4.355, \ \textrm{and} \ 4.360$ GHz. We show the experimental contour for the reflection coefficient and compare the resonance locations with the classical-approximation formula for $\omega_+$ in Eq. (\[eq:class\]) and with the numerical results with $N=5,7$ ($N=7$ is calculated also in the RWA).[]{data-label="fig:Boxes"}](fig4){width="1.0\linewidth"}

In Fig. \[fig:Boxes\](a), we further demonstrate the essential difference between the quantum and classical regimes and the role of the Bloch–Siegert shift by analyzing the dependence of the response on the qubit frequency at the two extremes of the drive power. As expected, at very high powers the resonance is located at the bare cavity frequency, irrespective of the qubit frequency. At low drive powers, the resonance location decreases with the qubit frequency, which is characteristic to a vacuum-Stark and BS shifted cavity. The inclusion of the vacuum-BS shift clearly gives a better fit to the resonance data. We have also studied in more detail the BS effect in the transition region. In addition to numerical results, we can also employ the classical model to calculate the normal modes $$\label{eq:class}
\omega_{\pm} = \sqrt{\frac{\Omega^2+\omega_{\rm c}^2}{2}
\pm\frac{1}{2}\sqrt{\left[\Omega^2-\omega_{\rm c}^2\right]^2+16g^2\Omega^2\omega_{\rm c}/\omega_0} },$$ where $\Omega^2 = J_0(\sqrt{P_{\rm in}/P_0})\omega_{0}^2$, $P_{\rm in}$ is the power fed into the system, and $P_0=1.0$ pW. These are 