# Damping modes of harmonic oscillator in open quantum systems

## TL;DR

This paper analyzes the damping modes of a harmonic oscillator in open quantum systems, identifying four modes that affect the covariance matrix and ensuring the uncertainty principle is maintained, using quantum master equations.

## Contribution

It introduces a set of generators that preserve hermiticity and trace, classifies damping modes, and demonstrates their effects using quantum master equations.

## Key findings

- Four damping modes identified with distinct effects on covariance matrix
- Damping modes can cause exponential decay or shift in covariance components
- Proper combination of modes maintains the uncertainty relation

## Abstract

Through a set of generators that preserves the hermiticity and trace of density matrices, we analyze the damping of harmonic oscillator in open quantum systems into four modes, distinguished by their specific effects on the covariance matrix of position and momentum of the oscillator. The damping modes could either cause exponential decay to the initial covariance matrix or shift its components. They have to act together properly in actual dynamics to ensure that the generalized uncertainty relation is satisfied. We use a few quantum master equations to illustrate the results.