectral properties of the Floquet-Bloch operator in other parameter
regimes.
In the last chapter 7, we introduce a microscopic model for the transport of a dilute
polarized fermionic gas across an optical lattice under static forcing, where the essential role
is played by a weak coupling to the admixture of an interacting gas of bosons loaded into the
1.2 Outline of the thesis
9
same optical lattice. The interaction-induced decoherence of fermionic Bloch oscillations
is analyzed within the master equation approach, and the current-voltage characteristics
obtained numerically for a small finite system is compared with the phenomenological
Esaki-Tsu prediction [47].
Appendix A supplies the present introductory chapter with the theory of dipoleoptical trapping potentials and with a description of concrete experimental realizations
of one-dimensional optical lattices addressed in the thesis. The diagonalization procedure
of the fermionic Hubbard Hamiltonian is given in the Appendix B as a complement to
chapter 4. In Appendix C, we derive the Floquet-Bloch operator in the regime of strong
static forcing – a result which is used in chapter 5.
10
1. Introduction
Chapter 2
Description of the system
The main subject of the present thesis is the transport properties of interacting cold atoms
initially confined by a periodic potential, and subsequently exposed to an additional linear
static potential.
For an accurate theoretical prediction of the system time evolution, we need an efficient representation of the fermionic many-particle Hamiltonian, of the unperturbed manyparticle initial state and of the time evolution operator. This will be the subject of chapters 2-4.
In the present chapter, starting from a general description of a many-particle system,
with a focus on cold interacting Fermi atoms in a deep optical lattice, we will derive an
effective lattice model, well known as the Hubbard model [20, 21], which will serve as a formal basis for the central parts of this thesis. On our way to that effective model, including
also additional static forcing, we will expose the hierarchy of necessary approximations,
and deduce matrix elements of the Hamiltonian in two different representations – Bloch
waves and Wannier states, – in which the kinetic and the atom-atom interaction terms turn
diagonal, respectively. Then we will reduce the general many-particle model by projection
on a subspace of low energy excitations well separated from the rest of the spectrum, at
strong atom-atom interactions.
We will also briefly recall the phenomenology of single particle behavior under linear
static forcing, but a more detailed analysis of the single particle dynamics in a ring-shaped
and in an elongated optical lattice with weak parabolic confinement is reserved for the
next chapter. There, we will also introduce an approach to describe the time-evolution of
a single as well as of a many-particle system under static forcing, in a ring-shaped geometry.
2.1
Cold atom Hamiltonian
Let us start with the general form of a many particle Hamiltonian,
H = H (1) + H (2) + ... ,
(2.1)
12
2. Description of the system
with single particle contributions
H (1) =
N
X
i=1
H (1) (ri ) =
N
X
p2i
+ V (ri ) ,
2M
i=1
(2.2)
two-particle interactions
H (2) =
1 X (2)
1X
H (ri − rj ) =
u(ri − rj ) ,
2
2
i6=j
(2.3)
i6=j
and many-body contributions which we do not spell out here.1 We denote the interaction
potential between two particles by u(ri −rj ). The character and the origin of the latter will
be discussed in much detail for cold neutral atoms in section 2.1.4. The potential V (ri ) in
the single particle contribution above includes the optical lattice, and possibly additional
external potentials. Since, we will focus on the effect of a linear static force, we write
V (ri ) = VOL (ri ) − F
N
X
xi ,
(2.4)
i=1
where F is the magnitude of the static force, and VOL (ri ) is a periodic function of x. Since
throughout the thesis we will stick to a one dimensional system geometry assuming strong
confinement along the y, z-axes, the static force is aligned along the x-axis.
Before we proceed further, it is useful to play a bit with the current form of the Hamiltonian (2.1).
2.1.1
Static tilt vs. lattice acceleration
Let us switch to the acceleration frame or, what is the same, to the interaction representation with respect to the static linear potential. By substitution of the new variable
!
N
i X
Ψ̃(ri ) = exp − F t
xi Ψ(ri )
(2.5)
~
i=1
into the Schrödinger equation generated by the Hamiltonian H, one obtains the new Hamiltonian H̃ for the variable Ψ̃(ri ), where only the single particle terms differ from their
original form,
H̃
(1)
(t) =
N
X
i=1
1
H̃
(1)
N
X
(pi + ex F t)2
(t)(ri ) =
+ VOL (ri ) .
2M
i=1
(2.6)
The many-body scattering cross section in cold atoms confined by an optical lattice is sufficiently small
as compared to two-body processes, and will anyway be neglected later.
2.1 Cold atom Hamiltonian
13
The result of this transformation is the emergence of a time-dependent vector potential
instead of the linear term – which was gauged away. By using the additional substitution:
x′ = x + F t2 /2M, which implies p′x = px − F t, it is seen that an effective linear tilt F x is
equivalent to an acceleration F/M of the optical lattice. Note that the interaction part of
the Hamiltonian remains unaffected, since only interparticle distances enter in H (2) .
This time-dependent representation of the Hamiltonian is translation invariant, and,
thus, compatible with periodic boundary conditions. The latter are preferable for the
treatment of dynamical, and, in particular, transport (see chapter 7) problems on periodic
lattices with a small number of sites. It is worth noting that both Hamiltonians, (2.2) and
(2.6), are unilaterally equivalent at Dirichlet and any open boundary conditions, by virtue
of (2.5). Though, when periodic boundary conditions are chosen, the time-dependence
cannot be gauged away from (2.6) any more [48].
Due to this remarkable correspondence, static forcing can be realized not only for atoms
in a one-dimensional optical lattice, but also in ring-shaped geometries (see Appendix A.2),
by rotating the lattice at linearly increasing angular velocity (Appendix A.3). We will
furthermore see in a moment that the time-dependent form (2.6) bears some advantages
to describe the low energy dynamics of particles in periodic potentials.
2.1.2
Fock representation
We proceed with a short recollection of many-body theory, such as to set the scene for our
subsequent analysis of many-particle dynamics.
A fundamental difference between classical and quantum mechanics in many particle
theory is that particles with the same intrinsic physical properties (e.g., charge, spin) are
indistinguishable in quantum mechanics. This leads to the distinction of two fundamental symmetry classes: bosonic and fermionic, together with the corresponding symmetry
properties of the associated many-particle basis states.
Let us consider Ns indistinguishable fermionic particles with the same spin component
s. Let the numbers p1 , p2 , . . . , pNs label one of the possible configurations of single particle states chosen from a complete and orthonormal basis {ψl (x), l = 1, . . . , L}. Then a
complete basis of the multi-particle system is spanned by the antisymmetrized products
(s)
Ψn1 ···nL (x1 , . . . , xNs ) of the single particle states, which are defined by the Slater determinant
(s)
(s)
ψp1 (x1 ) ψp(s)
)
(x
)
·
·
·
ψ
(x
p
2
N
1
1
s
(s)
(s)
(s)
ψ (x ) ψ (x ) · · · ψ (x ) 1
p
p
p
1
2
N
2
2
2
s
Ψ(s)
(2.7)
.
.
.
.
n1 ...nL (x1 , . . . , xNs ) = √
.
..
..
..
..
Ns ! ψ (s) (x ) ψ (s) (x ) · · · ψ (s) (x )
pNs
1
pNs
2
pNs
Ns
where the numbers ni = 0, 1 are the fermionic occupation numbers of a single particle state
ψi (x). The basis states are antisymmetric with respect to any permutation of the labels pi ,
(s)
or particle coordinates xi . Thus the basis states Ψn1 ···nL (x1 , · · · xNs ) are uniquely defined
by the set of occupation numbers. The wave-functions written as |nis = |n1 . . . nL is :=
14
2. Description of the system
(s)
Ψn1 ···nL (x1 , . . . , xNs ) are referred to as Fock states. The corresponding Fock space has
Ns =
L!
(L − Ns )!Ns !
(2.8)
orthogonal basis states given by the different possible distributions of Ns (fermionic!) particles over L lattice sites.
Later on, we will focus on fermionic systems with two spin components. The basis of
such a two-component Fermi gas in an optical lattice is given by the tensor product of the
basis states of the single component problem
|ni = |ni↑ ⊗ |ni↓ .
(2.9)
Consequently, the total Fock space is spanned by N = N↑ N↓ basis states.
The cold atom Hamiltonian in the Fock representation
Let us now rewrite the general many particle Hamiltonian (2.1) with single-particle term
(2.6) in the Fock basis.
Multiplication from left and right with the identity, both times written as a complete
sum over single particle basis states (or “orbitals”) |lα i, and subsequent rearrangement
result in the following expressions:
H̃
(1)
(t) =
L
N X
X
i=1 α=1
=
L
X
|lα i
(i)(i)
hlα |H̃ (1) (t)(ri )|lβ i
α,β=1
=
L
X
hlα |H̃
(1)
X
(t)(ri )
L
X
β=1
N
X
i=1
|lβ i(i)(i) hlβ |
|lα i(i)(i) hlβ |
(1)
H̃lα ς,lβ ς (t)c†lα ς clβ ς ,
(2.10)
α,β=1 ς=↑,↓
and
H
(2)
N
L
1X X
=
|lα i(i)(i) hlα ||lβ i(j)(j) hlβ |H (2) (ri − rj )|lγ i(j)(j) hlγ ||lδ i(i)(i) hlδ |
2
i6=j α,β,γ,δ
N
L
1X X
|lα i(i) |lβ i(j)(i) hlα |(j) hlβ |H (2) (ri − rj )|lγ i(j) |lδ i(i)(j) hlγ |(i) hlδ |
=
2 i6=j α,β,γ,δ
=
L
N
X
1 X (2)
Hlα ,lβ ,γ,lδ
|lα i(i) |lβ i(j)(j) hlγ |(i) hlδ |
2 α,β,γ,δ
i6=j
L
1 X X (2)
H
c† c† cl t cl ς ,
=
2 α,β,γ,δ ς,t=↑↓ lα ς,lβ t,γt,lδ ς lα ς lβ t γ δ
(2.11)
2.1 Cold atom Hamiltonian
15
where the fermionic creation and annihilation operators c†lς and clς , (l ∈ Z/LZ acting on
fermionic Fock states |ni were introduced. They satisfy the following (anti-)commutation
relations with respect to lattice site numbers and spin components, respectively, {clς , cmt } =
{c†lς , c†mt } = 0, and {clς , c†mt } = δlm δςt ({., .} the anti-commutator), and act in Fock space
with the pseudovacuum |0i defined by clς |0i = |0i. In the above expressions, we could
factorize out the single and two-particle matrix elements of H (1) and H (2) , since these are
(1)
identical for all particles. Thus, we have shown that the matrix element H̃lα ς,lβ ς (t) and
(2)
Hlα ς,lβ t,γt,lδ s can be defined by using a complete set of single particle orbitals.
At last, when evaluating the matrix elements of a fermionic Hamiltonian in the Fock
basis, one has to take care of some phase factors, which may emerge each time a product
c†l1 ς cl2 ς is applied on a non-vacuum Fock state:
c†l1 ς cl2 ς |n1 n2 ...nL iς = (−1)ν δnl1 ς ,0 δnl2 ς ,1 |...nl1 ς + 1...nl2 s − 1...iς ,
with
(2.12)
max(l1 ς,l2 ς)−1
ν=
X
ni ,
(2.13)
i=min(l1 ς,l2 ς)+1
what directly follows from the (anti-)commutation relations.
2.1.3
Boundary conditions
As we already pointed out at the very beginning of the introduction, we expect the emergence of universal dynamical behavior already for lattices with rather small numbers of
sites, L ≃ 10. Then dynamical properties will depend on occupation numbers, rather
than on the particular lattice size or on atom numbers. To minimize finite size effects
which would be unavoidable for such lattice sizes with Dirichlet boundary conditions, due
to reflection effects, we use periodic boundary conditions. Thus we will always use the
translationary invariant Hamiltonian H̃ (1) (t), where the static force acting on the atoms is
represented by a time-dependent vector potential. We expect our results to be observable
also in sufficiently long optical lattices, L ≫ 10, with Dirichlet or von Neumann boundary
conditions for time scales and initial conditions which do not probe the sample confinement
[49]. Moreover, our predictions can be directly tested in experiments with recently proposed, and partially realized, ring-shaped optical lattices. The quasimomentum formalism
outlined below for the calculation of the matrix elements of the system Hamiltonian with
static forcing is exact for such configurations.
As discussed in the Section 1.1.1, the specific experimental realization of the optical
lattice may induce amendments to the ideally strictly periodic potential: inhomogeneities
in the focusing beam, or an additional confinement through inhomogeneous magnetic fields
can often be approximated by a superimposed parabolic potential. The effect thereof on the
dynamics of cold atoms can be understood within a quantum pendulum model. Obviously,
in that case there is no appropriate acceleration frame to transform to, and one has to
operate with a translationary non-invariant form of the Hamiltonian in Section 3.3.
16
2.1.4
2. Description of the system
Character of atom-atom interactions
The physical origin and exact form of the Hamiltonian are now specified, except for the
atom-atom interaction. This is the subject of the present paragraph.
Since the ground states of all alkali atoms are spherically symmetric, at distances larger
than molecular dimensions (& 5Å), their interaction potentials can be well approximated by
the lowest order (i.e., the induced dipole-dipole interaction) of the van der Waals potential,
−C6 r −6 [50]. The corresponding van der Waals length r0 ≡ (MC6 /~2 )1/4 (C6 varies from
∼ 1600a0 (Li) to ∼ 6300a0 (Cs), where a0 = ~/(me c α) is the Bohr radius of hydrogen),
which is of the order of 50Å, gives the typical extension of the last molecular bound state in
the interatomic potential. If the atoms are cold enough, such that the relative kinetic energy
E ≡ ~2 k 2 /M (here k denotes the relative wave vector) of the two interacting particles is
smaller than the characteristic van der Waals’ energy Ec ∼ 2~2 /Mr02 , which is of the order
0.1 − 1 mK for alkali atoms, the contributions of non-zero angular momenta (l 6= 0) to the
scattering process are by a factor of order (kr0 )2l smaller than that of s-wave scattering
(l = 0), and, thus, can be neglected (see discussion in [51] and references therein).
The result is that, at sufficiently low temperatures, the true interaction potential u(xi −
xj ) for neutral atoms may be replaced by an effective delta potential
u(xi − xj ) =
4πas ~2
δ(xi − xj ) ,
M
(2.14)
where as is the low-energy s-wave “scattering length”.
Remarkably, the value of the s-wave scattering length, which depends not only on
the chemical and isotopic species involved, but also on the hyperfine indices of the two
atoms, can be tuned from positive to negative by appropriate adjustment of an external
additional constant magnetic field (in the vicinity of a Feshbach resonance). Thus, the
resulting atomic interaction can have repulsive, as well as attractive character.
2.2
Single band approximation
An additional static forcing complicates the dynamics even for a single particle, and a
general analytical solution is not yet available. Although the single particle evolution
of an initially localized wave-packet (in configuration or in quasimomentum space) can
still be calculated numerically exactly, the same numerical approach becomes already very
inefficient for few interacting particles. Fortunately, there is an approximation which significantly simplifies the model Hamiltonian and yet preserves the required accuracy of the
solution. We will introduce this approximation in this section, and show how it can be
applied to the single particle an to the interacting many particle system on a lattice.
Once the boundary conditions are fixed we can proceed with the choice of a specific
basis, for an explicit evaluation of the matrix elements of the Hamiltonian. Since we
know that the general many-particles basis states in the Fock representation are uniquely
defined through the single particle orbitals, we only need to define the basis for a single
∆
−1
−1
17
εα (κ)
E(k)
2.2 Single band approximation
1
∆
−1
k [πd ]
−1
κ [πd ]
1
Figure 2.1: Emergence of an energy band structure from the energy dispersion relation of a
free particle (in a one dimensional picture). Left panel: parabolic energy dispersion of the free
particle (red line) is splitted by a superimposed periodic potential at p = ~k = ± πd ~ (blue line).
Right panel: energy εα of a particle in a periodic potential, as a function of the quasimomentum
κ. Ground and first excited bands, α = 0 and α = 1, respectively, are shown.
particle wave-function obeying the periodic boundary conditions of the ring-shaped lattice
geometry.
The most natural choice for such a basis is the set of eigenstates of the Hamiltonian
H0 =
p2
+ VOL (x) .
2M
(2.15)
for a single particle in a one-dimensional optical lattice, VOL (x + d) = VOL (x), where all
single particle contributions to the total many-particle Hamiltonian take diagonal form.
An arbitrarily small amplitude V0 of the periodic potential VOL (x) induces at least one
splitting ∆ ∼ V0 [43] in the parabolic free particle dispersion relation ε(k) = ~2 k 2 /2M, at
the momentum ~k = ± πd ~ (Fig. 2.1a). The Bloch theorem [46] gives a general solution for
the eigenvalue problem
H0 φα,κ (x) = εα (κ)φα,κ (x) ,
(2.16)
in the form of Bloch waves
φα,κ (x) = exp(iκx)χα,κ (x) , χα,κ (x + d) = χα,κ (x),
(2.17)
with new quantum numbers κ and α referred to as the lattice quasimomentum and the
energy band index, respectively. Thus, the single valued function ε(k) is mapped on the
multivalued periodic function ε(κ + 2π/d) = ε(κ), which has different eigenenergy values
εα (κ) for a given quasimomentum κ (Fig. 2.1b).
Let us now use the explicit form of the optical lattice potential, in order to obtain the
single particle spectrum with the associated eigenstates.
2. Description of the system
1/2
n=2
0.4
0.2
n=3
n=4
0
−1
0
1
0
( ε0(κ)−s
|an / a1|
0.6
) / 2J
18
1
2
log s
3
4
−1
−1
−0.5
2
0
0.5
−1
κ [πd ]
1
Figure 2.2: The ground energy bands as a function of the quasimomentum. Left: coefficients an
of the Fourier expansion (2.25), as a function of the potential depth s. Right: actual form of the
ground band as a function of the quasimomentum, at scaled lattice depth s = 1/2, 1, 2, 4, 8, 16
(from blue to red in the plot), in comparison to the first Fourier mode (tight-binding) approximation (2.23) (red line).
2.2.1
Energy bands in a one dimensional optical lattice
The periodic potential seen by cold atoms is given by (see Appendix A)
πx ,
(2.18)
V (x) = V0 cos2
d
where the lattice constant d equals half the wave length λL of the laser beams creating
the lattice, d = λL /2, V0 is the lattice depth typically expressed through the dimensionless
parameter s in units of the recoil energy
~2 kL2
~2 π 2
=
.
(2.19)
2m
2md2
The latter is just the amount of energy gained by the atom due to the emission of a single
photon with wave vector kL = 2π/λL = π/d.
The eigenvalue problem for a particle in a periodic potential like (2.18) is equivalent to
solving the Mathieu equation, which does not have explicit analytical solutions. However,
the form of the ground band ε0 (κ), its width
V0 = sER ,
ER =
∆ε0 = ε0 (π/d) − ε0 (0) =: 4J(s) ,
(2.20)
and the gap ∆(s) (splitting between the ground and first excited bands), both as functions
of s can be analytically estimated at the limit of shallow [43] (s ≪ 1):
2 s 2
s2
|κ|
|κ|
−1 + 4
−1 −
,
(2.21)
ε0 (κ) =
π/d
π/d
16
"r
#
s2
s s 2
s
1
1
4+
1− +
, ∆(s) = s/2 ,
(2.22)
− −1 ≃
J(s) =
4
16 4
4
4
8
2.2 Single band approximation
and deep [52] (s ≫ 1):
19
√
s − 2J(s) cos(κd) ,
√
4
J(s) = √ (s)3/4 e−2 s , ∆(s) ∼ s1/2 ,
π
ε0 (κ) =
(2.23)
(2.24)
optical lattice.
The above expressions demonstrate different qualitative dependences of the width of
the ground energy band, parameterized by 4J(s), and the energy gap, ∆(s), on the depth
s of the periodic potential (2.18):
(i) J(s) decreases exponentially, as s increases (J(0) = 1/4);
(ii) ∆(s) monotonically increases as a power of s (∆(0) = 0).
Therefore, the ratio J(s)/∆(s) can be tuned over few orders of magnitude simply by
variation of the optical potential depth s.
The exact form of the energy bands can be found with high accuracy by available
numerical algorithms (see, e.g., [53]). Expanding the numerically obtained2 ground band
energy dependence ε0 (κ) in a Fourier series
Z
∞
X
1
d π/d
ε0 (κ) = a0 +
ε0 (κ) cos(nκd) dκ
an cos(nκd) , an =
2
π
−π/d
n=1
(2.25)
it is straightforward to justify the first Fourier mode approximation3 (2.23). Figure 2.2
illustrates the validity range of the latter. It is clearly seen that the relative amplitude an
of the higher modes exponentially decreases with the mode number n. The relative weight
|a2 /a1 | of the second mode with respect to the dominant one amounts to less than 10%
at s ⋍ 5 (Fig. 2.2, left panel). Therefore, together with higher corrections due to higher
modes it can be neglected at larger values of s. For example, the exact form of the ground
band ε0 (κ) at s = 16 is perfectly approximated by (2.23), and the difference cannot be
visually resolved any more (Fig. 2.2, right panel).
2.2.2
Bloch oscillations and Landau-Zener tunneling
The periodicity of the Bloch particle, i.e. a particle in a periodic potential, energy as a
function of quasimomentum and a non-vanishing energy gap between the ground the first
excited band, ε0 (κ) and ε1 (κ), respectively, results in a nontrivial system response to an
additional linear static force F .
2
Here, the method [54], based on a discrete Fourier transform and subsequent basis truncation, where
at the final step one has to solve an eigenvalue problem, is applied.
3
This is equivalent to the tight-binding model, which accounts for the tunneling between neighboring
wells of the periodic potential only.
20
2. Description of the system
Consider the Hamiltonian (2.15) garnished with an additional static potential gauged
away such as to appear as a time-dependent vector potential (our discussion in Sec. 2.1.1)
H̃(t) =
πx (p + F t)2
.
+ V0 cos2
2M
d
(2.26)
Its adiabatic (instantaneous, i.e., for fixed t) spectrum coincides with that of H0 , but for
a quasimomentum which is shifted linearly in time, κ(t) = κ + F t. Under the assumption
that the static force does not couple the first two bands, F d ≪ ∆, the energy of a particle
initially prepared in the ground band will adiabatically evolve as ε0 (κ+F t), until it reaches
the edge of the first Brillouin zone at κ = kL , where it will experience a Bragg reflection,
κ = kL → κ = −kL , and, then, continue its evolution, returning to its initial state at
each Bloch period TB = 2π~/F d. This causes a periodic modulation of the particle mean
velocity and position, known as Bloch oscillations [55]. If the static tilt is further increased,
transitions to the first and higher excited bands may be induced, mediated by LandauZener tunneling [44, 45]. The subsequent population of higher excited bands results in an
acceleration of the particle across the lattice.
To observe Bloch oscillations on long time scales, it is therefore preferable to confine
the particle dynamics to the ground Bloch band.
A detailed analysis [56] has shown that the transition rate
R=
1
|T |2 ,
TB
(2.27)
which quantifies the relative loss of ground band population, due to tunneling to the first
excited band, predicts the upper bounds for the maximum magnitude of the static force
F d very well4
s
[F d/ER ]max ≈ , s ≫ 1 ,
(2.28)
4
and
[F d/ER ]max ≈ s2 , s ≪ 1 ,
(2.29)
when evaluated with the Landau-Zener expression for the tunneling probability |T |2 [45],
2
π ∆(s)2
2
|T | ≈ exp −
.
(2.30)
8 F d/ER
Below which the tunneling probability is bounded, |T |2 6 0.01. The relations (2.22) and
(2.24) for ∆(s) were employed to deduce the estimates (2.28) and (2.29).
From now on, we will always assume the magnitude of the static force to be confined to
the interval [0, Fmax ], with Fmax defined by the above expressions (2.28) and (2.29). Under
4
For sufficiently strong static forcing F d ∼ ∆(s), the Landau-Zener prediction deviates non-monotonous
from the exact numerical results, due to a Stokes phenomenon (oscillations of the band populations)
[57]: This occurs when the Bloch period TB = 2π~/F d becomes too short for an independent adiabatic
description of subsequent tunneling events at the edge of the Brillouin zone.
2.2 Single band approximation
21
this assumptions, projection of the Hamiltonian (2.26) onto the ground band leads to the
following form, in quasimomentum representation (see Appendix I in Ref. [58])
Hκ = ε0 (κ + F t) ,
(2.31)
with no coupling to the second band.
2.2.3
Wannier basis and tight-binding model
The complete orthogonal basis in quasimomentum space, formed by the Bloch waves
φκ (x) := φ0,κ (x), has its counterpart in real space, where the basis states are Wannier
functions given by the Fourier coefficients of φκ (x),
ψl (x) =
π/d
X
exp(−iκld)φκ (x),
(2.32)
κ=−π/d
with l integer.
Both sets of states are invariant under translations: Bloch states in real space, x′ →
x + d, and Wannier states in κ-space, κ′ → κ + 2π/d, as follows from the expansion (2.32).
Therefore, they have essentially similar properties, but in different spaces: Wannier states
extend over the whole κ-space, equivalently to the Bloch waves considered as functions
of the coordinate, and they are well localized in coordinate space, as Bloch waves are in
κ-space.
The Hamiltonian operator Hκ rewritten in the Wannier basis takes a discrete matrix
form with elements
Z ∞
Hl,l′ =
dxψl∗ (x)Hκ ψl′ (x)
Z−∞
∞
XX
=
dx
exp(iκld)φ∗κ (x)ε0 (κ + F t) exp(−iκ′ l′ d)φκ′ (x)
−∞
=
κ
∞
X
κ′
1
a0 +
an eitF nd/~δl−n,l′ + e−itF nd/~δl+n,l′ ,
2
n=1
(2.33)
where an are the Fourier expansion coefficients (2.25) of the ground band energy.
As demonstrated in Sec. 2.2.1 (see Fig. 2.2), the coefficients an , which are now the
off-diagonal elements of Hl,l′ , decrease with n, and only two of them, a0 and a1 , contribute
in the regime of deep optical lattices, s > 5. Therefore, the Hamiltonian operator Hl,l′
has a tridiagonal matrix representation in the Wannier basis, that is the essence of the
celebrated tight-binding model (see, e.g., [59]), where tunneling of a particle to the nearest
sites of the lattice only is possible.
Denoting the Bloch states φκ (x) by |κi and the Wannier functions ψl (x) by |li, we
obtain for a single particle with static forcing
"
#
∞
X
Hκ (t) = ε0 − J
cos(κl d + tF d/~) |κl ihκl | ,
(2.34)
l=1
22
2. Description of the system
in quasimomentum representation, and
H(t) = ε0 |lihl| − 2J
∞
X
l=1
eitF d/~|l + 1ihl| + e−itF d/~|lihl + 1| .
(2.35)
in the Wannier basis. The constant energy shift ε0 = a0 /2 gives the energy at the center
of the ground band.
