produces an oscillatory response as a function of added photons.

[*Experimental results –*]{} The physical device \[Fig. \[fig:schema\](b)\] consists of a $\lambda/4$-waveguide-resonator cavity capacitively coupled to a transmon qubit with $E_{\rm J\Sigma}/E_{\rm C}=58$, flux-biased at $\omega_0/(2\pi)=5.16$ GHz. For the cavity, $\omega_{\rm c}/(2\pi)=4.376$ GHz and $\kappa/(2\pi) = 4.0$ MHz. The signals used for spectroscopy and for relaxation measurements are shown in Fig. \[fig:schema\](c),(d).

In Fig. \[fig:expBSshift\](a), we present the measured reflection coefficient $\Gamma$ as a function of the probe frequency $\omega_{\rm p}$ and the number of cavity quanta $N_{\rm c}$ at the off-resonant drive frequency $\omega_{\rm d}/(2\pi) = 4.35$ GHz. The spectrum clearly shows the quantum-to-classical transition and the BS effect, as expected from the previous theoretical considerations and they are well-matched by numerical simulations for the Hamiltonian in Eq. (\[eq:driven RabiHam\]), generalized to the case of a multilevel transmon [@Koch07; @CottetThesis; @supplement] and including the counter-rotating terms. The numerics have converged up to $N_{\rm c} \approx 100$ for $N=7$ transmon states. We have also performed a systematic experimental study of the spectrum at various transmon and drive frequencies [@supplement], obtaining similar features to those presented in Fig. \[fig:expBSshift\](a).

![Frequency renormalization and Bloch–Siegert effect. (a) Measured reflection coefficient $|\Gamma|$ as a 