Temporal non-equilibrium dynamics of a Bose Josephson junction in presence of incoherent excitations

The time-dependent non-equilibrium dynamics of a Bose-Einstein condensate (BEC) typically generates incoherent excitations out of the condensate due to the finite frequencies present in the time evolution. We present a detailed derivation of a general non-equilibrium Green's function technique which describes the coupled time evolution of an interacting BEC and its single-particle excitations in a trap, based on an expansion in terms of the exact eigenstates of the trap potential. We analyze the dynamics of a Bose system in a small double-well potential with initially all particles in the condensate. When the trap frequency is larger than the Josephson frequency, $\Delta > \omega_J$, the dynamics changes at a characteristic time $\tau_c$ abruptly from slow Josephson oscillations of the BEC to fast Rabi oscillations driven by quasiparticle excitations in the trap. For times $t<\tau_c$ the Josephson oscillations are undamped, in agreement with experiments. We analyze the physical origin of the finite scale $\tau_c$ as well as its dependence on the trap parameter $\Delta$.