Damping modes of harmonic oscillator in open quantum systems

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.