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We study the dynamics of the relative phase between two atomic Bose-Einstein condensates coupled via collisions and via a Josephson-like coupling. We derive the equations of the motion for the relative phase and the relative number operators from the second-quantized Hamiltonian of the system using a quantum field theoretical approach. We distinguish the cases in which the two condensates are in the same trap or in two different traps and study the influence of this difference on the first-order correlation function of atomic fields. In identical traps this function does not undergo dephasing. We calculate the dephasing time for the case of different traps.

We study the process of four-wave mixing (4WM) in ultracold degenerate atomic gases. In particular we address the problem of 4WM in boson-fermion mixtures. We develop an approximate description of such processes using asymptotic analysis of high order perturbation theory taking into account quantum statistics. We perform also numerical simulations of 4WM in boson-fermion mixtures and obtain an analytic and numerical estimate of the efficiency of the process.Comment: 4 page