Chapter 3.3, Problem 1E

Interpretation Introduction

Interpretation:

Assuming quasi-static approximation $\dot{\text{N}}\approx \text{0}$, the first-order equation for n is derived, and $\text{N (t)}$ is expressed in terms of $\text{n (t)}$, $p_c$ is determined when $n*=0$ becomes unstable for $p>p_c$, type of bifurcation at the laser threshold $p_c$ is determined. The range of parameter which makes quasi-static approximation $\dot{\text{N}}\approx \text{0}$ is determined.

Concept Introduction:

The first-order equation for the evolution of $\text{N(t)}$ can be derived by substituting quasi-static approximation $\dot{\text{N}}\approx \text{0}$ in the equation $\dot{\text{N}}\text{ = -GnN - fN +p }$

The pump strength can be evaluated by taking derivative of $\dot{\text{n}}$ with respect number of excited atoms $\text{n}$.

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

The range of parameter for the approximation can be obtained by considering the rate of change of $\dot{\text{n}}$ is very less than the rate of change of $\dot{\text{N}}$