Abstract:
The Bose-Hubbard Hamiltonian of spin-2 cold bosons with repulsive interaction in an optical lattice is proposed. After neglecting the hopping term, the site-independent Hamiltonian and its energy eigenvalues and eigenstates are obtained. We consider the hopping term as a perturbation to do the calculations in second order and draw the phase diagrams for different cases. The phase diagrams show that there is a phase transition from Mott insulator with integer number bosons to superfluid when the ratio $c_0/t$ ($c_0$ is the spin-independent on-site interaction and $t$ the hopping matrix element between adjacent lattice sites) is decreased to a critical value and that there is different phase boundary between superfluid and Mott insulator phase for different Zeeman level component in some ground states. We find that the position of phase boundary for different Zeeman level component is related to its average population in the Mott ground state.

Abstract:
We study the insulator-superfluid transition of spin-1 bosons in an optical lattice in a uniform magnetic field. Based on a mean-field approximation we obtained a zero-temperature phase diagram. We found that depending on the particle number the transition for bosons with antiferromagnetic interaction may occur into different superfluid phases with spins aligned along or opposite to the field direction. This is qualitatively different from the field-free transition for which the mean-field theory predicts a unique (polar) superfluid state for any particle number.

Abstract:
It is shown that, in a reasonable approximation, the quantum state of $p$-bosons in a bi-partite square two-dimensional optical lattice is governed by the nonlinear boson model describing tunneling of \textit{boson pairs} between two orthogonal degenerate quasi momenta on the edge of the first Brillouin zone. The interplay between the lattice anisotropy and the atomic interactions leads to the second-order phase transition between the number-squeezed and coherent phase states of the $p$-bosons. In the isotropic case of the recent experiment, Nature Physicis 7, 147 (2011), the $p$-bosons are in the coherent phase state, where the relative global phase between the two quasi momenta is defined only up to mod($\pi$): $\phi=\pm\pi/2$. The quantum phase diagram of the nonlinear boson model is given.

Abstract:
We have studied superfluid-Mott insulating transition of spin-1 bosons interacting antiferromagnetically in an optical lattice. We have obtained the zero-temperature phase diagram by a mean-field approximation and have found that the superfluid phase is to be a polar state as a usual trapped spin-1 Bose gas. More interestingly, we have found that the Mott-insulating phase is strongly stabilized only when the number of atoms per site is even.

Abstract:
We study the superfluid-Mott insulator transition of antiferromagnetic spin-1 bosons in an optical lattice described by a Bose-Hubbard model. Our variational study with the Gutzwiller-type trial wave function determines that the superfluid-Mott insulator transition is a first-order one at a part of the phase boundary curve, contrary to the spinless case. This first-order transition may be observed through an experiment, such as a Stern-Gerlach type, under a magnetic field.

Abstract:
We study the Mott phases and superfluid-insulator transition of spin-three bosons in an optical lattice with an anisotropic two dimensional optical trap. We chart out the phase diagrams for Mott states with $n=1$ and $n=2$ atoms per lattice site. It is shown that the long-range dipolar interaction stabilizes a state where the chains of the ferromagnetically aligned spins run along the longer trap direction while the spin ordering is staggered between nearby chains, leading to an antiferromagnetic ordering along the shorter trap direction. We also obtain the mean-field phase boundary for the superfluid-insulator transition in these systems and study the nature of spin ordering in the superfluid state near the transition. We show that, inside the superfluid phase and near the superfluid-insulator phase boundary, the system undergoes a first order antiferromagnetic-ferromagnetic spin ordering transition. We discuss implications of our results for $^{52}$Cr atoms and suggest possible experiments to detect different phases in such systems.

Abstract:
We investigate the nonlinear light-matter interaction of a Bose-Einstein condensate trapped in an external periodic potential inside an optical cavity which is weakly coupled to vacuum radiation modes and driven by a transverse pump field. Based on a generalized Bose-Hubbard model which incorporates a single cavity mode, we include the collective backaction of the atoms on the cavity light field and determine the nonequilibrium quantum phases within the nonperturbative bosonic dynamical mean-field theory.With the system parameters adapted to recent experiments, we find a quantum phase transition from a normal phase to a self-organized superfluid phase, which is related to the Hepp-Lieb-Dicke superradiance phase transition. For even stronger pumping, a self-organized Mott insulator phase arises.

Abstract:
To realize band structures with non-trivial topological properties in an optical lattice is an exciting topic in current studies on ultra cold atoms. Here we point out that this lofty goal can be achieved by using a simple scheme of shaking an optical lattice, which is directly applicable in current experiments. The photon-assistant band hybridization leads to the production of an effective spin-orbit coupling, in which the band index represents the pseudospin. When this spin-orbit coupling has finite strengths along multiple directions, non-trivial topological structures emerge in the Brillouin zone, such as topological defects with a winding number 1 or 2 in a shaken square lattice. The shaken lattice also allows one to study the transition between two band structures with distinct topological properties.

Abstract:
The Bose Hubbard model describing interacting bosons in an optical lattice is reduced to a simple spin-1 XY model with single-ion anisotropy in the vicinity of the Mott phase. In the strong coupling Mott insulating regime, we propose a mean field theory based on a constraint SU(3) pseudo-boson representation on the effective model and discuss the excitation spectra and the phase transition to the superfluid state. Further to the superfluid phase, we use the coherent-state approach to derive the collective excitation modes. It is found that the Mott phase has two degenerate gapped quadratic excitation spectra which graduate into two degenerate gapless linear ones at the transition point, and one gapless linear mode with one gapped quadraticmode in the superfluid phase.

Abstract:
The properties of interacting bosons in a weak, one-dimensional, and bichromatic optical with a rational ratio of the constituting wavelengths $\lambda_1$ and $\lambda_2$ are numerically examined along a broad range of the Lieb-Liniger interaction parameter $\gamma$ passing through the Sine-Gordon transition. It is argued that there should not be much difference in the results between those due to an irrational ratio $\lambda_1/\lambda_2$ and due to a rational approximation of the latter. For a weak bichromatic optical lattice, it is chiefly demonstrated that this transition is robust against the introduction of quasidisorder via a weaker, secondary, and incommensurate optical lattice superimposed on the primary one. The properties, such as the correlation function, Matsubara Green's function, and the single-particle density matrix, do not respond to changes in the depth of the secondary optical lattice $V_1$. For a stronger bichromatic optical lattice, however, a response is observed because of changes in $V_1$. It is found accordingly, that holes in the SG regime play an important role in the response of properties to changes in $\gamma$. The continuous-space worm algorithm Monte Carlo method [Boninsegni \ea, Phys. Rev. E $\mathbf{74}$, 036701 (2006)] is applied for the present examination. It is found that the worm algorithm is able to reproduce the Sine-Gordon transition that has been observed experimentally [Haller \ea, Nature $\mathbf{466}$, 597 (2010)].