A Dynamical Theory of Floquet Materials

This project investigates Floquet materials—engineered systems whose properties are dynamically altered using time-periodic forcing, such as shining light on graphene or modulating optical waveguides. This approach allows for reversible control of physical behavior and has gained traction in fields like quantum engineering, photonics, and acoustics. Although experimental studies have observed intriguing phenomena like edge conduction in these materials, the theoretical foundation—especially in continuum models described by partial differential equations—is incomplete and, at times, contradictory. The investigator develops a rigorous mathematical framework for understanding Floquet materials using continuum models, with the goal of explaining fundamental features such as wave localization and energy transport. The project serves the national interest by advancing foundational science in applied mathematics and mathematical physics, and contributing to emerging technologies that rely on wave manipulation. The project supports graduate education at the New Jersey Institute of Technology and promotes collaboration and dissemination of scientific knowledge through scientific workshops and seminars. The investigator studies time-periodic parametric forcing in periodic media, focusing on non-autonomous dispersive partial differential equations (PDE) — specifically Schrodinger and Dirac equations—in contrast to the prevalent use of discrete, tight-binding approximations. The project addresses four core challenges: (1) developing the theory of “effective gaps” for bulk Floquet Hamiltonians in continuum settings; (2) analyzing the long-time behavior and radiation damping of edge modes; (3) establishing new dispersive decay estimates for Floquet Hamiltonians; and (4) exploring the potential for a PDE-based analog of the Floquet topological bulk-edge correspondence. The project integrates tools from spectral theory, asymptotic analysis, homogenization, and infinite-dimensional dynamical systems, offering novel perspectives on Floquet engineering and contributing broadly to the analysis of non-autonomous PDEs. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.