  We study the relaxation of the Holstein model after a sudden switch-on of the
interaction by means of the nonequilibrium dynamical mean field theory, with
the self-consistent Migdal approximation as an impurity solver. We show that
there exists a qualitative change in the thermalization dynamics as the
interaction is varied in the weak-coupling regime. On the weaker interaction
side of this crossover, the phonon oscillations are damped more rapidly than
the electron thermalization timescale, as determined from the relaxation of the
electron momentum distribution function. On the stronger interaction side, the
relaxation of the electrons becomes faster than the phonon damping. In this
regime, despite long-lived phonon oscillations, a thermalized momentum
distribution is realized temporarily. The origin of the thermalization
crossover found here is traced back to different behaviors of the electron and
phonon self-energies as a function of the electron-phonon coupling. In
addition, the importance of the phonon dynamics is demonstrated by comparing
the self-consistent Migdal results with those obtained with a simpler
Hatree-Fock impurity solver that neglects the phonon self-energy. The latter
scheme does not properly describe the evolution and thermalization of isolated
electron-phonon systems.
