Current noise in a vibrating quantum dot array

TL;DR

This paper introduces methods to calculate zero-frequency noise in quantum shuttles, focusing on a three-dot array, and compares two formulations, providing insights into transport regimes through numerical and analytical analysis.

Contribution

It presents two equivalent formulations for noise calculation in quantum shuttles and applies them to a three-dot array, including analytical rate equations for weak coupling.

Findings

Quantum regression theorem and counting variable approach yield identical results.

Numerical calculations reveal distinct transport regimes, such as shuttling and cotunneling.

Analytical rate equations agree with full numerical results in weak coupling regime.

Abstract

We develop methods for calculating the zero-frequency noise for quantum shuttles, i.e. nanoelectromechanical devices where the mechanical motion is quantized. As a model system we consider a three-dot array, where the internal electronic coherence both complicates and enriches the physics. Two different formulations are presented: (i) quantum regression theorem, and (ii) the counting variable approach. It is demonstrated, both analytically and numerically, that the two formulations yield identical results, when the conditions of their respective applicability are fulfilled. We describe the results of extensive numerical calculations for current and current noise (Fano factor), based on a solution of a Markovian generalized master equation. The results for the current and noise are further analyzed in terms of Wigner functions, which help to distinguish different transport regimes (in particular, shuttling vs. cotunneling). In the case of weak inter-dot coupling, the electron transport proceeds via sequential tunneling between neighboring dots. A simple rate equation with the rates calculated analytically from the P(E)-theory is developed and shown to agree with the full numerics.