Covariance propagation

For a centred Gaussian state write

For quadratic Hamiltonian and localization coefficient $D(t)$,

Hence

These equations follow by moving the Hamiltonian and double commutators to the three quadratic observables. They preserve Gaussianity in the quadratic localization model. They also hold for second moments of non-Gaussian states under the regularized kick model, by Lemma lem:momentsG.

For an inverted stage with curvature $-M{\omega }^2$,

The curvature contains $M$, not $M^2$. The latter would have the wrong dimensions in the Hamiltonian.

For a flat interval of duration $t$, constant coefficient $D$ gives

Cooling can be represented by its own linear Gaussian covariance equation with drift $B_{\mathrm{c}}$ and diffusion $N_{\mathrm{c}}$. It is not reproduced by substituting a thermal output state into a closed-system harmonic stage. For the restricted reference diagnostic we use the declared post-cooling thermal covariance

This is an input conditional on the preparation characterization, not a new cooling-performance claim.