Lifetimes, Line Widths, and Line Shapes

The spontaneous-emission rate $A_{21}$ sets how fast an excited atom decays. That finite lifetime has a spectral consequence: a line of nonzero width. A state that lived forever would emit a pure sinusoid at a single frequency; a state that decays emits a truncated wave train, and a truncated wave train contains a spread of frequencies. This lesson works out the width and shape of a spectral line from three mechanisms — the natural width from the finite lifetime, the Doppler width from thermal motion, and the pressure width from collisions — and combines them into the observed profile. The shapes differ (Lorentzian, Gaussian, and their Voigt convolution), and so do their dependences on temperature, density, and driving intensity, which is what lets an experimenter diagnose and defeat them.

Lifetime and the exponential decay

An atom in an excited state $∣ 2 ⟩$ that can decay to several lower states has a total decay rate equal to the sum of the individual spontaneous rates,

Γ = k \sum A_{2 k},

and the excited-state population of an ensemble decays exponentially,

N_{2} (t) = N_{2} (0) e^{- Γ t} = N_{2} (0) e^{- t / τ}, τ = \frac{1}{Γ} .

The lifetime $τ$ is the reciprocal of the total decay rate, and a level with several decay channels lives no longer than its fastest channel allows. For a strong optical transition $Γ \sim 1 0^{8} s^{- 1}$ and $τ \sim 10 ns$ ; a metastable level with only forbidden channels can have $τ$ of seconds or longer.

The excited-state population decays exponentially at the total rate Gamma, with 1/e lifetime tau. The emitted wave train is correspondingly damped, and its finite duration is what gives the line a nonzero width.

The natural line width

Treat the emitted classical field as a damped oscillator. While the atom decays, the radiated field amplitude follows the excited-state amplitude, which decays at half the population rate (the population goes as amplitude squared):

E (t) = E_{0} e^{- i ω_{0} t} e^{- t /2 τ}, t \geq 0,

and $E (t) = 0$ for $t < 0$ . The spectrum of the emitted light is the squared modulus of the Fourier transform of $E (t)$ :

\tilde{E} (ω) = E_{0} 0 \int \infty e^{i (ω - ω_{0}) t} e^{- t /2 τ} d t = \frac{E _{0}}{\frac{1}{2 τ} - i ( ω - ω _{0} )} .

Taking the modulus squared gives the Lorentzian line shape,

I (ω) \propto \frac{1}{( ω - ω _{0} ) ^{2} + ( Γ/2 ) ^{2}}, Γ = \frac{1}{τ},

a symmetric peak centered at $ω_{0}$ with full width at half maximum in angular frequency equal to $Γ$ . In ordinary frequency the natural line width is

Δ ν_{nat} = \frac{Γ}{2 π} = \frac{1}{2 π τ} .

A quantum treatment (the Weisskopf-Wigner theory) reproduces the same Lorentzian with the same width; the classical damped oscillator is not a coincidence but the correct envelope. The natural width is the time-energy uncertainty relation made quantitative: a state of lifetime $τ$ has an energy uncertainty $Δ E = ℏΓ = ℏ/ τ$ , so

Δ E τ \approx ℏ.

For the sodium $D$ lines $τ \approx 16 ns$ , so $Δ ν_{nat} \approx 10 MHz$ — a few parts in $1 0^{8}$ of the optical frequency. This is narrow, but in a room-temperature vapor it is swamped by thermal motion.

Doppler broadening

Atoms in a gas move, and a moving atom radiating at rest-frame frequency $ω_{0}$ is seen shifted by the first-order Doppler effect. An atom with velocity component $v_{z}$ along the line of sight is observed at

ω = ω_{0} (1 + \frac{v _{z}}{c}) .

In thermal equilibrium the velocity component $v_{z}$ has a Maxwell-Boltzmann distribution,

P (v_{z}) d v_{z} = \frac{m}{2 π k _{B} T} e^{- m v_{z}^{2} /2 k_{B} T} d v_{z},

a Gaussian of width $k_{B} T / m$ . Mapping velocities to frequencies through the Doppler relation transfers the Gaussian to the frequency axis: the line shape is

I (ω) \propto exp [- \frac{m c ^{2} ( ω - ω _{0} ) ^{2}}{2 k _{B} T ω _{0}^{2}}],

a Gaussian centered at $ω_{0}$ . Its full width at half maximum, converted to frequency, is the Doppler width

Δ ν_{D} = \frac{ν _{0}}{c} \frac{8 k _{B} T ln 2}{m} .

Two dependences distinguish Doppler broadening from natural broadening: it scales as $T$ and as $1/ m$ , so it grows with temperature and shrinks for heavier atoms. For sodium at $500 K$ , $Δ ν_{D} \approx 1.7 GHz$ , more than a hundred times the natural width. In a vapor cell the observed line is Doppler-dominated and Gaussian, and its narrow Lorentzian core is hidden — the reason Doppler-free techniques exist.

Doppler width grows as the square root of temperature (and as one over the square root of atomic mass). A hotter or lighter gas gives a broader Gaussian line; cooling the sample narrows it.

Pressure (collisional) broadening

Collisions with other atoms interrupt the radiating wave train. Each collision randomizes the phase of the emitted field, so instead of one long damped sinusoid the atom emits a sequence of shorter uncorrelated segments. Shorter coherent segments mean a broader spectrum, by the same Fourier argument as the natural width. If the mean time between phase-interrupting collisions is $τ_{col}$ , the resulting profile is again Lorentzian, with an added width

Δ ν_{col} = \frac{1}{π τ _{col}} = \frac{n v ˉ σ}{π},

where $n$ is the perturber number density, $\overset{v}{ˉ}$ the mean relative speed, and $σ$ the collision cross section. The key signature is the linear dependence on density (hence pressure at fixed temperature): pressure broadening grows in proportion to how many perturbers there are. Extrapolating the width to zero pressure isolates the natural width, and the slope measures the collision cross section.

