In this thesis, we study the parametric excitations of a homogeneous Bose-Einstein condensate(BEC) at finite temperature obtained from modulating the interaction strength. In Bogoliubov theory, a result connecting the real part of the diagonal Bogoliubov coefficient and instability, i.e. exponential growth of excitations, is obtained for general periodic modulations. Consequently, an analytic solution for an arbitrary square wave modulation is derived that can be studied to understand similar drivings like the sinusoidal case. This adds to previous works studying only the case for small driving amplitudes. In the recent experimental and theoretical literature well-established nonclassicality criteria from the field of quantum optics have been directly applied to the case of excitations in matter-waves like those studied here. Among these are violations of Cauchy-Schwarz inequalities, Glauber-Sudarshan P-nonclassicality, sub-Poissonian number difference squeezing (also known as the two-mode variance) and the criterion of nonseparability. We review the strong connection of these criteria and their meaning in quantum optics, and point out differences in the interpretation between light and matter waves. By proposing a classical theory of a thermal BEC that gives the same predictions as the quantum theory in the high temperature limit, and using it as a reference, we can isolate real quantum effects like spontaneous emission caused by quantum noise. We conclude that to date in experiments relevant for this scenario nonclassical effects have not conclusively been observed and conjecture that additional, noncommuting, observables have to be measured to this end. Moreover this has important implications for proposed analog gravity models where the observation of nonclassical effects is a major goal. In the second part of the thesis, we present simulational methods with focus on the Truncated Wigner Approximation (TWA). We take a path integral approach inspired by [Polkovnikov, 2010] and derive a set of classical equations from the semi-classical limit that is more general than TWA. We add some proofs to the literature and identify problems in this approach. We also present a highly optimized software package as an extension of previous work [Jain, 2007]. It can carry out the relevant simulations and we use it to confirm the theoretical predictions of the first part.
In this thesis, we study the parametric excitations of a homogeneous Bose-Einstein condensate(BEC) at finite temperature obtained from modulating the interaction strength. In Bogoliubov theory, a result connecting the real part of the diagonal Bogoliubov coefficient and instability, i.e. exponential growth of excitations, is obtained for general periodic modulations. Consequently, an analytic solution for an arbitrary square wave modulation is derived that can be studied to understand similar drivings like the sinusoidal case. This adds to previous works studying only the case for small driving amplitudes. In the recent experimental and theoretical literature well-established nonclassicality criteria from the field of quantum optics have been directly applied to the case of excitations in matter-waves like those studied here. Among these are violations of Cauchy-Schwarz inequalities, Glauber-Sudarshan P-nonclassicality, sub-Poissonian number difference squeezing (also known as the two-mode variance) and the criterion of nonseparability. We review the strong connection of these criteria and their meaning in quantum optics, and point out differences in the interpretation between light and matter waves. By proposing a classical theory of a thermal BEC that gives the same predictions as the quantum theory in the high temperature limit, and using it as a reference, we can isolate real quantum effects like spontaneous emission caused by quantum noise. We conclude that to date in experiments relevant for this scenario nonclassical effects have not conclusively been observed and conjecture that additional, noncommuting, observables have to be measured to this end. Moreover this has important implications for proposed analog gravity models where the observation of nonclassical effects is a major goal. In the second part of the thesis, we present simulational methods with focus on the Truncated Wigner Approximation (TWA). We take a path integral approach inspired by [Polkovnikov, 2010] and derive a set of classical equations from the semi-classical limit that is more general than TWA. We add some proofs to the literature and identify problems in this approach. We also present a highly optimized software package as an extension of previous work [Jain, 2007]. It can carry out the relevant simulations and we use it to confirm the theoretical predictions of the first part.
