Five approaches to exact open-system dynamics: Complete positivity, divisibility and time-dependent observables

TL;DR

This paper explores various notions of divisibility in open quantum system dynamics through five approaches, analyzing a solvable fermionic model to understand how divisibility affects observable behaviors and transient currents.

Contribution

It provides a comprehensive comparison of five methods to analyze divisibility in quantum dynamics and reveals novel phenomena like reentrant occupation behavior.

Findings

Loss of semigroup-divisibility can cause temporary increase in level occupation.

Loss of complete positivity divisibility restricts current reversals.

Transient currents reveal signatures of divisibility properties.

Abstract

To extend the classical concept of Markovianity to an open quantum system, different notions of the divisibility of its dynamics have been introduced. Here we analyze this issue by five complementary approaches: equations of motion, real-time diagrammatics, Kraus-operator sums, as well as time-local (TCL) and nonlocal (Nakajima-Zwanzig) quantum master equations. As a case study featuring several types of divisible dynamics, we examine in detail an exactly solvable noninteracting fermionic resonant level coupled arbitrarily strongly to a fermionic bath at arbitrary temperature in the wideband limit. In particular, the impact of divisibility on the time-dependence of the observable level occupation is investigated and compared with typical Markovian approximations. We find that the loss of semigroup-divisibility is accompanied by a prominent reentrant behavior: Counter to intuition, the level occupation may temporarily \emph{increase} significantly in order to reach a stationary state with \emph{smaller} occupation, implying a reversal of the measurable transport current. In contrast, the loss of the so-called completely-positive divisibility is more subtly signaled by the \emph{prohibition} of such current reversals in specific time-intervals. Experimentally, it can be detected in the family of transient currents obtained by varying the initial occupation. To quantify the nonzero footprint left by the system in its effective environment, we determine the exact time-dependent state of the latter as well as related information measures such as entropy, exchange entropy and coherent information.