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We consider a time-dependent small quantum system weakly coupled to anenvironnement, whose effective dynamics we address by means of a Lindbladequation. We assume the Hamiltonian part of the Lindbladian is slowly varyingin time and the dissipator part has small amplitude. We study the properties ofthe evolved state of the small system as the adiabatic parameter and couplingconstant both go to zero, in various asymptotic regimes. In particular, weanalyse the deviations of the transition probabilities of the small systembetween the instantaneous eigenspaces of the Hamiltonian with respect to theirvalues in the purely Hamiltonian adiabatic setup, as a function of bothparameters.

Adiabatic Lindbladian Evolution with Small Dissipators