Local group velocity and path-delay: Semi-classical propagators for the time evolution of Wigner functions in deep tunneling and in dispersive media

Controlled semi-classical approximations for the evolution kernels (the propagators) of the Wigner function in the cases of tunneling and propagation in dispersive media are derived. The semi-classical propagators follow well defined trajectories determined by the local group velocity and a path delay which is a multi-dimensional generalization of Wigner phase-delay time. A comparison between exact and semi-classical time evolution gives good agreement for propagation in dispersive media if the harmonic term in the dispersion relation is the dominant term and excellent agreement for deep tunneling.