The spectrogram expansion of Wigner functions

TL;DR

This paper introduces an asymptotic expansion of Wigner functions using Hermite spectrograms, enabling accurate approximation of quantum expectations and efficient sampling methods, with applications demonstrated through numerical experiments.

Contribution

It presents a novel asymptotic expansion of Wigner functions in terms of probability densities, facilitating high-accuracy quantum expectation calculations.

Findings

Exact formulas for quantum expectations of polynomial observables.

High-frequency regime approximations with arbitrary accuracy.

A Markov Chain Monte Carlo method for sampling from the new densities.

Abstract

Wigner functions generically attain negative values and hence are not probability densities. We prove an asymptotic expansion of Wigner functions in terms of Hermite spectrograms, which are probability densities. The expansion provides exact formulas for the quantum expectations of polynomial observables. In the high frequency regime it allows to approximate quantum expectation values up to any order of accuracy in the high frequency parameter. We present a Markov Chain Monte Carlo method to sample from the new densities and illustrate our findings by numerical experiments.