A Solution of the Bloch-Maxwell’s Equations without the Use of the “Factorization Assumption”

Abstract

The dynamics of the interaction of an electromagnetic field with an ensemble of two-level resonant atoms is usually described by the coupled Maxwell-Schrödinger equations. When the electric field is written as a carrier wave with slowly varying phase and slowly varying amplitude, and if the medium through which this wave propagates is inhomogeneously broadened, then the Maxwell-Schrödinger equations for the steady state can be written as a set of five nonlinear coupled integro-differential equations in which the unknown functions are the three components of the Bloch vector and the envelope and phase of the field. A variety of solutions for this system have been found [1], but all satisfy the so-called “factorization assumption”, according to which the dependence of the absorption component of the Bloch vector on the detuning frequency is factored out from its time dependence. In this note we investigate the possibility of solving the above mentioned system of equations without the benefit of the factorization assumption.