Perturbation Theory for PT-Symmetric Sinusoidal Optical Lattices at the Symmetry-Breaking Threshold

TL;DR

This paper develops perturbation theory for PT-symmetric sinusoidal optical lattices at the symmetry-breaking threshold, confirming previous findings of amplitude suppression and revealing birefringence phenomena.

Contribution

It extends perturbation theory analysis to second order for PT-symmetric optical lattices at the threshold, confirming amplitude behavior and uncovering birefringence effects.

Findings

Amplitude growth is suppressed at the threshold.

Second-order perturbation theory confirms previous results.

Birefringence phenomena are identified in wave packet evolution.

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 give rise to 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 an initial wave packet this growth is suppressed, giving instead a constant maximum amplitude. We revisit this problem by developing the perturbation theory further. We verify that the results found by Longhi persist to second order, and with different input wave packets we are able to see the seeds in perturbation theory of the phenomenon of birefringence first discovered by Makris et al.