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We propose a theoretical framework based on $SU(3)$ coherent states as a convenient tool to describe the collective state of a Bose-Einstein condensate of spin 1 atoms at thermal equilibrium. We work within the single-mode approximation, which assumes that all atoms condense in the same spatial mode. In this system, the magnetization $m_z$ is conserved to a very good approximation. This conservation law is included by introducing a prior distribution for $m_z$ and constructing a generalized statistical ensemble that preserves its first moments. In the limit of large particle numbers, we construct the partition function at thermal equilibrium and use it to compute various quantities of experimental interest, such as the probability distribution function and moments of the population in each Zeeman state. When $N$ is large but finite (as in typical experiments, where $Nsim 10^3-10^5$), we find that fluctuations of the collective spin can be important.
We investigate the ground-state phases of a mixture of spin-1 and spin-2 Bose-Einstein condensates at zero magnetic field. In addition to the intra-spin interactions, two spin-dependent interaction coefficients are introduced to describe the inter-sp
We systematically investigate the weakly trapped spin-1 Bose-Einstein condensates with spin-orbit coupling in an external Zeeman field. We find that the mean-field ground state favors either a magnetized standing wave phase or plane wave phase when t
Solitons in multi-component Bose-Einstein condensates have been paid much attention, due to the stability and wide applications of them. The exact soliton solutions are usually obtained for integrable models. In this paper, we present four families o
We study quantum spin mixing in a binary mixture of spin-1 condensates including coherent interspecies mixing process, using the familiar spinor condensates of $^{87}$Rb and $^{23}$Na atoms in the ground lower hyperfine F=1 manifolds as prototype exa
We have computed phase diagrams for rotating spin-1 Bose-Einstein condensates with long-range magnetic dipole-dipole interactions. Spin textures including vortex sheets, staggered half-quantum- and skyrmion vortex lattices and higher order topologica