2.3
Hubbard model
In the previous section, we have introduced the single band approximation, defined appropriate bases for the single particle problem, and presented the single-particle Hamiltonian
in the corresponding representations. In the present section, we will derive the effective
many-particle Hubbard Hamiltonian,5 by using the Fock representation of many-particle
wave-functions introduced in Section 2.1.2.
In order to derive the Hubbard model on a one-dimensional, periodic, L-site lattice,
in coordinate or quasimomentum representation, one has to substitute the single particle
orbitals |lα i (see, e.g., Eq. (2.10)) by localized Wannier functions or Bloch waves, respectively.
With Eqs.(2.2) and (2.10), the single particle contribution in the Wannier representation
is given by
2
Z ∞
p
(1)
∗
Hlα ,lβ =
dxψlα (x)
+ V (x) ψlβ (x) .
(2.36)
2M
−∞
Assuming a deep periodic potential (s > 5), only elements next to the main diagonal
contribute within the single band approximation (see Eq. 2.33),
(1)
Hlα ,lβ ≃ −J eitF d/~δlα −1,lβ + e−itF d/~δlα +1,lβ
,
(2.37)
where J = ∆ε0 /4, see Eq. (2.20).
For atomic samples composed of two different fermionic components (ς, t =↑, ↓), the
interaction matrix elements in the Wannier representation are
(2)
Hlα ς,lβ t,γt,lδ ς
5
Z
Z ∞
4πas ~2 ∞
∗
=
dxi ψlα ς (xi )ψlδ ς (xi )
dxj ψl∗β t (xj )ψlγ t (xj )δ(xi − xj )
M
−∞
−∞
Z
4πas ~2 ∞
=
dxψl∗α ς (x)ψlδ ς (x)ψl∗β t (x)ψlγ t (x) ,
(2.38)
M
−∞
Historically, the Hubbard model was introduced in 1962 by M. Gutzwiller [20], as a simple effective
model to investigate electron correlation effects in the d-bands of transition metals. J. Hubbard’s work
appeared independently [21].
2.3 Hubbard model
23
where we replaced the interaction potential u(xi − xj ) in Eq. (2.11) by an effective delta
potential (2.14). Since each Wannier function is well localized within each site of the optical
lattice, the off-diagonal elements can be neglected. Hence
Z
4πas ~2 ∞
(2)
Hlα ς,lβ t,γt,lδ ς = Uδlα ς,lδ ς δlβ t,lγ t , U =
dx|ψl (x)|4 .
(2.39)
M
−∞
Summing up the single (2.37) and two-particle (2.39) terms, we obtain the simplest
form of the Hubbard Hamiltonian for two fermionic components of neutral atoms in a deep
optical lattice
H = −J
L X
X
(eitF d/~c†l,ς cl+1,ς
l=1 ς=↑,↓
+
e−itF d/~c†l+1,ς clς )
+U
L
X
nl↑ nl↓ ,
(2.40)
l=1
with the hopping matrix element J and the on-site interaction strength U, where nlς :=
c†lς clς are the particle number operators. Furthermore, we also assume periodicity of the
lattice, setting cL+1,ς = c1,ς .
Quasimomentum representation
Alternatively, we can also seek the quasimomentum representation of the fermionic Hubbard (FH-, in short) Hamiltonian. This can be derived from the above Fock representation
through the basis transformation
L
c†κ
1 X iκld †
e cl ,
=√
L l=1
L
1 X −iκld
cκ = √
e
cl ,
L l=1
(2.41)
or by direct representation in the quasimomentum basis composed of Bloch waves, see
section 2.2.3 above. For the derivation of the interaction part, one needs the Fourier
expansion of the interaction potential (2.14):
u(xi − xj ) =
4πas ~2 1 X iq(xi −xj )
e
.
M L q
(2.42)
The tunneling term comes in simple diagonal form, and the complete Hamiltonian thus
reads:
XX
U X †
cκ+q↑c†κ′ −q↓ cκ′ ↑ cκ↓ ,
(2.43)
cos(κς d + F t)c†κς cκς +
Hκ = −2J
L
′
κ ς=↑,↓
κ,κ ,q
where the sum runs over quasimomenta κ, κ′ with discrete increments q = 2πn/L, n =
0, ..., L − 1. It is therefore particularly easy in this representation to construct the eigenstates and the spectrum at vanishing interaction strength, since the total energy is then just
the sum of independent single particle contributions with their individual quasimomenta.
24
2. Description of the system
The interaction term represents two-body scattering processes: it couples all possible
pairs of Fock states with conserved total quasimomentum:
|..., κi , ..., κj , ...i ↔ |..., κi + q, ..., κj − q, ...i .
(2.44)
To close the consideration of the unperturbed FH-model (i.e., without static forcing),
we would like to describe an effective Hamiltonian approach used in the limiting case of
strong interaction strength.
2.3.1
Effective Hamiltonians at strong interactions
When the hopping or the interaction term dominate each other, the entire spectrum of the
FH-Hamiltonian at F = 0 splits into well separated energy subbands (see, e.g., Fig. 4.1 in
Sec. 4.2), and it may become preferable to describe the system independently within each
subband, unless non-zero static tilt induces interband transitions.
In what follows, we will see that projection of the original FH-Hamiltonian on the
subspace associated with one of the energy subbands reveals new characteristic energyscales which competitively influence the system time-evolution under static forcing.
We will here consider in some detail the case of strong interaction strength, |U| ≫ J,
and distinguish attractive and a repulsive interaction, U < 0 and U > 0, respectively.6
Strong attractive interactions
Let us start with the case of a strong attractive interaction. Then, the ground band consists
of states where fermions with different polarization pair up in one of the lattice sites (here
we restrict our discussion to the case N↑ = N↓ ≡ N). Each pair of the atoms has zero total
spin and, thus, the paired atoms (singlets, in what follows) can be considered as bosons.
Given the limit of strong interaction, we can treat the hopping term as a perturbation,
to derive an effective Hamiltonian for the singlet states [60]. Denoting by ã†l = c†l,↑ c†l,↓ and
ãl = cl,↑ cl,↓ the singlet creation and annihilation operators, the effective Hamiltonian up to
irrelevant energy shift reads [61]:
#
"
!
"
3 #
2 X
X
X
J
1
4J
(−)
ñl+1 ñl + O
ñl +
ei2tF d/~ã†l+1 ãl + h.c. −
Heff =
, (2.45)
U
2
|U|
l
l
l
with U > 0 and the restriction (ãl )2 = (ã†l )2 = 0 for hard-core bosons (only one boson per
site is allowed). The above effective Hamiltonian gives the low energy spectrum (associated
with paired atoms) of the original Hamiltonian (2.40) with very high accuracy even at
2J/|U| ∼ 0.1 (see Fig. 2.3).
When tunneling between adjacent lattice sites each particle accumulates the phase difference proportional to the Bloch frequency, and the simultaneous tunneling of two particles
6
Although, Fig. 4.1 shows the energy level evolution as a function of positive interaction strength, the
limit of strong, negative U can be obtained by mirror reflection of the right-hand side of the spectrum
with respect to the u-axis.
2.3 Hubbard model
25
−1.6
E
−1.7
−1.8
−1.9
−2
−1
−0.95
−0.9
u
−0.85
−0.8
Figure 2.3: Spectrum of the effective Hamiltonian (6.18) at strong attractive interaction as
compared to the lowest bunch of levels of that of the original Hamiltonian (2.40). In both cases
static force is set to zero, F d = 0. Solid lines: 15 lowest energy levels of the fermionic Hamiltonian
(2.40), for L = 6 and N↑ = N↓ = 2 (dimension of the Hilbert space: N = 225). Dashed lines:
energy spectrum of the bosonic Hamiltonian (6.18), for L = 6 and N = 2 (dimension of the Hilbert
space: N = 15). All energy levels are shown as a function of the parameter u = U/(U + 2J).
thereof implies a phase shift twice as large. Thus, the corresponding Bloch frequency associated with static forcing of singlets is doubled with respect to the original single-particle
frequency: ωB′ = 2ωB = 2F d/~. Note that this also follows from the definition of the
creation (annihilation) operators for singlets. The last term in square brackets, in (6.18),
represents an effective repulsive nearest-neighbor interaction between the bosons, which
(−)
may lead to an anti-clustering of the singlets in the ground state of Heff (0).
The effective Hamiltonian (6.18), similarly to the original FH-Hamiltonian (2.40), could
be rewritten in the Bloch basis. Then, the tunneling term would correspond to a diagonal
term with the dispersion relation E(κi ) = (4J 2 /U)[1 + cos(κi d)], for N non-interacting
hard-core bosons, while the repulsive nearest-neighbor singlet-singlet interactions would
be represented by an off-diagonal term with much more complicated structure than that
for the on-site atom-atom interactions, in (2.43).
The model described by the Hamiltonian (6.18) is also referred to as the soft-core bosons
model, wherein the term soft is introduced not only to emphasize the composite nature
of the singlets, but also to underline the physical origin of their coupling described by
the effective Hamiltonian. According to the derivation of an effective Hamiltonian (see,
e.g. [60]), the term of the first order in J/|U| represents the direct coupling between the
levels of the ground subband, while the next high-order corrections represent the indirect
coupling between these levels through all the levels of other subbands. This is, precisely,
26
2. Description of the system
where the soft-core nature of the model comes from. Since there are no products of the
fermionic operators in the original FH-Hamiltonian, which describe the tunneling of singlets
as compounds, the first order term in (6.18) is absent. Therefore, the tunneling of each
singlet may only occur indirectly through the levels of other subbands, whose eigenstates
have single fermions (which are the parts of indirectly dissociated singlets!) on some orbitals
in the Wannier basis.
Strong repulsive interactions
At strongly repulsive interactions U > 0, below half-filling (i.e., at N↑ + N↓ < L), the
ground band includes only those Fock states which have at most one fermion per lattice
site. Above half-filling, the same statement holds for holes – thus, we restrict our study
to situations below half-filling, without loss of generality. The effective Hamiltonian in the
leading order in J/U coincides with the tunneling term of the original Hamiltonian (2.40)
" #
L X
2
X
J
(+)
,
(2.46)
Heff = −J
(eitF d/~c†l,s cl+1,s + e−itF d/~c†l+1,s cls ) + O
U
l=1 s=↑,↓
while virtual processes of position exchange between neighboring atoms through energetically “forbidden” double site occupancies, which were relevant at strongly attractive interactions, give contributions of the next order in J/U [62].7 Since the static forcing effects
only the phase of the matrix elements of the tunneling term, the next-order contributions
can be neglected.
To summarize the present section, in the limit of strong, repulsive interactions, atoms of
both fermionic components behave like one single component of non-interacting fermions,
due to an effective Pauli principle which originates from strong, repulsive interactions in
coordinate space, i.e., cl,↑ cl,↓ = c†l,↑ c†l,↓ = 0, while in the limit of strong attractive interaction, atoms form singlets with, in general, non-negligible nearest-neighbor interaction.
Remember that our description by the effective Hamiltonian above is only valid in the
ground subband (the lowest set of closely bunched energy levels, for a finite system), which
in the case of strong interaction is separated from the next, first excited subband by a gap
≃ |U|.
7
The Hamiltonian (2.46) together with its higher-order correction (effective spin-spin and nearestneighbor interactions) referred to as t-J model (in standard solid state physics notations, t is usually used
instead of J, while J := 4t2 /U ) is widely used in the theory of high temperature superconductivity, where
parameters t and J are considered to be independent, and, thus, J can be chosen arbitrary large. This is
in contrast to our case, where the next order corrections are always J/U times smaller than the first order
– tunneling term in (2.46).
Chapter 3
Single atom dynamics
Some intuition on how interacting fermionic atoms in an optical lattice behave under
static forcing can be obtained from the analysis of the single particle dynamics, since
the latter approximates that of a weakly interacting gas at low densities. Moreover, Bloch
oscillations, which are one of the central topics of this thesis, are a pure single particle effect,
which is expected to survive in the many-particle system, though modified by atom-atom
interactions. Therefore, in the present chapter, we will start out with the formulation of a
general framework applicable to the many-particle problem, and will subsequently focus on
characteristic dynamical features of the single particle case. This also includes a discussion
of dynamical properties induced by an additional parabolic confinement1 of the lattice.
3.1
Floquet-Bloch formalism
A time-dependent Hamiltonian such as (2.35),
H(t) = ε0 |lihl| − J
∞
X
l=1
eitF d/~|l + 1ihl| + e−itF d/~|lihl + 1| ,
lacks stationary solutions. However, for a periodic time dependence, all relevant information on the system evolution for any initial state can be obtained from the “one-cycle
time evolution operator”, which propagates the system over one period of the Hamiltonian
oscillatory time dependence. In the above problem this period is given by the Bloch period
TB = 2π~/F d of the single particle problem treated earlier.
The general time evolution operator, which propagates any initial state of the system
from time t0 to t > t0 , can be written as
Z
i t
′
′
W (t, t0 ) = ed
xp −
(3.1)
H(t )dt ,
~ t0
1
A weak parabolic confinement, if not compensated by additional external fields, is always present in
“bright” elongated optical lattices (for details, see Appendix A).
28
3. Single atom dynamics
where the hat over the exponential denotes time ordering.
As a direct consequence of the periodicity of H, W (t, t0 ) has the following properties:
(i)
W (nT, 0) = W n (T, 0) ;
(ii) W (t + T, T ) = W (t, 0) ;
(iii) W (t + nT, 0) = W (t, 0)W n (T, 0) .
(3.2)
The propagator over one Bloch cycle, WTB (t0 ) = W (t0 + TB , t0 ) is usually referred to
as the Floquet-Bloch operator (FB-operator, in short). Since the propagator W (t, 0) with
t 6= nT, n ∈ Z, does not in general commute with W n (T, 0), the FB-operator depends on
choice of t0 . We will always assume t0 = 0, if not specified otherwise.
Before proceeding with the single particle problem, let us note that in the definition
of the FB-operator the only thing what matters is the periodicity of the Hamiltonian in
time. Thus wether dealing with single or many particle problems is immaterial for the
formulation of the problem in terms of Floquet theory.
To solve the FB eigenvalue problem for a single particle, it is most appropriate to use
the quasimomentum representation (2.34) of the Hamiltonian (2.35),
#
"
∞
X
cos(κl d + tF d/~) |κl ihκl | .
(3.3)
Hκ (t) = ε0 − 2J
l=1
where H(t) has diagonal form, and, thus, commutes with itself at arbitrary times t′ .
This renders time-ordering trivial and we can directly integrate the diagonal matrixelements
Z ∞
X
i t
t′ F d
exp −
Wk (t, t0 ) =
dt′ |κl ihκl |
ε0 − 2J cos κl d +
~
~
t0
l=1
∞
X
tF d
ε0 (t − t0 )
2J
exp −i
=
sin κl d +
−i
~
Fd
~
l=1
t0 F d
|κl ihκl | .
(3.4)
− sin κl d +
~
Hence, the FB-operator WTB (t0 ) becomes completely degenerate (independent of t0 ) with
quasienergies exp(iε0 TB ), which can be set equal to unity by setting ε0 to zero, without loss
of generality.2 As a direct consequence, any initial superposition of basis states completely
revives after each Bloch period. Indeed, this result is rather expected, since there is no
dephasing or decay mechanism for single particle Bloch oscillations within the single-band
approximation.
When considering the many-particle problem in chapter 5, we will show that this degeneracy is lifted by a non-zero interaction strength, thus indicating a dephasing of Bloch
oscillations.
2
The energy ε0 gives just a constant energy offset of the ground band center, which is irrelevant for the
particle time evolution.
3.2 Bloch oscillations
3.2
29
Bloch oscillations
Let us now find the time-evolution of the single particle observables during one Bloch
period.
The time-evolution of the quasimomentum κ and of the coordinate x = −i∂/∂κ has a
very simple form in the Heisenberg picture:
dκ
i
=
[Hκ , κ] =
dt
~
i
dx
=
[Hκ , x] =
υ≡
dt
~
i
[ε0 (κ + tF/~), κ] = 0 ,
~
∂
1 ∂ε0 (κ + tF/~)
i
ε0 (κ + tF/~), −i
=
.
~
∂κ
~
∂κ
(3.5)
(3.6)
The solution of the first equation, κ = κ0 , which shows conservation of the particle quasimomentum in the acceleration frame, together with the periodicity of ε0 (κ), results in a periodic time-dependence of the velocity operator υ(t), with the Bloch frequency ωB = F d/~.3
For a general initial state, which we decompose in the Bloch basis {φκ (x)},
ψ(x, t0 ) =
π/d
X
fκ (t0 )φκ (x) ,
(3.7)
κ=−π/d
the mean particle velocity, and the mean particle position, oscillate with the Bloch frequency ωB :
π/d
1 X
∂ε0 (κ + ωB t)
hυ(t)i =
,
|fκ (t0 )|2
~
∂κ
(3.8a)
κ=−π/d
hx(t)i =
Z
t
′
′
υ(t )dt
t0
1
= x0 +
F
π/d
X
κ=−π/d
|fκ (t0 )|2 [ε0 (κ + ωB t) − ε0 (κ)] . (3.8b)
Since we will extensively use the tight-binding model and its many-particle generalization in the sequel of this thesis, we specialize (3.8) by direct substitution of the explicit
expression (2.23) for ε(κ):
π/d
2Jd X
|fκ (t0 )|2 sin(κd + ωB t) ,
hυ(t)i =
~
κ=−π/d
s≫1;
π/d
2J X
hx(t)i = x0 −
|fκ (t0 )|2 [cos(κd + ωB t) − cos(κd)] ,
Fd
κ=−π/d
3
(3.9a)
s≫1.
(3.9b)
The lattice period d in the expression for the Bloch frequency stems from the definition of quasimomentum in the irreducible Brillouin zone, κ ∈ [−π/d; +π/d].
30
3. Single atom dynamics
The expressions (3.8,3.9) allow us to make at least two important observations: First,
very counterintuitively from the classical physics point of view, the amplitude of the Bloch
oscillations in configuration space is inversely proportional to the magnitude of the static
force. Secondly, the oscillation amplitude vanishes for uniform distribution fκ (t0 ) = d/2π,
and attains a maximum for singular quasimomentum distributions fκ (t0 ) = δκ,κ0 . Note
that, since a single Bloch wave φκ (x) spreads over the entire configuration space, a change
of the particle state due to Bloch oscillations will be unnoticeable in coordinate space, for
initial conditions with a narrow quasimomentum distribution. Thus, the latter should have
a finite width to render BO of the mean particle position observable.
In the next section, we will investigate the impact of a weak parabolic confinement
along the lattice axis on the Bloch dynamics. We will see that such an additional parabolic
potential component induces qualitatively different dynamical regimes: Bloch oscillations
on one of the “shoulders” of the confining potential and dipole oscillations in the vicinity
of the parabola minimum. Both regimes also exhibit a transient dispersion of the atomic
wave-packet due to a confinement-induced non-linearity, followed by possible subsequent
revivals [63]. Note that such aperiodic behavior is not in contradiction to the periodicity of
the FB-operator discussed above, since the parabolic confinement breaks the translationary
symmetry of the Hamiltonian, and thus destroys the periodicity of the system response to
an external tilt.
3.3
Effects due to an additional parabolic confinement
The system we have in mind here is represented by atoms loaded in an anisotropic optical
lattice with a weak modulation along the x direction,
Vopt (r) = −V0 cos2 (kLx) − V⊥ [cos2 (kLy) + cos2 (kL z)] , V0 ≪ V⊥ ,
(3.10)
amended by a harmonic confinement
Vtrap (r) =
2
Mω 2 x2 Mω⊥
(y 2 + z 2 )
+
.
2
2
(3.11)
The dynamics along the x-coordinate can be described by the one-dimensional Hamiltonian
p2
Mω 2 2
V0
2πx
H=
+
−
cos
x ,
(3.12)
2M
2
d
2
and our target is to characterize the dynamics of a single particle localized initially in the
trap center, and set into motion by a sudden shift of the trap origin.
First, we will analyze this problem in the tight-binding approximation (see Sec. 2.2.3
above), and then elaborate its exact solution for a general choice of system parameters.
3.3 Effects due to an additional parabolic confinement
3.3.1
31
Quantum pendulum model
The tight-binding approximation for the Hamiltonian (3.12) takes the form
Htb =
X
νX 2
l |lihl| − J
(|l + 1ihl| + h.c.) ,
2
l
(3.13)
l
where ν = Mω 2 d2 . P
Accordingly, the time evolution of the amplitudes al (t) of the wave
function ψ(x, t) =
l al (t)|li, represented in the “Wannier basis” |li, is given by the
coupled equations
ν
i~ȧl = l2 al − J (al+1 + al−1 ) ,
(3.14)
2
which immediately follow from application of Htb on ψ, and subsequent projection on |li.
Better insight into the dynamics generated by (3.13) can be obtained by mapping it
on the Hamiltonian of a quantum pendulum: multiplying both sides of lth equation (3.14)
by the phase exp(ilθ), then summing the
Pleft and right sides of all equations, respectively,
and introducing the function Φ(θ, t) = l al (t) exp(ilθ), the equations for the amplitudes
al (t) are reduced to those for the wave function Φ(θ, t) of a quantum pendulum, with
Hamiltonian
∂
ν
(3.15)
Hpen = L2 − 2J cos θ , L = −i
2
∂θ
where the hopping matrix element J plays the role of the gravitational field. Note that
the above Hamiltonian is also the exact representation of (3.13) in the Bloch basis. Thus,
the phase θ of the pendulum is associated with the atomic quasimomentum κ, θ = κd.
The dynamical and spectral properties of the pendulum Hamiltonian (3.15) were studied in [64, 65], and, in particular, in much detail in the context of quantum chaos, since the
same Hamiltonian serves as the universal tool to approximate behavior of a complex quantum system near each nonlinear resonance it may have in phase space [66]. For the effective
Planck constant4 ~eff ∼ (ν/2J)1/2 ≪ 1, the quantum dynamics can resolve the classical
phase space structure, and much of the classical dynamical features of the Hamiltonian
Hpen =
ν 2
L − 2J cos θ
2
(3.16)
are inherited by the quantum system.
This statement is illustrated in Fig. 3.1, which shows the 120 lowest eigenstates of
the quantum pendulum, as well as in Fig. 3.2, where the differences between neighboring
energy levels Ωq (n) = (En+1 − En )/~ are plotted as a function of the quantum number
n. The inset in Fig. 3.1 depicts the phase portrait of the pendulum, and the solid line
in Fig. 3.2 depicts the frequency Ω = Ω(n) of the classical pendulum as a function of the
dimensionless action n. The dimensional action n is defined as the area enclosed by a
phase-space trajectory, divided by 2π, and is a continues quantity in classical dynamics,
The effective Planck constant ~eff is defined as ~ rescaled over the characteristic classical action (~n∗ ,
in the text above) [67].
4
32
3. Single atom dynamics
120
θ
100
80
n
−40 −20
60
0
L
20 40
40
20
0
−60
−40
−20
0
l
20
40
60
Figure 3.1: Gray-scale plot of the probability density of the energy eigenstates (for variable
quantum number n) of the quantum pendulum (3.15), as a function of the site number l, at
ν = 0.008 × J. The inset shows the phase space structure of the associated classical pendulum.
The separatrix trajectory corresponds to the state with quantum number n∗ ≃ 40, according to
(3.17).
while above it is rescaled over ~. The separatrix trajectory, which separates librational and
rotational motion of the pendulum, corresponds to the action [66]
n∗ = (8/π)(2J/ν)1/2 .
(3.17)
In the quantum case, this quantity gives the number of extended states associated with
libration. The rotational states with n > n∗ appear in pairs, with vanishing energy splitting, what is a quantum manifestation of the twofold degeneracy of the rotational frequency
branch (i.e., clockwise and counterclockwise rotations). The perturbative expansion of Ω(n)
in small parameter n/n∗ ≪ 1 and its asymptotic dependence for n ≫ n∗ can be evaluated
exactly [68]:
n 1n
Ω(n) = Ω0 1 −
+O ∗
, n/n∗ ≪ 1 ,
(3.18)
∗
π n
n
∗ n
4 n
, n/n∗ ≫ 1 ,
(3.19)
+O
Ω(n) = Ω0
∗
π n
n
where Ω0 = (2νJ)1/2 /~ is the frequency of small, harmonic oscillations of the pendulum
around its equilibrium point.5
5
Note that the dimensionless classical action n, when identified with the quantum number, does not
account for possible degeneracies in the spectrum, and labels rather the eigenstates than the eigenvalues.
3.3 Effects due to an additional parabolic confinement
33
3
Ω(n) / Ω0
2
1
0
0
1
*
n/n
2
Figure 3.2: Frequency of the classical pendulum (3.16) vs. the classical action n, in units of
the separatrix action n∗ (see (3.17)). The differences Ωq (n)/Ω0 = (En+1 − En )/~Ω0 between
adjacent eigenvalues of the quantum pendulum (stars)
p faithfully follow the classical prediction
(solid line), for an effective Planck constant ~eff ∼ ν/2J ≃ 0.025 ≪ 1. The dash-dotted lines
indicate the asymptotic behavior (3.18-3.19). The two-fold degeneracy in the rotational regime
(n > n∗ ) generates the second branch of frequencies Ωq (n) with vanishing values (stars).
A quantity related to (3.17) is the maximal angular momentum along the separatrix
trajectory,
π
(3.20)
l∗ = 2(2J/ν)1/2 = n∗ .
4
This quantity distinguishes two different regimes of the atomic wave-packet dynamics in
our original problem. For an initial shift of the trap origin l0 < l∗ , setting an atom initially
placed at the trap origin into motion, the latter will perform oscillations around the trap
origin, while for l0 > l∗ the atom will oscillate in a restricted region, on one of the wings
of the parabolic potential. Clearly, this reflects the division of classical phase space into
regions of librational and rotational motion, as illustrated in the inset of Fig. 3.1. These two
regimes of the atomic dynamics are also termed dipole (DO) and Bloch (BO) oscillations,
respectively.
3.3.2
Dephasing and revivals of Bloch and dipole oscillations
After our analysis of the spectral structure in the previous section, we will now focus
on the wave-packet dynamics, and on the evolution of a relevant observable, the particle
mean momentum. The main distinction of the pendulum quantum dynamics as compared
to its classical dynamics is the possible revival of the initially localized wave-packet after
34
3. Single atom dynamics
1
0
−1
0
10
20
30
40
10
20
t/T0
30
40
p/p0
1
0
−1
0
Figure 3.3: Dephasing and revivals of dipole (top) and Bloch (bottom) oscillations of the mean
momentum of an atom in a parabolic lattice with l∗ = 10. Initial shifts are l0 = 3 (top) and
l0 = 20 (bottom). Time is measured in periods T0 = 2π/Ω0 of the pendulum ground state. Note
the differences of frequencies and revival times of BO and DO, which directly follow from the
asymptotic behavior of the frequency spectrum (3.18,3.19) and from the derivatives thereof with
respect to the quantum number n.
its non-linearity-induced dephasing (which is of classical origin). At the revival time, the
wave-packet restores its initial form with large fidelity.
