Non-Hermitian disorder in two-dimensional optical lattices

A. F. Tzortzakakis, K. G. Makris, and E. N. Economou

Phys. Rev. B 101, 014202 – Published 6 January 2020

Abstract

In this paper, we study the properties of two-dimensional lattices in the presence of non-Hermitian disorder. In the context of coupled mode theory, we consider random gain-loss distributions on every waveguide channel (on site disorder). Our work provides a systematic study of the interplay between disorder and non-Hermiticity. In particular, we study the eigenspectrum in the complex frequency plane and we examine the localization properties of the eigenstates, either by the participation ratio or the level spacing, defined in the complex plane. A modified level distribution function vs disorder seems to fit our computational results.

Received 1 October 2019

DOI:https://doi.org/10.1103/PhysRevB.101.014202

Physics Subject Headings (PhySH)

Anderson localization Localization Optical lattices & traps Photonics

Coupled mode theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

A. F. Tzortzakakis¹, K. G. Makris^1,2, and E. N. Economou^1,2

¹Physics Department, University of Crete, Heraklion, 71003, Greece
²Institute of Electronic Structure and Laser, FORTH, 71110 Heraklion, Crete, Greece

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Vol. 101, Iss. 1 — 1 January 2020

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Images

Figure 1
Eigenvalue spectra of a $50 \times 50$ waveguide lattice (2500 sites) in the complex frequency plane, for a particular realization of the random system and for various disorder strengths. In the left column, disorder is applied only to the imaginary part of the potential strength, while $Re (n_{p, q}) = 0$ ; in the right column, disorder of the same $W$ is applied in both the real and the imaginary part of $n_{p, q}$ . Note that the color of each eigenvalue is related to the participation ratio (spatial extension) of the corresponding eigenstate, as seen in the color bar of the figures.
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Figure 2
Density of states per unit area in the complex plane $Real (ω$ )- $Imag (ω$ ), for the case of disorder in both the real and the imaginary part of the diagonal matrix elements and disorder strength $W = 3$ . We have averaged over 1000 realizations of the disorder.
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Figure 3
Level spacing $S$ plotted along the real axis (a) and the imaginary axis (b) of the complex frequency plane $Real (ω$ )- $Imag (ω$ ) and for the case of disorder in both the real and the imaginary part of the diagonal matrix elements and disorder strength $W = 3$ . The blue dotted line represents results of calculations using the definition (9) of level spacing ( $S_{1}$ ), while the black line represents results using definition (10) ( $S_{2}$ ). An average over 1000 realization of disorder is performed in each case.
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Figure 4
Mean extent length $λ$ , averaged over the whole spectrum, as a function of the disorder strength W, for a lattice of $50 \times 50$ waveguides; uncorrelated disorder for all the three cases: Disorder only in the imaginary part of the diagonal matrix elements (blue dotted line), disorder only in the real part (red dash-dot line), and disorder of the same amplitude in both the real and the imaginary part (black dashed line).
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Figure 5
Probability density P of normalized level spacings S, as defined by Eq. (10), averaging over the whole spectrum. In this plot, disorder is applied in both the real and the imaginary part of the potential. The results for three values of disorder strength are shown: $W = 1$ (black dotted line), $W = 3$ (blue dashed line), and $W = 5$ (red dash-dot line). An ensemble of 50 realizations of disorder is used in each case.
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Figure 6
Probability densities $P$ of normalized level spacings $S$ , as defined in Eq. (10), averaged over the whole spectrum for disorder strength $W = 1$ and for the same size. (a) $P$ for a full complex matrix with random elements (blue) compared with Ginibre (red) and sub-Wigner (black). (b) $P$ for our model (blue) compared with Ginibre (red) and sub-Wigner (black). An ensemble of 50 realizations of disorder is used in each case.
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