Exact Floquet States Of A Two-component Bose-Einstein Condenste Driven By A Laser Standing Wave

In the mean-field theory, the macroscopic wavefunction of a Bose-Einstein condensate obeys Gross-Pitaevskii equation. To investigate the physical properties of BEC, we require to seek the exact or approximate solutions of GPE. Several exact stationary solutions of GPEs have been obatined for BEC held in some special potentials, and by using the exact solutions, many physical properties of BEC are revealed. And there are only few reports in the exact nonstationnary solutions. In this paper we constuct a family of exact nonstationary solutions-exact Floquet solutions(EFSs) in BEC system and investigate the various physical properties which the EFSs reveal.This paper consists of four chapters. In the first chapter, we shall give a simple introduction to the mean-field theory and the two-component Bose-Einstein condensate.In the second chapter, based on the idea of balance between the internal and external potentials, it is shown that for a two-component Bose-Einstein condensate (BEC) confined in an optical lattice the presence of a space-time periodic laser field can induce a family of exact Floquet states (EFSs). The atomic-number densities of the EFSs are illustrated and the atomic current, phase blowing-up and the quantum reflection are investigated. The balance conditions and blowing-up region on parameter space are found, and the influences of phase blowing-up to the velocity fields, flow densities are revealed. It is demonstrated that the BEC motions can be controlled by adjusting the laser frequency, wave vector and amplitude.In the third chapter, in order to investigate whether the system is stable or not, we apply Lyapunov stability criterion and the linear stability analysis and find that: if the strength of driving laser is smaller than twice of the lattice strength, the phenomena of phase blowing-up occurs and causes the weak-instability of the EFSs; and if the strength of laser is larger than twice of the lattice strength, the phenonema of phase blowing-up does't occur, so the system is possibly stable. By reinforcing the laser strength, we can avoid the instability and bring the system into possible stability. In the last part of this paper, we give a simple summary and discussion to the above-mentioned works. Here, our main works are involved in chpater two and three.