In Chapter 1 we set up the physical and mathematical framework for the description of interacting
bosons on a lattice. After a brief introduction to the basic concepts of Bose-Einstein condensation, and
how the condensate can be manipulated using optical lattices, we derive the Bose-Hubbard Hamiltonian
(BHH) which is the paradigm model as far as the quantum description is concerned. Then the semiclassical
limit of the BHH – the discrete nonlinear Schroedinger equation (DNLS) – as well as the mean-field
(classical) description, the so-called Gross-Pitaevskii equation (GPE), is discussed. We give an overview
of the relevant literature and of the various physical systems that are captured by the BHH including,
among others, arrays of Josephson junctions and bond vibrations in small molecules. The chapter ends
with an outline of the remaining thesis.
In Chapter 2, we look at some interesting experiments done in cold bosonic systems. Ostensibly,
done to realize the dc and ac Josephson effect in a system of cold bosons, it was soon found that the
system can show some rather interesting non-linear behavior. This includes features such as solitons
and breathers , but most interestingly, it shows the damping of the center-of-mass motion. Theoretical
explanations are given for both the experiments. The damping of center-of-mass of the system is seen as
a result of modulational instability. We give a different interpretation to this opening up lines for further
investigations. We outline these at the end of the chapter.
In Chapter 3, we discuss various experiments which contradicted the idea of a sharp modulational
instability. Rather, the loss of coherence is smeared out over a large domain. We then reproduce some
calculations from the literature showing how the dissipation can be thought of as arising from quantum
phase slips. We calculate the tunneling rates for quantum phase slips and show a phase diagram demarcating
stable and unstable regions of current which have been have obtained from mean-field calculations
via the Gutzwiller ansatz.
In Chapter 4, we look at the problem in details analyzing it as a Tomonaga-Luttinger liquid. The
framework of Tomonaga-Luttinger liquid is actually a non-perturbative tool through which we can investigate
an one-dimensional interacting bosonic fluid. We look at two possible mechanisms of decay
of current- We look at the effect of a weak link in a Josephson Junction array as well as local quantum
phase slips on a bosonic superfluid. We find that quantum phase slips are actually irrelevant under
renormalisation group in both cases and the supercurrent is revived.
In chapter 5, we look at the non-linear dynamics of the structures in greater detail, thus elucidating
chaos. This mechanism of chaos is different from all realizations of chaos known so far and is a result of
phase fluctuations, which are encoded by quantum phase slips.
In Chapter 1 we set up the physical and mathematical framework for the description of interacting
bosons on a lattice. After a brief introduction to the basic concepts of Bose-Einstein condensation, and
how the condensate can be manipulated using optical lattices, we derive the Bose-Hubbard Hamiltonian
(BHH) which is the paradigm model as far as the quantum description is concerned. Then the semiclassical
limit of the BHH – the discrete nonlinear Schroedinger equation (DNLS) – as well as the mean-field
(classical) description, the so-called Gross-Pitaevskii equation (GPE), is discussed. We give an overview
of the relevant literature and of the various physical systems that are captured by the BHH including,
among others, arrays of Josephson junctions and bond vibrations in small molecules. The chapter ends
with an outline of the remaining thesis.
In Chapter 2, we look at some interesting experiments done in cold bosonic systems. Ostensibly,
done to realize the dc and ac Josephson effect in a system of cold bosons, it was soon found that the
system can show some rather interesting non-linear behavior. This includes features such as solitons
and breathers , but most interestingly, it shows the damping of the center-of-mass motion. Theoretical
explanations are given for both the experiments. The damping of center-of-mass of the system is seen as
a result of modulational instability. We give a different interpretation to this opening up lines for further
investigations. We outline these at the end of the chapter.
In Chapter 3, we discuss various experiments which contradicted the idea of a sharp modulational
instability. Rather, the loss of coherence is smeared out over a large domain. We then reproduce some
calculations from the literature showing how the dissipation can be thought of as arising from quantum
phase slips. We calculate the tunneling rates for quantum phase slips and show a phase diagram demarcating
stable and unstable regions of current which have been have obtained from mean-field calculations
via the Gutzwiller ansatz.
