# Open quantum systems in Heisenberg picture

## TL;DR

This paper derives exact formulas for the reduced density matrix, characteristic function, and Wigner distribution of a driven dissipative quantum harmonic oscillator using the Heisenberg picture, with potential generalizations.

## Contribution

It provides a novel derivation of the reduced density matrix and related functions for open quantum systems in the Heisenberg picture, including generalizations via Magnus expansion.

## Key findings

- Exact expressions for the reduced density matrix are obtained.
- Formulas for the characteristic function and Wigner distribution are derived.
- A possible generalization using Magnus expansion is proposed.

## Abstract

In the framework of the Heisenberg picture, an alternative derivation of the reduced density matrix of a driven dissipative quantum harmonic oscillator as the prototype of an open quantum system is investigated. The reduced density matrix for different initial states of the combined system is obtained from a general formula, and different limiting cases are studied. Exact expressions for the corresponding characteristic function in quantum thermodynamics and Wigner quasi distribution function are found. A possible generalization based on the Magnus expansion of the evolution operator is presented.