## 3 Numerical Computation Conventional and Unconventional Photon Blockade
### Unconventional Photon Blockade with Atom-Cavity Detuning
When \(J\) is fixed at \(2.5\kappa_{a}\) and \(\gamma=0.6\kappa_{a}\), we can obtain the real solution of the optimal parameter pairs \(\{\{g\to-7.63415\kappa_{a},\ \Delta\to-6.89979\kappa_{a}\},\ \{g\to-7.63415\kappa_{a},\ \Delta\to 6.89979\kappa_{a}\},\ \{g\to-0.924508 \kappa_{a},\ \ \Delta\to-1.32075\kappa_{a}\},\ \{g\to-0.924508 \kappa_{a},\ \ \Delta\to 1.32075\kappa_{a}\},\ \{g\to 0.924508 \kappa_{a},\ \ \Delta\to 1.32075\kappa_{a}\},\ \{g\to 7.63415\kappa_{a},\ \ \Delta\to-6.89979 \kappa_{a}\},\ \{g\to 7.63415\kappa_{a},\ \Delta\to 6.89979 \kappa_{a}\}\}.\)  
We verify the solution by plotting the equal-time second-order correlation function \(g^{(2)}(0)\) as a function of the scaled detuning for different coupling strength \(g\) in Figure 8. As expected, the values of \(g^{(2)}(0)\) are much smaller when we use the \(\{g,\,\Delta\}\) pairs listed above than others, i.e., it shows two strong photon antibunching effects at \(\Delta\simeq 1.3\kappa_{a}\) with \(g=0.924508\kappa_{a}\) in Figure 8a and \(\Delta\simeq 6.9\kappa_{a}\) with \(g=7.63415\kappa_{a}\) in Figure 8b. We also see the symmetric property from optimal \(\{g,\,\Delta\}\) solutions, that is, when the coupling strength \(g\) is fixed at one of the optimal values, there are two symmetric optimal detunings for Cavity A, Cavity B, and the atom with respect to the driving laser, and vice versa. The underlying physical mechanism of the optimal solution for a single photon blockade lies in the destructive interference between different paths of two photon excitation in Cavity A. As illustrated in Figure 2a and Equation (29), the occupation of \(|2,0,g\rangle\) can be minimized by reducing the contribution from \(|1,0,g\rangle\) and \(|1,1,g\rangle\). When destructive interference between the direct excitation path \(|0,0,g\rangle\to|1,0,g\rangle\to|2,0,g\rangle\) and the indirect excitation paths \(|0,0,g\rangle\rightarrow|1,0,g\rangle\rightarrow|0,1,g\rangle\rightarrow|0,0,e\rangle \rightarrow(|1,0,e\rangle\leftrightarrow|0,1,e\rangle\leftrightarrow|0,2,g \rangle\leftrightarrow|1,1,g\rangle)\rightarrow|2,0,g\rangle\) occurs, the occupation of \(|2,0,0\rangle\) will be reduced; hence, the value of \(g^{(2)}(0)\) is decreased. Figure 9 shows several optimal \(\{g,\,\Delta\}\) pairs for different atom decay rate \(\gamma\). When \(\gamma=0.25\kappa_{a}\), one can get \(\{\{g\rightarrow\pm 0.782092\kappa_{a},\,\Delta\rightarrow\pm 1.19804\kappa_{a}\},\,\{g \rightarrow\pm 7.89333\kappa_{a},\,\Delta\rightarrow\pm 7.17599\kappa_{a}\}\}\), and \(\{\{g\rightarrow\pm 0.817538\kappa_{a},\,\Delta\rightarrow\pm 1.22938\kappa_{a}\},\,\{g \rightarrow\pm 7.8193\kappa_{a},\,\Delta\rightarrow\pm 7.09409\kappa_{a}\}\}\) with \(\gamma=0.35\kappa_{a}\), \(\{g\rightarrow\pm 0.85711\kappa_{a},\,\Delta\rightarrow\pm 1.26361\kappa_{a}\},\,\{g \rightarrow\pm 7.74537\kappa_{a},\,\Delta\rightarrow\pm 7.01522\kappa_{a}\}\}\) with \(\gamma=0.45\kappa_{a}\). As shown in Figure 9, with the increase in the atom lifetime, the optimal coupling strength \(g\) decreases in the UPB regime where \(g<\kappa_{a}\) and increases in the conventional photon blockade regime where \(g>\kappa_{a}\).