Radiative Lifetime and Natural Linewidth

Spontaneous emission hands this article a single number and a hard consequence. From the Einstein $A$ coefficient — the rate at which an isolated excited atom dumps its energy into the quantised vacuum, established in the two preceding AMO articles — comes the fact that an excited state is not stationary at all. Its population drains as $\dot N_2 = -\Gamma N_2$ with $\Gamma = A_{21}$ , so $N_2(t)=N_2(0)e^{-\Gamma t}$ , and the atom spends on average a radiative lifetime $\tau = 1/\Gamma$ in the upper level before a photon leaves. That much is kinematics of decay. The consequence is spectroscopic: because the emitted wave turns on at excitation and dies away over a time $\tau$ , it is a finite wave train, not an endless sinusoid, and a finite wave train is never monochromatic. Fourier-transform the damped dipole amplitude $\propto e^{-\Gamma t/2}e^{-i\omega_0 t}$ — the amplitude decays at half the population rate, a factor that trips up nearly everyone — and the power spectrum is a Lorentzian centred at $\omega_0$ with full width at half maximum $\Gamma$ in angular frequency, or $\Delta\nu = \Gamma/2\pi$ in ordinary frequency. This is the natural linewidth: the narrowest a transition can ever be. It is homogeneous broadening, identical for every atom regardless of velocity or environment, so it cannot be squeezed out by cooling, trapping, or better collimation — the price of coupling to the electromagnetic vacuum. Its inverse relationship with lifetime — a nanosecond ringdown smears a line by megahertz, a second-long metastable one by millihertz — is what makes forbidden clock transitions the sharpest rulers in physics.

Key takeaways

An excited state decays spontaneously at rate $\Gamma = A_{21}$ , giving exponential decay and a $1/e$ radiative lifetime $\tau = 1/\Gamma$ .
The finite lifetime truncates the emitted wave, so its Fourier transform is a Lorentzian of FWHM $\Delta\nu = \Gamma/2\pi = 1/(2\pi\tau)$ — the natural linewidth.
This is the intrinsic homogeneous linewidth associated with spontaneous emission alone: in the absence of other broadening mechanisms, an isolated atom has the corresponding natural Lorentzian line shape.
Decay channels add rates, $\Gamma_{\mathrm{tot}}=\sum_f A_{2f}$ , and the two levels a line connects add their widths: $\Delta\omega=\Gamma_2+\Gamma_1$ .
Allowed optical lines have $\tau\sim1$ – $30\,\mathrm{ns}$ , $\Delta\nu\sim5$ – $160\,\mathrm{MHz}$ ; metastable clock states live seconds and are ultra-narrow at the millihertz level.
Key idea: a state that decays exponentially at rate $\Gamma$ radiates a truncated wave whose Fourier transform is a Lorentzian of FWHM $\Delta\nu=\Gamma/2\pi=1/(2\pi\tau)$ ; lifetime and linewidth are Fourier conjugates, so $\Delta\nu\,\tau=1/2\pi$ .

Where this lives today

The natural linewidth is the design parameter behind many of the most precise instruments ever built. In a rubidium magneto-optical trap, the $6\,\mathrm{MHz}$ natural width of the cooling transition sets the Doppler cooling limit $T_D=\hbar\Gamma/(2k_B)$ — about $150\,\mu\mathrm{K}$ for the Rb D2 line — because the same $\Gamma$ that broadens the line determines the velocity-selective response to red- and blue-detuned light. Optical lattice clocks invert the logic: the transition is chosen to be highly forbidden so its natural width shrinks to the millihertz level of the second worked example, giving line quality factors above $10^{17}$ and enabling fractional frequency uncertainties at the $10^{-18}$ level — sufficient to resolve gravitational redshift over centimetre-scale height differences. Between these extremes, saturated-absorption spectroscopy exploits the natural width's homogeneous character: a strong pump saturates only the zero-velocity atoms, and a counter-propagating probe reads a Lamb dip at line centre whose width approaches the natural limit, exposing hyperfine structure the Doppler profile would smear away. The same $\Gamma$ appears in laser physics, where $g(\nu)$ sets the gain cross-section's frequency dependence, so a narrow line concentrates gain and lowers threshold. One number — the spontaneous decay rate — quietly sets the temperature of a cold-atom cloud, the accuracy of the world's best clocks, the resolution of precision spectroscopy, and the threshold of a laser.

