The emitted line: a Lorentzian from the photon amplitudes

SI units with ħ explicit; nothing here is relativistic, so no metric signature enters. Quoted from topic 12.05: in the one-excitation sector |ψ(t)⟩ = cₑ(t) e^(−iω₀t)|e, 0⟩ + Σₖ cₖ(t) e^(−iωₖt)|g, 1ₖ⟩, and in the rotating-wave approximation iċₖ = gₖ* cₑ e^{(i(ωₖ − ω₀)t)} exactly, with ħgₖ = ⟨e, 0|V|g, 1ₖ⟩. Quoted from topic 12.06: the Markov step gives cₑ(t) = e^{(−(Γ/2 + iΔ)t)}. Insert it, with no further approximation, and integrate from cₖ(0) = 0: cₖ(t) = gₖ* [1 − e^{(i(ωₖ − ω₀′)t − Γt/2)}]/(ωₖ − ω₀′ + iΓ/2), with ω₀′ = ω₀ + Δ. Dimensions: gₖ and the denominator are both s⁻¹, so cₖ is a pure number. Two features are fixed already. The centre is ω₀′, not ω₀, because the phase e^(−iΔt) of cₑ beats against e^{(i(ωₖ − ω₀)t)} in the integrand: the atom radiates at the frequency it actually oscillates at. The width is Γ because the envelope e^(−Γt/2) switches the source off, and a source lasting about 2/Γ cannot radiate a narrower line.

Take the modulus squared and write it with real functions: |cₖ(t)|² = |gₖ|² [1 − 2e^(−Γt/2) cos(δₖt) + e^(−Γt)]/(δₖ² + Γ²/4), with δₖ = ωₖ − ω₀′. Early-time limit: for Γt ≪ 1 the bracket becomes 2(1 − cos δₖt), giving |gₖ|² sin²(δₖt/2)/(δₖ/2)², the broad sinc² of Quantum Mechanics II's golden rule, growing as t² at resonance. Late-time limit: the bracket tends to 1 and the Lorentzian is left. In between, the centre fills as (1 − e^(−Γt/2))², the square of an amplitude built at Γ/2, so at t = 1/Γ it stands at 15% of its final height although the photon is already in the field with probability 0.63; at 2/Γ it reaches 40%, at 8/Γ 96%. Off resonance the fringes, 2π/t apart, overshoot the final curve by up to a factor (1 + e^(−Γt/2))², and that excess flows back through the atom to be re-emitted nearer ω₀′. Single-mode probabilities are not monotonic in time; only their sum is.

To sum over box modes, both polarisations and all directions, write Σₖ |gₖ|² F(ωₖ) = ∫ dω G(ω) F(ω), with G(ω) = Σₖ |gₖ|² δ(ω − ωₖ), the coupling-weighted mode density in s⁻¹; topic 12.06's rate is Γ = 2πG(ω₀). G changes on the scale ω₀ and the line on the scale Γ, tens of millions of times smaller for an optical line, so set G = Γ/2π across the line. Then ∫ dδ/(δ² + Γ²/4) = 2π/Γ and ∫ dδ cos(δt)/(δ² + Γ²/4) = (2π/Γ) e^(−Γt/2) give Σₖ |cₖ(t)|² = 1 − 2e^(−Γt) + e^(−Γt) = 1 − e^(−Γt), derived: at every instant the field holds exactly the population the atom has lost, the check that no probability leaks out of the one-excitation sector. At long times dP/dω = (Γ/2π)/[(ω − ω₀′)² + Γ²/4], in s, with area 1, peak 2/(πΓ), and half height where (ω − ω₀′)² = Γ²/4, so the full width is Γ. Half the photons lie inside that width and 3.2% beyond ten widths: the 1/δ² tails are heavy.

Quantum Mechanics II assumed an amplitude e^(−Γt/2) e^(−iω₀t), Fourier transformed it, took Γ from the Einstein A coefficient and put the centre at the unperturbed ω₀. Nothing in that recipe guaranteed that the line held one photon, that its width was the decay rate rather than merely resembling it, or where its centre sat. Here all three follow from one G(ω): Γ = 2πG(ω₀) both empties the level and sets the width, and the centre carries Δ = P∫ dω G(ω)/(ω₀ − ω), the principal value topic 12.06 found divergent and left for Unit 13. The line centre is where a radiative shift becomes observable. In a real atom the ground level shifts too, through the counter-rotating terms the rotating-wave approximation dropped, so a measured centre is the difference of two shifted levels. Measured check: sodium's D2 upper level has a lifetime of 16.25 ns, measured by time-resolved fluorescence, so Γ/2π = 9.79 MHz, and the width measured directly on the line in laser-cooled sodium agrees with it to a few tenths of a per cent.

For a line from level 2 to a lower level 1 that itself decays, add amplitudes c₁ₖ, atom in 1 with photon k, and c₀ₖₗ, atom in 0 with photons k and l. The Markov step applied to level 1 removes the l photons and leaves ċ₁ₖ = −i gₖ* c₂ e^{(i(ωₖ − ω₂₁)t)} − (Γ₁/2 + iΔ₁) c₁ₖ, with c₂ = e^{(−(Γ₂/2 + iΔ₂)t)}. Its solution is a difference of two exponentials, and the probability that the first photon went into mode k is |c₁ₖ|² plus what has already passed on, Γ₁∫|c₁ₖ|² dt. At long times that is |gₖ|² [(Γ₁ + Γ₂)/Γ₂]/[(ωₖ − ω₂₁′)² + (Γ₁ + Γ₂)²/4], derived in outline: summed with G = Γ₂/2π it has area 1, full width Γ₁ + Γ₂, and centre ω₂₁′ = ω₂₁ + Δ₂ − Δ₁, the difference of the shifted levels. Limits: Γ₁ → 0 returns the two-level line, and Γ₂ → 0 leaves a width Γ₁. If level 2 has other decay channels the line's area is its branching ratio and its width is unchanged. The figure's second slider is this result.

Before G was frozen the long-time line read dP/dω = G(ω)/[(ω − ω₀′)² + Γ²/4]. With Unit 11's A⋅p coupling, |gₖ|² ∝ |ε⋅⟨e|p|g⟩|²/ωₖ and the mode density grows as ωₖ², so G(ω) = (Γ/2π)(ω/ω₀) for ω ≥ 0; written with E⋅d it is (Γ/2π)(ω/ω₀)³. The two agree on resonance and part in the wings, a sign that the two-level truncation, not only the Lorentzian, fails there; Lamb met the same ambiguity in 1952 fitting hydrogen line shapes. Either way the error is of order |ω − ω₀|/ω₀: 4 × 10⁻⁸ one width from the centre of Lyman-α, about a per cent at 1 nm. And G vanishes for ω ≤ 0, while the Lorentzian puts Γ/(2πω₀) = 6.4 × 10⁻⁹ of its photons at negative frequency; that lower edge is what makes decay non-exponential at very late times (quoted, Khalfin 1958). Measured check: damped Lyman-α absorbers in quasar spectra are fitted with this Lorentzian and Γ = 6.27 × 10⁸ s⁻¹ several ångströms from the centre, some 10⁵ widths out.