Transport In Semiconductor Superlattice And Phase Transition Of BEC In Optical Lattice

This paper is mainly devoted to study of the quantum transport in semiconductor superlattice and phase transition of BEC in optical lattice. Firstly, I provide a brief review of the previous achievements and investigations on the low-dimensional quantum devices and semiconductor superlattice, in which some principal theories such as Bloch Oscillations, Wannier-Stark ladder, Zener tunneling and related progress in experiments are introduced. Secondly, based on the discussion of the dynamical characteristics of Bloch electron driven by two-mode AC electrical fields we find a localization condition of the Bloch electron on a localized state and then extend this result to the multi-field case. Thirdly we introduce the superlattice model with alternating site energies which is more complicated than the simple tight-binding model, and moreover we explore the influence of the alternating diagonal term in Hamiltonian on the quasi-energy bands and dynamics. According to Floquet theorem, the relations between the dynamical localization and quasi-energy bands are pointed out with both numerical and analytical calculations on quasi-energy bands and their corresponding time evolution. Then we investigate the phase transition of Bose-Einstein condensation in optical lattice. After a brief introduction of some experimental methods required in condensing ultracold bosons, and of the relevant fundamental theories, starting from the Bose-Hubbard model we study the excitationspectrum with Bogliubov transformation. The dispersion relation within the long-wave limit is linear, which is the same as that for the Bose gas of superfluid phase in the absence of the optical lattice. Based on the analysis of the excitation spectrum, we obtain the superfluid-Mott-insulator phase transition condition, and superfluid velocity of ultracold dilute gas of bosonic atoms in an optical lattice, and point out that the velocity of superfluid can be experimentally adjusted by controlling the parameters of optical lattice. Lastly, we discuss the energy-band structure of ultracold atoms in optical lattice by means of Green function method and in addition, procure the superfluid-Mott phase transition condition in mean-field approximation which is in agreement with the result in the literature.