# Photon Counting Statistics of a Microwave Cavity Coupled with Double Quantum Dots
## 2 Model and Theoretical Formalism
### Master Equation  
Considering the Born-Markov approximation and tracing out the electrode lead reservoirs, one can obtain the system equation of the DQDs and the microwave cavity as follows [39]:  
\[\frac{d}{dt}\rho_{S}=-i\big{[}H_{sys},\rho_{S}\big{]}+\mathcal{L}_{T}\rho+ \mathcal{L}_{D}\rho, \tag{10}\]  
where the middle-term \(\mathcal{L}_{T}\rho\) stands for the tunneling effect between DQDs and leads, and the last term \(\mathcal{L}_{D}\rho\) expresses the dissipation term of the microwave cavity induced by the thermal reservoir; i.e.,  
\[\mathcal{L}_{T}\rho=\Gamma_{r}\mathcal{D}[d_{2}]\rho_{S}+\Gamma_{I} \big{(}1-\eta^{2}\big{)}\mathcal{D}\Big{[}d_{1}^{\dagger}\big{]}\rho_{S}+ \Gamma_{I}\eta^{2}\Big{[}a^{\dagger}a\Big{[}d_{1},d_{1}^{\dagger}\rho_{S} \big{]}+\Big{[}\rho_{S}d_{1},d_{1}^{\dagger}\Big{]}a^{\dagger}a\Big{]} \tag{11}\] \[+\Gamma_{I}\eta^{2}\Big{[}\mathcal{D}\Big{[}a^{\dagger}d_{1}^{ \dagger}\Big{]}\rho_{S}+\mathcal{D}\Big{[}ad_{1}^{\dagger}\Big{]}\rho_{S} \big{]},\]  
\[\mathcal{L}_{D}\rho=\frac{\
u}{2}[1+\overline{n}]\Big{[}2a\rho_{S}a^{\dagger} -a^{\dagger}a\rho_{S}-\rho_{S}a^{\dagger}a\Big{]} \tag{12}\] \[+\frac{\
u}{2}\overline{n}\Big{[}2a^{\dagger}\rho_{S}a-aa^{ \dagger}\rho_{S}-\rho_{S}aa^{\dagger}\Big{]}\]  
Here, we take operator \(X\) referred to above to expand up to the second order in \(\eta\) and assume that the energy levels with one microwave cavity photon mediating tunneling every time are in the bias window. Additionally, \(\Gamma_{\alpha}=\sum_{k}2\pi\rho_{\alpha k}\Omega_{\alpha k}^{2}\), \(\Gamma_{\alpha}\). describes the tunneling rate of the electron from the lead \(\alpha\), while \(\rho_{\alpha k}\) describes the density of the electron state with momentum \(k\) at lead \(\alpha\). The latter \(\gamma\) is the dissipation of the microwave cavity field, where \(\overline{n}\) is the boson number in the thermal reservoir. Super-operator \(D\) has the following form: \(D[A]\rho=A\rho A^{\dagger}-\frac{1}{2}\Big{[}A^{\dagger}A\rho+\rho A^{\dagger} A\Big{]}\), \(A\) is the arbitrary operator. Assuming the dissipation of the cavity field is smaller than the decay of the DQDs, the freedom of the DQDs could be eliminated as it is considered as an environment. Since we are interested in the behavior of the system in the limit \(\mathrm{t}\rightarrow\infty\), applying the projecting operator method to calculate the density of the cavity, we can obtain the final master equation of the cavity field after taking the same approximation as used in [39]:  
\[ \begin{array}{ll}\frac{d}{dt}\mu=&i\delta_{m}\Big{[}a^{\dagger}a,\mu\Big{]}+ \frac{1}{2}[\gamma[1+\overline{n}]+A_{-}(\omega_{m})]\Big{[}2a\mu a^{\dagger} -a^{\dagger}a\mu-\mu a^{\dagger}a\Big{]}\\ &\hskip 14.226378 pt+\frac{1}{2}[\gamma\overline{n}+A_{+}(\omega_{m})]\Big{[}2 a^{\dagger}\mu a-aa^{\dagger}\mu-\mu aa^{\dagger}\Big{]}\end{array}  \tag{13}\]  
where  
\[ \begin{array}{ll}\delta_{m}=\eta^{2}\frac{\Omega_{\alpha}}{2}(\alpha_{x})- \mathrm{Im}[S(\omega_{m})+[S(-\omega_{m})]]\\ A_{-}(\omega_{m})=2\mathrm{Re}[S(\omega_{m})]+2D\\ A_{+}(\omega_{m})=A_{-}(-\omega_{m})\\ D=\frac{1}{2}\eta^{2}\Big{\langle}\rho_{d}^{0}\Big{\rangle}\frac{\sum}{\sigma }\Gamma_{\alpha}\end{array}  \tag{14}\]  
The operator \(\Big{\langle}\rho_{d}^{0}\Big{\rangle}\) describes the stationary probability of the DQDs in the empty state, \(\mu\) stands for density matrix of microwave cavity field, \(\delta_{m}\) is the driving-induced frequency shift, and \(A_{\pm}\) shows the rates induced by the interaction between DQDs and cavity; moreover,  
\[\mathrm{S}(\omega_{m})=\frac{\eta^{2}\Omega_{0}^{2}}{4}\!\int_{0}^{\infty}dte^ {i\omega_{m}t}\Big{\langle}\sigma_{y}(t)\sigma_{y}(0)\Big{\rangle}, \tag{15}\]  
where \(\Big{\langle}\sigma_{y}(t)\sigma_{y}(0)\Big{\rangle}\) can be obtained via the quantum regression theorem. \(\langle\cdots\rangle\) represents the quantum average of the DQDs degrees.