excited states of the transmon start to contribute to the observed resonance location when $N_{\rm c}\sim 1$, as seen from the deviation between the numerical $N=2$ and $N=7$ results in Fig. \[fig:BSshift\](a). This coincides with the breakdown of the dispersive approximation used in Eq. (\[eq:RWAtwolevel\]), which is expected to occur when $N_{\rm c}\ll N_{\rm crit}$ [@Boissonneault09], where $N_{\rm crit} = [(\omega_0-\omega_{\rm c})/(2g)]^2 \approx 30$ for our experimental values.

At the transition and further in the classical regime, the two subsystems become strongly hybridized. In this case $[\hat{H},\hat{\sigma}_{\rm z}] \neq 0$ and, therefore, the qubit population is no longer conserved. We have observed this effect by performing relaxation measurements: under continuous constant driving at $\omega_{\rm d}=4.366$ GHz (see Fig. \[fig:schema\](d)), we excite the system with a long pulse at the ac-Stark shifted qubit frequency and we monitor the response of the cavity after this pulse ends. We extract the relaxation time from fitting the relaxation traces with an exponential, see Fig. \[fig:expBSshift\](b). The results show an increase in the relaxation time as the power is ramped up to above a cavity photon number $N_{c}\approx 1$, coinciding with the onset of the quantum-to-classical transition. In the same power range the qubit frequency 