# Topological dynamics and excitations in lasers and condensates with saturable gain or loss

## TL;DR

This paper classifies and analyzes symmetry-protected and symmetry-breaking dynamical solutions in nonlinear bosonic systems with non-hermitian symmetry, revealing topological modes and phase transitions relevant to lasers and polariton condensates.

## Contribution

It introduces a classification of nonlinear topological solutions in bosonic systems with saturable gain or loss, highlighting new stable modes and phase transition phenomena.

## Key findings

- Existence of stable symmetry-protected modes in nonlinear systems
- Identification of symmetry-protected power-oscillations without linear counterparts
- Topological phase transitions characterized by bifurcations in nonlinear regimes

## Abstract

We classify symmetry-protected and symmetry-breaking dynamical solutions for nonlinear saturable bosonic systems that display a non-hermitian charge-conjugation symmetry, as realized in a series of recent groundbreaking experiments with lasers and exciton polaritons. In particular, we show that these systems support stable symmetry-protected modes that mirror the concept of zero-modes in topological quantum systems, as well as symmetry-protected power-oscillations with no counterpart in the linear case. In analogy to topological phases in linear systems, the number and nature of symmetry-protected solutions can change. The spectral degeneracies signalling phase transitions in linear counterparts extend to bifurcations in the nonlinear context. As bifurcations relate to qualitative changes in the linear stability against changes of the initial conditions, the symmetry-protected solutions and phase transitions can also be characterized by topological excitations, which set them apart from symmetry-breaking solutions. The stipulated symmetry appears naturally when one introduces nonlinear gain or loss into spectrally symmetric bosonic systems, as we illustrate for one-dimensional topological laser arrays with saturable gain and two-dimensional flat-band polariton condensates with density-dependent loss.