wire bond connection.

As a result, our estimation of the total attenuation from the ports 1a and 1b to the input of the cavity is 88.5 dB, i.e. $P_{\rm in, a} \approx P_{1{\rm a}}-88.5$ dBm in the spectroscopy experiment and similarly $P_{\rm in, b} \approx P_{1{\rm b}}-88.5$ dBm in the relaxation experiment. In the spectroscopy experiment, the power $P_{1{\rm a}}$ is obtained from the readout of the analog signal generator, from which we substract $3.5\ \textrm{dB}$, representing the insertion loss of the power combiner. In the case of relaxation measurements, the power $P_{1{\rm b}}$ is obtained experimentally from mixer calibration (measuring the power at 1b for different DC voltages applied to the I port in the linear regime of the mixer). The estimates above are remarkably consistent with results obtained from fitting the spectroscopy data in Fig. 3(a) and the ac Stark-shifted qubit frequency \[inset of Fig. 3(b)\] in the main text, which correspond to an attenuation of $\sim 85.5$ dB in the line 1a-1c (1b-1c). In this sense, any one of these plots can be regarded as a precise power calibration for the others.

Transmon characterization
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The classical Hamiltonian function for our driven transmon-cavity system can be written 