Resonance fluorescence: Mollow triplet and antibunching

Units are SI with ħ explicit; nothing here is relativistic, so the course's metric signature (+, −, −, −) never enters. Unit 11's mode equations, integrated with the atom as a source, split each mode operator into a free part and a part driven by σ₋ at earlier times. Summing the driven parts over modes, with the Markov step that treats σ₋e^(iω₀t) as slow, gives in the far zone the classical dipole field with d e^(−iω₀t) replaced by d σ₋(t − r/c): outlined here, not derived in full. The free part annihilates the vacuum, so a detector, which measures normally ordered products, sees only the atom: the intensity is 2ε₀c|ψ|²ρₑₑ, with |ψ| = ω₀²d sin θ/(4πε₀c²r) in V m⁻¹. Over a sphere the sin²θ pattern integrates to 8π/3 and the power to ω₀⁴d²ρₑₑ/(3πε₀c³) = ħω₀Γρₑₑ, so the photon flux is Γρₑₑ with Unit 12's Γ, a check on the normalisation. For sodium's D2 line the measured 16.25 ns lifetime gives Γ = 6.15 × 10⁷ s⁻¹, so one atom scatters at most Γ/2 = 3.1 × 10⁷ photons s⁻¹.

Write σ₋ = ⟨σ₋⟩ + δσ₋. The correlation splits into |⟨σ₋⟩|², which never decays and so transforms into a delta function at ω_L, and ⟨δσ₊(t + τ)δσ₋(t)⟩, which decays and gives lines of finite width. The first is elastic scattering by the atom's mean dipole, as sharp as the laser; the second is fluorescence proper. Topic 14.06's resonant steady state, used here, has ρₑₑ = Ω²/(Γ² + 2Ω²) and |⟨σ₋⟩|² = Ω²Γ²/(Γ² + 2Ω²)², so the elastic fraction is 1/(1 + s), derived, with s = 2Ω²/Γ² = I/Iₛₐₜ and Iₛₐₜ = πhcΓ/(3λ³) = 6.26 mW cm⁻² for sodium, computed from the measured Γ and λ. The limits check it. As Ω → 0 all the light is elastic: the atom responds linearly, like a driven classical dipole, and this is Rayleigh scattering. As s grows the atom scatters faster, ρₑₑ → ½, yet the Bloch vector shrinks to the centre of the ball and the mean dipole vanishes, so the elastic fraction falls as 1/s, to 1.9% at Ω = 5.1Γ. Inelastic light is the price of saturation, a nonlinearity no linear oscillator has.

To evaluate ⟨δσ₊(t + τ)δσ₋(t)⟩ we quote the quantum regression theorem: under the Born-Markov approximations a two-time average ⟨A(t + τ)B(t)⟩ obeys, in τ, the same linear equations as ⟨A(t + τ)⟩, starting from ⟨A(t)B(t)⟩. For fluctuations these are the homogeneous Bloch equations. On resonance, in Bloch components, du/dt = −Γu/2, dv/dt = −Γv/2 − Ωw and dw/dt = −Γw + Ωv, so the decay constants are −Γ/2 and the roots of (λ + Γ/2)(λ + Γ) + Ω² = 0, which are λ = −3Γ/4 ± iΩ′ with Ω′ = √(Ω² − Γ²/16), derived. Each constant becomes a line centred by its imaginary part, with half-width minus its real part: a Lorentzian at ω_L of full width Γ and a pair at ω_L ± Ω′ of full width 3Γ/2, which also carry a dispersive part, of relative size about 5Γ/(4Ω′) for Ω ≫ Γ, that pulls their maxima slightly inside ±Ω′. The initial values ⟨σᵢσ₋⟩ − ⟨σᵢ⟩⟨σ₋⟩ fix the weights, and the inelastic area is ρₑₑ − |⟨σ₋⟩|², as it must be. Carried through exactly, this gives the closed form the figure draws at every Ω, and a direct numerical integration of the master equation reproduces it to five decimal places.

