# Waiting Time Distributions of Transport through a Two-Channel Quantum System
## 2 Model and Theoretical Formalizations
### Non-Interacting Dots  
For the non-interacting dots, the states of the central QD system are presented as follows: \(|0\rangle\) (empty dot state), \(|1\rangle\) (E\({}_{1}\) state occupied with one electron), \(|2\rangle\) (E\({}_{2}\) state occupied with one electron), and \(|3\rangle\) (E\({}_{1}\) and E\({}_{2}\) states are occupied with two electrons). Thus, the density matrix of Equation (4) can be expressed as  
\[\frac{d}{dt}\rho=\left(\mathcal{L}_{0}+\mathcal{J}_{F}e^{ix}\right)\rho \tag{6}\]where the Liouvillian basis of \(\rho\) is \(\left(\rho_{00},\rho_{11},\rho_{22},\rho_{33},\rho_{12},\rho_{21}\right)^{T}\). Some off-diagonal elements are omitted because they are not involved in the calculations of the WTD and the Pearson correlation coefficient.  
\[\mathcal{L}_{0}=\left(\begin{array}{cccccc}-2\Gamma_{l}&0&0&0&0\\ \Gamma_{l}&-\Gamma_{r}-\Gamma_{l}&0&0&-\eta^{*}\Gamma_{r}/2&-\eta\Gamma_{r}/2 \\ \Gamma_{l}&0&-\Gamma_{r}-\Gamma_{l}&0&-\eta^{*}\Gamma_{r}/2&-\eta\Gamma_{r}/2 \\ 0&\Gamma_{l}&\Gamma_{l}&-2\Gamma_{r}&\Gamma_{l}&\Gamma_{l}\\ \Gamma_{l}&-(\Gamma_{l}+\eta\Gamma_{r})/2&-(\Gamma_{l}+\eta\Gamma_{r})/2&0&i \delta\varepsilon-\Gamma_{r}-\Gamma_{l}&0\\ \Gamma_{l}&-(\Gamma_{l}+\eta^{*}\Gamma_{r})/2&-(\Gamma_{l}+\eta^{*}\Gamma_{r} )/2&0&0&-i\delta\varepsilon-\Gamma_{r}-\Gamma_{l}\end{array}\right)\]  
and  
\[\mathcal{J}_{F}=\left(\begin{array}{cccccc}0&\Gamma_{r}&\Gamma_{r}&0&\eta^{* }\Gamma_{r}&\eta^{*}\Gamma_{r}\\ 0&0&0&\Gamma_{r}&0&0\\ 0&0&0&\Gamma_{r}&0&0\\ 0&0&0&0&0&0\\ 0&0&0&\eta\Gamma_{r}&0&0\\ 0&0&0&\eta^{*}\Gamma_{r}&0&0\end{array}\right)\]  
The standard expression for the WTD of the two-channel QD system in the time domain is [19; 25; 26; 34]:  
\[\mathrm{W}(\tau)=\frac{\left(\mathbf{I},\mathcal{J}_{F}e^{\mathcal{L}_{0}\tau }\mathcal{J}_{F}\overline{\mathbf{P}}\right)}{\left(\mathbf{I},\mathcal{J}_{F }\overline{\mathbf{P}}\right)} \tag{7}\]  
where \(\mathbf{I}=[1,1,1,0,0]\) for a strong Coulomb blockade case and \(\mathbf{I}=[1,1,1,1,0,0]\) for a non-Coulomb repulsion case. \(\overline{\mathbf{P}}\) is the steady state of the system in \(t=0\). The waiting time \(\tau\) is given by  
\[\langle\tau^{n}\rangle=n!(-1)^{(n+1)}\frac{\left(\mathbf{I},\mathcal{J}_{F} \left(\mathcal{L}_{0}\right)^{-(n+1)}\mathcal{J}_{F}\overline{\mathbf{P}} \right)}{\left(\mathbf{I},\mathcal{J}_{F}\overline{\mathbf{P}}\right)} \tag{8}\]  
In order to analyze the microscopic fluctuations, we can define two consecutive waiting times, \(W_{2}(\tau_{1},\tau_{2})\), which is the joint probability distribution that the first electron waits for time \(\tau_{1}\), and the next electron waits for time \(\tau_{2}\), before tunneling to the drain [19; 25; 26].  
\[\mathrm{W}_{2}(\tau_{1},\tau_{2})=\frac{\left(\mathbf{I},\mathcal{J}_{F}e^{ \mathcal{L}_{0}\tau_{2}}\mathcal{J}_{F}e^{\mathcal{L}_{0}\tau_{1}}\mathcal{J}_{F }\overline{\mathbf{P}}\right)}{\left(\mathbf{I},\mathcal{J}_{F}\overline{ \mathbf{P}}\right)} \tag{9}\]  
If \(\mathrm{W}_{2}(\tau_{1},\tau_{2})=\mathrm{W}(\tau_{1})\mathrm{W}(\tau_{2})\), the system is a renewal system, otherwise, it is a nonrenewal system. To further characterize the short-time correlated behavior of the system, the Pearson correlation coefficient is defined as  
\[p=\frac{\langle\tau_{1}\tau_{2}\rangle-\langle\tau^{2}\rangle}{\langle\tau^{ 2}\rangle-\langle\tau^{2}\rangle} \tag{10}\]