In the recent experimental and theoretical literature well-established nonclassicality criteria from the field of quantum optics have been directly applied to the case of excitations in matter-waves. Among these are violations of Cauchy–Schwarz inequalities, Glauber–Sudarshan P-nonclassicality, sub-Poissonian number-difference squeezing (also known as the two-mode variance) and the criterion of nonseparability. We review the strong connection of these criteria and their meaning in quantum optics, and point out differences in the interpretation between light and matter waves. We then calculate observables for a homogeneous Bose–Einstein condensate undergoing an arbitrary modulation in the interaction parameter at finite initial temperature, within both the quantum theory as well as a classical reference. We conclude that to date in experiments relevant for this scenario nonclassical effects have not conclusively been observed and conjecture that additional, noncommuting, observables have to be measured to this end. Moreover this has important implications for proposed analog gravity models where the observation of nonclassical effects is a major goal.
After the first experimental realization of a Bose-Einstein condensate (BEC) in 1995, BECs have become a subject of intense experimental and theoretical study. In this dissertation, I present our results on the classical and quantum dynamics of BECs at zero temperature under different scenarios. First, I consider the analog of slow light in the collision of two BECs near a Feshbach resonance. The scattering length then becomes a function of the collision energy. I derive a generalization of the Gross-Pitaevskii equation for incorporating this energy dependence. In certain parameter regimes, the group velocity of a BEC traveling through another BEC decreases. I also study the feasibility of an experimental realization of this phenomena. Second, I analyze an experiment in which a BEC in a ring-shaped trap is stirred by a rotating barrier. The phase drop across and current flow through the barrier is measured from spiral-shaped density profiles created by interfering the BEC in the ring-shaped trap and a concentric reference BEC after release from all trapping potentials. I show that a free-particle expansion is sufficient to explain the origin of the spiral pattern and relate the phase drop to the geometry of a spiral. I also bound the expansion times for which the phase drop can be accurately determined and study the effect of inter-atomic interactions on the expansion time scales. Third, I study the dynamics of few-mode BECs when they become dynamically unstable after preparing an initial state at a saddle point of the Hamiltonian. I study the dynamics within the truncated Wigner approximation (TWA) and find that, due to phase-space mixing, the expectation value of an observable relaxes to a steady-state value. Using the action-angle formalism, we derive analytical expressions for the steady-state value and the time evolution towards this value. I apply these general results to two systems: a condensate in a double-well potential and a spin-1 (spinor) condensate. Finally, I study quantum corrections beyond the TWA in the semiclassical limit. I derive general expressions for the dynamics of an observable by using the van Vleck-Gutzwiller propagator and find that the interference of classical paths leads to non-perturbative corrections. As a case study, I consider a single-mode nonlinear oscillator; this system displays collapse and revival of observables. I find that the interference of classical paths, which is absent in the TWA, leads to revivals.
The experimental discovery of dilute ultracold quantum gases in 1995 has instigated a broad interest in ultracold atoms and molecules, and paved the way for extensive studies of a wide range of both experimental and theoretical topics. Various theories were developed to describe Bose-Einstein condensates (BECs) for different ranges of temperature and interaction strengths. This thesis focuses on studying the mean-field theory of the Gross-Pitaevskii (GP) equation which provides a good description for the macroscopic wave function when both quantum and thermal fluctuations are negligible. This means that the range of validity is restricted to small interaction strengths at zero temperature. Within the GP theory we study analytically and numerically the nonlinear dynamics of BECs which are induced by a harmonic modulation of either the interaction strength or the harmonic trapping potential. At first a detailed numerical investigation reveals that solving the partial differential equation of Gross and Pitaevskii can be reduced to solving ordinary differential equations for the condensate widths within a Gaussian variational approach. Despite this tremendous simplification the prominent nonlinearity of the underlying GP equation is inherited by the variational equations, which thus allow to describe a rich variety of nonlinear phenomena in different experimental setups. In order to describe them analytically we transfer the Poincar\'e-Lindstedt method, which represents a well-established tool in the field of nonlinear dynamics, to the