Dicke Superradiance: Collective Quantum Coherence

Emitter Ensemble

Number of atoms N 60

Single-atom decay rate γ 0.30

Phases synchronised — collective burst

Live State

Time t (Γ⁻¹ units)

0.00

Intensity I(t)

0.00

Inversion W(t)

1.00

Per-atom coherence |ρ_eg|

0.00

N=60, γ=0.30 → Γ=Nγ=18.0 t_D = ln(N)/Γ = 0.23 I_peak = N²γ/4 = 270.0

How it works

Take N identical two-level atoms confined to a region smaller than the emission wavelength, all initially excited. Treated as independent emitters, they decay one by one with an ordinary exponential: I(t) = Nγ e^−γt. But their shared electromagnetic field correlates them — R. Dicke showed in 1954 that this coupling builds up a macroscopic transition dipole and the ensemble instead fires a single cooperative burst, called superradiance.

The ensemble is modelled as one collective Bloch vector of length N/2 (a "giant spin"). Its population-inversion component W (+1 = all excited, −1 = all ground) obeys the mean-field Dicke equation:

dW/dt = −(Γ/2)(1 − W²),   Γ = Nγ  (collective linewidth)
Solution: W(t) = −tanh[(Γ/2)(t − t_D)],   t_D = ln(N)/Γ

Radiated intensity: I(t) = (NΓ/4)(1 − W²) = (N²γ/4)·sech²[(Γ/2)(t − t_D)]
Peak intensity:      I_peak = N²γ/4   (at t = t_D)

The transverse Bloch component is exactly the quantum coherence of the ensemble — the magnitude of the off-diagonal density-matrix element |ρ_eg| = ½√(1 − W²). It is zero at W = ±1 (pure excited or pure ground, no coherence) and peaks precisely at t = t_D, the instant of maximum emission — the burst is the coherence discharging. A tiny seed tipping angle θ₀ ≈ 1/√N (standing in for the vacuum fluctuations that actually trigger the instability) is what lets the "fully inverted" state begin to tip at all.

N slider — more atoms means a stronger, faster, more sharply peaked burst: peak power grows as N², while the delay t_D and pulse width shrink like 1/N.
γ slider — the bare single-atom spontaneous-emission rate; it sets the absolute intensity scale.
Mode toggle — "Incoherent" forces each atom to decay independently with a random, uncorrelated phase (no shared dipole forms), reproducing plain exponential fluorescence I(t) = Nγe^−γt with no burst — the direct control experiment showing that the burst requires coherence, not just excited atoms.

Real-world relevance: Dicke superradiance has been observed in atomic vapours, quantum dot ensembles, and superconducting-qubit arrays, and the same coherence-vs-decoherence competition underlies why real quantum computers must be shielded from stray correlated coupling to the environment.

Excited atom Ground-state atom Collective dipole / burst

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