Coherent versus incoherent dynamics during Bose-Einstein condensation in atomic gases

We review and extend the theory of the dynamics of Bose-Einstein condensation in weakly inter acting atomic gases. We present in a unified way both the semiclassical theory as well as the full quantum theory. This is achieved by deriving a Fokker-Planck equation that incorporates both the coherent and incoherent effects Of the interactions in a dilute Bose gas. In first instance we focus our attention on the nonequilibrium dynamics of a homogeneous Bose gas with a positive interatomic scattering length. After that we discuss how our results can be generalized to the inhomogeneous situation that exists in the present experiments with magnetically trapped alkali gases, and how we can deal with a negative interatomic scattering length in that case as well. We also show how to arrive at a discription of the collective modes of the gas that obeys the Kohn theorem at all temperatures. The theory is based on the many-body T-matrix approximation throughout, since this approximation has the correct physical behavior near the critical temperature and also treats the coherent and incoherent processes taking place in the gas on an equal footing.