COLLECTIVE LIGHT / THE QUIET SIDE

Subradiant Atomic Array

Share one excitation across an atomic chain. Turn neighboring amplitudes out of phase, watch radiation cancel, then break the protection by increasing the spacing. Inspect the emission lobes and timestamp statistics.

A finite one-dimensional chain of fixed two-level emitters in the single-excitation, free-space vector-dipole Markov model. Parallel dipoles are perpendicular to the chain; coherent exchange and collective decay are both included. No motion, disorder average, extra loss or preparation apparatus. Subradiance is finite-lived, not perfect storage.

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. RECORD THE LIGHT.

How quiet is quiet?

Each trial starts fresh and records the full time window. Ideal full-solid-angle photon counting, followed by the selected efficiency. No-click trials stay 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 uses the isolated-emitter population lifetime. The counter has full solid-angle coverage and independent detection loss. No background, dead time, timing jitter or technical preparation error. Intervals describe 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: Change the phase. Keep the excitation.
One-sentence intuition: The single-excitation amplitudes interfere in every emission direction. Their geometry and phase set both the radiation pattern and the survival dynamics.

Core mathematical model

Amplitude dynamics

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

The vector Green tensor supplies dissipative and coherent couplings. Frequencies are in isolated-atom decay units.

Excitation survives

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

The lost excitation is radiated into the full solid angle. A finite chain does not have a perfectly dark initial phase ramp.

Common difficulties

A quiet counter is not proof of storage

Typical misconceptionNo click means the excitation is definitely still present.

Better mental modelThe detector may miss an emitted photon. The model survival and observed no-click fraction are separate quantities; only sampling uncertainty is shown.

Run the experiment

01
Compare the cause
Compare In phase and Alternate at two single-atom lifetimes.
What to observe: The alternating preparation retains most excitation while the in-phase preparation radiates rapidly.
02
Break the effect
Increase the spacing or displace one site. Inspect both survival and directionality.
What to observe: The result follows the selected state and geometry, not a fixed animation.
03
Count photons
Collect 200 full decay trials. Drag the time gate, pin the data, change the preparation and restore. Export timestamps, including no-click trials.
What to observe: Detection inefficiency can also produce silence. Compare the measured click probability with the efficiency-aware model.