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

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.