# Photon Counting Statistics of a Microwave Cavity Coupled with Double Quantum Dots
## 3 Discussion  
From Equation (17), one can find that the parameters \(A_{+}\) and \(A_{-}\) represent the influence of DQDs on the density matrix of the microwave cavity. \(A_{+}\) is the rate of the photon emitted from DQDs to the cavity and \(A_{-}\) is the rate of the photon absorbed by DQDs from the cavity. It can be found that, if \(\gamma+W<0\), the average phonon occupancy of the cavity can be increased to infinity and has no stationary solution. In contrast, for \(\gamma+\mathrm{W}>0\), one can reach a steady state. The average photon number of the cavity in a steady state is \(\overline{n}_{c}=(\gamma\overline{n}+A_{+})/(\gamma+W)\). Firstly, we investigate the variation range of the parameters \(A_{+}\) and \(A_{-}\), and the influence on the average photon number \(\overline{n}_{c}\). The parameters in this paper are assumed with reference to the typical experimental parameters [43; 44]. In Figure 2, the parameters \(W=A_{-}-A_{+}\), \(A_{+}\) and \(\overline{n}_{c}\) are plotted as a function of the driving detuning \(\Delta=\omega_{d}-\varepsilon\) for different driving Rabi frequencies of the external microwave field. This shows that the sign of the \(W\) exhibits a dependence on the detuning. \(W\) is at maximum when \(\Delta<0\) and at minimum when \(\Delta>0\), while \(A_{+}\) is at a maximum when \(\Delta>0\). One can also find that the average photon number of the microwave cavity in a steady state is at minimum when \(W\) is at maximum and it increases to infinity when \(W<0\). Additionally, \(\overline{n}_{c}\) could be less than the photon number \(\overline{n}\) of the thermal reservoir when \(W>0\), while it could be more than \(\overline{n}\) when \(W<0\). Figure 2 also shows that the parameters \(W\), \(A_{+}\) and \(\overline{n}_{c}\) are dependent on the driving Rabi frequency. Figure 3 demonstrates the WTDs for different parameters. The black line exhibits the WTD of the cavity without the DQD system. It can be found that all the WTDs start off at a finite value, \(W(\tau)\approx 2\overline{\Gamma}e^{-[6\frac{\overline{n}_{c}(1+\overline{n})}{ \gamma+W}+1](\gamma+W)\tau/2}\) (\(\overline{\Gamma}\tau\ll 1\)), and then decay exponentially to zero over long times, \(W(\tau)\approx\frac{4\overline{\Gamma}\overline{\gamma}(\gamma+W)}{\left(\gamma +W+\overline{\gamma}+2\overline{\Gamma}\right)^{2}}e^{-\left(\sqrt{1+4\overline {n}_{c}\gamma(1+\overline{n})}/(\gamma+W)-1\right)\ (\gamma+W)\tau/2}\) (\(\overline{\Gamma}\tau\gg 1\)). Comparing the green line to the black line, one can find that, when detuning \(\Delta=-4\Gamma\), the WTD of the cavity coupled with DQDs is very close to that of a cavity without DQDs because the average photon number of the cavity \(\overline{n}_{c}=0.85\) is very close to that of the thermal reservoir. The cyan line and red line show that the decay rate of \(\Delta=5.5\Gamma\) is very different from that of the DQDs replaced by a thermal bath, with \(\overline{n}_{h}=10\) for the long time limit, although these two decay rates are very close to each other for a short time range. Figure 4 shows the first-order, second-order, and third-order cumulants of waiting time as a function of detuning with different driving Rabi frequencies and microwave cavity frequencies. From Figures 2 and 4, it can be found that the cumulants of waiting time vary abruptly in the vicinity of the maximum of \(W\), and the cumulants vary much more steeply if \(W\) has a larger maximum as detuning varies. Overall, the waiting time statistical properties can be tuned by the parameters of the DQD system. Figure 5 demonstrates the large-deviation statistics of the photon emission current from the cavity to the thermal reservoir as a function of the photon emission current \(J\). From Figure 5, we can find that the distributions become strongly non-Poissonian and large emission currents are more likely as the average photon number of the cavity \(\overline{n}_{c}\) increases when adjusting the parameters of the DQD system.