Why three lines, and why 1:2:1, is clearest for Ω ≫ Γ with dressed states. In the frame rotating at ω_L the resonant Hamiltonian (ħΩ/2)(σ₊ + σ₋) has eigenstates |±⟩ = (|g⟩ ± |e⟩)/√2 split by ħΩ; counting laser photons, each manifold of N photons is such a doublet, the next ħω_L lower. Emission takes the atom from manifold N to N − 1 through σ₋, whose four elements ⟨±|σ₋|±⟩ and ⟨∓|σ₋|±⟩ all have modulus ½, so each transition runs at Γ/4 per unit population: |+⟩ → |+⟩ and |−⟩ → |−⟩ at ω_L, |+⟩ → |−⟩ at ω_L + Ω, |−⟩ → |+⟩ at ω_L − Ω. The cross rates balance, so the populations are ½ each and the lines carry Γ/8, Γ/4 and Γ/8: weights 1:2:1 summing to Γρₑₑ = Γ/2, derived. Divided by the widths 3Γ/2, Γ and 3Γ/2 they give heights 1:3:1. For sodium at Ω = 2π × 100 MHz the lines lie 100 MHz apart, with full widths 9.8 MHz and 14.7 MHz. Measured, quoted: Schuda, Stroud and Hercher saw the triplet in a sodium beam in 1974, and Wu, Grove and Ezekiel in 1975, crossing laser and beam at right angles to remove the Doppler width, found its spacing and heights in good agreement with Mollow's theory.

Two detectors measure Glauber's coincidence rate G⁽²⁾(τ) ∝ ⟨E⁽⁻⁾(t)E⁽⁻⁾(t + τ)E⁽⁺⁾(t + τ)E⁽⁺⁾(t)⟩, which the source field turns into ⟨σ₊(t)σ₊(t + τ)σ₋(t + τ)σ₋(t)⟩. Read it as a measurement. The first detection applies σ₋, and σ₋ρₛₛσ₊ = ρₑₑ|g⟩⟨g| whatever ρₛₛ was: a photon has just left, so the atom is certainly in its ground state. In the Markov approximation that photon never returns, so the master equation evolves |g⟩⟨g| for a time τ and the second detection comes at rate Γρₑₑ(τ), with ρₑₑ(0) = 0. So G⁽²⁾ ∝ Γρₑₑ × Γρₑₑ(τ), and dividing by (Γρₑₑ)², the rate of uncorrelated pairs, gives g⁽²⁾(τ) = ρₑₑ(τ)/ρₑₑ(∞), derived; the regression theorem returns the same expression. On resonance the Bloch equations solved from w = −1 give the closed form 1 − e^(−3Γτ/4)[cos Ω′τ + 3Γ/(4Ω′) sin Ω′τ]. Checks: g⁽²⁾(0) = 0 at every Ω; g⁽²⁾ → 1 as τ → ∞, independent detections; for Ω ≪ Γ the roots turn real and g⁽²⁾ = (1 − e^(−Γτ/2))²; for Ω ≫ Γ it rings, first peaking at Ω′τ = π at 1 + e^{(−3πΓ/(4Ω′))}, which is 1.45 at Ω = 3Γ.

A classical field gives the detectors a non-negative intensity I(t). Then g⁽²⁾(0) = ⟨I²⟩/⟨I⟩² = 1 + Var(I)/⟨I⟩² ≥ 1, shown, and for stationary light Cauchy-Schwarz, ⟨I(t)I(t + τ)⟩ ≤ √(⟨I(t)²⟩⟨I(t + τ)²⟩) = ⟨I²⟩, gives g⁽²⁾(τ) ≤ g⁽²⁾(0). Thermal light sits at 2 and an ideal laser at 1. One atom breaks both bounds: g⁽²⁾(0) = 0 is sub-Poissonian, and g⁽²⁾ rising from it is antibunching, photons spaced by the time to re-excite, about half a Rabi period under strong drive and a few lifetimes under weak. Measured, quoted: in 1977 Kimble, Dagenais and Mandel crossed a sodium beam with a resonant laser and saw coincidences rise from a minimum at τ = 0 over roughly the 16 ns lifetime, the first light shown to need a quantum description. Their beam held a fluctuating number of atoms, whose independent photons pull g⁽²⁾(0) up towards 1, so sub-Poissonian counts waited for Short and Mandel in 1983, and a single stored ion in 1987 removed the averaging over atoms altogether.