# Semiclassical propagation of Wigner functions

## TL;DR

This paper develops and tests semiclassical methods for propagating Wigner functions in phase space, demonstrating their effectiveness in modeling quantum phenomena like tunneling and chaos in molecular systems.

## Contribution

It introduces two semiclassical approximation schemes for Wigner function propagation and evaluates their performance on nonlinear molecular models.

## Key findings

- Semiclassical methods accurately reproduce quantum effects like tunneling.
- Performance remains robust in chaotic and high-dimensional systems.
- Proposed numerical strategies facilitate application to complex systems.

## Abstract

We present a comprehensive study of semiclassical phase-space propagation in the Wigner representation, emphasizing numerical applications, in particular as an initial-value representation. Two semiclassical approximation schemes are discussed: The propagator of the Wigner function based on van Vleck's approximation replaces the Liouville propagator by a quantum spot with an oscillatory pattern reflecting the interference between pairs of classical trajectories. Employing phase-space path integration instead, caustics in the quantum spot are resolved in terms of Airy functions. We apply both to two benchmark models of nonlinear molecular potentials, the Morse oscillator and the quartic double well, to test them in standard tasks such as computing autocorrelation functions and propagating coherent states. The performance of semiclassical Wigner propagation is very good even in the presence of marked quantum effects, e.g., in coherent tunneling and in propagating Schr\"odinger cat states, and of classical chaos in four-dimensional phase space. We suggest options for an effective numerical implementation of our method and for integrating it in Monte-Carlo--Metropolis algorithms suitable for high-dimensional systems.