Efficient solution of the Wigner-Liouville equation using a spectral decomposition of the force field

TL;DR

This paper introduces a spectral decomposition approach to solve the Wigner-Liouville equation more efficiently by reformulating it in terms of the force field, simplifying numerical evaluation and enabling stable, deterministic simulations.

Contribution

The paper presents a novel spectral force-based reformulation of the Wigner-Liouville equation, reducing complexity and improving numerical stability for quantum dynamical simulations.

Findings

Avoids oscillatory kernel evaluation

Enables stable deterministic numerical implementation

Demonstrates effectiveness with a tunneling diode simulation

Abstract

The Wigner-Liouville equation is reformulated using a spectral decomposition of the classical force field instead of the potential energy. The latter is shown to simplify the Wigner-Liouville kernel both conceptually and numerically as the spectral force Wigner-Liouville equation avoids the numerical evaluation of the highly oscillatory Wigner kernel which is nonlocal in both position and momentum. The quantum mechanical evolution is instead governed by a term local in space and non-local in momentum, where the non-locality in momentum has only a limited range. An interpretation of the time evolution in terms of two processes is presented; a classical evolution under the influence of the averaged driving field, and a probability-preserving quantum-mechanical generation and annihilation term. Using the inherent stability and reduced complexity, a direct deterministic numerical implementation using Chebyshev and Fourier pseudo-spectral methods is detailed. For the purpose of illustration, we present results for the time-evolution of a one-dimensional resonant tunneling diode driven out of equilibrium.