to a nearly-zero value given by $\Xi_{\rm ground}\approx 2g/(\omega_0+\omega_{\rm c}) \approx 10^{-2}$, indicating the quantum regime. As the power increases, the order parameter starts to increase at $N_{\rm c}\approx 1$ and the fluctuations become large as well. At high enough power (approximately $N_{\rm c}=10$), the system enters the classical phase, where a large number of states participate in the dynamics. Interestingly, the BS effect is reproduced by the order parameter which is seen as the nonmonotonic deviation between the RWA and non-RWA order parameters.

![The Bloch–Siegert effect in the quantum and classical regimes. (a) Measured reflection coefficient $|\Gamma|$ as a function of $\omega_0$ and $\omega_{\rm p}$, taken at high power (upper panel) and low power (lower panel). The strongest resonance minima extracted from the data are marked with dots. The lower panel demonstrates the vacuum BS shift of $\chi_{\rm BS} = 0.7$ MHz, displayed as the difference between the calculated cavity frequency with only the analytic vacuum-Stark shift ($\omega_{\rm c}-\chi_0$, dashed white line) and with the analytic vacuum-BS shift included ($\omega_{\rm c}-\chi_0-\chi_{\rm BS}$, solid black line). (b) Bloch–Siegert shift as a function of driving power (corresponding to $N_{\rm c}=10\ldots 1650$) in the classical regime for $\omega_{\rm d}/(2\pi) = 4.350, \ 