Short-time counting statistics of charge transfer in Coulomb-blockade systems

We study full counting statistics of electron tunneling in Coulomb-blockade systems in the limit of short measuring-time intervals. This limit is particularly suited to identify correlations among tunneling events, but only when analyzing the charge-transfer statistics in terms of factorial cumulants $C_{\mathrm{F},m}\left(t\right)$ rather than ordinary ones commonly used in literature. In the absence of correlations, the short-time behavior of the factorial cumulants is given by $C_{\mathrm{F},m}\left(t\right)\propto {(-1)}^{m-1}{t}^m$. A different sign and/or a different power law of the time dependence indicates correlations. We illustrate this for sequential and Andreev tunneling in a metallic single-electron box.

Figure 1

(a) Normal-state metallic island (blue) weakly tunnel coupled to a superconducting lead (red). Via the gate voltage $V_{\text{g}}$, the equilibrium charge $n_{\text{g}}$ of the island is tuned to zero. The current through a single-electron transistor (green) monitors the island charge $n$ as a function of time. (b) Sketch of the states and transition rates.

Figure 2

Factorial cumulants as a function of time (a) without and (b) with Andreev tunneling. The sign of ${(-1)}^{m-1}{C}_{1,m}$ is positive for continuous and negative for dashed lines.

Figure 3

Generalized factorial cumulants $C_{s,m}$ normalized by $C_{1,m}$ as a function of $s$ (a) without and (b) with Andreev tunneling at time (a) $t=0.7\text{ms}$ and (b) $t=0.1\text{ms}$. In (b), factorial cumulants of odd order $m$ display an $s$ dependence.