been predicted and studied in the resonant qubit-cavity case [@Alsing92; @Carmichael15; @Fink10; @Fink17]. In the dispersive limit, the transition region is no longer abrupt, but it is expected to soften into a Kerr-type nonlinearity [@Carmichael15], in agreement with our observations.

As the system approaches and enters the classical phase, the frequency renormalization due to the BS effect results in a strongly nonmonotonic dependence on the number of photons in the cavity. The effect escapes the scope of the simple RWA, but can be modeled numerically using the Floquet formalism [@Shirley65] for the driven cavity coupled with a multilevel transmon. The measured deviation from the RWA result is in agreement with the simulations and constitutes the first experimental observation of the BS effect in a driven quantum-to-classical transition. [*Theoretical predictions for a two-level system –* ]{} We develop a simple physical picture of these phenomena by considering the two lowest transmon states coupled with the cavity which is driven by a tone of amplitude $A$ and frequency $\omega_d$. The resulting driven Rabi Hamiltonian (scaled with $\hbar$) is written as $$\label{eq:driven RabiHam}
\hat{H}= \omega_{\rm c} \hat{a}^{\dag}\hat{a} + \frac{\omega_0}{2}\hat{\sigma}_{\rm z} + g(\hat{a}^{\dag}+\hat{a})\hat{\sigma}_{\rm x} + A\cos(\omega_{\rm d} t)(\hat{a}^{\dag}+\hat{a}),$$ where $\omega_{\rm c}$ and $\omega_{0}$ are 