$\mathcal{P}\mathcal{T}$-Symmetric Wave Chaos

We study a new class of chaotic systems with dynamical localization, where gain or loss mechanisms break the Hermiticity, while allowing for parity-time ($\mathcal{P}\mathcal{T}$) symmetry. For a value ${\gamma }_{\mathcal{P}\mathcal{T}}$ of the gain or loss parameter the spectrum undergoes a spontaneous phase transition from real (exact phase) to complex values (broken phase). We develop a one parameter scaling theory for ${\gamma }_{\mathcal{P}\mathcal{T}}$, and show that chaos assists the exact $\mathcal{P}\mathcal{T}$ phase. Our results have applications to the design of optical elements with $\mathcal{P}\mathcal{T}$ symmetry.

Figure 1

Scaling behavior of ${\gamma }_{\mathcal{P}\mathcal{T}}$ for the $\mathcal{P}\mathcal{T}\mathrm{KR}$ defined by Eq. (3) for various $M$, and $K_0>5$ with $N=63$ (pink triangles), 127 (red circles), 225 (green squares), and 511 (blue diamonds). All data show a nice collapse to the theoretical prediction in Eq. (6) (dashed black line). Inset: Parametric evolution of eigenvalue magnitudes for chaos (DL) in the upper (lower) figure with $N=127$, and $\Lambda =0.05$ (10). The first branching pair is responsible for the transition to broken $\mathcal{P}\mathcal{T}$ phase, and in both cases follows the predicted functional form (see text) shown with blue circles.

Figure 2

(a) Distributions of ${\Delta }_{\min }$ for localized eigenfunctions displaying a convergence to log-normal behavior. Centers are shifted for ease of comparison. Inset: Linear relation between ${\Delta }_{\min }$ and ${\gamma }_{\mathrm{PT}}$ over roughly 12 orders of magnitude. Parameters and symbols are the same as in Fig. 1. (b) Distribution of minimum level spacings in the chaotic regime in which Wignerian behavior (blue line) is observed. Inset: The integrated distribution $I({\Delta }_{\min })={\int }_0^{{\Delta }_{\min }}\mathcal{P}(x)dx$ in a log-log plot. The best fit (blue line) has power two. In both cases, $N=127$.