Lattice Wigner equation

TL;DR

This paper introduces a lattice-based numerical scheme for solving the Wigner equation, accurately capturing moments and ensuring stability, validated on quantum potentials and applied to quantum transport in multiple dimensions.

Contribution

The paper presents a novel lattice discretization method for the Wigner equation that guarantees moment accuracy and numerical stability, applicable to multi-dimensional quantum systems.

Findings

Accurate moment recovery up to desired order.

Good agreement with theoretical results for quantum potentials.

Demonstrated applicability to 3D open quantum systems.

Abstract

We present a numerical scheme to solve the Wigner equation, based on a lattice discretization of momentum space. The moments of the Wigner function are recovered exactly, up to the desired order given by the number of discrete momenta retained in the discretisation, which also determines the accuracy of the method. The Wigner equation is equipped with an additional collision operator, designed in such a way as to ensure numerical stability without affecting the evolution of the relevant moments of the Wigner function. The lattice Wigner scheme is validated for the case of quantum harmonic and anharmonic potentials, showing good agreement with theoretical results. It is further applied to the study of the transport properties of one and two dimensional open quantum systems with potential barriers. Finally, the computational viability of the scheme for the case of three- dimensional open systems is also illustrated.