Non-Hermitian off-diagonal disordered optical lattices

TL;DR

This work investigates 1D and 2D optical lattices with off-diagonal disorder in the non-Hermitian regime, focusing on spectral properties, localization of eigenmodes, and single-channel transport. It identifies three coupling scenarios (Hermitian, real non-Hermitian, and complex non-Hermitian) and reveals that real non-Hermitian systems are spectrally real under open boundaries due to a similarity transform, while complex non-Hermitian systems exhibit genuinely complex spectra and unconventional transport. The study uncovers unconventional phenomena such as abrupt Anderson jumps in real-spectrum 1D systems and, for 2D, jumps induced by complex spectra, highlighting the role of chiral symmetry and non-Hermitian topology in localization. The results provide a reference framework for non-Hermitian off-diagonal disorder and point to future work on mobility edges, scaling, and experimental realizations in photonic lattices.

Abstract

Within the framework of non-Hermitian photonics, we investigate the spectral and dynamical properties of one- and two-dimensional non-Hermitian off-diagonal disordered optical lattices, where randomness is applied to the couplings rather than to the on-site potential terms. We analyze eigenvalue distributions and the localization properties of the eigenmodes, comparing them with those of the corresponding Hermitian lattices. Furthermore, we study their transport behavior under single-channel excitation and identify unconventional phenomena such as jumps between distant lattice regions in systems with a purely real spectrum, as well as complex spectrum-induced Anderson jumps, reported here for the first time in two dimensions. Our results establish a reference framework for non-Hermitian off-diagonal disorder and open new directions for future studies of localization phenomena.

Figures (11)

• Figure 1: Schematic representation of the one-dimensional lattice models considered in this work. (a) Random nearest-neighbor couplings are identical in the forward and backward directions, leading to a Hermitian Hamiltonian. (b) Random nearest-neighbor couplings differ between forward and backward directions, giving rise to a non-Hermitian Hamiltonian. In both cases, variations in the color and width of the arrows indicate different coupling strengths, while in case (b) the couplings may be either real or complex.
• Figure 2: Density of states for 1D lattices ($N=500$ sites): (a)–(c) Comparison of the DOS of real eigenvalues, $\rho(\epsilon)$, between the Hermitian (blue-shaded surface) and real non-Hermitian (black line) cases for (a) $W=0.75$, (b) $W=1.25$, and (c) $W=2$. (d)–(f) DOS in the complex plane (colormap) for the complex non-Hermitian case. Insets in bottom row panels show the real-projected DOS $\tilde{\rho}_R$ (blue lines) and the imaginary-projected DOS $\tilde{\rho}_I$ (red lines) for (d) $W=0.75$, (e) $W=1.25$, and (f) $W=2$.
• Figure 3: Averaged participation ratios for 1D lattices($N=500$ sites): (a)–(c) Comparison of $\overline{P}(\epsilon)$ for eigenmodes in the Hermitian (blue-shaded surface) and real non-Hermitian (black line) cases for (a) $W=0.75$, (b) $W=1.25$, and (c) $W=2$. (d)–(f) $\overline{P}(\epsilon)$ in the complex plane (colormap) for the complex non-Hermitian case with (d) $W=0.75$, (e) $W=1.25$, and (f) $W=2$.
• Figure 4: Propagation dynamics for 1D lattices ($N=500$ sites). Left panels: Evolution of the normalized wavefunction, initially localized at site $n=250$, for weak/strong disorder realizations ($W=0.75$ / $W=2$) in the (a)/(e) Hermitian and (c)/(g) real non-Hermitian cases. Right panels: For each lattice site $n$, the bar height gives the number $\mu_n$ of right eigenstates whose center of mass lies within the corresponding spatial bar. The color encodes the averaged projection coefficients $\langle |c_{j,0}|\rangle$ of the eigenmodes contained in each bin, for weak/strong disorder realizations ($W=0.75$ / $W=2$) in the (b)/(f) Hermitian and (d)/(h) real non-Hermitian cases.
• Figure 5: Evolution of optical power: (a) Comparison of the optical power $\mathcal{P}(z)$ for a single realization of weak disorder in the Hermitian case (black dotted line) and the real non-Hermitian case (black solid line). The evolution of the pseudopower$\tilde{\mathcal{P}}$ [defined in Eq. (13)] for the latter case is also shown (blue solid line), shown to the right axis. (b) Same as in (a), but for single realizations of strong disorder ($W=2$). The Hermitian / non-Hermitian realizations used in panels (a) and (b) are the same as those in Figs. 4(a)/(c) and 4(e)/(g), respectively.