Mean field theory for the domain formation in a spin-1 condensate

Spin domains were recently observed in the off-equilibrium dynamics of a large condensate of $^{87}$Rb atoms confined in a cigar shaped optical trap\footnote{Private communications with M. -S. Chang and M. S. Chapman.}. For a two component condensate, the number of atoms within each component is conserved. The dynamics of domain formation has been investigated in detail using the mean field theory\footnote{K. Kasamatsu and M. Tsubota, Phys. Rev. Lett. {\bf 93}, 100402 (2004).}. For a spin-1 condensate (with $^ {87}$Rb or $^{23}$Na atoms), the number of atoms for each component varies due to spin exchange collisions $2|F=1,m_F=0\rangle \leftrightarrow |F=1,m_F=-1\rangle+|F=1,m_F=1\rangle$. In this study, we investigate the effect of such exchange interactions on the dynamics of domain formation in a spin-1 condensate. Using both analytic calculations for a homogeneous condensate and numerical simulations for a trapped condensate, we provide a detailed understanding of the stable and unstable regions of the off- equilibrium dynamics. We also address the important role of an external magnetic field.