# Photonic Topological Insulators (17w5144)

Arriving in Banff, Alberta Sunday, September 10 and departing Friday September 15, 2017

## Organizers

Mikael Rechtsman (Pennsylvania State University)

Michael Weinstein (Columbia University)

Marin Soljacic (Massachusetts Institute of Technology)

## Objectives

The goal of the proposed workshop is to bring together the leading researchers in the fields of photonic topological insulators and mathematical research in topology, analysis, and partial differential equations, in order to stimulate new insights and advances in the field of topological photonics, a field which spans questions of deep interest to the mathematics community, as well as to the applied and fundamental physics. We will also aim to bring together graduate students from these usually disparate fields in a highly interdisciplinary setting. This will enable mathematics students to learn about the possibilities and limitations of available experimental techniques, and will spur conversations about known theory that could lead to new experiments. There will be a large tutorial component to the workshop to better facilitate understanding and discussion between fields.

The organizers have made significant contributions in complementary realms of photonic topological materials: (1) the group of Marin Soljacic made the first observation of the topological protection of photons in the microwave regime; (2) Mikael Rechtsman made the first observation of topological protection of light (photons at optical frequencies); and (3) Michael Weinstein and collaborators have pioneered the rigorous mathematical investigation of the origins of topological insulation and protection (with an emphasis on graphene-like structures).

We expect the workshop to be very productive due to the large overlap between research interests of the invitees, spanning both the mathematics and physics communities. Examples include the work of Michael Weinstein (Columbia Mathematics / Applied Mathematics) and Charles Fefferman (Princeton Mathematics), who work on mathematical aspects of “photonic graphene,” which has been implemented experimentally by Mikael Rechtsman (Penn State Physics) and Marin Soljacic (MIT Physics). Jeremy Marzuola (UNC Mathematics) and Maciej Zworski (Caltech Mathematics) have worked extensively on the nonlinear Schrödinger equation in complex potentials, an essential ingredient for understanding the nonlinear optics of topological structures. Braxton Osting (Utah Mathematics) has worked on optimized optical resonator structures, the basis for the work of Mohammad Hafezi (University of Maryland Electrical Engineering) on silicon photonic topological insulators. Indeed, the potential for sparking new collaborations (that would not have otherwise taken place) is great.

There is a large and growing community of physicists and mathematicians (as well as electrical engineers) that work on topological photonics. One piece of evidence for this is our list of invitees (many of whom have confirmed to us that they will be attending if the workshop should take place). Invitees come from leading theoretical physics groups, theoretical electrical engineering groups, leading experimental groups in both of these fields, as well as of course leading mathematicians. At major physics and engineering conferences there are already subsections and many talks devoted to Topological Photonics (for example, at: CLEO – the conference on lasers and electro-optics; FiO – Frontiers in Optics; the March Meeting of the American Physical Society, the meeting of the Division of Atomic, Molecular and Optical Physics of the American Physical Society; PQE- Progress in Quantum Electronics, among others). From the point of view of mathematicians, there is significant and growing interest, as evidenced by meetings being planned within the next year, for example: “Mathematical aspects of topological protection,” at Columbia (Spring, 2016 – planning in progress), and at the “Special Year in Mathematics and Optics” at IMA / University of Minnesota, where there will be workshops will be devoted to “novel optical materials”, such as photonic topological insulators, and nonlinear optics. Hence, the fall of 2017 is a natural timeframe for a workshop on this topic at Banff.

Specific questions to be addressed at the workshop:

i) Most understanding of topological photonics is in the regime of linear propagation. However, a very important direction of applications is nonlinear optical devices (e.g., parametric amplifiers and oscillators, supercontinuum sources, complex lasers), which exploit such meta-materials. These require description using the nonlinear dispersive equations, such as the nonlinear Schrodinger / Gross-Pitaevskii equation. What is the interplay between nonlinearity and photonic topological behavior of the underlying linear metamaterial? What kinds of nonlinear wave structures are persistent when the background microstructure is a linear topological insulator?

ii) What are the rigorous underpinnings of the generalized bulk-boundary correspondence? It is strongly-held common wisdom that a topological interface state is always present between regions of different topological invariants. Is this always true? Can this be proved/disproved for more generalized topological systems?

iii) What are the limitations on current experimental systems, and can we find new ways to achieve protected topological insulators that have stronger protection? Given the limitations of experiments, what are the minimum necessary conditions for achieving topological protection?

