Chaotic Shock Wave And Vortices In Bose-Einstein Condensation

Since the realization of Bose-Einstein condensation in a dilute atomic gas, the relevant investigations have attracted enormous attention. It not only provides per-feet macroscopic quantum systems to investigate many fundamental problems in quantum mechanics, but also has a wide range of applications such as in atom laser, quantum computation and the exactitude measure. In recent years, great progress has been made in the theoretical and experimental studies of Bose-Einstein con-densation. There are also many investigations about the nonlinear structures in Bose-Einstein condensates, such as dark soliton, bright soliton, collapsing wave, vortices and shock wave, which are hot research topics nowadays. In the frame-work of mean-field theory, based on the theoretical analysis and numerical method we discuss the dynamic behavior of shock wave and vortices in one-dimensional and two-dimensional Bose-Einstein condensation existing in various external po-tentials. We obtain some exact solutions of the system under certain parameter conditions and propose a scheme to suppress chaos, to generate and control the movement of vortices. Some meaningful results are discovered in our study.This thesis consists of the following six parts. In the first chapter, we give a brief introduction about the experimental realization of Bose-Einstein conden-sation and relevant fundamental concepts, the mean-field theory and the Gross-Pitaevskii equation. From the mean-field theory, we investigate the dynamic char-acteristic of shock wave and vortices in Bose-Einstein condensation. Finally, a brief introduction on the application prospects and research significances of Bose-Einstein condensation is also provided.In the second chapter, we investigate a one-dimensional open Bose-Einstein condensation with attractive interaction, by considering the effect of feeding from nonequilibrium thermal cloud and applying the time-periodic inverted-harmonic potential. Using the direct perturbation method and the exact shock wave solution of the stationary Gross-Pitaevskii equation, we obtain the chaotic perturbed solu-tion and the Melnikov chaotic regions. Based on the analytical and the numerical methods, the influence of the feeding strength on the chaotie motion is revealed. In the case of "nonpropagated" shock wave with fixed boundary, the number of condensed atoms increases faster as the feeding strength increases. However, for the free boundary the metastable shock wave with fixed front density oscillates its front position and atomic number aperiodically, and their amplitudes decay with the increase of the feeding strength.In chapter three, we investigate the phase effects of a periodically driven Bose-Einstein condensate held in a spatially two-dimensional harmonic or inverted-harmonic potential. A formally exact solution of the time-dependent Gross-Pitaevskii equation is found, which depicts the shock wave with chaotic or periodic ampli-tude and phase. The atomic densities are illustrated numerically, and the circularly symmetric distributions in the separable phase and the axially symmetric Bose-Einstein condensate clusters in the inseparable phase are shown. It is demonstrated that the periodic driving may lead to chaos for both phases, which plays a role in avoiding the escape of the solution and restraining the Bose-Einstein conden-sate collapse and blast. The results suggest a method for controlling the directed transports of the two-dimensional Bose-Einstein condensate.In chapter four, we investigate vortex dynamics of a periodically driven Bose-Einstein condensate confined in a spatially two-dimensional optical lattice. An exact Floquet solution of the Gross-Pitaevskii equation is obtained for a certain pa-rameter region which can be divided into the phase-jumping and phase-continuing regions. In the former region, for a small ratio of driving strength to optical lat-tice depth the vortices keep nearly unmoved. With the increase of the ratio, the vortices undergo an effective interaction and periodically evolve along some fixed circular orbits that leads the vortex dipoles and quadrupoles to produce and break alternatively. There is a critical ratio in the phase-jumping region beyond which the vortices generate and melt periodically. In the phase-continuing region, the condensate in the exact Floquet state evolves periodically without zero-density nodes. The stability and instability of the exact solution are illustrated numer-ically. It is demonstrated that the exact solution is stable for both parameter regions, except for a subregion of the phase-jumping region in which stability of the condensate is lost. However, in the phase-jumping region stability of the solu-tion can be destroyed by a small parameter perturbation. In the phase-continuing region the solution is structurally stable under a small parameter perturbation. The results suggest a scheme for creating and controlling matter-wave vortices.In chapter five, we investigate vortex dynamics behavior of Bose-Einstein con-densate in a spatially two-dimensional harmonic potential with spatial dependent or independent interaction. The stationary and non-stationary vortex solutions can be obtained by designing various laser potentials which can be experimentally realized. For the stationary vortex solutions, vortices remain unmoved. The non-stationary vortex solutions contain stable and unstable vortices clusters. When vortices move periodically along some fixed and closed orbits, vortices clusters are stable. However, for the unstable clusters, vortices disappear and appear period-ically with the evolution of time. The results suggest a scheme for creating and controlling matter-wave vortices.In chapter six, we give a brief summary of the work and a outlook of the applications of the Bose-Einstein condensate system.