We consider a composite open quantum system consisting of a fast subsystemcoupled to a slow one. Using the time-scale separation, we develop an adiabaticelimination technique to derive at any order the reduced model describing theslow subsystem. The method, based on an asymptotic expansion and geometricsingular perturbation theory, ensures the physical interpretation of thereduced second-order model by giving the reduced dynamics in a Lindblad formand the state reduction in Kraus map form. We give explicit second-orderformulas for Hamiltonian or cascade coupling between the two subsystems. Theseformulas can be used to engineer, via a careful choice of the fast subsystem,the Hamiltonian and Lindbald operators governing the dissipative dynamics ofthe slow subsystem.
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