Narrow Results

The deviation effect of spinor mode from the single-mode for a spin-1 Bose gas of trapped atoms is studied beyond the mean field theory. Based on the effective Hamiltonian with nondegenerated level of the collective spin states, the splitting level of the system energy due to the deviation effect has been calculated. For the large condensates of ⁸⁷Rb and ²³Na with atom number N>10⁵, the splitting fraction of the energy, arising from the magnetization exhibited by the trapped Bose gas, is found to have a typical order of (10^-4~10^-8), decreasing as N^-2 for ⁸⁷Rb and increasing as -N^-2 for ²³Na, respectively.

In this paper, we study the properties of information entropy in spin–orbit coupled spin-1 Bose–Einstein condensates. Our discussion is divided into two parts, stationary state and dynamics. For the former, we find that the coordinate component S_r and total entropy S increase with the strength of the spin–orbit coupling $\kappa$. The momentum component S_k shows a discontinuity behavior, it increases with $\kappa$ when $\kappa$ is lesser than a critical value ${\kappa }_c$, while it decreases with $\kappa$ when $\kappa$ is greater than ${\kappa }_c$. In order to characterize the disorder-to-order transition, we introduce an order parameter $\eta$. Numerical results display that $\eta$ gradually increases with increasing $\kappa$ and reaches its saturation, which implies that the system becomes more and more ordered. For the latter, the S_r increases and S_k decreases with increasing $\kappa$ at the same time. The entropic uncertainty relation is well obeyed at fixed $\kappa$. Moreover, $\eta$ decreases as a function of t. The strength effect of spin–orbit coupling on order-to-disorder transition becomes weak.

In this paper, we study the vortex formation of spin-orbit coupled (SOC) spin-1 dipolar Bose–Einstein condensates by numerically solving the Gross–Pitaevskii equation. We focus the effects of dipole–dipole interaction (DDI) on vortex structures and dynamics behavior. We find that the symmetry of density profiles in vortex lattice formation process and time evolution of the angular momentum is broken in the presence of DDI. This is due to the symmetry breaking of anisotropy DDI. Also, the total angular momentum decreases with increasing the relative dipole strength. However, it is shown that increasing SOC strength can improve the rotation efficiency. This result reflects the SOC dominates the DDI.

In this study, we investigate the dynamics properties of spin-1 ferromagnetic Bose–Einstein condensates (BECs) with dipole–dipole interaction (DDI). Numerical results are obtained by solving the mean-field nonlocal Gross–Pitaevskii equation. We find that the population of the information entropy in momentum space $S_k$ is greatly affected by geometric structure and dipole strength. The prolate condensate is favorable to obtain bigger $S_k$ than oblate condensate. The density distribution verifies that the formation of magnetic domain is associated with the variation of $S_k$. In addition, the kinetic energy displays that dramatical oscillation appears for strong dipolar interaction and it is larger in prolate condensate. Also, we use the order parameter to describe the order–disorder crossover. It is shown that BECs develop toward the disordered state with prolate condensate under strong DDI.

We develop a simple mean-field theory for studying spin-1 Bose-Hubbard model for cold atoms in an optical lattice model and obtain its zero temperature phase diagram. The superfluid phase can be polar (ferro) for antiferromagnetic (ferromagnetic) interaction. The superfluid to Mott insulator transition is continuous for ferromagnetic interaction. However, the polar superfluid to Mott insulator phase is a first order transition for even densities and continuous for odd densities.