## 3 Results
### Time Domain Transmon Characterization  
The time domain qubit characterization is conducted using the setup described in Section 2.2. We performed Rabi and Ramsey spectroscopy to measure the relaxation time \(T_{1}\) and the decoherence time \(T_{2}\). Figure 5 reports the Chevron plot for the Rabi frequency, while panel c shows the Rabi oscillations dependence from the excitation pulse power in the on-resonance case. Rabi spectroscopy is essential in studying the time domain behavior of the qubit. This technique allows for determining the temporal length of the excitation pulse that brings the qubit from \(|0\rangle\) to \(|1\rangle\) (\(\pi\) pulse) or into a superposition state (\(\pi/2\) pulse), where \(|0\rangle\) and \(|1\rangle\) contribute equally. Panel c of Figure 5 reports the power dependence of the Rabi oscillations. \begin{table}
\begin{tabular}{c c} \hline \hline
**Variables** & **Values** \\ \hline \(\chi/2\pi\) [MHz] & \(-3.41\pm\) 0.08 \\ \(\chi_{01}/2\pi\) [MHz] & \(-10.2\pm\) 0.2 \\ \(\chi_{12}/2\pi\) [MHz] & \(-13.6\pm\) 0.3 \\ \(\alpha\) [MHz] & \(421\pm 84\) \\ \(g_{01}/2\pi\) [MHz] & \(92.5\pm 1\); \(75\pm 12\) \\ \(C\) [fF] & \(46\pm 5\) \\ \(T_{1}\) [us] & \(8.68\pm 0.72\) \\ \(T_{2}\) [us] & \(2.30\pm 0.11\) \\ \(T_{\phi}\) [us] & \(2.65\pm 0.15\) \\ \(L_{J}\) [nH] & \(13\pm 2\) \\ \(l_{C}\) [nA] & \(24.7\pm 1.3\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the experimental qubit-resonator parameters. From this map, we extract the Rabi frequency dependence from the excitation power (Figure 5d). Adopting the semi-classical approach of [28], where the excitation field is treated as classical, the Rabi frequency dependence from the average number of photons \(\bar{n}\) is:  
\[\Omega_{R}=2g_{01}\sqrt{\bar{n}} \tag{7}\]  
By fitting the data of Figure 5d with a straight line, we obtain an independent estimation of \(g_{01}/2\pi\) = \(75\pm 12\) MHz, which is in relatively good agreement with our previous result of 92.5 MHz. We calculated the power entering the cavity in dBm as the sum of the excitation power (generator output power plus the line attenuation) and an additional attenuation factor due to the detuning from the excitation frequency and the resonator frequency. We estimate the latter from simulations to be \(-104\) dB. The excitation power is then converted into watts and we calculate the average number of photons as:  
\[\bar{n}=\frac{P}{\bar{h}\
u\gamma} \tag{8}\]  
where \(P\) is the excitation power, \(h\) is the Planck constant, \(\
u\) is the excitation frequency, and \(\gamma\) is the cavity dissipation rate, which is estimated to be about 200 kHz from the line width of the low power feature of Figure 3. The large uncertainty value on our \(g_{01}/2\pi\) estimation is mainly due to an uncertainty on the power attenuation values that we estimated as \(\pm 2\) dB. To extract the qubit lifetime \(T_{1}\), we measured the ground state population (Figure 6, left). By fitting with an exponential function, we obtain \(T_{1}\) = 8.68 \(\pm\) 0.72 us. We performed Ramsey spectroscopy by sending two off-resonance \(\pi/2\) pulses separated by a delay \(\Delta t\)  
Figure 5: (**a**) measurement scheme for the Rabi spectroscopy. (**b**) Chevron plot, acquired with excitation power P = \(-93\) dBm. The y-axis is given as detuning from the resonance frequency of 6.4194 GHz. (**c**) Rabi oscillation dependence from the excitation tone power. Excitation frequency = 6.4194 GHz. (**d**) Linear dependence of the Rabi frequency from the square root of the average photon number. The data have been fit with a straight line angular coefficient \(6.6\times 10^{-3}\) MHz\({}^{-1}\) and intercept \(2.6\times 10^{-4}\) MHz\({}^{-1}\). Ramsey oscillations of the ground state population are reported in Figure 6 (right panel) for a detuning of 600 KHz. We also reproduced the same measurements with a detuning of 200 and 400 KHz (not shown), and we estimate \(T_{2}=2.30\pm 0.11\) us.