In Chapter 4, we look at the problem in details analyzing it as a Tomonaga-Luttinger liquid. The
framework of Tomonaga-Luttinger liquid is actually a non-perturbative tool through which we can investigate
an one-dimensional interacting bosonic fluid. We look at two possible mechanisms of decay
of current- We look at the effect of a weak link in a Josephson Junction array as well as local quantum
phase slips on a bosonic superfluid. We find that quantum phase slips are actually irrelevant under
renormalisation group in both cases and the supercurrent is revived.
In chapter 5, we look at the non-linear dynamics of the structures in greater detail, thus elucidating
chaos. This mechanism of chaos is different from all realizations of chaos known so far and is a result of
phase fluctuations, which are encoded by quantum phase slips.
Bose-Einstein condensate (BEC) is a new state of matter, experimentally realized in a dilute alkali gas of weakly interacting bosons. The gas is cooled to very low temperatures, of the order of nano-Kelvin, at which point a large fraction of the atoms occupy the lowest quantum state. Under such conditions, the quantum effects become apparent on a macroscopic scale. Many fundamental ideas
of quantum mechanics have been realized and proven to be true in experiments,
by studying the Bose-Einstein condensates. Amongst them, the observation of
macroscopic matter wave interference, the creation of vortices, dark and bright
solitons etc, find special mention. The dynamics of this system in presence of an
optical lattice reveals a superfluid to Mott insulator quantum phase transition.
This has opened a new avenue for research on BEC and its applications. From
a theoretical standpoint, the dynamics of BEC can be described by means of an
effective mean-field theory. This formalism is much simpler than the many-body
Schr¨odinger equation and based on a classical nonlinear evolution equation, known
as Gross-Pitaevskii equation. The nonlinear nature of the GPE leads to several interesting phenomena like soliton propagation, vortices, four wave mixing etc.,
which are closely related to nonlinear fiber optics.
The present thesis deals with nonlinear excitations and different phases of BEC
and is organized as follows: after a brief introduction, in chapter two, we start
with BEC in a one-dimensional shallow optical lattice, experiencing both cubic
and quintic nonlinearity. A superfluid density wave, of twice the periodicity of
lattice potential, is identified. This indicates matter redistribution, where atoms
from neighbouring sites have been depleted leading to a density wave behaviour.
Interestingly, it ceases to exist when only one of these interactions is operative.
We predict the loss of superfluidity through a classical dynamical phase transition,
where modulational instability leads to the loss of phase coherence. In presence
of pure two-body interaction, the superfluid density shows a periodicity commensurate
with the lattice potential. Apart from the superfluid state, a density wave
insulating phase is also found to exist, possessing two frequency modulations commensurate
with the lattice potential. A dynamical phase transition connects the
superfluid phase with an insulating phase, which in turn, is connected with the
density wave insulating phase. The solutions are found to be marginally stable, as
per the Vakhitov-Kolokolov (VK) criterion.
Chapter three describes the dynamics of the atomic-molecular Bose-Einstein
condensate (AMBEC) by mean-field Gross-Pitaevskii equation (GPE), taking into
account the atom-molecular interaction, as well as atom-molecule inter-conversion.
The two-body atom-atom, atom-molecule and molecule-molecule interactions are
analogous to the cubic nonlinearity of the standard GPE, whereas, the quadratic
nonlinear term arising from the inter-conversion between atoms and molecules is
identical to the scenario in strongly coupled BEC in one dimension. The presence
of both weak and strong interactions adds further richness to this system. We
find the molecular BEC possesses a complex dark (grey) soliton, while the atomic condensate is a bright one. The energy and momentum of this system is computed
in order to see the effect of the both weak and strong interactions on Lieb mode
profile and compare it with the pure atomic condensate profile. Another class
of solutions exist in this system, where both atomic and molecular condensates
possess bright solitons.
In chapter four, we investigate the effect of a harmonic trap on the dynamics
of grey solitons in Bose-Einstein condensates. It is observed that, these nonlinear
excitations of the cigar-shaped Bose-Einstein condensate have strong coupling
with the trap at low momenta and hence can be effectively isolated from the
Bogoliubov sound modes, which respond weakly to harmonic confinement. This
strong coupling with the trap also makes the grey soliton amenable for control and
manipulation through trap modulation and temporal variation of the two-body interaction.
