digitalized by a fast data aquisition card Agilent Acqiris U1084A. For each value of the resonator drive power, the transmon state preparation and resonator response measurement are repeated $13\times 10^6$ times and averaged. This scheme allows us to monitor in time domain the variation of the reflected signal over several microseconds. After the trailing edge of the pulse, we record the relaxation traces of the system, and we extract the $T_1$ time by fitting with an exponential. In all these traces, small oscillations exist immediately after the trailing edge of the pulse, leading to the fitting errors shown in Fig. 3(b) in the main text. This procedure is repeated at 95 drive power values. For tuning the qubit transition frequency, in both cases depicted in Figs. \[fig:mschema\](a) and \[fig:mschema\](b) the dc flux bias is provided by an Agilent 33500B waveform generator. This is combined with the qubit drive using a bias-tee and applied to the transmon gate, see Fig. \[fig:mschema\](c). The signal from the cavity output line of the dilution refrigerator is separated from the input signal using a cryogenic circulator, and then amplified by cryogenic and room-temperature amplifiers. DC blocks (not shown) are used to break all possible ground 