  We consider a quantum dot, affected by a local vibrational mode and contacted
to macroscopic leads, in the non-equilibrium steady-state regime. We apply a
variational Lang-Firsov transformation and solve the equations of motion of the
Green functions in the Kadanoff-Baym formalism up to second order in the
interaction coefficients. The variational determination of the transformation
parameter through minimization of the thermodynamic potential allows us to
calculate the electron/polaron spectral function and conductance for adiabatic
to anti-adiabatic phonon frequencies and weak to strong electron-phonon
couplings. We investigate the qualitative impact of the quasi-particle
renormalization on the inelastic electron tunneling spectroscopy signatures and
discuss the possibility of a polaron induced negative differential conductance.
In the high-voltage regime we find that the polaron level follows the lead
chemical potential to enhance resonant transport.
