# Inverse scattering transform for two-level systems with nonzero background

## TL;DR

This paper develops an inverse scattering transform for the Maxwell-Bloch system with nonzero background, explicitly constructs soliton solutions, and analyzes their properties and stability under various initial conditions.

## Contribution

It introduces a formalism for the inverse scattering transform with nonzero background and derives explicit soliton, periodic, and rational solutions for the Maxwell-Bloch system.

## Key findings

- Pure background states generally do not exist with nonzero background.
- Explicit one-soliton solutions are derived in determinant form.
- Solutions are stable only when initial population inversion is negative.

## Abstract

We formulate the inverse scattering transform for the scalar Maxwell-Bloch system of equations describing the resonant interaction of light and active optical media in the case when the light intensity does not vanish at infinity. We show that pure background states in general do not exist with a nonzero background field. We then use the formalism to compute explicitly the soliton solutions of this system. We discuss the initial population of atoms and show that the pure soliton solutions do not correspond to a pure state initially. We obtain a representation for the soliton solutions in determinant form, and explicitly write down the one-soliton solutions. We next derive periodic solutions and rational solutions from the one-soliton solutions. We then analyze the properties of these solutions, including discussion of the sharp-line and small-amplitude limits, and thereafter show that the two limits do not commute. Finally, we investigate the behavior of general solutions, showing that solutions are stable (i.e., the radiative parts of solutions decay) only when initially atoms in the ground state dominant, i.e., initial population inversion is negative.