Field theory for coherent optical pulse propagation

Abstract

We introduce a notion of “matrix potential” to nonlinear optical systems. In terms of a matrix potential $g$, we present a gauge-field-theoretic formulation of the Maxwell-Bloch equation that provides a semiclassical description of the propagation of optical pulses through resonant multilevel media. We show that the Bloch part of the equation can be solved identically through $g$ and the remaining Maxwell equation becomes a second-order differential equation with a reduced set of variables due to the gauge invariance of the system. Our formulation clarifies the (non-Abelian) symmetry structure of the Maxwell-Bloch equations for various multilevel media in association with symmetric spaces $G/H$. In particular, we associate the nondegenerate two-level system for self-induced transparency with $G/H=\mathrm{SU}\left(2\right)/\mathrm{U}\left(1\right)\mathrm{}$ and three-level $\Lambda$ or $V$ systems with $G/H=\mathrm{SU}\left(3\right)/\mathrm{U}\left(2\right)\mathrm{}$. We give a detailed analysis for the two-level case in the matrix potential formalism, and address various properties of the system including soliton numbers, effective potential energy, gauge and discrete symmetries, modified pulse area, conserved topological, and nontopological charges. The nontopological charge measures the amount of self-detuning of each pulse. Its conservation law leads to a different type of pulse stability analysis that explains earlier numerical results.