Bose-Einstein condensates (BECs) have been one of the most active areas of research since their experimental birth in 1995. The complicated nature of the experiments on BECs suggests to observe them in reduced dimensions. The dependence of the collective excitations of the systems on the spatial degrees of freedom allows the study in lower dimensions. In this thesis, we first study two effectively one-dimensional parallel linearly coupled BECs in the presence of external potentials. The system is modelled by linearly coupled Gross-Pitaevskii (GP) equations. In particular, we discuss the dark solitary waves and the grey-soliton-like solutions representing analogues of superconducting Josephson fluxons which we refer to as the fluxon analogue (FA) solutions. We analyze the existence, stability and time dynamics of FA solutions and coupled dark solitons in the presence of a harmonic trap. We observe that the presence of the harmonic trap destabilizes the FA solutions. However, stabilization is possible by controlling the effective linear coupling between the condensates. We also derive theoretical approximations based on variational formulations to study the dynamics of the solutions semi-analytically.\n\nWe then study multiple FA solutions and coupled dark solitons in the same settings. We examine the effects of trapping strength on the existence and stability of the localized solutions. We also consider the interactions of multiple FA solutions as well as coupled dark solitons. In addition, we determine the oscillation frequencies of the prototypical structures of two and three FA solutions using a variational approach.\n\nFinally, we consider two effectively two-dimensional parallel coupled BECs enclosed in a double well potential. The system is modelled by two GP equations coupled by linear and nonlinear cross-phase-modulations. We study a large set of radially symmetric nonlinear solutions of the system in the focusing and defocusing cases. The relevant three principal branches, i.e. the ground state and the first two excited states, are continued as a function of either linear or nonlinear couplings. We investigate the linear stability and time evolution of these solutions in the absence and presence of a topological charge. We notice that only the chargeless or charged ground states can be stabilized by adjusting the linear or nonlinear coupling between the condensates.
Bose-Einstein condensates (BECs) have been one of the most active areas of research since their experimental birth in 1995. The complicated nature of the experiments on BECs suggests to observe them in reduced dimensions. The dependence of the collective excitations of the systems on the spatial degrees of freedom allows the study in lower dimensions. In this thesis, we first study two effectively one-dimensional parallel linearly coupled BECs in the presence of external potentials. The system is modelled by linearly coupled Gross-Pitaevskii (GP) equations. In particular, we discuss the dark solitary waves and the grey-soliton-like solutions representing analogues of superconducting Josephson fluxons which we refer to as the fluxon analogue (FA) solutions. We analyze the existence, stability and time dynamics of FA solutions and coupled dark solitons in the presence of a harmonic trap. We observe that the presence of the harmonic trap destabilizes the FA solutions. However, stabilization is possible by controlling the effective linear coupling between the condensates. We also derive theoretical approximations based on variational formulations to study the dynamics of the solutions semi-analytically.\n\nWe then study multiple FA solutions and coupled dark solitons in the same settings. We examine the effects of trapping strength on the existence and stability of the localized solutions. We also consider the interactions of multiple FA solutions as well as coupled dark solitons. In addition, we determine the oscillation frequencies of the prototypical structures of two and three FA solutions using a variational approach.\n\nFinally, we consider two effectively two-dimensional parallel coupled BECs enclosed in a double well potential. The system is modelled by two GP equations coupled by linear and nonlinear cross-phase-modulations. We study a large set of radially symmetric nonlinear solutions of the system in the focusing and defocusing cases. The relevant three principal branches, i.e. the ground state and the first two excited states, are continued as a function of either linear or nonlinear couplings. We investigate the linear stability and time evolution of these solutions in the absence and presence of a topological charge. We notice that only the chargeless or charged ground states can be stabilized by adjusting the linear or nonlinear coupling between the condensates.
Two effectively one-dimensional parallel coupled Bose–Einstein condensates in the presence of external potentials are studied. The system is modelled by linearly coupled Gross–Pitaevskii equations. In particular, grey-soliton-like solutions representing analogues of superconducting Josephson fluxons as well as coupled dark solitons are discussed. Theoretical approximations based on variational formulations are derived. It is found that the presence of a magnetic trap can destabilize the fluxon analogues. However, stabilization is possible by controlling the effective linear coupling between the condensates.
