The phenomenon of interest is the resonance frequency shift and the larger noise in the measurements inside the intermediate-powered region as shown below (from Fig. 3.3 of Characterisation of Transmon qubit chips, L. M. Janssen.
If I understand correctly, in the low-powered regime we can employ dispersive approximation to simplify the JC model into (cf. A Quantum Engineer’s Guide to Superconducting Qubits Eq. 144 -- Eq. 147)
$$ H_\mathrm{disp}=\omega_r\left(a^\dagger a+\frac{1}{2}\right)+\frac{1}{2}\left(\omega_q+\frac{g^2}{\Delta}+\frac{2g^2}{\Delta}a^\dagger a\right)\sigma_z $$
But when the photon number $n>n_c=\Delta^2/(4g^2)$, the approximation breaks down and we go into the region of interest. Literature seems to suggest that we need to go beyond RWA (e.g., Eq. 2 of Measurement-induced state transitions in a superconducting qubit: Beyond the rotating wave approximation) but I am not entirely sure.
Also, can we combine such a model with the notch-typed resonator model, i.e., Eq. 1 in Efficient and robust analysis of complex scattering data under noise in microwave resonators?
(There seems to be a relevant question on QC@SE but it gets no answer: Many-photon limit of dispersive shift Hamiltionian .)
