PT-symmetry enabled stable modes in multi-core fiber

TL;DR

The study addresses PT-symmetry breaking in a two-dimensional bounded continuum by modeling a disk with two parity-symmetric gain/loss disks. It formulates the problem as the eigenproblem $\mathcal{A}=-\Delta+V(x)$ on $\Omega$ with hard-wall boundaries and piecewise imaginary potential, solving it with interface-conforming finite elements and the FEAST solver to track eigenvalues as the gain–loss strength $\gamma$ varies. Key findings include multiple PT-breaking and PT-restoring transitions, a non-monotonic dependence of the breaking threshold $γ_{PT}$ on the disk separation $d$, and an inverse scaling with disk radius $\rho$, indicating that stable propagating modes can exist in a multi-core fiber with distributed gain and loss. The results suggest threshold engineering opportunities in bounded two-dimensional PT-symmetric photonic systems and provide a computational framework for analyzing complex PT dynamics in non-symmetric 2D domains.

Abstract

Open systems with balanced gain and loss, described by parity-time PT-symmetric Hamiltonians have been deeply explored over the past decade. Most explorations are limited to finite discrete models (in real or reciprocal spaces) or continuum problems in one dimension. As a result, these models do not leverage the complexity and variability of two-dimensional continuum problems on a compact support. Here, we investigate eigenvalues of the Schrodinger equation on a disk with zero boundary condition, in the presence of constant, PT-symmetric, gain-loss potential that is confined to two mirror-symmetric disks. We find a rich variety of exceptional points, re-entrant PT-symmetric phases, and a non-monotonic dependence of the PT-symmetry breaking threshold on the system parameters. By comparing results of two model variations, we show that this simple model of a multi-core fiber supports propagating modes in the presence of gain and loss.

Figures (5)

• Figure 1: Schematic cross-section of cylindrical, multi-core fiber. The radius $R=1$ sets the length-scale. The pink region $D_R$, centered at $x_1=d/2$ with radius $\rho$ denotes the gain region, and the green region $D_L$, centered at mirror-symmetric point $x_1=-d/2$ with the same radius denotes the loss region. When the gain-loss regions have no real-part for the index-contrast with the rest of the fiber (purple), i.e. $V_0=0$, the eigenmodes are not just confined to the regions $D_L$ and $D_R$.
• Figure 2: Flow of eigenvalues $\lambda_{mp}(\gamma)$ for the first seven eigenvalues. All of them except $m=0$ cases are doubly degenerate at $\gamma=0$ and this degeneracy is lifted with increasing $\gamma$. The first $\mathcal{PT}$-symmetry breaking transition occurs at $\gamma=\gamma_\mathrm{PT}\approx 97$ immediately followed by $\mathcal{PT}$-restoring transition near $\gamma_\mathrm{PT}=102$ (detailed in the second plot). This is followed by a significantly broad $\mathcal{PT}$-broken region, and another small $\mathcal{PT}$-breaking and restoring transition. This re-entrant $\mathcal{PT}$-symmetric phase in a model with single gain-loss parameter is uncommon. These results are independent of the background potential value $V_B=1$, changing which uniformly shifts all eigenvalues $\lambda_n$ while leaving the flow-diagram unchanged.
• Figure 3: Mode-intensity evolution for pair of eigenvalues in Fig. \ref{['fig:ptbreaking']}b that become complex and then real again. The black circles denote the gain-loss regions with $d=0.3$ and $\rho=0.1$. Mode intensities in (a)-(b), at $\gamma=96$, are $\mathcal{PT}$-symmetric. Modes in (c)-(d), at $\gamma=99$, are in the $\mathcal{PT}$-symmetry-broken region: the intensities of the two modes are mirror-images of each other, while each intensity, by itself, shows a broken $\mathcal{PT}$-symmetry. In (e)-(f), gain-loss strength is increased further to $\gamma=102$, the eigenvalues become real again, leading to mode intensities that are individually mirror symmetric.
• Figure 4: Dependence of $\gamma_\mathrm{PT}$, where first complex-conjugate eigenvalues emerge, on the dimensionless radius $\rho$ of the gain-loss domain. (a) At distance $d/R=0.5$, $\gamma_\mathrm{PT}$ varies inversely with the $\rho\leq 2d$. (b) This inverse behavior, expected from the effective-strength model with $\delta$-function gain-loss points, is valid for different values of $d$.
• Figure 5: Dependence of the first threshold $\gamma_\mathrm{PT}$ on the distance $d$ for a fixed radius $\rho=0.1$ of the gain-loss domains. After an initial decay, the threshold is recovered even as the gain and loss regions move farther away from each other.