Chapter 3.3, Problem 2E

Interpretation Introduction

Interpretation:

Assuming $\dot{\text{P}}\approx \text{0}$, $\dot{\text{D}}\approx \text{0}$, the Maxwell–Bloch equations, i.e., first-order equation for the evolution of $\text{E}$ is to be derived, where $\text{E}$ is the electric field, $\text{P}$ is the mean polarization of the atoms, and $\text{D}$ is the population inversion. All the fixed points for the equation of $\text{E}$ are to be found. The bifurcation diagram of ${\text{E}}^{*}$ vs. $\text{λ}$ is to be drawn.

Concept Introduction:

The first-order equation for the evolution of $\text{E}$ can be derived by substituting the given conditions in Maxwell–Bloch equations and solving the equations.

Fixed points are the points where $\dot{\text{E}}\text{ = 0}$.

Stability can be found by differentiating the derived first order equation with respect to $\text{E}$.

The bifurcation diagram can be drawn using the fixed points.