## 3 Numerical Computation Conventional and Unconventional Photon Blockade
### Quantum Master Equation
Thus, we can analyze the peaks and dips of the second correlation curve based on the profiles of the \(\rho_{33}\) and \(\rho_{88}\) curves. As shown in Figure 4, \(\rho_{33}\gg\rho_{88}\) with \(\rho_{33}=\langle 1,0,g|\rho_{ss}|1,0,g\rangle\) and \(\rho_{88}=\langle 2,0,g|\rho_{ss}|2,0,g\rangle\). There are three peaks in the \(\rho_{33}\) curve that are caused by the resonance transitions of the system. As shown in Figure 2b, there are three resonant transition channels for a single photon, i.e., \(\ket{\psi_{0}}\rightarrow\ket{\psi_{1}^{-}}\), \(\ket{\psi_{0}}\rightarrow\ket{\psi_{1}^{0}}\), and \(\ket{\psi_{0}}\rightarrow\ket{\psi_{1}^{+}}\). The positions of these three peaks can be calculated from the eigenvalues of these states. The positions of the left peak, the middle one, and the right one are obtained based on the resonant transition conditions in the rotating frame \(E_{1}^{+}-E_{0}=0\), \(E_{1}^{0}-E_{0}=0\), and \(E_{1}^{-}-E_{0}=0\). According to Equations (5)-(8) and based on parameters used in Figure 4, one can obtain the positions of these three peaks: \(-9.1782\), \(0\), and \(9.1782\). The five peaks in the \(\rho_{88}\) curve are caused by the resonant transition of two photons, whose position can be calculated in the same way as those in the \(\rho_{33}\) curve, and the values are \(-6.6446\), \(-4.2683\), \(0\), \(4.2683\), and \(6.6446\), respectively. The other two peaks in the \(\rho_{88}\) curve, whose positions are the same as the left one and right one in the \(\rho_{33}\) curve, are induced by a single photon resonant transition. The dips in the \(\rho_{33}\) and \(\rho_{88}\) curves are induced by the destructive quantum interference effect between different transition paths, as shown in Figure 2b. The locations of the two peaks in the correlation function \(g^{(2)}(0)\) correspond to the two dips in the \(\rho_{33}\) curve, where the excitation of the single photon resonant transition is low. There is a single photon resonant transition and a two-photon resonant transition at \(\Delta/\kappa_{a}=0\) in the \(\rho_{33}\) and \(\rho_{88}\) curves. The dip at \(\Delta/\kappa_{a}=0\) in the correlation function \(g^{(2)}(0)\) curve means the single photon resonant transition is far stronger than the two-photon resonant transition at that position. The locations of the other dips in the \(g^{(2)}(0)\) curve correspond to those in the \(\rho_{88}\) curve. To investigate the influence of the coupling strength \(g\) and the hopping interaction strength \(J\) on the correlation function, in Figure 5, we plot the equal-time second correlation function as the two scaled parameters, \(g/\kappa_{a}\) and \(J/\kappa_{a}\), with the fixed detuning \(\Delta=0\), which is a detuning of the single photon resonant transition. From Figure 5, we can see that, when the coupling strength between Cavity B and the atom is larger than \(\kappa_{a}\), i.e., \(g/\kappa_{a}>1\), one can obtain a minimum value of \(g^{(2)}(0)\) at the ratio between the hopping interaction strength and the coupling strength \(J/g\approx 1.25\). At this ratio, the two photon transition is suppressed and the single photon transition is far stronger. As illustrated in Figure 5, two-photon antibunching can also be obtained when the coupling strength between Cavity B and the atom is weaker than the decay rate of Cavity A. That is, the single photon blockade effect can be obtained in our system under weak nonlinearity, which is induced by the atom in Cavity B in our system with \(g<\kappa_{a}\). Around the other region at \(g\approx 0.5\kappa_{a}\), one can obtain a low value of \(g^{(2)}(0)\) as described in Figure 5. However, if we continue to decrease the coupling strength below \(0.3\kappa_{a}\), the photon blockade effect will disappear, and a two-photon bunching effect will become obvious. Figure 4: Plots of the density matrix element \(\rho_{33}\), \(\rho_{88}\) and the second-order correlation function \(g^{(2)}(0)\) as functions of the scaled detuning of Cavity A for \(\varepsilon=0.1\kappa_{a}\) and \(g=1.8\kappa_{a}\).