Natural, Doppler, and collision linewidths

Let optical coherence decay as $\exp(-\gamma_2t)$ for positive time. Its Fourier response has a denominator $\gamma_2-i(\omega-\omega_0)$; the absorptive line or emitted spectral intensity has a Lorentzian dependence with angular-frequency half width $\gamma_2$. Thus its angular-frequency FWHM is $2\gamma_2$, while ordinary-frequency FWHM is $\gamma_2/\pi$. Keeping both definitions visible prevents a factor-of-two error from mixing lifetime and amplitude decay.

If the upper and lower levels have population loss rates $\Gamma_u$ and $\Gamma_l$, spontaneous loss contributes half their sum to coherence decay. Independent Markovian pure dephasing adds another term without necessarily changing populations. For a stable lower level with no pure dephasing, an upper population lifetime tau gives ordinary-frequency natural FWHM $1/(2\pi\tau)$. Lifetime broadening is not a statement that every individual emitted photon has a randomly assigned classical energy before detection.

For a gas in thermal equilibrium, the velocity component along the observation axis is Gaussian with standard deviation $\sqrt{k_BT/m}$. In the nonrelativistic limit, the first-order Doppler shift is proportional to that component. The frequency profile therefore inherits a Gaussian shape, whose full width is $2\sqrt{2\ln2}$ times its standard deviation. The emitter mass is the whole atom or ion, not the electron mass used in the electronic Schrödinger equation.

Doppler broadening is inhomogeneous: atoms with different velocities contribute different center frequencies. Lowering temperature reduces its width as the square root of temperature, while increasing mass reduces it as the inverse square root. A directed beam or a nonequilibrium gas can have a different velocity distribution, so a Maxwellian formula then needs justification. Transforming a narrow frequency width into a wavelength width uses the magnitude of the local derivative; broad asymmetric features require transforming the entire profile and its density convention.

In the impact approximation, short independent collisions randomize phase and can produce approximately exponential coherence decay and a Lorentzian contribution. Their rates then add to other homogeneous coherence rates. Collisions can also shift the center, change populations, and leave the impact regime at high density or far into line wings. Treating every pressure-broadened line as one universal Lorentzian would ignore the actual perturber dynamics and the possibility of quasistatic or correlated effects.

Independent Gaussian Doppler and Lorentzian homogeneous profiles convolve to a Voigt profile. Their FWHM values do not simply add. Independent Lorentzian widths do add, while Gaussian variances add for independent Gaussian convolutions. An instrument contributes its own response, and strong probing introduces power broadening or saturation. A fitted Voigt width therefore does not automatically reveal a natural lifetime. The oscillator lab illustrates deterministic damping of a coordinate, with a damping ratio rather than a spectral linewidth control; it does not produce a Doppler distribution or a fitted Voigt spectrum.