Phenomenological memory-kernel master equations and time-dependent Markovian processes

Do phenomenological master equations with a memory kernel always describe a non-Markovian quantum dynamics characterized by reverse flow of information? Is the integration over the past states of the system an unmistakable signature of non-Markovianity? We show by a counterexample that this is not always the case. We consider two commonly used phenomenological integro-differential master equations describing the dynamics of a spin $1/2$ in a thermal bath. By using a recently introduced measure to quantify non-Markovianity [Breuer et al., Phys. Rev. Lett. 103, 210401 (2009)] we demonstrate that as far as the equations retain their physical sense, the key feature of non-Markovian behavior does not appear in the considered memory kernel master equations. Namely, there is no reverse flow of information from the environment to the open system. Therefore, the assumption that the integration over a memory kernel always leads to a non-Markovian dynamics turns out to be vulnerable to phenomenological approximations. Instead, the considered phenomenological equations are able to describe time-dependent and unidirectional information flow from the system to the reservoir associated with time-dependent Markovian processes.