Using the tight-binding Hamiltonian (3.13), or, equivalently, the quantum pendulum
Hamiltonian (3.15), any solution of the time dependent Schrödinger equation can be decomposed as
X
X (n)
bn exp(−iEn t/~)φn (x) , φn (x) =
al ψl (x) ,
(3.21)
ψ(x, t) =
n
l
R
where the bn = φn (x)ψini (x − l0 d)dx are the expansion coefficients of some initial state
ψini (x) displaced by a sudden shift of the trap origin over l0 lattice sites, in the Wannier
basis ψl (x).
On the quantum level, dephasing is due to the non-equidistance of the energy levels of
the quantum pendulum. Denoting by σ the characteristic width of the initial wave-packet
ψ0 in the pendulum eigenstates, general arguments lead to the following estimate for the
dephasing time [63]:
τσ = ~/σΩ′ (n(l0 )) ,
(3.22)
where Ω′ (n) = s(n)∂Ω(n)/∂n is the anharmonicity of the spectrum, and s(n) is the degeneracy of the level n (alternatively, each degenerate level in (3.19) can be counted only
3.3 Effects due to an additional parabolic confinement
35
once). In the asymptotic regions (3.18-3.19), the anharmonicity is proportional to ν, and
the corresponding dephasing times are τσ = 8~/σν, if l0 ± σ/2 ≪ l∗ , and τσ = ~/σν, if
l0 ± σ/2 ≫ l∗ .
While the dispersion of the wave-packet can actually be studied by considering an appropriate ensemble of classical trajectories (which will evolve with different frequencies),
revivals are a pure quantum effect, due to the discreteness of the energy spectrum. When
the relative phases accumulated by the eigenstates in an initial superposition are all multiples of 2π, the wave packet exhibits complete revivals. In the asymptotic spectral region of
the pendulum, where the difference between eigenvalues is just an integer multiple of the
(BO)
(DO)
same anharmonicity, any wave packet will revive after Tν
= 2π~/ν and Tν
= 16π~/ν,
in the regime of Bloch and dipole oscillations, respectively. In particular, the wave function
(3.21) resumes its initial form up to an irrelevant, global phase factor.
To demonstrate this pronounced behavior we plot the time evolution of the atomic
mean momentum in the parabolic lattice in Figure 3.3, with ν = 0.08J (l∗ = 10), for initial
shifts l0 = 3 (upper panel, DO) and l0 = 2l∗ = 20 (lower panel, BO). As initial wave
function, we choose the ground state of the system, which gives σ ≃ (J/2ν)1/4 . It is seen
that the initial decay of the momentum oscillations [63],
p(t) = p0 exp(−t2 /2τσ2 ) sin(Ωt) ,
t ≪ Tν(DO) , Tν(BO) ,
(3.23)
is followed by almost perfect revivals – which, however spoiled on sufficiently long time
(DO)
(BO)
scales t ≫ Tν , Tν
due to high order corrections in (3.18,3.19). The dynamics of the
mean coordinate (not shown) is quite similar, and can be obtained from p(t) by integration.
3.3.3
Beyond the single band approximation
The tight-binding approximation is a particular case (deep optical lattice) of the more
general single band approximation (see section 2.2). A natural question emerges: does the
latter remain a good approximation at any shift l0 , or for arbitrary non-linearity ν of the
lattice?
To answer the above question let us analyze the eigenvalues of the exact Hamiltonian
(3.12). Figure 3.4 shows the eigenstates of this Hamiltonian in coordinate representation,
at four different sets of the system parameters, on the same gray scale as in Fig. 3.1.
The tight-binding model possesses a single parabolic band, whereas the spectrum of the
original system forms a multi-band structure. The emergence of the second band on the
energy scale of the first 120 eigenvalues under variation of the lattice depth, at constant
J/ν, is illustrated in Fig 3.4. Since lowering the lattice depth below the deep optical lattice
threshold V0 ∼ 5ER (see section 2.2.3) also enhances the tunneling amplitudes which couple
to non-neighbouring sites, also the eigenstates of the first parabolic band differ from those
in the tight-binding approximation.
Thus, in the presence of higher energy bands, the single band, and, in particular, the
more restricted tight-binding approximation are justified only in the case of negligible
Landau-Zener tunneling, i.e., when the potential difference νl0 between neighboring sites
n
36
3. Single atom dynamics
120
120
100
100
80
80
60
60
40
40
20
20
n
0
−60 −40 −20
0
20
40
60
0
−60 −40 −20
120
120
100
100
80
80
60
60
40
40
20
20
0
−60 −40 −20
0
l
20
40
60
0
−60 −40 −20
0
20
40
60
0
l
20
40
60
Figure 3.4: Gray-scale plot of the probability density of the energy eigenstates (for variable
quantum number n) of the original Hamiltonian (3.12), as a function of the site number l as in
Fig. 3.1, at the fixed ratio ν/J = 0.008. The amplitudes and frequencies of the optical potential
are changed accordingly from V0 = 4 × ER , ν = 0.0007 × ER (top left), over V0 = 2 × ER ,
ν = 0.0012 × ER (top right) and V0 = 1 × ER , ν = 0.0015 × ER (bottom left) to V0 = 0.5 × ER ,
ν = 0.0018 × ER (bottom right). Note the progressive intrusion of levels of the second parabolic
band into the excited energy range of the ground band of the original Hamiltonian, with increasing
relative strength of the parabolic confinement, ν/V0 . Since the tight-binding Hamiltonian (3.13)
only describes the ground band levels (see Fig. 3.1), that approximation remains valid only within
a finite range of initial shifts of the trap center. Above the critical value, defined by the bottom
of the second band, Landau-Zener tunneling starts to compete with the Bloch oscillations (also
see Fig. 3.5).
3.3 Effects due to an additional parabolic confinement
37
Figure 3.5: Center of mass oscillations of an atom in a parabolic lattice, with ν = 0.008 × J =
0.0007 × ER , and V0 = 0.5 × ER , for an initial shift l0 = 60d. The gray-scale plot shows the
probability density as a function of time, in units of the ground state period T0 = 2π/Ω0 of the
pendulum. The tunneling-induced transition between the ground and first excited band is clearly
reflected by the decay of Bloch oscillations on the left, and their reappearance on the right wing
of the parabolic potential. Dashed vertical lines indicate the critical shift l∗ ≃ 31, which separates
the regimes of BO and of DO.
at a distance l0 from the trap center is smaller than the energy gap ∆′ (which, due to
parabolic confinement, differs from the gap ∆ for Bloch particles, discussed in Sec. 2.2)
separating the ground parabolic band from the rest of the spectrum. Note that, at a given
set of system parameters, this condition imposes a restriction on the maximal shift l0 from
the trap origin. If this condition is violated, the atomic dynamics turns into a superposition
of BO in the lowest parabolic band, and of DO in the first excited band (see Fig. 3.5).
In the above, we collected all features of single particle dynamics in tilted, and possibly
harmonically confined one dimensional optical lattices. Let us now inspect the amendments
brought about by many-particle interactions. We shall start out with fermionic many
particle dynamics in periodic lattices, in the absence of further external perturbations.
38
3. Single atom dynamics
Chapter 4
Interacting fermions in the absence
of static forcing
In this chapter we focus on the symmetry and integrability properties of the FH-model
derived in Sec. 2.3. We characterize the spectrum of the system by the density of states
and by the associated spectral statistics, for periodic and twisted boundary conditions.
The latter case is important for our further analysis of the dynamical problem in chapters
5 and 6, since the functional dependence of the energy eigenvalues on the phase twist
coincides with that of the instantaneous spectrum of the FH-Hamiltonian under static
forcing written in the acceleration frame (see our previous discussion in Sec. 2.1.1).
The eigenvalue problem defined by the unperturbed (F = 0) FH-Hamiltonian (2.40),
H = −J
L X
X
(c†l,s cl+1,s + c†l+1,s cls ) + U
l=1 s=↑,↓
L
X
nl↑ nl↓ ,
l=1
was shown to be equivalent to finding the solution of a set of coupled nonlinear equations by Lieb and Wu [22], by use of the nested Bethe ansatz [69, 70].1 The LiebWu equations couple two sets of quantum numbers, the charge momenta k = {ki|i =
1, 2, . . . , N↑ + N↓ }, (Re(ki ) ∈ R/2πR), and the spin rapidities of the atoms with down-spins
λ = {λα |α = 1, 2, . . . , N↓ }, as well as the parameters J and U.2 The quantum numbers
{ki } and {λα }, parameterize the eigenenergies, and define the associated eigenfunctions
through spin dependent amplitudes ψk,λ (x; s) (s = {si |i = 1, 2, . . . , N↑ + N↓ } denotes a
given spin configuration with N↑ atoms in one and N↓ in the other component).
1
Importantly, the solution of the Lieb-Wu equations gives the ground, but not all excited eigenstates:
to obtain the complete set of eigenfunctions spanning the entire Hilbert space of the model, a hidden
SO(4) symmetry (see Sec. 4.1) of the FH-model has to be taken into account [71].
2
Momentum is used to refer to the quantum number k, since the energy eigenvalues as a function of the
ki are given by the standard relation for non-interacting fermions (which typically
are charge carriers, e.g.
P
electrons or holes, in solid state physics) in a periodic potential: Ei = −2J i cos ki . The name “rapidity”
for the quantum number λ is borrowed from the special theory of relativity, since the way the λα are
coupled in the Lieb-Wu equations is reminiscent of the summation principle for relativistic velocities.
40
4. Interacting fermions in the absence of static forcing
Despite the integrability of the FH-model, spelled out by the explicit expression for
the eigenfunctions as functions of ki and λα [72], no general analytical solution of the
Lieb-Wu equations in terms of the model parameters J and U is available. Moreover,
the solution of the complete eigenvalue problem as expressed by the Lieb-Wu equations
requires a nontrivial numerical treatment, even for small system sizes [73].
In our present approach, we therefore prefer to diagonalize the FH-Hamiltonian numerically, and achieve a considerable reduction of the computational efforts by careful
consideration of the symmetry properties of the model.
4.1
Symmetries
The existence of a certain symmetry of a model is equivalent to a non-trivial integral
of motion I (i.e., [I, H] = 0). If the associated eigenfunctions (eigenvalues) are known
explicitly, the Hamiltonian can be easily transformed in the representation associated with
that symmetry, with a block diagonal matrix form with respect to the different values
of the corresponding quantum number (the eigenvalues of I). Such representation will
considerably speed up the remaining numerical diagonalization (taking care of those degrees
of freedom where no symmetry can be revealed).
In this section, we list all the U, J-independent (discrete) symmetries of the FH-model
one by one, together with the associated quantum numbers to label the blocks of the
FH-Hamiltonian in the induced representation.
4.1.1
Invariance under spatial transformations
The first class of symmetries is generated by spatial transformations of the periodic lattice.
Obviously, the reflection operator σ, which inverts the order of the occupation numbers,
σ|n1 , n2 , ..., nL i↑ ⊗ |m1 , m2 , ..., mL i↓ =
|nL , nL−1 , ..., n1 i↑ ⊗ |mL , mL−1 , ..., m1 i↓
(4.1)
preserves the system energy. Since σ 2 = id, its eigenvalues are σ1,2 = ±1.
Periodic boundary conditions imply translational invariance
τ |n1 , n2 , ..., nL i↑ ⊗ |m1 , m2 , ..., mL i↓ =
|nL , n1 , n2 , ..., nL−1 i↑ ⊗ |mL , m1 , m2 , ..., mL−1 i↓ ,
(4.2)
where the translation operator τ L = 1 induces a discrete set of quasimomenta κi =
2πl/Ld, l = 1, . . . , L. Suitable combinations of translation and reflection operators, e.g.
σ p τ n with n = 1, . . . , L and p = 0, 1, generates all possible invariant spatial transformations
of the periodic lattice.
Remember that we already used the eigenstates of the translation operator to rewrite
the Hamiltonian in the quasimomentum representation, see Eq. (2.41). Also the operator
4.1 Symmetries
41
σ takes a very simple form in this representation,
!
!
!
!
Y
Y †
Y
Y †
cκ l s σ
cκ′ s |0i = h0|
cκ l s
c−κ′ s σ|0i = δκ,−κ′ ,
σκ,κ′ = h0|
l′ ,s
l,s
l
l
l′ ,s
l,s
(4.3)
where we used basis transformation c†κl → c†l (2.41), and the relation σc†l = c†L−l σ, which
follows immediately from the definition (4.1) of σ. Thus, the reflection operator connects
states with opposite total quasimomenta,
σ|κi = | − κi ,
(4.4)
and σ and τ do not commute, i.e., cannot be diagonalized simultaneously by the same
basis transformation. Only for κ = π (at even L) and κ = 2π does the reflection operator
map the corresponding subspaces onto themselves, which therefore can be factorized and
labeled with a well-defined reflection parity.3
Both operators can equally be expressed through creation and annihilation operators
in coordinate space,
σ=
⌊L/2⌋−1
Y
l=1
where
Y
Pls,(L−l)s ,
τ=
Y
s=↑,↓
s=↑,↓
P1s,2s P2s,3s · · · P(L−1)s,Ls ,
Pls,ms′ := 1 − (c†ls − c†ms′ )(cls − cms′ ) , l 6= m
(4.5)
(4.6)
are permutation operators.
4.1.2
Spin symmetry
The next type of symmetry is inherited from the conservation of the total spin, as well as
of its projection, associated with the operators
S2 =
S− S+ + S+ S−
+ Sz2 ,
2
(4.7)
– which is, in general, referred to as the Casimir operator – and
Sz =
n↑ − n↓
2
(4.8)
respectively, with raising and lowering operators (also known as ladder operators),
†
S+ = (S− ) =
L
X
j=1
c†j↑ cj↓
,
ns =
L
X
njs .
(4.9)
j=1
This becomes obvious by considering semicyclic, τ L/2 , and cyclic, τ L , lattice “translations”, which
commute with the reflection operator, [τ L/2 , σ] = [τ L , σ] = 0, and whose action on the relevant subspaces
associated with κ = π (at even L) and κ = 2π, respectively, is equivalent to τ .
3
42
4. Interacting fermions in the absence of static forcing
The eigenvalue equations
S2 |ψs i = S(S + 1)|ψs i ,
Sz |ψs i = Sz |ψs i
(4.10)
imply the following sets of eigenvalues:4
|N↑ − N↓ |/2 ≤ S ≤ (N↑ + N↓ )/2 ,
Sz = (N↑ − N↓ )/2 ,
(4.11)
where the total spin S ∈ Z, if Sz ∈ Z, and S ∈ Z + 1/2 otherwise.
4.1.3
Conditional symmetries
The symmetries listed above are general, i.e., valid for any choice of N↑ , N↓ , L. There are
two additional symmetries which emerge only for special values thereof: if the two fermionic
components have the same total occupation numbers N↑ = N↓ , one may exchange one
component with the other, without changing the energy of the system. This operation is
generated, similarly to reflection, by the spin-flip operator
π=
L
Y
Pl↑,l↓ ,
(4.12)
l=1
with eigenvalues π = ±1, and represents the first additional symmetry type.
Another symmetry type is known as the quasi-spin symmetry, and emerges only for
an even number L of lattice sites. Then we can define an operator which anticommutes
with the Hamiltonian, and, thus, can be used to map the original spin algebra on a new
η-pairing (quasi-spin) algebra. The new algebra of operators reads
n↑ + n↓
ηz =
−n
2
, η− =
2n
X
j=1
(−1)j cj↑ cj↓
, η+ = (η− )† ,
(4.13)
with the Casimir operator
η− η+ + η+ η−
+ ηz2 .
(4.14)
2
Similarly to the spin eigenvalues, the Casimir operator (4.14) has eigenvalues η(η + 1),
with quasi-spin quantum numbers η ∈ Z, if ηz = L − |N↑ + N↓ |)/2 ∈ Z, and η ∈ Z + 1/2
otherwise, and (L − |N↑ + N↓ |)/2 ≤ η ≤ (L − |N↑ − N↓ |)/2. The generators (the Casimir
operator and the (quasi-)spin projection operator) of both, spin and quasi-spin algebras
commute with each other, and their associated symmetry group SU(2) × SU(2) yields
on R4 the complete SO(4) symmetry of the FH-Hamiltonian [75]. This symmetry is not
only necessary to construct the complete set of the eigenstates of the FH-model from the
incomplete Lieb-Wu solution (using the associated raising operators) [71]; but also at half
η2 =
4
Angular momenta (or spin) eigenvalues of quantum can be found purely algebraically, using only the
commutation relations of the corresponding operators [74].
4.2 Integrability and spectral properties
43
filling (i.e., for N = N↑ = N↓ , L = 2N), due to this symmetry the entire spectrum
comprises of four elementary excitations [76].
In summary, the spectrum of the FH-Hamiltonian is classified by the set of quantum
numbers
{κ; σ, if κ = π or 2π; S; π, if N↑ = N↓ ; η, if L ∈ Z} .
(4.15)
The (quasi)-spin projection Sz (ηz ) takes the same, unique value for all eigenenergies of
the Hamiltonian, for a fixed number of particles in each spin component, and is therefore
omitted.
4.2
Integrability and spectral properties
In the present section, we focus on the universal spectral properties of the FH-model, as a
nontrivial case of the general class of integrable quantum systems. To set the scene, let us
start with a general discussion of integrability in the context of spectral theory.
The formal definition of integrability (Liouville integrability), as the existence of a
complete set of integrals of motion including the Hamiltonian itself, is widely used for
both, quantum and classical mechanics, although the meaning of a “complete set” has to
be defined more specifically.
Let us shortly recall the classical definition. The symplectic structure of Hamilton
equations of motion is incorporated by the Poisson brackets, and to form a complete set
the number of mutually independent, conserved quantities has to be the same as the number
n of degrees of freedom. The last statement is based on the Liouville theorem (theorem 1.2
in [77]), where it is mathematically proven that the presence of a complete set of integrals
of motion ensures that all possible trajectories of a general solution lie on an n-dimensional
torus, and that there exists a set of new, canonically conjugate coordinates – action-angle
variables – with the following properties: the first half of them, the actions Ii , remain
constant, labeling the invariant tori, while the second half, the canonically conjugated
angles φi , evolve periodically in time with frequencies, ωi , along topologically irreducible
contours on the torus. In fact, integrability itself does not imply separability, i.e., one
cannot always find a transformation to pairwise decoupled action–angle variables, even if
the complete set of conserved quantities is known (see, for example, the discussion on the
Toda chain in [78], and related references therein). Integrability only guarantees, that a
system trajectory sticks to invariant tori in phase space. Thus, its temporal behavior is
defined by the same set of n frequencies, which are analytical functions of the Hamiltonian
parameters, and depends smoothly on infinitesimal changes of the initial conditions. In
contrast, a non-integrable system is expected to be irregular or chaotic, since there are not
enough constraints to fix a complete set of n “eigen” frequencies. As a result, the evolution
of a chaotic system becomes very sensitive to infinitesimal changes of the initial conditions,
what is widely used as a very general characterization of the non-integrability of a given
system [68]. The latter is numerically testable for systems with few but also with many
44
4. Interacting fermions in the absence of static forcing
number degrees of freedom. Unfortunately, there is no general analytical criterion to infer
(non-)integrability, except for specific classes of model systems [79].
What do we know about (non-)integrability in quantum mechanics? It is clear that,
within the canonical quantization procedure applied to integrable classical systems, one
arrives at an integrable quantum model. Moreover, if the classical dynamics is separable,
its quantum version is separable too, and the number of conserved quantities in a complete
set of quantum observables coincides with the number of degrees of freedom of the corresponding classical system. The most famous example of non-trivial integrable dynamics is
provided by a particle in a three-dimensional Coulomb potential, where the three degrees
of freedom in the classical, as well as in the quantum case have their corresponding three
integrals of motion: the angular momentum (L), the Runge-Lenz vector (A), and energy
(H) itself.5
However, there are also quantum systems, including the FH-model, which lack a classical analogue. For such models, in particular for the FH-model, there is no precise definition
of the number of degrees of freedom. It was shown [80], that algebraic origin of the LiebWu equations is an infinite number of mutually commuting, system parameters-dependent
integrals of motion – the “currents” Ii (U, J), with the first one I0 given by the system
Hamiltonian. In fact, it is not yet clear [81] how many of these are sufficient to ensure integrability for a given finite lattice size and particle number. Therefore, there is no rigorous
mathematical theorem which would provide an integrability criterion similar to Liouville
integrability, for quantum systems which lack a unique classical origin.
Another way to characterize “non-integrable” quantum systems is through the statistical description of their spectral properties. The Bohigas-Giannoni-Schmit-Pechukas
(BGSP) conjecture [82, 83] predicts universal fluctuations of quantum spectra, depending
on their integrability as well as on the fundamental symmetry properties: systems with
chaotic classical origin exhibit Wigner-Dyson statistics of the nearest neighbor level spacing
distribution (LSD),

2
4
(sπ/2)e−s /π ,
(orthogonal)

2 /4π
2
−s
PWD (s) =
(4.16)
(s 32/π)e
,
(unitary)
 4 18 6 3 −s2 64/9π
(s 2 /3 π )e
, (symplectic symmetry class) ,
while systems with integrable classical counterpart are well described by the Poisson distribution [84]
P (s) = e−s .
(4.17)
The BGSP conjecture is based on a comparison of random matrix theory (RMT) [67] with
a large amount of numerically and experimentally generated spectral data on complex
systems as quantum billiards, compound nuclei, and some other quantum many-particle
models. The conjecture is convincingly corroborated by these comparisons.6
The Poisson distribution is the characteristic feature of completely uncorrelated spectra. Therefore, if the model remains integrable under changes of one of the parameters
5
Note that in quantum case, although the four operators L, Lz , A, Az commute with the Hamiltonian
45
4
4
2
2
E
E
4.2 Integrability and spectral properties
0
−2
−4
0
0
−2
0.2
0.4
0.6
u
0.8
1
−4
0
0.2
0.4
0.6
0.8
1
u
Figure 4.1: Energy level dynamics of the Fermi-Hubbard model in units of U + 2J, as function of
u = U/(U + 2J), with J > 0, for N = 6 (N↑ = 3, N↓ = 3) atoms in L = 6 lattice sites. While the
left panel shows the complete spectrum, the right plot selects those levels with quasimomentum
κ = 0, total spin S = 0, quasispin η = 0, and even parities π = 1, σ = 1. Although all discrete
symmetries were divided out, there are few points of accidental degeneracies (levels crossing in
the right panel) left. This, in fact, implies additional, interaction-dependent integral(s) of motion
(see discussion in the main text).
of the Hamiltonian, then the energy levels are expected to evolve independently under
such changes and thus will cross at particular values. Moreover, the level crossing are allowed within each irreducible subspace defined by different quantum numbers of parameterindependent (discrete) symmetries.
To see such Poisson behavior in the FH-model, we take a finite lattice with L = 6,
at half filling. Then one can choose an arbitrary irreducible block of the complete matrix
representation of the Hamiltonian. Here we take the subspace with quasimomentum κ = 0,
total spin S = 0, quasispin η = 0, and even parities, π = 1 and σ = 1. We vary
dimensionless interaction parameter u = U/(2J + U) (J > 0) such as to effectively scan
the entire parameter range of the model, from non-interacting (J 6= 0, U = 0) to infinitely
heavy, interacting (J = 0, U 6= 0) particles. Indeed, the expected crossings are present in
Fig. 4.1 for repulsive interaction (U > 0), i.e., 0 6 u 6 1.
It is very easy to prove the general statement that the presence of a non-trivial, udependent integral of motion implies at least one crossing:
Let H(u) = H0 + uV be a Hamiltonian that, in a certain basis, can be represented by an s × s matrix. For example, H(u) can be one of the blocks of the
Hubbard Hamiltonian. Clearly, any matrix M that is an analytical function
of a particle in a Coulomb potential, only three of them, including the energy itself, mutually commute.
6
For quantum systems with a small Hilbert space, the problem of statistical sampling can be often
overcome by superimposing many, statistically independent spectra, obtained under variation of those
system parameters which do not affect the system symmetry.
46
4. Interacting fermions in the absence of static forcing
1
n(E)
data
fit
0.5
0
−2
−1
0
1
2
3
E
I(s)
1
data
0.5
Poisson
WD
0
0
1
2
3
4
s
Figure 4.2: Statistical analysis of the FH-model spectrum. The density of states (histogram) is
fitted by a normal distribution (solid line), in the top panel. As expected for an integrable system,
the integrated level spacing distribution (solid line) tightly follows the Poisson law (4.17) (dashed
line), which is compared to the Wigner-Dyson distribution for the orthogonal random matrix
ensemble (4.16) (dashed-dotted line). The statistics are based on the subset of eigenenergies with
quasimomentum κ = 2π/L, zero spin, and even parity π = 1. The system parameters are u = 0.6,
L = 11, and N = 6, N↑ = N↓ = 3.
of H(u) and u, M = f (u, H(u)), commutes with H. Obviously, M is not an
independent conserved quantity. Therefore, we say that J(u) is a nontrivial
integral of motion if J(u) is hermitian, [J(u), H(u)] = 0 and at the same time
J(u) cannot be written as an analytical function of H(u) and u.
. . . Since J(u) and H(u) commute, there is a basis where both these operators
are diagonal. Let E1 (u), . . . , Es (u) and J1 (u), . . . , Js (u) be the eigenvalues of
H(u) and J(u), respectively. Consider the following set of algebraic equations:
a1 E1s−1 (u) + · · · + as−1 E1 (u) + as = J1 (u)
..
..
.
.
s−1
a1 Es (u) + · · · + as−1 Es (u) + as = Js (u)
(4.18)
If equations (4.18) have a solution, J(u) and H(u) are not independent. Namely,
J(u) = a1 H(u)s−1 + · · · + as . The system (4.18) has no solutions if and only if
Ei (u∗ ) = Ej (u∗ ) and Ji (u∗ ) 6= Jj (u∗ ) for some i, j, and u∗ . Thus, the Hamiltonian can have a nontrivial integral of motion only if it has a level crossing.7
7
We reproduce here the original quote from [85].
4.3 Spectrum under twisted boundary conditions
47
To demonstrate that the FH-model exhibits Poisson energy levels statistics we enlarge
the lattice size to L = 11, and consider the largest irreducible block with quasimomentum
κ = 2π/L, zero spin and even parity π = 1.8 In Fig. 4.2, we compare the integrated level
spacing distributions (an equivalent characteristics)
Z s
I(s) =
P (s′ )ds′
(4.19)
0
instead of P (s) themselves. Also note that, to extract the universal statistical features of
the spectrum one has to perform spectral unfolding [67], i.e., rescale the energy levels by
multiplication with the local density of states n(E), which, in our present case of fermions
at intermediate interaction strength (here we choose u=0.6), can be approximated by a
Gaussian (see the upper panel of Fig. 4.2). As also spelled out by Fig. 4.1, the spectrum
splits into well separated bands at small and large interaction strengths: thus, the density
of states is no more Gaussian, and needs to be fitted by an appropriate function.
