Finite-frequency counting statistics of electron transport: Markovian Theory

TL;DR

This paper develops a Markovian quantum master equation approach to calculate frequency-dependent counting statistics of electron transport, extending existing zero-frequency methods to finite frequencies and higher cumulants, with applications to quantum dot models.

Contribution

It introduces a generalized formula for finite-frequency current cumulants within Markovian theory, extending MacDonald's formula beyond shot noise and enabling analysis at arbitrary bias voltages.

Findings

Derived explicit formulas for second- and third-order cumulants.

Demonstrated the method on a single resonant level model.

Discussed the limits and applicability of the Markovian approximation.

Abstract

We present a theory of frequency-dependent counting statistics of electron transport through nanostructures within the framework of Markovian quantum master equations. Our method allows the calculation of finite-frequency current cumulants of arbitrary order, as we explicitly show for the second- and third-order cumulants. Our formulae generalize previous zero-frequency expressions in the literature and can be viewed as an extension of MacDonald's formula beyond shot noise. When combined with an appropriate treatment of tunneling, using, e.g. Liouvillian perturbation theory in Laplace space, our method can deal with arbitrary bias voltages and frequencies, as we illustrate with the paradigmatic example of transport through a single resonant level model. We discuss various interesting limits, including the recovery of the fluctuation-dissipation theorem near linear response, as well as some drawbacks inherent of the Markovian description arising from the neglect of quantum fluctuations.