Gaussian variational approach. Modulation of some of the parameters of a BEC can give rise to prominent nonlinear features, such as shifts in the frequencies of collective oscillation modes, higher harmonics generation, and resonant mode coupling. As a first application we consider a recent Bose-Einstein condensate of Li-7, which has been excited experimentally by a harmonic modulation of the atomic s-wave scattering length with the help of a Feshbach resonance. Combining an analytical perturbative approach with extensive numerical simulations, we analyze the emerging nonlinear dynamics. In particular, we present the resulting shift of collective excitation spectra close to parametric resonances where the driving frequency coincides with a collective mode frequency. Another application of experimental importance is to study geometric resonances in BECs with two-body and three-body contact interactions, where changing the geometry of the trapping potential leads to commensurate collective excitation frequencies with emerging nonlinear effects such as frequency shifts and resonant mode coupling. In this context we also show that a small repulsive three-body interaction is able to extend the stability region of the condensate for an attractive two-body interaction as it increases the critical number of atoms in the trap. Apart from a periodic modulation of a system parameter the dipole mode frequency can be excited by introducing an abrupt change in the potential, where the center of mass oscillates according to the Kohn theorem back and forth in the trapping potential with the natural trap frequency irrespective of both the strength and the type of the two-particle interaction. To this end, we study the collective excitation modes of a Bose-Einstein condensate in a harmonic Ioffe- Pritchard in the vicinity of a Feshbach resonance for experimentally realistic parameters of a Rb-85 BEC, where the dipole mode is excited in z-direction. A linearization of the underlying equations of motions shows that the dipole mode frequency decreases when the bias magnetic field approaches the Feshbach resonance, so the Kohn theorem is violated. Finally within a variational approach we discuss the physical properties of a BEC in an axially-symmetric harmonic trap with a single vortex in the center. At first we analyze the equilibrium configuration and determine the vortex size as well as the Thomas- Fermi radii of the condensate in the Thomas-Fermi limit. Then we calculate the critical rotation frequency for the emergence of the vortex and compare our findings with the literature. Finally, we investigate how the presence of the vortex changes the collective excitation frequencies and we discuss the free expansion of the BEC. All these results indicate how nonlinear dynamics properties could be made clearly observable in future experiments on the basis of our results.
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 examines two aspects of the non-equilibrium dynamics of Bose-Einstein condensates (BECs). Our particular interests are the relaxation of near-integrable systems, and the inter-particle correlations present in BECs, both in equilibrium and following an abrupt change in the Hamiltonian. In the first part of this thesis we investigate the stability of solitary wave solutions to a system of two coupled one-dimensional non-linear Schrodinger equations (CNLSE). This system has an integrable point known as the Manakov[1] model which has analytic dark-bright soliton solutions[2]. We break the integrability of this model by varying the inter-species interaction strength, and employ variational and numerical methods in order to find dark-bright solitary wave solutions, away from the integrable point. The time evolution of these states is calculated via numerical integration of the CNLSE, which allows us to assess the stability of these solutions in isolation and to quantify their robustness against collisions with a dark soliton. We find that there is a broad region of the parameter space in which solutions for black-bright (stationary) solitary waves can be found. In this domain, the integrability breaking is revealed only during collisions. Prior to a collision, the numerical solutions are indistinguishable from true solitons, however, the collisional stability of these solitary waves is significantly affected by the extent to which integrability is broken. We find that there is a smooth transition between the non-dispersive particle-like nature of soliton interactions for integrable systems, and the destructive collisions which take place when the system is perturbed far from the integrable point. In the case of moving (grey-bright) solitary waves, we find a more restricted region of the parameter space which admits solitary wave solutions. In this domain we are able to find approximately stable solitary wave solutions using a variational ansatz. The collisional stability of these grey-bright solitary waves is also found to depend upon the extent to which integrability is broken. We conclude that the observation of stable, long-lived dark-bright solitary waves is not sufficient to indicate near-integrability of the two-component