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**Observation of the Bloch–Siegert shift in a driven quantum-to-classical transition: supplementary information**

Experimental methods {#experimental-methods .unnumbered}
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![Simplified circuit schematic of the measurement setup. We show (a) the room-temperature control electronics for the reflection measurement and; (b) the relaxation time measurement. (c) Low-temperature circuit. Here $\omega_{\rm q}$ refers generically to the frequency of the microwave tone applied to the qubit (port 3c of the low temperature setup), while $\omega_{\rm d}$ and $\omega_{\rm p}$ are the drive and probe frequencies.[]{data-label="fig:mschema"}](figS1_v1){width="0.8\linewidth"}

In Fig. \[fig:mschema\], we present the electronic measurement set up at the room temperature used for the reflection spectrometry, and for the measurements of the relaxation time. We also show show the electrical circuit at low temperature (including the transmon). For the spectroscopy measurements \[Fig. \[fig:mschema\](a)\], the cavity probe signal with frequency $\omega_{\rm p}$ is generated by an Agilent N5230C PNA-L network analyzer and the cavity drive with frequency $\omega_{\rm d}$ is provided by an Agilent E8257D analog signal generator. These are combined in 