The physical picture

Picture the excited atom as a bell that has just been struck. Its dipole oscillates near the natural frequency $\omega_0$ , while energy radiates away, so the amplitude decays exponentially and the note fades. A bell ringing forever makes a perfectly pure tone — a single spectral spike; a bell that fades over a time $\tau$ cannot, because building a note that starts and stops means superposing a band of frequencies spanning roughly $1/\tau$ . That unavoidable spread is the natural linewidth. The tighter the pitch, the longer the bell must ring; long life and sharp colour are two faces of one Fourier coin.

Figure

A prepared excited atom radiates like a struck bell ringing down — the dipole field E(t) ∝ e^(−Γt/2) cos ω₀t decays as its energy leaks away, and the intensity ∝ e^−Γt falls with lifetime τ = 26.2 ns for the ⁸⁷Rb 5P₃/₂ state, a finite wave train rather than an endless sinusoid

Because spontaneous emission is an intrinsic property of an isolated atom, identical atoms with the same transition frequency have the same natural linewidth. In an ideal ensemble with no inhomogeneous broadening, this produces a common homogeneous Lorentzian line. This is homogeneous broadening, and it is why the natural line behaves so differently from the Gaussian Doppler profile, where each atom is monochromatic but the crowd of velocities smears the centre. Ask what colours the ringdown of the first figure contains, and the answer is the second: a Lorentzian peaked at $\omega_0$ whose full width at half maximum is exactly the decay rate $\Gamma$ .

Figure

Fourier-transforming that finite ringdown gives a Lorentzian power spectrum centred at ω₀ with full width at half maximum Γ — for ⁸⁷Rb the natural width is Δν = Γ/2π = 6.07 MHz, the irreducible floor set by spontaneous emission alone

The relation $\Gamma\tau = 1$ is often dressed as the energy–time uncertainty relation $\Delta E\,\Delta t \gtrsim \hbar$ , with $\Delta E = \hbar\Gamma$ and $\Delta t = \tau$ — correct, but it undersells the physics: the exponential-decay derivation is an exact Fourier statement, an equality, not the inequality of the uncertainty principle. The line has the width it has because the wave has the length it has.

The core equations

The chain begins with how the population empties. Spontaneous emission gives every excited atom the same constant decay probability per unit time, and a constant per-atom rate produces exponential decay; the decay constant is the Einstein $A$ coefficient, and its reciprocal is the lifetime.

N_2(t) = N_2(0)\,e^{-\Gamma t}, \qquad \Gamma = A_{21}, \qquad \tau = \frac{1}{\Gamma} \qquad (1)

where: $N_2$ — population (number density) of the excited state; $\Gamma$ — total spontaneous decay rate in $\mathrm{s^{-1}}$ ; $A_{21}$ — Einstein spontaneous-emission coefficient; $\tau$ — radiative lifetime, the $1/e$ decay time of the population. Read it as: after one lifetime a fraction $1/e\approx37\%$ of the atoms remain excited. What an observer detects is the radiated field, and here lies the factor that causes most errors. Radiated power (intensity) tracks the population, falling as $e^{-\Gamma t}$ ; the field amplitude is its square root, falling at half that rate — a cosine at $\omega_0$ inside an envelope decaying at $\Gamma/2$ , switched on at excitation.

E(t) = E_0\,e^{-\Gamma t/2}\,e^{-i\omega_0 t}\;\Theta(t) \qquad (2)

where: $E_0$ — initial field amplitude; $\omega_0$ — transition angular frequency; $\Gamma/2$ — amplitude decay rate, half the intensity rate; $\Theta(t)$ — Heaviside step, emission begins at excitation. The physical reading: a truncated, damped sinusoid — the wave train of the first figure. Ask what frequencies this wave contains and you Fourier-transform it; the modulus-squared of the transform is the observed spectrum, and it is a Lorentzian centred on $\omega_0$ . This is the natural line shape, derived in full below.

