Waiting Time Distributions for the Transport through a Quantum-Dot Tunnel Coupled to One Normal and One Superconducting Lead

We have studied the waiting time distributions (WTDs) for subgap transport through a single-level quantum-dot tunnel coupled to one normal and one superconducting lead. The WTDs reveal the internal dynamics of the system, in particular, the coherent transfer of Cooper pairs between the dot and the superconductor. The WTDs exhibit oscillations that can be directly associated to the coherent oscillation between the empty and doubly occupied dot. The oscillation frequency is equal to the energy splitting between the Andreev bound states. These effects are more pronounced when the empty state and double-occupied state are in resonance.

Figure 1

(a) Schematic setup: quantum-dot (QD) tunnel coupled to one normal lead ($N$) with tunnel-coupling strength ${\Gamma }_N$ and one superconducting lead ($S$) with tunnel-coupling strength ${\Gamma }_S$. Superconducting lead and QD are considered as a combined hybrid system. The normal lead is model as an electron bath. (b) Time line with jumps processes, a jump ${\mathcal{J}}_o$ to the single-occupation sector always has to be followed by a jump ${\mathcal{J}}_e$ to the even-occupation sector. The general waiting time $w(\tau )$ is the distribution of all $\Delta t_m$, $m\in \mathbb{N}$. In the exemplary sequence of jumps shown, the waiting time matrix entries $w_{e,o}(t)$ and $w_{o,e}(t)$ are the distribution of all $\Delta t_{2m-1}$ and $\Delta t_{2m}$. The two curves in time intervals $\Delta t_1$ and $\Delta t_4$, are sketches of the WTDs.

Figure 2

The waiting time distributions $w_{e,o}(\tau )$, $w_{o,e}(\tau )$, and $w(\tau )$ for $\delta =0U$, $0.4U$, $1U$. For ease of visualization, $w_{e,o}(\tau )$, $w_{o,e}(\tau )$ have been multiplied by a factor $1/2$. For comparison, the dashed curves for $w_{o,e}(\tau )$ and $w(\tau )$ are computed without the inclusion of the off diagonal elements of the reduced density matrix between the Andreev bound states. Parameters: ${\Gamma }_N=0.01U$ and ${\Gamma }_S=0.2U$.

Figure 3

Time evolution of ${\overline{\rho }}_{DD}+{\overline{\rho }}_{00}$, ${\overline{\rho }}_{DD}$, and ${\overline{\rho }}_{00}$ obtained by the time evolution with the Liouvillian without jumps ${\mathcal{W}}_0$ when the system is initially prepared in the doubly occupied state for $\delta =0U$, $0.1U$, $1U$. Parameters: ${\Gamma }_N=0.01U$ and ${\Gamma }_S=0.2U$.

Figure 4

Conditional waiting times ${\widehat{w}}_{C,|0⟩}(z)$ (system initially empty $|0⟩$) and ${\widehat{w}}_{C,|D⟩}(z)$ (system initially double occupied $|D⟩$) for $\delta =0U$ and $\delta =0.4U$ at ${\Gamma }_N=20{\Gamma }_S$. The dashed curves are calculated without the inclusion of the coherent dynamics.