Einstein A and B Coefficients

Every rate-equation model of a laser, amplifier, LED, or atomic clock rests on Einstein's two relations. In a laser gain medium the small-signal gain coefficient is $\gamma(\nu)=\sigma(\nu)\,[N_2-(g_2/g_1)N_1]$, and the threshold condition $g_1N_2>g_2N_1$ is a direct restatement of $g_1B_{12}=g_2B_{21}$: without inversion the same transition that could amplify instead absorbs. Fiber amplifiers (erbium-doped, the backbone of the internet) are designed by balancing exactly these coefficients across the gain band. The relations also drive precision physics in the opposite direction. In cold-atom and Rydberg experiments the ambient 300 K blackbody field, weak as it is, still stimulates real transitions at rate $B_{21}\rho(\nu)$; because $\rho\propto\nu^2$ at low frequency and Rydberg transitions lie in the microwave, blackbody-stimulated redistribution and ionization limit state lifetimes and must be modelled quantitatively. Radiative-transfer codes for stellar atmospheres and nebulae invert observed line strengths for temperature, density, and abundance using the very same $A$'s and $B$'s — Einstein's oven, now the size of a galaxy.

Put a huge ensemble of identical two-level atoms in a sealed oven filled with blackbody light. Each atom flickers up and down, absorbing and emitting, yet the level populations $N_1,N_2$ hold perfectly steady. Einstein's move was to insist on a microscopic balance sheet: every process that lifts an atom to the upper level must be paid back by a process that drops it. Three elementary processes suffice. Absorption removes a photon from the field and promotes the atom. Stimulated emission is triggered by an existing photon and produces a second photon identical to it — same direction, frequency, phase, and polarization. Spontaneous emission is the atom's private clock: it fires whether or not any light is present, sending a photon off in a random direction. The two field-driven rates scale with how much light is already there, the spectral energy density $\rho(\nu)$; the spontaneous rate does not. That single asymmetry is the whole story. Balancing the sheet and demanding that the light in the box be Planckian forces the spontaneous-to-stimulated ratio $A_{21}/B_{21}$ to equal the number of electromagnetic modes per unit volume per unit frequency, $8\pi\nu^2/c^3$, times the photon energy $h\nu$. In other words, spontaneous emission is nothing but stimulated emission driven by the one quantum of zero-point field that every mode carries. The companion relation $g_1B_{12}=g_2B_{21}$ says absorption and stimulated emission are microscopically reversible, weighted only by how many sublevels each level offers. This is why you cannot get net amplification from a two-level system in thermal equilibrium: Boltzmann always keeps the lower level more populated per sublevel, so absorption wins. Only by pumping the system into a population inversion — more atoms per sublevel upstairs than down — does stimulated emission overtake absorption and the medium amplifies. Every laser is a machine for engineering that inversion.

The key contrast lives in the second figure. Because $A_{21}/(B_{21}\rho)=e^{h\nu/k_BT}-1$, the balance between spontaneous and stimulated emission is a pure function of frequency at fixed temperature. Plotting it reveals a sharp crossover that separates the microwave regime, where a thermal field already carries many photons per mode, from the optical regime, where spontaneous emission is overwhelming.

The three processes each change the upper-level population, but only two of them see the field. Written as rate contributions, spontaneous emission depletes level 2 at a fixed per-atom rate, while stimulated emission and absorption scale with $\rho(\nu)$ evaluated at the transition frequency.

where: $N_1,N_2$ — number densities in the lower and upper levels; $A_{21}$ — spontaneous-emission coefficient (units $\mathrm{s^{-1}}$); $B_{12},B_{21}$ — absorption and stimulated-emission coefficients; $\rho(\nu)$ — spectral energy density per unit frequency ($\mathrm{J\,s\,m^{-3}}$). Read physically: $A_{21}$ is the inverse radiative lifetime; the $B$'s are cross-sections in disguise, telling you how strongly the transition couples to whatever light is present. Balancing (1) in equilibrium with a Boltzmann population ratio and demanding the field be Planckian collapses to two temperature-free identities — the heart of the article.

where: $g_1,g_2$ — degeneracies (sublevel counts) of the two levels; $h$ — Planck constant; $c$ — speed of light; $\nu$ — transition frequency. Read physically: the first identity makes absorption and stimulated emission mirror images weighted by degeneracy; the second is the free-space mode density $8\pi\nu^2/c^3$ times $h\nu$, so it is fixed by the geometry of the vacuum, not by the atom's internal details. The single most useful rearrangement compares the two downward channels at a given field. Dividing the spontaneous by the stimulated rate gives a pure number that depends only on the photon occupancy of the mode.

where: $\bar n$ — mean number of photons in a resonant mode at temperature $T$ (the Bose–Einstein occupancy). Read physically: stimulated emission beats spontaneous emission exactly when there is more than one photon per mode. The two are equal at $\bar n=1$, i.e. $h\nu=k_BT\ln 2$ — the crossover frequency. Finally, the spontaneous coefficient sets the radiative lifetime, the bridge to the neighbouring articles on decay and linewidth.

where: $\tau$ — natural lifetime of the upper level; $\omega_{21}=2\pi\nu$; $\langle 1|\hat{\mathbf d}|2\rangle$ — transition dipole moment. The explicit dipole formula is the quantum-electrodynamic result owned by aq-qm-spontaneous-emission-einstein-coefficients; here it simply tells us $A_{21}$ is a fixed atomic number that Einstein's ratios then convert into $B_{21}$.

