Controlling Chaos And Analysing Stability Of Bose-Einstein Condensates In Optical Lattices

The realization of Bose-Einstein condensation in a dilute atomic gas has attracted enormous attention. The Bose-Einstein condensate (BEC) not only offers the perfect macrosciopic quantum system to investigate many foundmental problems in quantum mechanics, but also has extensive application foregrounds such as in atom laser, quantum computation and quantum information. In the framework of mean-field theory the Bose-Einstein condensates is govonered by the Gross-Pitaeviskii equation. In this paper, based on the Gross-Pitaeviskii equation, we have studied the chaotic property and stability of the Bose-Einstein condensates held in the optical lattices.Our paper is organized as the following four parts. In the first chapter, we shall give a simple introduction to mean-field theory and the research status, improvements and applications of chaos and stability in Bose-Einstein condensates.In the second chapter, for a Bose-Einstein condensate (BEC) confined in a double lattice consisting of two weak laser standing waves we find the Melnikov chaotic solution and chaotic region on the parameter space by using the direct perturbation method. In the chaotic region, spatial evolutions of the chaotic solution and the corresponding distribution of particle number density are bounded but unpredictable between their superior and inferior limits. It is illustrated that when the relation k₁≈k₂ between the two laser wave vectors is kept, the adjustment from k₂ < k₁ to k₂≥k₁ can transform the chaotic region into regular one or on the contrary. This suggests a feasible scheme for generating and controlling chaos, which could lead to an experimental observation in the near future.In the third chapter, we have discussed BEC trapped in superlattices consisted of a strong laser standing wave and a weak laser standing wave. Treating the weak laser standing wave as a perturbation, the zeroth order solution and the first order corrected solution are obtained. We find the first order correction of solution satisfies the one dimensional spatial Mathieu equation. We have also analyzed the stability of BEC trapped in one optical lattice and found the dynamical stability region, instability region and indefinite stability region on the parameter space. By modulating the experimental parameter, we can always keep the BEC in the stability region.In the last chapter of this paper, we give a simple summary and discussion to the above-mentioned works. Here, our main works are involved in chapter two and three.