  We develop the nonequilibrium extension of bosonic dynamical mean field
theory (BDMFT) and a Nambu real-time strong-coupling perturbative impurity
solver. In contrast to Gutzwiller mean-field theory and strong coupling
perturbative approaches, nonequilibrium BDMFT captures not only dynamical
transitions, but also damping and thermalization effects at finite temperature.
We apply the formalism to quenches in the Bose-Hubbard model, starting both
from the normal and Bose-condensed phases. Depending on the parameter regime,
one observes qualitatively different dynamical properties, such as rapid
thermalization, trapping in metastable superfluid or normal states, as well as
long-lived or strongly damped amplitude oscillations. We summarize our results
in non-equilibrium "phase diagrams" which map out the different dynamical
regimes.
