Wigner transport in linear electromagnetic fields

TL;DR

This paper develops a gauge-invariant Wigner equation for linear electromagnetic fields in two-dimensional electron transport, simplifying the complex equation to enable Monte Carlo solutions and quantum transport simulations.

Contribution

It introduces a simplified, gauge-invariant Wigner equation for 2D electron transport in linear electromagnetic fields and proposes Monte Carlo algorithms for its numerical solution.

Findings

Reformulation into a Fredholm integral equation facilitates Monte Carlo methods.

Two stochastic algorithms enable evaluation of physical quantities and the Wigner function.

The approach provides a heuristic quantum particle model for transport analysis.

Abstract

Applying a Weyl-Stratonovich transform to the evolution equation of the Wigner function in an electromagnetic field yields a multidimensional gauge-invariant equation which is numerically very challenging to solve. In this work, we apply simplifying assumptions for linear electromagnetic fields and the evolution of an electron in a plane (two-dimensional transport), which reduces the complexity and enables to gain first experiences with a gauge-invariant Wigner equation. We present an equation analysis and show that a finite difference approach for solving the high-order derivatives allows for reformulation into a Fredholm integral equation. The resolvent expansion of the latter contains consecutive integrals, which is favorable for Monte Carlo solution approaches. To that end, we present two stochastic (Monte Carlo) algorithms that evaluate averages of generic physical quantities or directly the Wigner function. The algorithms give rise to a quantum particle model, which interprets quantum transport in heuristic terms.