Transitions and selection rules
Not every pair of levels can exchange a photon. The rate of a transition is governed, to leading order, by the electric-dipole matrix element \langle f|\,\mathbf{d}\,|i\rangle of the atom coupling to the light's electric field. When this integral vanishes by symmetry the transition is `forbidden`; when it survives it is `allowed`. The conditions for it to survive are the selection rules.
Electric-dipole selection rules. \Delta\ell=\pm1 because a photon carries one unit of angular momentum and has odd parity; \Delta S=0 because the dipole operator does not touch spin. Violating them makes a line orders of magnitude weaker, not strictly impossible.
Einstein's A and B coefficients
In 1917 Einstein found the whole logic of light–atom exchange by pure thermodynamics, before quantum mechanics existed. He identified three processes between two levels 1 and 2: absorption (rate B_{12}\rho, driven by the radiation density \rho), stimulated emission (rate B_{21}\rho, an incoming photon triggering an identical one), and spontaneous emission (rate A_{21}, decay with no prompting). Demanding thermal equilibrium with a blackbody field forces exact relations among the three.
The Einstein relations. The first says spontaneous emission grows as \nu^3 — a deep reason why it is hard to make lasers at short (UV, X-ray) wavelengths. The second ties absorption to stimulated emission through the level degeneracies g_1,g_2.
The decisive insight is that stimulated emission produces a photon identical to the one that triggered it — same frequency, direction, phase and polarization. Absorption removes photons; stimulated emission clones them. Whether a beam grows or shrinks passing through a gas is a race between the two, and that race is decided by which level is more populated. This is also the physics behind Rabi oscillations, where a resonant field drives an atom coherently back and forth between the two levels.
Population inversion and gain
In ordinary thermal matter the lower level is always more populated (the Boltzmann factor guarantees it), so a beam is absorbed. To make light amplify, you must invert this — get more atoms in the upper level than the lower, a population inversion. Then stimulated emission wins, and one photon becomes two, becomes four: coherent amplification.
The laser: light in lockstep
A laser is three things together: a gain medium with a population inversion, a pump that maintains it, and an optical cavity — two mirrors that send photons back through the medium again and again so the avalanche builds. The device oscillates once the round-trip gain exceeds the round-trip losses (mirror leakage, absorption). At that threshold the output leaps from a feeble glow to a bright, directional beam.
The laser threshold condition — gain per pass equals loss per pass. Below it there is no coherent beam; cross it and the output turns on sharply. R_1,R_2 are the mirror reflectivities.
Because every photon is a clone, laser light is coherent (fixed phase relationship), monochromatic (one sharp frequency), and directional (a tight beam). Those three properties — impossible for a thermal source like a bulb — are why the laser reshaped science and technology, and, as Guide 5 shows, gave us a tool precise enough to grab an atom and cool it to a stop.