Kinetic theory of a spin-1/2 Bose-condensed gas

We derive a kinetic theory for a spin-1/2 Bose-condensed gas of two-level atoms at finite temperatures. The condensate dynamics is described by a generalized Gross-Pitaevskii equation for the two-component spinor order parameter, which includes the interaction with the uncondensed fraction. The noncondensate atoms are described by a quantum kinetic equation, which is a generalization of the spin kinetic equation for spin-polarized quantum gases to include couplings to the condensate degree of freedom. The kinetic equation is used to derive hydrodynamic equations for the noncondensate spin density. The condensate and noncondensate spins are coupled directly through the exchange mean field. Collisions between the condensate and noncondensate atoms give rise to an additional contribution to the spin diffusion relaxation rate. In addition, they give rise to mutual relaxation of the condensate and noncondensate due to lack of local equilibrium between the two components.