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Number and spin densities in the ground state of a trapped mixture of two pseudospin-1/2 Bose gases with interspecies spin-exchange interaction

We consider the ground state of a mixture of two pseudospin-$\1/2$ Bose gases with interspecies spin exchange in a trapping potential. In the mean field approach, the ground state can be described in terms of four wave functions governed by a set of coupled Gross-Pitaevskii-like (GP-like) equations, which differ from the usual GP equations in the existence of an interference term due to spin-exchange coupling between the two species. Using these GP-like equations, we calculate such ground state properties as chemical potentials, density profiles and spin density profiles, which are directly observable in experiments. We compare the cases with and without spin exchange. It is demonstrated that the spin exchange between the two species lowers the chemical potentials, tends to equalize the wave functions of the two pseudospin components of each species, and thus homogenizes the spin density. The novel features of the density and spin density profiles can serve as experimental probes of this novel Bose system.