Abstract The soliton solutions of the two-dimensional homogeneous Bose–Einstein Condensates Model that is related to the atomic interactions strength and the initial momentum have been discussed through this article to explain the vision when the initial momentum is sufficiently high; the thickness of the bubble-shaped BEC undergoes a counterintuitive thinning. In quantum statistics, the suggested model indicates to two available ways in which a group of noninteracting similar particles may occupy a collection of obtainable possible discrete energy states at thermodynamic equilibrium has been investigated. Many new types of soliton solutions that weren’t achieved before can be constructed, and its descriptions have been documented in the conclusion. In order to derive these new soliton perceptions of the perturbed dynamics of rotating as well as the norm square of the obtaining solution that physically denote to the envelop of these dynamics, we will apply five various schemas. The five distinct schemas that are implemented for the first time to this target are the generalized Kudryashov Schema, the Paul-Painleve Approach schema, the (G′/G)-Expansion Schema, the Extended Simple Equation Schema, and the Riccati–Bernoulli Sub Order Schema are implemented individually, independently in consecutive stages. The consistency for all explored analytical solutions are cleared although these schemas have different tools in prepared. The 2-kind and 3-kind graphs simulations as well as the norm square, which denote physically to envelop of these dynamics, have been documented. The configurator figures that detect the Dynamics’s behaviors as well as all mathematical treatment have been defined via Mathematica program.
Abstract The soliton solutions of the two-dimensional homogeneous Bose–Einstein Condensates Model that is related to the atomic interactions strength and the initial momentum have been discussed through this article to explain the vision when the initial momentum is sufficiently high; the thickness of the bubble-shaped BEC undergoes a counterintuitive thinning. In quantum statistics, the suggested model indicates to two available ways in which a group of noninteracting similar particles may occupy a collection of obtainable possible discrete energy states at thermodynamic equilibrium has been investigated. Many new types of soliton solutions that weren’t achieved before can be constructed, and its descriptions have been documented in the conclusion. In order to derive these new soliton perceptions of the perturbed dynamics of rotating as well as the norm square of the obtaining solution that physically denote to the envelop of these dynamics, we will apply five various schemas. The five distinct schemas that are implemented for the first time to this target are the generalized Kudryashov Schema, the Paul-Painleve Approach schema, the (G′/G)-Expansion Schema, the Extended Simple Equation Schema, and the Riccati–Bernoulli Sub Order Schema are implemented individually, independently in consecutive stages. The consistency for all explored analytical solutions are cleared although these schemas have different tools in prepared. The 2-kind and 3-kind graphs simulations as well as the norm square, which denote physically to envelop of these dynamics, have been documented. The configurator figures that detect the Dynamics’s behaviors as well as all mathematical treatment have been defined via Mathematica program.
We numerically study the dynamics of rotating ring-shaped perturbation on two-dimensional homogeneous Bose–Einstein condensates, where a new ring-shaped structure with reverse rotation appears. The reversely rotating mode is directly caused by the existence of the plane wave (namely the homogeneous background). By the modified linear stability analysis method, we quantitatively predict the influence of the background's density on perturbation dynamics, including the velocity, amplitude, and frequency of the two rings. We construct an approximative solution to describe the short-lived dynamics of initial perturbation, which agrees well with our numerical results. Also, after the two rings separate, the transfer of atom number between them becomes linear, and the rate of transfer is impacted by the radial momentum of initial perturbation.
Bose–Einstein Condensation (BEC) of atomic gases has attracted a renewed theoretical and experimental interest in quantum many-body systems at extremely low temperatures (Pethick & Smith; 2002). This excitement stems from two favorable features: (1) by applying magnetic fields and lasers, most of the system parameters, such as the shape, dimensionality, internal states of the condensates, and even the strength of the interatomic interactions, are controllable; (2) due to the diluteness, the mean-field theory explains experiments quite well. In particular, the Gross–Pitaevskii (GP) equation demonstrates its validity as a basic equation for the condensate dynamics. The GP equation is the counterpart of the nonlinear Schrodinger (NLS) equation in nonlinear optics. Thus, a study based on nonlinear analysis is possible and important. In nonlinear physics, a soliton is remarkable object not only for the fact that exact solutions can be obtained but also for its usefulness as a communications tool due to its robustness. In general, solitons are formed under the balance between nonlinearity and dispersion. For atomic condensates, the former is attributed to the interatomic interactions, while the latter comes from the kinetic energy. Either dark or bright solitons are allowable depending on the positive or negative sign of the interatomic coupling constants g, respectively, and indeed have been observed in a quasione-dimensional (q1D) optically constructed waveguide (Strecker et al.; 2002) (Khaykovich et al.; 2002). Such matter-wave solitons are expected in atom optics for applications in atom laser, atom interferometry, and coherent atom transport (Meystre; 2002). In this chapter, we extend the analysis of the matter-wave solitons to a multicomponent case by considering the so-called spinor condensate (Stenger et al.; 1998) whose spin degrees of freedom are liberated under optical traps. Based on theoretical and experimental results, we introduce a new integrable model which describes the dynamical properties of the matter-wave soliton of spinor condensates (Ieda et al.; 2004a). We employ the inverse scattering method to solve this model exactly. As a result, we predict the occurrence of undiscovered physical phenomena such as macroscopic spin precession and spin switching. The chapter is organized as follows. In Sec. 2, the mean field theory of condensate is briefly reviewed. Section 3 introduces an effective interatomic coupling in a q1D condensate. Using these results, we consider a spinor condensate in q1D regime in Sec. 4. Then, in Sec. 5, we
We investigated the dynamic evolution and interference phenomena of bubble-shaped Bose–Einstein condensates achievable in a micro-gravity environment. Using numerical solutions of the Gross–Pitaevskii equation describing the dynamic evolution of the bubble-shaped Bose–Einstein condensates, we plotted the evolution of the wave function density distribution on its two-dimensional (2D) cross-section and analysed the resulting patterns. We found that changes in the strength of atomic interactions and initial momentum can affect the dynamic evolution of the bubble-shaped Bose–Einstein condensates and their interference fringes. Notably, we have observed that when the initial momentum is sufficiently high, the thickness of the bubble-shaped Bose–Einstein condensate undergoes a counterintuitive thinning, which is a counterintuitive result that requires further investigation. Our findings are poised to advance our comprehension of the physical essence of bubble-shaped Bose–Einstein condensates and to facilitate the development of relevant experiments in micro-gravity environments.
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 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.
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.