New classes of non-parity-time-symmetric optical potentials with all-real spectra and exceptional-point-free phase transition

Abstract

Paraxial linear propagation of light in an optical waveguide with material gain and loss is governed by a Schrödinger equation with a complex potential. Properties of parity-time-symmetric complex potentials have been heavily studied before. In this article, new classes of non-parity-time-symmetric complex potentials featuring conjugate-pair eigenvalue symmetry in its spectrum are constructed by operator symmetry methods. Due to this eigenvalue symmetry, it is shown that the spectrum of these complex potentials is often all-real. Under parameter tuning in these potentials, phase transition can also occur, where pairs of complex eigenvalues appear in the spectrum. A peculiar feature of the phase transition here is that, the complex eigenvalues may bifurcate out from an interior continuous eigenvalue inside the continuous spectrum, in which case a phase transition takes place without going through an exceptional point. In one spatial dimension, this class of non-parity-time-symmetric complex potentials is of the form $V(x)=h^{\prime }(x)-h^2(x)$, where $h(x)$ is an arbitrary parity-time-symmetric complex function. These potentials in two spatial dimensions are also derived. Diffraction patterns in these complex potentials are further examined, and unidirectional propagation behaviors are demonstrated.

arXiv comment5 pages, 3 figures

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• Spacing gain and absorption in a simple $\mathcal{P}\mathcal{T}$-symmetric model: spectral singularities and ladders of eigenvalues and resonances
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• Stability restoration by asymmetric nonlinear states in non-Hermitian double-well potentials
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