Long-time Markovianity of Multi-level Systems in the Rotating Wave Approximation

Abstract

There is intense discussion in literature about different approaches to definition and characterization of quantum Markovianity (see [1] for review). This is important due to modern both theoretical and applied interest in the non-Markovian phenomena in the open quantum systems (see e.g. [2, 3, 4] for recent reviews). Most of the known measures of non-Markovianity think about Markovianity [5, 6, 7, 8, 9] as of some property which is global in time. But a few works [11, 10] suggest that it is more natural to speak about some initial time (Zeno time) of order of bath correlation time before which the dynamics is surely highly non-Markovian and only after that it becomes Markovian. In [10] we have called such a behaviour long-time Markovian and have shown that it could be naturally captured by perturbation theory with Bogolubov-van Hove scaling. Bogolubov-van Hove scaling does not only insert the small parameter λ before the coupling constant but also rescales the time as t→ λt. It allows one to separate the time-scale on which the Markovian behavior occurs from the time scale of order of the bath correlation time which becomes of order of λ after the scaling. In [10] we have considered the simplest model, namely, the spin-boson in the rotating wave approximation (RWA). Here we generalize the main results of [10] to the multi-level model considered in [12] and [13]. In Section 2 we recall the results from [12, 13] in such a manner which is useful for the further parts of the article. In Section 3 we obtain the first asymptotic correction to the dynamics obtained in the Bogolubov-van Hove limit. Inspired by the unified Gorini–Kossakowski–Sudarshan– Lindblad (GKSL) quantum master equation approach [14] we do not only directly generalize the results of [10] here, but also take into account terms of order λ in the system Hamiltonian. We show that the corrected dynamics of the reduced density matrix after the correlation time could be described by a semigroup, but the initial condition should be renormalized. This leads to the corrected master equation with a time-independent generator, which is of the GKSL form for sufficiently small λ. In Section 4 we show that if one defines Markovianity in terms of the system correlation functions, then it leads to the semigroup property for the dynamical map describing the reduced density matrix. Thus, strictly speaking, our dynamics is not Markovian in the sense