# Waiting Time Distributions of Transport through a Two-Channel Quantum System
## 2 Model and Theoretical Formalizations  
The Hamiltonian of a two-channel Coulomb blockade system is governed by \((\hbar=\mathrm{e}=1)\)[27]:  
\[\mathrm{H}=\mathrm{H}_{d}+\mathrm{H}_{leads}+\mathrm{H}_{T}\]  
\[\mathrm{H}_{d}=E_{1}d_{1}^{\dagger}d_{1}+E_{2}d_{2}^{\dagger}d_{2}+Ud_{1}^{ \dagger}d_{1}d_{2}^{\dagger}d_{2} \tag{1a}\] \[\mathrm{H}_{leads}=\sum_{k}\bigl{(}\varepsilon_{k}c_{lk}^{ \dagger}c_{lk}+\varepsilon_{rk}c_{nk}^{\dagger}c_{rk}\bigr{)}\] (1b) \[H_{T}=\sum_{kj(=1,2)}\Omega_{jl}d_{j}^{\dagger}c_{lk}+\Omega_{jp}d_{j}^{ \dagger}c_{rk}+H.c. \tag{1c}\]where \(d_{1}(d_{2})\) stands for destroying an electron in the central QD state 1(2), \(c_{ak}^{\dagger}(c_{ak})\) creates (destroys) a lead electron with momentum \(k\) in lead \(\alpha=l\) or \(r\) (\(l\) = left, \(r\) = right); \(E_{1}\) and \(E_{2}\) are the respective energy levels of the two channels; and \(\Omega_{ja}\) describes the coupling constant between central QD state \(j\) and lead \(\alpha\). The third term on the left of Equation (1a) describes the Coulomb interactions between the electrons in the different dot states. In the above Hamiltonian, we omitted the spin indices. Considering the Born-Markov approximation, and tracing out the lead reservoirs, one can obtain a reduced central QD system master equation as [28]  
\[\frac{d}{dt}\rho=-i[H_{d},\rho]+\mathcal{L}_{l}\rho+\mathcal{L}_{r}\rho \tag{2}\]  
\[\mathcal{L}_{l}\rho=\Gamma_{l}\Big{(}\sum\
olimits_{j=1,2}\mathcal{D}\big{[} d_{j}^{\dagger}\big{]}\rho+\big{[}d_{2}^{\dagger}\rho d_{1}-\big{\{}d_{1}d_{2}^{ \dagger},\rho\big{\}}/2\big{]}+\big{[}d_{1}^{\dagger}\rho d_{2}-\big{\{}d_{2} d_{1}^{\dagger},\rho\big{\}}/2\big{]}\Big{)},\]  
\[\mathcal{L}_{r}\rho=\Gamma_{r}\Big{(}\sum\
olimits_{j=1,2}\mathcal{D}\big{[} d_{j}\big{]}\rho+\eta^{\dagger}\Big{[}d_{1}\rho d_{2}^{\dagger}-\big{\{}d_{2}^{ \dagger}d_{1},\rho\big{\}}/2\big{]}+\eta\big{[}d_{2}\rho d_{1}^{\dagger}-\big{\{} d_{1}^{\dagger}d_{2},\rho\big{\}}/2\big{]}\Big{)},\]  
where \(\mathcal{D}[A]\rho=A\rho A^{\dagger}-\frac{1}{2}\big{[}A^{\dagger}A\rho+\rho A ^{\dagger}A\big{]}\), \(\{A,\rho\}=A\rho+\rho A\), and \(A\) is an arbitrary operator. Here, it is assumed that \(E_{1,2}\) are inside the window of the bias voltage and the leads are at zero temperature. \(\Gamma_{\alpha}^{j}=\sum\
olimits_{k}2\pi\rho_{ak}\big{[}\Omega_{\alpha l} \big{]}^{2}\), \(\Gamma_{\alpha}^{j}\) describes the \(j\)th channel electron tunneling rate between the central QD and lead \(\alpha\), while \(\rho_{ak}\) describes the density of the electron state with momentum \(k\) at lead \(\alpha\). Since we focus on the quantum interference effect here, the two transmission channels are set to be identical. Thus, we assume that the coupling strengths of the two QD states with the left and right electrodes are equal and energy independent, i.e., \(|\Omega_{\alpha 1}|^{2}=|\Omega_{\alpha 2}|^{2}\) and \(\Gamma_{\alpha}^{1}=\Gamma_{\alpha}^{2}=\Gamma_{\alpha}\). In this paper, we adopt a phenomenological relative phase factor by choosing \(\Omega_{l1}=\Omega_{l2},\ \eta=\Omega_{r1}/\Omega_{r2}\), where \(\eta=e^{i\theta}\) is a relative phase parameter [27]. A phase difference between the two channels can be induced by the phase accumulation of spatial motion from the electrode to the dot, particularly in the presence of magnetic vector potential (i.e., the Aharanov-Bohm effect), as well as the phase changes associated with transmission through QDs [29; 30; 31]. Equation (2) does not include the effects of Coulomb interactions and is valid for a noninteracting case. This allows for double-electron occupation of the dot. If the dot is small, and the Coulomb repulsion is the largest energy scale, only one electron can inhabit the dot at a time, and sequential tunneling occurs.