Heisenberg formulation of adiabatic elimination for open quantum systems with two timescales

Consider an open quantum system governed by a Gorini-Kossakowski-Sudarshan-Lindblad master equation with two times-scales: a fast one, exponentially converging towards a linear subspace of quasi-equilibria; a slow one resulting from small decoherence and Hamiltonian dynamics. Usually adiabatic elimination is performed in the Schrodinger picture. We propose here a Heisenberg formulation where the invariant operators attached to the fast decay dynamics towards the quasi-equilibria subspace play a key role. Based on geometric singular perturbations, asymptotic expansions of the Heisenberg slow dynamics and of the fast invariant linear subspaces are proposed. They exploit Carr's approximation lemma from center-manifold and bifurcation theory. Second-order expansions are detailed and shown to ensure preservation, up to second-order terms, of the complete positivity for the slow propagator on a slow timescale. Such expansions can be exploited numerically to derive reduced-order dynamical models.