[Paper Review] Random matrix approaches to open quantum systems

Henning Schomerus|arXiv (Cornell University)|Oct 18, 2016

Topological and Geometric Data Analysis4 citations

TL;DR

This paper presents a comprehensive review of random matrix theory applications to open quantum systems, focusing on scattering, decay, and transport in disordered or chaotic systems. It establishes connections between universal statistical behaviors—such as scattering poles, delay times, and transport properties—and underlying ensembles like Wigner-Dyson, Wishart-Laguerre, and Ginibre, demonstrating how symmetries and non-hermitian dynamics govern system-wide observables.

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.

Motivation & Objective

• To provide a unified framework linking random matrix ensembles to physical observables in open quantum systems.
• To clarify the interplay between fundamental symmetries (time-reversal, chiral, charge-conjugation) and statistical behavior in scattering and transport.
• To explain how universal predictions emerge from effective random matrix models despite complex underlying dynamics.
• To explore connections between ergodicity, thermalization, and localization in many-body and disordered systems.
• To present analytical tools—such as non-crossing approximations and Green’s function techniques—for computing eigenvalue densities and scattering properties.

Proposed method

• Uses Gaussian and Wishart-Laguerre ensembles to model random Hamiltonians and scattering matrices in systems with and without time-reversal symmetry.
• Applies the non-crossing approximation to compute averaged Green’s functions and eigenvalue densities for non-hermitian matrices.
• Employs block matrix formalism and resolvent techniques to derive spectral densities for scattering systems with random impurities.
• Utilizes free probability and Blue function composition to solve for eigenvalue distributions in complex Ginibre ensembles.
• Derives the Petermann factor and scattering pole distribution via trace relations in non-hermitian random matrix models.
• Applies the stroboscopic and continuous-time scattering approaches to connect microscopic models to macroscopic transport and decay observables.

Experimental results

Research questions

• RQ1 How do the ten Wigner-Dyson universality classes emerge from symmetries in open quantum systems?
• RQ2 What is the statistical distribution of scattering poles and delay times in random scattering matrices?
• RQ3 How do non-hermitian random matrix ensembles describe the spectral properties of open quantum systems with gain or loss?
• RQ4 What is the role of the Petermann factor in non-orthogonal mode dynamics and its relation to eigenvalue density?
• RQ5 How does ergodicity break down in many-body systems, and what role does localization play in this transition?

Key findings

• The eigenvalue density for non-hermitian Ginibre matrices follows the circular law: ρ(z) = M/π for |z| < 1, with a uniform distribution in the complex plane.
• The average Petermann factor within the spectrum is given by O(z)/ρ(z) = M(1 - |z|²), directly linking non-orthogonality to spectral radius.
• For Wishart-Laguerre ensembles, the eigenvalue density is ρ(λ) = M(c_x - 1)√[(λ - λ₋)(λ₊ - λ)] / [2πλ(1 + λ)], with λ± determined by M_x, M_y, and M.
• The non-crossing approximation yields self-consistent equations for the Green’s function, enabling analytical computation of spectral densities.
• The scattering matrix in random systems leads to universal delay time and transmission statistics governed by the underlying symmetry class.
• Non-orthogonal modes in non-hermitian systems result in enhanced sensitivity and non-universal transport, quantified by the Petermann factor.