Rabi model in the dispersive regime

The dispersive regime of circuit QED is the main workhorse for today’s quantum computing prototypes based on superconducting qubits. Analytic descriptions of this model typically rely on the rotating wave approximation of the interaction between qubits and resonator fields, using the Jaynes-Cummings model as starting point for the dispersive transformation.
Here we present our results on the dispersive regime of the 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 [1], we derived simple analytic expressions for all experimentally relevant parameters like dispersive shift and resonator induced Purcell decay rate, both for idealized two-level systems as well as for realistic multi-level and weakly anharmonic qubits.
The analytical equations are easily tractable and reduce to the known Jaynes-Cummings results in the relevant limit, but show qualitative differences at large detuning, allowing for more accurate modelling of the interaction between superconducting qubits and resonators.

[1] C. Müller, and T. M. Stace, Phys. Rev. A 95, 013847 (2017)