We have shown that in presence of an expulsive oscillator trap, the grey
soliton gets amplified, while having a compression in its width. Exact solutions are
also obtained in case of the two component Bose-Einstein condensates (TBECs),
where one component is a grey soliton and the other a bright one. The presence
of the trap and inter-species interaction significantly affects the grey soliton dynamics.
Similar to the single component, in the present case, we have shown that
both grey and bright soliton can be compressed simultaneously. A detailed study
of the energy and momentum of the soliton is reported, in order to find the effect
of the trap on the Lieb mode.
In chapter five, we study a BEC-defect complex in a quasi-one dimensional
scenario, where the defect atom moves in the condensate with constant velocity.
Explicit solutions of the coupled mean-field equations, describing defect-grey soliton
dynamics are obtained, demonstrating the coexistence of grey soliton and a
localized defect. Unlike the case of dark soliton, where the defect trapping center
has vanishing superfluid density, the moving grey soliton necessarily possesses a finite superfluid component at the defect location. The presence of the defect can
lower the grey soliton’s maximum velocity, as compared to the defect free case,
where the maximum velocity is the sound velocity. We have also carried out the
stability analysis of the obtained solutions using renormalized momentum formalism
and Vakhitov-Kolokolov criteria and find that the obtained solutions are stable
through both the methods. The grey soliton’s energy gets substantially modified
through its interaction with the defect, opening up the possibility of its control
through defect dynamics.
Chapter six describes a parametrically forced Bose-Einstein condensate in the
presence of a general moving optical lattice. The interaction between the atoms in
the condensate and the time dependent lattice potential leads to a novel propagating
superfluid matter wave, which can be controlled through chirp management.
This system, when placed in a trap, accelerates and undergoes rapid nonlinear
compression, controlled by the chirp. The density achieves its maximum, when
the matter wave changes direction. It is also shown that this system exhibits dynamical
superfluid insulator transition, where superfluidity breaks down and the
condensate transits to an insulating phase. The stability has been investigated
using Vakhitov-Kolokolov criterion and found that it depends on the chirp, as well
as on the nature of the atom-atom interaction. The exact expression for energy
is obtained and analyzed in detail to gain physical understanding of the chirp
management of the sinusoidal excitations and also the dynamical phase transition.
This thesis covers most of my work in the field of ultracold atoms loaded in optical lattices. It makes a route through the physics of cold atoms in periodic potentials starting from the simple noninteracting system and going into the many-body physics that describes the strongly correlated Mott insulator regime. Even though this thesis is a theoretical work all the chapters are linked either with experiments already done or with ongoing experimental efforts. This thesis can be divided into four different parts. The first part comprises chapters 1 to 3. In these chapters, after a brief introduction to the field of optical lattices I review the fundamental aspects pertaining to the physics of systems in periodic potentials. The second part deals with the superfluid weakly interacting regime where standard mean field techniques can be applied. This is covered in chapters 4 and 5. Specifically, chapter 4 introduces the discrete nonlinear Schroedinger equation (DNLSE) and uses it to model some experiments. In chapter 5 I go one step further and include the small quantum fluctuations neglected in the DNLSE by studying quadratic approximations of the Bose-Hubbard Hamiltonian. Chapters 6 to 8 can be grouped as the third part of the thesis. In them I adopt an effective action formalism, the so called two particle irreducible effective action (2PI) together with the closed time path (CTP) formalism to study far-from-equilibrium dynamics. The many-body techniques discussed in these chapters systematically include higher order quantum corrections, not included in the quadratic approximations of the Hamiltonian, which we show are crucial for a correct description of the quantum dynamics outside the very weakly interacting regime. Finally, chapter 9 to 11 are devoted to study the Mott insulator phase. In these chapters using perturbation theory I study the Mott insulator ground state and its excitation spectrum, the response of the system to Bragg spectroscopy, and propose a mechanism to correct for the residual quantum coherences inherent to the Mott insulator ground state. Even though small these are not ideal for the use of neutral atoms in optical lattice as a tool for quantum computation.
