Completely-positive non-signalling non-Markovian dynamics

TL;DR

This paper characterizes non-Markovian quantum dynamics under complete positivity and non-signalling, providing a new integro-differential framework that captures memory effects and enables advanced quantum control.

Contribution

It introduces a comprehensive formalism for non-Markovian quantum dynamics that generalizes the Lindblad equation with memory effects and removes the need for a regression theorem.

Findings

Derived the emission spectrum of a driven two-level system with non-Markovian bath

Showed the Mollow triplet linewidth becomes frequency-dependent due to memory effects

Established a formalism for multi-time correlations without regression theorem

Abstract

We define non-Markovian quantum dynamics as evolution in which the current state depends on all past states, and completely characterize its structure under the assumptions of complete positivity and non-signalling. The resulting continuous-time dynamics is an integro-differential equation that augments the Gorini-Kossakowski-Sudarshan-Lindblad equation with a memory integral, and is capable of describing the quantum state of systems exposed to noise with any integrable power spectral density with no further approximations. We then establish a formalism to evaluate multi-time correlations of measurement outcomes in this general setting, obviating the need for a regression theorem. As an application, we derive the emission spectrum of a driven two-level system coupled to a non-Markovian bath: the familiar Mollow triplet acquires a frequency-dependent linewidth that encodes the memory of the bath. Our work provides a rigorous yet transparent description of the quantum state of non-Markovian systems, opening the door for state estimation and state-based quantum control beyond the Markovian regime.