Emission depends on the modes: the Purcell factor

SI units throughout, with every rate and linewidth an angular frequency in s⁻¹; nothing here is relativistic, so no metric signature enters. Topic 12.02 summed the golden rule, Γ = 2π Σ|gₖ|² δ(ωₖ − ω₀), over free-space modes and found Γ₀ = ω³d²/(3πε₀ħc³). Nothing in that sum belonged to the atom alone: it needed the coupling to each mode, ħgₖ = d⋅E_vac, and the number of modes at ω₀. A cavity changes both. One mode of volume V, with the dipole along its field at the maximum, has ħg = d√(ħω/(2ε₀V)), Unit 11's Jaynes-Cummings coupling. Its mirrors leak, so its energy rings down as e^(−κt) with κ = ω_c/Q, and its spectrum is the Lorentzian ρ_c(ω), normalised to one mode. If the emitter's own line is narrower than κ, ρ_c is smooth across it and the golden rule gives Γ_cav = 2πg²ρ_c(ω₀). At resonance ρ_c = 2/(πκ), so Γ_cav = 4g²/κ, derived. Dimensions: s⁻² over s⁻¹ is a rate. The rate grows as κ falls, because the same one mode is squeezed into a narrower band.

Divide 4g²/κ by Γ₀ and the dipole cancels: F_P = 6πc³Q/(ω³V) = (3/(4π²))(λ³/V)Q, derived; λ³/V and Q are pure numbers, as a ratio of rates needs. Read it as two factors. Per unit volume the cavity offers a peak density 2Q/(πωV) where free space offers ω²/(π²c³), a ratio of 2πQc³/(ω³V). The remaining 3 is orientation: averaged over directions and both polarisations, |ε̂⋅d̂|² is 1/3, so a free-space dipole collects a third of each mode's g², while the aligned cavity dipole collects all of it. Now the free-field limit. In a box far larger than λ the modes lie closer together than κ, their Lorentzians merge into the smooth ω² density, and summing 2πg²ρ_c(ω₀) over all of them, with each g² averaged over the box and over orientation, returns exactly Γ₀ whatever Q is. Enhancement therefore comes from isolating one mode in a small volume, not from mirrors as such. For hydrogen's 2p, Γ₀ = 6.27 × 10⁸ s⁻¹ matches the measured 1.60 ns lifetime. A mode at 121.6 nm with V = λ³ and Q = 10³, if one could be built, gives F_P = 76 and a lifetime near 21 ps, with Γ_cav/κ = 3 × 10⁻³, safely a rate. At Q = 10⁴ the formula promises F_P = 760 and 2.1 ps, but there Γ_cav/κ = 0.31 and g = 0.28κ is past κ/4, so the decay oscillates and no such lifetime exists.

The emitter samples ρ_c at its own frequency, so a detuning Δ = ω₀ − ω_c gives Γ_cav(Δ) = (4g²/κ)/(1 + 4Δ²/κ²), derived: half a linewidth off, the rate halves. And g² is the square of d⋅E at the emitter, so a dipole tilted by θ from the local field, at a point where the field is a fraction of its peak, carries a further |E(r)/Eₘₐₓ|² cos²θ. F_P is a ceiling. What the total rate does depends on what else the emitter can radiate into. An open cavity, such as two small mirrors, leaves most directions untouched, so the emitter keeps about Γ₀ through the sides and Γ = Γ₀ + Γ_cav: the cavity adds a channel and takes none away. A closed cavity offers nothing else, and an emitter detuned by many linewidths sees only the Lorentzian's wing, g²κ/Δ², which can lie far below Γ₀: emission is inhibited. The fraction of photons that enter the mode, β = Γ_cav/Γ, is what a single-photon source is built to push towards 1.

Suppression can be total. Between perfectly conducting plates a distance L apart, the part of E parallel to the plates must vanish on both, so a mode carrying such a field needs k_z = nπ/L with n ≥ 1, and ω ≥ πc/L. With L below λ/2 the transition lies under that cutoff, a dipole parallel to the plates has no mode at all, and Γ_∥ = 0, derived. A perpendicular dipole couples to the n = 0 mode, whose field is normal to the plates and uniform across the gap: with E_vac² = ħω/(2ε₀AL) over an area A and Aω/(2πc²) modes per unit ω, the golden rule gives Γ_⊥ = (3λ/(4L))Γ₀, derived, 1.5Γ₀ just under the cutoff. Both enhancement and inhibition have been measured, and the results are quoted. In 1983 sodium Rydberg atoms in a superconducting cavity at 340 GHz with Q near 10⁶ emitted about 500 times faster than in free space; the Purcell formula gives that with V ≈ 1.5 × 10² λ³, a mode about five wavelengths across. In 1985 caesium atoms in circular Rydberg states, between plates just under half a wavelength apart, lived at least 20 times longer.

Topic 12.05's equations hold unchanged with one leaking mode in place of the free-space continuum. Eliminating the photon amplitude leaves, at resonance, the kernel g² e^(−κτ/2), which remembers for 2/κ, and topic 12.06's Markov step needs that memory short beside the decay time 1/Γ_cav. Since Γ_cav/κ = 4g²/κ² = (2g/κ)², the rate is self-consistent only while 2g ≪ κ. Neglecting Γ₀, differentiating once turns the memory integral into c̈ₑ + (κ/2)ċₑ + g²cₑ = 0, with roots s = −κ/4 ± √(κ²/16 − g²), derived. For g ≪ κ/4 the slow root is −2g²/κ, and the population falls at 4g²/κ: the golden rule recovered. Past g = κ/4 the roots turn complex and cₑ oscillates; as κ → 0 it becomes cos(gt), so Pₑ = cos²(gt), Unit 11's vacuum Rabi oscillation at 2g, the one-mode limit. The photon is now re-absorbed before it escapes, and the excitation, shared between atom and mode, drains at κ/2 however large g is. Lowering κ further makes emission slower, not faster.

Sampling ρ_c at one frequency assumed the emitter's line, of full width Γₑ, much narrower than κ. A free atom meets that easily, with Γ₀/2π of a few MHz against cavity widths of tens of MHz or more. An emitter in a solid at room temperature, its line broadened by phonons, may not. For a spread of emitter frequencies with a normalised Lorentzian distribution ρₑ, the rate is 2πg²∫ρ_c ρₑ dω, and a convolution of two Lorentzians is a Lorentzian whose widths add, as topic 12.07 found for two decaying levels. So Γ_cav = 4g²/(κ + Γₑ) at resonance, derived for that case; that a dephasing-broadened line gives the same is quoted. In F_P the cavity's Q becomes an effective ω/(κ + Γₑ). A mode at 637 nm with Q = 1.0 × 10⁴ has κ/2π = 47 GHz; an emitter line 1.0 THz wide drops the effective Q to 449 and F_P by a factor of 22, and better mirrors then buy almost nothing. The limit Γₑ → 0 returns 4g²/κ.