Dissipation of the Rabi model beyond the rotating wave approximation: Quasi-degenerate qubit and ultra-strong coupling

Environmental influences on the dynamics of a coupled qubit-oscillator system are studied analytically. We investigate the case of a quasi-degenerate qubit within the ultra-strong coupling regime for which the qubit frequency is much smaller than the frequency of the oscillator, and the coupling between the qubit and the oscillator is large, both of which invalidate the usually employed rotating wave approximation. In contrast to the standard quantum optics master equation, we explicitly take the qubit-oscillator coupling into account while microscopically deriving a dressed state master equation. Using the derived master equation, we discuss a spectroscopic technique which can be used to probe the dressed energy level structure of the qubit-oscillator system.