4.3
Spectrum under twisted boundary conditions
It is also known that the integrability of the FH-model persists under twisted boundary
conditions c†L+1,s cL,s = eiLφs (−1)Ns −1 c†1,s cL,s [86], or, what is equivalent, that the following
FH-model with a modified tunneling term
H = −J
L X
X
(eiφs c†l,s cl+1,s
l=1 s=↑,↓
+
e−iφs c†l+1,s cls )
+U
L
X
nl↑ nl↓ ,
(4.20)
l=1
is integrable under standard periodic boundary conditions c†L+1,s cL,s = (−1)Ns −1 c†1,s cL,s .
The above Hamiltonian is not a mathematical abstraction. In particular at φ↑ = φ↓ ,
it is used to describe the behavior of charged particles on a ring, with non-zero constant
magnetic flux across it. Our main reason to discuss here this specific model is its direct
relation to the transport problems we will be dealing with further down in this thesis, in
particular Bloch oscillations (see chapter 6). Indeed, a time dependent phase induces a
potential difference between any two different points on the ring, thus inducing a current
of particles across the lattice. Moreover, the effect of a linearly time-dependent phase on
the particles is the same as that of a linear static force, and we already encountered the
above representation of the Hamiltonian with a time-dependent phase when considering
tight-binding model for single and many particle systems in an optical lattice amended by
a linear potential, in the acceleration frame (see Eq. (2.35) in Sec. 2.2.3, and Eq. (2.40) in
Sec. 2.3, respectively).
For any constant phase, as for vanishing phase, the eigenenergies and eigenfunctions of
the Hamiltonian (4.20) can be defined explicitly through the momenta ki and rapidities
8
The parity σ cannot be specified, since reflection does not commute with the elementary translation
over κ = 2π/L (see Sec. 4.1).
48
4. Interacting fermions in the absence of static forcing
1
data, φ=π/2
n(E)
fit, φ=π/2
fit, φ=0
0.5
0
−2
−1
0
1
2
3
E
1
I(s)
φ=π/2
Poisson
0.5
WD
φ=0
0
0
1
2
3
4
s
Figure 4.3: Same as Fig. 4.2, but with twisted boundary conditions, see Eq. (4.20). The phase
difference φ between adjacent sites of the Hamiltonian (4.20) is chosen as φ↑ = φd n = π/2. The
blue line in the upper panel corresponds to the continuous red line in Fig. 4.2. As expected
from the persistence of the Fermi-Hubbard model integrability under phase twist, we didn’t find
any deviation of the level spacing distribution from the universal Poisson distribution (4.17) for
integrable systems. Note, however, the twist-induced slight deformation of the density of states
in the top panel.
λj . These quantum numbers satisfy the generalized Lieb-Wu equations [86], obtained by
extension of the nested Bethe-Yang ansatz [69]. Therefore, a nonvanishing phase does not
break the integrability of the Fermi-Hubbard model, and one expects to find a complete
set of integrals of motion, as a function of both parameters, the dimensionless interaction u
and the phase φ. As a result, the energy levels may cross as a function of u and φ, and there
should be no avoided level crossings in the level dynamics. Indeed, as we have checked for
arbitrary choice of the phase φ, the integrated level spacing distribution (Fig. 4.3, red line,
for φ = π/2) always closely follows the Poisson distribution, while the density of states is
typically noticeably deformed by a non-zero phase shift (compare the dashed blue (φ = 0),
and the continuous red (φ = π/2) lines in Fig. 4.3).
4.4
Summary
We have seen that the Fermi-Hubbard Hamiltonian exhibits universal spectral properties
common to all integrable systems, at any values of the phase φ, the interaction strength
U, and the tunneling coupling J.
Given the correspondence between a constant phase-shift and a time-dependent term
4.4 Summary
49
in the Hubbard Hamiltonian with tilt in the acceleration frame, it is natural to ask the
following question: Does the system remain integrable under variations of its phase in
time? In the next chapter, section 5.2.3 will deliver the answer to this intriguing question,
for the case of a linear time dependence of φ. There, in particular, we reserve some space
for a detailed discussion of the properties of chaotic quantum systems.
50
4. Interacting fermions in the absence of static forcing
Chapter 5
Spectral properties of interacting
fermions under linear static forcing
The phenomenon of Bloch oscillations is completely understood both theoretically and
experimentally as a single particle quantum effect (see chapter 3), while its analog in
strongly correlated many-particle quantum systems was not studied until recently [87].
There, for the case of indistinguishable cold bosonic atoms with on-site interaction in a onedimensional optical lattice, it was shown that nonvanishing on-site interaction can induce
periodic modulations or strong dephasing with complete decay of the Bloch oscillations,
depending on the strength of the static force.
Not only the particle-particle interaction, but also the underlying statistics define the
system-response to the static forcing. Indeed, due to the Pauli principle, systems with
different statistics live on Hilbert spaces with different size. In contrast to fermions, there
is no restriction on the bosonic occupation number of a lattice site. Therefore, we expect
differences in the dynamical behavior of bosons and of fermions, and the purpose of the
present thesis is to shed some light on these issues – in the, also experimentally realized,
context of fermionic or of mixed fermionic and bosonic atoms in optical lattices.
Let us recall of what we already discussed. The basic derivation of the Hubbard model
– a many particle generalization of the tight-binding or narrow single band approximation
[20, 21] was presented in chapter 2. We also pointed out that, in the limit of strong
interactions, a (sub)band structure emerges in the spectrum and an effective Hamiltonian
can be derived to describe the system dynamics within the ground subband. Chapter 4
was devoted to the properties of interacting fermions in a periodic potential in the absence
of any additional external forcing. In particular, we saw that exploring the symmetry
properties of the system lead to a considerable simplification of the diagonalization of the
corresponding Hamiltonian, and how characteristic spectral properties of the model reflect
the model integrability.
In the present chapter, the effect of an additional-static potential is treated within
the formalism of the many-particle Floquet-Bloch (in short, FB-) operator. We identify its
spectral properties, and then, in the following chapter 6, employ them to explain the system
dynamics. There, we will also monitor the time-dependence of some system observables:
52
5. Spectral properties of interacting fermions under linear static forcing
the mean atomic energy and the mean atomic momentum (the latter is an easily accessible
quantity in recent experiments [88]).
The problem we address here can be formulated as follows: two distinct sorts of fermions
are placed in a finite periodic lattice with L sites. The particle numbers in each component,
N↑ and N↓ , are fixed. We impose the periodic boundary conditions,1 , and introduce the
effective static force represented in the form of a time-dependent phase shift associated
with the acceleration frame (see Secs. 2.1.1 and 2.2.3), to be consistent with the periodic
boundary conditions. This phase action on neutral particles is mediated by a linear increase
of the rotation frequency of the lattice. Thus, our system can be described by the timedependent Hamiltonian
X
X X
eitF d/~c†l+1,s cl,s + e−itF d/~c†l,s cl+1,s + U
nl,↑ nl,↓ ,
(5.1)
H(t) = −J
s=↑,↓
l
l
in the coordinate, and by
Hκ (t) = −2J
XX
κ
s=↑,↓
cos(κs d + tF d/~)c†κs cκs +
U X †
c
c† ′ cκ′ ↑ cκ↓ ,
L ′ κ+q↑ κ−q ↓
(5.2)
κ,κ ,q
in the quasimomentum representation, respectively. Note that the phase passes 2π once
per Bloch period TB = 2~π/F d, with F d the amplitude of the effective linear potential.
Our goal is to find the time evolution of relevant observables such as the mean kinetic
energy and the mean momentum under static forcing, in comparison to the single particle
problem, such as to understand possible effects due to Fermi statistics and on-site particleparticle interactions.
For that purpose, we will first define the FB-operator, which propagates the system
over the its shortest characteristic time scale, and investigate its spectral properties in
different parameter regimes. Subsequently, we will use the Heisenberg picture to evolve
the system kinetic energy and momentum.
5.1
The Floquet-Bloch operator
Let us apply the many-particle approach of our above treatment of the FH-model, where
effects due to the particle statistics and their interactions were already analyzed, to the
dynamical problem of Bloch oscillations. The single particle manifestations thereof were
already discussed in detail in chapter 1. To define the many-particle FB-operator we use
the same definition introduced in Sec. 3.1,
Z
i t0 +τ
Wτ (t0 ) = W (t0 + τ, t0 ) = ed
xp −
H(t)dt ,
(5.3)
~ t0
1
The role of different boundary conditions was discussed at the very end of Sec. 2.2.3 and in Sec. 2.1.3.
For overview of possible experimental realizations, see Sec. 1.1.1.
5.1 The Floquet-Bloch operator
53
In contrast to the single particle case, the the many particle FH-Hamiltonian has a
much more complicated algebraic structure. Therefore, in general, the integration in the
above definition can not be performed exactly, except for at least two limits: vanishing
interaction strengths U → 0, or strong static forcing F d ≫ J, U.
5.1.1
Limit of vanishing interaction strength (U → 0)
Indeed, the Hamiltonian in the quasimomentum representations (5.2) becomes diagonal
for U = 0, and the evolution operator
"
tF d
2J X X
sin κs d +
Wκ (t, t0 )U →0 = exp −i
F d κ s=↑,↓
~
t0 F d
c†κs cκs
(5.4)
− sin κs d +
~
reduces to the identity at t = t0 +TB , i.e., WTB (t0 ) = id. This proves that Bloch oscillations
of non-interacting fermions do not decay, in complete analogy to the single particle case.2
The fermionic nature of the many particle system is reflected by the restrictions on the
quasimomentum occupation numbers imposed by the Pauli principle for fermions.
5.1.2
Limit of strong static forcing (F d ≫ J)
In the limit of a large amplitude of the static force in comparison to the tunneling matrix
element, F d ≫ J, the Bloch period TB = 2π~/F d becomes the shortest time scale of
the system evolution. Therefore, we can use the argument of time scale separation, to
integrate out the rapidly oscillating, non-diagonal matrix elements of the FB-operator (see
Appendix B). This leaves us with a diagonal form of the FB-operator in Wannier basis:
!
UTB X
′
hn|WTB (0)|n i ≃ exp −i
(5.5)
nl,↑ nl,↓ δn,n′ ,
~
l
where nl = hn|nl |ni is the occupation number of site l, and off-diagonal elements of the
order O (J/F d) have been neglected.
This result can be qualitatively explained by the following decomposition of the evolution operator in kinetic and interaction terms on short time scales (first order in ∆t):
W (∆t, 0) ≃ W (∆t, 0)U =0 W (∆t, 0)J=0 .
(5.6)
If we substitute on the left-hand side the shortest time scale – the Bloch period TB – instead
of ∆t, then the first factor becomes just the FB-operator of non-interacting fermions, which
2
It is also clear that this result does not depend on the statistics of the particles, and holds for bosons
as well.
54
5. Spectral properties of interacting fermions under linear static forcing
is the identity matrix (see our discussion in Sec. 5.1.1 above), while the second factor
coincides with the above result (5.5).
If the FB-operator tends to a diagonal, butP
not a completely degenerate matrix, what
kind of dynamics does it generate? Since hn| l nl,↑ nl,↓ |ni is always an integer number,
WTB (0)F/U turns into the identity for an integer ratio F/U. Thus, although the Bloch oscillations are expected to dephase, as a direct manifestation of particle-particle interactions,
the initial state of the system will periodically revive exactly, with an interaction-dependent
revival frequency ΩU = U/~.
5.2
Spectrum of the Floquet-Bloch operator: general
case
In the general situation, the matrix representation of the FB-operator is very dense (i.e.,
most off-diagonal elements are non-zero and of the same order), in both the coordinate and
the quasimomentum representation, in contrast to the two limits discussed above, where
it takes diagonal form. To obtain its matrix elements we employ the numerical stepwise
integration algorithm based on a Padé approximation with scaling and squaring [89]:3 The
Bloch period [t0 , TB + t0 ] is divided into N equal, small intervals ∆T = TB /N, and the
FB-operator is approximated by the product of N discrete time evolution operators over
intervals [ti , ti+1 = ti + ∆T ]
N
Y
i
WTB (t0 ) =
exp − H(ti )∆T
~
i=1
(5.7)
Convergence of this algorithm was shown to be sufficient to perform accurate subsequent
statistical eigenvalue analysis of the FB-operator, and also compared with a Runge-Kutta
algorithm (see, e. g., [90]).
For a characterization of the spectral properties and their statistical analysis, we first
have to extract an irreducible block with well defined quantum numbers. Therefore, as in
the case of the FH-Hamiltonian, it is necessary to identify all those symmetries, associated
with time-independent observables. Clearly such construction of the FB-operator block by
block, exploring its decomposition on invariant subspaces, is also much preferable from the
numerical point of view.
5.2.1
Symmetries of the Floquet-Bloch operator
As a direct consequence of definition (5.3), if the time-dependent Hamiltonian H(t) has
block diagonal structure in a time independent, fixed basis at any moment of time, then
the FB-operator will inherit this property. Thus, the FB-operator owns all permanent (i.e.,
time independent) symmetries of the Hamiltonian H(t).
3
This method is implemented in by the expm() function in Matlab.
5.2 Spectrum of the Floquet-Bloch operator: general case
55
Since we know all symmetries of the time-independent FH-model, we can find out using
commutation relations: which of these symmetries survive, and which are broken, by the
forcing-induced phase twist between adjacent sites.
A direct calculation leads to the following result: the FB-operator commutes with the
lattice shift operator τ , the total spin operator S2 , and, in the particular case of an equal
number of particles in both components (N↑ = N↓ ), with the spin-flip operator π. Hence,
the total quasimomentum κ, the spin-flip parity π, and the total spin S are these good
quantum numbers which can be used to label the quasienergies of the FB-operator.
In addition to the discrete symmetries of the Hamiltonian, the FB-operator may posses
a time-reversal symmetry. The latter implies that for a given solution |ψ(−t)i of the
Schrödinger equation, there is an independent solution |ψ̃(t)i = A|ψ(−t)i, with A2 = ±1
[67]. Equivalently to the above condition, the Hamiltonian has to satisfy
H(t)A = AH(−t) .
(5.8)
Note that, in general, for the time-reversible FB-operator, the operator A does not coincide
with the conventional time-reversal operator T (which is defined as: T xT −1 = x, T pT −1 =
−p, and T 2 = ±1). In particular, the Hamiltonian of our present model commutes with T ,
thus, violating (5.8), while the same equality holds with the nonconventional time reversal
operator A given by the reflection operator σ (introduced in Sec. 4.1) on the subspaces
Fκ=0(π/d) with total quasimomentum zero at any integer L and additionally π/d at even L.
Indeed, we can prove that
H(t)σ = σH(−t) on Fκ=0(π/d) .
(5.9)
Since the interaction part commutes with σ and does not depend on time, to prove the
above statement one only has to deal with the tunneling term. The actions of the leftand right-hand
sides of Eq. (5.9) associated with the tunneling term on general solution
P
|φ(t)i = κ aκ (t)|κi of the Schrödinger equation, respectively, coincide:
#
"
X
X
aκ (t)|κi =
cos(κs d + ωB t)nκs σ
κ
κs ,s
σ
"
X
κs ,s
cos(κs d − ωB t)nκs
#
X
κ
X
κ
a−κ (t)
X
cos(κs d + ωB t)n−κ(κs ) |κi ,
(5.10)
X
cos(−κs d − ωB t)n−κ(κs ) |κi .
(5.11)
κs ,s
aκ (t)|κi =
X
κ
a−κ (t)
κs ,s
Here we used that the reflection symmetry (which is responsible for the twofold degeneracies
of the Hamiltonian H(t = 0)) pairwise connects the Bloch basis vectors with opposite total
quasimomenta: σ|κi = |−κi (see Sec. 4.1). Since the action of σ does not change the total
56
5. Spectral properties of interacting fermions under linear static forcing
quasimomentum only of those states which are composed from the Bloch basis states with
total quasimomentum zero and π/d, the time-reversal symmetry holds independently only
on the associated subspaces Fκ=0(π/d) .
Thus, although the discrete, reflection symmetry of the Hamiltonian is broken at (t
mod TB ) 6= 0, the associated reflection operator represents a nonconventional time-reversal
symmetry of the FB-operator on Fκ=0(π/d) . This symmetry indicates that the group of
invariant unitary transformations of the Hamiltonian is reduced to two isomorphic orthogonality classes [67], which may be mapped onto each other by the operator σ. For the
FB-operator, this means that its canonical class of invariant transformations on Fκ=0(π/d)
is formed by all special (i.e., norm preserving) orthogonal transformations. The type of
the fundamental transformation group defines the spectral properties of the corresponding
random matrix ensemble in the random matrix theory [67]. Due to the BGSP conjecture [82, 83] one expects to witness the universal statistics of the energy eigenvalues for the
non-integrable system after identification of its canonical class of invariant transformations.
A few pages further down, we will see that the time-dependent FH-model may serve
as a demonstration example, where the universal statistics defined by the canonical class
emerge.
5.2.2
Parametric dynamics of the quasienergies
In this section we numerically explore a wide parameter range, thus merging analytical
predictions for the limiting parameter values discussed in Sec. 5.1.1, 5.1.2. Our numerical
approach run on an ordinary PC is able to perform the complete diagonalization of systems
with Hilbert space dimensions N comparable to that of N↑ = N↓ = 5 atoms on L = 10
lattice sites (i.e., according to (2.8), N = 63504), on the time scale of several hours.
Furthermore, we will see that systems of much smaller size already contain essential, if
not complete, information on general, qualitative properties of larger systems in the same
parameter regime.
First, we choose a relatively small system, with N1 = N2 = 2 atoms in L = 5 lattice
sites, to monitor the evolution of the FB-operator spectrum under variation of the driving
force F . Figure 5.1 shows the result of the computation for equal interaction and tunneling
strengths, J = U, for κ = 0, S = 0, and π = 1. Note that because of its unitary, the
eigenphases of the FB-operator are defined modulo 2π. The elementary interval [−π, π] is
denoted by the red dashed line. The scaling of the quasienergies εα /U of the FB-operator
(standardly defined as the argument of the eigenphases divided by modulo TB ) in the left
panel is done to emphasize the levels behavior at strong static forcing (F d ≫ J), where we
predict an equidistant
P ladder (5.5) formed by highly degenerate levels. Indeed, all three
possible values of l nl↑ nl↓ = 0, 1, 2 for the case of N1 = N2 = 2 are present.
The unscaled version of the quasienergies presented in right panel of the Fig. 5.1 reveals
the complicated level “dynamics” as a function of the static tilt comparable or smaller than
the tunneling (interaction) constant. There are only avoided crossings and no levels cross
at all, though not all avoided crossings are well resolved on the scale of the figure. This is,
in fact, the typical spectral manifestation of non-integrability, i.e., of the incompleteness of
5.2 Spectrum of the Floquet-Bloch operator: general case
0.5
2
ε / Fd
1
0
α
εα / U
57
0
−1
−2
−0.5
−1
0
1
log (Fd / J)
2
1
2
3
J / Fd
4
5
Figure 5.1: Quasienergies of an irreducible block of the Floquet-Bloch operator, as a function
of the static tilt F d, in units of the tunneling matrix element (left panel), and of the inverse
quantity (F d/J)−1 (right panel). Here we singled out the Hilbert subspace with κ = 0, S = 0,
and π = 1 for N↑ = N↓ = 2 fermions on L = 5 lattice sites. Furthermore, we fixed J = U (see
further description in the text). The red dashed lines depict the borders of the first Floquet zone
±F d/2.
set of conserved quantities. It also indicates that, in comparison to the non-perturbed FHmodel, which has a complete set of non-trivial U-dependent integrals (see Sec. 4.3 above),
the FH-model under static forcing does not have such integrals left, i.e., non-trivial, F dependent operators commuting with the FB-operator are absent.
It is also worth noting, that the elementary interval [−π, π] (red dashed line) on the
left of Fig. 5.1 will be shifted to the left (right) when decreasing (increasing) the ratio
of interaction to tunneling strength, while the level dynamics will remain qualitatively
unchanged. Nevertheless, the limit of vanishing interaction strength cannot be identified
on the left panel, since both εα and U tend to zero. On the right of Fig. 5.1, the average
slope of the level “spaghetti” will increase (decrease) when increasing (decreasing) the
interaction strength, and, therefore, the limit of U → 0 will be easily seen, since the entire
level dynamics will shrink to the abscissa axis.
5.2.3
Spectral statistics
To confirm our above qualitative observation, we performed a statistical analysis of the
spectrum at a slightly larger system size, to achieve a better statistical sampling. For
weak and moderate forcing (and n̄ ∼ 1), we observe that the FB-operator can be identified
with a random matrix of either the Circular Orthogonal (COE) or the Circular Unitary
Ensemble (CUE) [67]. The supporting numerical evidence is provided in Fig. 5.2, which
shows the integrated level spacing distribution for the eigenphases εα of the FB-operator.
58
5. Spectral properties of interacting fermions under linear static forcing
1
90
60
120
30
I(s)
150
0.5
180
0
210
330
240
300
270
0
0
1
2
ξ
i
3
4
s
Figure 5.2: Integrated level spacing distributions for the Floquet-Bloch operator WTB (0),
Eq. (5.3) (blue solid lines and histogram), as compared to the corresponding random matrix
predictions: CUE for κ = 2π/Ld and F = 0.06 (red dotted line); COE for κ = 0 and F = 0.06
(red dash-dotted line); Poisson (red dashed line). In all cases, the interaction parameter was
set to u = 0.6, for spin S = 0 and even parity π = 1. Lattice length L = 11, particle number
N = 6, with equal occupations N↑ = N↓ = 3 of both spin components. The emergence of a
regular spectrum of the FB-operator for F d > J (here F d = 1) is shown by red crosses forming
an equidistant ladder of the eigenphases.
The numerical data are seen to reliably follow the Wigner-Dyson distribution for CUE
32 2
4s2
P (s) = 2 s exp −
,
(5.12)
π
π
and COE (for levels from subspace with κ = 0, π/d),
πs2
π
.
P (s) = s exp −
2
4
(5.13)
respectively. The emergence of the two different statistical universality classes, CUE and
COE, was already anticipated by our analysis of the system symmetries, in the preceding
Sec. 5.2.1, where we found the nonconventional time reversal symmetry of the FB-operator
when projected on the subspaces with κ = 0, π/d.
In contrast to the almost uniform distribution of phases εα observed at moderate static
forcing F d ∼ J for κ = 2π/Ld (Fig. 5.2, outer blue circle) and κ = 0 (Fig. 5.2, inner blue
5.3 Summary
59
circle), the quasienergy spectrum start to split into a number of bunches, which merge in
highly degenerate equidistant levels (Fig. 5.2, red crosses), with increased static forcing.
Here we found four such discrete levels in agreement with diagonal
PLapproximation (5.5)
for the given system configuration of N↑ = N↓ = 3 particles ( l=1 nl↑ nl↓ = 0, 1, 2, 3,
independently on number of lattice sites L). As it follows from the perturbative expansion
of the FB-operator in small parameter J/F d (see Appendix C), the width of each band
is approximately J/F d times smaller than the interaction constant U. Therefore, the
structure of each bunch of levels may dynamically manifest only at sufficiently large times
in comparison to the revival time 2π~/U ≫ TB .
It is important to note that, since the discussed symmetry properties are independent
on the sign of the interaction constant, all statistical results shown for repulsive interaction
(U > 0), also hold for the attractive case (U < 0). Of course, differences still may appear
in the dynamical response of the ground state, which is fundamentally different for strong
repulsive and attractive interaction constants (|U| ≫ J), respectively.
5.3
Summary
Let us recall the “instantaneous” (i.e., at a fixed moment of time) integrability of the
time-dependent FH-model, which was briefly discussed at the end of Sec. 4.3, and the
question posed in the summary there: whether this system, with linearly time-dependent
phases remains also integrable. The answer found in the present chapter is negative. The
FH-model under static forcing reveals itself as a unique example where non-integrability
and level correlations only result from the non-stationarity of the system.
In the next chapter, we will discuss in much detail the dynamical manifestations of the
spectral properties of the FH-model under static forcing, in various parameter regimes.
60
5. Spectral properties of interacting fermions under linear static forcing
Chapter 6
Bloch oscillations of interacting
fermions
In this section, we will investigate the time-dependence of relevant observables of the
Fermi-Hubbard model under static tilt. In particular, we will witness the dynamical manifestations of the statistical properties of the FB-operator discussed above. Furthermore,
we will explore the relevance of the instantaneous spectrum of the Hamiltonian H(t) for
a prediction of the dynamics, in the limit of strong (diabatic) and weak (adiabatic) static
forcing, respectively, and discuss the exponential (and irreversible) decay of Bloch oscillations at intermediate forcing strengths. Analytical expressions will be obtained in the case
of strong static forcing, using the asymptotic form of the FB-operator.
If the FB-operator exhibits the statistical properties of a random matrix, the exact
system dynamics are hardly tractable analytically, but can be monitored, for a finite system
size, with an accurate numerical treatment. We use the following scheme in our numerical
calculations: The time dependence over short time intervals up to one Bloch cycle is
inferred by direct propagation of the total wave function,
Ψ(t) =
N
X
j=1
cj (t)|nj i ,
(6.1)
with the the Schrödinger equation. When interested in the long time behavior t ≫ TB ,
it is less time consuming to first construct the FB-operator(s) WTB (ti ) = W (ti + TB , ti ),
simultaneously storing the grid of coefficients cj (ti ) at different moments of time ti ∈
[t0 , ..., t0 + TB ], and then perform the long-time propagation over an interval ∆t = nTB , n
integer, by n-fold matrix multiplication with WTB .
Let us now specify the relevant observables. Due to the periodic time dependence of the
62
6. Bloch oscillations of interacting fermions
total Hamiltonian, certainly the time-dependent total energy is a characteristic quantity,
#
"
X †
E(t) = hΨ(t)|H(t)|Ψ(t)i = −2JRe hΨ(t) |
cl+1,s cl,s | Ψ(t)ieitF d/~
+UhΨ(t)|
X
l
s,l
nl,↑ nl,↓ |Ψ(t)i .
(6.2)
An observable which is easily accessible in typical experiments is the atomic mean velocity
X
2Jd
eitF d/~c†l+1,s cl,s − e−itF d/~c†l,s cl+1,s |Ψ(t)i
hΨ(t)|
~
l
#
"
X †
2Jd
=
cl+1,s cl,s | Ψ(t)ieitF d/~ ,
Im hΨ(t) |
~
s,l
υ(t) = i
(6.3)
which is experimentally monitored by time of flight measurements [88]. The expression
above is just a many-particle generalization of Eq. (3.9b), where, due to atom-atom interactions, the initial quasimomentum distribution fκ (t0 ) does not preserve its form any more,
and evolves according to fκ (t) = hΨ(t)|nκ |Ψ(t)i. Equivalently, the mean atomic velocity
up to a constant prefactor (which is the number of fermions per unit length of the lattice)
can also be considered as the mean atomic current across the lattice.
