## 3 Results  
### Transmon Spectroscopic Characterization  
The transmon characterization consists in the extraction of the Hamiltonian parameters from the experimental data. The Hamiltonian of a dispesively coupled transmon resonator system (with \(\hbar=1\)) is [18]:  
\[H_{JC}=(\omega_{r}-\frac{\chi_{12}}{2}+\chi\sigma^{z})a^{\dagger}a+\frac{1}{2}( \omega_{q}+\chi_{01})\sigma^{z}, \tag{1}\]  
where \(\omega_{r}\) is the resonator angular frequency, \(\sigma^{z}\) is the Pauli matrix, \(a^{\dagger}\) (_a_) is the creation (annihilation) operator, and \(\omega_{q}\) is the qubit angular frequency. \(\chi\) is the total dispersive shift which is defined as:  
\[\chi=\chi_{01}-\frac{\chi_{12}}{2} \tag{2}\]  
while \(\chi_{n,n+1}\) are defined as:  
\[\chi_{n,n+1}=\frac{g_{n,n+1}^{2}}{\Delta_{n,n+1}} \tag{3}\]  
where \(g_{n,n+1}\) is the coupling strength between the energy level \(n\) and \(n+1\) of the qubit, and \(\Delta_{n,n+1}\) is defined as \(\omega_{n,n+1}-\omega_{r}\). \(\omega_{n,n+1}\) is the qubit \(\left|n\right\rangle\rightarrow\left|n+1\right\rangle\) transition frequency. In this notation, \(\omega_{q}=\omega_{01}\). Among the different parameters of interest, \(g_{01},\chi_{01},\chi_{12},\chi\), and the anharmonicity \(\alpha\) are of particular relevance. Figure 3 reports the resonator absorption spectrum (S21) acquired as a function of the probing power. Two features are clearly observable, with one appearing at low power and one appearing at high power. The low-power feature accounts for the frequency of the dressed system when the transmon is in the ground state. The sharp high power peak appearing around -75 dbm is given by the bare resonator frequency [26]. For the Al cavity  
Figure 2: Scheme of the experimental setup used for the transmon characterization. without the silicon chip, we measure a \(Q_{0}\) of 217,000 \(\pm\) 16,000. These two features of Figure 3 are separated by  
\[\omega_{r}-\omega_{r}^{{}^{\prime}}=\chi+\frac{\chi_{12}}{2}, \tag{4}\]  
where \(\omega_{r}\) is the bare resonator frequency, while \(\omega_{r}^{{}^{\prime}}\) is the frequency of the dressed resonator-qubit system. From our data, we obtain that \((\chi+\frac{\chi_{12}}{2})/2\pi=-10.2\pm 0.1\) MHz. This power scan of the resonator is an extremely useful preliminary measurement and allows for choosing the most suitable readout power for the qubit characterization in the time domain. We performed two-tone qubit spectroscopy in order to resolve single photon number peaks inside the cavity [27]. The cavity is coherently probed with a tone resonant with \(\omega_{r}/2\pi=v_{r}\). At the same time with a second tone, we excite the \(|0\rangle\rightarrow|1\rangle\) qubit transition. The qubit state and the photon number inside the cavity are coupled (see Equation (1)). Thus, every time the qubit is excited, the cavity absorption peak undergoes a dispersive shift. We detect this as a dip in the cavity absorption spectrum. The qubit absorption spectra acquired for different powers of the cavity probe tone are reported in Figure 4.  
The peaks corresponding to individual photon number states are clearly observable and are separated by \(2\chi/2\pi=-6.82\)\(\pm\) 0.16 MHz. Combining \(\chi\) with Equation (4), we  
Figure 4: Qubit spectroscopy of individually resolved photon numbers inside the cavity, for an average photon population of \(N=1.8\) (**left**, \(P_{probe}=-102\) dbm) and \(N=4.1\) (**right**, \(P_{probe}=-98\) dbm). Each peak is separated by \(2\chi/2\pi=-6.82\pm 0.16\) MHz. Figure 3: Cavity power scan. The high power feature (B) corresponds to the bare cavity transmission peak. The low power feature (A) is the transmission peak of the dressed cavity-qubit system with the qubit in the ground state. estimate \(\chi_{12}/2\pi=-13.6\pm 0.3\) MHz. Using (2), we obtain that \(\chi_{01}/2\pi=-10.2\pm 0.2\) MHz. We then extract the bare resonance of the qubit. The frequency position of the peak corresponding to zero photons in the cavity is equal to \((\omega_{01}+\chi_{01})/2\pi=6.4194\) GHz.