Dicke Superradiance Ladder: Exact Quantum Master Equation

Emitter Ensemble

Number of atoms N 60

Single-atom decay rate γ 0.30

Dicke ladder — cooperative cascade

Live State

Time t (Γ⁻¹ units)

0.00

Exact I(t)

0.00

Mean-field I_mf(t)

0.00

Inversion W(t)=⟨m⟩/j

1.00

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How it works

This is the exact-quantum companion to the 3D Dicke sim, which animates the mean-field Bloch-vector approximation. Here the state is never reduced to a single classical vector: the full permutation-symmetric ensemble is a probability distribution p_k(t) over the Dicke ladder, where k = number of atoms already de-excited (k = 0…N). The exact master equation is a birth-death chain,

dp_k/dt = Γ_k−1·p_k−1 − Γ_k·p_k
Coherent:   Γ_k = γ(N−k)(k+1)     ← cooperative Dicke rate
Incoherent: Γ_k = γ(N−k)          ← independent atoms

Radiated intensity: I(t) = Σ_k Γ_k·p_k(t)   (exact photon flux)

This ladder equation is solved numerically every frame with RK4 — a completely independent computation from the 3D sim's closed-form tanh/sech² solution. Because the Dicke rate Γ_k peaks sharply near the middle of the ladder (k≈N/2, where Γ_k≈γN²/4), probability piles up and cascades down that peak, producing the same qualitative burst — but the exact peak height and timing differ measurably from the mean-field formulas I_peak=N²γ/4, t_D=ln(N)/Γ shown for comparison (dashed curve). Numerically the exact peak converges toward roughly 0.78–0.80 of the mean-field prediction as N grows, not 1 — a genuine quantum-fluctuation correction the classical Bloch vector cannot see, because it starts from an infinitely sharp "seed tipping angle" hack whereas the ladder equation already has a nonzero decay rate Γ₀=Nγ at the fully-inverted top rung.

N slider — ladder length N+1; more atoms sharpen and speed up the cascade and shift the exact/mean-field gap.
γ slider — single-atom decay rate; sets the absolute rate scale of every rung.
Mode toggle — "Incoherent" switches every rung's rate to the independent-atom form Γ_k=γ(N−k), reproducing plain exponential fluorescence I(t)=Nγe^−γt exactly (verified numerically) with no collective burst.
Ladder panel — the bottom strip is the population p_k(t) itself, binned across the ladder and redrawn every frame: watch probability start piled at the fully-excited rung (k=0) and cascade down through the high-rate middle, exactly where the burst fires.

Real-world relevance: this birth-death treatment is the same formalism used to compute exact photon-counting statistics in superradiance experiments (atomic vapours, quantum dot ensembles, superconducting-qubit arrays), where the mean-field peak intensity is a leading-order estimate but the measured peak is reliably somewhat lower and later — precisely the correction this ladder reproduces.

Exact I(t) — ladder master equation Mean-field I(t) — classical Bloch vector Ladder population p_k(t)

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