Dimensionalcrossover Of A Bose-einstein Condensate In Three-dimensional Optical Lattice

The ground state and the collective excitations of a Bose-Einstein Condensate (BEC) in an optical lattice are investigated in this thesis. We pay more attention on the relations of ground state and its collections for a BEC from a two-dimensional (2D) optical lattice and an anisotropic3D optical lattice, transition to a ID Lieb—iniger modle respectively. Firstly, a brief introduction on the fundamental concepts of BEC, specially its experimental realization and the corresponding mean-field theory. Considering a homogeneous weakly interacting Bose gas, its ground state energy and collective excitation spectrum can be obtained, using the Gross-Pitaevskii (GP) equation and the Bogoliubov’s approach, respectively. And in the framework of the mean-field theory, those two results coincide with each other. The quantum depletion and chemical potential, as well as the pressure at the ground state can also be calculated from ground state energy. With the help of Bogoliubov’s approximation, the ground state formula for a BEC trapped in an optical lattice, specifically, a two-dimensional (2D) optical lattice and an anisotropic3D optical lattice, are presented respectively. We found the ground state energy depends on the parameters of the optical lattice. Our results for the2D optical lattice case can be recovered those in the ID Lieb-Liniger model in the limit of2t/gn=0,means we turn off the2D optical lattice. Considering an anisotropic3D optical lattice, the ground state energy can also be recovered to by adjusting the optical lattice parameters2t1/gn=0to zaro in x, y direction, and after redefining effective mass as m*=9π2/16d2g1Dn1D.