GKLS Vector Field Dynamics for Gaussian States

TL;DR

This paper constructs a vector field framework for GKLS dynamics in Gaussian states, decomposing it into Hamiltonian, gradient-like, and jump components, with applications to dissipative harmonic oscillators.

Contribution

It introduces a novel vector field decomposition for GKLS generators acting on Gaussian states, highlighting the structure of dissipative quantum dynamics.

Findings

Decomposition of GKLS vector field into three parts.

Application to harmonic oscillator with dissipation.

Illustration of the perturbative nature of dissipative terms.

Abstract

We construct the vector field associated to the GKLS generator for systems described by Gaussian states. This vector field is defined on the dual space of the algebra of operators, restricted to operators quadratic in position and momentum. It is shown that the GKLS dynamics accepts a decomposition principle, that is, this vector field can be decomposed in three parts, a conservative Hamiltonian component, a gradient-like, and a Choi-Kraus or jump vector field. The two last terms are considered a "perturbation" associated with dissipation. Examples are presented for a harmonic oscillator with different dissipation terms.