which is swept within the window $[4.36, 4.39]$ GHz. (d) In the $T_1$ measurements, the cavity is driven continuously at $\omega_{\rm d}$. In the steady state, a long $2 \mu$s microwave pulse at the ac-Stark shifted qubit frequency $\omega_q \gg \omega_{\rm c}$ is applied to the gate line of the qubit. When the pulse ends, the decay traces of the system relaxing back to the steady state are monitored in the time domain using a homodyne detection scheme. []{data-label="fig:schema"}](fig1){width="1\linewidth"}

In this work we take a different route. We recognize that the counter-rotating terms do not conserve the excitation number. Therefore, the natural framework for their experimental demonstration is that of driven-dissipative systems [@Hartmann16; @Noh16]. Guided by this intuition, we realize a setup consisting of a transmon [@Koch07] dispersively coupled to a cavity, where the cavity is driven at a fixed off-resonance microwave tone, while at the same time the spectrum is scanned by a comparatively weaker probe field, see Fig. \[fig:schema\]. At low driving powers, we observe the expected vacuum ac-Stark shift [@Reed10; @Boissonneault10; @Bishop10; @Suri15; @Paik2011]. This is followed by a transition regime dominated by nonlinear effects as the power is increased. For the Jaynes–Cummings model, such transition has 