I(\omega) \propto \frac{1}{(\omega-\omega_0)^2 + (\Gamma/2)^2} \qquad (3)

where: $I(\omega)$ — spectral power density; $\omega-\omega_0$ — detuning from line centre; $\Gamma$ — FWHM in angular frequency. The physical reading: a sharp central peak with slowly-decaying $1/(\omega-\omega_0)^2$ wings, far heavier in the tails than a Gaussian. For quantitative work the line shape is normalised to unit area, so it serves as a probability density over frequency and slots into absorption and gain cross-sections.

g(\omega) = \frac{\Gamma/2\pi}{(\omega-\omega_0)^2 + (\Gamma/2)^2}, \qquad \int_{-\infty}^{\infty} g(\omega)\,\mathrm{d}\omega = 1 \qquad (4)

where: $g(\omega)$ — area-normalised lineshape, units of time; $\Gamma$ — FWHM in angular frequency. The physical reading: $g(\omega_0)=2/\pi\Gamma$ at the peak, so a narrower line is not just thinner but taller — the same photons in a smaller band. The width is read straight off equation (3): the intensity halves where the detuning equals $\pm\Gamma/2$ , so the full width is $\Gamma$ . Converting from angular to ordinary frequency divides by $2\pi$ — where the working number lives.

\Delta\omega_{\mathrm{FWHM}} = \Gamma, \qquad \Delta\nu_{\mathrm{FWHM}} = \frac{\Gamma}{2\pi} = \frac{1}{2\pi\tau} \qquad (5)

where: $\Delta\omega$ — FWHM in angular frequency; $\Delta\nu$ — FWHM in ordinary frequency (Hz); $\tau$ — radiative lifetime. The physical reading: quote widths in $\Delta\nu$ (megahertz), never forgetting the $2\pi$ that separates it from $\Gamma$ . Rearranged, this is the lifetime–linewidth relation, and multiplying by $\hbar$ gives the natural energy width $\Delta E=\hbar\Gamma=\hbar/\tau$ . This is the exact linewidth result for an exponentially decaying state, rather than an uncertainty-principle inequality.

\Delta\nu\,\tau = \frac{1}{2\pi}, \qquad \Delta E = \hbar\Gamma = \frac{\hbar}{\tau} \qquad (6)

where: $\Delta E$ — energy width of the excited level; $\hbar$ — reduced Planck constant. The physical reading: shorter-lived means broader, with the product pinned to $1/2\pi$ — an equality, not a bound. A real line connects two levels, and if either is unstable it contributes to the width. The decay rates of the two states add, each being the sum of that state's partial decay channels.

\Delta\omega = \Gamma_2 + \Gamma_1, \qquad \Gamma_i = \sum_{f} A_{if} \qquad (7)

where: $\Gamma_2,\Gamma_1$ — total decay rates of the upper and lower levels; $A_{if}$ — partial rate from level $i$ to final state $f$ . The physical reading: decay to a stable ground state sets $\Gamma_1=0$ and recovers the single-level width; branching to several finals is why an upper-state lifetime and a specific line's strength differ. Finally, the spontaneous-emission rate is determined by the transition dipole matrix element together with the transition frequency, linking the spectral linewidth to the underlying atomic wavefunctions.

A_{21} = \frac{\omega_0^3\,|\mathbf{d}_{21}|^2}{3\pi\varepsilon_0\hbar c^3} \qquad (8)

where: $\mathbf{d}_{21}$ — transition electric-dipole matrix element; $\varepsilon_0$ — vacuum permittivity; $c$ — speed of light. The physical reading: the steep $\omega_0^3$ dependence makes high-frequency transitions decay far faster for the same dipole — ultraviolet lines broad, forbidden infrared lines narrow.

Derivation AThe Lorentzian line shape from the damped ringdown

The natural line shape is the frequency content of a finite wave train, obtained by Fourier-transforming the radiated field of a dipole that switches on at excitation and rings down. The goal is to show explicitly that an exponentially damped oscillation at $\omega_0$ produces a Lorentzian whose full width at half maximum is exactly the population decay rate $\Gamma$ — tying the line shape rigorously to the lifetime, not to uncertainty-principle hand-waving. We start from the field, transform it, square it, and read off the width.

Step 1. Write the radiated field as a cosine at $\omega_0$ inside an exponentially falling envelope, switched on at $t=0$ : $E(t)=E_0\,e^{-\Gamma t/2}e^{-i\omega_0 t}$ for $t\ge0$ and zero before. Why: the amplitude must decay at $\Gamma/2$ so that the intensity $|E|^2\propto e^{-\Gamma t}$ follows the excited population, which is the only rate the physics actually fixes.

