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

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.

Supplementary Videos S1-S5 are included as ancillary files, or see http://www.youtube.com/watch?v=2HsIo1OgZFA&list=PL3IrfT-_t1GQvk8LYYyXvdFkp8ymv1Slc ; corrected numbering of references in SI; reorganized SI

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• Field Theory of Many-Body Lindbladian Dynamics
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• Homotopy, Symmetry, and Non-Hermitian Band Topology
• Braid Protected Topological Band Structures with Unpaired Exceptional Points
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• Unsupervised Learning of Topological Non-Abelian Braiding in Non-Hermitian Bands
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• Third-order exceptional point in an ion-cavity system
• Braids and Higher-order Exceptional Points from the Interplay Between Lossy Defects and Topological Boundary States
• Implementing non-Abelian Hatano-Nelson model in electric circuits
• Gain-loss-induced non-Abelian Bloch braids
• Non-Bloch Theory for Spatiotemporal Photonic Crystals Assisted by Continuum Effective Medium
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• Realization of geometric phase topology induced by multiple exceptional points
• Dynamically encircling an exceptional point through phase-tracked closed-loop control
• Observation of Braid-Protected Unpaired Exceptional Points
• Topological eigenvalues braiding and quantum state transfer near a third-order exceptional point
• Topological Operations Around Exceptional Points via Shortcuts to Adiabaticity
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• Exceptional Non-Hermitian Topology Associated with Non-Toroidal Brillouin Zones
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• Non-hermitian topology and entanglement in an optomechanical superlattice
• Complex energy structures of exceptional point pairs in two level systems
• Entanglement manifestation of knot topology in a non-Hermitian lattice
• Machine learning topological energy braiding of non-Bloch bands
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• Ergodic Hysteresis of the Kerr black hole spectrum
• Emergence of Nodal-Knot Transitions by Disorder
• Exceptional topology on nonorientable manifolds
• Braiding of dynamical eigenvalues of Hermitian bosonic Kitaev chains
• Non-orientable Exceptional Points in Twisted Boundary Systems
• Higher-order exceptional lines in a non-Hermitian JaynesCummings triangle
• Emergence of Hermitian topology from non-Hermitian knots