Brownian motion ensembles and parametric correlations of the transmission eigenvalues: Application to coupled quantum billiards and to disordered wires

TL;DR

This paper investigates the statistical correlations of transmission eigenvalues in quantum systems using two Brownian motion models, providing insights into coupled quantum dots and disordered wires through mathematical analysis.

Contribution

It introduces and compares two Brownian motion ensembles for modeling transmission eigenvalue correlations in quantum scatterers, highlighting their differences and applications.

Findings

Different Brownian motion models yield distinct correlation behaviors.

Mathematical analogies between models clarify physical differences.

Applications to quantum dots and disordered wires demonstrate model relevance.

Abstract

The parametric correlations of the transmission eigenvalues $T_i$ of a $N$-channel quantum scatterer are calculated assuming two different Brownian motion ensembles. The first one is the original ensemble introduced by Dyson and assumes an isotropic diffusion for the $S$-matrix. The second Brownian motion ensemble assumes for the transfer matrix $M$ an isotropic diffusion yielded by a multiplicative combination law. We review the qualitative differences between transmission through two weakly coupled quantum dots and through a disordered line and we discuss the mathematical analogies between the Fokker-Planck equations of the two Brownian motion models.