The competition between the three processes shifts dramatically with frequency, temperature, and field strength.

Example 1 — the A/B ratio and why a sodium lamp is incoherent.

Step 1. The Na D line sits at $\lambda=589\ \mathrm{nm}$, so $\nu=c/\lambda=5.09\times10^{14}\ \mathrm{Hz}$. Compute the mode-density prefactor $A_{21}/B_{21}=8\pi h\nu^3/c^3$.

Step 2. Numerically $8\pi h\nu^3/c^3=8\pi(6.626\times10^{-34})(5.09\times10^{14})^3/(3.00\times10^8)^3=\mathbf{8.15\times10^{-14}\ J\,s\,m^{-3}}$. This is the spontaneous-to-stimulated conversion factor per unit spectral energy density.

Step 3. At $T=300\ \mathrm{K}$, $h\nu/k_BT=(6.626\times10^{-34})(5.09\times10^{14})/[(1.381\times10^{-23})(300)]=\mathbf{81.5}$. The stimulated-to-spontaneous ratio is $\bar n=1/(e^{81.5}-1)\approx e^{-81.5}\approx10^{-36}$.

Step 4. So in a 300 K thermal field the Na atom emits spontaneously about $10^{36}$ times more often than it is stimulated: essentially every D-line photon is spontaneous, random in direction and phase. A sodium street lamp is incoherent for a purely thermodynamic reason — at optical frequencies there is far less than one photon per mode.

Example 2 — locating the maser window.

Step 1. Stimulated and spontaneous emission are equal when there is one photon per mode, $\bar n=1$, i.e. $e^{h\nu/k_BT}=2$, so $\nu_\times=(k_BT/h)\ln2$.

Step 2. At $T=300\ \mathrm{K}$: $k_BT/h=(1.381\times10^{-23})(300)/(6.626\times10^{-34})=6.25\times10^{12}\ \mathrm{Hz}$, and $\nu_\times=6.25\times10^{12}\times0.693=\mathbf{4.33\times10^{12}\ Hz}=4.33\ \mathrm{THz}$, a wavelength $\lambda_\times=c/\nu_\times=\mathbf{69\ \mu m}$.

Step 3. Below 4.33 THz (microwave and radio) a room-temperature thermal field already holds more than one photon per mode, so stimulated emission dominates. Above it (far-IR through visible), spontaneous emission takes over.

Step 4. This is exactly why the maser (1954, microwave) preceded the laser (1960, optical): building gain against spontaneous loss is far easier where spontaneous emission is intrinsically weak. The $\nu^3$ growth of $A_{21}$ is the obstacle every short-wavelength laser must overcome.

The spontaneous-emission coefficient $A_{21}$ is the quantity Einstein could not avoid inventing. His 1917 balance sheet needed a downward process that runs with no field present; without it, absorption and stimulated emission alone cannot reproduce the Planck spectrum's $-1$ in the denominator. So $A_{21}$ entered physics as a thermodynamic necessity, its absolute value unknown until Dirac quantized the field a decade later. It is the archetype of a coefficient whose ratios are pinned by symmetry long before its magnitude is computable.

References. Foot, Atomic Physics, Ch. 1 · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 1 · Demtröder, Laser Spectroscopy, Vol. 1, Ch. 2 · Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 14 · Hecht, Optics, 5th ed.

Provenance. All coefficient relations follow the standard Einstein (1917) thermodynamic derivation as presented in Foot Ch. 1 and Loudon Ch. 1. The A/B ratio $8\pi h\nu^3/c^3$ and $g_1B_{12}=g_2B_{21}$ are exact and textbook-standard; they are reproduced here by matching the balance equation to the Planck spectrum (verify_aq-optics-einstein-a-b-coefficients.py). The quantum expression $A_{21}=\omega^3|d|^2/(3\pi\varepsilon_0\hbar c^3)$ is the QED result owned by aq-qm-spontaneous-emission-einstein-coefficients (recalled by name); the $B$ perturbation rates are owned by aq-qm-absorption-stimulated-emission. No fabricated numerical values are used.

Builds on: Spontaneous Emission · Stimulated Emission · Resonant Absorption · Detailed Balance and the Planck Spectrum Leads to: Radiative Lifetime and Natural Linewidth · Oscillator Strength · Transition Dipole Moment · Lorentzian Line Shape