decreases according to the standard quantum theory of ac-Stark shifted transitions [@Boissonneault09; @Boissonneault12a; @Boissonneault12b] which also includes effects caused by the higher transmon energy levels. At about the same number of photons, discrepancies with respect to this model start to appear. A classical theory [@supplement; @Irwin; @Irwin2] which treats the transmon as an anharmonic oscillator provides a good fitting with $T_1$ up to the power level when the cavity frequency can no longer be taken as constant (about $N_c \approx 2$, after which the measured $T_1$ drops again).

In Fig. \[fig:expBSshift\](c), we demonstrate that the fluctuations $\langle \delta N^2\rangle = \langle N^2\rangle - \langle N \rangle^2$ of the transmon occupation number change rather abruptly when the drive power is swept over the transition regime. Thus, distinctly from the transition predicted by Carmichael [@Carmichael15], also the transmon becomes a multilevel classical object. In the laboratory frame, we characterize the quantum-to-classical transition by defining a dimensionless order parameter $\Xi=|\langle (\hat{a}+\hat{a}^{\dag})\hat{\Pi}_{\rm x}\rangle|$, where $\hat{\Pi}_{\rm x}=\hat{c}+\hat{c}^{\dag}$ is the analog of $\hat{\sigma}_{\rm x}$ and $\hat{c}$ is the annihilation operator for the multilevel transmon [@BishopPhD]. Note that this order parameter includes the correlations between the cavity and the transmon. At low powers the order parameter saturates 