drive amplitude $\tilde{G}/2 \equiv G(1-\xi)$. The parameter $\xi$ is determined by solving $G(1-\xi)=\omega_0 J_1(2G\xi/\omega_{\rm d})$ [@supplement].

![Numerical simulation of the spectrum for a two-level system, corresponding to experimental values $\omega_{\rm c}/(2\pi)=4.376$ GHz, $\omega_0/(2\pi)=5.16$ GHz, $g/(2\pi)=80$ MHz, $\omega_{\rm d}/(2\pi) = 4.350$ GHz, and $\kappa/(2\pi) = 4.0$ MHz. (a) Probe reflection coefficient $|\Gamma|$ as a function of probe frequency $\omega_{\rm p}$ and cavity occupation $N_{\rm c}$. The locations of resonance in the RWA (blue) and in the CHRW (green) approximation are shown as well. (b) Reflection coefficients for the three different probe frequencies indicated in (a) with dashed vertical lines.[]{data-label="fig:BSshift"}](fig2){width="1.0\linewidth"}

We show the effective cavity resonance frequencies in the RWA and CHRW approximation in Fig. \[fig:BSshift\](a). We compare the results with the numerical simulations [@supplement] of the probe reflection coefficient $\Gamma(\omega_{\rm p})=(Z(\omega_{\rm p})-Z_0)/(Z(\omega_{\rm p})+Z_0)$, where $Z(\omega_{\rm p})$ and $Z_0$ are the impedances of the driven cavity coupled to a two-level transmon and the transmission line, respectively. The impedance $Z(\omega_{\rm p})$ was calculated from the probe induced transition rates between the quasienergy states of the driven cavity-transmon system by employing the Kramers-Kronig relation [@Silveri13; @Tuorila13; @LL; @supplement; @Blumel89; @Blumel91; @Grifoni98; @Gasparinetti13; @Hausinger11; @HausingerPhD]. We see that the RWA gives the overall qualitative behaviour 