Dissipative Rabi model in the dispersive regime

The dispersive regime of circuit QED is the main workhorse for todays quantum computing prototypes based on superconducting qubits. Analytic descriptions of this model typically rely on the rotating wave approximation of the interaction between the qubits and resonators, using the Jaynes-Cummings model as starting point for the dispersive transformation. Here we present analytic results on the dispersive regime of the dissipative Rabi model, without taking the rotating wave approximation of the underlying Hamiltonian. Using a recently developed hybrid perturbation theory based on the expansion of the time evolution on the Keldysh contour [Phys. Rev. A 95, 013847 (2017)], we derive simple analytic expressions for all experimentally relevant dynamical parameters like dispersive shift and resonator induced Purcell decay rate, focussing our analysis on a generic multi-level qubit. The analytical equations are easily tractable and reduce to the known Jaynes-Cummings results in the relevant limit. They however show qualitative differences at intermediate and large detuning, allowing for more accurate modelling of the interaction between superconducting qubits and resonators. In the limit of strong resonator driving, our results additionally predict new types of drive induced qubit dissipation and dephasing, not present in previous theories.