Stability And Chaotic Behavior Of A Two-component Bose-Einstein Condensate

Bose-Einstein condensation has been an attractive subjects in recent decades. It not only offers the perfect macroscopic quantum systems to investigate many fundamental problems in quantum mechanics but also has extensively application foregrounds such as in atom laser and quantum computation. Its intrinsic nonlin-earity and the interaction with externally applied fields make it a kind of classical chaotic system. In the framework of mean-field theory the Bose-Einstein condensates are governed by the Gross-Pitaeviskii equation. We investigate the stability, macroscopic quantum self-trapping(MQST) and chaotic properties of a periodically driven Bose-Einstein Condensate (BEC) with two hyperfine states in a single well. We also give a scheme for controlling the chaotic region of the system by using a laser field.This paper consists of four parts. In the first chapter, we shall give a simple introduction to the mean-field theory and its history, research status and applications of chaos in BEC. In the second part, we study the stability of stationary states and chaotic behavior of the two-component BEC system. The stabilities of the steady-state solutions are analyzed with linear stability theorem. The result shows that the steady-state relative population will appear the tuning-fork bifurcation, when the physical parameters hold a certain relation. And the two bifurcations of tuning-fork denote the two steady-states of relative population. They are critically stable, that is associated with the metastable stationary MQST. The dependence of the MQST on the initial conditions, the population transfer and the relative energy is revealed, and the periodical MQST are found. Finally, we numerically find that the relative population oscillation will undergo a process from order to chaos, through a series of period-doubling bifurcations. The oscillating period of the relative population increases gradually with the increase of the time-dependent relative energy. When the relative energy is equal to or greater than a critical value, the oscillating period of the relative population tends to infinity, chaos emerges. Meanwhile, we find the chaotic MQST.In the third chapter, the Melnikov chaotic solution and boundedness conditions are derived from a direct perturbation theory that leads to the chaotic regions in...