= Solution
A <three-level laser> uses a pump transition from ground level1 to a higher level3, rapid relaxation $3\to2$, and a long-lived upper lasing level2. The laser transition is $2\to1$, so its lower level is the ground state. Fast feeding and slow upper-state decay allow excitation to accumulate, unlike steady resonant pumping of a simple two-level transition.
<Stimulated emission> adds photons to the stimulating optical mode, but ground-state <photon absorption> competes with it. The <population inversion> condition is $N_2/g_2>N_1/g_1$ when the relevant transition strengths obey the usual Einstein relations. For equal degeneracies and negligible $N_3$, it reduces to \b[$N_2>N_1$, requiring more than half the active population in level2].
For a simple pump rate R per ground-state emitter, fast $3\to2$ relaxation and negligible laser intensity, the <three-level laser rate equation> reduces to
$$
\dot N_2=RN_1-\frac{N_2}{\tau_{21}},\qquad N_1+N_2\simeq N_{\rm tot},\qquad
\boxed{\frac{N_2}{N_1}=R\tau_{21},\quad R\tau_{21}>1\ \text{for inversion}.}
$$
This <three-level ground-state inversion threshold> is only transparency; positive gain must also overcome optical losses. For an illustrative uniformly filled cavity of length $\ell$, equal absorption/emission cross-section $\sigma$, internal intensity-loss coefficient $\alpha_i$ and mirror power reflectivities $R_1,R_2$, the threshold condition is
$$
\boxed{g=\sigma(n_2-n_1)\geq\alpha_i+\frac1{2\ell}\ln\frac1{R_1R_2}.}
$$
Here $n_i$ are number densities, unlike the total populations $N_i$. The <optical resonator> supplies feedback and selects modes; a partially transmitting mirror couples out light. Spontaneous emission seeds the field, and stimulated transitions deplete the inversion until saturated gain balances loss. A <ruby laser> is the standard example. Real level degeneracies, branching losses and pump saturation modify the simple equations. <Fundamentals of Photonics and Quantum Electronics lecture notes by Franz X. Kärtner>[Chapter7§§7.1–7.2.1, printed pp.293–298, optical feedback and three-/four-level media; original reduced ground-state model specified above]{ref}
\Image[/past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2012/paper-1-aqp2012-three-level-laser.png]
{title=Original three-level laser scheme with fast feeding of a metastable upper laser state}
{source=paper-1-aqp2012-three-level-laser.py}
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