Electron Waiting Times of a Cooper Pair Splitter

Electron waiting times are an important concept in the analysis of quantum transport in nanoscale conductors. Here we show that the statistics of electron waiting times can be used to characterize Cooper pair splitters that create spatially separated spin-entangled electrons. A short waiting time between electrons tunneling into different leads is associated with the fast emission of a split Cooper pair, while long waiting times are governed by the slow injection of Cooper pairs from a superconductor. Experimentally, the waiting time distributions can be measured using real-time single-electron detectors in the regime of slow tunneling, where conventional current measurements are demanding. Our work is important for understanding the fundamental transport processes in Cooper pair splitters and the predictions may be verified using current technology.

Figure 1

Electron waiting times of a Cooper pair splitter. (a) Two QDs are coupled to a superconducting source of Cooper pairs and two normal-metal drains. A tunneling event (star) starts the clock, which symbolizes the measurement scheme based on single-electron detectors [23, 52, 53, 54]. A subsequent tunneling event stops it. WTDs for tunneling into the same or different leads are shown in (b) and (c). The WTDs ${\mathcal{W}}_{ji}(\tau )$ $[{\mathcal{W}}_{ji}^{\mathrm{ex}}(\tau )]$ are evaluated using Eq. (4) [(5)]. Parameters are $\xi ≔{\gamma }_L={\gamma }_R=10\gamma$, ${\gamma }_{\mathrm{CPS}}={\gamma }_{\mathrm{EC}}=\gamma$, and ${\epsilon }_L={\epsilon }_R=0$. Dashed lines are exponentials with decay rates $\xi$ (gray) and $2{\gamma }_{\mathrm{CPS}}^2/\xi$ (black). Corresponding to the recent experiments, the rate $\gamma$ would be on the order of kilohertz and the waiting times would be in the millisecond range [52, 53, 54].

Figure 2

Spin-resolved WTDs. (a) Spin-resolved WTDs for tunneling into the same lead. (b) Spin-resolved WTDs for tunneling into different leads. In (a) and (b), the parameters are ${\gamma }_L={\gamma }_R\equiv 10\gamma$, ${\gamma }_{\mathrm{CPS}}={\gamma }_{\mathrm{EC}}=\gamma$, and ${\epsilon }_L=-{\epsilon }_R=10\gamma$. (c) Spin-resolved WTDs for tunneling into the same lead with same parameters except that ${\epsilon }_L={\epsilon }_R=0$. (d) The branching ratio in Eq. (7) corresponding to the WTDs in (b).

Figure 3

Coherent oscillations. (a) Spin-resolved WTDs for tunneling into the same lead. (b) Spin-resolved WTDs for tunneling into different leads. In both (a) and (b), the parameters are ${\gamma }_L={\gamma }_R\equiv 0.1\gamma$, ${\gamma }_{\mathrm{CPS}}={\gamma }_{\mathrm{EC}}=\gamma$, and ${\epsilon }_L=-{\epsilon }_R=10\gamma$.

Figure 4

Joint WTDs and correlation functions. In (a) and (b), the parameters are ${\gamma }_L={\gamma }_R=10\gamma$, ${\gamma }_{\mathrm{CPS}}={\gamma }_{\mathrm{EC}}\equiv \gamma$, and ${\epsilon }_L={\epsilon }_R=0$. In (c) and (d), the parameters are ${\gamma }_L={\gamma }_R=0.1\gamma$, ${\gamma }_{\mathrm{CPS}}={\gamma }_{\mathrm{EC}}=\gamma$, and ${\epsilon }_L={\epsilon }_R=0$.