Cold atoms trapped in optical lattices (crystals of light) provide a pristine platform for exploring quantum many body physics. Motivated by several recent experiments, this thesis examines the equilibrium and non-equilibrium dynamics of a Bose-Einstein condensate (BEC) loaded in a low dimensional optical lattice in order to realize novel quantum phases. There are two main research directions in this thesis. The first one involves the possibility that exotic order spontaneously forms when two-component bosons are trapped in a honeycomb lattice. My studies on this theme is motivated by the observation of a “twisted superfluid” state in Prof. Klaus Sengstock’s group at Hamburg (Soltan-Panahi et al., Nat. Phys. 8, 71 (2012)). A twisted superfluid involves Bose-Einstein condensation into a state whose order parameter has a spatially varying phase. In chapter 3, I study the stability of a Bose-Einstein condensate towards forming a twisted superfluid within the framework of mean field theory. Despite a exhaustive numerical search I do not find a parameter regime with a twisted superfluid. This search involved all experimentally relevant parameter regimes and therefore mean field theory predicted that the experimentalists should not observe a twisted superfluid. I conclude that the experimental observations were either a manifestation of counter superfluidity or due to interactions during time-of-flight. Subsequent experiments showed that the observations were an artifact of the measurement process. The second research direction in this thesis is an exploration of the stability of periodically driven quantum systems (also known as Floquet systems). Floquet systems can be used to realize exotic non-equilibrium quantum phases which do not have a counterpart in static systems. However, the driving can cause these systems to heat up which presents a major obstacle to creating exotic states. To explore this issue in a concrete example, I model an experiment (Parker, Ha, and Chin, Nat. Phys. 9, 769 (2013)) where a Bose-Einstein condensate loaded in an optical lattice is subjected to periodic shaking. I investigate the stability of this Floquet BEC to interactions. This research direction consists of 3 studies. In chapter 4, I first do this analysis for a purely one-dimensional system and identify a large parameter regime where the BEC is stable. In the next two chapters, I go beyond 1D and consider the role of transverse degrees of freedom. This is because the shaken lattice experiments that I model involves a 1D array of pancakes. I find that this geometry leads to much more dissipation than a purely 1D system. This extra dissipation arises because interactions can transfer energy between different directions. In chapter 5, I consider the extreme case where there is no transverse confinement. I find that in the absence of transverse confinement, a one-dimensional Floquet BEC is generically unstable. Finally, in chapter 6, I consider harmonic transverse confinement modeling the crossover between chapters 4 and 5. I find that as the transverse confinement is made stronger, the atom loss rate initially increases, but beyond a critical transverse confinement, the atom loss disappears due to unavailability of phase space for scattering. I also predict that if the transverse confinement is tuned to the vicinity of certain magic values, the heating rate exhibits a sharp drop. I perform similar analyses for a shaken square lattice and find that generically a low-dimensional Floquet BEC can be stabilized by suitably designing the transverse confinement.
Since the first experimental observation of the Bose-Einstein condensation of a gas of alkali atoms, six years ago, there has been an increasing interest on this new, highly interdisciplinary, branch of physics. It was realized, from the very beginning that, despite of the diluteness of such systems, the nonlinearity was crucial on the understanding of both the equilibrium and dynamical properties. In this Thesis we point out that there is a further key aspect that must be taken into account when the condensates are trapped in deep optical lattices: discreteness. The interplay between discreteness and nonlinearity raises a new class of deeply non-trivial phenomena which disappear in the continuum limit of the Gross-Pitaevskii equation (GPE). Recognizing and exploiting such interplay is the main topic of this Thesis. We think that the dynamics of a Bose-Einstein condensate (BEC) in optical lattices is going to become of central interest in the next future. Optical potentials provide BEC's waveguides, which prevent the spatial spreading of the BEC wave packets and preserve their coherence over long distances. This has allowed to build high sensitive interferometers, including gyroscopes and gravitometers. From the theoretical point of view, the big challenge is to reconsider the meaning of phenomena which have been well understood in homogeneous systems: superfluidity, for instance, is in the top list. A further example is given by quantum phase transitions. The connection and the interplay with the language and insights of the nonlinear physics community is a second challenge. We believe that the results obtained in the BEC context can provide a new prospective view into old problems. The understanding of the modulational instability in terms of phase randomization and a (classical) dynamical phase transition are among this line, as well as the possibility to generalize the Landau criteria for superfluidity to discrete systems. On the other hand, it is clear that nonlinear physicists can have much to say about BEC. With this Thesis we hope we have provided, together with original results, some links between these different languages and views.
