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

Abstract

Do phenomenological master equations with 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 integrodifferential master equations describing the dynamics of a spin 1/2 in a thermal bath. By using a recently introduced measure to quantify non-Markovianity [H.-P. Breuer, E.-M. Laine, and J. Piilo, 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 uni-directional information flow from the system to the reservoir associated to time-dependent Markovian processes.

Measures of non-Markovianity: Divisibility versus backflow of information

“…Despite this kind of equations is used as a simple model of non-Markovian dynamics, it was showed in [19] that N BLP (Λ) = 0. Howeverσ(∆; t) is not always decreasing for any Helstrom matrix.…”

“…We performed explicit construction showing that in general these two measures do not agree supporting [17][18][19].…”

“…In a recent paper [17] Haikka, Cresser and Maniscalco performed detailed analysis of these approaches studying the dynamics of the driven qubit in a structured environment. It is indicated [17] that the concepts of RHP and BLP need not agree, as was also conjectured in [18] and checked in [19]. BLP measure was recently analyzed for the dynamics of a qubit coupled to a spin environment via an energy-exchange mechanism [20].…”

“…The converse is in general not true. This problem was carefully analyzed in the paper [19] for phenomenological integro-differential master equations and in the recent paper [17], where the authors have studied the non-Markovian character of a driven qubit in a structured reservoir in different dynamical regime using N RHP (Λ) and N BLP (Λ).…”

Non-Markovian effects on the nonlocality of a qubit-oscillator system

“…We believe that the reason for such a behavior, though, should be researched in the multifaceted nature of non-Markovian dynamics. In fact, the Shabani-Lidar ME can be recast into the form of a time-dependent, time-local ME whose coefficients are not all positive, therefore signaling the breakdown of divisibility [28] and thus the impossibility to describe the evolution in Markovian terms, according to the criterion put forward by Rivas et al [16], although no kickback of information is possible (as witnessed by the fact that the measure proposed in Ref.[17] is strictly null, for such a dynamical map). It is just intriguing that the Bell-CHSH function is able to experience the subtleties of such differences in such a striking way.…”