As for the initial conditions |Ψ(0)i, we will always choose the ground state of the
system in the absence of external forcing, if not specified otherwise. The physical nature
of the ground state depends strongly on the ratio between the hopping matrix element
J and the interaction constant U. Referring to the emergent band structure at strong
interaction, discussed in Sec. 2.3.1, we will distinguish moderate |U| . J, and strong,
|U| ≫ J, interactions, and analyze them separately. To explore the complete variety of
possible dynamical regimes of the driven FH-model, we will scan over different amplitudes
of the static forcing, F d, in all these cases.
6.1
Dynamics at moderate interactions (U . J)
Let us start with trivial case of non-interacting particles. Since the FB-operator equals the
identity operator, Eq. (5.4), the time dependence of any observable is periodic with the
Bloch period TB . In particular, for the system initially prepared in an eigenstate κ with
single particle quasimomenta κl,s , the mean energy and velocity evolve according to
Eκ (t) = −2J
X
cos(κl,s d + tF d/~) ,
(6.4a)
2Jd X
υκ (t) =
sin(κl,s d + tF d/~) .
~ l,s
(6.4b)
l,s
6.1 Dynamics at moderate interactions (U . J)
63
E(t)/2JN
1
0.5
0
−0.5
−1
0
4
8
10
20
12
16
20
30
40
50
E(t)/2JN
1
0.5
0
−0.5
−1
0
t/TB
Figure 6.1: Decay (top) and revivals (bottom) of Bloch oscillations of the mean energy E(t) =
hΨ(t)|H(t)|Ψ(t)i of non-polarized (N↑ = 3, N↓ = 3) fermions (solid lines) under weak F/J =
0.3J (top) and strong F/J = 50 (bottom) static forcing, in comparison to the dynamics of
a polarized sample with the same number of fermions (top, dashed line). The interaction-totunneling strength ratio and the number of lattice sites were in all cases fixed to U/J = 3) and
L = 11, respectively. Note that the time scale determined by TB = 2π/F (Bloch period) is
different in the upper and lower plot.
In the case of non-vanishing interaction strengths, the FB-matrix, in general, becomes
a typical representative of the random matrix ensemble with Wigner-Dyson level-spacing
distribution (see Sec. 5.2.3), in both Wannier and Bloch bases. Therefore, the dynamics
of the system observables generated by this FB-operator, in general, cannot be periodic
with the Bloch period as in Eqs. (6.4). In particular, BO may become quasiperiodic, or
irregular, or irreversibly decay in time.
An example of such time-evolutions of the mean energy, for week static forcing, F d ≪ J,
is shown on the top panel in Fig. 6.1: the BO of interacting atoms irreversibly decay (solid
line), in contrast to the non-decaying BO (dashed line) of the same number of identical
(i.e. polarized) non-interacting fermionic atoms, see also the experiment reported in [91].
Although, presently, we don’t know the decay rate as a function of the system parameter,
we will suggest a promising approach to deduce this dependence in Sec. 6.3.1
In addition to the general scenario above, in the limit of strong static forcing, F d ≫ J,
the appropriate power of the FB-operator associated with the time-evolution of the system
over a time interval ≃ 2π~/U tends to the identity operator (see Sec. 5.1.2). Thus the
decaying BO should revive after 2π~/U. See an example of such revivals on the bottom of
1
For the solution to the problem of finding the decay rate of BO for a dilute, one-component, fermionic
gas coupled to the bosonic thermal bath, see chapter 7.
64
6. Bloch oscillations of interacting fermions
Fig. 6.1, and the subsequent discussion in the next section 6.1.1.
In the next section, using the asymptotic form of the FB-operator, we will evaluate the
envelope function for the revivals of BO of the mean atomic energy and velocity.
6.1.1
Interaction-induced amplitude modulations at strong static
forcing (F ≫ J)
If the magnitude of the tilt dominates the tunneling constant, i.e. F d ≫ J (F d & |U|),
the FB-operator takes diagonal form in Wannier basis, Eq. (5.5),
!
X
UT
B
hn|WTB (0)|n′ i = exp −i
(6.5)
nl,↑ nl,↓ δn,n′ .
~
l
The mean fermionic energy after an integer number n of the Bloch periods can be written
as
#
"
X †
n†
n
cl+1,s cl,s | WTB (0)Ψ(0)i + U0 ,
(6.6)
E(nTB ) = −2JRe hΨ(0)WTB (0) |
s,l
with the constant
U0 = UhΨ(0) |
X
l
nl,↑ nl,↓ | Ψ(0)i ,
(6.7)
which accounts for the interaction energy of the initial state |Ψ(0)i. The mean atomic
velocity according to (6.3) is given as
#
"
X
2Jd
c†l+1,s cl,s | TBn (0)Ψ(0)i .
(6.8)
υ(nTB ) = −
Im hΨ(0)WTn†B (0) |
~
s,l
Note that the expression in the square brackets in (6.6) and in (6.8) can be simplified
employing the diagonal form of the FB-operator and the fermionic commutation relations.
Indeed, since
!
X
X
ni,↑ ni,↓ c†l+1,↓ cl,↓ =
ni,↑ c†i,↓ ci,↓ c†l+1,↓ cl,↓
i
i
ni,↑ c†i,↓ δi,l+1 − c†i,↓ c†l+1,↓ ci,↓ cl,↓
i
h
X
=
ni,↑ δi,l+1 δl,i − cl,↓ c†i,↓ − c†l+1,↓ δl,i − cl,↓ c†i,↓ ci,↓
X
= nl+1,↑ c†l+1,↓ cl,↓ − nl,↑ c†l+1,↓ cl,↓ + c†l+1,↓ cl,↓
ni,↑ ni,↓
=
X
= c†l+1,↓ cl,↓
X
i
ni,↑ ni,↓ + nl+1,↑ − nl,↑
!
i
,
(↑↔↓) ,
(6.9)
6.1 Dynamics at moderate interactions (U . J)
65
where (↑↔↓) denotes the equality (or expression) obtained from the preceding it equality
(or expression) by substitutions: ↑→↓ and ↓→↑, we can commute any power m of the
interaction term
!m
!m
X
X
ni,↑ ni,↓
c†l+1,↓ cl,↓ = c†l+1,↓ cl,↓
ni,↑ ni,↓ + nl+1,↑ − nl,↑
, (↑↔↓) . (6.10)
i
i
Therefore, the tunneling term and the FB-operator WTB (0), which is given by an exponential function of the interaction term (6.5), up to a constant prefactor, commute as well.
Correspondingly, expressions (6.6) and (6.8) can be rewritten in the following form
"
#
X †
E(nTB ) = −2JRe hΨ|
cl+1,↑ cl,↑ e−iU (nl+1,↓ −nl,↓ )nTB /~|Ψi + (↑↔↓) + U0 ,
"
2Jd
Im hΨ|
υ(nTB ) =
~
l
X
l
c†l+1,↑ cl,↑ e−iU (nl+1,↓ −nl,↓ )nTB /~|Ψi + (↑↔↓)
#
Since the differences hn|nl+1,s − nl,s |ni in the exponentials above can take only three values
0 and ±1, we finally arrive at the general expressions
E(nTB ) = E0 [A(N↓ , N↑ , L) + (1 − A(N↓ , N↑ , L)) cos(nUTB /~)] + U0 , (6.11a)
υ(nTB ) = υ0 [A(N↓ , N↑ , L) + (1 − A(N↓ , N↑ , L)) sin(nUTB /~)] ,
(6.11b)
where E0 and υ0 are the ground state kinetic energy (i.e, the ground energy minus the
energy of atom-atom interactions) and the mean velocity, respectively, and A(N↓ , N↑ , L) is
a time-independent, dimensionless coefficient. Thus, the BO amplitude of both observables
are modulated with a new interaction dependent frequency ΩU = U/~ (see an example in
Fig. 6.1 (bottom) with two fermionic components, N↑ = N↓ = 3, L = 11).
The above result can also be understood with an analysis of the distribution of eigenenergies of the FB-operator. Indeed, as shown in Sec. 5.1.2, the spectrum of the FB-operator
splits into a number of closely bunched levels (“subband”), with increasing static force,
until, at F d/J → ∞, each of these subbands squeezes to a highly degenerate quasienergy
level. These levels form an equidistant ladder2 with the spacing determined by the interaction constant U. Since the FB-matrix is diagonal in Wannier basis, its eigenstates
coincide the basis states which, in turn, are the eigenstates of the system in the limit of
strong interactions, |U| ≫ J. The general initial state |Ψi has finite overlap with the
eigenstates of the different subbands, and, thus, the time period required to rebuild an
initial superposition of the eigenphases in |Ψi is defined by the interaction constant U.3
2
The
P number of levels in this ladder is equal to the number of distinct matrix elements of the interaction
term l nl↑ nl↓ in the Wannier basis.
3
In the particular case when the initial state |Ψi is given by a superposition of Wannier basis states
associated with a single subband of the FB-operator, |Ψi acquires only some phase shift after each Bloch
cycle. This obviously doesn’t affect the evolution of the system observables, i.e., A(N↓ , N↑ , L) = 1.
66
6. Bloch oscillations of interacting fermions
In general, it is an extremely complicated task to write down an explicit expression for
the ground or thermal initial state of the system, and consequently calculate the coefficient
A(N↓ , N↑ , L). However, the problem becomes much easier at weak interactions, |U| ≪ J,
with the system initially loaded in its ground state. Although zero temperature transport
properties require the determination of the exact ground state, in the case of dynamical
phenomena such as BO, where the system is diabatically transfered from the ground state
into the excited energy range, the ground state of the system can be well approximated by
the ground state of non-interacting fermions. The latter is given by the Bloch basis state
!
Ns
L
Ns
Y Y
X
Y Y
1
†
c†κi,s |0i = N/2
|Ψ0 i =
(6.12)
exp(ilκi )cl |0i ,
L
s=↑,↓ i=1
l=1
s=↑,↓ i=1
with the single-particle quasimomenta κi,s = 2πi/L (i = 1, . . . , L), which minimize the
total energy:
!
Ns
XX
(U ≪J)
cos(κi,s ) .
(6.13)
E0
≃ −2J min
κi,s
s=↑,↓ i=1
Then the velocity amplitude in (6.11b) reads
(U ≪J)
υ0
2Jd
≃
min
~ κi,s
Ns
XX
s=↑,↓ i=1
sin(κi,s )
!
.
(6.14)
In the particular case of systems with N↑ = N↓ = 1, and N↑ = N↓ = L − 1, the ground
state (6.12) simplifies to an equally weighted superposition of all Wannier states. Then one
can analytically show that A(N↑ , N↓ , L) = 1 − 4/L. Thus, for a very dilute gas of fermions,
i.e., for (N↑ + N↓ )/L ≪ 1, we expect that the modulation amplitude will vanish as 1/L,
with increasing system size.
In general, one can explicitly (i.e., symbolically) obtain the ground state (6.12) in
Wannier basis, for relatively large system sizes (see below), with the help of a personal
computer, and then substitute it into Eq. (6.11) to find the modulation amplitude. Our
subsequent discussion is based on such calculations, for systems with N↑ + N↓ 6 16 and
L 6 20.
The first important observation is that, when increasing the lattice size, the modulation
amplitude becomes a function of only two parameters: of the mean occupation numbers
n̄s = Ns /L, s =↑, ↓ (i.e., the particle density of a single component s), rather than of
the particle numbers Ns , s =↑, ↓, and of the lattice size L. As an example, on the left of
Fig. 6.2, the saturation of the coefficient A(N↑ , N↓ , L), with increasing L, is demonstrated
for the samples with n̄↑ = n̄↓ = 1/2 (blue circles) and n↑ = 1/2 − 1/2L, n↓ = 1/2 + 1/2L
(red circles).
Therefore we expect that the saturation values of the coefficients A(N↑ , N↓ , L) obtained
for finite, relatively small systems (see the right panel of Fig. 6.2), characterize also all larger
(i.e., with L > 20) systems with identical occupation numbers.
6.2 Dynamics at strong interactions (|U| ≫ J)
67
0.5
1
N=N↓=N↑, L=2N
1
N=N =N −1, L=2N+1
↑
0.4
↓
0.35
0.3
0.5
0.6
0
1
0.25
0.2
2
0.8
↓
↑
A(n , n )
A(N↑,N↓,L)
0.45
3
4
5
6
7
8
0.5
n
↓
1
0 0
0.4
0.5
n
↑
N
Figure 6.2: Interaction-induced modulation of Bloch oscillations. Left panel: the coefficient
A(N↓ , N↑ , L) (see (6.11a) and (6.11b)) at half-filling for L = 2N , N = N↓ = N↑ (blue circles) and
in the vicinity of half-filling for an odd L = 2N + 1, N = N↑ = N↓ − 1 (red circles) as a function
of N (to guide the eye, the data points are connected by solid lines with the same color). Right
panel: the same coefficient A(n̄↓ , n̄↑ ) as a function of the mean occupation numbers n̄s = Ns /L
with s =↑, ↓ in a wide range of fillings is shown as a mesh plot (right). The latter is based on the
data from all possible configurations with N↑ + N↓ 6 16 particles in L 6 20 lattice sites.
The results of our calculations of these saturation values A(n̄↓ , n̄↑ ) := A(N↑ , N↓ , L)|L→∞ ,
for various occupation numbers, are summarized in the right plot of Fig. 6.2. We find
that A(n̄↓ , n̄↑ ) has a global minimum as a function of the mean occupation numbers, at
n↑ = n↓ = 1/2, for even L, and at n↑ = 1/2 − 1/2L, n↓ = 1/2 + 1/2L, for odd L, i.e. at
half filling), and is also symmetric with respect to particle-hole substitution: ns ↔ 1 − ns ,
s =↑, ↓. As accurately indicated by the large-N asymptotic behavior on the left of Fig. 6.2,
this minimum lies between 0.295 and 0.317 for L > 15. According to Eq. (6.11), this defines
a maximum of the modulation amplitude 1 − A(n̄↓ , n̄↑ ) = 0.694 ± 0.011, for L > 16.
6.2
Dynamics at strong interactions (|U | ≫ J)
At moderate interaction strengths, |U| . J, the statistical properties of the FB-operator, as
well as the dynamical response do not depend on the sign of the interaction constant, since
the ground state of the unperturbed system (Bloch basis state) has a non-vanishing overlap
with all eigenstates (Wannier basis states) of the FB-operator (see discussion above).
The situation changes a lot at strong interactions, where the spectrum of the Hamiltonian splits into well-separated “subbands”, which are not coupled by the BO up to some
value of the static forcing. That can be seen from the instantaneous picture, which was
already used once for the single particle spectrum in Sec. 2.2.2.4 Oscillations in time of the
4
For the role of the instantaneous spectrum in the evolution of the mean energy and velocity, see our
68
6. Bloch oscillations of interacting fermions
instantaneous energies H(t) are proportional to the tunneling constant, which is a small
J ≪ U. Therefore, levels from different subbands stay always at a distance of subband
separation ∼ U, and do not talk to each other, unless the amplitude of the static forcing
F d becomes comparable to the interaction constant U. Thus, the associated separation in
the spectrum of the FB-operator for a weak static forcing F d ≪ |U| is present as well.
We assume again that the system is initially found in its ground state. Therefore we can
employ the effective Hamiltonians (see Sec. 2.3.1), which are the projections of the original
Hamiltonian onto the lowest energy subband. Since the ground state and the spectral
properties of the associated subband depend on the sign of the interaction constant, we
shall analyze repulsive and attractive interactions separately.
6.2.1
Retrieved coherence of oscillations (U > 0)
We begin with the simpler case of repulsive interactions. As shown in Sec. 2.3.1, there is
effectively no interaction between the fermions, therefore the effective Hamiltonian up to
the first order in J/U is given by the tunneling term of the FH-Hamiltonian
(+)
Heff
= −J
L X
X
(eitF d/~c†l,s cl+1,s + e−itF d/~c†l+1,s cls ) ,
(6.15)
l=1 s=↑,↓
but the Hilbert space is spanned only by those states which have only one fermion per
lattice site, i.e., cl,↑ cl,↓ = c†l,↑ c†l,↓ = 0.
Thus, the expressions for the mean energy and velocity coincide with those for N =
N↑ + N↓ non-interacting polarized (i.e., identical, since only one fermion per lattice site is
allowed) fermions:
N
X
E(t) = −2J
cos(κi d + tF d/~) ,
(6.16)
i=1
N
2Jd X
υ(t) =
sin(κi d + tF d/~) .
~ i=1
(6.17)
An example of such oscillations of Fermi atoms is shown in the upper panel of Fig. 6.3,
for U/J = 18 and F d/J = 0.2. The lower panel of Fig. 6.3 is intended to illustrate the
importance of the condition F ≪ U, which allows us to eliminate the coupling to the
higher subbands. If this condition is violated, the higher energy subbands come into play,
and the Bloch dynamics turns very complicated. The only (but important) exception from
this rule is the case of filling factor unity (N = L), where E(t) ≡ 0 in Eq. (6.16). For
strongly interacting Bose atoms, this problem was analyzed in Ref. [92]. It was found that
the initial Mott insulator state of Bose atoms (which is the ground state of the system for
an integer filling factor) responds resonantly to the static force, showing oscillations of the
mean energy with the characteristic frequency ∼ J/~. We expect that a similar result can
subsequent discussion in Sec. 6.3.
6.2 Dynamics at strong interactions (|U| ≫ J)
69
E(t)/2JN
1
0.5
0
−0.5
−1
0
10
20
30
20
30
E(t)/2JN
3
2
1
0
−1
0
10
t/T
B
Figure 6.3: Bloch oscillations of Fermi atoms in the regime of strong repulsive interactions for
F d/J = 0.2 (top) and F d/J = 18 (bottom), for N↑ = N↓ = 3, U/J = 18, L = 11. If the static tilt
is comparable to the interaction constant (e.g. for F d = U (bottom)), the static force resonantly
couples the equidistant energy subbands formed at strong interaction U ≫ J. That causes the
failure of the effective description (6.15) in capturing non-regular complex oscillations behavior
(bottom).
be obtained for the antiferromagnetic initial state (i.e. for Ψ(0)i = | . . . ↑↓↑ . . . i, in the
Wannier basis) of Fermi atoms.
6.2.2
Fermionic pairing: dynamics of bosonic compounds (U < 0)
We turn to the case of strong attractive interactions. As shown in Sec. 2.3.1, a system
with an equal number of particles in both components can be described by an effective
Hamiltonian (2.46) for compounds (singlets) formed by pairs of bound attractive fermions.
Up to the second order in J/U, the effective Hamiltonian reads
"
!
#
2 X
X
X
4J
1
(−)
Heff =
ñl +
ei2tF d/~ã†l+1 ãl + h.c. −
ñl+1 ñl ,
(6.18)
U
2
l
l
l
with (ãl )2 = (ã†l )2 = 0, where ã†l = c†l,↑ c†l,↓ and ãl = cl,↑ cl,↓ the creation and annihilation
operators for the singlets. The doubling of the Bloch frequency ωB′ = ωB for singlets
directly follows from the definition of the singlet operator ã†l+1 (ãl ), which describes the
simultaneous tunneling of two fermions.
The impact of periodic driving is very similar to that in the FH-model, although the
interaction part couples neighboring sites in comparison to on-site interactions among
70
6. Bloch oscillations of interacting fermions
(E−U)/2JN
0
−0.02
0
10
20
30
(E−U)/2JN
0
−0.02
0
4
8
t/TB
12
16
Figure 6.4: Bloch oscillations of Fermi atoms in the regime of strong attractive interactions, for
F d/Jeff = 11 (top), F d/Jeff = 0.22 (bottom), and Jeff ≡ Ueff = 4J 2 /U . The other parameters are
U/J = 22, L = 12, and N↑ = N↓ = 4. Note the doubling of the Bloch frequency (top) due to the
pairing of fermions from different components in configuration space, and the qualitatively similar
behavior to that in the regime of intermediate interaction strengths U ∼ J shown in Fig. 6.1. This
is a result of an effective rescaling of the tunneling amplitude J and of the interaction constant
U to one and the same amplitude 4J 2 /U .
fermions in the original Hamiltonian. The interaction energy of the singlets, as evident
from Eq. (6.18), is of the same order as the kinetic energy, and one can reproduce the
results of Sec. 6.1 for this system as well. Thus, we expect to observe decaying BO for
static tilts with F d < J 2 /|U|, and a modulation of the BO frequency Ωeff = ~U/4J 2
for static tilts |U| ≫ F > 4J 2 /|U|. The numerical simulations of the system dynamics
presented in Fig. 6.4 confirm these theoretical predictions.
At a static force comparable to the subband spacing |U|, the effective Hamiltonian
(6.18) is not applicable any more. In this case, the BO are strongly affected by resonant
tunneling between the subbands of the total Hamiltonian, and, thus, we expect a quasiregular behavior of the system observables, alike the resonant tunneling scenario encountered
in the case of strong repulsive interactions (see Fig. 6.3).
So far, to obtain the time-evolution of the system observables we used the FB-operator
and direct numerical simulations. In the next section, we will focus on those situations
where the instantaneous spectrum of the Hamiltonian H(t) may acquire some physical
meaning and provide us with additional information.
E(t) / 2JN
6.3 Instantaneous spectrum in (non-)adiabatic evolution
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
71
υ(t) / υ0
−1
−1
−1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
−1
−1
−1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
t/TB
t/T
t/TB
B
Figure 6.5:
Bloch oscillations of the mean atomic energy, Eq. (6.2), and the mean atomic
velocity, Eq. (6.3), (solid blue lines) of non-interacting (left column) and weakly interacting
(middle and right column) fermions under weak (adiabatic) static forcing F d ≪ |U |, for N↑ = 1,
N↓ = 2, L = 4, U = J (middle column), and for N↑ = N↓ = 2, L = 4, U = J (right column).
The dashed red lines in the upper row plots represent the instantaneous spectrum Eκ of H(t).
Clearly, the mean energy coincides with single levels of the instantaneous spectrum under weak
static forcing. Note the period TB /L of the single level oscillations in the instantaneous spectrum
at half filling: n↑ = n↓ = 1/2 (right column).
6.3
Instantaneous spectrum in (non-)adiabatic evolution
Let us demonstrate that the effect of a non-vanishing interaction on BO can be also fruitfully analyzed in terms of the instantaneous spectrum E(t) of H(t). For U = 0, the mean
energies Eκ (t) are precisely the instantaneous eigenvalues of H(t) (which are distinct from
the quasienergies of the Floquet eigenvalues of H(t): e.g., they all are equal to zero for
U = 0, since WTB (t0 ) = id, see Eq. (5.4)!), which cross without coupling (left column of
Fig. 6.5), while nonvanishing interaction strengths U > 0 induce isolated avoided crossings
(middle and right columns of Fig. 6.5). These, in general, cause deviations of the mean energy from the instantaneous energy levels, determined by the size of the avoided crossings
and the “angular velocity” F d/~ (Bloch frequency ωB ) in (6.4) – which sets the passage
time of the particle across the various avoided crossings, and thus the adiabatic to diabatic
transition ratios (see our discussion on Landau-Zener tunneling in Sec. 2.2.2).
6. Bloch oscillations of interacting fermions
E(t) / 2JN
72
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
−1
−1
−1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
υ(t) / υ
0
1
−1
−1
−1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
1
1
0.5
0.5
0.5
|Cn|
2
1
0
0
0.5
t/T
B
1
0
0
0.5
t/T
B
1
0
0
0.5
t/TB
1
Figure 6.6: Bloch oscillations of the mean atomic energy, Eq. (6.2) (upper row, solid blue line),
the mean atomic velocity, Eq. (6.3) (center row, solid blue line) and population |cn |2 of the
instantaneous eigenstates of H(t) (bottom row) of interacting fermions for N↑ = 1, N↓ = 2,
L = 4 and U = J. Adiabatic F d ≪ |U | (left), intermediate F d ∼ |U | (middle), and diabatic
F d ≫ |U | (right) static forcing. The evolution of the instantaneous spectrum of H(t) is shown
by the colored dashed lines in the upper row.
Let us now focus on the limit of weak, i.e. adiabatic, static forcing, which was not
discussed in much detail before. Due to the integrability of the model at any instant, see
Sec. 4.3), we expect to exist such F d ≪ U for any U, that dynamics remain adiabatic, i.e.,
the system assumed to be initially found in one of the eigenstates E(0) of the Hamiltonian
H(t = 0) has time to adopt to time variations H(t). Indeed, as illustrated in Fig. 6.5,
despite the dramatic changes in the instantaneous spectrum of a non-interacting system
with introduction of non-vanishing interactions, the mean energy calculated numerically
for sufficiently small static forcing closely follows the associated instantaneous level E(t).
Adiabatic evolution of the system assumed initially to be found in the ground state
strongly depends on the number of particles and the lattice size. It is known that, at half
filling, N↑ = N↓ = L/2, the instantaneous level associated to the ground energy level at
t = 0 is well separated from the rest of the spectrum, i.e., it does not cross any other
instantaneous level, for arbitrary non-vanishing interaction strengths (see, for example,
E(t) / 2JN
6.3 Instantaneous spectrum in (non-)adiabatic evolution
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
73
υ(t) / υ0
−1
−1
−1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
−1
−1
−1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
1
1
0.5
0.5
0.5
n
|C |
2
1
0
0
0.5
t/T
B
1
0
0
0.5
t/T
B
1
0
0
0.5
t/T
1
B
Figure 6.7:
Bloch oscillations of the mean atomic energy, Eq. (6.2) (upper row, solid blue
line), the mean atomic velocity, Eq. (6.3) (center row, solid blue line) and population |cn |2 of the
instantaneous eigenstates of H(t) (bottom row) of interacting fermions for L = 2N↑ = 2N↓ = 4
(half filling), and the identical parameters as in Fig. 6.6. Note that in the present case, due to
sufficient separation between single levels of the instantaneous spectrum of H(t) at half filling [93],
the evolution of the mean energy Eκ (t) has much smaller deviation from the single instantaneous
levels in the adiabatic regime than below (see Fig. 6.6) or above half-filling, i.e., for N↑ + N↓ > L
or N↑ + N↓ < L, respectively.
[94]). Furthermore, the level dynamics is periodic with a period τs = TB /(N↑ + N↓ ) =
TB /L [93] shorter than the Bloch period (see Fig. 6.5)!5 Thus, in addition to the limit
of strong static forcing with respect to the tunneling coupling F d ≫ J, the system also
exhibits regular behavior in the limit of weak (adiabatic) static forcing F d ≪ |U|, at half
filling. Note that, the minimal energy separation between the ground and the next, excited
instantaneous energy level is linearly proportional to the interaction constant U, for small
U ≪ J. Thus, the maximal Bloch frequency, ωB = F d/~, at which the time-evolution
of the mean energy is still adiabatic (i.e., where Landau-Zener tunneling is negligible),
5
With that, in particular, one can suggest a very beautiful experiment – to count the number of atoms
(or to identify half filling in the system) by detecting the short period τs of BO.
