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## Optical waveguide arrays: quantum effects and PT symmetry breaking

Yogesh N. Joglekar, Clinton Thompson, Derek D. Scott, Gautam Vemuri

Over the last two decades, advances in fabrication have led to significant progress in creating patterned heterostructures that support either carriers, such as electrons or holes, with specific band structure or electromagnetic waves with a given mode structure and dispersion. In this article, we review the properties of light in coupled optical waveguides that support specific energy spectra, with or without the effects of disorder, that are well-described by a Hermitian tight-binding model. We show that with a judicious choice of the initial wave packet, this system displays the characteristics of a quantum particle, including transverse photonic transport and localization, and that of a classical particle. We extend the analysis to non-Hermitian, parity and time-reversal ($$\mathcal{PT}$$) symmetric Hamiltonians which physically represent waveguide arrays with spatially separated, balanced absorption or amplification. We show that coupled waveguides are an ideal candidate to simulate $$\mathcal{PT}$$-symmetric Hamiltonians and the transition from a purely real energy spectrum to a spectrum with complex conjugate eigenvalues that occurs in them.

http://arxiv.org/abs/1305.3565
Optics (physics.optics); Disordered Systems and Neural Networks (cond-mat.dis-nn); Quantum Physics (quant-ph)

## Determination of S-curves with applications to the theory of nonhermitian orthogonal polynomials

Gabriel Álvarez, Luis Martinez Alonso, Elena Medina

This paper deals with the determination of the S-curves in the theory of non-hermitian orthogonal polynomials with respect to exponential weights along suitable paths in the complex plane. It is known that the corresponding complex equilibrium potential can be written as a combination of Abelian integrals on a suitable Riemann surface whose branch points can be taken as the main parameters of the problem. Equations for these branch points can be written in terms of periods of Abelian differentials and are known in several equivalent forms. We select one of these forms and use a combination of analytic an numerical methods to investigate the phase structure of asymptotic zero densities of orthogonal polynomials and of asymptotic eigenvalue densities of random matrix models. As an application we give a complete description of the phases and critical processes of the standard cubic model.

http://arxiv.org/abs/1305.3028
Mathematical Physics (math-ph)