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

The $PT$ symmetric potential $V_0[\cos (2\pi x\mathrm{/}a)+i\lambda \sin (2\pi x\mathrm{/}a)]$ 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.