74
6. Bloch oscillations of interacting fermions
increases with increasing interaction strength U.6
Let us also shed some light on the case of moderate and of strong static forcing with
respect to the interaction strength F d & U, where BO, in general, decay (see Secs. 6.1,
6.2.2). In terms of the instantaneous spectrum, the decay of BO can be viewed as a net
diffusion of the energy with subsequent population of the nearest-neighbor instantaneous
energy levels due to tunneling, when these levels come close enough to each other. To
illustrate this, we plot the occupation probabilities |c2l (t)| of the instantaneous energy
levels for two different configurations of the system together with the mean energy and
velocity for N1 = 1, N2 = 2, L = 4 (Fig. 6.7), and for N1 = N2 = 2, L = 4 (Fig. 6.6).
It is evident that during the time-evolution under intermediate static forcing F d ∼ |U|
(middle columns in Fig. 6.7, 6.6) all instantaneous eigenstates become populated during
one Bloch cycle in agreement with the randomness of the FB matrix elements, pointed out
in Sec. 5.2.3.
Remember that for a static tilt F d ≫ U and F d ∼ J, the FB-operator does note exhibit
the simple asymptotic form (5.5), obtained in the limit F d ≫ J. In this situation, the above
interpretation of decaying BO when furnished with an accurate diffusive model may provide
us with an approach to find the decay rate of BO. Indeed for F d . U and electronic onedimensional systems driven by a time-dependent magnetic flux, the energy net transport
across the instantaneous spectrum (and also the emergence of a short period τs in the
dynamics of the lowest instantaneous energy level) was already discussed in [95, 96], and
a mapping of the associated diffusive regime on a quantum walk was suggested in [97].
6.4
Summary
In the framework of the FH-model, we have discussed the signatures of on-site interactions
among spinless, two-component fermions in fermionic BO excited by static forcing. As a
consequence of the quantum chaotic spectral properties of the FB-operator (see previous
Sec. 5), the corresponding BO of the system observables generally dephase, and decay
exponentially on time scales of several Bloch periods. Yet, we also identified few limiting
parameter regimes where the system exhibits regular oscillations, since its FB-operator
approaches an analytically given diagonal form. In the latter situation, the BO reveal a
variety of modifications: an interaction-induced periodic amplitude modulation, as well as
doubling of the Bloch period. We thus have reached a complete, general classification of
the different dynamical regimes. Our results are summarized in the following table.
6
To shorten the Bloch period may be necessary for those experiments with cold atoms where the
time-evolution of the system is accessible only over a finite time-interval.
6.4 Summary
75
Repulsive interaction: U > 0
Attractive interaction: U < 0
F ≪ U – regular BO (adiabatic evolution in the instantaneous picture)
reduction of the Bloch period to TB /N at half filling (N↑ = N↓ = L/2)
F < J – decaying BO (signature of the underlying spectral nonintegrability)
|U | ∼ J
F ≫ J – modulations with frequency Ω = ~/U
(diabatic evolution in the instantaneous picture)
Atoms with different polarizations
repel each other in configuration
space ⇒ Eff. single component
dynamics:
Atoms with different polarizations
form pairs in configuration space
at N↑ = N↓ ⇒ Eff. soft-core bosons:
F < J 2 /U – decaying BO
|U | ≫ J
F < |U | – non-decaying BO
J 2 /U < F < |U | – period-doubling
and amplitude modulation of BO
with frequency Ωeff = ~U/J 2 ;
maximal modulation
amplitude at half filling
F ≫ |U | – Quasiregular BO (Tunneling to higher subbands.)
Table 6.1: Classification of the dynamical regimes for a two-component fermionic gas in a
one-dimensional optical lattice under static forcing.
76
6. Bloch oscillations of interacting fermions
Chapter 7
Directed atomic current across an
optical lattice
In the preceding chapters, we have shown that a single particle, as well as an interacting
many particle system, exhibits Bloch oscillations as a dynamical response to static forcing,
unless the latter is sufficiently strong to induce Landau-Zener tunneling to higher bands.
This can be reformulated in terms of the single- or many-particle transport properties
across an optical lattice: for large magnitudes of the static force, F d > ∆, the particles
are ballistically accelerated, due to tunneling to higher bands, while for small magnitudes,
F d ≪ ∆, the system is adequately described using the single band approximation, which
predicts Bloch oscillations of the mean particle momentum around zero, with constant amplitude, in the single particle case, and, in general, with vanishing amplitude for interacting
particles. Thus, neither for single nor interacting many particle systems can one expect a
directed current across a lattice.1 In the present chapter of the resent thesis, we want to
find out how an atomic net current may emerge under static forcing, on time scales which
are large compared to the Bloch frequency.
Normal transport behavior, as a drift of carriers with constant velocity through a periodic potential, is one of the fundamental problems in solid state physics. Its solution relies
on a somewhat ambivalent ingredient: dissipation processes [1] are required to allow the
particles to monotonically loose their potential energy, in response to a static force. In
fact, this is the only reason why a normal metal becomes a good conductor. It is worth
noting that, without any environment, represented, e.g., by phonon modes and defects in
a crystal lattice, all carriers will be localized under static forcing, and will perform Bloch
oscillations.
Unfortunately, due to the high complexity of dissipation processes in bulk crystals,
the emergence of a directed current is hardly treatable within exact microscopic models,
1
To be formally correct, we have to comment on the relevant time scales in the case of interacting
atoms. The entire atomic sample may move over some distance along the direction of the static force,
during a short transient equilibration time (when the mean atom-atom interaction energy increases), in the
parameter regime of decaying Bloch oscillations (Sec. 6.3), leading to some finite drift velocity. However,
on larger time scales, the Bloch oscillations fade out, and so does any nonvanishing drift velocity.
78
7. Directed atomic current across an optical lattice
and a phenomenological approach is usually employed to introduce the relevant relaxation
mechanism [47].
Here, we devise a microscopical model with one dissipation channel for the non-interacting
carriers (fermions), realized by the coupling to a finite bath of another sort of interacting
particles (bosons). Although this model may serve only as the extreme simplification of real
electronic systems, it gives us a proper, accurate description of the response of ultracold
atoms to a static forcing in a one-dimensional optical lattice, in the presence of dissipation. With our approach we are able to express the relaxation time scale as a function of
the microscopic system parameters, and to identify the conditions for the appearance of a
non-vanishing drift velocity across the lattice [98].
The present chapter is organized as follows. In the first section, we introduce the model
ingredients and the total Hamiltonian, and discuss the properties of interacting bosons in
the framework of the Hubbard model. Then we set up the initial conditions for the carriers
and the environment, and perform numerical simulations for particular sets of parameters,
to demonstrate the role of initial conditions for the subsystem dynamics. Then, in the
subsequent section 7.3, we prove that an initially completely thermalized bath is sufficient
to use the Markov approximation to derive the master equation for the fermionic degree of
freedom, and to derive explicit expression for the relaxation time scales. Finally, in Sec. 7.4,
we extract the current-voltage characteristics – the drift velocity as a function of the static
tilt – from numerical simulations and compare this result with the phenomenological EsakiTsu prediction, which we parameterize with the relaxation time scales obtained from the
master equation approach.
7.1
Model ingredients
We consider a single fermionic2 particle on a lattice with L sites, and periodic boundary
conditions. The particle together with a bath of NB interacting bosons are loaded into the
same optical lattice. All particles interact with each other by means of s-wave collisions,
such that on the scale of the lattice constant only on-site interactions become relevant.
For simplicity, we assume that the static force acts only on the fermion, and that it does
not induce appreciable Landau-Zener tunneling, on the time scale of several Bloch cycles
(see Sec. 2.2.2, for details). The periodic potential should be deep enough to justify the
tight-binding model (for details, see Sec. 2.2.3).
Under these assumptions, our model Hamiltonian reads
Htot = HF + HB + Hint ,
with the single particle fermionic component,
!
L
L
X
JF X
HF = −
|l + 1ihl| + H.c. + F d
l|lihl|,
2
l=1
l=1
2
(7.1)
|L + 1i = |1i
(7.2)
We choose a fermion as carrier, since the single particle model is expected to provide proper results
also for a single component fermionic gas at sufficiently low occupation numbers, nF ≪ 1.
E(u)
7.1 Model ingredients
10
8
6
4
2
0
−2
−4
−6
0
0.2 0.4 0.6 0.8
79
1
10
8
6
4
2
0
−2
−4
−6
0
3
2.5
2
1.5
1
0.2 0.4 0.6 0.8
u
1
0.5
0.4
u
0.6
0.8
u
Figure 7.1: Evolution of the energy levels of the BH-Hamiltonian (7.3), under variation of the
parameter u = WB /(WB + JB ), with JB > 0. NB = 5 bosons on L = 4 lattice sites. Left: all
levels; middle: levels with quasimomentum κ = 1; right: zoom into the region marked by the
dashed red frame in the middle plot. Note the absence of any level crossing, and the presence of
a large number of anticrossings.
and its many particle, bosonic component
JB
HB = −
2
X
!
a†l+1 al + H.c.
l
+
WB X
nl (nl − 1) .
2 l
The on-site interaction between fermionic and bosonic component is given by
X
Hint = WF B
nl |lihl|,
(7.3)
(7.4)
l
where al (a†l ) represents the bosonic creation (annihilation) operator, n = a†l al the bosonic
number operator, JB the bosonic hopping matrix element, WB the bosonic on-site interaction constant, and WF B the on-site interaction energy between a fermion and a
boson. The dimension of the complete Hilbert space is N = LNLNB , where NLNB =
(L − 1 + NB )!/(L − 1)!/NB ! is the dimension of the bosonic subspace.
Our central goal is to identify the response of the fermion to a static force, in the
presence of the dissipation mechanism introduced by the coupling to the finite bath of
bosons. We are also interested in the role of the finiteness of the bath for the system
behavior. Therefore, we will address properties of the bosonic bath in the next section,
before analyzing the dynamics of the total system.
7.1.1
Bosonic bath
We have already discussed in much detail the standard Hubbard model for fermions. The
main formal difference between bosonic and fermionic models lies in the commutation relations between the associated creation and annihilation operators. In contrast to fermions,
80
7. Directed atomic current across an optical lattice
n(E)
0.2
90
0.1
n(E)
0
0.2
70
0
0.2
n(E)
E(u)
0.1
50
0.1
0
−10
0
10
E
20
30
40
1
I(s)
30
10
0
1
2
3
4
5
0
0
1
u
2
3
4
s
Figure 7.2: Evolution of the energy levels of the BH-model, with NB = 7 bosons on L = 9 lattice
sites, under variation of the dimensionless parameter u = WB /JB (left). One easily recognizes
the emergence of a band structure along the red solid lines given by the asymptotic dependence
P
Ei (u) = uJB j nj (nj − 1)/2 of the eigenvalues, merging into a bunch of highly degenerate levels
as u → ∞. The displayed integrated level spacing distributions (bottom right), together with the
density of states (top and middle right) at small u = 0.1, large u = 2, and intermediate u = 0.5
interaction strengths were sampled along the the green dash-dotted line, the green dashed line
and the black dashed line in the left plot, respectively.
the bosonic many particle wave-function is symmetric in respect to permutations of its
constituents. Therefore, indistinguishable bosons may occupy the same single particle
Wannier function. Such configurations with more than two particles on the same site of
the lattice render the application of the Bethe ansatz impossible [99], and the existence
of an exact (in the sense of integrability discussed in Chap. 4) solution for the bosonic
Hubbard (in short, BH-) model is therefore very unlikely.
As a numerical proof of the non-integrability of the BH-model, one can involve the complete absence of level crossings in the parametric energy level dynamics. Indeed, once all
discrete symmetries are factored out, levels within one irreducible subspace repel each other
(see Fig. 7.1), and the nearest-neighbor level spacing distribution of the BH-Hamiltonian
obeys GOE Wigner-Dyson statistics in a wide range of intermediate interaction strengths
(see Fig. 7.2)
I(s) = 1 − e−πs
2 /4
.
(7.5)
7.1 Model ingredients
81
Note that, by tuning the parameters of the BH-model, one can switch from the regime
of noninteracting particles at vanishing on-site interaction constant, |WB | ≪ JB , to the
intermediate case, |WB | ∼ JB , and to strongly interacting hard-core bosons, |WB | ≫ JB .
In the first case, the system description reduces to that of the single particle problem, and
in the third one it can be mapped on the model of non-interacting fermions. Therefore,
the BH-model does have an explicit analytical solution in the limits JB = 0 or WB = 0.
It is also important to note that, although the nearest neighbor level statistics coincides
well with the theoretical prediction in a wide range of intermediate values of the interaction
constant (Fig. 7.2), there are differences in the density of states: when the interaction
(b)
(t)
constant WB decreases below, or exceeds, some critical value, Wcr ≪ J or Wcr ≫ J,
respectively, we observe a density of states which consists of a number of bands, rather
then a localized distribution. These bands turns into highly degenerate levels in the limits
of vanishing tunneling JB = 0 or interaction strength WB = 0 (see Figs. 7.1 and 7.2).
7.1.2
Initial conditions
We assume that the fermion initially has a completely smeared out quasimomentum, and
that the bosonic bath is in a thermal superposition of its eigenstates, given by the Boltzmann distribution at initial temperature T :
L
1 X
|Ψ(0)i = |ΨB (0)i ⊗ |ΨF (0)i = √
L l=1
PNLNB −βE B /2 iφj
j
e |ψij
j=0 e
q
⊗ eiθl |li .
PNLNB −βE B
i
i=0 e
(7.6)
Here, β −1 = kB T sets the thermal energy scale; |ψii represent the eigenstates of the bosonic
subsystem, and φj and θl are uniformly distributed random phases on the unit circle, which
define the total quasimomentum of the bosonic subsystem, and the quasimomentum of a
single fermion, respectively.
The width of the bosonic spectrum ∆EB , and, thus, the heat capacity – the amount
of energy which can be transfered to the bosons – depends on the number of particles,
the tunneling matrix element JB , and the interaction constant WB . At small interaction
strengths WB . JB , the width of the spectrum is approximately given by that of the noninteracting bosons (WB = 0), i.e., by ∆EB ≃ 2JB ×NB . The heat capacity can be estimated
as the difference between the ground state energy (corresponding to zero temperature) and
P (B)
the average i Ei /NBL for a thermalized bath, at kB T ≫ ∆EB . Then, for the Gaussianlike symmetric density of state at WB . JB (see Fig. 7.2), the heat capacity is half of the
spectral width, ∆EB /2 = JB NB .
The initial temperature uniquely defines the initial mean energy of the bath, and the
unused heat capacity, at fixed parameters of the bosonic Hamiltonian. Figure 7.3 shows
that dependence for the same system as in Fig. 7.2, at interaction constant WB /JB = 3/7.
We also chose three different values of “low”, and one value of “high” (thermalized bath)
temperature, which will be used to set the initial conditions for our subsequent numerical
simulations, where we expect to observe the energy transfer from the fermionic degree of
82
7. Directed atomic current across an optical lattice
3
2
1
−1
−2
B
E (0)/J
B
0
−3
−4
−5
−6
−7 −1
10
0
10
1
kBT/JB
10
2
10
Figure 7.3: Mean bosonic bath energy, as a function of temperature: kB T = 0.71 × JB (blue
circle), 1.43×JB (red triangle), 2.86×JB (magenta square), 150×JB (black star), and at complete
∞ = E (k T → ∞) (red dashed line). The energy differences between the
thermalization EB
B B
latter and the former gives the bath heat capacity as a function of temperature, thus limiting the
maximal amount of energy which can be absorbed from the fermionic degree of freedom through
collisional on-site interactions (7.4).
freedom to the bath, mediated by on-site Fermi-Bose interactions. Due to thermalization
process, the bosonic subsystem should approach an equally weighted superposition of its
energy eigenstates (dashed line in Fig. 7.3).
7.2
Numerical experiment
We start with the discussion of our numerical results for the total system evolution, with the
bath prepared in qualitatively different initial states: at “low” and “high” temperatures.
In our opinion, it is very natural that the numerical experiment proceeds and motivates the
subsequent theoretical analysis. Similar to real experiments, numerical simulations help us
to build some intuition on a phenomenon, and to formulate the relevant questions.
We simulated the time evolution3 of the mean energy of the bosonic subsystem, and of
the mean velocity of the fermions for the bosons initially prepared in the thermal equilibrium at low, kB T = 2.86 × JB (blue circle in Fig. 7.3), and high, 150 × JB (black star in
Fig. 7.3) temperatures. The fermionic initial state is chosen in the form of a superposition
of Bloch waves with arbitrary quasimomenta, uniformly distributed over the first Brillouin
3
We use the same numerical procedure as in Chap. 6.
7.2 Numerical experiment
83
v(t)/v0
1
0
−1
0
2
2
EB (t)/JF
4
6
8
6
8
t/TB
1
0
−1
0
2
4
t/TB
Figure 7.4: Mean velocity of a single fermion (top) vs. mean energy of the bosonic bath (bottom).
Dashed lines correspond to an initially completely thermalized bath, when no energy transfer
from the fermionic degree to the bath is possible. While the particle-bath coupling induces the
decoherence of fermionic Bloch oscillations, there is no net current in the system (see dashed line
in the top panel). In contrast, when the bosonic bath is initially prepared at low temperatures,
such that it has a finite heat capacity, a non-vanishing drift velocity emerges (compare solid to
dashed lines). The dash-dotted line is obtained by direct time integration of the drift velocity,
multiplied by the magnitude of the static force. This yields the dependence of the fermionic
potential energy on time, the drop of which equals (up to a constant shift associated with the
difference of the initial and instant fermionic, kinetic energies) the work performed by the linear
potential to drag the fermion across the lattice, and can be associated with the heating of the
bath (compare solid and dash-dotted lines in the bottom panel).
zone, as specified in the previous section, see Eq. (7.6).
Inspection of our results displayed in Fig. 7.4 leads to the first observation that atomic
collisions cause an exponential decay of the mean fermionic velocity, with the same rate γ,
υ(t) ≃ υ0 e−γt sin(ωB t) ,
(7.7)
for both initial temperatures (upper plot in Fig. 7.4). The heating of the bath from its
lower initial temperature is clearly seen in the lower plot in Fig. 7.4 (solid line). Starting
from its initial value, the mean bosonic energy EB increases monotonically in time, to
its maximum value at EBtherm ≃ EBground + JB N. Since such behavior is observed over a
time scale much longer than the decay time ~/γ of the BO, it turns out to be the result
of a continuous energy transfer at the constant rate, resulting a finite drift velocity of
the fermion: the fermionic potential energy is transfered to the fermionic kinetic energy,
84
7. Directed atomic current across an optical lattice
then to the energy of the bosonic bath through the collisionally induced interactions of
the fermion with bosons. The drift velocity ῡ(t) = υ(t) − υ0 e−γt sin(ωB t) would ideally
saturate in the stationary regime tstat > ~/γ for an infinite bath, but for the finite bosonic
subsystem the fermionic drift velocities vanishes as soon as the bosonic subsystem is fully
thermalized, due to its finite heat capacity. Therefore, the associated system behavior will
be referred further as to that in the quasistationary regime. Note that the mean velocity
of the fermion decays to zero on the relaxation time scale ~γ −1 for the initially completely
thermalized bosonic bath, see the dashed line in the upper plot of Fig. 7.4).
To validate the complete transfer of fermionic potential energy to heat energy of the
bath, in the quasistationary regime, one can compare
the loss of the fermionic potential
Rt
(−)
energy, given by the integral EF (t) = F x̄(t) = 0 ῡ(t)dt, with the energy EB (t) − EB (0)
gained by the bosons over the same period of time. It turns out that these energies (up to
a constant shift associated with the difference of the initial and instant fermionic, kinetic
(−)
energies) perfectly coincide with each other, at t > ~/γ (dash-dotted line, EF (t) + EB (0),
and solid line, EB (t), in Fig. 7.4). Thus, one can extract the drift velocity the fermionic ῡ
not only from mean velocity, but also, indirectly, from the time derivative of the bosonic
energy EB (t) (see Sec. 7.4.1).
The relaxation constant γ characterizes the decay of BO due to interaction-induced
decoherence, on the one hand, and controls the rate of energy transfer from the fermion to
the bath, on the other. Thus, finding the functional dependence of γ on the microscopic
parameters of the total Hamiltonian becomes ultimate goal. Together with the knowledge
of the heat capacity of the bath, it will provide the complete information to predict the
dynamics of the fermionic subsystem.
In the next section, we will discuss conditions that allow us to integrate out the environment, within a standard perturbative approach, and to formulate a master equation for
the fermionic degree of freedom, where decoherence due to particle-bath interaction will
effectively enter through the single parameter γ.
7.3
Master equation approach
Here we present the complete derivation of the master equation for the single particle
density matrix of the fermion, by means of the standard perturbative approach [60]. Although we shall follow a standard procedure, an important distinction of our case with
respect to others will arise at the final step, due to the local on-site interactions between
the subsystems and the environment formed by a finite ensemble of locally interacting
particles.
Let ρ(t) be the density matrix of entire system. Then the associated evolution equation
reads:
dρ(t)
i
= − [Htot , ρ(t)] .
(7.8)
dt
~
In the interaction representation with respect to the sum of the unperturbed Bose and
7.3 Master equation approach
85
Fermi subsystems,
ρ̃(t) = ei(HB +HF )t/~ρ(t)e−i(HB +HF )t/~,
(7.9)
the density matrix equation becomes
dρ̃(t)
i
= − [Hint (t), ρ̃(t)]
dt
~
(7.10)
with
Hint (t) = ei(HB +HF )t/~Hint e−i(HB +HF )t/~ = WF B
X
l(t)nl (t) ,
(7.11)
l
where
l(t) = eiHF t/~|l >< l|e−iHF t/~, nl (t) = eiHB t/~nl e−iHB t/~ .
P
(7.12)
To simplify further analysis, we extract the mean coupling energy WF B TrB { l l(t)nl (t)ρ̃(t)} =
nWF B from the Hint (t) (by adding the same constant energy to the fermionic Hamiltonian),
and deal only with fluctuations of the bosonic particle occupation number around its average value
X
X
l(t)(nl (t) − n) = WF B
Hint (t) = WF B
l(t)∆n(t) .
(7.13)
l
l
Let us proceed further by integrating equation (7.10) between t and ∆t
i
ρ̃(t + ∆t) = ρ̃(t) −
~
Z
t+∆t
dt′ [Hint (t′ ), ρ̃(t′ )] .
(7.14)
t
Iteration of (7.14) results in the following expansion
Z
i t+∆t ′
ρ̃(t + ∆t) − ρ̃(t) = −
dt [Hint (t′ ), ρ̃(t)]
~ t
2 Z t+∆t Z t′
i
dt′
dt′′ [Hint (t′ ), [Hint (t′′ ), ρ̃(t′′ )]]
−
~
t
" t
3 #
i∆tWF B
+O
.
~
(7.15)
By tracing over the bosonic subsystem in (7.15), we obtain for the first term on the right
hand:
Z t+∆t
Z t+∆t X
′
′
TrB
dt [Hint (t ), ρ̃(t)] = WF B
dt′
[l(t′ ), ρ̃F (t)TrB {∆nl (t′ )ρ̃B (t)}] ,
t
t
l
(7.16)
where we employed the standard assumption of factorizability of the total density matrix
into a product of the density matrices of the system constituents ρ̃(t) = ρ̃F (t) ⊗ ρ̃B (t). This
requires that the correlations between the subsystems at time t′ decay on the short time
scale τc ≪ ∆t, and do not contribute to the evolution of ρ̃F (t) over the interval [t′ , t′ + ∆t]
86
7. Directed atomic current across an optical lattice
(see coarse graining condition (7.23) for the time-scale separation below) [60]. For a large
size of the bath, and a small coupling constant between the subsystems, the density matrix
of the bosons evolves slower than that of the fermion.4 Therefore, we can assume that
ρ̃B (t) ∼
= ρ̃B (t + ∆t) , and [ρ̃B (t), HB ] = 0 .
(7.17)
This, together with a delocalized initial state of the fermion in the form of an arbitrary
superposition of Bloch waves, makes the first term (7.16) negligibly small.
For the second term in the expansion (7.15), we have
(Z
)
Z ′
t+∆t
t
= TrB
=
t
dt′
TrB
(Z
WF2 B
dt′′ [Hint (t′ ), [Hint (t′′ ), ρ̃(t′′ )]]
t
t+∆t
′
dt
t
XZ
l,m
Z
t+∆t
t′
t
′
dt
t
dt′′ [Hint (t′ )Hint (t′′ )ρ̃(t′′ ) − Hint (t′′ )ρ̃(t′′ )Hint (t′ )
Z
− Hint (t′ )ρ̃(t′′ )Hint (t′′ ) + ρ̃(t′′ )Hint (t′′ )Hint (t′ )]}
t′
t
dt′′ [Rl,m (t′ , t′′ ) (l(t′ )m(t′′ )ρ̃F (t′′ ) − m(t′′ )ρ̃F (t′′ )l(t′ ))
− Rm,l (t′′ , t′ ) (l(t′ )ρ̃F (t′′ )m(t′′ ) − ρ̃F (t′′ )m(t′′ )l(t′ ))] ,
(7.18)
where we have introduced the time dependent, bosonic density-density correlation function
Rl,m (t′ , t′′ ) = TrB {∆nl (t′ )∆nm (t′′ )ρ̃B (t′′ )} .
(7.19)
As it follow from the definition of the operator nl (t)
′
′
′′
′′
Rl,m (t′ , t′′ ) = TrB {eiHB t /~∆nl eiHB (t −t )/~∆nm e−iHB t /~ρ̃B (t′′ )}
= TrB {∆nl (t′ − t′′ )∆nm (0)ρ̃B (t′′ )} = Rl,m (t′ − t′′ ) .
P
Then, for a thermal superposition of bath eigenstates i ci (kB T )|ψi i, we have
X
Rl,m (t′ − t′′ ) =
|ci (kB T )|2 |ψi ihψi |∆nl (t′ − t′′ )∆nm (0)ρ̃B (t′′ )
(7.20)
i
=
X
i,j
′
′′ )/~
|ci (kB T )|2 hψi |∆nl |ψj ihψj |∆nm |ψi iei(Ei −Ej )(t −t
.
(7.21)
For a large number of non-zero coefficients ci (kB T ) in this superposition, as in the case of a
thermal Boltzmann distribution at finite temperature T , and for an irregular spectrum Ei
with a vanishing number of degeneracies, as in the case of the bosonic bath with WignerDyson level spacing distribution, it is very natural to expect that the above correlation
4
Consistently with the numerical simulation in the previous section.
7.3 Master equation approach
87
functions have short decorrelation times τ (which we assume to be the same for any l and
m, since they all are given by the sum over the same set of frequencies (Ei − Ej )/~), such
that they can be approximated by a delta function
′
t − t′′
′
′′
Rl,m (t − t ) ≈ Rl,m δ
,
(7.22)
τ
and τ satisfies the usual, coarse graining condition for the time-scale separation of system
and bath time-evolution,
τ ≪ ∆t ≪ τBath ,
(7.23)
where τBath is the characteristic time scale on which the environment increases its temperature, due to its collisional coupling to the fermionic subsystem.
