How Lasers Work — Question 2

Question 2

Steady resonant pumping cannot invert an ideal two-level atom

An equal-degeneracy two-level ensemble has total population N. Resonant incoherent light drives upward and downward stimulated transitions at rate W per atom, and spontaneous decay has rate A. Use $\dot N_2=WN_1-WN_2-AN_2$.

1. Find the steady upper-level fraction.

2. Find the steady population difference and its limit as $W\to\infty$.

Question 2 - Solution

Steady resonant pumping cannot invert an ideal two-level atom

(a) Method and calculation

$$N_1=N-N_2:\quad0=WN-(2W+A)N_2.$$$$\boxed{N_2/N=W/(2W+A)<1/2}\quad(A>0).$$

(b) Interpretation and further calculation

$$\boxed{(N_2-N_1)/N=2W/(2W+A)-1=-A/(2W+A)}.$$$$W\to\infty:\quad\boxed{N_2/N\to1/2,\quad N_2-N_1\to0^-}.$$

Physical check

This impossibility result concerns steady incoherent resonant pumping in the stated closed two-level model. A coherent pi pulse can transiently invert a two-level system and is a different preparation.