Dynamical Behaviors In Bose-Einstein Condensates In Optical Lattice

Dynamical behaviors of Bose-Einstein condensate (BEC) in optical lattices have attracted enormous attention both experimentally and theo-retically. A lot of research shows the atoms interactions play a important role for dynamic properties of BEC in optical lattices. The energy band structure, tunneling dynamics, the stabilities, self-trapped states of BEC are affected dramatically by atoms interactions. The self-trapped phenom-ena of BEC in optical lattice belong to the fields of nonlinear quantum, it is relevant to the stabilities of cold atoms system and quantum phase tran-sition. However, most present studies are only focused on considering the two-body interaction for dynamic behaviors of BEC in optical lattice. In fact, The three-body interaction can not be ignored for the real behaviors of BEC. It have influenced dramatically for Dynamic behaviors of BEC. So, one aim of this article is to study the dynamic behavior of BEC in op-tical lattices with two-body and three-body interactions. The three-body interaction have important affect on the stabilities and the macroscopic quantum self-trapping of the system. On the other hand, the aim of this article is to investigate the influence of the relative phase of the two lat-tice harmonics on the band structure, the tunneling dynamics of BEC in Fourier-Synthesized optical. The major result are listed as follows:Chapterâ… :The history of research, the realization technique, and application prospect and research meaning of BEC are reviewed briefly.Chapterâ…¡:We present on the basic theory of BEC.Chapterâ…¢:We study the dynamical behaviors of BEC with two- and three-atom interactions in optical lattices with analytical and numeri-cal methods. It is found that the steady-state relative population appears tuning-fork bifurcation when the system parameters are changed to certain critical values. In particular, the existence of three-body interaction not only transforms the bifurcation point of the system but also affects greatly on the macroscopic quantum self-trapping behaviors associated with the critically stable steady-state solution. In addition, we also investigated the influence of the initial conditions, three-body interaction, and the energy bias on the macroscopic quantum self-trapping. Finally, by applying the periodic modulation on the energy bias, we observed that the relative pop-ulation oscillation exhibits a process from order to chaos, via a series of period-doubling bifurcations.Chapter IV:By employing a nonlinear three-mode model, the band structure of BEC in Fourier-Synthesized optical lattices is studied, where the nonlinearity is coming from the mean field treatment of interaction be-tween atoms. In linear case, we obtain the band structure for certain poten-tial values. It is demonstrated that the energy band structure is strongly dependent on the value of relative phase of the two lattice harmonics. In the nonlinear case, we show that the eigenenergies as the functions of the quasi-momentum have a novel bowl structure in the middle energy level . We find that there exist four critical values of interaction strength at which the band structure will undergo interesting changes. Furthermore, the stability of eigenstate is also investigated.Chapter V:By applying the mean field theory, we study the band structure and tunneling dynamics of BEC in accelerate Fourier-Synthesized optical lattices. It is found that the band structure undergo interesting changes. Because of these topological distortions on the energy levels, we expect that the tunneling dynamics appear dramatic change. The tunnel-ing probability is nonzero when the nonlinear term is very small and the energy levels keep the same topological structure as that in the linear case. Especially, the tunneling probability as a function of sweeping rate shows an irregular oscillation for the state on the middle level.At last, we make a conclusion and outlook of this thesis.