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We consider a well-known exactly solvable model of an open quantum systemwith pure decoherence. The aim of this paper is twofold. Firstly, decoherenceis a property of open quantum systems important for both quantum technologiesand the fundamental question of quantum-classical transition. It is worthwhileto study how the long-term rate of decoherence depends on the spectral densitycharacterizing the system-bath interaction in this exactly solvable model.Secondly, we address a more general problem of the Markovian embedding of anon-Markovian open system dynamics. It is often assumed that a non-Markovianopen quantum system can be embedded into a larger Markovian system. However, weshow that such embedding is possible only for the Ohmic spectral densities (forthe case of a positive bath temperature) and is impossible for both the sub-and super-Ohmic spectral densities. From the other side, for the Ohmic spectraldensities, an asymptotic large-time Markovianity (in terms of the quantumregression formula) takes place.

Long-term behaviour in an exactly solvable model of pure decoherence and the problem of Markovian embedding