Effective (Floquet) Lindblad generators from spectral unwinding

A mathematical description of the reduced dynamics of an open quantum system can often be given in terms of a completely positive and trace preserving (CPTP) map, also known as quantum channel. In a seminal work by Wolf et al. [Phys. Rev. Lett. 101, 150402 (2008)], it was shown that deciding whether a given quantum channel was generated from an underlying effective Markovian dynamics, with time-independent generator of Lindblad form, is generally an NP-hard problem. The difficulty is related to the fact that one has to search through all possible branches of the operator logarithm of the map, in order to identify, if any of the resulting effective generators is of Lindblad form. In this work we show that in cases where one has access to the full reduced dynamics at all previous times (the dynamical map) one can significantly facilitate the search for an effective generator by making use of Floquet theory. By performing a spectral unwinding such that the effective micromotion is minimized, the effective Floquet generator is often an excellent candidate for an effective generator of Lindblad form. This significantly reduces the complexity of the search for an effective generator of Lindblad form in many (though not all) cases. Our results are relevant for engineering Floquet Lindbladians in complex many-body systems.