The Optical Lattice Super-cold Bose Gas Superfluidity

In 1995, Ketterle, Wieman and Cornell et al. first realized the Bose-Einstein condensations (BEC) in the dilute trapped alkali-metal atom gases which were predicted by Einstein eighty years ago. It was a new milestone in the research history of the atomic physics at low temperature. Because of atom's quantum effect at low temperature, the atoms of condensates can tunnel in optical lattices, this is in superfluid phase. As the potential depth of the lattice is increased, the atoms are confined, a transition is observed from a superfluid to a Mott insulator phase. In the superfluid phase, each atom is spread out over the entire lattices, with long-range phase coherence. But in the insulating phase, atoms are localized at individual lattice site, with no phase coherence across the lattice. The excitation spectrum is gapless in superfluid phase whereas there is a gap in the Mott insulator phase. The critical velocities which are nonzero are relation of the different spin component. When the velocity of the liquid exceeds the critical velocity, the superfluid phase will be destroyed.In this paper, we calculate the excitation spectrum and the critical velocities of the spin-1 ultra-cold bosons trapped in the optical lattices and external magnetic field by the Bogliubov method. Using the same method, we calculate the excitation spectrum and critical velocities of the spin-2 ultra-cold bosons trapped in the optical lattices. We find some significant conclusions as follows: the spin freedom of the spin-1 ultra-cold bosons trapped in the optical lattices is frozen by the external magnetic field, there exists no superfluid phase for spin components±1, while the critical velocities are zero. Whereas there exists superfluid phase for spin components 0, while the critical velocity is not zero. For the spin-2 ultra-cold bosons trapped in the optical lattices, there exists superfluid phase for five spin component and the critical velocities are different for five spin component; the excitation spectrum and the critical velocities of the bosons increase as the tunneling matrix J , the interaction matrix U and the average atoms of the optical lattice increase.The thesis consists of five parts. The first part is an introduction of the basic theory and experiment research of the BEC. The second part is an introduction of optical lattice and its physical properties, Bose-Hubbard model and the Landau theory of superfluidity. In third part, we calculate the excitation spectrum and the critical velocities of the spin-1 ultra-cold bosons trapped in the optical lattices and external magnetic field by the Bogliubov method. Using the same method, we calculate the excitation spectrum and critical velocities of the spin-2 ultra-cold bosons trapped in the optical lattices. We analyze the dependence of the excitation spectrum and critical velocities on the physical parameters. We notice that the critical velocities of the superfluid are spin-component dependent and can be controlled by adjusting the laser lights that form the optical lattices. Possible experiments to detect the superfluid phase and component separation of spinor BEC are also discussed. The last part of the thesis is a brief summary, and also shows the outlook of the future work.