Substitution of the correlation function (7.22) into (7.18) gives
ρ̃F (t + ∆t) − ρ̃F (t) =
Z t+∆t
WF2 B X
τ Rl,m
dt′ (l(t′ )m(t′ )ρ̃F (t′ ) − m(t′ )ρ̃F (t′ )l(t′ )) −
2 l,m
t
Z
t+∆t
WF2 B X ∗
τ Rl,m
dt′ (m(t′ )ρ̃F (t′ )l(t′ ) − ρ̃F (t′ )l(t′ )m(t′ ))
2 l,m
t
(7.24)
(7.25)
∗
where we used the relation Rl,m (−t) = Rl,m
(t), which directly follows from the expression
(7.21).
After taking the derivative and recollecting the terms we end up with
W2 X
dρ̃F (t)
τ [Rl,m l(t)m(t)ρ̃F (t)
= − F2B
dt
2~ l,m
∗
−2Re(Rl,m )m(t)ρ̃F (t)l(t) + Rl,m
ρ̃F (t)l(t)m(t)
(7.26)
Hence, the evolution of the matrix elements ρ̃F i,j (t) = hi(t)|ρ̃F (t)|j(t)i in the interaction
representation satisfies
∂ ρ̃F i,j (t)
= −γi,j ρ̃F i,j (t) ,
(7.27)
∂t
while in the time independent Fock basis |i >= e−iHF t/~|i(t) >,
i
∂ρF i,j (t)
= − [HF (t), ρF i,j (t)] − γi,j ρF i,j (t) ,
∂t
~
(7.28)
where the relaxation constants are expressed through bosonic correlation functions
γi,j =
WF2 B
ζi,j (0)τ ,
~2
(7.29)
88
7. Directed atomic current across an optical lattice
where
ζi,j (t) = Re [Ri,i (t) − Ri,j (t)]
X
=
|cn (kB T )|2 hψn |∆ni |ψm ihψm |∆ni − ∆nj |ψn i cos [t(En − Em )/~] . (7.30)
n,m
As follows from (7.28) the off-diagonal elements of the fermionic density matrix decay
exponentially fast to zero, in agreement with a preliminary numerical simulation performed
in Sec. 7.2, where we had observed decaying Bloch oscillations of the mean fermionic
velocity.
In section 7.3.2, we will discuss solutions of the master equation (7.28), while in the next
section we will first focus on finding the correlation function amplitudes (7.30), together
with the associated decorrelation time scales.
7.3.1
Bosonic density-density correlation functions
As follows from the definition (7.30), the correlation functions cannot decay on time scales
shorter than the inverse of the largest energy difference in the bosonic spectrum, i.e.,
τ > ~/(NJB ) = τB /N. The exact values at different system parameters depend on the
initial distribution of the level populations |c(kT )i |2 , and on the density of states. In our
approach we are able to extract the latter from our numerical simulations. Together with
the initial amplitudes ζi,j (0), the decorrelation time τ gives us the necessary information
about the bosonic subsystem to define the relaxation constants γi,j .
The decorrelation time τ and the amplitudes ζi,j (0) can be related by means of the
following integration:
Z t
Z ∞ t
2
′
′
τ=
δ
dt ≃ τ (t ≫ τi,j ) =
Re
(7.31)
ζi,j (t )dt .
τ
ζi,j (0)
−∞
0
Let us analyze the results for the numerically evaluated above integral, shown in Fig. 7.5:
There we plot the correlation function ζi,i±1(t), and the decorrelation time scale τ (t), at
four different temperatures introduced through the Boltzmann distribution (7.6): kB T =
0.71 × JB (blue lines), kB T = 0.143 × JB (magenta lines), kB T = 2.86 × JB (red lines),
and kB T = 150 × JB (black lines)(as in Fig. 7.3), and at three different sets of the bosonic
parameters: WB /JB = 0.4, 0.7, 1 (from left to right).
Remarkably, the monitored quantities behave qualitatively equally in all cases: the
correlation functions decay to zero, and the corresponding decorrelation times approach
approximately the same constant value, for all three values of the interaction constant and
of the lower temperatures, except the case of completely thermalized bosons at kB T =
150 × JB (black lines).
Since the corresponding decorrelation time is inversely proportional to the width of the
spectrum, and a finite temperature effectively reduces the part of the spectrum involved,
one may expect τ (t) to be larger at lower temperatures. This would be certainly true for a
ζi, i± 1 (t) / ζi, i± 1(0)
7.3 Master equation approach
89
1
1
1
0.75
0.75
0.75
0.5
0.5
0.5
0.25
0.25
0.25
0
0
0
−0.25
−0.25
−0.25
τi, i± 1 (t) / τB
−0.5
0
2
4
6
8 10
−0.5
0
2
4
6
8 10
−0.5
0
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
0
2
4
6
t / τB
8 10
0
0
2
4
6
t / τB
8 10
0
0
2
4
6
8 10
2
t / τB
4
6
8 10
Figure 7.5: Top: Bosonic correlation functions ζi,i±1 (t), normalized to their maximum value
(7.39), at four different temperatures chosen as in Fig. 7.3. Bottom: Decorrelation time scales
τi,i±1 (t) deduced from the integral (7.31) over the finite time interval [0, t]. The interaction
constant was varied in the range WB /JB = 0.4, 0.7, 1 (from left to right).
homogeneous density of states, but in the present case of the BH-Hamiltonian the density
of states is localized, in the form of a Gaussian. Moreover, well separated bands emerge
one after the other, with increasing interaction strength (see Fig. 7.2). This latter effective
discreteness of the spectrum may cause the those frequencies, given by the band separation
to dominate among in (7.30), at equal population of bosonic energy levels. Therefore,
we tend to attribute the divergence of the decorrelation time at complete thermalization
(Fig. 7.5, middle and right rows of panels) to the emergence of the subband structure, at
large interaction strengths.
It turns out that also the amplitudes of the correlation functions ζi,i±1(0) are systematically smaller for lower temperatures, at WB < JB . Therefore, the relaxation time scale
γ will exhibits the same temperature dependence, i.e., the decay “constant” will increase
during the time evolution, due to the thermalization of the bath.
The amplitudes ζi,j (0) in the high temperature limit can be calculated analytically, for
an arbitrary number of bosons N, on a lattice of arbitrary length L. Indeed, since we
90
7. Directed atomic current across an optical lattice
assume that the density matrix of the bosonic subsystem is given by the thermal state
−Ei β
δi,j
δi,j e
= NB ,
(7.32)
hi|ρB |ji = P NB
NL
N
−E
β
m
L
1
m=1 e
≫JB N
β
where the indices i, j label the eigenstates of the bosonic subsystem, and NLNB = (L −
1 + NB )!/(L − 1)!/NB !, ρB is represented by the identity matrix in the “high-temperature”
limit kB T ≫ JB N, in any complete orthonormal basis. Hence,
(k T ≫JB N/2)
Ri,j := Ri,jB
(t = 0) = Tr{ni nj } − n̄2
NB
NL
1 X
δl,m {ni }l,m{nj }l,m − n̄2 .
=
NB
NL l,m=1
(7.33)
It is most appropriate to perform calculations in Wannier basis, where the bosonic particle
number operators are represented by the diagonal matrices {ni }l,m δl,m (the lth element on
the main diagonal of the operator ni is given by the number of bosons on the ith site for
the lth basis state).
With the following summation rules
NB
X
n=0
N
B
X
n=0
N
B
X
n=0
n
NL−1
= NLNB ,
(7.34)
NB −1
n
nNL−1
= (L − 1)NL+1
,
(7.35)
NB −1
NB −2
n
n2 NL−1
= (L − 1)NL+1
+ L(L − 1)NL+2
,
(7.36)
and some ordinary combinatorics we obtain:
Ri,i
Ri,j
N
1 X
1 − 1/L
NB
,
≡ h∆n i = NB
(N − n)2 NL−1
− n̄2 = n̄(n̄ + 1)
1 + 1/L
NL n=0
2
N
1 X
1/L
NB
=
(N − n)(n − k)NL−2
− n̄2 = −n̄(n̄ + 1)
,
NB
1 + 1/L
NL n=0
Using the definition of ζi,j (t) (7.30), we have:
0,
for
ζi,j (0) =
n̄(n̄ + 1)/(1 + 1/L) , for
i=j
i 6= j .
(7.37)
i 6= j.
(7.38)
(7.39)
Then, finally, the relaxation constants (7.29) take the form:
γi,j = γ (1 − δi,j )
(7.40)
v(t)/v0
7.3 Master equation approach
1
0
v(t)/v0
−1
0
1
v(t)/v0
91
2
4
6
8
2
4
6
8
4
6
8
0
−1
0
1
0
−1
0
2
t/TB
Figure 7.6:
Normalized mean velocity of a single fermion in units of υ0 = JF d/~, obtained
numerically for
√ F d = 0.57 × JF , at three different values of the interaction constant WF B =
0.1 × JF , 0.1 2 × JF , 0.2 × JF (from top to bottom). The relaxation time ~/γ predicted by
Eq. (7.41) (vertical dashed lines) perfectly captures the decay of Bloch oscillations in agreement
with (7.43).
with
τ WF2 B n̄(n̄ + 1)
γ=
,
~2 1 + 1/L
(7.41)
now explicitly expressed through the experimentally accessible, microscopic model parameters. Note that the finite size effect included in (7.41) by the dependence on the finite
number L of lattice sites vanishes with increasing L as 1/L.
7.3.2
Interaction-induced decoherence
As follows from (7.28), atomic collisions introduced by the effective on-site interaction
induce an exponential decay of the off-diagonal elements of the fermionic single particle
density matrix to zero. Since the mean fermionic velocity according to (3.8a) is given by
"
#
"
#
L
X
X (F )
υ(t) = υ0 Im hψF (t)|
|lihl + 1|eiωB t |ψF (t)i = υ0 Im
(7.42)
ρl,l+1 (t)eiωB t ,
l=1
l
with υ0 = JF d/~, it will decay exponentially on the same time scale:
υ(t) = υ0 e−γt sin(ωB t) .
(7.43)
92
7. Directed atomic current across an optical lattice
Note the agreement with the direct numerical simulations presented in Fig. 7.6, for different
values of the interaction constant WF B , where the decorrelation time τ ≃ 2τB , for a
completely thermalized bosonic bath and WB /JB = 3/7, is very close to τ̃ = 1.46τB –
the value extracted numerically from the simulations shown in Fig. 7.6, with a best fit to
Eq. (7.41).
Taking into account the coupling between the fluctuating off-diagonal elements ρF i,j (t),
i 6= j, and elements from the main diagonal ρF i,i (t) in the quasistationary regime t ≫ 1/γ,
one can analytically derive a normal diffusion across the lattice [100],
h∆x2 (t)i ∼ Dt ,
(7.44)
γ
.
+ γ2
(7.45)
with the diffusion coefficient
D = υ02
ωB2
However, since we assumed the initial fermionic wave function to be given by a random
superposition of Bloch waves, which are delocalized over the whole lattice, normal diffusion
is not evident from our simulations.5
7.4
Beyond the Markov approximation
In the above, we derive the appropriate quantitative description of the collisionally induced
decoherence of Bloch oscillations in the framework of the short decorrelation times, the
realm of the Markov approximation (see, e.g., [60]). The obtained master equation (7.28)
remains valid at any time, for the completely thermalized bath, however, for the bath at
low temperatures, it neglects the energy transfer from the fermionic degree of freedom to
the bath (due to the assumption of the quasistationarity of the initial state of the bosonic
environment (see Sec. 7.3)) and time-dependence (through increase of the bosonic bath
temperature) of the relaxation constant, on large time scales t > τBath . According to the
master equation (7.28), independently of the initial temperature, the mean atomic velocity
decays exponentially to zero over finite transient times ∼ 1/γ, and no nonvanishing drift
emerges at the stationary regime t ≫ 1/γ. Thus, the standard Markov approximation fails
to describe fermionic transport.
Formally, to incorporate the drift velocity and the finite heat capacity of the bosonic
bath, one can introduce a time dependent, imaginary shift of the off-diagonal elements of
the density matrix in Wannier basis (see Eq. (7.42)): ρl,m 7→ ρl,m±1 − i exp(−ξt)∆ρl,m ,
l 6= m, on the right-hand side of the master equation (7.28) with ξ > 0, which would
cause the fermionic drift velocity to vanish at t ≫ 1/ξ accordingly to finiteness of the heat
capacity of the environment. Also a small variation of the relaxation constants with time
can be phenomenologically included: γ 7→ γ(t). Although such modification of the master
equation turns out to be quite logical from the phenomenological point of view, a correct
and complete derivation from first principles remains an open problem.
5
Note that for the initially localized wave packet, the above expression predicts a suppression of the
diffusion with increasing Bloch frequency!
7.4 Beyond the Markov approximation
93
3
2
1
0
−1
EB(t) /JF
0.2
−2
B
E (t) /J
F
0.4
−3
0
0.2
−4
0.4
−5
−0.6
0
200
−6
−7
0
1000
2000
3000
4000
400
t /τB
5000
6000
600
7000
800
8000
t /τB
Figure 7.7: The mean energy of the bosonic subsystem EB (t), for forty initial state realizations
with different random quasimomentum of the single fermion, and at three different temperatures
of the bosonic bath (blue solid lines). Dashed vertical lines indicate the mean energy of the
∞ (red). The time
bath with zero temperature (green) and of the completely thermalized bath EB
∞ averaged over many realizations (not shown) is
dependence of the energy difference EB (t) − EB
successfully fitted by an exponential function, Eq. (7.46) (magenta solid lines).
In this section, we will describe the emergence of a directed current within a systematic
numerical approach beyond the Markov approximation. We shall prepare the bosonic
environment initially at low temperatures, and then monitor the heating process until
complete thermalization. The main part of the discussion will concern the dependence
of the fermionic drift velocity on the magnitude of the static force, and on the initial
temperature of the bath. In particular, we shall numerically extract those values of the
rate ξ(F d), which are required for the phenomenologically extended form of the master
equation proposed above.
7.4.1
Thermalization of the environment
The initial temperature (which enters as β = kB T ) completely defines the initial mean
energy EB (t = 0) of the bosonic subsystem, but the time evolution of EB (t) during the
transient time ~γ −1 , and equally so its time derivative, strongly depend on the particular
94
7. Directed atomic current across an optical lattice
−4
5
x 10
4
ξ τB
3
2
1
0
0
0.2
0.4
0.6
0.8
Fd /JF
Figure 7.8: The decrement ξ (dots with standard deviation bars) of the energy difference (EB (t)−
∞ ), Eq. (7.46), as a function of the static tilt, at four different initial temperatures of the bosonic
EB
bath: kB T = 0 (green), kB T = 0.71 × JF (blue), 1.43 × JF (red) and 2.86 × JF (magenta). Note
that ξ shows universal, temperature independent behavior.
choice of the random phases φj and θl , although further propagation of EB (t) at times
t > ~γ −1 gives almost the same result (see Fig. 7.7); independent of the particular realization. To overcome this transient effect, we average the characteristic quantities, which we
extract from the time dependence EB (t), over a large set of K > 10 different initial state
realizations, e. g., the results shown in Figs. 7.7-7.9 were obtained for K = 40.
Since the fermionic drift velocity is directly related to the rate of monotonic energy
transfer to the bosonic subsystem (see discussion in Sec. 7.2), we can extract its value by
a linear fit to the time dependence of the bosonic energy EB (t). In fact, due to the finite
size of the lattice (here L = 9), the bosonic bath has a finite heat capacity. Thus, the
bosonic energy EB (t) approaches its maximum energy value EB∞ = EB (kB T → ∞) rather
exponentially
EB (t) − EB∞ = (EB (0) − EB∞ ) exp(−ξt)
(7.46)
than linearly, on sufficiently large time scales (see Fig. 7.7). This exponential behavior
becomes more noticeable at larger amplitudes of the static force, corresponding to Bloch
frequencies ωB > γ, at which the decrement ξ reaches its plateau with maximum values
(see Fig. 7.8 for details).
We can conclude from analysis of Fig. 7.8 that the long time behavior of the bosonic
mean energy EB (t) can be well interpolated by the exponential function. The decrement
7.4 Beyond the Markov approximation
95
ξ depends strongly on the magnitude of the static force, but is practically independent on
the initial temperature of the bosonic bath. It takes very close values at three different
low temperatures kB T = 0.71, 1.43, 2.86 × JF .6 This, in fact, justifies that the state of
the bath can be well approximated by a Boltzmann distribution at each moment of time,
and that the bath time scale τBath as the largest time scale can be used to define the
quasistationary regime at low temperatures (see Sec. 7.2). Furthermore, since the fermionic
drift velocity is proportional to the time derivative of the mean bosonic bath energy (7.46),
it exponentially decays in time, while its initial amplitude scales linearly with the initial
difference EB (0) − EB∞ determined by the initial temperature of the bath (for EB (0) as a
function of the bath temperature, see Fig. 7.3).
7.4.2
Current-voltage characteristics
Let us analyze the “current-voltage” characteristics of the atomic current, i.e, the dependence of the drift velocity on the magnitude of the static force. The result of our numerical
calculations obtained at three different temperatures is shown in Fig. 7.9: All curves exhibit a maximum precisely when the Bloch frequency ωB = F d/~ equals the decay rate γ of
Bloch oscillations evaluated at the particular temperature. The maximal achievable current
under variation of the static tilt decreases with increasing temperature, due to thermalization of the bosonic bath, and slightly shifts to larger values: ~γ = 0.02, 0.023, 0.027 × JF
at temperatures kB T = 0.71, 1.43, 2.86 × JF .
Such behavior, with vanishing current at small, F d/~ ≪ γ, and large F d/~ ≫ γ,
magnitudes of the static force is a result of the interplay between Bloch oscillations and
collision-induced relaxation processes: high fermion-boson collision rate in comparison to
the Bloch frequency induce diffusive transport, and a large Bloch frequency with respect
to the collision rate reestablishes Bloch oscillations. Thus, in both cases, directed current
is suppressed.
Phenomenological Esaki-Tsu model
More quantitatively, the “current-voltage” characteristics can be derived in the framework
of the phenomenological model first introduced by L. Esaki and R. Tsu, to describe the
transport of electrons under static forcing in a semiconductor superlattice [47]. Starting
from
κ̇ = F/~ , υ = ~−1 ∂ǫ(κ)/∂κ ,
(7.47)
the velocity increment over a time interval dt is
dυ = F ~−2 (∂ 2 ǫ(κ)/∂κ2 )dt .
6
(7.48)
Notable deviations of the decrement ξ(F d) at zero initial temperature of the bath from its behavior
at finite temperatures (see Fig. 7.8) may be attribute to finiteness of the bosonic bath and discreteness of
its energy spectrum resolved by the static tilt, which, resonantly to transition on Bloch frequency, couples
the ground energy level with a few excited states.
96
7. Directed atomic current across an optical lattice
0.3
0.3
0.25
0.25
υ/υ
0
0.2
0.2
0.1
0
υ/υ
0.15
0.15
0.05
0
0
0.1
0.01
0.02
0.03
0.04
Fd /J
F
0.05
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fd /J
F
Figure 7.9: “Current-voltage” characteristics – the dependence of the fermionic drift velocity
on the static tilt, at different temperatures of the bosonic bath (chosen temperatures as in the
previous figures). The numerical simulation (dots with standard deviation bars) is compared to
the phenomenological Esaki-Tsu prediction (7.49) with a relaxation constant γ obtained from
Eq. (7.41) and a maximum amplitude max(υ) extracted numerically (dashed lines) or fixed at
the value max(υ) = υ0 /8 defined by the original Eq. (7.49) (dashed-dotted line). The inset zooms
into the low-F d range of the figure.
The dispersion relation of the tight-binding approximation reads ǫ(κ) = −JF cos(κd), and
we can furthermore introduce a phenomenological scattering rate γ̃. Then the drift velocity
in the stationary regime t ≫ ~/γ̃ is given by
Z ∞
Z ∞
υ0
ωB /γ̃
−2
(∂ 2 ǫ(κ)/∂κ2 ) exp(−tγ̃)dt =
υd =
exp(−tγ̃)dv = F ~
. (7.49)
4 1 + (ωB /γ̃)2
0
0
The above equation (7.49) expresses the fact that the maximum drift velocity emerges
at the border-line between the linear response regime (“Ohmic” behavior) and the regime
of negative differential conductivity is completely determined by the scattering rate γ̃,
under changes of the static force, or, correspondently, of the Bloch frequency ωB .
In our case, the unknown phenomenological constant γ̃ is replaced by the fermionboson collision-rate γ, fully controlled by the microscopic model parameters, according to
Eq. (7.41). Moreover, our calculations based on the dynamics of a very finite number of
atoms result in the same optimal potential gradient to achieve the maximal current. The
functional dependence (7.49), with γ̃ replaced by the expression (7.41), is represented by
the dash-dotted line in Fig. 7.9. The only difference between this phenomenological model
and our present results lies in the prefactor, which fixes the absolute value of the drift
7.5 Summary
97
velocity, and depends on the temperature of the bosonic subsystem: it is maximal at zero
temperature, and decreases linearly with the energy difference EB (t) − EB (∞) between the
initial mean energy of the bosonic bath and its value at complete thermalization.
Such lowering of the drift velocity with increasing bath temperature is consistent with
the analysis of a similar model [101], where a single particle in a tilted periodic potential
is coupled to an Ohmic bath. In the latter case, the coupling to the bath was modeled
with an imaginary time functional approach, by a non-local, dissipative term included in
the effective classical action.
Recently, the Esaki-Tsu dependence of the net current of non-interacting spinless electrons as a function of the static forcing (magnetic flux for the electrons) was shown to
emerge in various electronic ring chains due to a local coupling to a particle reservoir
modeled by a lead attached to one of lattice sites [102]. Although, the origin of the transition from Ohmic to negative differential conductivity in these systems was qualitatively
explained in the framework of the non-equilibrium Green’s function formalism, the functional form of the current-voltage characteristics was not obtained.7
7.5
Summary
In the present chapter, we analyzed the dynamics of a dilute (i.e., n̄ ≪ 1), non-interacting
gas of particles (e.g., spin polarized fermions) under static forcing in a finite, one-dimensional
periodic optical lattice. We have shown that the collisionally induced interactions with an
admixture of a finite number of mutually interacting Bose particles loaded into the same
optical lattice cause the decoherence of the fermionic Bloch oscillations, and may induce a
directed fermionic current across the optical lattice.
Assuming deep optical lattices, we used the tight-binding approximation for the noninteracting fermions, and the BH-model for the finite bosonic environment, respectively. A
master equation formulation was applied to describe the collisionally induced decoherence
process of the fermionic Bloch oscillations. With that, the relevant time scales such as the
decorrelation time of the bosonic environment τ and the decay constant of Bloch oscillations
γ were obtained. Both were found to be independent on magnitude of the static forcing.
The Markov approach to derive the master equation is shown to be valid for times larger
than the decorrelation time of the bosonic density-density correlations, for an initially fully
thermalized bosonic bath, as well as for finite temperatures of the environment, but over
times which are much smaller than the characteristic time scales of the bath response due
to the heating process: 2π/ξ ≫ t > τ . Solutions of the master equation capture well
the decoherence of Bloch oscillations obtained by exact numerical simulations, but since
the initial state of the bosonic bath is assumed to be unchangeable within the Markov
approximation, and, thus, no energy transfer from the fermions to the bath is possible, the
drift velocity at finite temperatures was a priori excluded. Nevertheless a non-vanishing
directed fermionic current observed numerically at finite temperatures can be incorporated
7
Occasionally, the phenomenological Esaki-Tsu (7.49) prediction was not mentioned in [102] at all.
98
7. Directed atomic current across an optical lattice
by appropriate changes of the obtained equations for the off-diagonal density matrix elements, however the explicit derivation of such a modified master equation remains an open
problem.
Our numerical simulations revealed that the conditions for the appearance of a directed
fermionic current and its visibility depend on the temperature, the density of state, and
the size of the environment. These parameters completely determine the amount of the
total energy which can be absorbed by the bath from the fermionic degree of freedom, and
also the dissipation rate ξ of the total fermionic energy. The amplitude of the fermionic
current decreases, with increasing temperature, for a Gaussian density of state of the
environment, and vanishes as soon as the bosonic bath is completely thermalized. The
usage of temperature for the characterization of the bosonic energy is, indeed, approved,
since even for a bath made of NB = 7 bosonic particles in L = 9 lattice sites, the population
of the bath energy levels remains perfectly described by the Boltzmann distribution, during
the entire thermalization process.
The current amplitude passes through a global maximum for a static forcing which
makes the Bloch frequency to match the collision rate of the fermions with the bosons.
Below this value, the current exhibits linear Ohmic behavior, while above that value the
current is inversely proportional to the magnitude of the static force. Such current-voltage
characteristics emerge in good agreement with the phenomenological Esaki-Tsu prediction,
which accounts for the interplay between Bloch oscillations, and a dissipation introduced
by hand, through the relaxation constant γ. In contrast to the phenomenological approach,
we were able to express the dissipation rate through the microscopic model parameters.
Apart from that, due to the finiteness of the bosonic bath, we observe a strong temperature
dependence of the maximal current, and a weak temperature dependence of the decay
constant.
In all examples studied in the previous sections, the crossover from Ohmic to negative differential conductivity occurs at small magnitudes of the static force (F d . JF ).
This ensures that Landau-Zener tunneling remains negligible, and, together with relaxation constant fully controllable by the microscopic model parameters, makes a possible
experimental observation of negative differential conductivity with cold atoms in optical
lattices more accessible and flexible than that with electrons in solid state materials [103].
7.5 Summary
99
Conclusions, remarks and new open problems
In the present thesis, we analyzed how the presence of non-vanishing interactions in cold
Fermi gases or in Fermi-Bose mixtures modifies the dynamical response of non-interacting
fermions to static forcing.
In chapter 3, we recalled (and generalized for an additional parabolic confinement) the
single particle results, which also hold for the dilute gas, i.e., N↑(↓) ≪ 1, of non-interacting
fermions. In the framework of the single band approximation, we demonstrated the emergence of non-decaying Bloch oscillations of the momentum and position of a single particle
in a ring-shaped optical lattice under static forcing. This was accomplished in two steps:
First, we demonstrated that the Floquet-Bloch operator defined as the time evolution operator over one Bloch period equals the identity – indicating the return of any initial state
of the system when evolved over an integer number of Bloch periods. Second, we found
that the continuous time evolution of relevant system observables during one Bloch cycle
is given by simple harmonic functions, at the Bloch frequency, or by a sum of harmonic
functions evolving at multiples of the Bloch frequency, in deep (tight-binding limit) and
shallow optical lattices, respectively. Also we considered the related problem of single
particle dynamics in an open elongated optical lattice with a weak parabolic confinement
applied along the lattice. In the tight-binding limit, we mapped this problem on that of
the quantum pendulum. Then, by using semiclassical arguments for the center of mass
motion, we were able to associate Bloch oscillations on one of the shoulders of the parabolic
confinement and dipole oscillations around the trap center, with rotational and librational
modes of a classical pendulum, respectively. This mapping gives not only the most adequate explanation of the emergence of these two qualitatively different types of oscillations,
but it also allows us to find the associated frequencies as functions of the system parameters, and of the energy level number. In contrast to homogeneous, elongated or ring-shaped
optical lattices, the single particle oscillations in parabolic lattices dephase, with possible
further revivals. Both the dephasing and revival times are inversely proportional to the
strength of parabolic confinement, while the dephasing time is also inversely proportional
to the initial width of the single particle wave-packet.
