Full-counting statistics of time-dependent conductors

Abstract

We develop a scheme for the computation of the full-counting statistics of transport described by Markovian master equations with an arbitrary time dependence. It is based on a hierarchy of generalized density operators, where the trace of each operator yields one cumulant. This direct relation offers a better numerical efficiency than the equivalent number-resolved master equation. The proposed method is particularly useful for conductors with an elaborate time dependence stemming, e.g., from pulses or combinations of slow and fast parameter switching. As a test bench for the evaluation of the numerical stability, we consider time-independent problems for which the full-counting statistics can be computed by other means. As applications, we study cumulants of higher order for two time-dependent transport problems of recent interest, namely steady-state coherent transfer by adiabatic passage (CTAP) and Landau-Zener-Stückelberg-Majorana (LZSM) interference in an open double quantum dot.

Waiting time distributions in a two-level fluctuator coupled to a superconducting charge detector

“…We partition the rate matrix as L = L 0 + J + + J − with jump operators J ± describing charge transfers to and from the island, respectively. 57 We resolve the probability vector p(n, t) such that it accounts for the number of tunneling events n. The n-resolved equations of motion, d dt p(n, t) = L 0 p(n, t) + J + p(n − 1, t) + J − p(n + 1, t), are decoupled by introducing the counting field χ via the definition p(χ, t) ≡ n p(n, t)e inχ . We then arrive at a modified master equation for p(χ, t)…”

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].…”

Electron pumping in the strong coupling and non-Markovian regime: A reaction coordinate mapping approach

“…In order to accurately obtain the current and current fluctuations we follow the full counting statistics formal-ism [64][65][66][67]69] and introduce a counting field ξ associated with reservoir ν. Let us define the modified density matrix…”

“…with time dependent superoperator L(t) and jump superoperator J (ξ, t), which satisfies J (0, t) = 0. An auxiliary equation method [65][66][67] may be used to access high order cumulants and moments directly. For the current I(t) and noise S(t) it follows [66] where…”