Truncated Wigner approximation as a non-positive Kraus map

TL;DR

This paper demonstrates that the Truncated Wigner Approximation can be represented as a Lindblad-type evolution with a non-positive operator, revealing limitations in its physical interpretability and providing methods to estimate the emergence of non-physical states.

Contribution

It establishes a connection between the Truncated Wigner Approximation and non-positive Kraus maps, highlighting its non-physical aspects and offering a way to estimate negative eigenvalues.

Findings

The Wigner function evolution corresponds to a non-positive operator.

Negative eigenvalues of the operator can be efficiently estimated.

Short-time dynamics of Kerr and second harmonic generation are analyzed.

Abstract

We show that the Truncated Wigner Approximation developed in the flat phase-space is mapped into a Lindblad-type evolution with an indefinite metric in the space of linear operators. As a result, the classically evolved Wigner function corresponds to a non-positive operator $\hat{R}(t)$, which does not describe a physical state. The rate of appearance of negative eigenvalues of $\hat{R}(t)$ can be efficiently estimated. The short-time dynamics of the Kerr and second harmonic generation Hamiltonains are discussed.