The organizers have made significant contributions in complementary realms of photonic topological materials: (1) the group of Marin Soljacic made the first observation of the topological protection of photons in the microwave regime; (2) Mikael Rechtsman made the first observation of topological protection of light (photons at optical frequencies); and (3) Michael Weinstein and collaborators have pioneered the rigorous mathematical investigation of the origins of topological insulation and protection (with an emphasis on graphene-like structures).

We expect the workshop to be very productive due to the large overlap between research interests of the invitees, spanning both the mathematics and physics communities. Examples include the work of Michael Weinstein (Columbia Mathematics / Applied Mathematics) and Charles Fefferman (Princeton Mathematics), who work on mathematical aspects of “photonic graphene,” which has been implemented experimentally by Mikael Rechtsman (Penn State Physics) and Marin Soljacic (MIT Physics). Jeremy Marzuola (UNC Mathematics) and Maciej Zworski (Caltech Mathematics) have worked extensively on the nonlinear Schrödinger equation in complex potentials, an essential ingredient for understanding the nonlinear optics of topological structures. Braxton Osting (Utah Mathematics) has worked on optimized optical resonator structures, the basis for the work of Mohammad Hafezi (University of Maryland Electrical Engineering) on silicon photonic topological insulators. Indeed, the potential for sparking new collaborations (that would not have otherwise taken place) is great.

There is a large and growing community of physicists and mathematicians (as well as electrical engineers) that work on topological photonics. One piece of evidence for this is our list of invitees (many of whom have confirmed to us that they will be attending if the workshop should take place). Invitees come from leading theoretical physics groups, theoretical electrical engineering groups, leading experimental groups in both of these fields, as well as of course leading mathematicians. At major physics and engineering conferences there are already subsections and many talks devoted to Topological Photonics (for example, at: CLEO – the conference on lasers and electro-optics; FiO – Frontiers in Optics; the March Meeting of the American Physical Society, the meeting of the Division of Atomic, Molecular and Optical Physics of the American Physical Society; PQE- Progress in Quantum Electronics, among others). From the point of view of mathematicians, there is significant and growing interest, as evidenced by meetings being planned within the next year, for example: “Mathematical aspects of topological protection,” at Columbia (Spring, 2016 – planning in progress), and at the “Special Year in Mathematics and Optics” at IMA / University of Minnesota, where there will be workshops will be devoted to “novel optical materials”, such as photonic topological insulators, and nonlinear optics. Hence, the fall of 2017 is a natural timeframe for a workshop on this topic at Banff.

Specific questions to be addressed at the workshop:

i) Most understanding of topological photonics is in the regime of linear propagation. However, a very important direction of applications is nonlinear optical devices (e.g., parametric amplifiers and oscillators, supercontinuum sources, complex lasers), which exploit such meta-materials. These require description using the nonlinear dispersive equations, such as the nonlinear Schrodinger / Gross-Pitaevskii equation. What is the interplay between nonlinearity and photonic topological behavior of the underlying linear metamaterial? What kinds of nonlinear wave structures are persistent when the background microstructure is a linear topological insulator?

ii) What are the rigorous underpinnings of the generalized bulk-boundary correspondence? It is strongly-held common wisdom that a topological interface state is always present between regions of different topological invariants. Is this always true? Can this be proved/disproved for more generalized topological systems?

iii) What are the limitations on current experimental systems, and can we find new ways to achieve protected topological insulators that have stronger protection? Given the limitations of experiments, what are the minimum necessary conditions for achieving topological protection?