GKLS Vector Field Dynamics for Gaussian States

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.