This dissertation focuses on the properties of nonlinear waves in Bose-Einstein condensates (BECs). The fundamental model here is the nonlinear Schrodinger equation, the so-called Gross-Pitaevskii (GP) equation, which is a mean-field description of BECs. The systematic analysis begins by considering the dark-bright (DB)-soliton interactions and multiple-dark-bright-soliton complexes in atomic two-component BECs. The interaction between two DB solitons in a homogeneous condensate and at the presence of the trap are both considered. Our analytical approximation relies in a Hamiltonian perturbation theory, which leads to an equation of motion of the centers of DB-soliton interacting pairs. Employing this equation, we demonstrate the existence of robust DB-soliton molecules, in the form of stationary two- and three-DB-soliton states. Also the equilibrium distance of the constituent solitons and the corresponding oscillation frequencies are found semianalytically, where the latter corresponds to the characteristic anomalous modes' eigenfrequencies that we numerically computed via a so called Bogoliubov-de Gennes (BdG) linearization analysis. Those studies are discussed in Chapter 2. Then, we extend our studies to the dynamics of dark-bright (DB) solitons in binary BECs at finite temperature using a system of two-coupled dissipative GPs. We show that the effect of the bright soliton is to partially stabilize dark solitons against temperature-induced dissipation, thus providing longer lifetimes in Chapter 3. Furthermore, the dark-dark (DD) solitons as a prototypical coherent structure that emerges in two-component BECs are studied and are connected to dark-bright (DB) solitons via SO(2) rotation. We obtained their beating frequency and their frequency of oscillation inside a parabolic trap. They are identified as exact periodic orbits in the Manakov limit of equal inter- and intra- species nonlinearity strengths with and without the trap and we showcase the persistence of such states upon weak deviations from this limit. Also we investigated in detail the effect of the deviation from the Manakov case by considering different from unity scattering length ratios in Chapter 4. Next, we revisited Hamiltonian eigenvalue problems that typically arise in the linearization around a stationary state of a Hamiltonian nonlinear PDE. Also we presented a overview of the known facts for the eigenvalue counts of the corresponding unstable spectra. In particular, we focused on a straightforward plan to implement finite-dimensional techniques for locating this spectrum via the singular points of the meromorphic Krein Matrix and illustrated the value of the approach by considering realistic problems for recently observed experimentally multivortex and multisoliton solutions in atomic Bose-Einstein condensates in Chapter 5. In the two dimensional scenario, we also examine the stability and dynamics of vortices under the effect of dissipation used as a simplified model for the inclusion of the effect of finite temperatures in atomic BECs, which enables an analytical prediction that can be compared directly to numerical results in Chapter 6. In all the above studies, our analytical prediction from the equation of motion are in good agreement with the numerical results from the BdG analysis.
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.
This chapter focuses on exploring localized excitations in the variable coefficient nonlinear Schrodinger (NLS) equation and its generalizations. The motivation comes from the fact that the NLS equation and its variants appear in several branches/topics of physics, including nonlinear optics and Bose-Einstein condensates (BECs). The chapter reviews critically the variety of matter wave solitons and their interactions in single- and multi-component Bose-Einstein condensates. It provides a brief overview of the mean Gross- Pitaevskii (GP) equation, derivation of the one-dimensional GP equation, and mapping of the 1D GP equation to the standard NLS equation. The chapter discusses matter wave single and multi bright and dark solitons of Bose– Einstein condensates with varying trap potentials and atomic interactions. It shows various solitons solutions of multi-component BECs described by the two coupled GP equations. In this case, the coupled GP equation has been mapped onto a two coupled NLS equation.
This thesis includes theoretical studies regarding stability and manipulation of Bose-Einstein condensates (BECs) of ultra-cold atoms in 2D trapping geometry, as well as formation of steady states of exciton-polariton Bose-Einstein condensates created in solid states. We analyze and numerically model the dynamics and localization of the condensates using mean-field model. Chapter 1 contains an introduction to the physics of ultra-cold atom BEC and excitonpolariton BEC which provides a framework for the work presented in later chapters. In Chapter 2, we consider a method for achieving dynamically controllable transport of highly mobile matter-wave solitons in an ultra-cold atom BEC with attractive interparticle interaction loaded into a driven two-dimensional optical lattice. Our numerical analysis based on the mean-field model and the theory based on the effective particle approach demonstrate that fast, time-periodic rocking of the two-dimensional optical lattice enables efficient stabilization and manipulation of spatially localized matter wave packets via induced reconfigurable mobility channels. Chapter 3 consists of an investigation of the instability - collapse of a BEC with attractive interactions. In this chapter we explore the influence of an orbital angular momentum on the collapse of vortex-free elliptic clouds of Bose-Einstein condensates trapped in a radially symmetric harmonic potential or a rotating elliptic potential. The results of our analysis show that the number of trapped ultracold atoms corresponding to the collapse threshold can be radically increased for such rotating nonlinear matter waves in a radially harmonic trap. The results corresponding to a BEC cloud confined in a rotating elliptic trap show that the elongated stationary states can be parallel or perpendicular to the long axis of the trap and display bistable nature. In Chapter 4, we examine spatial localization and dynamical stability of Bose-Einstein condensates of exciton-polaritons in microcavities under the condition of off-resonant spatially inhomogeneous optical pumping both with and without a harmonic trapping potential. We employ the open-dissipative Gross-Pitaevskii model for describing an incoherently pumped polariton condensate coupled to an exciton reservoir. We reveal that spatial localization of the steady-state condensate occurs due to effective self-trapping created by the polariton flows, regardless of the presence of the external potential. A ground state of the polariton condensate with repulsive interactions between the quasiparticles represents a dynamically stable bright dissipative soliton. We also investigate the conditions for sustaining spatially localized structures, with nonzero angular momentum, in the form of single-charge vortices. Chapter 5 consider the existence of novel spatially localized states of exciton-polariton Bose-Einstein condensates in semiconductor microcavities with fabricated periodic inplane potentials. Our theory shows that, under the conditions of continuous nonresonant pumping, localization is observed for a wide range of optical pump parameters due to effective potentials self-induced by the polariton flows in the spatially periodic system. We show that the self-localization of exciton-polaritons in the lattice may occur both in the gaps and bands of the single-particle linear spectrum, and is dominated by the effects of gain and dissipation rather than the structured potential, in sharp contrast to the conservative condensates of ultra-cold alkali atoms.
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.