  The coherent dynamics of a Holstein polaron in strong electric fields is
considered under different regimes. Using analytical and numerical analysis, we
show that even for small hopping constant and weak electron-phonon interaction,
the original discrete Wannier-Stark (WS) ladder electronic states are each
replaced by a semi-continuous band if a resonance condition is satisfied
between the phonon frequency and the ladder spacing. In this regime, the
original localized WS states can become {\em delocalized}, yielding both
`tunneling' and `stretched' polarons. The transport properties of such a system
would exhibit a modulation of the phonon replicas in typical tunneling
experiments. The modulation will reflect the complex spectra with
nearly-fractal structure of the semi-continuous band. In the off-resonance
regime, the WS ladder is strongly deformed, although the states are still
localized to a degree which depends on the detuning: Both the spacing between
the levels in the deformed ladder and the localization length of the resulting
eigenfunctions can be adjusted by the applied electric field. We also discuss
the regime beyond small hopping constant and weak coupling, and find an
interesting mapping to that limit via the Lang-Firsov transformation, which
allows one to extend the region of validity of the analysis.
