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

Quantum Physics;Optimization and Control