The Born-Markov master equation for an atom in vacuum

Units are SI throughout; nothing here is relativistic, and where the course's covariant forms are recalled the metric is diag(1, −1, −1, −1). The model is Unit 11's: a two-level atom at the origin, H_A = ħω₀σ₊σ₋, coupled to every mode k, a wavevector with one of its two polarisations, of a box of volume L³ through H_I = −d⋅E(0), counter-rotating terms kept. In the interaction picture the joint state obeys dρ̃ₜₒₜ/dt = −(i/ħ)[H_I(t), ρ̃ₜₒₜ(t)]. Integrate from 0 to t, substitute back, and trace over the field: dρ̃/dt = −(i/ħ)Tr_F[H_I(t), ρₜₒₜ(0)] − (1/ħ²)∫₀^t Tr_F[H_I(t), [H_I(t′), ρ̃ₜₒₜ(t′)]] dt′. That is still exact. If the atom starts uncorrelated with an empty field, ρₜₒₜ(0) = ρ(0) ⊗ |0⟩⟨0|, the first term carries ⟨0|E|0⟩ = 0 and vanishes. The field then enters only through ⟨0|E(t)E(t′)|0⟩ = Σₖ (ħωₖ/(2ε₀L³)) εₖεₖ e^{(−iωₖ(t − t′))}: the only ordering to survive is a photon created at t′ and destroyed at t.

The equation is not closed yet, because ρ̃ₜₒₜ(t′) is the joint state at earlier times. The Born approximation replaces it by ρ̃(t′) ⊗ |0⟩⟨0|. That does not claim that no photon is emitted. It claims that whatever the atom emitted has gone: a photon created at the atom is a wave packet moving off at c, 30 cm away after 1 ns, and the modes the atom couples to later are, to the order kept, still empty. The factorised equation is already second order in d, and the atom-field correlations it drops change it only at higher order, by a relative amount of order Γ/ω₀. The approximation fails when the reservoir is small enough to remember. A single high-finesse cavity mode hands its photon back to the atom, and that is Unit 11's Jaynes-Cummings problem, a reversible exchange, not a decay.

Put τ = t − t′. After the trace, the kernel multiplying σ₊σ₋ρ̃ is Σₖ |gₖ|² e^{(i(ω₀ − ωₖ)τ)}. Its spectrum, |gₖ|² times the mode density, rises as ω³ and is smooth through ω₀, so the kernel is a spike no more than about 1/ω₀ wide, as topic 12.05 found. Now the interaction picture matters: ρ̃ has had the fast phase e^(−iω₀t) taken out and changes only at the rate Γ, so across the spike ρ̃(t − τ) → ρ̃(t) with a relative error of order Γ/ω₀. For hydrogen's 2p level 1/ω₀ = 6.45 × 10⁻¹⁷ s and 1/Γ = 1.60 ns, a ratio of 4.0 × 10⁻⁸. For t ≫ 1/ω₀ the upper limit goes to infinity, and ∫₀^∞ e^(ixτ) dτ = πδ(x) + iP(1/x), quoted from topic 12.06, splits the kernel into Γ/2 plus iδₑ. The same step in the Schrödinger picture would be wrong: there ρ_eg turns over in 1/ω₀, exactly the width of the spike.

Put d(t) = d_eg σ₊ e^(iω₀t) + d_eg* σ₋ e^(−iω₀t) into the double commutator. Products such as σ₊σ₊ vanish for two levels, but the sandwiched terms do not. σ₋ρ̃σ₊ comes with e^(iω₀τ) alone and is slow; σ₋ρ̃σ₋ and σ₊ρ̃σ₊ carry e^{(∓2iω₀t)}. Over one hydrogen 2p lifetime those turn about 8 × 10⁶ times, and their integrated effect is of relative size Γ/(2ω₀) = 2.0 × 10⁻⁸, so the secular approximation drops them. The counter-rotating part of the coupling survives only as a shift of |g⟩. Transform back with H_A and the formula card's equation follows, derived. Two checks cost a line each. Trace: d(ρₑₑ + ρ_gg)/dt = −Γρₑₑ + Γρₑₑ = 0. Dimensions: ω₀³|d_eg|²/(ε₀ħc³) is s⁻³ (C m)² over (C² J⁻¹ m⁻¹)(J s)(m³ s⁻³), which is s⁻¹, as a rate must be. Keep the 2ω₀ terms and the equation is of Redfield type, which need not keep ρ positive.

The real part of the kernel gives Γ = 2π Σₖ |gₖ|² δ(ωₖ − ω₀), topic 12.02's golden-rule sum. Turn Σₖ into (L³/8π³)∫d³k, sum |d_eg⋅ε|² over the two polarisations and average it over all directions to get (2/3)|d_eg|², set k = ω₀/c, and L³ cancels: Γ = ω₀³|d_eg|²/(3πε₀ħc³), the A coefficient. For hydrogen 2p → 1s, ħω₀ = (3/4)(13.606 eV) = 10.204 eV and the integral between the 1s and 2p orbitals gives |⟨g|r|e⟩|² = (2¹⁵/3¹⁰)a₀² = 0.555 a₀², so Γ = 6.27 × 10⁸ s⁻¹ and 1/Γ = 1.60 ns, the measured lifetime. The imaginary parts give δₑ = PΣₖ|gₖ|²/(ω₀ − ωₖ) and, from the counter-rotating terms, δ_g = −PΣₖ|gₖ|²/(ω₀ + ωₖ). Both grow without bound with the mode cutoff, as Unit 13 found; that part is absorbed into the observed ω₀′. The finite remainder for a real atom needs every level, not two: Bethe's logarithm, cut off near mc², gives 1040 MHz for the splitting of hydrogen's 2s₁/₂ and 2p₁/₂ levels against the measured 1058 MHz, an estimate quoted from Unit 13 and not the full QED value.

In the basis |e⟩, |g⟩ the master equation is three linear equations: dρₑₑ/dt = −Γρₑₑ, dρ_gg/dt = +Γρₑₑ and dρ_eg/dt = −(Γ/2 + iω₀′)ρ_eg. Population flows from |e⟩ to |g⟩ at Γ, while the coherence, reached only by the anticommutator at half strength, decays at Γ/2: T₁ = 1/Γ and T₂ = 2/Γ, the radiative limit of topic 14.02. Nothing in the solution asked whether ρ(0) was pure. The limits check out. Γ → 0 returns the von Neumann equation. ρ(0) = |e⟩⟨e| gives ρₑₑ = e^(−Γt), the |cₑ|² of topic 12.06. Every initial state ends in |g⟩⟨g|, the only steady state, as it must in a vacuum at zero temperature. Positivity survives too: ρₑₑρ_gg − |ρ_eg|² = e^(−Γt)[ρₑₑ(0)(1 − ρₑₑ(t)) − |ρ_eg(0)|²], which is never below its starting value times e^(−Γt). A pure start other than |g⟩ does not stay pure: the Bloch vector leaves the sphere and comes back to it only at |g⟩.