Spontaneous emission ($P_e(t)=P_e(0)e^{-\Gamma t}$)

Spontaneous emission is radiative decay of an excited matter quantum state by emission of a photon, without an applied resonant drive. In a simple Markov decay model the excited-level population obeys ${\dot{P}}_e=-\Gamma P_e$. During driven control, the expected emission count is governed by $\Gamma \int P_e(t)dt$; suppressing the lossy level's population is therefore the purpose of dark-state adiabatic transfer.

Wide perfectly conducting plates with $d\ll c/{\omega }_0$ admit only a normal-electric-field transverse electromagnetic mode at the transition frequency. The electric-dipole interaction therefore couples to the normal component of the atomic transition dipole. A normal dipole has enhanced available mode density; an in-plane dipole has no allowed electric-dipole radiation channel below the higher-mode cutoffs. The conclusion assumes negligible changes to the atom and negligible absorption, leakage and symmetry-breaking mixing.

For a normal transition dipole of magnitude $D$, the Fermi's golden rule, one-polarization mode density $\omega /(2\pi c^{2}d)$ and single-photon squared field $\hslash \omega /(2{ϵ}_{0}V)$ give $A_{\perp }=D^2{\omega }_0^2/(2{ϵ}_0\hslash c^{2}d)$. Dividing by the free-space electric-dipole spontaneous emission rate gives the title. The factor three relative to the scalar density-of-states ratio comes from the isotropic free-space polarization average. This ideal weak coupling formula assumes wide plates and unperturbed atomic structure, with atomic size much smaller than $d$.

For a specified electric-dipole transition, summing free radiation modes gives $A_{21}={\omega }_{21}^3|{\mathbf{d}}_{12}{|}^2/(3\pi {ϵ}_0\hslash c^3)$. Unresolved degenerate levels require sums over final substates and appropriate initial-state averages. Modified radiation environments alter the available modes.

An excited state can decay by emitting two photons whose energies sum to the transition energy. In hydrogen, this permits the metastable $2s$ state to reach the ground state during cosmological recombination.

For a closed two-level system with spontaneous emission population decay $P_e(t)=P_e(0)e^{-\Gamma t}$ and no additional dephasing, the natural linewidth has full width at half maximum $\Gamma$ in angular frequency, or $\Gamma /(2\pi )$ in hertz. The excited-state lifetime is $1/\Gamma$.

A radiative cascade is a sequence of downward photon-emitting transitions following excitation or recombination. Cascades feed lower excited levels and change their emitted line intensities. An effective recombination coefficient includes all relevant cascade paths under its stated optical-depth and density assumptions.

The critical density of an atomic transition is the electron density at which its collisional de-excitation rate equals its spontaneous radiative decay rate. A multilevel atom requires summed competing rates rather than a single-transition ratio. This atomic density scale is distinct from cosmological critical density.

For two independently excited optically thin upper levels sharing a lower state, statistical equilibrium gives the displayed ratio with critical densities $n_{c,j}=A_{j1}/C_{j1}$. When $n_{c,2}<n_{c,3}$, the fractional sensitivity $|d\ln R/d\ln n|$ is greatest at $n=\sqrt{n_{c,2}{n}_{c,3}}$. With constant absolute ratio error, the largest $|dR/d\ln n|$ occurs instead at $n=n_{c,2}$. Thus a density diagnostic's optimum requires both an atomic model and a measurement-error convention. Additional collisional coupling changes the intermediate-density curve.

At fixed density below critical density, optically thin two-level cooling scales as $\mathcal{C}\propto T^{-1/2}{e}^{-\theta /T}$, with $\theta =\Delta E/k_B$. Under temperature-independent heating, $d\ln \mathcal{C}/d\ln T=-1/2+\theta /T$ gives restoring thermal balance for $T<2\theta$ and instability for $T>2\theta$. This boundary assumes fixed ionic abundance and does not describe a constant-pressure perturbation.

At the maximum of a cooling curve with constant heating set to that maximum, the linear restoring term vanishes. Because cooling falls on both sides of the maximum, the equilibrium is not generally stable to perturbations of both signs even though a first-order eigenvalue is zero.