This dissertation discusses a number of theoretical models of coupled bosonic modes, all closely related to the Bose-Hubbard dimer. In studying these models, we will repeatedly return to two unifying themes: the classical structure underlying quantum dynamics and the impact of weakly coupling a system to an environment. Or, more succinctly, semiclassical methods and open quantum systems. \n \nOur primary motivation for studying models such as the Bose-Hubbard is their relevance to ongoing ultracold atom experiments. We review these experiments, derive the Bose-Hubbard model in their context and briefly discuss its limitations in the first half of Chapter 1. In its second half, we review the theory of open quantum systems and the master equation description of the dissipative Bose-Hubbard model. This opening chapter constitutes a survey of existing results, rather than original work. \n \nIn Chapter 2, we turn to the mean-field limit of the Bose-Hubbard model. After reviewing the striking localization phenomena predicted by the mean-field (and confirmed by experiment), we identify the first corrections to this picture for the dimer. The most interesting of these is the dynamical tunneling between the self-trapping points of the mean-field. We derive an accurate analytical expression for the tunneling rate using semiclassical techniques. \n \nWe continue studying the dynamics near the self-trapping fixed points in Chapter 3, focusing on corrections to the mean-field that arise at larger nonlinearities and on shorter time scales than dynamical tunneling. We study the impact of dissipation on coherence and entanglement near the fixed points, and explain it in terms of the structure of the classical phase space. \n \nThe last chapter of the dissertation is also devoted to a dissipative bosonic dimer model, but one arising in a very different physical context. Abandoning optical lattices, we consider the problem of formulating a quantum model of operation of the cylindrical anode magnetron, a vacuum tube crossed-field microwave amplifier. We derive an effective dissipative dimer model and study its relationship to the classical description. Our dimer model is a first step towards the analysis of solid-state analogs of such devices.
The first successful experimental realization of a Bose-Einstein condensate (BEC) in 1995 sparked a still growing research activity that in the meantime has led to beautiful and surprising results, such as the interference of matter waves [4], the Mott-superfluid phase transition [49] or the direct observation of Anderson localization of matter waves [10]. The strong interest in BECs arises, among others, from the possibility to investigate known but hard to study quantum systems on a rather macroscopic scale: for example one can load ultra-cold bosons into an optical lattice created by a standing light wave, and thereby simulate a solid state device. When it comes to a measurement on the BEC, however, essentially all measurement techniques result in the destruction of the condensate: for example in the so called “time of flight method” the optical lattice is switched off to observe the expansion of the trapped atoms and thereby directly visualize their momentum distribution, which highlights the condensed component [11]. In this thesis we elaborate on a means to probe a Bose-Einstein condensate from outside, with a scattering particle. The interaction between the probe particle and the bosons allows energy exchange between them, i.e., this setup includes inelastic scattering. After the scattering process, the condensate is still confined in the optical lattice and can therefore be used for further investigations. Of course, this method of probing a BEC is only then a serious alternative if the properties of the BEC can be inferred from the scattering signal. Besides outlining the scattering formalism, the central aim of this work is to show how the properties of the BEC can be deduced from those quantities accessible in a scattering experiment, like for example the transmission or the inelastic cross section. The other, equally strong motivation of this work concerns the signature of the interaction between the bosons in the scattering quantities. The dynamics of an analog classical system depends strongly on the interaction strength, which can induce a complex structure of the underlying phase space like, e.g., the appearance of a separatrix or even the emergence of chaotic dynamics in a globally mixed regular-chaotic phase space. We are interested to which extent such phase-space metamorphosis is reflected in the properties of the scattering particle. Currently light scattering (Bragg spectroscopy) on an optical lattice [78, 77] or rather on an optical lattice embedded into a cavity [62,68,19] is under intense investigation. These efforts predominantly aim at a characterization of the Mott-superfluid transition by specific scattering properties. In the present thesis we focus on condensates loaded in two and three lattice sites, with a large number of bosons. These systems exhibit a diversity of important physical effects (as we will see below) and thus are of paradigmatic relevance for a detailed understanding of multiparticle quantum dynamics in periodic potentials.