A new Phase Space Density for Quantum Expectations

TL;DR

This paper introduces a novel phase space density for quantum states that improves expectation value approximations and is more accessible for sampling than traditional Wigner functions, with applications in quantum dynamics simulations.

Contribution

It presents a new phase space density combining Husimi functions and Hermite spectrograms, enabling more accurate quantum expectation estimates and practical sampling methods.

Findings

Provides a density that approximates quantum expectations to second order.

Demonstrates improved accuracy over Husimi function in semiclassical regimes.

Supports numerical simulations of quantum dynamics using classical trajectories.

Abstract

We introduce a new density for the representation of quantum states on phase space. It is constructed as a weighted difference of two smooth probability densities using the Husimi function and first-order Hermite spectrograms. In contrast to the Wigner function, it is accessible by sampling strategies for positive densities. In the semiclassical regime, the new density allows to approximate expectation values to second order with respect to the high frequency parameter and is thus more accurate than the uncorrected Husimi function. As an application, we combine the new phase space density with Egorov's theorem for the numerical simulation of time-evolved quantum expectations by an ensemble of classical trajectories. We present supporting numerical experiments in different settings and dimensions.