Directly computing Wigner functions for open quantum systems

TL;DR

The paper tackles the problem of evolving phase-space distributions for open quantum systems by directly computing the time-dependent Wigner function $W_S(x;p;t)$ from its initial values, avoiding the need to solve the Wigner equation. It develops a general, perturbative framework for a non-relativistic system interacting with an arbitrary environment, where the environment is traced out and the evolution is encoded in a kernel built from system propagators and the influence Hamiltonian $H_{IF}$. The method is demonstrated on a Yukawa-type model with a non-relativistic particle coupled to a relativistic scalar environment via $H_I^{SE} = -g ∫ d^3x \, Ψ(x) Ψ†(x) ⊗ φ(x)$, with a second-order expansion in the coupling $g$ that makes use of environmental propagators $Δ^F$, $Δ^D$, $Δ^<$ and system propagators $G^F$, $G^D$. The resulting time-dependent Wigner function is a computable expression built from the initial Wigner data and environment-induced correlations, providing a useful tool to study decoherence and the quantum-to-classical transition in phase space, and opening avenues for numerical implementations and more realistic models.

Abstract

The Wigner function is a well-known phase space distribution function with many applications in quantum mechanics. In this article, we consider an open quantum system consisting of a non-relativistic single particle interacting with a general, possibly relativistic environment. For this system, we derive an expression for directly computing the time-dependent Wigner function from its initial values. This result renders time-dependent Wigner functions more applicable without having to make additional approximations that would otherwise be required in order to solve the corresponding equation of motion. As an illustration of our findings, we discuss the example of a non-relativistic single scalar particle interacting via a Yukawa interaction with an environment comprising another type of scalar field that is treated relativistically.

Figures (1)

• Figure 1: Diagrammatic representation of the terms contributing to $W_\mathcal{S}(\vec{x};\vec{p};t)$ as given in Eq. (\ref{['eq:WignerExFinal']}); a crossed box represents the Wigner function at the initial time $0$, a solid line stands for a propagator of the non-relativistic particle, and a dashed line is a $\phi$-propagator. (a): free evolution of the non-relativistic particle corresponding to the zeroth order term in Eq. (\ref{['eq:WignerExFinal']}); (b), (c) and (d) represent the first, second and third term at order $g^2$, respectively.