Step 2. Take the Fourier transform, an elementary exponential integral that converges because of the damping: $\tilde E(\omega)=\int_0^\infty E_0\,e^{-\Gamma t/2}e^{-i\omega_0 t}e^{i\omega t}\,\mathrm{d}t = \dfrac{E_0}{\Gamma/2 + i(\omega_0-\omega)}$ . Why: the finite wave train has a definite, convergent spectrum; without damping the integral diverges and the spectrum collapses to a delta spike, the monochromatic idealisation.

Step 3. Form the power spectrum as the modulus squared: $I(\omega)=|\tilde E(\omega)|^2 = \dfrac{E_0^2}{(\omega-\omega_0)^2 + (\Gamma/2)^2}$ . Why: a detector registers power, not field, and squaring the complex Lorentzian amplitude gives the real Lorentzian intensity.

Step 4. Locate the half-maximum: $I$ falls to half its peak where $(\omega-\omega_0)^2=(\Gamma/2)^2$ , i.e. at $\omega=\omega_0\pm\Gamma/2$ , so the full width is $\Gamma$ and $\Delta\nu=\Gamma/2\pi$ . Why: the payoff — the spectral width equals the decay rate exactly, converting a measured linewidth directly into a lifetime.

Go deeper · Derivation B · 1 minFrom the transition dipole moment to the lifetime

Derivation BFrom the transition dipole moment to the lifetime

Everything numerical about a transition — its lifetime, natural width, and oscillator strength — is set by one quantity, the transition dipole matrix element $\mathbf{d}_{21}$ . The goal is to see where the decay rate $A_{21}$ comes from and why it scales as $\omega_0^3|\mathbf{d}_{21}|^2$ , then to invert it: a measured fluorescence lifetime delivers $A_{21}=1/\tau$ and hence the dipole moment — the primary experimental route to tabulated line strengths. We build the rate from Fermi's golden rule, track the frequency powers, and read the chain backwards.

Step 1. Apply Fermi's golden rule to the atom coupled to the continuum of vacuum modes in the electric-dipole approximation: the rate is proportional to $|\mathbf{d}_{21}|^2$ times the photon mode density at $\omega_0$ times a coupling factor. Why: spontaneous emission is stimulated emission into the empty vacuum modes, so the same dipole coupling that drives absorption sets the decay, weighted by the modes available to receive the photon.

Step 2. Count the frequency powers: the density of states rises as $\omega^2$ and the field-per-mode carries one more factor of $\omega$ , giving $A_{21}=\omega_0^3|\mathbf{d}_{21}|^2/3\pi\varepsilon_0\hbar c^3$ . Why: the $\omega_0^3$ is why ultraviolet transitions decay in nanoseconds while same-dipole infrared ones crawl — the single strongest lever on the lifetime.

Step 3. Invert to the lifetime, $\tau=1/A_{21}=3\pi\varepsilon_0\hbar c^3/(\omega_0^3|\mathbf{d}_{21}|^2)$ , so a time-resolved fluorescence measurement of $\tau$ yields $|\mathbf{d}_{21}|$ once $\omega_0$ is known. Why: this closes the loop between an easily measured decay time and the dipole matrix element that otherwise needs a full atomic-structure calculation — how transition-strength tables are built.

Limiting cases

Case	Result	Meaning
Allowed optical transition	$\tau\sim1$ – $30\,\mathrm{ns}\;\Rightarrow\;\Delta\nu\sim5$ – $160\,\mathrm{MHz}$	Strong electric-dipole lines (alkali resonance lines) decay in nanoseconds; the natural width is real but usually far below the room-temperature Doppler width of $\sim1\,\mathrm{GHz}$ .
Metastable / forbidden state	$\tau\gtrsim1\,\mathrm{s}\;\Rightarrow\;\Delta\nu\lesssim1\,\mathrm{Hz}$	Dipole-forbidden transitions have tiny $A_{21}$ ; the resulting ultra-narrow lines are the basis of optical clocks, where other broadenings dominate entirely.
Two decaying levels	$\Delta\omega=\Gamma_2+\Gamma_1$	The natural homogeneous widths add; for a stable ground state $\Gamma_1=0$ and only the upper-state width appears, while a short-lived lower level contributes additional natural broadening.
Doppler-dominated regime	$\Delta\nu_{\mathrm D}\gg\Delta\nu_{\mathrm{nat}}$	In a warm vapour the Gaussian Doppler width buries the Lorentzian core; the natural width survives only in the far wings and is recovered by sub-Doppler techniques.
Very high frequency	$A_{21}\propto\omega_0^3$	For a fixed dipole, ultraviolet and X-ray lines decay far faster than infrared ones, so their natural widths are larger.

