The Department of Applied Mathematics is pleased to host this series of colloquium lectures, funded in part by a generous gift from the Boeing Company. This series will bring to campus prominent applied mathematicians from around the world.
Title: Magnetic fields, Pseudo-magnetic fields and Applications
Abstract: Quantum tunneling is a phenomenon which plays an important role in, for example, physical, chemical and biological processes. Further, the control of tunneling is central to quantum technologies. Its mathematical paradigm is the Schroedinger operator with a symmetric double-well potential.
It has been understood since the early days of quantum mechanics that, in the absence of a magnetic field, the quantum particle’s wave function always tunnels from one well into the neighboring well through a ``classically forbidden'' region. Further, the tunneling-time is related to the reciprocal of the (non-zero but exponentially small) “eigenvalue splitting”.
In the first part of this talk, I'll present new results on quantum tunneling in 2D systems in the presence of a strong, constant and perpendicular magnetic field.
We construct a family of double well potentials containing examples for which the eigenvalue splitting vanishes, and hence quantum tunneling is completely eliminated. I’ll remark on possible implications of this phenomenon. In contrast, for typical double-well potentials, magnetic tunneling does occur and we prove an upper bound on its tunneling time. This is joint work with C.L. Fefferman and J. Shapiro (to appear in PNAS; https://arxiv.org/abs/2412.21100.
In the second part of this talk, I'll turn to an emergent magnetic effect of interest in condensed matter physics and engineered metamatrials. I’ll explain how non-uniform deformations of a wave-propagating medium with honeycomb symmetry (e.g. graphene or a non-magnetic photonic crystal) induce effective (pseudo-) magnetic and electric fields, described by a (homogenized) Dirac Hamiltonian. The theory demonstates that choosing a deformation which corresponds to a constant perpendicular effective magnetic field, induces a ``flat band’’ (zero group velocity) energy spectrum, which has applications to enhancing light-matter interactions and inducing nonlinear effects. Recent experiments, in the setting of photonic crystal slabs, confirm our theory. This is joint work with group of M.C. Rechtsman in Penn State Physics. (Phys. Rev. A 103, 013505 and Nature Photonics 18 580-585 (2024).
Video: Watch the talk on Dropbox here