Lifetime, Linewidth, and Natural Broadening

The natural linewidth is the currency of precision measurement. Optical atomic clocks run on transitions with pathologically small $\Gamma$: the $^{87}$Sr clock line at 698 nm has a natural width near one millihertz ($Q \approx 4\times10^{17}$), and lattice clocks resolving fractions of it reach $10^{-18}$ fractional uncertainty. The interrogation crossover derived above is their daily engineering reality: seconds of coherent probing are needed before the measured line approaches the floor — which is why clock lasers must themselves be stabilized to hertz-level linewidths. The same physics runs both directions in the quantum-optics lab: the natural width sets the bandwidth and temporal shape of single photons from atoms and quantum dots, and a cavity changes $\Gamma$ (the Purcell effect) — the linewidth belongs to the atom plus its electromagnetic environment. In astrophysics the Lorentzian damping wings of Lyman-α shape galaxy absorption spectra; in nuclear metrology the 4.7 neV Mössbauer width of ⁵⁷Fe registers a building's worth of gravitational redshift.

Strike a bell and listen. The pitch is definite while the tone lasts, but the tone does not last. A Fourier analyzer reports a distribution of frequencies, centred on the pitch, whose width is the reciprocal of the ring-down time: a short clang is broad, a sustained hum narrow. An excited atom is a bell struck once — its dipole oscillates at $\omega_0$ while the excited amplitude drains at rate $\Gamma/2$, and the emitted light carries exactly that damped waveform.

The shape matters as much as the width. A single exponential decay produces a Lorentzian — algebraic $1/(\omega-\omega_0)^2$ wings that outlive any Gaussian: far from line centre, where the Doppler core has vanished, the damping wings of strong stellar lines are Lorentzian physics. And the And the line shape warns against interpreting the heuristic relation $\Delta E \sim \hbar/(2\tau)$ as the standard deviation of a Lorentzian energy distribution: the Lorentzian is so heavy-tailed that its second moment, and hence its standard deviation, diverges — the appropriate quantitative statements are the FWHM relations derived below, together with the orthogonalization-time bounds associated with the Mandelstam–Tamm relation. There is a second, more modern picture: the linewidth as an interrogation budget. The Lorentzian is what you record after watching forever; a finite window $T$ gives a ridged, broad profile narrowing as $1/T$ until, around ten lifetimes, it settles onto the floor. The figure below is computed from the finite-time Fourier transform: the same emitter has an FWHM of $11.2\Gamma$ at $T=\tau/2$ and $1.01\Gamma$ at $T=10\tau$, where $\Gamma$ denotes the natural FWHM. Clock builders live on this curve — a millihertz line is useless without the seconds of coherent watching that resolve it.

Finally, the ladder of magnitudes. Natural widths are fractionally tiny — of order parts in $10^7$ for some allowed optical lines — and can become dramatically smaller for weak or forbidden electric-dipole transitions. For an electric-dipole transition, the spontaneous-emission rate scales as $\Gamma \propto \omega_0^3 |\mathbf{d}|^2$, making it strongly suppressed by both small transition dipole moments and low transition frequencies. That is a resource: precision measurement is the art of choosing states that refuse to decay.

Everything hangs on one empirical law and the amplitude behind it. An undisturbed excited population decays exponentially — what a delayed-coincidence photon counter actually records — with a rate built from the spontaneous rates of every open channel. The deep reason the decay is exponential (and when it is not) belongs to Survival Amplitude and Decay of Unstable States; here we spend the working law.

where: $c_e(t)$ is the probability amplitude for finding the atom in the excited state, $\Gamma$ is the total population decay rate of the excited state, and $A_{e\rightarrow f}$ is the Einstein $A$ coefficient for spontaneous emission into the final state $f$, with $\Gamma=\sum_f A_{e\rightarrow f}$. Physically, the total decay rate is determined by all open decay channels, so a fast ultraviolet decay channel can broaden the level even when the transition being observed is in the red. The radiating amplitude carries half the population-decay exponent because $c_e(t)\propto e^{-\Gamma t/2}$, while the excited-state probability $|c_e(t)|^2$ decays as $e^{-\Gamma t}$. This factor of two is a common source of errors in linewidth calculations:

where: $\omega_0 = (E_e - E_g)/\hbar$ — the Bohr frequency of the transition being watched; $\theta(t)$ — the decay starts when the state is prepared. This damped phasor is the Weisskopf-Wigner ansatz: in the quantum theory of radiation it emerges as the resummed solution of the atom-field coupling, repairing the naive first-order probability that grows linearly forever. Its rigorous credentials — when a spectrum yields exact exponential decay, and where the quadratic and power-law eras hide — are Survival Amplitude and Decay of Unstable States' property; we use the ansatz as every laser lab does. Experimentalists quote linewidths in hertz — and $2\pi$ slips are endemic — so keep the dictionary at hand:

where: the three widths are one Lorentzian measured in angular frequency, ordinary frequency, and energy. A 1.6 ns lifetime is $\Gamma = 6.3\times10^8\ \mathrm{s^{-1}}$ but $\Delta\nu \approx 100$ MHz — a factor-6.28 slip that has survived referees. The energy version, $\Delta E\,\tau = \hbar$, is the honest "energy-time uncertainty" of a decaying state: an equality about a distribution's FWHM, not a bound involving a time operator (there is none — The Energy-Time Uncertainty Relation owns that argument, and Pauli's proof of it).