In the chapters 4-7, we applied the model introduced in chapter 2 and methods employed
in chapter 3 to the many-particle problems (see Outline of the thesis, on page 10). For the
interacting, two-component fermionic gas under static forcing in a one-dimensional optical
lattice with ring-shaped geometry, we obtained the spectral characteristics of the associated FB-operator and the time-evolution of the relevant system’s observables: the mean
atomic fermionic energy and velocity, in various parameter regimes (Chaps. 5-6). With
the help of a statistical analysis, we have shown that, in general, the FB-operator has the
typical properties of the evolution operator of a non-integrable quantum system, whereas
the Fermi-Hubbard Hamiltonian at each instant remains integrable, since the level spacings
of the quasienergies of the FB-operator obey the Wigner-Dyson distribution, whereas the
instantaneous spectrum of the time-dependent Hamiltonian exhibits a Poisson distribution
(Chap. 4). We also confirmed that, due to non-integrability, the quasienergies as functions
of the static force (driving parameter) repel each other, thus demonstrating a large number
100
7. Directed atomic current across an optical lattice
of avoided level crossings, unless these levels correspond to a different time-independent
symmetry of the FB-operator. In addition, we identified the non-conventional time reversal
symmetry generated by the reflection operator, for the FB-operator projected onto the two
subspaces associated with the total quasimomentum zero and π. The level-spacings for
these projections have the Wigner-Dyson distribution for the Circular Orthogonal Ensemble of random matrices, while for the other subspaces, the spectral statistics is determined
by the Wigner-Dyson distribution for the more general, Circular Unitary Ensemble. We
also analytically found that, in the limit of strong static tilt with respect to the tunneling
coupling, the FB-matrix has a diagonal form in the Wannier basis, and its spectrum forms
a regular equidistant ladder. This results in the amplitude modulation of BO of the system
observables. Employing the FB-formalism together with direct numerical simulation, we
classified the various dynamical regimes of the time-evolution of mean atomic energy and
velocity (Chap. 6). In particular, we identified decaying, modulated and quasiregular BO.
Let us note the relevance of the these results to possible experimental implementations.
The observation of the modulated BO may reveal an alternative way (i) to measure the
interaction constant from the interaction-induced modulation frequency: U = ~ΩU , and
(ii) to find out how far or close the system is from or to half filling by extracting the
modulation amplitude of BO. In addition, the magnitude of the static forcing can be easily
obtained by identifying the Bloch period. The latter, as the shortest time scale in the
system has been already used to measure the Casimir-Polder force [104], gravity [91], and
the fine structure [105, 106] constant, with unprecedented precision.
Importantly, in both the single and the many-particle problem, the mean atomic velocity was found to exhibit BO. Thus no directed atomic current across the lattice was
generated by the static forcing. In the last chapter 7 of the present thesis, we suggested
and solved the realistic model where a quasistationary atomic drift emerges due to coupling
to the finite bosonic bath. To describe the decoherence of a fermionic carrier induced by
such coupling we derived a master equation within the Markov approximation. Although
this master equation approach a priori excludes the emergence of non-vanishing fermionic
drift velocity, it provided us with the fermion-boson collision rate, which is fully controlled
by the microscopic model parameters and weekly depends on the bath temperature. This
collision rate gives us the decay rate of BO, and together with the initial temperature of the
bosonic bath completely quantifies the “current-voltage characteristics” – the dependence
of the atomic current on the strength of the static forcing: The maximal current at a given
temperature is reached for the magnitude of the static force when the associated Bloch
frequency matches collision rate, while the current amplitude decreases with increasing
the bath temperature. The value of maximal current separates the Ohmic regime from
the regime of negative differential conductivity, in agreement with the phenomenological
Esaki-Tsu model [47] for the electronic transport in semiconductor superlattices.
There were also some open problems, which emerged and remained unsolved in the
present thesis. We recall here the two most significant of them:
• The first is to find the decay rate of BO as a function of system parameters, for the
interacting two-component fermions.
7.5 Summary
101
Due to the universal spectral properties of the associated FB-operator, we expect that
the decay rate has to exhibit some universal properties too. Likely, for relatively large
systems (i.e., for N↑(↓) > 5, L > 10), this rate has a simple functional dependence on
the system parameters. We incline to view that a diffusive model based on the tunneling between the instantaneous energy levels of the FH-Hamiltonian (see Sec. 6.3)
is very promising for finding the decay rate of BO.
• The second is to derive an extended master equation for the fermionic degree of
freedom coupled to the finite bosonic bath, which is able to capture the emergence
of non-vanishing, quasistationary fermionic drift velocity and also takes into account
the finiteness of the bosonic bath.
As recently noted without derivation in [107], there is a simple master equation
(without any additional parameters), which gives both the decoherence of BO and a
nonvanishing atomic drift. One of the possible way to include the finite capacity of
the bath was shortly discussed in the present thesis (see Sec. 7.4.1), and evidently
requires further development.
102
7. Directed atomic current across an optical lattice
Appendix A
Realization of optical lattices
The appearance of a conservative optical potential (“dipole trap”) is a “second order effect”
due to the interaction between a laser-induced dipole moment with the same laser field.
Let us consider a neutral atom, in some electronic “ground” (metastable) state |gi with
energy εg , continuously exposed to a monochromatic laser field with electric field vector
E(r, t) = E(r) exp(−iωt), and frequency ω tuned close to some atomic resonance frequency
ω0 = (εe − εg )/~ between the ground and some excited state |ei with energy εe . If the
electric field E(r, t) does not change its amplitude on time scales of the order of the laser
frequency ω −1 and on length scales comparable to the size of the atom, then the following
atom-field interaction Hamiltonian is valid
Hdip = −µE(r, t) + h.c. ,
(A.1)
where µ = er is the electric dipole moment operator (“dipole approximation”).
The non-diagonal elements, he|Hdip|gi, couple the two atomic levels and induce a periodic population exchange with Rabi frequency
ΩR (r) = −2E(r)he|µe|gi~−1 ,
(A.2)
where e is the polarization vector of the electric light component.
If the detuning δ = ω − ω0 is large in comparison to the Rabi frequency δ ≫ ΩR , but
still bounded from above by the resonance frequency, δ ≪ ω0 , the excited state can be
adiabatically eliminated within the rotating wave approximation (See, e.g., [108]). The
ground state energy is shifted by the ac-Stark shift
V (x) = ~Ω2R (r)/4δ ,
(A.3)
which depends on the position of the atomic center-of-mass.1 Thus the atom experiences
an optically induced effective potential V (x).
1
The same result can also be obtained with the non-degenerate time-independent second order perturbative theory.
104
A. Realization of optical lattices
To emphasize the role of the eliminated excited state, |ei, let us rewrite (A.3) by
inserting the expression (A.2) for the Rabi frequency:
V (r) =
3πc2 γ
I(r) .
2ω03 δ
(A.4)
Here, we used I(r) = 2ε0 c|E(r)|2 for the light intensity,2 and
γ=
ω02
|he|µ|gi|2 ,
2
3πε0 ~c
(A.5)
for the natural width of the excited level.
Note that the potential V (r) given by the intensity pattern, I(r), can be inverted by
changing the sign of detuning δ,3 V (r) → −V (r). Therefore, one distinguishes the case
of negative detuning, δ < 0 (“red detuning”), with the potential minima at the positions
of maximal intensity (“bright” trapping potential), from that of positive detuning, δ > 0
(“blue detuning”), when the potential and the intensity minima coincide (“dark” trapping
potential).
The coherent response of the induced dipole-moment, based on electronic transitions
with induced photon absorption from the laser mode, and subsequent stimulated reemission
back into the laser mode, is accompanied by the incoherent process of spontaneous emission
into the modes of the electromagnetic environment. The related decay rate γsc (r) over one
Rabi cycle can be estimated from the product of the natural linewidth, γ, and the average
excited state population, he|b
ρ(r)|ei ≈ (ΩR (r)/δ)2 . With expressions (A.2) and (A.5), the
decay rate takes the form
3πc2 γ 2
γsc (r) = ~−1 3
I(r) .
(A.6)
2ω0 δ
Thus the degree of “imperfection” ν of the dipole trapping potential V (r), due to spontaneous emission,
γ
~γsc (r)
= ,
(A.7)
ν=
V (r)
δ
can be usually reduced to negligible values by the appropriate choice of the detuning δ.
In typical experimental, the three time scales defined by the laser frequency ω ≃ 600
THz, detuning δ ≃ 100 GHz, and decay rate γ ≃ 100 MHz, are well separated by orders
of magnitude, such that the rotating-wave approximation is valid, and dissipation due to
spontaneous emission can be neglected.
2
For a multi-level atom, if the ground and the excited state resolve the subelectronic energy level
structure, |gi i and |ej i, the expression (A.4) includes a sum over module squared dipole moment matrix
elements µi,j = Ci,j |µ|, where Ci,j are real dimensionless coefficients, which depend on the light polarization
e and the electronic and the nuclear angular momenta of the specified sublevels, and |µ| is the reduced
matrix element defined by the width of the excited state and the resonance frequency ω0 .
3
Since δ ≪ ω, the change in the frequency over ∆ω = 2δ, and, thus, in the intensity magnitude due to
frequency-dependent polarizability, is negligible.
A.1 One-dimensional elongated geometry
A.1
105
One-dimensional elongated geometry
If a focused Gaussian laser beam with wave length λL is superimposed with its retroreflection, the resulting standing wave intensity is given by the periodic, cylindrically symmetric
(along the propagation direction) function
I(r, z) = 4 ×
2
2P
−2 2r
w (z) cos2 (2πz/λ ) ,
e
L
πw 2(z)
(A.8)
p
where w(z) = w0 1 + (z/zR )2 is size of the beam waist, and zR = πw02 /λL is the Rayleigh
length (which is the distance from the beam focus at which the beam area doubles). The
prefactor 4 is due to constructive interference between both beams. Consequently, near
the trap center, z ≪ zR and r ≪ w0 , the resulting optical potential can be approximated
by
"
2 2 #
z
r
cos2 (2πz/λL ) ,
(A.9)
−
V (r, z) ≃ V0 1 − 2
w0
zR
where the amplitude, V0 , using (A.4) is defined as
V0 =
6πc2 γ 2P
.
ω03 δ πw02
(A.10)
The two-beams configuration produces an optical lattice along the z-axis, with period
d = λL /2.4 The red-detuned laser automatically provides central confinement in the radial
direction, while the blue-detuned one, in contrast, repels atoms from the beam center and
thus requires some trapping potentials in addition.
The characteristic energy scales of the radial and longitudinal (without and with optical
lattice) confinement potentials can be expressed through their harmonic frequencies:
√ 1
1
1
2 2V0 2
1 2V0 2
π 2V0 2
(OL)
ωr =
, ωz =
, ωz
=
,
(A.11)
w0
m
zR m
d m
correspondingly.
In typical experiments, zR ≃ 30w0 , w0 ≃ 20d, and d ≃ 400 nm. Therefore, the
confinement frequencies are ordered as follows:
ωz(OL) ≫ ωr ≫ ωz .
(A.12)
This results in a pancake-like shape of the surfaces of constant potential (Fig. 1.1, on the
left).
Note that, if the extension of the atomic cloud is comparable to the Rayleigh length
zR , such that the parabolic longitudinal confinement becomes noticeable, it can modify the
properties of the trapped cold atoms significantly. In particular, we analyze the effect of a
non-vanishing parabolic potential on the dynamics of cold atoms in Section 3.3.
4
The lattice constant d can be additionally adjusted by changing the angle θ between the counterpropagating beams, d(θ) = d/ sin(θ/2).
106
A. Realization of optical lattices
Figure A.1: Ring-shaped optical potential formed by a plane wave and a LG mode with L = 14,
p = 0. The picture is taken from [14].
A.2
Ring-shaped geometry
The radial intensity dependence of a single laser beam can be modified arbitrarily by
spatial light modulators (SLMs) acting as reconfigurable diffractive optical components,
i.e. holograms [15]. In particular, Laguerre-Gaussian (LG) laser modes with discrete
rotational symmetry around the propagation direction are produced using SLMs. A single
LG mode with frequency ω and azimuthal and radial indices l and p respectively, has the
form
2π
r2
LG
Elp (r, ϕ, z) = E0 fpl (r) exp i ωt + lϕ −
z−
+ Φ|l|
,
(A.13)
λL
2R
s
−ξ2
2p!
2
ξ |l| L|l|
ξ
e ,
(A.14)
fpl (r) = (−1)p
p
π (p + |l|)!
√
P
p+l (−ξ)m
with ξ = 2r/w(z), where Llp (ξ) = pm=0 p−m
are associate Laguerre polynomials,
m!
Φ|l| = (|l|+p+1)arctan(z/zR ) is the Gouy phase (which is the difference in phases acquired
along the propagation of the LG beam and a plane wave with the same optical frequency),
and R = z (1 + (zR /z)2 ) is the radius of the front wave curvature.
Recently, it was proposed theoretically [14] to create a periodic angular modulation of
the intensity pattern, by interference between a single mode, (A.13), and a plane wave,
E0 ei(ωt−2πz/λL ) . Finally, a standing wave along the z-axis is formed by superimposing the
original configuration with its retroreflection. The resulting optical potential close to the
beam focus (z ≪ zR ) then takes the form
V (r, ϕ, z) = 4V0 1 + fpl2 (r) + 2fpl (r) cos(lϕ) cos2 (kz) ,
(A.15)
where V0 is the potential amplitude defined by the intensity maximum of a single LG mode.
Intensity
A.2 Ring-shaped geometry
107
HaL
HbL
LG-5
LG5 +LG-5
LG3
LG11
LG3 +LG11
Phase
LG5
Figure A.2: Generation of bright (a) and dark (b) lattices from the interfering LG beams with
different l values on an area of 6w × 6w. Note that the dark lattice sites are positioned at phase
singularities. The picture together with its description is taken from [109].
Thus the interior of the ordinary Gaussian confinement is structured into separated,
ring-shaped traps, concentric with respect to the z-axis, with L = l angularly equidistant
energy minima each. The energy barrier between the rings can be enhanced as compared to
that between neighboring lattice sites of each ring, by adjusting r0 /λL (i.e., focusing the LG
beam). For L & 15, relative tunneling rates tz /tφ . 10−2 can be achieved with r0 /λL ∼ 100
An example of the ring-shaped periodic potential (A.15) at fixed axial coordinate z = 0,
for LG beams with p = 0,5 and l = 14, is shown in Figure A.1.
An alternative experimental setup, which was already partially6 realized [13], consists
in two collinear LG modes with the same indices p, but different indexes l, δl = l2 − l1 ,
and a non-zero frequency difference δω. In this case, the optical potential reads
V (r, ϕ, z) = V0 f0l2 1 (r) + f0l2 2 (r) + 2f0l1 (r)f0l2 (r) cos(ϕδl − tδω + δΦ(z)) ,
(A.16)
where the term δω(z − r 2 /2R)/c, being negligible for the experimental parameters [13], has
been omitted.
The potential (A.16) has δl minima and maxima as a function of φ, and rotates at
an angular frequency δω/δl. The bright potentials can be formed by the LG modes with
an equal radii (i.e. l = l1 = −l2 , δ = 2l), while dark potentials by an appropriate
combinations of l1 6= l2 (see Fig. 1.2). The twist of ring minima (maxima), which form
a twisted “spaghetti-like” form along the z-axis, induced by the Gouy phase difference,
δΦ = ||l1 | − |l2 ||arctan(z/zR ), can be neglected by operating far apart from the beam focal
point, z ≫ zR (what is in the opposite regime in respect to that in the first proposal).
5
|l|
Note that the Laguerre polynomial Lp=0 is constant function equal to unity.
6
The desired interference pattern was created, but loading of cold atoms into the trap has not been
performed yet.
108
A. Realization of optical lattices
The conservative confinement along the axial direction cannot be induced by counterpropagating LG beams with a non-zero frequency difference (compare with first proposal),
since it will induce a drift of the standing wave with speed λL δω/4π. Therefore, one should
employ an additional confinement potential. For instance, a pair of counterpropagating
Gaussian beams can be applied to produce a lattice of “rings” along the axial direction,
and than one of the “rings” can be selected by a magnetic confinement.
A.3
Static forcing of cold atoms in an optical lattice
From the experimental point of view an additional linear potential required to induce a tilt
was already realized by two different techniques in the elongated optical lattices: either
by tilting the optical lattice with respect to its, in the ideal case, horizontal position,
such that the gravitational field induces a potential energy shift along the lattice, or by
introducing a time-dependent frequency shift, δω = 2kL at, between the counterpropagating
laser beams. In the latter case, the standing wave accelerates due to the phase difference,
V (z, t) = V0 cos2 (d−1 (z − at2 /2)), thus effective constant force, F = Ma, acting on the
atoms.
Since in the acceleration frame a linear static tilt is equivalent to a phase shift of the
tunneling amplitude linear in time (see, for details, Sec. 2.1.1), in the case of a ring-shaped
geometry an effective linear tilt can be induced by a time-dependent twist of the periodic
boundary conditions. For the first realization discussed in details in Sec. A.2 the latter
can be achieved by an additional, cone-shaped magnetic field B = Bϕ eϕ + Bz ez . Then
the wave function will acquire the phase factor Ψ → exp(imF π cos θ)Ψ, on an interval
2π in the azimuth, with tan θ = Bϕ /Bz and mF the magnetic
pquantum number of the
atom. The realization of the time-dependent ratio Bϕ /Bz = 1/(t/T )2 − 1 on interval
(0, T ), will be equivalent to the static force driving with F d = ~mF π/LT acting over the
n = mF /2L Bloch periods. Since magnetic quantum number of alkaline atoms |mF | < 10,
the observation time of the Bloch oscillations is strongly limited to the one Bloch circle
already for the L = 5 lattice sites.
To induce an effective static force F = ma along the second realization of a ring-shaped
lattice discussed in Sec. A.2, one has to apply the same trick as in the case of elongated
lattice geometry. In contrast to the previous proposal, with a specific external magnetic
field configuration, no additional components are required. Solely, the frequency difference
between two collinear LG modes has to be changed linearly in time, δω = 4πat/λL .
Appendix B
Diagonalization procedure
The diagonalization procedure is realized in two steps: analytical decomposition of the
total Hamiltonian into a direct sum of the “irreducible” blocks labeled by complete sets
of the quantum numbers associated with the discrete symmetries of the Hamiltonian, and
their subsequent direct numerical diagonalization.
Suppose we want to evaluate the matrix elements of the block with labels κ, S, η, σ, π.
First, we construct the Wannier basis for given lattice size L and particle numbers N↑ , N↓ .
After that we obtain those eigenstates of the translation operator τ which correspond to a
given eigenvalue κ:
1 X i2πκl l
|κ, ni = √
e
τ |ni ,
(B.1)
M l=1
where M is the number of different Wannier basis states generated by repeated application
of τ on the “seed” state [110]. With these eigenvectors, we calculate the matrix elements
of the Hamiltonian (i.e., κ-block) and of all remaining symmetry operators, in subspace
associated to κ. Since there are L different quasimomenta, the average size of the κ-block
is approximately L times smaller than that of the entire Hilbert space.
Next to dividing out the translation symmetry, we turn to the operator of total spin.
Although, in general, we do not know explicitly the eigenvectors of the total spin operator,
the knowledge of its eigenvalues allow us to write a system of linear equations for its eigenvectors, with a straight forward algebraic analytical solution, for any given eigenvalue S.
As soon as the spin symmetry has been utilized, we apply the same procedure to the
conditional discrete symmetries, in the following order: quasispin, spin-flip and reflection.
In the ideal case (in the sense of a sufficient reduction of the average block size), i.e., at
half filling: N↑ = N↓ = L/2 when all mentioned above symmetries are present, the average
size of “irreducible” blocks decreases by the factor ∼ L3 /4 as compared to the size of the
entire Hilbert space N , e.g., for for N1 = N2 = L/2 = 8, the average size of the blocks is
∼ 104 as compared to N = 165636900 > 108 .
110
B. Diagonalization procedure
Appendix C
The FB-operator at strong static
forcing (F d ≫ J)
By integrating once the Schrödinger equation for the evolution operator and then iterating,
the FB-operator can be written as [111]
1
W (TB ) = id +
i~
|
Z
TB
0
1
dtH(t) +
i~
{z
} |
X1 (TB )
= id + X1 (TB ) +
∞
X
1
(i~)
m=2
Z
Z
TB
0
1
dtH(t)
i~
{z
Z
t
0
X2 (TB )
TB
dt′ H(t′ ) + . . . .
}
dtH(t)Xm−1 (t) .
(C.1)
0
Let us subsequently evaluate the contribution of each term to the entire expansion.
To proceed, we introduce short notations for the kinetic and interaction terms of the
Hamiltonian (5.3):
H(t) = −J(Kt + Kt† ) + UI ,
with
Kt :=
XX
s=↑,↓
l
eitF d/~c†l+1,s cl,s
and I :=
(C.2)
X
nl,↑ nl,↓ .
(C.3)
l
Then the first non-trivial term
Z
1 t
†
−J(Kt + Kt ) + UI
X1 (t) =
i~ 0
i
†
†
= ǫ K0 − Kt − K0 + Kt − tUI ,
~
X
i
nl,↑ nl,↓ ;
X1 (TB ) = − UTB
~
l
ǫ :=
J
Fd
,
(C.4)
(C.5)
112
C. The Floquet-Bloch operator at strong static forcing
the second term
Z
i
i 1 t ′h
†
†
†
′
X2 (t) =
dt −J Kt′ + Kt′ + UI × ǫ K0 − Kt′ − K0 + Kt′ − t UI
i~ 0
~
= −ǫ2 Kt K0 − K02 − Kt K0† + K0 K0† − Kt† K0 + K0† K0 + Kt† K0† − (K0† )2
ǫU ǫ2 2
Kt − K02 + (Kt† )2 − (K0† )2 +
I Kt† − K0† + Kt − K0
+
2
Fd
ǫtU ǫtU ǫtJ †
†
†
†
−
Kt K t − Kt K t −
Kt − Kt I −
I K0 − K0
i~
i~
i~
2
1 tU
ǫU Kt − K0 − Kt† + K0† I +
I2 ,
(C.6)
−
i~
2 i~
i
1 2
ǫTB h X2 (TB ) =
J K0 K0† − K0† K0 + UI K0 − K0† − U K0 − K0† I
X1 (TB ) −
2
i~
i
i
h ǫTB h
1 2
;
(C.7)
J K0 , K0† + U I, K0 − K0†
X1 (TB ) −
=
2
i~
and the next terms are of the higher order in ǫ.
In the entire expansion (C.1), the diagonal operator X1 and its powers are the only
contributions with non-zero elements on the main diagonal. They can be combined back
into the operator exponential:
!
UTB X
W (TB ) = exp −i
nl,↑ nl,↓ + O(ǫ) ,
(C.8)
~
l
while the rest of the terms gives the only non-diagonal contribution (note that, TB =
2π/F d) of the first or of the second order in ǫ = J/F d ≪ 1, for |U| ∼ J and for |U| ∼ F d,
respectively.
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Acknowledgements
This thesis would hardly emerge without:
ideas and methods suggested by Prof. Andrey R. Kolovsky. I’m grateful to
Andrey R., who has lunched my PhD project, for introducing me to the theory
of cold matter and of quantum chaos, and for kind support during my stay in
Dresden;
the patience and scientific school of Prof. Andreas Buchleiter, who has guided
my work until “the thesis took its shape”. I’m very thankful to Andreas –
Doktorvater, who has always time for his students, for his care, support and
encouragement. I’m also grateful to Andreas for open, exciting discussions not
only on scientific topics, and for german lessons which helped me a lot to get
to know Germany;
the numerical schemes provided by Dr. Javier Madroñero which sufficiently
speeded up our calculation of the time evolution for the fermi-bose mixture;
belief of family in me, and especially, warm and incredibly wise words of my
grandmother.
That was a great pleasure to work among friends in open and stimulating atmosphere
in the Max Planck Institute for the Physics of Complex Systems in Dresden, and in the
Institute of Physics in Albert-Ludwigs University of Freiburg.
I wish to thank all of my colleagues and friends, especially: Thomas Wellens for critical
reading of this thesis, Sandro Wimberger for his care and hospitality during my visit in Pisa;
Dmitry Kovrizhin for very useful discussion on condense matter physics, his hospitality
during my stay in Paris and Tango classes; Evgueni Starostine for beautiful mathematical
models I got to know; Ming-Chuang Chung for useful remarks concerning different boundary conditions and for his delicious exotic meals; Hannah Venzl, Torsten Scholak, Moritz
Hiller and Florian Mintert for assisting me to compose the “Zusammenfassung”; Jochen
Mikosch for providing me with excellent TEX - template; Andrei Lyubonko and Alexey
Mikaberidze for visiting me in Freiburg; and also Alejo Salles, Adriano Aragão, Fernando
de Melo, Celsus Bouri, Leandro Aolita, Carlos Viviescas, Scott Sanders, Artem Dudarev,
Olivier Brodier, Marc Busse, Alexei Schelle, Boris Fine, Slava Schatokhin, Ilja Eremin,
Maxim Korshunov, Mikhail Titov, Sergey Denisov, Sergey Flach, Alexander Donkov, Nikolaj Korabel, Viktor Bezugly, and Alexander Cherny for the wonderful moments we shared.
Last but not least, I want to acknowledge those I met beyond the Ivory Tower: Katrin,
Yevgeniya, Natasha, Egle, Lina, Gaile, Jeanette and Lena.
Lebenslauf
Name
Geburtsdatum/ort
Staatsangehörigkeit
Adresse
Alexey V. Ponomarev
11. April 1981 in Krasnojarsk (Russland)
russisch
Fehrenbachallee 8, N116
D-79106 Freiburg im Breisgau
Schulbildung
1988-1998
Schule N41 in Krasnojarsk (Russland)
Studium der Physik
1998-2003
Staatliche Universität Krasnojarsk (Russland)
27. Juni 2003
Physik-Diplom (mit Auszeichnung)
Diplomarbeit: “Gedämpfte Bloch-Oszillationen
kalter Atome in optischen Gittern”
Betreuer: Prof. Andrey R. Kolovsky (Krasnojarsk)
Promotion in Physik
2005-2007
an der Fakultät für Physik der LMU München
Betreuer: PD Andreas Buchleitner (Dresden)
Stipendien
1998-2003
2000-2001
2001-2002
2001-2003
2004-2006
Stipendium der Staatlichen Universität Krasnojarsk
Stipendium von Coca-Cola für begabte Stundenten
Stipendium des Bezirksgouverneurs Krasnojarsk
Stipendium der Regierung der Russische Föderation
Promotionsstipendium der Max-Planck-Gesellschaft
Freiburg, den 31. März 2008