hbar \omega_{ge}\sigma _{ee} \\  \nonumber
&&+\hbar g_{0}\left( a^{\dag}\sigma _{ef}+a\sigma _{fe}\right) + 2\hbar \Omega(\sigma_{fg}+\sigma_{gf}) \cos\omega_p t.\end{aligned}$$ Here, the atomic operator $\sigma_{jk}$ is defined as $\sigma_{jk}=|j\rangle\langle k|$ with $\{|j\rangle,|k\rangle\}=\{|g\rangle,|e\rangle, |f\rangle\}$ and $a^\dag$ and $a$ are the photon creation and annihilation operators in the single-mode resonator, respectively. The first four terms of Eq. (1) represent the Jaynes-Cummings Hamiltonian for the two-level system with the following specific features: it presents the interaction of two excited states ($|e\rangle$ and $|f\rangle$) with the resonator rather than the ground and excited states. Nevertheless, the Jaynes-Cummings physics can be applied here [@JC1963; @Haroche2006; @Rempe1987]. Namely, in the vicinity of the $|e\rangle \leftrightarrow |f\rangle$ resonance with the resonator, $|f\rangle$ level splits due to the interaction with the resonator vacuum mode, which can be described in terms of zero-photon dressed states as $|v_{0}\rangle=\cos\theta|f0\rangle-\sin\theta|e1\rangle$ and $|u_{0}\rangle=\sin\theta|f0\rangle+\cos\theta|e1\rangle$, where $\tan{2\theta} = 2g_{0}/(\omega_{r}-\omega_{f}+\omega_{e})$ and the state $|jn\rangle=|j\rangle\otimes|n\rangle$ with the two quantum numbers $j$ and $n$, denoting the three-level atomic states $|j\rangle$ and the Fock states of resonator $|n\rangle=\lbrace|0\rangle,|1\rangle,|2\rangle,...\rbrace$, respectively. The dressed states are schematically shown in the left panel of Fig. \[picture\](d), while the right panel shows the general case of a system with $n$ photons in the resonator. The last term represents the probing field which couples the ground and the second excited states ($|g\rangle \leftrightarrow |f\rangle$, where the vacuum mode $|0\rangle$ is omitted for simplification).

![(a) Spectrum of the artificial atom vs biased flux plotted as a reflection coefficient $|r|$. The transition $|e\rangle \leftrightarrow |f\rangle$ is v