Lifetime, Linewidth, and Natural Broadening

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 shape warns against the folklore "the energy is uncertain by $\hbar/2\tau$": the Lorentzian is so heavy-tailed that its standard deviation diverges — the honest statements are the FWHM dictionary derived below and the orthogonalization bounds owned by The Energy-Time Uncertainty Relation and Mandelstam-Tamm Bound and Evolution Timescales. 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 exactly: the same emitter looks eleven natural widths wide at $T=\tau/2$ and within one percent of the floor at $T=10\tau$. 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 — parts in $10^7$ for allowed optical lines — and plummet for forbidden transitions, because $\Gamma \propto \omega_0^3 |\mathbf d|^2$ is merciless to small matrix elements and low 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: $c_e(t)$ — amplitude to still find the atom excited; $\Gamma$ — total decay rate of the level; $A_{e\to f}$ — the Einstein spontaneous rate into final state $f$, computed in Spontaneous Emission and the Einstein A Coefficient. Read the sum physically: the level does not know which line you are watching — every open exit door drains the same reservoir, so a level with a fast ultraviolet escape route is broad even in its feeble red line. The radiating amplitude carries the oscillation and half the decay exponent — half, because the probability $|c_e|^2$ must decay at the full rate $\Gamma$. This factor of two is the most common slip in linewidth arithmetic:

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: $\textbf{88.2\%}$ of decays take the Lyman-β door, $\textbf{11.8\%}$ the Hα door.

Step 3. Lyman-β width: $1s$ is stable, so $\Delta\nu = \Gamma_{3p}/2\pi = \textbf{30.2 MHz}$.

Step 4. Hα-component width: $2s$ adds $8.23/2\pi = \textbf{1.3 Hz}$ — seven orders down — so this line is also 30.2 MHz wide. The minority line is exactly as broad as the majority line: the width belongs to the level, not the branch. 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.

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.

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.

Provenance. Lineshape and FWHM derivations follow Loudon (Quantum Theory of Light, Ch. 2) and Cohen-Tannoudji Vol. II; 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 (verify against Sakurai Ch. 5, Griffiths Ch. 11). A verifier should recompute: (1) the Fourier transform giving the Lorentzian and FWHM $=\Gamma$; (2) the energy distribution $P(E)$ and $\Delta E\,\tau=\hbar$; (3) the numerical estimate $\Gamma\sim10^8\,\mathrm{s^{-1}}$, $\Delta\nu\sim10\,\mathrm{MHz}$ for $\lambda\approx600\,\mathrm{nm}$, $|d_{eg}|\approx ea_0$, and compare with the standard sodium-D lifetime $\approx16\,\mathrm{ns}$.

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.