Collective light / The bright side

Dicke Superradiant Burst

Why can a group of excited atoms release its light in one bright burst? Because atoms that share one decay channel emit as a group. A fully excited ensemble starts out at the ordinary rate of independent atoms, then speeds up and peaks when about half the atoms are still excited. The light comes late, as a burst far brighter than the same atoms would give on their own.

Let a group of excited atoms decay through one shared channel and watch a delayed burst of light build up. Compare it with atoms that emit independently, and record the arrival time of each photon over repeated runs.

The model is ideal symmetric Dicke decay, with independent emitters as a separate comparison. A partly excited starting state is a symmetric Dicke state, not a random set of excited atoms. The angular surface shows the dipole radiation pattern; it is not a travelling wavefront or a piece of apparatus. Coherent dipole shifts, pumping, dephasing and any specific hardware are left out.

ANGULAR EMISSIONMODEL

EMISSION RATE

0

0

Unconditional ensemble mean

Radius: cube-root intensity · fixed gain within each model

Dots: equal mean excitation, not individual trajectories. Surface: dipole angular intensity; no propagation or physical detector surface.

0

Repeat the preparation and record the light

How quiet is quiet?

Each trial starts from a fresh preparation and records the full time window. Photons are counted ideally over the full solid angle, then kept with the selected efficiency. Trials with no click still count in the denominator.

Analyze clicks before this time

0

Recorded clicks per trial

\text{—}

Trials with a click

\text{—}

Model click probability

\text{—}

Model excitation at the gate

\text{—}

01 / PHOTON TIMES

The burst has a shape.

Cyan: efficiency-scaled model rate. Gold: timestamp histogram per trial. Drag the gold gate; arrows move it. Dashed: independent-emitter reference.

$\Gamma_0t$ · $\text{clicks}/(\text{trial}\cdot\Gamma_0t)$

02 / SURVIVAL

Where the excitation went.

Cyan: remaining excitation fraction. Coral: independent-emitter reference. Pale vertical: selected scene time. Gold: analysis gate.

$\Gamma_0t$ · $\langle m\rangle/m_0$

03 / DIRECTIONAL DIAGNOSTIC

Where light can escape.

Equatorial cut of the 3D surface, in the plane perpendicular to the dipoles. Radius is linear here. This is a model diagnostic; the timestamp counter collects all directions.

$\frac{1}{\Gamma_0}\frac{dR}{d\Omega}$

04 / RAW TRIALS

Every row begins again.

Last 40 trials. Each mark is a detected photon; empty rows are retained. The full ledger and all timestamps are exported.

$\Gamma_0t$

Last 20 raw trials

Trial	Clicks	Timestamps · $\Gamma_0t$

Model, units & sources

Time is measured in isolated-emitter population lifetimes. The counter covers the full solid angle and loses detections independently. There is no background, dead time, timing jitter or technical preparation error. The intervals show sampling uncertainty only.

\tau=\Gamma_0t,\qquad \lambda_0=2\pi/k_0

The ideal symmetric collective channel is a model limit. It does not describe a real dense cloud, whose near-field interactions cannot be ignored without consequence. Independent emission is a separate comparison. A partial initial excitation is rounded to an integer symmetric Dicke state, and uniformization propagates its population probabilities.

r_m/\Gamma_0=m(N-m+1),\qquad R=\sum_mr_mP_m

The default 3D radius uses a cube-root gain and is an angular diagnostic, not a distance. Enlarge shape explicitly normalizes each frame to reveal weak lobes; compare the numerical rate for actual strength. Its fixed intensity reference is the maximum Dicke ladder rate at 32 atoms, or 12 times the single-dipole maximum for the array. The equatorial 2D plot is linearly scaled with its maximum printed. Atom dots are a separate schematic state diagram, not a detector surface.

Actual preparation and numerical diagnostic:

0

The diagnostic is the population-normalization error. Photon waiting times are exponential draws, not samples read from the plotted mean curve.

Holzinger et al. — Dicke decay, Eqs. 1–6 & quantum trajectories
Piovella — vector Green tensor, Eqs. 24–27 & 38
Lehmberg — radiation from an N-atom system

Start with the essentials

Focus question: Why can a group of excited atoms release its light in one bright burst?
One-sentence intuition: On the collective ladder, emission is fastest when about half of the excitation remains. A fully excited ensemble starts at the independent-atom rate, and the burst comes later.

Core mathematical model

Emission rate at each step of the ladder

r_m=\Gamma_0m(N-m+1)

Every emitted photon, whether or not it is detected, lowers the number of excitations by one.

How the populations change

\dot P_m=r_{m+1}P_{m+1}-r_mP_m

The curve is the average over all ideal emission histories. Each recorded run of photons is one independent realization of this process.

Common difficulties

A silent counter does not prove the excitation is stored

Typical misconceptionIf there is no click, the excitation must still be there.

Better mental modelThe detector can miss a photon that was emitted. The survival probability from the model and the observed fraction of no-click trials are different quantities, and the uncertainty shown covers sampling only.

Run the experiment

01
Compare two cases
Keep the emitter count fixed and switch between Release together and Emit independently.
What to observe: The collective pulse rises before it falls. Independent emission falls from the start.
02
Break the effect
Lower the initial Dicke excitation. The delayed burst of a fully excited ensemble does not appear for every starting state.
What to observe: The result changes with the chosen state and geometry, because it is computed from them each time.
03
Count photons
Collect 200 complete decay trials. Drag the time gate, pin the data, change the preparation, then restore it. Export the timestamps; trials with no click are included.
What to observe: A detector that misses photons also produces silence. Compare the measured click probability with the model that includes detector efficiency.