Nonlinear currents in a ring-shaped waveguide with balanced gain and dissipation

TL;DR

This paper analyzes linear and nonlinear modes in a parity-time symmetric ring waveguide with localized gain and loss, revealing exceptional points, exact solutions, and stability properties relevant to various condensate and optical systems.

Contribution

It introduces a model with symmetric gain and dissipation in a ring waveguide, providing exact solutions and stability analysis for nonlinear modes with current flow and exceptional points.

Findings

System exhibits exceptional points with asymmetric gain-loss placement.

Exact analytical solutions for nonlinear modes with current flow.

Stable currentless and current-carrying nonlinear solutions identified.

Abstract

We describe linear and nonlinear modes in a ring-shaped waveguide with localized gain and dissipation modeled by two Dirac $\delta$ functions located symmetrically. The strengths of the gain and dissipation are equal, i.e., the system obeys the parity-time symmetry. This configuration describes atomic Bose-Einstein condensates with local loading and local elimination of atoms, polaritonic condensates, or optical ring resonators with local pump and absorption. We discuss the linear spectrum of such a system and show if the location of the $\delta$ functions is slightly asymmetric, then the system can be driven through a series of exceptional points by the change of the gain-and-loss coefficient. In the nonlinear case, the system admits solutions with spatially constant and periodic densities which are presented in the exact analytical form. These solutions are supported by the current directed from the gain potential towards the absorbing potential. The system also admits currentless solutions. Stability and bifurcations of nonlinear solutions are also described.