What is Absorption, spontaneous and stimulated emission?

Consider atoms with two energy levels $E_1 < E_2$, populations (number densities) $N_1$, $N_2$, and light of frequency $\nu$ with $h\nu = E_2 - E_1$ and spectral energy density $u(\nu)$ (energy per unit volume per unit frequency). Three processes exchange energy:

• absorption: an atom in $E_1$ absorbs a photon and rises to $E_2$, at the rate $B_{12}\,u(\nu)\,N_1$ per unit volume and time;
• spontaneous emission: an atom in $E_2$ falls to $E_1$ on its own, emitting a photon in a random direction with a random phase, at the rate $A_{21}N_2$ — $1/A_{21} = \tau$ is the lifetime of the level;
• stimulated emission: a photon of frequency $\nu$ passing an atom in $E_2$ induces it to fall, emitting a second photon identical to the first (same frequency, direction, phase and polarization), at the rate $B_{21}\,u(\nu)\,N_2$.

$A_{21}$, $B_{12}$, $B_{21}$ are the Einstein coefficients of the transition.