Soliton And Landau-zener Tunneling Dynamics Of Bose-einstein Condensates Trapped In Optical Lattices

Bose-Einstein condensates (BEC) have been successfully loaded in optical lattices, which is similar to the electrons bounded in the lattices of solid from solid-state physics. Some physical properties of BEC trapped in optical lattices are similar to that of interaction between electrons and lattices. The BEC atoms confined in periodic potential wells induced by optics share not only many properties of electrons in solid but also hydrodynamics properties of nonlinear materials and other nonlinear systems. Therefore, the field of matter wave attracts extensive research interests of investigators. Recently, ultracold Boses have been successfully loaded in a dichromatic optical lattice potential which was interferenced by two pairs of lasers, the wavelength of the first pair of lasers is different with the second one. Both the amplitude of laser can be modulated from weak to strong. So, the investigation of the solitons and Landau-Zener tunneling properties of the BEC confined in an optical lattice or a dichromatic optical lattice possess definite foundational significance. In this thesis, beginning from GP equation which was based on mean-field theory, we first study analytically soliton dynamics behaviors of the BEC confined in a combined trap which is consist of an optical lattice and a harmonic oscillator trap, and then investigate numerically Landau-Zener tunneling properties of BEC loaded in a dichromatic optical lattice. The thesis is organized as follows:In chapter one, we illuminate the foundational knowledge of BEC and obtain the GP equation based on the mean-field theory. Subsequently, we introduce the research trend of BEC loaded in an optical lattice as well as its research significance.Based on the BEC can be realized in the experiments, in chapter two, by developing multiple-scale method combined with Wentzel-Kramer-Brillouin expansion, we analytically study the soliton dynamics behaviors of BEC trapped in a combined trap which is consist of an optical lattice potential and a harmonic oscillator trap. It is shown that a black soliton of the BEC can be transformed to a grey soliton due to the shallow periodic potential modulating effect. Meanwhile, when the periodic potential strength increases slightly, the velocity of the soliton decreases, however, the width of the soliton slowly decreases at initial stage, goes through a minimum, and then slowly increases.Similarly, one can obtain a dichromatic optical lattice through interferenced between two pairs of lasers. So in chapter three, under the consideration of BEC confined in a periodic dichromatic optical lattice potential, we study its Bloch energy band structure and Landau-Zener tunneling behaviors of the first and second band gap. It is shown that when the relative phase of the two lattice increases from 0 toπ, the width of the first band gap in the Bloch energy band increases, while that of the second one decreases. Meanwhile, we find that the depth and the relative phase of the two lattices have an important effect on Landau-Zener tunneling properties.In chapter four, we make a summary of our work and some expectations for future.