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We consider an open quantum system described by a Lindblad-type masterequation with two times-scales. The fast time-scale is strongly dissipative anddrives the system towards a low-dimensional decoherence-free space. To performthe adiabatic elimination of this fast relaxation, we propose a geometricasymptotic expansion based on the small positive parameter describing thetime-scale separation. This expansion exploits geometric singular perturbationtheory and center-manifold techniques. We conjecture that, at any order, itprovides an effective slow Lindblad master equation and a completely positiveparameterization of the slow invariant sub-manifold associated to thelow-dimensional decoherence-free space. By preserving complete positivity andtrace, two important structural properties attached to open quantum dynamics,we obtain a reduced-order model that directly conveys a physical interpretationsince it relies on effective Lindbladian descriptions of the slow evolution. Atthe first order, we derive simple formulae for the effective Lindblad masterequation. For a specific type of fast dissipation, we show how any Hamiltonianperturbation yields Lindbladian second-order corrections to the first-orderslow evolution governed by the Zeno-Hamiltonian. These results are illustratedon a composite system made of a strongly dissipative harmonic oscillator, theancilla, weakly coupled to another quantum system.

Adiabatic elimination for open quantum systems with effective Lindblad master equations