  We study the open relaxation dynamics of an asymmetric dipole that is
ultrastrongly coupled to a single electromagnetic cavity mode. By using a
thermalizing master equation for the whole interacting system we derive a phase
diagram of the Liouvillian gap. It emerges that the ultrastrong coupling
inhibits the system relaxation toward the equilibrium state due to an
exponential suppression of the dipole tunneling rate. However, we find that
polaronic multi-photon resonances restore fast relaxation by a cavity-mediated
dipole resonant tunneling process. Aside of the numerical evidences, we develop
a fully analytical description by diagonalizing the Rabi model through a
generalized rotating-wave approximation, valid in the so-called polaron frame.
The relaxation physics of such ultrastrong-coupling systems is then reduced to
a multi-photon polaron version of the standard text-book dressed states
picture. At the end we discuss an extension to a multi-well dipole that can set
the basis of a cascaded resonant tunnelling setup in the ultrastrong coupling
regime.
