We develop a projection operator formalism for studying both the zero temperature equilibrium phase diagram and the non-equilibrium dynamics of the Bose-Hubbard model. Our work, which constitutes an extension of Phys. Rev. Lett. {\bf 106}, 095702 (2011), shows that the method provides an accurate description of the equilibrium zero temperature phase diagram of the Bose-Hubbard model for several lattices in two- and three-dimensions (2D and 3D). We show that the accuracy of this method increases with the coordination number $z_0$ of the lattice and reaches to within 0.5% of quantum Monte Carlo data for lattices with $z_0=6$. We compute the excitation spectra of the bosons using this method in the Mott and the superfluid phases and compare our results with mean-field theory. We also show that the same method may be used to analyze the non-equilibrium dynamics of the model both in the Mott phase and near the superfluid-insulator quantum critical point where the hopping amplitude $J$ and the on-site interaction $U$ satisfy $z_0J/U \ll 1$. In particular, we study the non-equilibrium dynamics of the model both subsequent to a sudden quench of the hopping amplitude $J$ and during a ramp from $J_i$ to $J_f$ characterized by a ramp time $τ$ and exponent $α$: $J(t)=J_i +(J_f-J_i) (t/τ)^α$. We compute the wavefunction overlap $F$, the residual energy $Q$, the superfluid order parameter $Δ(t)$, the equal-time order parameter correlation function $C(t)$, and the defect formation probability $P$ for the above-mentioned protocols and provide a comparison of our results to their mean-field counterparts. We find that $Q$, $F$, and $P$ do not exhibit the expected universal scaling. We explain this absence of universality and show that our results for linear ramps compare well with the recent experimental observations.
Phase diagram of ultracold atoms on optical lattice: Comparative study to slave fermion and slave boson for Bose Hubbard modelPhase diagram of ultracold atoms in optical lattices: Comparative study of slave fermion and slave boson approaches to Bose-Hubbard model
Extended Bose-Hubbard model with incompressible states at fractional numbers
将玻色-哈伯德模型扩展到包括最近邻和远邻相互作用，并与分数量子霍尔效应（Fractional Hall effect，简称FHE）有关。这两种模型都可以在含有量子气体的光晶格中进行研究。计算了具有强排斥作用（弱跳跃）的扩展Bose-哈伯德模型的基态.不可压缩的莫特绝缘体状态被发现在合理的填充分数与实验观察到的最低朗道水平的主要和次要的朗道填充分数兼容。讨论了这些状态在分数填充生存或经历一个莫特绝缘体过渡到超流体的跳跃项包括在何种程度上。
