quilibrium background radiation from the control lines. The $|e\rangle \leftrightarrow |f\rangle$ line is ‘brightest’ at the degeneracy point ($\delta\Phi = 0$), where $\hbar\omega_{eg}$ is minimal. It is difficult to avoid the unwanted population of the $|e\rangle$ level with an energy gap of about 1 GHz, but in our experiment it is useful as it allows to reveal the full spectroscopy picture. The calibrated system effective temperature is around 75 mK (details in Supplementary Material  [@Suppl]), which is similar to the results of other groups [@Geerlings2013; @Jin2015]. The transitions $|g\rangle \leftrightarrow |e\rangle$ and $|e\rangle \leftrightarrow |f\rangle$ are bright due to large transition matrix elements. However, the $|g\rangle \leftrightarrow |f\rangle$ and $|g0\rangle \leftrightarrow |e1\rangle$ lines vanish at $\delta\Phi = 0$ as they are prohibited due to selection rules [@Abdumalikov2010; @Inomata2012]. The dashed lines in Fig. \[ATS\](b) represent the calculations from the full-system Hamiltonian with Josephson energy $E_{J}=h \times242\,\rm GHz$ of the junctions, charging energy $E_{c}= h \times4.2\,\rm GHz$, and the geometrical factor $\alpha=0.8$ of the SQUID Josephson junctions [@Inomata2012]. Our artificial atom is coupled to the resonator through capacitance $C_{c} \approx 2.0\, \rm fF$.

Note that the $|e\rangle \leftrightarrow |f\rangle$ transition between the two excited states is resonant with the resonator at $\delta\Phi\sim2.15\times10^{-3}\Phi_{0}$. Next, we measure the reflection from the system in the vicinity of the resonance $\omega_{ef} \approx \omega_r$. The result ($|r|$ in the frequency range from 6 to 12.5 GHz vs $\delta\Phi$) is shown in Figs. \[ATS\](c) and (d). The anticrossings are clearly observed in the two spectroscopic lines corresponding to the transitions $|g\rangle \leftrightarrow |f\rangle$ in Fig. \[ATS\](c) and $|e\rangle \leftrightarrow |f\rangle$ in Fig. \[ATS\](d), respectively. This confirms the strong coupling between the excited state tra