Unravelling the master equation into quantum jumps

Units are SI with ħ kept; the course's metric (+, −, −, −) plays no part in this non-relativistic topic. Quoted from topic 14.04: dρ/dt = −(i/ħ)[H, ρ] + Σₖ[LₖρLₖ† − ½(Lₖ†Lₖρ + ρLₖ†Lₖ)], each Lₖ carrying units s^(−1/2), and for a two-level atom in the vacuum L = √Γ σ₋ with Γ the Weisskopf-Wigner rate of Unit 12. Gather the anticommutator into the commutator and the same equation reads dρ/dt = −(i/ħ)(H_eff ρ − ρH_eff†) + Σₖ LₖρLₖ†, with H_eff = H − (iħ/2)Σₖ Lₖ†Lₖ. The first term alone removes trace at the rate Σₖ⟨Lₖ†Lₖ⟩; the second puts exactly that much back, sandwiched between Lₖ and Lₖ†. That bookkeeping is the whole idea: the first term is Schrödinger evolution with a leak, the second a sudden replacement of the state, and each acts on a pure state as easily as on ρ. For an undriven atom H_eff = (ħω₀ − iħΓ/2)|e⟩⟨e|: the complex energy of Weisskopf-Wigner decay returns as the generator of the smooth part.

Take a pure state |ψ⟩ and one step dt. With probability dpₖ = ⟨ψ|Lₖ†Lₖ|ψ⟩dt channel k fires and the state becomes Lₖ|ψ⟩/√(dpₖ/dt); with probability 1 − Σₖdpₖ nothing fires and it becomes (1 − iH_eff dt/ħ)|ψ⟩/√(1 − Σₖdpₖ). Average the projector over the branches. A jump branch contributes dpₖ × Lₖ|ψ⟩⟨ψ|Lₖ†/(dpₖ/dt) = Lₖ|ψ⟩⟨ψ|Lₖ† dt: its probability cancels its normalisation. The no-click branch contributes |ψ⟩⟨ψ| − (i dt/ħ)(H_eff|ψ⟩⟨ψ| − |ψ⟩⟨ψ|H_eff†), its normalisation cancelled the same way. Together they are the rewritten master equation to first order in dt, derived; with every Lₖ set to zero the rule collapses to Schrödinger evolution. Linearity extends the result from one pure state to any mixture, so trajectories started from the eigenstates of ρ(0), drawn with its eigenvalues as weights, reproduce ρ(t). The price is statistical: N runs fix a population to within at most 0.5/√N, ±0.005 for 10⁴ runs. The gain is size: a d-level system needs d amplitudes instead of d² matrix elements.

Chain the no-click factors over many steps and the normalisations telescope: the probability of no click from 0 to τ is the squared norm S(τ) of the unnormalised state e^{(−iH_eff τ/ħ)}|ψ(0)⟩, and the density of the first click is w(τ) = −dS/dτ = Σₖ‖Lₖe^{(−iH_eff τ/ħ)}|ψ(0)⟩‖², both derived. This is also the efficient algorithm: draw a uniform r, evolve under H_eff until S = r, then jump. Two undriven cases check it. Started in |e⟩, S = e^(−Γτ) and w = Γe^(−Γτ): the one-atom, free-field limit returns Unit 12's exponential decay. Started in (|g⟩ + |e⟩)/√2, in the frame rotating at ω₀ the unnormalised state is (|g⟩ + e^(−Γτ/2)|e⟩)/√2, and S = (1 + e^(−Γτ))/2, which tends to ½: half the runs never click. Those silent runs are steered towards |g⟩, with ρₑₑ = e^(−Γτ)/(1 + e^(−Γτ)) = 0.27 at Γτ = 1, although no photon was seen, because a detector that stays dark makes |e⟩ less likely. Weighting 0.27 by S = 0.68 restores the ensemble value ½e⁽⁻¹⁾ = 0.18, as the master equation demands.

Drive on resonance, H = (ħΩ/2)(σ₊ + σ₋) in the rotating frame. Every click applies σ₋ and leaves |g⟩, so every wait starts from the same state and the record is a renewal process. Writing the no-click state α|g⟩ + β|e⟩, dα/dτ = −i(Ω/2)β and dβ/dτ = −i(Ω/2)α − (Γ/2)β, a damped oscillator with β(0) = 0, solved by β = −i(Ω/2κ)e^(−Γτ/4) sin κτ with κ = ½√(Ω² − Γ²/4). So w = Γ|β|² = [ΓΩ²/(Ω² − Γ²/4)]e^(−Γτ/2) sin²κτ, derived: Γ times a pure number, a rate. At τ = 0 it vanishes; for Ω ≫ Γ it rings as sin²(Ωτ/2) under e^(−Γτ/2). Because Γ²/4 + 4κ² = Ω², the integrals collapse: ∫w dτ = 1 and ⟨τ⟩ = 2/Γ + Γ/Ω², exactly 1/(Γρₑₑ) for topic 14.06's steady state Ω²/(Γ² + 2Ω²). Near τ = 0, w/(Γρₑₑ) = (Γ² + 2Ω²)τ²/4, the small-τ form of topic 14.07's g²(τ). Later they part, since g² counts any later click and w only the next: with R = Γρₑₑ, Rg²(τ) = w(τ) + ∫₀^τ w(τ′)Rg²(τ − τ′)dτ′, derived from the reset. A start-stop histogram measures w only if every photon is detected; at low efficiency it approaches g².

The split into H_eff and jumps is not unique. Replace L by L + ξ, with ξ a complex number in s^(−1/2), and H by H − (iħ/2)(ξ*L − ξL†): expanding the two terms shows the extra pieces cancel, so the master equation and every average it predicts are unchanged, derived. The trajectories change completely. Physically, ξ is a coherent local-oscillator field mixed with the fluorescence on a beam splitter before the counter. As |ξ| grows, clicks come faster and each moves the state less; in the limit the record is a continuous photocurrent and the state diffuses, obeying a stochastic Schrödinger equation driven by white noise. That homodyne unravelling is named here, not derived. A photon counter sees an atom that jumps to |g⟩; a homodyne detector sees the same atom drift continuously over the Bloch sphere; both average to the same ρ(t). The conditioned state is therefore relative to a chosen measurement, and it stays pure only if every photon is recorded: with detection efficiency below one the missed jumps must be averaged over, and it becomes mixed.

One photon from one atom is rarely detected, so a single jump would pass unseen. Dehmelt's electron shelving makes it visible. Drive a strong cycling transition, Γ near 10⁸ s⁻¹, so the ion scatters some 10⁷ photons s⁻¹ and even a detection efficiency of 10⁻³ gives 10⁴ counts s⁻¹; add a weak transition to a metastable shelf. While shelved the ion cannot scatter, and the fluorescence stops until the shelf decays. On the shelf the only channel is its decay, so the jump formalism gives a no-decay probability e^(−t/τ_D) with τ_D the shelf lifetime: dark periods are exponentially distributed with mean τ_D, derived. In 1986 three groups recorded such telegraph signals from single ions, quoted: barium in Seattle (Nagourney, Sandberg and Dehmelt) and Hamburg (Sauter, Neuhauser, Blatt and Toschek), and mercury in Boulder (Bergquist, Hulet, Itano and Wineland). Dark-period statistics then became lifetime measurements, also quoted: 86 ± 3 ms for the ²D₅/₂ level of Hg⁺ in 1987, and 31.2 ± 0.9 s for the 5D₅/₂ level of Ba⁺ in 2014.