Transport and spectral features in non-Hermitian open systems

A F Tzortzakakis,A Szameit,K G Makris,E N Economou

doi:10.1103/physrevresearch.3.013208

Abstract

We study the transport and spectral properties of a non-Hermitian one-dimensional disordered lattice, the diagonal matrix elements of which are random complex variables taking both positive (loss) and negative (gain) imaginary values: Their distribution is either the usual rectangular one or a binary pair-correlated one possessing, in its Hermitian version, delocalized states, and unusual transport properties. Contrary to the Hermitian case, all states in our non-Hermitian system are localized. In addition, the eigenvalue spectrum, for the binary pair-correlated case, exhibits an unexpected intricate fractallike structure on the complex plane and with increasing non-Hermitian disorder, the eigenvalues tend to coalesce in particular small areas of the complex plane, a feature termed "eigenvalue condensation". Despite the strong Anderson localization of all eigenstates, the system appears to exhibit transport not by diffusion but by a new mechanism through sudden jumps between states located even at distant sites. This seems to be a general feature of open non-Hermitian random systems. The relation of our findings to recent experimental results is also discussed.

Highlights

Anderson localization raising the possibility of suppression of diffusion in disordered media [1] is a fundamental phenomenon of wave physics and has been extensively studied in both quantum and classical domain [2,3,4,5,6,7,8,9]
In this work we study for the first time the spectral and dynamic properties of one-dimensional (1D) waveguide lattices, which are characterized by non-Hermitian disorder in the diagonal matrix elements n = R,n + i I,n; the imaginary part, I,n, of the latter has either a rectangular distribution, defined in Eq (2a), usually centered around zero with a total width equal to 2W
The most interesting and surprising among our findings is the new kind of apparent transport in spite of strong localization of the eigenstates not by diffusion but through the sudden jumps even to distant sites; it seems that the nonorthogonality of the eigenmodes and their quite different amplitude play a significant role for what seems to be a new mechanism for transport in open non-Hermitian systems

Summary

INTRODUCTION

Anderson localization raising the possibility of suppression of diffusion in disordered media [1] is a fundamental phenomenon of wave physics and has been extensively studied in both quantum and classical domain [2,3,4,5,6,7,8,9]. Quite recently there has been a renewed interest for nonHermitian Anderson localization problems [49,50,51,52,53,54], since it was realized that in the context of optical physics one can experimentally realize linear random non-Hermitian Hamiltonians, away from the highly nonlinear regime of random lasers and the majority of abstract non-Hermitian random matrices. The proposed complex random discrete models can be considered the most relevant non-Hermitian analog of the Anderson original problem. In this case, the non-Hermiticity is a direct consequence of the complex nature of the index of refraction, whereas the coupling between nearest neighbors is real and fixed. We emphasize that quite different non-Hermitian Anderson models have been previously examined in a number of related theoretical works [56,57,58,59] (see Appendix A)

FORMULATION

LOCALIZATION IN OUR SYSTEMS

NEW KIND OF TRANSPORT BY JUMPS

CONCLUSIONS