## 2 Model  
As shown in Figure 1, we consider a coupled cavity system that consists of two cascaded cavities. The first cavity (Cavity A) is driven by a classical light field with frequency \(\omega_{L}\) and amplitude \(\varepsilon\). The Hamiltonian of Cavity A with the driven lesser is given by  
\[H_{A}=\hbar\omega_{a}a^{\dagger}a+\hbar\varepsilon(a^{\dagger}e^{-i\omega_{L} t}+ae^{i\omega_{L}t}) \tag{1}\]  
where \(a\) is the annihilation operator of Cavity A whose resonant frequency is \(\omega_{a}\), and \(a^{\dagger}\) is the creation operator. We apply the rotating wave approximation. The Hamiltonian of the second cavity (Cavity B) reads as  
\[H_{\rm B}=\hbar\omega_{b}b^{\dagger}b+\hbar\omega_{1}\sigma^{\dagger}\sigma+ \hbar g(\sigma^{\dagger}b+\sigma b^{\dagger}) \tag{2}\]  
Here, \(b\) and \(b^{\dagger}\) are the annihilation operator and creation operator of the mode in Cavity B, whose resonant frequency is \(\omega_{b}\). The lowering (up) operator of the two-level atom in Cavity B is denoted by \(\sigma\left(\sigma^{\dagger}\right)\), and the transition frequency of the atom is \(\omega_{1}\). \(g\) describes the coupling strength between the atom and Cavity B. These two cavities are coupled through photon hopping interactions with strength of \(J\). In a rotating frame with respect to \(H_{0}=\hbar\omega_{L}a^{\dagger}a+\hbar\omega_{L}b^{\dagger}b+\hbar\omega_{L} \sigma^{\dagger}\sigma\), the Hamiltonian of the whole system is given by (\(\hbar=1\)):  
\[H=\Delta_{a}a^{\dagger}a+\Delta_{b}b^{\dagger}b+\Delta_{1}\sigma^{\dagger} \sigma+g(\sigma^{\dagger}b+\sigma b^{\dagger})+J(a^{\dagger}b+ab^{\dagger})+ \varepsilon(a^{\dagger}+a). \tag{3}\]  
\(\Delta_{a}\), \(\Delta_{b}\) and \(\Delta_{1}\) are, respectively, the frequency detuning of Cavity A, Cavity B, and the atom with respect to the driving field. In the weak-driving regime, where the driving amplitude \(\varepsilon\) is far less than the dissipation rate of Cavity A, we consider the total excitation number \(N\leq 2\) in our system. Then the bare states of the system are \(|0,0,g\rangle\), \(|0,0,e\rangle\), \(|0,1,g\rangle\), \(|1,0,g\rangle\), \(|0,1,e\rangle\), \(|1,0,e\rangle\), \(|1,1,g\rangle\), \(|0,2,g\rangle\), and \(|2,0,g\rangle\), with the first number denoting the photon number in Cavity A, the second number denoting the photon number in Cavity B, and the third part denoting the status of the atom in Cavity B. For simplification, we denote these nine bare states by \(|0\rangle\), \(|1\rangle\), \(\cdots\), \(|8\rangle\), respectively. The Hamiltonian in Equation (3) without the driving field can be expressed in the following matrix:  
\[\left[ \begin{array}{cccccccc}0&0&0&0&0&0&0&0&0\\ 0&\Delta_{1}&g&0&0&0&0&0&0\\ 0&g&\Delta_{b}&J&0&0&0&0&0\\ 0&0&J&\Delta_{a}&0&0&0&0&0\\ 0&0&0&0&\Delta_{b}+\Delta_{1}&J&0&g&0\\ 0&0&0&0&J&\Delta_{a}+\Delta_{1}&g&0&0\\ 0&0&0&0&0&g&\Delta_{a}+\Delta_{b}&J&J\\ 0&0&0&0&g&0&J&2\Delta_{b}&0\\ 0&0&0&0&0&J&0&2\Delta_{a}\end{array} \right] \tag{4}\]  
which is defined based on the bare states mentioned above. Assuming \(\Delta_{a}=\Delta_{b}=\Delta_{1}=\Delta\), the eigenvalues and eigenstates of the system within the two excitation subspace are given by  
\[E_{0}=0,|\psi_{0}\rangle=|0,0,g\rangle \tag{5}\]  
\[E_{1}^{0}=\Delta_{e},\,\left|\psi_{1}^{0}\right\rangle=-\frac{J}{\sqrt{g^{2}+ J^{2}}}|0,0,e\rangle+\frac{g}{\sqrt{g^{2}+J^{2}}}|1,0,g\rangle \tag{6}\]  
\[E_{1}^{-}=-\sqrt{g^{2}+J^{2}}+\Delta_{e},\left|\psi_{1}^{-}\right\rangle=\frac {g}{\sqrt{2J^{2}+2g^{2}}}|0,0,e\rangle-\frac{1}{\sqrt{2}}|0,1,g\rangle+\frac{ J}{\sqrt{2J^{2}+2g^{2}}}|1,0,g\rangle \tag{7}\]  
\[E_{1}^{+}=\sqrt{g^{2}+J^{2}}+\Delta_{e},\left|\psi_{1}^{+}\right\rangle=\frac {g}{\sqrt{2J^{2}+2g^{2}}}|0,0,e\rangle+\frac{1}{\sqrt{2}}|0,1,g\rangle+\frac{ J}{\sqrt{2J^{2}+2g^{2}}}|1,0,g\rangle \tag{8}\]  
\[E_{2}^{0}=2\Delta \tag{9}\]  
Figure 1: Schematic of the studied system. The system consists of two cavities, which are coupled to each other through a photon-hopping interaction. The first cavity is a linear cavity driven by a continuous laser.