  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 $\Delta$.
