While in classical physics the notion of what is Markovian or not is well defined, this is not the case when it comes to open quantum systems. What makes a quantum dynamics Markovian or non-Markovian (NM)? Traditional answers to this question involve the well-known Lindblad master equation (ME) and/or the ability of an open dynamics to be governed by a time-local ME.
The last few years, yet, witnessed a thorough revision of such concepts in the attempt to establish in a rigorous way exact criteria for assessing whether or not, and even quantifying, a dynamics is NM. This resulted in a number of NM "measures" that have been put forward. Based on these and a number of further theoretical progresses, the lack of a Lindblad ME or time-local ME turns out not to be a reliable criterion. For instance: there are Markovian dynamics that are not described by a Lindblad ME. On the other hand, open dynamics that are strongly NM can often be shown to fulfill a time-local ME.
After reviewing all these new concepts in depth, I will discuss a specific instance of application of non-Markovianity measures from my latest research work: an atom emitting into a disordered medium. In such a case, the occurrence of localized field modes due to Anderson localization makes the atom dynamics non-Markovian, as confirmed by the behavior of non-Markovianity measures against the amount of disorder. The functional shape of this relationship is well reproduced by an effective phenomenological model.