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

Simon Malzard,Henning Schomerus,Emiliano Cancellieri

doi:10.1364/oe.26.022506

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.

Highlights

A wide range of topological quantum effects manifest themselves in symmetries of an excitation spectrum
That these states enjoy topological protection can be further ascertained by identifying topological excitations in the stability spectrum, which we find to govern the spectral phase transitions between the different types of symmetry-protected solutions—corresponding to different topological phases of the dynamical system
We showed that spectral symmetries underpinning topological quantum systems can be extended to nonlinear complex-wave equations, where they lead to robust constraints of the dynamics

Summary

Introduction

A wide range of topological quantum effects manifest themselves in symmetries of an excitation spectrum. Our general strategy is as follows: Topological states in linear systems are protected by symmetry, but their number can change discretely in phase transitions, which are generally linked to degeneracies (such as when a band gap closes) This notion is underpinned by the continuity of the spectrum under smooth parameter changes (deformations of the system), a feature at the heart of linear spectral analysis. As a notable feature without analogue in the linear case, the symmetry protects a twisted variant of time-dependent power-oscillating states with Ψ(t + T/2) = XΨ∗(t), for which periodicity Ψ(t + T) = Ψ(t) is enforced by symmetry—suggesting that the nonlinear setting admits for a richer notion of protected states than the linear case That these states enjoy topological protection can be further ascertained by identifying topological excitations in the stability spectrum, which we find to govern the spectral phase transitions between the different types of symmetry-protected solutions—corresponding to different topological phases of the dynamical system. We illustrate our findings for two model systems representing one-dimensional topological laser arrays with saturable gain and two-dimensional flat-band polariton condensates with density-dependent loss

Results

Realization in lasers and condensates

Phase transitions and topological features of the excitation spectrum

Detailed derivation of the symmetry-constrained excitation spectrum

Conclusion