  Recent works on quantum resource theories of non-Gaussianity, which are based
upon the type of tools available in contemporary experimental settings, put
Gaussian states and their convex combinations on equal footing. Motivated by
this, in this article, we derive a new model of dissipative time evolution
based on unitary Lindblad operators which, while does not preserve the set of
Gaussian states, preserves the set of their convex combinations, i.e. so-called
quantum Gaussian states. As we demonstrate, the considered evolution proves
useful both as a description for random scattering and as a tool in dissipator
engineering.
