  We develop a theoretical description of non-adiabatic Josephson dynamics in
superconducting junctions containing low energy quasiparticles. Within this
approach we investigate the effects of midgap states in junctions of
unconventional d-wave superconductors. We identify a reentrance effect in the
transition between thermal activation and macroscopic quantum tunneling, and
connect this phenomenon to the experimental observations in Phys. Rev. Lett.
94, 087003 (2005). It is also shown that nonlinear Josephson dynamics can be
defined by resonant interaction with midgap states reminiscent to nonlinear
optical phenomena in media of two-level atoms.
