= Population inversion
{title2=$N_2>N_1$}
{wiki}
A <population inversion> puts more emitters in the upper than in the lower <quantum state> of a transition in the nondegenerate two-level model. Let $N_2,N_1$ be their number densities. Equal stimulated-transition strengths make the emission rate proportional to $N_2$ and the competing <photon absorption> rate proportional to $N_1$, so net <optical gain> is proportional to $N_2-N_1$. For $N_2=6\times10^{20}\,\mathrm{m^{-3}}$ and $N_1=4\times10^{20}\,\mathrm{m^{-3}}$, the difference is positive; swapping them makes the medium absorb. Equal populations give zero net stimulated contribution. With degenerate levels, the relevant comparison uses populations per <statistical weight of an atomic level>, rather than simply $N_2>N_1$.
Why pump through extra levels? Ordinary steady resonant pumping of a nondegenerate two-level system drives both <photon absorption> and <stimulated emission>. With equal induced rate $W$ and upper-state decay rate $\Gamma$, its elementary population equation is
$$
\dot N_2=W(N_1-N_2)-\Gamma N_2,
\qquad
\frac{N_2}{N_1}=\frac{W}{W+\Gamma}\leq1
$$
at a stationary population. Increasing this pump saturates towards $N_2=N_1$; it cannot sustain <population inversion> in this model. This is a steady rate-equation conclusion, not a universal prohibition on all driven two-level dynamics or all possible <laser> gain mechanisms. https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/2641db6431ea70989c29e8512d5d91a7_chapter6.pdf[Franz Kärtner, MIT 6.974, Chapter 6, §§6.7–6.8: saturation and rate equations]{ref}.
In a familiar three-level path, pumping excites a higher level which relaxes into the upper lasing level; the lower lasing level is the ground state, so substantial ground-state depletion is needed. A four-level path also provides rapid emptying of the lower lasing level into a separate ground state. Suitable lifetimes then allow accumulation above and depletion below the lasing transition. Pumping still supplies the <energy>: extra levels provide a route to <population inversion>, not free <energy>. https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/18f30fad63a62ef4dd894d3752b55a60_chapter7.pdf[Franz Kärtner, MIT 6.974, §7.2.1: three- and four-level laser media]{ref}.
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