Common trapThe field amplitude and the intensity decay at different rates — but both give the same linewidth Γ

The single most common numerical error here is a stray factor of two. The excited-state population and the radiated intensity both decay at the full rate $\Gamma$ , but the field amplitude decays at only $\Gamma/2$ , because intensity is amplitude squared. Put $\Gamma$ into the field envelope and the line comes out twice too wide; put $\Gamma/2$ into the intensity and it comes out twice too narrow. The correct chain: population and intensity $\propto e^{-\Gamma t}$ , amplitude $\propto e^{-\Gamma t/2}$ , Lorentzian FWHM $=\Gamma$ (angular). The second stray factor is the $2\pi$ between $\Gamma$ and $\Delta\nu=\Gamma/2\pi$ : a lifetime of $10\,\mathrm{ns}$ gives $16\,\mathrm{MHz}$ , not $100\,\mathrm{MHz}$ . Keep the two rates and the $2\pi$ straight and the arithmetic is trivial.

Worked examples

1 · The natural linewidth of the rubidium-87 D2 line (the workhorse of laser cooling)

The $5^2S_{1/2}\to5^2P_{3/2}$ transition of $^{87}\mathrm{Rb}$ at $780\,\mathrm{nm}$ has lifetime $\tau=26.2\,\mathrm{ns}$ . Find its decay rate, natural linewidth in MHz, and energy width in eV, and compare with the room-temperature Doppler width.

Step 1. Invert the lifetime: $\Gamma = 1/\tau = 1/(26.2\times10^{-9}\,\mathrm{s}) = 3.82\times10^{7}\,\mathrm{s^{-1}}$ — this is also the Lorentzian FWHM in angular frequency.

Step 2. Divide by $2\pi$ for the ordinary-frequency FWHM: $\Delta\nu = \Gamma/2\pi = 3.82\times10^{7}/(2\pi)\,\mathrm{Hz} = \mathbf{6.07\ MHz}$ , the natural linewidth every laser-cooling experiment quotes.

Step 3. Energy width: $\Delta E = \hbar\Gamma = (1.055\times10^{-34})(3.82\times10^{7})\,\mathrm{J} = 4.0\times10^{-27}\,\mathrm{J} = \mathbf{2.5\times10^{-8}\ eV}$ — utterly tiny beside the $1.6\,\mathrm{eV}$ photon energy, so the level is extraordinarily sharp.

Step 4. Compare to Doppler: at $300\,\mathrm{K}$ this line's Gaussian Doppler FWHM is about $510\,\mathrm{MHz}$ , roughly $84$ times the natural width. In a warm vapour the $6.07\,\mathrm{MHz}$ natural line is completely buried under the $\sim0.5\,\mathrm{GHz}$ Doppler profile, recovered only by sub-Doppler methods.

2 · The line quality factor of a strontium optical clock

The $^{1}S_0\to{}^{3}P_0$ clock transition of $^{87}\mathrm{Sr}$ at $698\,\mathrm{nm}$ ( $\nu_0=4.29\times10^{14}\,\mathrm{Hz}$ ) is doubly forbidden, with a natural lifetime of about $150\,\mathrm{s}$ . Find its natural linewidth and line quality factor $Q=\nu_0/\Delta\nu$ , and say why clocks use such transitions.

Step 1. The natural linewidth is $\Delta\nu=1/(2\pi\tau)=1/(2\pi\times150\,\mathrm{s})=1.06\times10^{-3}\,\mathrm{Hz}=1.06\,\mathrm{mHz}$ — a millihertz-wide optical line, about $5.7\times10^9$ times narrower than the Rb D2 natural linewidth of $6.07\,\mathrm{MHz}$ .

Step 2. The line quality factor is $Q = \nu_0/\Delta\nu = (4.29\times10^{14})/(1.06\times10^{-3}) = \mathbf{4.0\times10^{17}}$ . For comparison, the Rb D2 line has $Q\approx3.84\times10^{14}/6.07\times10^{6}\approx6\times10^{7}$ — ten orders of magnitude worse.