Pressure broadening grows linearly with perturber density. The intercept at zero density is the pressure-independent width (natural plus Doppler); the slope encodes the collision cross section.

The Voigt profile

A real line carries both kinds of broadening at once: the Lorentzian natural and pressure widths, and the Gaussian Doppler width. Because the Doppler shift and the lifetime broadening are statistically independent, the observed profile is their convolution,

I_{V} (ω) = - \infty \int \infty G (ω^{'}) L (ω - ω^{'}) d ω^{'},

the Voigt profile. It has no elementary closed form, but its limits are clear: where the Gaussian dominates (low pressure, light atom, high temperature) the core is Gaussian; the Lorentzian always wins in the far wings, because a Gaussian falls off as $e^{- x^{2}}$ while a Lorentzian falls only as $1/ x^{2}$ . The wings of any real line are therefore Lorentzian, and the natural or pressure width can be read from them even when the Gaussian Doppler core hides it near the center.

Three line profiles of equal width and height. The Gaussian (Doppler) falls off fastest; the Lorentzian (natural, pressure) has heavy far wings; the Voigt convolution has a Gaussian-like core and Lorentzian wings.

Saturation and power broadening

The mechanisms above are properties of a weakly probed atom. A strong driving field adds one more. When the field is intense enough to cycle population between the levels faster than they can decay, the transition saturates, and the line broadens. The saturated width is

Δ ν = Δ ν_{nat} 1 + s, s = \frac{I}{I _{sat}},

where $s$ is the saturation parameter, $I$ the intensity, and $I_{sat}$ the saturation intensity of the transition. Power broadening is the driven-transition version of the width: the more strongly the atom is interrogated, the broader the line it presents, which sets a floor on how hard a precision measurement can push before the resolution it seeks is destroyed. The effect connects directly to the Rabi picture of the first lesson: a large Rabi frequency $Ω_{R}$ broadens the response, and $s = 2 Ω_{R}^{2} / Γ^{2}$ .

Transit-time and instrumental widths

Two further contributions limit real measurements. Transit-time broadening arises when an atom crosses a finite laser beam of diameter $d$ in a time $τ_{t} \sim d / \overset{v}{ˉ}$ ; the interaction is a truncated wave train just as a collision-limited one is, so it contributes a width $Δ ν_{t} \sim \overset{v}{ˉ} / d$ . Widening the beam or slowing the atoms reduces it, which is one more reason cold samples sharpen spectra. Instrumental broadening is the finite resolution of the spectrometer or the finite linewidth of the probe laser; it convolves with the atomic profile exactly as Doppler broadening does and must be measured separately (often with a reference line of known width) and deconvolved.

Because the Lorentzian and Gaussian contributions convolve rather than add, their widths do not combine linearly. The Lorentzian widths (natural, pressure, transit-time) add directly,

Γ_{L} = Γ_{nat} + Γ_{col} + Γ_{t},

while independent Gaussian widths (Doppler, some instrumental) add in quadrature,

Γ_{G}^{2} = Γ_{D}^{2} + Γ_{inst}^{2} .

The observed Voigt profile carries both $Γ_{L}$ and $Γ_{G}$ as independent parameters, and fitting the full shape recovers each: the Gaussian width reports the temperature, and the Lorentzian width, extrapolated to zero pressure and zero probe power, isolates the natural width and thereby the lifetime. Reading a line shape is a small inverse problem, and the distinct parameter dependences in the table below are what make it solvable.

The mechanisms compared

The four broadening mechanisms are distinguished by how their widths depend on the controllable parameters — temperature, density, intensity — and by their shapes.

Mechanism	Shape	Width scales as	Controlled by
Natural	Lorentzian	$1/ τ$ , fixed	the transition itself
Doppler	Gaussian	$T / m$	cooling, heavier isotope
Pressure	Lorentzian	$n \overset{v}{ˉ} σ$	lowering density
Power	Lorentzian	$1 + I / I_{sat}$	weaker probe

The hierarchy in a typical room-temperature vapor cell runs Doppler ( $GHz$ ) $≫$ pressure ( $MHz$ to $GHz$ , depending on pressure) $≳$ natural ( $MHz$ ). Reaching the natural width therefore means removing the Doppler and pressure contributions: work at low pressure to kill collisions, and use a Doppler-free method (saturated absorption, two-photon excitation, or a cold atomic sample) to eliminate the Gaussian. Those techniques are the subject of the spectroscopy lesson, and driving the natural width to its floor is the whole design goal of an optical clock.

The width hierarchy in a room-temperature vapor. Doppler broadening dominates at the gigahertz scale; pressure broadening is intermediate; the natural width is the megahertz-scale floor reached only after the others are removed.

A spectral line is a measurement of everything that limited the coherence of the emission that produced it. Its center gives the transition energy, corrected by fine, hyperfine, and QED shifts; its width and shape give the lifetime, the temperature, the pressure, and the probe intensity. Reading a line is reading all of them at once, and separating the contributions — Lorentzian from Gaussian, natural from collisional, intrinsic from instrumental — is the daily work of precision spectroscopy. This closes the module: the golden rule gave the rate, the dipole approximation gave the coupling, the selection rules gave which lines appear, and the line shape gives what each one carries.