Equilibration and asymptotic stationarity for non-Markovian master equations

Abstract

We show that certain positivity-preserving non-Markovian generalizations of the Kossakowski-Lindblad master equation can exhibit equilibration to an asymptotic state which is stationary with respect to the shifted system Hamiltonian for general system-bath coupling. This is in sharp contrast to results for Markovian forms which require strong relations between these operators (e.g. commutation of isolated system Hamiltonian and coupling operator). We also expand the list of sufficient conditions for positivity of non-Markovian master equations.

Optimally chosen Nakajima–Zwanzig master equation for mean field approximation

“…Note that Ref. [8] required that λ 2 /4 = V (0) which we will see is unnecessary. To avoid dealing with specific initial conditions ρ(0) we employ the Hadamard representation [9].…”

“…[8] to the case of inhomogeneous master equations. We also relax one unnecessary restriction imposed in Ref.…”

“…Non-Markovian generalizations of the well known Kossakowski-Lindblad master equation [1] have been a focus of many recent studies [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] in quantum open systems theory [19]. Such structures can preserve complete positivity [2,3,4,5,6,7] or positivity [8,9,10].…”

“…Such structures can preserve complete positivity [2,3,4,5,6,7] or positivity [8,9,10]. Efforts to quantify the non-Markovian character of such dynamics [16,17,18] are also underway, as are explorations of relations between integrodifferential and time local forms [11,14].…”

“…Semi-classical methods [23] and mean field approaches [24] have been explored. Here we examine a modification of the mean field method to connect a specific positivity preserving master equation which correctly equilibrates [8] to data for an electronic spin S = 1 associated with a Nitrogen-Vacancy center in diamond interacting with the I = 1/2 nuclear spins of 13 C impurities. We start in Section 2 with a form of the exact Nakajima-Zwanzig master equation [25] (NZME) which is valid for cases in which the bath operator Λ, associated with the NZME projection operator…”