Dynamics of Bose-Einstein Condensate in Linear and Non-Linear Regime

Abstract

The fascinating eld of exploring non-linear systems exhibits rich dynamics and these systems are found in almost every part of nature, whether it is the periodic beating of the heart, the ripples of sand dunes, complex shapes in snow akes, or the existence of stripes on Jupiter and on animals. Beyond rich spatial and temporal patterns, nonlinearity present in any system can lead to the formation and evolution of localized structures which have unusual features such as solitons and vortices. Solitons are spatially localized structures or waves which arise due to the balance of dispersion and nonlinearity in a system. These structures maintain their shape during propagation, are remarkably stable against any perturbations or mutual collision, and show particle-like properties. In general, solitons underlie the understanding of tidal bores, cyclones, massive ocean waves like tsunamis, signal conduction in neurons, and natural phenomena such as \Morning Glory" (hundreds of kilometers of cloud waves). Solitons have been actively studied in many di erent domains including in mathematics and physics (such as in the context of the solution of the Korteweg-de Vries equation, in shallow water, in magnetic thin lms and optical bers). Solitons have given a massive impetus to today's telecommunications industry due to the ability of optical pulses to propagate as solitons for vast distances without signi cant loss or dispersion. In this thesis, we explored, through experiments and numerical simulations, the behaviour and dynamics of the Bose-Einstein Condensate (BEC) in two di erent regimes: linear (interactions in the system are negligible) and nonlinear regime (interactions in the system becomes dominant). A BEC of dilute atomic vapor not only provides a clean and well-controlled environment to study a variety of physics problems of super uid systems including solitons and vortices but also opens up avenues to investigate the interface between the quantum and the classical systems. While probing the nonlinear regime of the condensate (through numerical simulations), we focused on the formation and dynamics of dark solitons in a 2D Rb BEC. Solitons in a BEC can be of two types: bright (for attractive inter-particle interaction) and dark (for repulsive inter-particle interaction). A matter wave dark soliton formed in a BEC is a dip in atom density with a phase gradient across the dip. This phase across the dip determines the propagation speed of the solitons in the condensate. Stationary solitons are called dark solitons while moving solitons are called gray solitons. Within the framework of mean eld approximation, we used a non-linear Schr odinger equation called Gross-Pitaevskii equation to numerically realise the formation of dark solitons in a 2D Rb BEC using various techniques such as phase imprinting and density engineering. In this part of our numerical study, we imprinted a smooth phase gradient (experimentally likely condition) and contrasted it with imprinting a sharp phase gradient (usually found in literature) onto the condensate. We studied the outcomes of these imprintings on the generation and instability dynamics of solitons and were able to highlight the e ects of the imprinting on long-term vortex decay dynamics. We pointed out the rich dynamics exhibited by the vortex dipoles stemming from the unstable dark soliton and consolidated an alternate method to generate dark solitons in double well potential under periodic modulation of interactions. We also explored the formation of a transient soliton lattice and the existence of an intriguing phase wherein the Faraday pattern and soliton lattice were found to coexist under certain conditions. It needs to be pointed out that the limitation of this study arose from the fact that we had access only to a 3D Rb condensate in our lab. Therefore, in the nal phase of the current research, we investigated the limitations of experimentally realizing the above-mentioned numerical study of phase imprinting of a 2D Rb condensate. In the same vein, we studied the possibilities of modifying our experimental set-up to overcome this limitation. We also conducted an experimental and numerical study to probe the linear regime of the condensate. To investigate the linear regime, bouncing dynamics of the 3D spherical Rb condensate from a Gaussian barrier were studied; this is popularly known as the `Quantum Bouncer' problem. This part of the study resulted in the emergence of unusual fringe patterns in the condensate. We also looked at the possibility of transforming the fringe pattern into solitons, which was numerically con rmed by changing the interaction strength in the condensate.