# Waiting Time Distributions of Transport through a Two-Channel Quantum System
## 3 Results and Discussion
### Non-Interacting Dots  
Figure 4 shows \(1/\langle\tau\rangle\) and the randomness parameter, \(\langle\langle\tau^{2}\rangle\rangle/\langle\tau^{2}\rangle\), as a function of the asymmetry of dot-state couplings to the left and right leads, respectively. This figure demonstrates that the phase difference between the two channel electrons jumping to the right lead has an obvious effect on the randomness parameter, while it has little effect on the average current; notably, the average currents are the same for the constructive interference (\(\eta=1\)) case and the destructive interference (\(\eta=-1\)) case. Comparing Figure 4b with the corresponding Fano factor in the other work [27], we observe that the randomness parameter is different from the Fano factor of the system. Thus, the system is a nonrenewal system for non-interacting dots.  
Figure 5 exhibits the WTDs of the system for three different phase differences between the two channels as a function of time \(\tau\). Although the parameter values of Figure 5a are the same as those of Figure 3a, the oscillatory behaviors of the WTDs in Figure 3a disappear. This occurs because the presence of one excess electron inside the central dot system leads to no possibility for more than one electron at the same time to occupy the two channel states and destroys the coherent superposition of the two channel states. Figure 5b shows that, if we decrease the value of the electron tunneling rate \(\Gamma_{l}\) between the central QD and the left lead, the oscillatory behaviors of the WTDs appear because of the decrease in the probability that a double electron will occupy the central QD system. Figure 5 also shows that the probable density of the WTDs at time \(\tau=0\) is nonzero, which suggests that it is now a multiple reset system because two electrons can be detected in the drain lead directly following one  
another for non-interacting dots. Comparing the blue line with the cyan line in Figure 5a, we observe that the WTD recovers some features of the oscillatory behavior as the level spacing \(\delta\varepsilon\) increases.  
Figure 6 demonstrates the Pearson correlation coefficient \(p\) of the system for three different phase differences between the two channels as a function of the asymmetry of the dot-state couplings to the left and right leads with the same parameter values as those in Figure 5. Figure 6 shows that the electron waiting times are negatively correlated in the current case, in which a short waiting time is more likely to be followed by a long waiting time, and vice versa, because of the anti-bunching properties of fermion. Since we obtain the master equation under the Born-Markov approximation, the nonrenewal behavior of the system in the current case does not result from non-Markovian behavior.  
Comparing Figure 6a,b, we also observe that the phase differences between the two channels obviously enhance the Pearson correlation coefficient. Furthermore, the Pearson correlation coefficient can be decreased by decreasing the value of the electron tunneling rate \(\Gamma_{l}\), which can lead to a decrease in the probability of a double electron occupying the central QD system. This implies that the Pearson correlation coefficient is relative to the multiple reset characteristics. If the two channels of the QD system are doubly occupied, both electrons are likely to subsequently tunnel out, which leads to a short waiting time; then, the system will fill and empty again, which results in a long waiting time.