Superfluid-Mott Insulator Phase Transition At Finite Temperature

Optical lattices provided possibility for the realization of many physical problems, such as quantum tunneling model and quantum phase transitions. The phase transition of ultra cold atoms in the optical lattices from superfluid phase to Mott insulator phase opened a new door for the investigation of strong correlation mechanism under well controlled conditions. Currently the fluorescence imaging technique allows us to observe space- and time-resolved characterization of single-atom and single-site across the quantum phase transition. In this thesis, we first give a brief introduction of Bose-Hubbard model, and extended-Bose-Hubbard model with two-or three-body on-site interactions, and calculate the phase boundary of the superfluid state and the Mott-insulator state combining the mean-field approximation with the perturbation theory. Meanwhile we give details on how to evaluate the interaction parameter U and the hopping matrix t. Then starting from the spin-1 Bose-Hubbard model under the tight-binding approximation, we study the MI phase in the limit of t=0, calculate the phase boundary of the SF-MI transition, and discuss the phase diagram with the even-odd dependence of the MI phase. Finally, based on the above knowledge and the theory of thermodynamics, we study the number density, the superfluid density and the entropy of the atoms for the SF and MI states in two and three- dimensional optical lattices at finite temperature. Our results can be readily verified in the current experiments.