Nonlinear Quantum Characters In Ultra-cold Degenerate Atomic Gas

The nonlinear atomic interaction is an important character for the ultra-cold degenerate atomic gas. It is one of the hot points in current research to deeply understand the quantum behaviors induced by the nonlinear interaction, not only for extending the quantum theory, but also for coherent manipulation and potential application of the ultra-cold degenerate atomic gas. In this thesis, we have suggested one set of orthogonal basis for describing the nonuniform BECs, explored the coherence decay law enhanced by the nonlinear interaction and proposed one experiment to simulate one famous nonlinear problem.It is important to exactly describe the quantum system based on one orthogonal and complete basis in the frame of linear quantum theory. Due to the nonlinearity and external trap, it is difficult to find this kind basis for nonlinear system. With the help of one exactly stationary solution for G-P with1-D infinite square well, a set of orthogonal basis was presented, which is one to one corresponding to the basis of the linear case. Combined the orthogonal basis and the Bogoliubov theory, we have investigated the stability of the stationary states for G-P with the one-dimensional infinite square well, and found that:Whether the positive or negative nonlinear interaction, the ground state is always stable; the first excited state is, however, instable for the negative large enough nonlinear interaction.As one of the fundamental properties of BECs, the coherence has attracted more attentions both for theorists and experimenters. We have investigated the interference phenomena from spatially imbalanced BECs. An asymmetric interference pattern is reported by the asymmetry of the initial states. Considering the special character of the atom and this asymmetric interference pattern, we have suggested one novel scheme to detect weak gravity-like force, which benefits from the nonlinear interaction. The sensitivity can be around δg～10-6.It is one of the crucial problems to be solved for the realization of the atom lasers, due to the negative factors from unavoidable environments and other factors. Using quantum Loschmidt echo, the evolution of the coherence of BECs with an anharmonic trap, and subjected to the surrounding perturbation is investigated. Our results have shown that the coherence loss can be described by a Fermi-decay law, in which there are three indispensable factors:the anharmonic trap, the external perturbation and the nonlinear interaction. The anharmonic trap creates the anharmonic oscillations, and the weak random perturbation causes the loss of coherence by disturbing their coherent oscillations, while the nonlinear interaction enhances the loss to the Fermi-decay law. Based on the Fermi-decay law, some suggestions are presented to prolong the coherent time during coherently manipulating condensates.Finally, we have shown that the evolution of multicomponent BECs with a harmonic trap can be understood as the Fermi-Pasta-Ulam problem (FPU). The single-component BEC performs a nonlinear harmonic oscillation, proved by the experiment, and can be mode for FPU. The nonlinear couple between different modes in FPU, is realized by atomic nonlinear interaction. We have calculated the energy and the Husimi distribution of each component and found the energy recurrence and the energy sharing for different regimes of nonlinear interaction. For the two components case, the strong energy sharing is found, once nonlinear interaction is large enough. For the three components case, the situation is more complex but still energy sharing is observed.