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We provide model reduction formulas for open quantum systems consisting of atarget component which weakly interacts with a strongly dissipativeenvironment. The time-scale separation between the uncoupled dynamics and theinteraction allows to employ tools from center manifold theory and geometricsingular perturbation theory to eliminate the variables associated to theenvironment (adiabatic elimination) with high-order accuracy. An importantspecificity is to preserve the quantum structure: reduced dynamics in(positive) Lindblad form and coordinate mappings in Kraus form. We provideformulas of the reduced dynamics. The main contributions of this paper are (i)to show how the decomposition of the environment into $K$ components enablesits efficient treatment, avoiding the quantum curse of dimension; and (ii) toextend the results to the case where the target component is subject toHamiltonian evolution at the fast time-scale. We apply our theory to amicrowave superconducting quantum resonator subject to material losses, and weshow that our reduced-order model can explain the transmission spectrumobserved in a recent pump probe experiment.

Adiabatic elimination for multi-partite open quantum systems with non-trivial zero-order dynamics