## 3 Numerical Computation Conventional and Unconventional Photon Blockade
### Quantum Master Equation  
We assume our system satisfies the Born approximation and is in a \"short-memory environment\" such that it also satisfies the Markov approximation. The time-evolution of the system density matrix is then governed by the Lindblad master equation [26; 27]:  
\[ \begin{array}{c}\stackrel{{\bullet}}{{\rho}}=i[\rho,H]+\frac{ \kappa_{b}}{2}(\overline{n}_{tha}+1)\Im[a]+\frac{\kappa_{b}}{2}\overline{n}_{ tha}\Im[a^{\dagger}]+\frac{\kappa_{b}}{2}(\overline{n}_{thb}+1)\Im[b]\\ \qquad\qquad+\frac{\kappa_{b}}{2}\overline{n}_{thb}\Im[b^{\dagger}]+\frac{ \gamma}{2}(\overline{n}_{th1}+1)\Im[\sigma]+\frac{\gamma}{2}\overline{n}_{th1 }\Im[\sigma^{\dagger}]\end{array}  \tag{14}\]  
The super operator \(\Im[o]=2\rho o^{\dagger}-\rho o^{\dagger}o-o^{\dagger}o\rho\) is the Lindblad term [28; 29] accounting for the losses to the environment. \(\overline{n}_{thx}=[\exp(\hbar\omega_{x}/K_{B}T_{x})-1]^{-1}\), \(x=\{a,b,1\}\) are the average thermal excitation numbers of the bath at temperature \(T_{x}\), with \(K_{B}\) as the Boltzmann constant. \(\kappa_{a}\) and \(\kappa_{b}\) are the  
Figure 2: (a) Energy levels within two excitation subspaces and the corresponding excitation paths; (b) the eigenenergy spectrum of the dressed states within two excitation subspaces. dissipation rates of Cavity A and Cavity B, respectively. The transition rate from \(|e\rangle\) to \(|g\rangle\) of the atom is described by the term proportional to \(\frac{\gamma}{2}(\overline{n}_{th1}+1)\), which contains a rate for spontaneous transitions and a rate for stimulated transitions induced by thermal photons. The last term in Equation (14) describes the transition rate from \(|g\rangle\) to \(|e\rangle\), which is obtained by absorbing thermal photons from the cavity field. At optical frequencies and laboratory temperatures, the thermal photon number \(\overline{n}_{thx}\) can be completely negligible [26]. The time evolution of the mean photon numbers in Cavity A can be obtained by solving the master Equation (14). The result is shown in Figure 3 in which Cavity A, Cavity B, and the atom have the same detuning with respect to the driving laser. The photon-number statistics in Cavity A can be evaluated by the result of the photon correlation function, which is defined in the steady-time limit as  
\[g^{(2)}(\tau)=\lim_{t->\infty}\frac{\left\langle a^{\dagger}(t)a^{\dagger}(t+ \tau)a(t+\tau)a(t)\right\rangle}{\left\langle a^{\dagger}(t)a(t)\right\rangle^ {2}}. \tag{15}\]  
We assume that there is zero delay between photons, i.e., \(\tau=0\), and Equation (15) can then be simplified to  
\[g^{(2)}(0)=\frac{\left\langle a^{\dagger}a^{\dagger}aa\right\rangle_{ss}}{ \left\langle a^{\dagger}a\right\rangle_{ss}^{2}} \tag{16}\]  
where'ss' means steady state. In the weak excitation limit, \(g^{(2)}(0)\) can be roughly estimated by the system density matrix element expanded in the bare state subspace. That is,  
\[\left\langle a^{\dagger}a^{\dagger}aa\right\rangle_{ss}=Tr(a^{\dagger}a^{ \dagger}aa\rho_{ss})\approx\sum_{k=0}^{8}\langle k|a^{\dagger}a^{\dagger}aa\sum _{m,n=0}^{8}\rho_{mn}|m\rangle\langle n|k\rangle=2\rho_{88} \tag{17}\]  
\[\left\langle a^{\dagger}a\right\rangle_{ss} =Tr(a^{\dagger}a\rho_{ss})\approx\sum_{k=0}^{8}\langle k|a^{ \dagger}a\sum_{m,n=0}^{8}\rho_{mn}|m\rangle\langle n|k\rangle\] \[=\rho_{33}+\rho_{55}+\rho_{66}+2\rho_{88}\approx\rho_{33} \tag{18}\]  
where \(k\) in \(|k\rangle\) is the index corresponding to the bare states listed in Section 2. The detuning with respect to the driving light is an important factor affecting the photon blockade effect. We plot the equal-time second-order correlation function \(g^{(2)}(0)\) as a function of the scaled detuning \(\Delta/\kappa_{a}\) in Figure 4, where \(\Delta=\Delta_{a}=\Delta_{b}=\Delta_{1}\). The red dashed curve in Figure 4 is calculated according to Equations (16)-(18), and the blue curve is obtained by solving Equation (14) numerically. These two curves match well in the parameter domain.