Transmission eigenvalues and the bare conductance in the crossover to Anderson localization

TL;DR

This study measures the transmission matrix of microwave waves in random waveguides to understand the transition from diffusion to Anderson localization, revealing how eigenvalues relate to conductance.

Contribution

It provides experimental insights into the behavior of transmission eigenvalues and conductance during the crossover to Anderson localization in microwave systems.

Findings

Eigenvalue $ au_1$ approaches unity in diffusive regime.

Average $ au_1$ is nearly equal to g in localized waves.

Spacing between $ ext{ln} au_n$ is constant and related to bare conductance.

Abstract

We measure the field transmission matrix t for microwave radiation propagating through random waveguides in the crossover to Anderson localization. From these measurements, we determine the dimensionless conductance, g, and the individual eigenvalues $\tau_n$ of the transmission matrix $tt^\dagger$ whose sum equals g. In diffusive samples, the highest eigenvalue, $\tau_1$, is close to unity corresponding to a transmission of nearly 100%, while for localized waves, the average of $\tau_1$, is nearly equal to g. We find that the spacing between average values of $\ln\tau_n$ is constant and demonstrate that when surface interactions are taken into account it is equal to the inverse of the bare conductance.