Universal density matrix for the phase space

TL;DR

This paper introduces a universal density matrix for phase space representation of quantum systems, enabling a unified approach to describe different systems with explicit elements and properties.

Contribution

It proposes a universal density matrix $ ext{W}$ for phase space, with explicitly derived elements and analysis of their properties, applicable to any quantum system.

Findings

Universal matrix $ ext{W}$ explicitly derived.

Diagonal elements are Wigner functions of harmonic oscillators.

Off-diagonal elements encode dissipation via frequency oscillations.

Abstract

In this paper, a new representation of the Wigner function for a quantum system in the phase space is proposed. The new representation is of the form $W=\operatorname{Sp}\left[ \rho \mathcal{W} \right]$, where $\rho$ is the density matrix, and $\mathcal{W}$ is the universal density matrix. The density matrix $\rho$ for each quantum system is different, and the universal matrix $\mathcal{W}$ is the same for any quantum system. Thus, the matrix $\mathcal{W}$ has a fundamental character. In the work, the elements of the universal matrix $\mathcal{W}$ were found explicitly and their properties were investigated. The diagonal elements of the matrix $\mathcal{W}$ are the Wigner functions of the harmonic oscillator, which do not introduce dissipation into the quantum system. The off-diagonal elements of the matrix $\mathcal{W}$ contain frequency oscillations responsible for dissipations in the quantum systems.