Inverse counting statistics based on generalized factorial cumulants

Abstract

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 parameters. 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 parameters.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...

Nonequilibrium phase transition in a single-electron micromaser

“…As we will see in the following, the factorial moments and cumulants provide a convenient description of the single-electron micromaser. In particular, while ordinary cumulants are useful to describe continuous random variables, factorial cumulants are in some cases better suited to characterize discrete random variables, such as the number of counted electrons or photons [56][57][58][59][60][61][62][63][64][65][66][67].…”

“…Here, by contrast, we show how the phase transitions can be directly observed from accurate measurements of the full counting statistics. In particular, we will see that factorial moments and factorial cumulants [56][57][58][59][60][61][62][63][64][65][66][67] are particularly useful to reveal the non-equilibrium phase transition, which may be observed in future experiments.…”

Counting quantum jumps: A summary and comparison of fixed-time and fluctuating-time statistics in electron transport

“…so we see that the long-time limit necessarily restricts cumulants to the zero-frequency power regime, where they potentially miss important short-time physics. Counting statistics at finite times, or rather finite frequencies, remain an active theoretical [87][88][89][90][91][92][93][94] and experimental [67] research area; non-Poissonian behavior of higherorder cumulants, for example, has been shown to depend on frequency [95]. Time-dependent current cumulants are able to identify short-time correlations between electrons [96,97]; in fact it has been proposed that higherorder factorial cumulants can be used as a detection technique for electron-electron interactions [98][99][100].…”

Random-walk topological transition revealed via electron counting

“…3(a) with a solid line, which shows its decaying character. Even though there is an underlying but complex relation between the exponent spectrum and the cumulants [40], the moments µ m = ∞ 0 t m P e (t)dt and the associated cumulants κ m [41] depicted in Fig. 3(b) do not provide direct information about the topology.…”