Measuring the knot of non-Hermitian degeneracies and non-commuting braids

TL;DR

This paper explores how non-Hermitian systems exhibit complex spectral braiding phenomena when control parameters are looped around degeneracies, extending understanding from two to multiple oscillators and demonstrating experimental results for three.

Contribution

It generalizes the topological analysis of spectral braids from two to arbitrary N oscillators and experimentally verifies these phenomena in a three-oscillator optomechanical system.

Findings

Control loops produce eigenfrequency braids with non-Abelian group structure for N>2

Spectral braiding is experimentally observed in a three-oscillator cavity optomechanical system

Non-trivial topology of degeneracies affects the spectral flow in non-Hermitian systems

Abstract

Any system of coupled oscillators may be characterized by its spectrum of resonance frequencies (or eigenfrequencies), which can be tuned by varying the system's parameters. The relationship between control parameters and the eigenfrequency spectrum is central to a range of applications. However, fundamental aspects of this relationship remain poorly understood. For example, if the controls are varied along a path that returns to its starting point (i.e., around a "loop"), the system's spectrum must return to itself. In systems that are Hermitian (i.e., lossless and reciprocal) this process is trivial, and each resonance frequency returns to its original value. However, in non-Hermitian systems, where the eigenfrequencies are complex, the spectrum may return to itself in a topologically non-trivial manner, a phenomenon known as spectral flow. The spectral flow is determined by how the control loop encircles degeneracies, and this relationship is well understood for $N=2$ (where $N$ is the number of oscillators in the system). Here we extend this description to arbitrary $N$. We show that control loops generically produce braids of eigenfrequencies, and for $N>2$ these braids form a non-Abelian group which reflects the non-trivial geometry of the space of degeneracies. We demonstrate these features experimentally for $N=3$ using a cavity optomechanical system.