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

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 while the dynamical map becomes noninvertible. 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.