## 3 Numerical Computation Conventional and Unconventional Photon Blockade
### Unconventional Photon Blockade with Atom-Cavity Detuning  
In this section, we focus on the unconventional photon blockade when Cavity A, Cavity B, and the atom are out-of resonance based on weak nonlinearities. By solving Equation (14) numerically, we plot the equal-time second order correlation function \(\log_{10}g^{(2)}(0)\) as a function of the scaled cavity detuning \(\Delta/\kappa_{a}\) and the scaled atom detuning \(\Delta_{1}/\kappa_{a}\) in Figure 6, where Cavity B is resonant with Cavity A. Based on the parameters of Figure 6, when the atom is weakly coupled to Cavity B by \(g=0.67\kappa_{a}\) and the interaction strength between the two cavities is \(J=2\kappa_{a}\), one can obtain an irregular single photon blockade region at the center of the figure while the detuning of Cavity A, Cavity B, and the atom are all within \([-2\kappa_{a},+2\kappa_{a}]\). By increasing the coupling strength and the hopping strength to \(g=0.8\kappa_{a}\) and \(J=4\kappa_{a}\), we can obtain the other two symmetric photon block detuning regimes located at about \([-\kappa_{a},-3\kappa_{a}]\) and \([+\kappa_{a},+3\kappa_{a}]\), as shown in Figure 6b. With the fixed coupling strength \(g=0.67\kappa_{a}\) and based on the condition that the atom in Cavity B is resonant with the driving laser, i.e., \(\Delta_{1}=0\), Figure 7 shows the correlation function \(\log_{10}g^{(2)}(0)\) as a function of different detunings of Cavities A and B for photon hopping strength between the two cavities \(J=3\kappa_{a}\) in Figure 7a and \(J=6\kappa_{a}\) in Figure 7b. One can see from Figure 7 that the single photon blocked region (\(\log_{10}g^{(2)}(0)<0\)) is symmetrical and centered on \(\Delta_{a}=0\) and \(\Delta_{b}=0\). As the moderate increase of the hopping strength \(J\), the single photon blocked region stretches to both the red sideband and blue sideband of Cavity A.  
Optimal Parameters for Sub-Poisson Characters Derived by the Non-Hermitian Effective Hamiltonian Method  
In this section, we focus on the analytical optimal conditions for the system parameters that maximize the sub-Poissonian character of Cavity A. In the weak driving limit, the system states can be expressed as an expansion on the bare states mentioned in Section 2:  
\[ \begin{array}{ll}|\psi(t)\rangle&=c_{000}|00g\rangle+c_{001}|00e\rangle+c_{010 }|01g\rangle+c_{100}|10g\rangle\\ &+c_{011}|01e\rangle+c_{101}|10e\rangle+c_{110}|11g\rangle+c_{020}|02g\rangle+c_ {200}|20g\rangle\end{array}  \tag{19}\]  
The system is governed by a stochastic Schrodinger equation as (assuming \(\hbar=1\))  
\[i\frac{d}{dt}|\psi(t)\rangle=H_{eff}|\psi(t)\rangle \tag{20}\]  
where the non-Hermitian effective Hamiltonian [17; 18; 26] can be written as  
\[ \begin{array}{ll}H_{eff}&=(\Delta_{a}-i\frac{\kappa_{b}}{2})a^{\dagger}a+( \Delta_{b}-i\frac{\kappa_{b}}{2})b^{\dagger}b+(\Delta_{1}-i\frac{\gamma}{2}) \sigma^{\dagger}\sigma\\ &+g(\sigma^{\dagger}b+\sigma b^{\dagger})+J(a^{\dagger}b+ab^{\dagger})+ \varepsilon(a^{\dagger}+a)\end{array}  \tag{21}\]  
Here, we consider the system evolution under the condition that the three components Cavity A, Cavity B, and the atom are resonant, i.e., \(\Delta_{a}=\Delta_{b}=\Delta_{1}=\Delta\), and the bandwidth of Cavity A equals that of Cavity B, i.e., \(\kappa_{a}=\kappa_{b}=\kappa\). The time-dependent coefficients \(c_{mnp}(t)\) can be obtained from Equations (20) and (21).