the frequencies of the cavity and the qubit, respectively, and $g$ is the strength of the coupling, see Fig. \[fig:schema\]. We displace the Hamiltonian in Eq. (\[eq:driven RabiHam\]) into the vacuum and apply the RWA. After a Schrieffer–Wolff transformation, we obtain the driven JC Hamiltonian [@supplement] $$\begin{aligned}
\label{eq:diagHam}
\hat{H} &=& (\omega_{\rm c} + \chi_0\hat{\sigma}_{\rm z})\hat{a}^{\dag}\hat{a} + \frac{1}{2}(\omega_0+\chi_0)\hat{\sigma}_{\rm z}\nonumber\\
&& + \frac{G}{2}\left(e^{i\omega_{\rm d} t}\hat{\sigma}_- + e^{-i\omega_{\rm d} t}\hat{\sigma}_+\right), \label{eq:RWAtwolevel}\end{aligned}$$ where $\chi_{0} = g^2/(\omega_{0}-\omega_{\rm c})$ is the vacuum ac-Stark shift. Thus, the cavity frequency $\omega_{\rm c} + \chi_0\hat{\sigma}_{\rm z}$ depends on the qubit state, which is modulated by the the drive with the effective strength $G=gA/\sqrt{(\omega_{\rm c}-\omega_{\rm d})^2+\kappa^2/4}$ where $\kappa$ is the cavity dissipation. Consequently at high powers, the effective cavity frequency becomes a weighted average over the qubit state [@Abragam; @Li13], i.e. simply $\omega_{\rm c}$.

However, $G$ diverges with increasing $A$, thus we expect that the RWA is not valid at large drive powers. To capture the behaviour in this regime, we again transform the cavity in Eq. (\[eq:driven RabiHam\]) into the vacuum but then apply the counter-rotating hybridized rotating-wave (CHRW) approximation [@Lu12; @Yan15]. After a Schrieffer–Wolff transformation, we obtain Eq. (\[eq:RWAtwolevel\]), but with the renormalized qubit frequency $\tilde{\omega}_0\equiv\omega_0J_0(2G\xi/\omega_{\rm d})$ and 