Repeated Interaction Quantum Systems: van Hove Limits and Asymptotic States

TL;DR

This paper analyzes the long-term behavior of open quantum systems with repeated interactions, establishing effective dynamics in weak coupling regimes and demonstrating their accuracy in predicting asymptotic states, including spin systems.

Contribution

It generalizes existing results to systems where von Neumann algebras are not finite type and links effective dynamics to the exact dynamics for long-term behavior.

Findings

Existence of two weak coupling effective dynamics for repeated interaction systems.

Effective dynamics accurately predict the unique asymptotic state of the system.

Application to spin systems demonstrating the theoretical results.

Abstract

We establish the existence of two weak coupling regime effective dynamics for an open quantum system of repeated interactions (vanishing strength and individual interaction duration, respectively). This generalizes known results in that the von Neumann algebras describing the system and the chain element may not be of finite type. Then (but now assuming that the small system is of finite type), we prove that both effective dynamics capture the long-term behaviour of the system: existence of a unique asymptotic state for them implies the same property for the respective exact dynamics--provided that the perturbation parameter is sufficiently small. The zero-th order term in a power series expansion in the perturbation parameter of such an asymptotic state is given by the asymptotic state of the effective dynamics. We conclude by working out the case in which the small system and the chain element are spins.