Random-matrix theory of amplifying and absorbing resonators with PT or PTT' symmetry

TL;DR

This paper develops random-matrix models for coupled amplifying and absorbing resonators with PT or PTT' symmetry, analyzing eigenvalue distributions and transitions between real and complex eigenvalues.

Contribution

It introduces Gaussian and circular random-matrix models for PT and PTT' symmetric systems, providing eigenvalue distributions and transition behaviors.

Findings

Fraction of real eigenvalues vanishes with fixed classical scales.

Transition from real to complex eigenvalues depends on coupling strength.

Differences in eigenvalue transitions relate to level spacing statistics.

Abstract

We formulate gaussian and circular random-matrix models representing a coupled system consisting of an absorbing and an amplifying resonator, which are mutually related by a generalized time-reversal symmetry. Motivated by optical realizations of such systems we consider a PT or a PTT' time-reversal symmetry, which impose different constraints on magneto-optical effects, and then focus on five common settings. For each of these, we determine the eigenvalue distribution in the complex plane in the short-wavelength limit, which reveals that the fraction of real eigenvalues among all eigenvalues in the spectrum vanishes if all classical scales are kept fixed. Numerically, we find that the transition from real to complex eigenvalues in the various ensembles display a different dependence on the coupling strength between the two resonators. These differences can be linked to the level spacing statistics in the hermitian limit of the considered models.