  This work represents a fundamental study of a Holstein polaron in one
dimension driven away from the ground state by a constant electric field.
Taking fully into account quantum effects we follow the time-evolution of the
system from its ground state as the constant electric field is switched on at t
= 0, until it reaches a steady state. At weak electron phonon coupling (EP) the
system experiences damped Bloch oscillations (BO) characteristic for
noninteracting electron band. An analytic expression of the steady state
current is proposed in terms of weak EP coupling and large electric field. For
moderate values of EP coupling the oscillations are almost critically damped
and the system reaches the steady state after a short time. In the strong
coupling limit weakly damped BO, consistent with nearly adiabatic evolution
within the polaron band, persist up to extremely large electric fields. A
traveling polaron under the influence of the electric field leaves behind a
trail of phonon excitations absorbing the excess energy gained from the
electric field. The shape of the traveling polaron is investigated in details.
