Eigenvalues bifurcating from the continuum in two-dimensional potentials generating non-Hermitian gauge fields

Abstract

It has been recently shown that complex two-dimensional (2D) potentials $V_{\epsilon }(x,y)=V(y+\mathrm{i}\epsilon \eta (x))$ can be used to emulate non-Hermitian matrix gauge fields in optical waveguides. Here $x$ and $y$ are the transverse coordinates, $V(y)$ and $\eta (x)$ are real functions, $\epsilon >0$ is a small parameter, and $\mathrm{i}$ is the imaginary unit. The real potential $V(y)$ is required to have at least two discrete eigenvalues in the corresponding 1D Schrödinger operator. When both transverse directions are taken into account, these eigenvalues become thresholds embedded in the continuous spectrum of the 2D operator. Small nonzero $\epsilon$ corresponds to a non-Hermitian perturbation which can result in a bifurcation of each threshold into an eigenvalue. Accurate analysis of these eigenvalues is important for understanding the behavior and stability of optical waves propagating in the artificial non-Hermitian gauge potential. Bifurcations of complex eigenvalues out of the continuum is the main object of the present study. We obtain simple asymptotic expansions in $\epsilon$ that describe the behavior of bifurcating eigenvalues. The lowest threshold can bifurcate into a single eigenvalue, while every other threshold can bifurcate into a pair of complex eigenvalues. These bifurcations can be controlled by the Fourier transform of function $\eta (x)$ evaluated at certain isolated points of the reciprocal space. When the bifurcation does not occur, the continuous spectrum of 2D operator contains a quasi-bound-state which is characterized by a strongly localized central peak coupled to small-amplitude but nondecaying tails. The analysis is applied to the case examples of parabolic and double-well potentials $V(y)$. In the latter case, the bifurcation of complex eigenvalues can be dampened if the two wells are widely separated.

24 pages, 6 figures; accepted for Annals of Physics

References in corpus (4)

• Electrically tunable artificial gauge potential for polaritons
• Giant amplification of modes in parity-time symmetric waveguides
• Phase transition through the splitting of self-dual spectral singularity in optical potentials
• New classes of non-parity-time-symmetric optical potentials with all-real spectra and exceptional-point-free phase transition