Population and phase dynamics of F=1 spinor condensates in an external magnetic field

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We show that the classical dynamics underlying the mean-field description of homogeneous mixtures of spinor F=1 Bose-Einstein condensates in an external magnetic field is integrable as a consequence of number conservation and axial symmetry in spin space. The population dynamics depends only on the quadratic term of the Zeeman energy and on the strength of the spin-dependent term of the atom-atom interaction. We determine the equilibrium populations as function of the ratio of these two quantities and the miscibility of the hyperfine components in the ground state spinors are thoroughly discussed. Outside the equilibrium, the populations are always a periodic function of time where the periodic motion can be a libration or a rotation. Our studies also indicates the absence of metastability.