Melnikov Chaos Of Bose-Einstein Condensates In Optical Lettices

The realization of atomic gases Bose-Einstein condensation in the experimentation has very importent purports of the theory and application. Bose-Einstein condensates not only offer the perfect macrosciopic quantum systems to investigate many foundmental problems in quantum mechanics but also have extensively application foregrounds such as in atom laser, CMOS and the exactitude measure. In the framework of the mean-field theory, the Bose-Einstein condensates is govonered by the Gross-Pitaeviskii equation. Based on the Gross-Pitaeviskii equation we had been study of the Melnikov chaos in Bose-Einstein condensates by theoretical analysis and numerical method. Meanwhile, we have also discussed the problem of controlling the chaotic motion of the condensated atoms, and a series of significant results are obtained in this paper. Our paper is organized as the following six parts:In chapter 1, we have introduced the conceptions and the laws of Bose-Einstein condensation and chaos.Their actuality and expectation are retrospected in this chaper. We also studied the chaotic behavior of Bose-Einstein condensates, and discussed the motheds of controlling chaos.In chapter 2, we have studied the chaotic Josephson effects in two weakly coupled Bose-Einstein condensates. Using the time-dependent self-consistent field method and the macroscopic one-body wave function, we investigated the BJJ equations that de- scribe time evolutions of the phase difference in symmetric and asymmetric trapped case. Wehave obtained the chaotic solution of the BJJ equation for a small-amplitude oscillation by using the direct perturbation method. The results reveal that the motion of the condensate atoms is regular without the time-dependent drive and the periodical drive causes the chaotic motion of the Bose-Einstein condensates in the asymmetric trap. Using the chaotic solution we have demonstrated the analytical boundedness and numerical unboundedness of the chaotic orbits that implies the incomputabilityof the system.In chapter 3, we have considered a trapped BEC in a tilted potential of the Wannier-Stark framework. Under the Melnikov's chaos criterion, we investigated the chaotic solutions based on the GP equation with trivial and non-trivial phases are constructed by using the direct perturbation technique. The chaotic regions on the parameter space are illustrated for the two different phases. The chaotic solutions describe the chaotic spatial evolutions of the atomic number density and energy density, and the chaotic regions supply a method for producing or eliminating chaos, through the adjustments of the controllable parameters. The instability of chaotic solution is discussed and the qualitative agreements with the work reported recently is found. It is well known that the chaos emerges in the processions of the BEC collapsing which plays a destructive role for the BEC system. Therefore, predicting and controlling chaos are quite important for the formation and application of a BEC.In chapter 4, we investigated the nonlinear transport of BEC atoms in Gaussian potential and ratchet potential. We have obtained a stationary nonlinear Schro from GP equation, and sloved this equation theoretically and numerically. The analytical results showed that the condensed system has a steady perturbation solution, and reveal that the nonlinear transport of the condensate atoms in Gaussian potential case. But in ratchet potential case, there is chaotic analytical solution in this system, and show that the chaotic transport of BEC atoms. The corresponding numerical results agree well with the theoretical analytical results. Besides, we can control the chaotic transport of condensate atoms by change the system parameters or boundary conditions.In chapter 5, we studied the Melnikov chaos of two-component Bose-Einstein condensates in tilted optical lattices. The analytical results show that the stationary perturbation solutions of the two coupled nonlinear Schrodinger equations are the instable chaotic solutions in Melnikov criterion for chaos, and there are Smale horseshoes chaos in the two condensates. The corresponding numerical results show that the phase orbits of the two condensates are identical when the system parameters and initial conditions or boundary conditions, which means that the two condensates are chaotic synchronized. However, they are different and asynchronous when there is any very small difference between the system parameters, which show that the system parameters play a very important role in controlling chaos and chaos synchronization.In chapter 6, we give a simple summary and discussion to the above-mentioned works. We also give a expectaton in the study for the chaotic motion of BEC,the stability of BEC system and the applications of the condensates. Here, our main works are involved in chapters two, three, four and five.