Step 3. The physics of the choice: fractional frequency stability improves with $Q$ , so a clock benefits from a very narrow transition, and equation (5) shows that a long lifetime produces a narrow natural linewidth. The $\sim150\,\mathrm{s}$ metastable lifetime of the clock state gives $Q\sim4\times10^{17}$ , providing an exceptionally sharp optical frequency reference and enabling lattice clocks to reach fractional frequency uncertainties at the $10^{-18}$ level.

Physics of the portrait

The natural linewidth is the first place the quantum vacuum wrote itself into a spectrum, and the person who first computed its shape from quantum electrodynamics gives this topic its face.

Origins · 1 minVictor Weisskopf and how this was discovered

Victor Weisskopf

1908–2002 · the quantum theory of natural line width

Victor Weisskopf was a twenty-two-year-old at Göttingen in 1930 when he and Eugene Wigner published the calculation that still bears both their names: the Weisskopf–Wigner theory of spontaneous emission. Dirac had shown in 1927 that quantising the electromagnetic field forces an excited atom to decay even in perfect vacuum, but it was Weisskopf and Wigner who solved the coupled equations for the atom radiating into the continuum of vacuum modes and proved that the upper-state amplitude decays exponentially — a clean $e^{-\Gamma t/2}$ — so the emitted spectrum is a Lorentzian of width $\Gamma$ . Their result turned the natural linewidth from an empirical fact into a theorem: the line has the width it has because the state decays the way it does, both fixed by the coupling to the vacuum. Weisskopf went on to compute the electron's self-energy (pointing toward renormalisation), work at Los Alamos, and direct CERN — but the ringdown-into-Lorentzian argument he set down as a student remains the standard derivation in every quantum-optics course, and it is why a measured lifetime and a measured linewidth are, to a spectroscopist, two ways of quoting the same number.

History

1896

— Lorentz models the emitting atom as a damped oscillator, predicting a Lorentzian line shape

1916

— Einstein introduces the A and B coefficients, fixing the spontaneous decay rate that sets the lifetime

1927

— Dirac quantizes the electromagnetic field and derives spontaneous emission from first principles

1930

— Weisskopf and Wigner obtain the exponential decay and the natural Lorentzian line width from quantum electrodynamics

1947

— Lamb and Retherford measure the vacuum-induced level shift in hydrogen, confirming the atom's coupling to the vacuum

Going further

Texts. Foot, Atomic Physics (2005), §7.1–7.4 · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 2 & 8 · Demtröder, Laser Spectroscopy, Vol. 1, §3.1 · Saleh & Teich, Fundamentals of Photonics, 3rd ed., §13.3 · Hecht, Optics, 5th ed., §7.4.3 (line broadening). Key numerical anchors: the Fourier transform of $e^{-\Gamma t/2}e^{-i\omega_0 t}\Theta(t)$ yields a Lorentzian of FWHM $\Gamma$ ; applying $\Delta\nu=1/(2\pi\tau)$ gives $\tau=10\,\mathrm{ns}\to\Delta\nu\approx16\,\mathrm{MHz}$ , $\tau=26.2\,\mathrm{ns}\to6.07\,\mathrm{MHz}$ (Rb-87 D2), and $\tau=150\,\mathrm{s}\to1.06\,\mathrm{mHz}$ (Sr-87 clock, $Q\approx4\times10^{17}$ ); the decay-rate formula is $A_{21}=\omega_0^3|\mathbf{d}_{21}|^2/3\pi\varepsilon_0\hbar c^3$ . Note on attribution: the widely repeated " $\tau=26\,\mathrm{ns}$ sodium D line" is inaccurate — sodium D2 lives $16.2\,\mathrm{ns}$ ( $\Delta\nu\approx9.8\,\mathrm{MHz}$ ); the $26.2\,\mathrm{ns}$ value is the rubidium-87 D2 line, used throughout this article.

Prerequisites. Einstein A and B Coefficients · Spontaneous Emission · Lorentzian Line Shape · Transition Dipole Moment. Leads to. Doppler Broadening and Gaussian Line Shape · Oscillator Strength · Resonant Absorption · Selection Rules for Dipole Transitions.

Ask the tutor: Give me any excited-state lifetime and I will turn it into a natural linewidth, an energy width, and a line quality factor — and tell you whether Doppler broadening would bury it.

Why is the natural linewidth called the "irreducible" limit?

