The University of Arizona

Lax Representations of the Maxwell-Bloch Equations for Two-Level Media with Permanent Dipoles

Lax Representations of the Maxwell-Bloch Equations for Two-Level Media with Permanent Dipoles

Series: PDE Seminar
Location: Math 402
Presenter: Scott Glasgow, Arizona Center for Mathematical Sciences, University of Arizona

We generalize the usual integrable two-level system to include permanent dipole moments. We demonstrate several Lax pairs. One version uses the spectral parameter natural to the system without permanent dipoles, and the L-A operators are non hermitian (neither hermitian nor skew hermitian) for any value of the spectral parameter. Another version is skew hermitian for most of the real axis but possess a pair of branch points on that axis--a property that precludes a straight forward implementation of the Zakharov-Shabat dressing method to find special solutions. Fortunately, these branch points can be removed without introducing new branches elsewhere, yielding a third pair in a new spectral parameter in which it is amenable to the dressing method. Unfortunately, there are six poles--a computational nightmare in the dressing method! We emphasize the intrinsic amenability of this model to the Lax representation and extrapolate to a conjecture on the N-level atom.

Department of Mathematics, The University of Arizona 617 N. Santa Rita Ave. P.O. Box 210089 Tucson, AZ 85721-0089 USA Voice: (520) 621-6892 Fax: (520) 621-8322 Contact Us © Copyright 2018 Arizona Board of Regents All rights reserved