Canonical form of master equations and characterization of non-Markovianity

Abstract

Master equations govern the time evolution of a quantum system interacting with an environment, and may be written in a variety of forms. Time-independent or memoryless master equations, in particular, can be cast in the well-known Lindblad form. Any time-local master equation, Markovian or non-Markovian, may in fact also be written in a Lindblad-like form. A diagonalisation procedure results in a unique, and in this sense canonical, representation of the equation, which may be used to fully characterize the non-Markovianity of the time evolution. Recently, several different measures of non-Markovianity have been presented which reflect, to varying degrees, the appearance of negative decoherence rates in the Lindblad-like form of the master equation. We therefore propose using the negative decoherence rates themselves, as they appear in the canonical form of the master equation, to completely characterize non-Markovianity. The advantages of this are especially apparent when more than one decoherence channel is present. We show that a measure proposed by Rivas et al. is a surprisingly simple function of the canonical decoherence rates, and give an example of a master equation that is non-Markovian for all times t > 0, but to which nearly all proposed measures are blind. We also give necessary and sufficient conditions for trace distance and volume measures to witness non-Markovianity, in terms of the Bloch damping matrix.

Analysis of the non-Markovianity for electron transfer reactions in an oligothiophene-fullerene heterojunction

“…However, this measure could fail to detect non-Markovianity because it depends on the sum of the canonical rates that appear in the canonical form of the master equation (see below) [31]. The sum may be positive even if some individual rate is negative.…”

“…However, this measure requires a sampling of pairs of initial states. Other measures based on the volume of dynamically accessible states in the system [36] or the time-dependent decoherence canonical rates [31] require propagation of basis operators only.…”

“…Different measures have been proposed to quantify the nonMarkovianity [31][32][33]. Breuer's non-Markovianity measure [35,63,64] is based on the trace distance of pairs of initial states…”

“…In order to understand the relation between the structure of the environment and the nonMarkovianity and to quantity these effects, several measures have been proposed recently. A review can be found in references [31][32][33][34]. The measure used by Breuer [35] is based on the quantum trace distance measuring the distinguishability between two quantum states.…”

“…Non-Markovianity is then detected by a temporary increase of this volume while a monotonous decrease is expected in the Markovian case. Another interesting approach analyzes the canonical relaxation rate constants [31] that are constant in a Markovian process and calibrate a canonical Lindblad Master equation [38,6]. The rates become time dependent and some can be temporarily negative in a non-Markovian evolution.…”