We introduce time-dependent, generalized factorial cumulants C m s (t) of the full counting statistics of electron transfer as a tool to detect interactions in nanostructures. The violation of the sign criterion (−1) m−1 C m s (t) ≥ 0 for any time t, order m, and parameter s proves the presence of interactions. For given system parameters, there is a minimal time span tmin and a minimal order m to observe the violation of the sign criterion. We demonstrate that generalized factorial cumulants are more sensitive to interactions than ordinary ones and can detect interactions even in regimes where ordinary factorial cumulants fail. We illustrate our findings with the example of a quantum dot tunnel coupled to electronic reservoirs either in or out of equilibrium.
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 CF,m(t) rather than ordinary ones commonly used in literature. In the absence of correlations, the short-time behavior of the factorial cumulants is given by CF,m(t) ∝ (−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.
Recent experiments (2015 Nature 521 196; 2017 Nat. Commun. 8 395) have presented evidence for electron pairing in a quantum dot beyond the superconducting regime. Here, we show that the impact of an attractive electron-electron interaction on the full counting statistics of electron transfer through a quantum dot is qualitatively different from the case of a repulsive interaction. In particular, the sign of higher-order (generalized) factorial cumulants reveals more pronounced correlations, which even survive in the limit of fast spin relaxation.
We propose a procedure to reconstruct characteristic features of an unknown stochastic system from the long-time full counting statistics of some of the system's transitions that are monitored by a detector. The full counting statistics is conveniently parametrized by so-called generalized factorial cumulants. Taking only a few of them as input information is sufficient to reconstruct important features such as the lower bound of the system dimension and the full spectrum of relaxation rates. The use of generalized factorial cumulants reveals system dimensions and rates that are hidden for ordinary cumulants. We illustrate the inverse counting-statistics procedure for two model systems: a single-level quantum dot in a Zeeman field and a single-electron box subjected to sequential and Andreev tunneling. following scenario. First, only steady-state counting statistics is considered, i.e., we assume that the system has already relaxed before counting starts. This excludes studying transient behavior after a perturbation of the system. The latter would, on the one hand, offer a direct access to some relaxation rate of the system [50-54]. On the other hand, the determination of the full spectrum of relaxation rates would require the knowledge of how to perturb the system in order to probe a specific relaxation rate.Second, we concentrate on the limit of long measuring-time intervals [ ] t 0; , for which the system dynamics is dominated by the slowest relaxation rate only. Nevertheless, the procedure of inverse counting statistics yields the full spectrum of all relaxation rates, as explained below.Given a measured distribution P N (t), what are the input data for the inverse counting statistics? In [43], it was suggested to use the (long-time) cumulants of the distribution function. Here, we propose to use generalized factorial cumulants [55,56] instead. The advantage of the latter is that they depend on an arbitrarily chosen parameter s. The outcome of the inverse counting statistics (such as the number of system states or the spectrum of relaxation rates) should, however, not depend on this parameters. Therefore, the s-independence of the results defines a powerful consistency criterion. Furthermore, as we will see in section 5, there are special cases in which part of the relaxation-rate spectrum is not accessible by inverse counting statistics with ordinary cumulants but is detectable by using generalized factorial cumulants with properly chosen parameters.As another difference to[43], we allow for a more general system-detector coupling by introducing the counting power m. In[43], the detector is assumed to be sensitive to only a single transition between two specific states increasing the detector counter just by one, the counting power is m=1. If this transition increases the detector counter by k, the counting power is m=k. If several transitions are counted by the detector, the counting power can be even larger. We allow for detectors counting arbitrarily many transitions between arbitrarily many sta...
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