# Waiting Time Distributions of Transport through a Two-Channel Quantum System
## 3 Results and Discussion
### Strong Coulomb Blockade Regime  
In the strong Coulomb blockade regime, only one electron is transported through the QD system every time, and the dot is empty after an electron tunnels out. Thus, the system is a single reset system. Figure 1 shows \(1/\langle\tau\rangle\) and the randomness parameter [26], \(\langle\langle\tau^{2}\rangle\rangle/\langle\tau^{2}\rangle\), as a function of the asymmetry of the dot-state couplings to the left and right leads, respectively. Comparing Figure 1b with the corresponding Fano factor in the work of [PERSON] and co-workers [27], we find that the randomness parameter is the same as the Fano factor of the system. Thus, the system is a renewal system when there exists a one-to-one relationship between the WTD and FCS [25; 26; 34]. The calculation of \(\langle\tau_{1}\tau_{2}\rangle\) also verifies that the system is a renewal system, and the system's average current is equal to \(1/\langle\tau\rangle\), which is illustrated in Figure 1a.  
The results show that the constructive interference (\(\eta=1\)) between the two channels leads to a super-Poisson noise, and the destructive interference (\(\eta=-1\)) leads to a sub-Poisson noise, while the \(\pi/2\) phase difference results in a sub-Poisson noise in a certain parameter range and a super-Poisson noise in another parameter range. This demonstrates that quantum interference can move the system back and forth between the sub-Poisson noise regime and the super-Poisson noise regime. Furthermore, the destructive interference (\(\eta=-1\)) can lead to a small average current and a long average waiting time between two electron jumping events to the right lead.  
Figure 2a exhibits the WTDs of the system for three different phase differences between two channels and the corresponding \(\left|\rho_{12}\right|\) was shown as a function of time \(\tau\) in Figure 2b. It shows that, for the constructive interference (\(\eta=1\)) case, the most probable waiting time and full width of WTDs are smaller than that of the other two cases, which leads to small average waiting time and large average current.  
Comparing Figure 2a,b, one can find that the smaller the decay rate of \(\left|\rho_{12}\right|\), the longer the average waiting time. The term \(\left|\rho_{12}\right|\) represents the interference between the two channel states and is known as the coherence [35]. In order to enhance the interference between the two channel states, we decrease the value of \(\Gamma_{r}\), which can be understood from Equation (5).  
Figure 3a exhibits the WTDs of the system for three different phase differences between the two channels; the corresponding \(\left|\rho_{12}\right|\) values are shown as a function of time \(\tau\) in Figure 3b. The parameter values of Figure 3 are the same as those of Figure 2, except \(\alpha=0.2\). It is very interesting that the WTDs manifest oscillatory behavior with an increase in waiting time, which is an analog of the optical double-slit interference fringe. The oscillatory behavior of WTDs could not be found in the FCS results in the work of [PERSON] and colleagues [27], which indicates that the WTDs can characterize the short-time physics and correlations in mesoscopic systems that are not accessible for FCS. These results also demonstrate that the phase differences between the two channels (the phase of \(\eta\)) are clearly reflected in the oscillation phase of the WTDs. Comparing the blue lines with the cyan lines, we observe that the oscillations of the WTDs decrease as the level spacing \(\delta\varepsilon\) decreases. The effect of the level spacing \(\delta\varepsilon\) is an analog of the slit spacing in optical double-slit interference.