= Solution
In the ideal <three-level laser>, pumping excites atoms out of the ground state to a higher auxiliary level, rapid relaxation fills a relatively long-lived upper <laser> level, and the <laser transition> returns to the ground state. A <ruby laser> is a standard example of this classification. Since the lower <laser> state initially holds most atoms, substantial pumping is needed to reverse the population per sublevel. With upper and lower degeneracies $g_2,g_1$ and $N_1+N_2\simeq N_T$, the positive-gain condition is
$$
N_2/g_2>N_1/g_1,\qquad
\boxed{N_2>\frac{g_2}{g_1+g_2}N_T.}
$$
For equal degeneracies more than half the active atoms must be excited before <optical gain> is even positive.
In a <four-level laser> the lower <laser> level is above the ground reservoir and empties rapidly after <laser> emission. Pumping again feeds a higher auxiliary level which relaxes into the upper <laser> state, but the lower laser-state population can remain small. The <helium-neon laser> provides a multilevel physical realization often represented by this scheme for a <four-level laser>. Positive <population inversion> then requires much less depletion of the ground reservoir, giving a much lower <laser pump threshold> for comparable lifetimes, <emission cross sections> and cavity losses.
The threshold of a <four-level laser> is not literally zero. Oscillation requires round-trip <optical gain> to overcome output coupling and other losses, and competing decay or incomplete lower-level removal raises the required pump. Real <laser> media contain more atomic states than ideal diagrams for a <three-level laser> or a <four-level laser>; the labels describe the relevant pumping and laser-transition cycle rather than the total number of states in the material.
\Image[/past-exam-of-the-quantum-atomic-and-molecular-physics-course-of-the-university-of-oxford/2016-three-and-four-level-laser-schemes.png]
{title=Why ground-state and rapidly emptied lower laser levels give different thresholds}
{description=Original three- and four-level pumping cycles. The diagrams distinguish pump absorption, rapid relaxation and stimulated laser emission; real media can contain additional levels.}
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