We review our recent theoretical advances in quantum information and many body physics with cold atoms in various external potential, such as harmonic potential, kagome optical lattice, triangular optical lattice, and honeycomb lattice. The many body physics of cold atom in harmonic potential is investigated in the frame of mean-field Gross-Pitaevskii equation. Then the quantum phase transition and strongly correlated effect of cold atoms in triangular optical lattice, and the interacting Dirac fermions on honeycomb lattice, are investigated by using cluster dynamical mean-field theory and continuous time quantum Monte Carlo method. We also study the quantum spin Hall effect in the kagome optical lattice.
We consider two coupled Gross Pitaevskii equations describing a two-component Bose Einstein condensate with time-dependent atomic interactions loaded in an external harmonic potential, and investigate the dynamics of vector solitons. By using a direct method, we construct a novel family of vector soliton solutions, which are the linear combination between dark and bright solitons in each component. Our results show that due to the superposition between dark and bright solitons, such vector solitons possess many novel and interesting properties. The dynamics of vector solitons can be controlled by the Feshbach resonance technique, and the vector solitons can keep the dynamic stability against the variation of the scattering length.
We theoretically investigate a three-dimensional Fermi gas with Rashba spin-orbit coupling in the presence of both out-of-plane and in-plane Zeeman fields. We show that, driven by a sufficiently large Zeeman field, either out-of-plane or in-plane, the superfluid phase of this system exhibits a number of interesting features, including inhomogeneous Fulde- Ferrell pairing, gapped or gapless topological order, and exotic quasi-particle excitations known as Weyl fermions that have linear energy dispersions in momentum space （i.e., massless Dirac fermions）. The topological superfluid phase can have either four or two topologically protected Weyl nodes. We present the phase diagrams at both zero and finite temperatures and discuss the possibility of their observation in an atomic Fermi gas with synthetic spin-orbit coupling. In this context, topological superfluid phase with an imperfect Rashba spin-orbit coupling is also studied.
