# Photon Counting Statistics of a Microwave Cavity Coupled with Double Quantum Dots
## 2 Model and Theoretical Formalism  
### Hamiltonian Model  
Figure 1 shows the system schematic of the DQDs with metal leads and the left quantum dot (QD) coupled to a one-mode microwave cavity, and each QD is modeled as a one-level system. An external driving microwave field has been applied between the two QDs; the vital advantage of this system is that one can adjust the gate voltage to tune both the energy-level splitting and the decay rate of the DQDs. The loss of the microwave cavity field is considered here by adding the interaction of the thermal bath and microwave cavity field. In our DQD system, the distance between the two QDs is so small that only one electron can occupy the DQDs at a time. Thus, the states of DQDs can be presented as follows: \(|0\rangle\) (vacuum state), \(|1\rangle\) (the left dot occupied by one electron), \(|2\rangle\) (the right dot occupied by one electron). The energy level difference of the two dots is \(\hbar\) when the electron jumps from the left dot to the right induced by an external ac microwave field of frequency \(\omega_{d}\) between the QDs. Adjusting the level difference \(\hbar\) by the bias voltage to satisfy the condition \(\omega_{m}=\varepsilon-\omega_{d}\) (\(\omega_{m}\) is the frequency of microwave cavity field), the DQDs would absorb a photon of energy \(\hbar(\varepsilon-\omega_{d})\) when an electron is excited from state \(|1\rangle\) to state \(|2\rangle\). As shown in Figure 1b, the left chemical potential of lead is higher than that of the lead on the right. The total Hamiltonian of the whole system can be written as (\(\hbar=1\)) [39]  
\[H=H_{\rm J}+H_{\rm m}+H_{\rm D}+H_{T}+H_{\rm I}. \tag{1}\]  
Figure 1: **(a)** The diagram shows the double quantum dots (DQDs) connected to symmetric leads: the left electrode is the electron source, and the right is the drain; between the two dots, a bias voltage is added. The left quantum dot is coupled with a microwave cavity by capacitive coupling. The level difference of the two dots is \(\hbar\), and the electron jumping from the left dot to the right is driven by an external alternating current (AC) microwave field of frequency \(\omega_{d}\) between the two dots. **(b)** An electron tunnels from the source to the left quantum dot and is driven to the right dot by the microwave field, finally tunneling to the drain (\(\mu_{\rm L}>\mu_{R}\)). The first term, \(H_{I}\), describes the left and the right leads with a bias potential,  
\[H_{I}=\sum_{\alpha k}E_{\alpha k}c_{\alpha k}^{\dagger}c_{\alpha k}, \tag{2}\]  
where the operator \(c_{\alpha k}^{\dagger}\) (\(c_{\alpha k}\)) creates (destroys) an electrode electron with momentum \(k\) in lead \(\alpha=l\) or \(r\) (\(l\) = left, \(r\) = right), and \(E_{\alpha k}\) describes the dispersion energy. The second term, \(H_{m}\), describes the energy of the microwave cavity field, which can be written as  
\[H_{m}=\omega_{m}a^{\dagger}a, \tag{3}\]  
where \(a^{\dagger}(a)\) is the microwave photon creation (annihilation) operator with frequency \(\omega_{m}\). The third term, \(H_{D}\), describes the DQDs driven by the microwave photon field; i.e.,  
\[H_{D}=\frac{\varepsilon}{2}\sigma_{z}+\Omega\sigma_{x}+\Omega_{0}\cos(\omega_ {d}t)\sigma_{x}, \tag{4}\]  
where the operator \(\sigma_{z}=d_{2}^{\dagger}d_{2}-d_{1}^{\dagger}d_{1}\) and \(\sigma_{x}=d_{2}^{\dagger}d_{1}+d_{1}^{\dagger}d_{2}\) mean the Pauli matrices of the left or the right dot. The operator \(d_{1}(d_{2})\) stands for destroying an electron in the left (right) dot, \(\varepsilon\) is the energy-level difference of DQDs, \(\Omega\) is the tunneling strength between the two QDs [40], and \(\Omega_{0}\) is the Rabi frequency of the driving microwave field with driving frequency \(\omega_{d}\)[40].