Random matrix approaches to open quantum systems

TL;DR

This paper reviews how random matrix theory models are used to understand open quantum systems, focusing on physical origins, mathematical structures, and key predictions related to scattering, decay, and localization phenomena.

Contribution

It provides a comprehensive overview of random matrix approaches to open quantum systems, linking theoretical models with physical insights and including recent developments in scattering and localization.

Findings

Random matrix models capture typical system behaviors in open quantum systems.

Signatures of random scattering are observed in decay, dynamics, and transport.

Connections between random matrix theory and phenomena like Anderson localization are discussed.

Abstract

Over the past decades, a great body of theoretical and mathematical work has been devoted to random-matrix descriptions of open quantum systems. In these notes, based on lectures delivered at the Les Houches Summer School "Stochastic Processes and Random Matrices" in July 2015, we review the physical origins and mathematical structures of the underlying models, and collect key predictions which give insight into the typical system behaviour. In particular, we aim to give an idea how the different features are interlinked. The notes mainly focus on elastic scattering but also include a short detour to interacting systems, which we motivate by the overarching question of ergodicity. The first chapters introduce general notions from random matrix theory, such as the ten universality classes and ensembles of hermitian, unitary, positive-definite and non-hermitian matrices. We then review microscopic scattering models that form the basis for statistical descriptions, and consider signatures of random scattering in decay, dynamics and transport. The last chapter briefly touches on Anderson localization and localization in interacting systems.