A spectral line is also an energy histogram: expand the decaying state over the exact stationary states of atom-plus-field, and the survival amplitude is the Fourier transform of the energy distribution — the Krylov-Fock master formula that Survival Amplitude and Decay of Unstable States owns — so the observed decay determines the distribution:

where: $P(E)$ — normalized energy distribution of the decaying state; $E_e$ — the nominal level position; $\hbar\Gamma$ — its FWHM. An unstable "level" is a Lorentzian band of true stationary states, and the emitted line photographs the band.

Two pieces of bookkeeping make the theorem laboratory-grade. Levels: independent channel rates add — equation (1); the branching ratio says how often each door is taken, but every observed line reflects the sum. Lines: if the lower level is itself unstable, the dipole coherence decays at the sum of the two half-rates.

where: $\Gamma_e$, $\Gamma_g$ — total decay rates of upper and lower level. A transition between two broad levels is broader than either. This is why the strontium clock line pairs a ground state with a doubly forbidden excited state — both widths near zero — and why Forbidden Transitions: Magnetic Dipole and Electric Quadrupole is this article's natural sequel.

Example 1 — Lyman-α end to end: from a rate to a wavelength smear. The hydrogen $2p$ level decays only to $1s$, at $A(2p{\to}1s) = 6.27\times10^{8}\ \mathrm{s^{-1}}$ — the flagship number of Spontaneous Emission and the Einstein A Coefficient, re-derived from the radial integral in this article's verification script. Find the lifetime, linewidth, wavelength width, and the Doppler comparison.

Step 1. Lifetime: $\tau = 1/A = \textbf{1.60 ns}$ — set entirely by $\omega_0^3\,|\langle 1s|\mathbf r|2p\rangle|^2$.

Step 2. Natural width: $\Delta\nu = A/2\pi = \textbf{99.8 MHz}$. Against $\nu_0 = c/\lambda = 2.47\times10^{15}$ Hz that is $Q = \nu_0/\Delta\nu = \textbf{2.5}\times\textbf{10}^{7}$.

Step 3. Wavelength width at $\lambda = 121.567$ nm: $\Delta\lambda = \lambda^2\Delta\nu/c = \textbf{4.9}\times\textbf{10}^{-5}\ \textbf{Å}$. No grating spectrograph resolves that.

Step 4. Doppler at 300 K: $\Delta\nu_D = \nu_0\sqrt{8\ln 2\,k_BT/m_Hc^2} = \textbf{3.0}\times\textbf{10}^{10}\ \textbf{Hz}$ — a factor ≈ 305 above natural. The natural line hides under the thermal Gaussian until you cool, trap, or Doppler-cancel.

Example 2 — the hydrogen 3p ledger: one level, two lines, one width. The $3p$ level has two E1 exit doors: $3p \to 1s$ (Lyman-β) at $A_1 = 1.672\times10^{8}\ \mathrm{s^{-1}}$ and $3p \to 2s$ (an Hα component) at $A_2 = 2.245\times10^{7}\ \mathrm{s^{-1}}$ — both computed from first-principles radial integrals in the verification script (matching NIST to 0.1%). Find the level width, branching, and both line widths, given that $2s$ decays only by two-photon emission at $8.23\ \mathrm{s^{-1}}$.

Step 1. Total rate: $\Gamma_{3p} = A_1 + A_2 = 1.897\times10^{8}\ \mathrm{s^{-1}}$, so $\tau_{3p} = \textbf{5.27 ns}$.

Step 2. Branching: $88.2\%$ of decays produce the Lyman-$\beta$ line, while $11.8\%$ produce the H$\alpha$ line.

Step 3. Lyman-$\beta$ linewidth: the $1s$ level is stable on this timescale, so $\Delta\nu_{\mathrm{FWHM}}=\Gamma_{3p}/(2\pi)=30.2\,\mathrm{MHz}$.

Step 4. H$\alpha$-component width: the $2s$ level adds $8.23/(2\pi)\approx1.3\,\mathrm{Hz}$, which is negligible compared with $30.2\,\mathrm{MHz}$, so this component has essentially the same linewidth, $30.2\,\mathrm{MHz}$. The minority line is essentially as broad as the majority line: the linewidth is determined primarily by the decay rates of the participating levels, not by the branching ratio. And the freakish narrowness of the $2s$ door (0.12 s lifetime) is forbidden-transition physics — why $2s$ cannot take a one-photon exit at all is owned by Forbidden Transitions: Magnetic Dipole and Electric Quadrupole.

References. Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., Ch. 11 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Ch. 5 · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 2 · Cohen-Tannoudji, Diu & Laloë, Vol. II, Ch. XIII · Weisskopf & Wigner, Z. Phys. 63, 54 (1930) · Foot, Atomic Physics, Ch. 7-8. The decay-rate formula $\Gamma=\omega_0^3|d_{eg}|^2/3\pi\varepsilon_0\hbar c^3$ is the standard Einstein-$A$/Weisskopf–Wigner result. Key checks: the Fourier transform yields the Lorentzian with FWHM $=\Gamma$; the energy distribution $P(E)$ satisfies $\Delta E\,\tau=\hbar$; and for $\lambda\approx600\,\mathrm{nm}$ with $|d_{eg}|\approx ea_0$ one obtains $\Gamma\sim10^8\,\mathrm{s^{-1}}$ and $\Delta\nu\sim10\,\mathrm{MHz}$, consistent with the standard sodium-D lifetime of $\approx16\,\mathrm{ns}$.

Builds on: Fermi's Golden Rule · First-Order Transition Amplitudes and Probabilities · Absorption and Stimulated Emission · The Dipole Approximation. Leads to: Semiclassical Theory of Atom–Radiation Interaction · Harmonic (Sinusoidal) Perturbations and Resonance.