Because it comes from spontaneous emission, which no experimental trick can switch off. Every other broadening can in principle be removed: Doppler broadening vanishes as you cool the atoms, collision broadening as you lower the density, transit-time broadening as you enlarge the beam. But spontaneous emission is the atom's coupling to the electromagnetic vacuum, which is everywhere and always present, so the Lorentzian of width $\Gamma$ is the floor beneath every real line. The only way to narrow it is to choose a transition with a longer lifetime — a physically different transition — not to change the conditions. That is exactly why optical clocks are built on forbidden transitions rather than well-cooled allowed ones.

Why does the field amplitude decay at Γ/2 rather than Γ?

Because $\Gamma$ is defined on the population, and what you observe as brightness is the intensity, which is the field amplitude squared. If the population — and the radiated power $\propto|E|^2$ — falls as $e^{-\Gamma t}$ , then the square root gives $|E|\propto e^{-\Gamma t/2}$ . The factor of two is not a subtlety about two decay processes; it is just the arithmetic of squaring. The two descriptions agree on the observable: the amplitude's $\Gamma/2$ envelope Fourier-transforms to a Lorentzian of FWHM $\Gamma$ , because a Lorentzian's width is twice the amplitude decay rate. The factor of two in the envelope is exactly cancelled by the factor of two in the width formula.

Is the energy–time uncertainty relation the real reason lines have width?

It is a correct summary but a misleading explanation. The relation $\Delta E\,\Delta t\gtrsim\hbar$ with $\Delta E=\hbar\Gamma$ and $\Delta t=\tau$ gives the right order of magnitude and captures the intuition that a state which does not last forever cannot have a perfectly sharp energy. But energy–time uncertainty is an inequality with a fuzzy "greater than," whereas the natural linewidth is an equality: the exponential-decay derivation pins $\Delta\nu\,\tau$ to exactly $1/2\pi$ . The width is not a lower bound the line happens to saturate; it is the precise Fourier transform of a precisely exponential decay. Treat the uncertainty relation as a mnemonic for the scale, and the Fourier argument as the actual physics.

How is the natural linewidth different from Doppler broadening?

They are opposite in character. Natural broadening is homogeneous: every atom, whatever its velocity, produces the identical Lorentzian, because all couple to the same vacuum. Doppler broadening is inhomogeneous: each atom is essentially monochromatic, but the spread of thermal velocities shifts different atoms' centre frequencies, and the observed line is the Gaussian envelope of that distribution. The distinction is not academic: because natural broadening is homogeneous, a saturating pump affects the whole line at once and a Lamb dip of near-natural width can be burned into the Doppler profile, whereas the inhomogeneous Doppler case lets you address one velocity class at a time. Their shapes differ too — the Lorentzian's far heavier wings dominate the far detunings even when the Doppler width is larger at the core.

If a transition can decay to several lower levels, which lifetime sets the linewidth?

The upper level's width is set by its total decay rate, $\Gamma_2=\sum_f A_{2f}$ , summed over every channel — the reciprocal of the measured fluorescence lifetime. That total width broadens every line from the level equally, whatever final state it goes to; what differs between lines is their strength (the branching ratio $A_{2f}/\Gamma_2$ ), not their width. If the lower level is itself unstable, its width $\Gamma_1$ adds on top, giving $\Delta\omega=\Gamma_2+\Gamma_1$ . So linewidth is a property of the two levels' total lifetimes, while brightness is a property of the specific branching — worth keeping straight when reading lifetime tables.

Why do optical atomic clocks deliberately use forbidden transitions?

Because a clock's precision scales with the line quality factor $Q=\nu_0/\Delta\nu$ , and the only way to make $\Delta\nu$ tiny at a fixed optical frequency is to make the lifetime enormous — which means a transition the dipole selection rules forbid, so $A_{21}$ is minuscule. The strontium clock transition of the second example lives about $150\,\mathrm{s}$ , giving a natural width near a millihertz and $Q\sim4\times10^{17}$ , ten orders of magnitude better than an allowed line. The cost is that it is hard to drive at all, needing intense, ultra-stable clock lasers and often field- or lattice-induced mixing to give it enough transition probability to interrogate. The natural linewidth is thus not a nuisance but the very resource the clock exploits. (It can in fact be reshaped by engineering the vacuum — a cavity alters the mode density and hence $A_{21}$ , the Purcell effect — but in free space it is the fixed floor.)