system. However, near-integrability may inferred if dark-bright solitary waves are observed to survive multiple collisions. These results are not only of theoretical interest, but may also inform future experiments which explore the behaviour of solitary waves in two-component Bose-Einstein condensates. Following this we investigate the properties of magnetic solitary wave solutions to the CNLSE. We find that the analytic ‘magnetic soliton’ solutions derived in reference [3] are dynamically unstable and relax to non-stationary magnetic solitary wave states over a short timescale. We investigate the collisional dynamics of these magnetic solitary waves and find that these excitations are remarkably robust. In the second part of this thesis we calculate the fluctuations in the number difference between two sub-regions of a Bose-Einstein condensate. This study is motivated by recent experiments performed by the Truscott group at the Australian National University (ANU), which indicate sub- Poissonian fluctuations even for condensates with large thermal depletion. This result cannot easily be reconciled with existing literature, which predicts strongly super-Poissonian fluctuations below the critical temperature (Tc). We develop one-dimensional, and three-dimensional, models to describe these experiments and employ the Bogoliubov formalism to explore the quantum fluctuations of the condensate, which have not previously been studied in detail. We calculate the relative number fluctuations between two halves of the condensate, and also between multiple pairs of spatial bins. For harmonically confined condensates at zero temperature, quantum correlations result in a suppression of the relative number fluctuations between the two halves of the condensate below the shot-noise level. However, the presence of even a small thermal component results in strongly super-Poissonian fluctuations below Tc; this is in agreement with the existing theoretical and experimental results. In addition, we find that the fluctuations of the harmonically confined condensate are non-uniform, with the relative number fluctuations being peaked towards the edges of the condensate. The recent studies of the Truscott group measure the fluctuations after free expansion. We therefore calculate the propagation of correlations following a sudden removal of the harmonic confinement. We find that free expansion results in a strong decrease in the relative number fluctuations. Condensates which exhibit super-Poissonian fluctuations when harmonically confined may therefore display number- squeezing after expansion. A second set of experiments by the Truscott group measured the density fluctuations present in small ‘atom laser’ pulses of atoms out-coupled from their BEC. We also model this procedure and find that immediately after the out-coupling process, the number fluctuations of the out-coupled atomic pulse are approximately Poissonian, irrespective of the correlations present in the atomic reservoir. However, the competing effects of interactions between the atomic pulse and the condensate reservoir, and free expansion of the pulse after out-coupling, can result in the measured fluctuations being either number squeezed or super-Poissonian. Our results will guide future experiments planned by the Truscott group.
Recent experimental observations of collective excitations of Bose condensed atomic vapours have stimulated interest in the microscopic description of the dynamics of a Bose-Einstein condensate confined in an external potential. We present a finite temperature field theory for collective excitations of trapped Bose-Einstein condensates and use a finite-temperature linear response formalism, which goes beyond the simple mean-field approximation of the Gross-Pitaevskii equation. The effect of the non-condensed thermal atoms we include using perturbation theory in a quasiparticle basis. This presents a simple scheme to understand the interaction between condensate and non-condensed atoms and enables us to include the effect the condensate has on collision dynamics. At first we limit our treatment to the case of a spatially homogeneous Bose gas. We include the effect of pair and triplet anomalous averages and thus obtain a gapless theory for the excitations of a weakly interacting system, which we can link to well known results for Landau and Beliaev damping rates. A gapless theory for trapped systems with a static thermal component follows straightforwardly. We then investigate finite temperature excitations of a condensate in a spherically symmetric harmonic trap. We avoid approximations to the density of states and thus emphasise finite size aspects of the problem. We show that excitations couple strongly to a restricted number of modes, giving rise to resonance structure in their frequency spectra. Where possible we derive energy shifts and lifetimes of excitations. For one particular mode, the breathing mode, the effects of the discreteness of the system are sufficiently pronounced that the simple picture of an energy shift and width fails. Experiments in spherical traps have recently become feasible and should be able to test our detailed quantitative predictions.