Abstract
The machine learning technique of persistent homology classifies complex systems or datasets by computing their topological features over a range of characteristic scales. There is growing interest in applying persistent homology to characterize physical systems such as spin models and multiqubit entangled states. Here, we propose persistent homology as a tool for characterizing and optimizing band structures of periodic photonic media. Using the honeycomb photonic lattice Haldane model as an example, we show how persistent homology is able to reliably classify a variety of band structures falling outside the usual paradigms of topological band theory, including “moat band” and multi-valley dispersion relations, and thereby control the properties of quantum emitters embedded in the lattice. The method is promising for the automated design of more complex systems such as photonic crystals and Moiré superlattices.
Highlights
Given the high degree of control and flexibility we have over the design of photonic band structures and synthetic gauge fields, with many different degrees of freedom, a natural question that arises is whether there exist interesting and useful topological phenomena falling outside the well-established paradigm of topological band theory, which is concerned with the “shape” of Bloch wave eigenstates, i.e., whether they exhibit vortices or gauge discontinuities within the Brillouin zone, independent of the energy dispersion
Our aim in this study is to introduce persistent homology to photonics and show how it may be useful for optimization of synthetic gauge fields and discovery of novel classes of photonic band structures, which merit more detailed studies
II, we provide a brief introduction to the technique of persistent homology and discuss how it may be applied to band structure optimization by using suitable distance metrics
Summary
There is growing interest in the design of synthetic gauge fields and topological band structures for light, motivated both by their exotic behavior and potential applications as disorderrobust photonic devices. The most common approach has been to emulate well-known topological phases from condensed matter physics. recent advances in synthetic dimensions in photonic systems provide platforms for observing, for the first time, topological effects in higher dimensions. Given the high degree of control and flexibility we have over the design of photonic band structures and synthetic gauge fields, with many different degrees of freedom, a natural question that arises is whether there exist interesting and useful topological phenomena falling outside the well-established paradigm of topological band theory, which is concerned with the “shape” of Bloch wave eigenstates, i.e., whether they exhibit vortices or gauge discontinuities within the Brillouin zone, independent of the energy dispersion.Controlling the shape of the energy dispersion landscape can be important. There is growing interest in the design of synthetic gauge fields and topological band structures for light, motivated both by their exotic behavior and potential applications as disorderrobust photonic devices.. The shape of a medium’s isofrequency contours or surfaces dictates the far-field radiation profile of localized emitters, and flat dispersion relations are highly desirable for realizing strong light–matter interactions.. The shape of a medium’s isofrequency contours or surfaces dictates the far-field radiation profile of localized emitters, and flat dispersion relations are highly desirable for realizing strong light–matter interactions.14 Both of these examples are based on the band structure at a specific energy The shape of a medium’s isofrequency contours or surfaces dictates the far-field radiation profile of localized emitters, and flat dispersion relations are highly desirable for realizing strong light–matter interactions. Both of these examples are based on the band structure at a specific energy
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