## 3 Numerical Computation Conventional and Unconventional Photon Blockade
### Unconventional Photon Blockade with Atom-Cavity Detuning
According to the idea of performing a consistent expansion of the elements to dominant order in powers of \(\varepsilon/\kappa_{a}\)[30], we neglect the subleading order in the driving laser amplitude and obtain the equations of \(c_{mnp}(t)\):  
\[ic_{001}^{\ \bullet}=(\Delta-i\frac{\gamma}{2})c_{001}+gc_{010} \tag{22}\]  
\[ic_{010}^{\ \bullet}=(\Delta-i\frac{\kappa}{2})c_{010}+gc_{001}+Jc_{100} \tag{23}\]  
\[ic_{100}^{\ \bullet}=(\Delta-i\frac{\kappa}{2})c_{100}+Jc_{010}+\varepsilon c_{000} \tag{24}\]  
\[ic_{011}^{\ \bullet}=(\Delta-i\frac{\kappa}{2})c_{011}+(\Delta-i\frac{\gamma}{2} )c_{011}+\sqrt{2}gc_{020}+Jc_{101} \tag{25}\]  
\[ic_{101}^{\ \bullet}=(\Delta-i\frac{\kappa}{2})c_{101}+(\Delta-i\frac{\gamma}{2} )c_{101}+gc_{110}+Jc_{011}+\varepsilon c_{001} \tag{26}\]  
Figure 7: The equal-time second-order correlation function \(\log_{10}g^{(2)}(0)\) as a function of scaled detuning \(\Delta_{a}/\kappa_{a}\) and \(\Delta_{b}/\kappa_{b}\) under the condition of the atom resonant with the driving laser \(\Delta_{1}=0\) for (**a**) \(J=3\kappa_{a}\) and (**b**) \(J=6\kappa_{a}\). Other parameters are the same as in Figure 6a. \[ic_{110}^{\ \bullet} =(2\Delta-i\kappa)c_{110}+gc_{101}+\sqrt{2}fc_{020}+\sqrt{2}fc_{200}+ \varepsilon c_{010} \tag{27}\] \[ic_{020}^{\ \bullet} =(2\Delta-i\kappa)c_{020}+\sqrt{2}gc_{011}+\sqrt{2}fc_{110}\] (28) \[ic_{200}^{\ \bullet} =(2\Delta-i\kappa)c_{200}+\sqrt{2}fc_{110}+\sqrt{2}\varepsilon c_{ 100}. \tag{29}\]  
The relationship between different \(c_{mnp}(t)\) in Equations (22)-(29) once again illustrates the multi-transition channels between different system states, which is consistent with that in Figure 2a. The amplitudes of the coefficient \(c_{mnp}(t)\) in Equation (19) can be divided into three levels with respect to \(\varepsilon/\kappa_{a}\). That is, the amplitudes of \(c_{000}(t)\) is of the zero order of \(\varepsilon/\kappa_{a}\), \(\{c_{001}(t)\), \(c_{010}(t)\), and \(c_{100}(t)\) of the first order and \(\{c_{011}(t)\), \(c_{101}(t)\), \(c_{110}(t)\), \(c_{020}(t)\), \(c_{200}(t)\}\) of the second order, i.e., \(c_{000}(t)\gg c_{001}(t),c_{010}(t),c_{100}(t)\gg c_{011}(t),c_{101}(t),c_{110}( t),c_{020}(t),c_{200}(t)\). The average photon occupations in Cavity A and the equal-time second-order correlations then approximate to  
\[n_{a}=\left\langle a^{\dagger}a\right\rangle= \Big{|}c_{100}\Big{|}_{2}+\Big{|}c_{101}\Big{|}_{2}+\Big{|}c_{110} \Big{|}_{2}+2\Big{|}c_{200}\Big{|}_{2}\simeq\Big{|}c_{100}\Big{|}^{2} \tag{30}\] \[ic_{010}^{\ \bullet}=(\Delta-i\frac{\kappa}{2})c_{010}+gc_{001}+Jc_{100} \tag{23}\]0  
The expressions of \(c_{100}\) and \(c_{200}\) can be obtained by solving Equations (22)-(29) for the steady state \(\left|\psi\right\rangle_{\rm ss}\) with \(c_{mnp}^{\ \bullet}=0\) and \(c_{000}=1\). \[ic_{010}^{\ \bullet}=(\Delta-i\frac{\kappa}{2})c_{010}+gc_{001}+Jc_{100} \tag{23}\]1  
\[ic_{010}^{\ \bullet}=(\Delta-i\frac{\kappa}{2})c_{010}+gc_{001}+Jc_{100} \tag{23}\]2  
\[ \begin{array}{ll}M&=32g^{6}X_{1}^{2}+2g^{2}(4J^{2}+X_{1}^{2})(X_{1}^{2}X_{2} +4J^{2}X_{4})\\ &+8g^{4}X_{1}[X_{1}^{2}X_{3}+J^{2}(8\gamma-4\kappa+8i\Delta)]+X_{5}[4J^{2}+X_ {1}^{2}]^{2}X_{1}\end{array}  \tag{34}\]  
where, \(X_{1}=\kappa+2i\gamma\Delta\), \(X_{2}=2\kappa^{2}+7\kappa\gamma+5\gamma^{2}+22i\kappa\Delta+34i\gamma\Delta-56 \Delta^{2}\), \(X_{3}=3\kappa+4\gamma+14i\Delta\), \(X_{4}=2\kappa^{2}-\kappa\gamma+\gamma^{2}+6i\kappa\Delta+2i\gamma\Delta-8 \Delta^{2}\) and \(X_{5}=[4J^{2}+(\kappa+\gamma+4i\Delta)^{2}](\gamma+2i\Delta)\). The optimal parameter for single photon blockade in Cavity A can be obtained by set \(c_{200}=0\) under the condition \(c_{100}\
eq 0\). Based on Equation (33), one can obtain the optimal parameter pairs of the coupling strength \(g\) and the detuning \(\Delta\) with fixed hopping strength \(J\) and atom decay rate \(\gamma\).