Line profiles and Doppler-free spectroscopy

Spontaneous decay truncates the excited-state amplitude as exp(−Γt/2), and the Fourier transform of a truncated exponential is a Lorentzian of FWHM Γ = 1/τ in angular frequency, or 1/2πτ in Hz. Every atom in the sample carries that profile identically, which is what homogeneous means: there is no subset of atoms with a narrower line. The Doppler width is a different kind of object. An atom with axial velocity v_z is resonant with a laser tuned to ν₀(1 + v_z/c), so the Maxwell–Boltzmann distribution over v_z relabels the resonance frequency atom by atom, producing a Gaussian of FWHM (ν₀/c)√(8 ln2 k_BT/M). Nothing about any individual atom has been broadened; the ensemble is simply a set of differently tuned absorbers. For sodium at 300 K the two numbers are 9.8 MHz and 1.32 GHz. The whole of Doppler-free spectroscopy exists because you can address an inhomogeneous distribution one class at a time and can never do the same to Γ.

The measured profile is the Lorentzian convolved with the Gaussian: every velocity class contributes a full Γ-wide line centred on its own shifted frequency. That convolution is the Voigt function, and it has a closed form in terms of the Faddeeva function w(z) = exp(−z²)erfc(−iz), namely V(ν) = Re[w(z)]/(σ√(2π)) with z = (ν − ν₀ + iγ)/(σ√2). In Python that is scipy.special.wofz, evaluated pointwise at whatever frequencies you like. Resist the temptation to FFT-convolve two sampled arrays instead. With γ/σ ≈ 1/114 you need a grid fine enough to resolve a 10 MHz core across a ±5 GHz span — some 2000 points before the core is sampled even twice, and of order 10⁴ to resolve its shape — and the periodic wrap-around of a discrete convolution contaminates exactly the far wings you were trying to measure. Use wofz for the model; keep the FFT for the data.

Far from line centre the Gaussian falls as exp(−x²/2σ²)/(σ√(2π)) and the Lorentzian only as γ/πx², so however small γ is, the Lorentzian eventually wins. For sodium at 300 K, with σ = 559 MHz and γ = 4.9 MHz, the crossing sits at 3.92σ — about 2.19 GHz from centre, where the profile has dropped to 1.1 × 10⁻³ of its peak. Two consequences follow for fitting. The Olivero–Longbothum width f_V ≈ 0.5346 f_L + √(0.2166 f_L² + f_G²) says a 9.8 MHz Lorentzian widens a 1317 MHz Gaussian to 1322 MHz, a change of 0.4 per cent: a Voigt fitted to the core alone cannot determine γ, because the fit trades it against σ and against your baseline. To pull γ out you either need three or four decades of dynamic range in the wings, or you fix f_G from a measured temperature and let the fit carry f_L alone.

Split one laser into a strong pump and a weak counterpropagating probe. At detuning δ the pump is resonant with atoms moving at v_z = +δ/k and the probe with atoms at v_z = −δ/k, so for any δ ≠ 0 the two beams talk to different halves of the velocity distribution and the probe reports the undisturbed Doppler profile. At δ = 0 both address the same class, v_z = 0. The pump has already driven a fraction (S/2)/(1+S) of that class into the excited state, so the probe finds fewer atoms to absorb and more to stimulate emission: the population difference falls to 1/(1+S) of its unpumped value and the probe absorption dips. The hole is Γ√(1+S) wide, but the probe brings its own Γ-wide response, and tuning the laser sweeps hole and probe class through v_z in opposite directions, so the dip traced out in detuning is (Γ/2)[1 + √(1+S)] wide — Γ at low power — and its depth, 1 − 1/√(1+S), creeps towards the full pedestal as the pump is raised. Watch for crossover resonances: when one lower level feeds two upper levels separated by Δ, the single class v_z = Δ/2k is pumped on one transition and probed on the other, giving a dip midway between the two real ones — often the strongest feature in the trace, and belonging to no level pair at all.

Saturated absorption throws away all but one velocity class. Two-photon spectroscopy keeps them all. Let an atom absorb one photon from each of two counterpropagating beams: the resonance condition is (ω₁ + ω₂) − (k₁ + k₂)⋅v = ω_eg, and with k₂ = −k₁ the velocity term vanishes identically, for every v. Every atom in the cell contributes to one line at ω_eg/2, so there is no pedestal to sit in — only a weak broad background from photon pairs taken out of the same beam. What survives is second order: time dilation shifts the resonance by −ν₀v²/2c², which for a 1 km s⁻¹ hydrogen beam is −13.7 kHz on the 1S–2S line, ten thousand natural widths, a systematic to be modelled rather than averaged away. The selection rules change too: two dipole steps means no net parity change, so ΔL = 0 or ±2, and the one-photon-forbidden 1S–2S transition becomes the whole point.

At this level the rate equations run out: the pump's coherence ρ_eg matters, and the honest model is the optical Bloch equations, ρ̇ = −(i/ħ)[H, ρ] + relaxation, with H carrying the Rabi frequency Ω = d⋅E/ħ. Their steady state is ρₑₑ = (S/2)/(1 + S + 4δ²/Γ²) with S = 2Ω²/Γ², which is where Γ√(1+S) comes from; push Ω past Γ and the same equations give the Autler–Townes splitting that no rate equation contains. Then count the remaining floors. Transit time: an atom crossing a waist w is driven for only w/v, giving roughly 0.4 v/w — 0.21 MHz for sodium at 526 m s⁻¹ through a 1 mm waist, but 400 kHz for a 1 km s⁻¹ hydrogen beam. Pressure broadening runs at tens of MHz per torr. And the laser's own linewidth convolves in directly, which is why a 9.8 MHz dip needs a laser locked well below it.