Collective light / The quiet side

Subradiant Atomic Array

How can neighboring atoms keep an excitation from radiating away so quickly? By sharing it with opposite phases. Their emission then cancels in every direction, so the excitation lasts much longer. Spread the atoms farther apart and the cancellation breaks down. Subradiance only stretches the lifetime; it cannot store light forever.

Spread one excitation over a chain of atoms. Put neighboring amplitudes out of phase and watch their radiation cancel, then pull the atoms apart and watch the protection fail. Look at the emission lobes and the statistics of photon arrival times.

The chain is finite and one-dimensional, made of fixed two-level emitters, and is computed with a single excitation in free space using the vector-dipole Markov model. The dipoles are parallel to each other and perpendicular to the chain, and both coherent exchange and collective decay are included. Atomic motion, disorder averaging, extra losses and the preparation apparatus are left out. A subradiant state still decays after a finite time; it does not store the excitation perfectly.

ANGULAR EMISSIONMODEL

EXCITATION REMAINING

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Unconditional ensemble mean

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

Dots: site excitation; hand direction: complex-amplitude phase. Physical chain lies on x, dipoles on z. Surface: angular intensity, not a spatial wavefront.

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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

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Recorded clicks per trial

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Trials with a click

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Model click probability

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Model excitation at the gate

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01 / PHOTON TIMES

A long, faint tail.

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

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

02 / SURVIVAL

Where the excitation went.

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

$\Gamma_0t$ · $S(t)$

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

Finite fixed chain along x; parallel dipoles along z. The vector Green tensor includes coherent exchange and collective radiative decay. A matrix exponential evolves the unnormalized single-excitation state; norm loss is emitted probability. Zero diagonal coherent shift defines the rotating frame. No nonradiative loss, motion or retardation beyond the Markov Green tensor.

\dot{\boldsymbol c}=\tfrac12(-\boldsymbol\Gamma+i\boldsymbol\Omega)\boldsymbol c

\frac{dR}{d\Omega}=\frac{3\Gamma_0}{8\pi}(1-n_z^2)\left|\sum_jc_je^{-ik_0n_xx_j}\right|^2

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:

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The diagnostic compares survival loss with the time integral of emission, using a finite trapezoidal step. It is a quadrature residual, not a physical loss. Photon times invert the survival curve on a fine grid.

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: How can neighboring atoms keep an excitation from radiating away so quickly?
One-sentence intuition: The amplitudes of the single shared excitation interfere in every direction of emission. Where the atoms sit and how their phases are set decide both where the light goes and how long the excitation survives.

Core mathematical model

How the amplitudes evolve

\dot{\boldsymbol c}=\tfrac12(-\boldsymbol\Gamma+i\boldsymbol\Omega)\boldsymbol c

The vector Green tensor gives both the dissipative and the coherent couplings. Frequencies are in units of the decay rate of an isolated atom.

Surviving excitation and its loss rate

S=\sum_j|c_j|^2,\quad -\dot S=\boldsymbol c^\dagger\boldsymbol\Gamma\boldsymbol c

The excitation that is lost leaves as light over the full solid angle. On a finite chain, the initial phase ramp is never perfectly dark.

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
Compare In phase and Alternate after two single-atom lifetimes.
What to observe: The alternating state keeps most of its excitation, and the in-phase state radiates quickly.
02
Break the effect
Increase the spacing or move one atom off its site. Watch how both the survival and the emission directions change.
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.