Applied Math Seminar: Multiscale analysis and high-order schemes for nonlinear multilevel Maxwell-Bloch equations given by Prof. Qing Xia (Rensselaer Polytechnic Institute)

In this talk, we will present a recent study of the
Maxwell-Bloch equations that model the nonlinear interactions of light and
matter, where the light is modeled classically by the Maxwell’s equations
with dispersions and the medium is modeled quantum-mechanically by the
multilevel rate equations. We will show the connection between rate
equations and the density matrix, where the former formulation is widely
used in the engineering community and the latter in the Physics
literature. A multiscale analysis of the Maxwell-Bloch equations based on
asymptotic expansions will also be discussed. The resulting reduced
envelope equations (or slow equations) capture amplitude dynamics of the
underlying solutions accurately and efficiently. In addition, we will talk
about high-order accurate numerical schemes based on finite difference
approximations in space and modified equation approach in time. The
proposed schemes allow point-wise update of the solutions for both single
domains and domains with material interfaces, thus enabling superb
parallelism for arbitrary geometry. This is joint work with A. V.
Kildishev and L. J. Prokopeva from Purdue’s Birck Nanotechnology Center,
and J. W. Banks, W. D. Henshaw, G. Kovacic and D. W. Schwendeman from RPI.