  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.
