PT-Symmetric Sinusoidal Optical Lattices at the Symmetry-Breaking Threshold

TL;DR

This paper investigates the behavior of PT-symmetric sinusoidal optical lattices at the symmetry-breaking threshold, revealing how wave amplitude saturation occurs due to the interplay of Jordan functions and Bloch waves.

Contribution

It explicitly constructs Bloch wave-functions and Jordan functions, providing detailed insight into wave amplitude saturation at the symmetry-breaking point.

Findings

Wave amplitude saturation is explained by the interaction of Jordan functions and Bloch waves.

Explicit construction of eigenfunctions clarifies the mechanism behind amplitude stabilization.

The study confirms previous numerical results with analytical detail.

Abstract

The $PT$ symmetric potential $V_0[\cos(2\pi x/a)+i\lambda\sin(2\pi x/a)]$ has a completely real spectrum for $\lambda\le 1$, and begins to develop complex eigenvalues for $\lambda>1$. At the symmetry-breaking threshold $\lambda=1$ some of the eigenvectors become degenerate, giving rise to a Jordan-block structure for each degenerate eigenvector. In general this is expected to result in a secular growth in the amplitude of the wave. However, it has been shown in a recent paper by Longhi, by numerical simulation and by the use of perturbation theory, that for a broad initial wave packet this growth is suppressed, and instead a saturation leading to a constant maximum amplitude is observed. We revisit this problem by explicitly constructing the Bloch wave-functions and the associated Jordan functions and using the method of stationary states to find the dependence on the longitudinal distance $z$ for a variety of different initial wave packets. This allows us to show in detail how the saturation of the linear growth arises from the close connection between the contributions of the Jordan functions and those of the neighbouring Bloch waves.