Phase space flow in the Husimi representation

TL;DR

This paper derives and analyzes the phase space flow in the Husimi representation, revealing features similar to Wigner flow but with distinct properties, tested on a double well potential.

Contribution

It provides explicit formulas for Husimi flow, including semiclassical corrections, and clarifies the nature of flow features like stagnation points and zeros.

Findings

Zeros of Husimi function are saddle points followed by centers.

Flow features like stagnation points are isolated and do not merge or split.

Husimi flow shares features with Wigner flow but is not affected by negativity.

Abstract

We derive a continuity equation for the Husimi function evolving under a general non-hermitian Hamiltonian and identify the phase space flow associated with it. For the case of unitary evolution we obtain explicit formulas for the quantum flow, which can be written as a classical part plus semiclassical corrections. These equations are the analogue of the Wigner flow, which displays several non-intuitive features like momentum inversion and motion of stagnation points. Many of these features also appear in the Husimi flow and, therefore, are not related to the negativity of the Wigner function as previously suggested. We test the exact and semiclassical formulas for a particle in a double well potential. We find that the zeros of the Husimi function are saddle points of the flow, and are always followed by a center. Merging or splitting of stagnation points, observed in the Wigner flow,does not occur because of the isolation of the Husimi zeros.