Collective Excitations And Nonlinear Dynamics In Bose-Einstein Condensates

In recent years, Bose-Einstein condensation (BEC) in dilute atomic gases has become an interesting study subject of physics. Not only can it provide a macro-system, by which quantum mechanics can be studied in another way, but also it will open a wide perspective of application, such as atomic laser, quantum information processing, and precision measurements. Nonlinearity and external potential make the system disinte-grable and complex, so the exploration about this macro quantum object has also become one of main tasks in the field of nonlinear dynamics. In this thesis, by mean-field frame of the Gross-Pitaevskii equation, the collective behaviors of the BEC are investigated analytically and numerically.In the first part, the collective excitations of a quasi-one-dimensional BEC in a an-harmonic trap V(x) =1/2(x~2 +λx~2) are investigated. Two low-energy collective modes are derived analytically by using standard variational method, and the effects of anhar-monicity of the trap on excitations of a BEC are studied. It is found that the low-energy collective frequencies are blue or red shifted depending onλ> 0 orλ< 0. Meanwhile, the variation of width and the motion of the center-of-mass are discussed. The result show that the beatings corresponding to variation of width, which indicate coupling between the modes, occur at different driving amplitudes. Finally, the collapse and revival of collective excitation induced by mode coupling is also demonstrated.Secondly, the parametric excitations in two-component BECs described by two coupled Gross-Pitaevskii equations (GPEs) are investigated analytically and numerically. By using the variational approximation, we derive the coupled equations of motion for the center-of-mass coordinates of the two condensates and their widths. The harmonic generation and nonlinear coupling of oscillation modes associated with the nonlinear dynamics of binary mixtures are revealed by exciting the internal parameter of the system. Meanwhile, we obtain analytically the resonance condition and resonance boundaries in terms of the modulation frequency and the strength of intercomponent interaction. In particular, we demonstrate a novel method of formation of multisoliton configurations (including soliton trains, soliton pairs and domain walls) through parametric resonance in two-component BECs by direct numerical simulation for the one-dimensional coupled GPEs.Finally, the evolution of BECs loaded into a periodic ring optical lattices (OL) trap is studied. By means of the variational method and direct numerical simulations of the Gross-Pitaevskii (GP) equation, the ground state properties and the vortex stabilities of the condensates for both repulsive and attractive cases are investigated. The results show that the bound states exist for determinate OL strength and interatomic interaction. However, the ground states of BECs undergo delocalizing-localizing transition for both attractive and repulsive cases as the strength of the OL or the interatomic interaction is decreased below the critical value. The ring OL can suppress the delocalizing transition efficiently. The physics behind this phenomenon is similar with the mechanism of the self-trapping in double well potential.The results can help us to realize the dynamical properties of BECs and are significant for real application in future. In addition, We hope our theoretical results will stimulate the experiments in the direction, and the corresponding results can be tested in future experiments.