obtained by treating the Josephson junction as a classical inductance and by averaging over the drive period [@supplement; @Saira16]. In Fig. \[fig:Boxes\](b) we present the results of experiments at three different drive frequencies. Because the transition occurs at a fixed value $N_{\rm c}\sim 1$ for all drive frequencies [@supplement], we show the responses in the same interval of cavity quanta, ranging from $N_{\rm c}=10\ldots 1650$. One notices that in all three measurements the BS effect is present. There is very good agreement between the experimental data and the numerics, and also in the high-power region the classical mode $\omega_+$ provides a good match to the data.

To conclude, we have studied a driven-dissipative quantum-to-classical transition in a circuit-QED setup consisting of a transmon coupled to a cavity. The response of the system to a weak probe reveals a clear signature of the Bloch–Siegert shift, which is nonmonotonous in the drive power. Our experiment paves the way toward future experiments of quantum simulations of dissipative phase transitions in one-dimensional circuit-QED lattices.

Discussions with D. Angelakis, E. Thuneberg, M. Silveri, and S. Laine on theory, and M. Sillanpää and J. Pirkkalainen on sample design and fabrication are gratefully acknowledged. We are grateful 