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 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 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.
We study the dynamics of dark-bright (DB) solitons in binary mixtures of Bose gases at finite temperature using a system of two coupled dissipative Gross–Pitaevskii equations. We develop a perturbation theory for the two-component system to derive an equation of motion for the soliton centers and identify different temperature-dependent damping regimes. We show that the effect of the bright ('filling') soliton component is to partially stabilize 'bare' dark solitons against temperature-induced dissipation, thus providing longer lifetimes. We also study analytically thermal effects on DB soliton 'molecules' (i.e. two in-phase and out-of-phase DB solitons), showing that they undergo expanding oscillations while interacting. Our analytical findings are in good agreement with results obtained via a Bogoliubov–de Gennes analysis and direct numerical simulations.
Motivated by recent experimental results, we study beating dark–dark (DD) solitons as a prototypical coherent structure that emerges in two-component Bose–Einstein condensates. We showcase their connection to dark–bright solitons via SO (2) rotation, and infer from it both their intrinsic beating frequency and their frequency of oscillation inside a parabolic trap. We identify them as exact periodic orbits in the Manakov limit of equal inter- and intra-species nonlinearity strengths with and without the trap and showcase the persistence of such states upon weak deviations from this limit. We also consider large deviations from the Manakov limit illustrating that this breathing state may be broken apart into dark–anti-dark soliton states. Finally, we consider the dynamics and interactions of two beating DD solitons in the absence and in the presence of the trap, inferring their typically repulsive interaction.
In the present contribution, we explore a host of different stationary states, namely dark-bright solitons and their lattices, that arise in the context of multicomponent atomic Bose-Einstein condensates. The latter are modeled by systems of coupled Gross-Pitaevskii equations with general interaction (nonlinearity) coefficients gij. It is found that in some particular parameter ranges such solutions can be obtained in analytical form, however, numerically they are computed as existing in a far wider parametric range. Many features of the solutions under study, such as their analytical form without the trap or the stability and dynamical properties of one dark-bright soliton even in the presence of the trap are obtained analytically and corroborated numerically. Additional features, such as the stability of soliton lattice homogeneous states or their existence and stability in the presence of the trap, are examined numerically.
Motivated by recent experimental results, we present a systematic theoretical analysis of dark-bright-soliton interactions and multiple-dark-bright-soliton complexes in atomic two-component Bose-Einstein condensates. We study analytically the interactions between two dark-bright solitons in a homogeneous condensate and then extend our considerations to the presence of the trap. We illustrate the existence of robust stationary dark-bright-soliton ``molecules,'' composed of two or more solitons, which are formed due to the competition of the interaction forces between the dark- and bright-soliton components and the trap force. Our analysis is based on an effective equation of motion, derived for the distance between two dark-bright solitons. This equation provides equilibrium positions and characteristic oscillation frequencies of the solitons, which are found to be in good agreement with the eigenfrequencies of the anomalous modes of the system.
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.