the appearance of ac-Stark shifts in the energy levels of both the qubit and the cavity. The inclusion of counter-rotating terms produces an additional displacement of the energy levels. This Bloch–Siegert (BS) shift [@BlochSiegert40] is usually very small on standard experimental platforms since it depends on the ratio between the coupling and the sum of the Larmor and cavity frequencies. Recently however, significant experimental effort has been put into increasing the coupling to values comparable with the Larmor frequency [@Ciuti2006; @Devoret2007; @Bourassa2009], most notable in semiconductor dots [@Gunter2009; @Todorov2010] and superconducting circuits [@FornDiaz10; @Niemczyk2010; @FornDiaz16; @Yoshihara16; @Baust2016; @Bishop09]. Other approaches for observing the Bloch–Siegert shift include the detailed analysis of two-level Landau–Zener spectra of Rydberg atoms [@Fregenal04; @Forre04] and Cooper-pair boxes [@Goorden03; @Tuorila10; @Tuorila13], as well as the simulation of the Rabi model in rotating frames [@Li13; @Braumuller16].

![Schematic of the experimental setup. (a) Energy levels of a dispersive JC system. The vacuum ac-Stark shift is given by $\chi_0=g^2/(\omega_0-\omega_{\rm c})$. (b) Optical micrograph of the main elements of the sample (false colors). (c) We drive the cavity with a detuned drive frequency $\omega_{\rm d}<\omega_{\rm c}$. The spectrum of the system is monitored by a weak probe with frequency $\omega_{\rm p}$, 