Open quantum dynamics with singularities: Master equations and degree of non-Markovianity

TL;DR

This paper addresses singularities in open quantum system master equations, proposing a higher-order reformulation to handle divergences and introducing a new measure for non-Markovianity based on information flow.

Contribution

It introduces a method to regularize singular master equations using higher-order derivatives and proposes a novel non-Markovianity measure based on information inflow.

Findings

Method successfully eliminates generator divergences in examples.

Reformulation extends the validity of quantum dynamics beyond singular points.

Average information inflow effectively characterizes non-Markovian behavior.

Abstract

Master equations describing open quantum dynamics are typically first order differential equations. When such dynamics brings the trajectories in state space of more than one initial state to the same point at finite instants in time, the generator of the corresponding master equation becomes singular. The first-order, time-local, homogeneous master equations then fail to describe the dynamics beyond the singular point. Retaining time-locality in the master equation necessitates a reformulation in terms of higher-order differential equations. We formulate a method to eliminate the divergent behavior of the generator by using a combination of higher-order derivatives of the generator with suitable weights and illustrate it with several examples. We also present a detailed study of the central spin model and we propose the average rate of information inflow in non-Markovian processes as a quantity that captures a different aspect of non-Markovian dynamics.