At fixed pressure and fixed composition, density varies as $T^{-1}$. With temperature-independent heating per particle, low-density cooling per particle scales as $T^{-3/2}{e}^{-\theta /T}$ and is restoring for $T<2\theta /3$. For constant heating per volume instead, the cooling scales as $T^{-5/2}{e}^{-\theta /T}$ and the threshold is $2\theta /5$. The heating normalization and perturbation constraint must be specified.

When collisions thermalize an optically thin two-level atom, the upper fraction is $f_u=g{e}^{-\theta /T}/(1+g{e}^{-\theta /T})$ and the emitted power per atom is $A_{21}\Delta E{f}_u$. Its logarithmic temperature slope is $(\theta /T)(1-f_u)>0$. Radiative cooling still occurs, but collision frequency no longer sets the emitted power as in the low-density limit.

A schematic statistical equilibrium is $n_l{n}_e{q}_{lu}=n_u(A_{ul}+n_e{q}_{ul})$. The radiative yield per excitation is $A_{ul}/(A_{ul}+n_e{q}_{ul})$. It is near unity below critical density $A_{ul}/q_{ul}$ and decreases above it. Quenching suppresses emission relative to naive low-density excitation scaling; it need not make the absolute emissivity decrease when density rises.

Einstein coefficients describe spontaneous emission, stimulated emission and absorption between atomic levels. With $B$ multiplying radiation energy density per frequency, thermal detailed balance gives $g_i{B}_{ij}=g_j{B}_{ji}$ and $A_{ji}/B_{ji}=8\pi h{\nu }^3/c^3$. Coefficients defined using mean specific intensity instead have a different normalization; specify the radiation convention.

Absorption of incident radiation that promotes an emitter from a lower to an upper state. In an Einstein-rate description the rate is proportional to the relevant spectral radiation energy density. The degeneracy-weighted absorption coefficient and stimulated-emission coefficient satisfy detailed balance.

For an ideal emitter able to exchange one fermion with a zero-chemical potential bath, Pauli blocking gives downward rate $A(1-f)$ rather than the bosonic $A(1+f)$. If $u_{\nu }=h\nu {\mathcal{D}}_{\nu }f$ and $C=A/(h\nu {\mathcal{D}}_{\nu })>0$, the rate is $A-C{u}_{\nu }$. Detailed balance with a Fermi-Dirac distribution implies $g_1{B}_{12}=g_{2}C$ and $A/C=h\nu {\mathcal{D}}_{\nu }$. For $d$ massless helicity states, ${\mathcal{D}}_{\nu }=d4\pi {\nu }^2/c^3$. Thus an additive induced-emission coefficient would be negative, expressing blocking rather than a negative total rate. The emitter must change its fermion parity; two ordinary electronic energy levels cannot exchange a single neutrino this way.

When $B_{12}$ and $B_{21}$ multiply radiation energy density per unit angular frequency, they differ by a factor $2\pi$ from coefficients multiplying the density per ordinary frequency. For nondegenerate levels, Boltzmann distribution populations and thermal equilibrium give $B_{12}=B_{21}$ and the free-space ratio in the title. Specifically $B_{\omega }=2\pi B_{\nu }$ while $A$ is unchanged. Polarization restrictions and level degeneracy must be specified separately.

Weak incoherent frequency components contribute probabilities. The long-time sinc-squared kernel gives the displayed stimulated absorption rate for a fixed polarization and spectral energy density per angular frequency. Isotropic unpolarized radiation instead contributes $|{\mathbf{d}}_{21}{|}^2/3$. Converting to density per ordinary frequency changes the coefficient by $2\pi$.

Let $A_{21}$ be the spontaneous emission coefficient and let $B\rho$ be the induced transition rate per atom under a resonant radiation spectral energy density $\rho$. For nondegenerate levels the Einstein coefficients obey $B_{12}=B_{21}=B$. In a closed two-level system with no collisional or other population transfer, steady state requiresFor nonzero upper population and drive, division gives the ratio of total spontaneous and stimulated emission rates in the title. In particular $N_1>N_2$ when $A_{21}>0$; continuous driving of this closed model approaches equal populations but cannot create a population inversion. No thermal population law is assumed.