Spontaneous and Stimulated Emission

When an atom interacts with light, three elementary processes are possible: absorption, stimulated emission, and spontaneous emission. The first two follow directly from the harmonic-perturbation theory of this module; the third requires quantizing the field but is tied to the others by a beautiful thermodynamic argument due to Einstein.

The three processes

Consider two atomic levels, lower $|a\rangle$ with energy $E_a$ and upper $|b\rangle$ with $E_b$ , separated by $\hbar\omega_0 = E_b - E_a$ .

Absorption. An atom in $|a\rangle$ absorbs a photon of energy $\hbar\omega_0$ and jumps to $|b\rangle$ . This is the co-rotating term of a harmonic drive at resonance.
Stimulated emission. An atom in $|b\rangle$ is hit by a photon of energy $\hbar\omega_0$ and is induced to drop to $|a\rangle$ , emitting a second photon identical in phase, frequency, and direction to the first. This is the counter-rotating term.
Spontaneous emission. An atom in $|b\rangle$ decays to $|a\rangle$ emitting a photon even with no applied field. This has no classical-field cause; it is driven by the vacuum fluctuations of the quantized electromagnetic field.

Stimulated processes from perturbation theory

For a monochromatic field, our RWA analysis showed that absorption and stimulated emission have equal rates per atom when the field is resonant — the matrix element $|V_{ba}| = |V_{ab}|$ is the same because $\hat V$ is Hermitian. Both rates are proportional to the intensity (energy density) of the driving field. This symmetry is the heart of the laser: shine resonant light on a population of excited atoms and each stimulated photon is a clone of the incoming one, producing coherent amplification.

The Einstein coefficients

Einstein parametrized the three processes with rate coefficients. With $\rho(\omega_0)$ the spectral energy density of the radiation and $N_a$ , $N_b$ the populations:

\left(\frac{dN_b}{dt}\right)_{\text{abs}} = +B_{ab}\,\rho(\omega_0)\,N_a,\qquad \left(\frac{dN_b}{dt}\right)_{\text{stim}} = -B_{ba}\,\rho(\omega_0)\,N_b,

\left(\frac{dN_b}{dt}\right)_{\text{spont}} = -A_{ba}\,N_b .

Here $B_{ab}, B_{ba}$ are the stimulated coefficients (set by the perturbation matrix element) and $A_{ba}$ is the spontaneous-emission rate.

Detailed balance fixes the relations

In thermal equilibrium the populations are fixed and follow the Boltzmann ratio $N_b/N_a = e^{-\hbar\omega_0/k_B T}$ (for nondegenerate levels), while the field is the Planck blackbody spectrum. Demanding that absorption balance total emission forces two exact relations:

B_{ab} = B_{ba}, \qquad \frac{A_{ba}}{B_{ba}} = \frac{\hbar\omega_0^3}{\pi^2 c^3} .

The first says absorption and stimulated emission are equally strong — confirming the perturbative result. The second is remarkable: it pins the spontaneous rate to the stimulated one without ever computing $A$ directly, and shows that spontaneous emission grows steeply with frequency ( $\propto \omega_0^3$ ). That is why excited states decay faster for higher-energy (e.g. ultraviolet) transitions than for low-energy ones.

Why spontaneous emission needs the quantized field

Semiclassically, an atom sitting alone in $|b\rangle$ with no applied field has $\hat H'=0$ , so first-order theory predicts it stays put forever. Yet excited atoms do decay. The resolution is that the electromagnetic field is itself a quantum system with a nonzero vacuum state; its zero-point fluctuations act as an ever-present perturbation. Treating the field modes as the continuum of final states and applying Fermi's golden rule yields a finite spontaneous-emission rate — the natural linewidth and finite lifetime of every excited atomic state.

The big picture

Absorption and stimulated emission are the two terms of a classical harmonic drive; spontaneous emission is the same physics with the field's vacuum playing the role of the perturbation. All three are unified by Fermi's golden rule, with the differences living entirely in what provides the perturbation and which continuum of final states is available.