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Q035 · Split / time / correlate

HBT Photon Correlation

Send a source to two independent detectors. Collect timestamps and build a correlation curve for coherent light, thermal temporal modes and a single emitter.

Interactive modelHBT Photon Correlation
Recorded clicksN=0N=0
Sampled central correlation—\text{—}

02 / FOLLOW THE EVIDENCE

From two clocks to one correlation curve

HOW A PAIR ENTERS THE CURVE

This bin, pooled over every recorded A click:

g^AB=pairsaccidentals\widehat g_{AB}=\frac{\text{pairs}}{\text{accidentals}}

Click the curve to inspect a delay bin. One means independent arrivals.

Blue: sampled A–B pairs; whiskers: one record-jackknife standard error. Gray: separate detector reference, omitted with dead time. Orange: optional A–A correlation. All values average over finite delay bins.

Numerical results & model checks

Central correlation / one standard error

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Central pairs / estimated accidentals

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Measured A / B count rates

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Detector model central reference

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Raw records & scientific boundary

Thermal modes are rectangular in time with independent exponential intensities, not a broadband blackbody spectrum. The single-emitter reference has equal incoherent pump and decay rates. Timestamps receive independent uniform jitter after each detector’s dead-time decision. Finite-window overlap is included; estimated rates still introduce finite-record bias. The error bars are approximate sampling errors, not significance certificates.

Independent records — the latest is replayed above
Latest 12 clicks; CSV includes every click and each exposure

Primary references: Wineland · Nobel lecture · Kimble et al. · 1977 · quED · HBT

03 / SAME BRIGHTNESS, DIFFERENT ARRIVALS

Put three sources side by side

Eight independent records per source. The flux and detector settings are held fixed. Open any result to inspect and export its actual timestamps.

Single emitter

Needs time to excite again

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Coherent light

Independent arrivals

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Thermal modes

Fluctuating intensity groups arrivals

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Ready to compare using the current detector settings.

Physics tutorial

Build an HBT correlation

BackgroundEqual brightness can hide different arrival statistics.

Why it mattersConnect detector events to an explicit inference.

Start with the essentials

Focus question
Do photons prefer arriving together?
One-sentence intuition
Use the record first; reveal the model separately.

Core mathematical model

Units

τ=t/t∗,r,γ,s,k,b,Φ in t∗−1\tau=t/t_*,\qquad r,\gamma,s,k,b,\Phi\ \text{in}\ t_*^{-1}

The reference time is arbitrary; no atomic species or physical lifetime is calibrated. All times, rates and CSV values use this reference.

Normally ordered correlation

g(2)(τ)=⟨:I(t)I(t+τ):⟩⟨I⟩2g^{(2)}(\tau)=\frac{\langle :I(t)I(t+\tau):\rangle}{\langle I\rangle^2}

Self-pairs are excluded. An HBT splitter routes each photon into exactly one output; the two detectors reveal pairs of different photons.

Three declared sources

gcoherent(2)=1,gthermal(2)=1+max⁡(0,1−∣τ∣/tc),gemitter(2)=1−e−4Φ∣τ∣g^{(2)}_{\rm coherent}=1,\quad g^{(2)}_{\rm thermal}=1+\max(0,1-|\tau|/t_c),\quad g^{(2)}_{\rm emitter}=1-e^{-4\Phi|\tau|}

Coherent photons form a Poisson process. Independent flat thermal modes have exponential intensities and geometric full-mode counts, with a random time origin. The two-level emitter has equal pump and decay rates, each twice the mean flux. The thermal peak shape is specific to this mode model.

Pairs from timestamps

Cj=∑a∈A∑b∈B1{tb−ta∈Bj},Ej=r^Ar^B∫Bj(T−∣τ∣)+ dτC_j=\sum_{a\in A}\sum_{b\in B}\mathbf1\{t_b-t_a\in B_j\},\quad E_j=\widehat r_A\widehat r_B\int_{B_j}(T-|\tau|)_+\,d\tau

Each independent record has its own overlap and observed-rate accidental estimate. Never pair events across record boundaries. The central bin includes negative and positive delays.

Pooled estimate and uncertainty

g^j=∑mCjm∑mEjm,SEjack2=M−1M∑m(g^j,(−m)−g‾j,(−))2\widehat g_j=\frac{\sum_m C_{jm}}{\sum_m E_{jm}},\qquad {\rm SE}_{\rm jack}^2=\frac{M-1}{M}\sum_m(\widehat g_{j,(-m)}-\overline g_{j,(-)})^2

Use at least three independent records for the delete-one-record jackknife standard error. This is an approximate sampling error, not a confidence guarantee. Rate estimation has finite-record bias; shared photon pairs are not independent Poisson observations.

Background and timing response

gAB(2)−1=SASB(SA+b)(SB+b)(gsource(2)−1),SA=RηΦ, SB=(1−R)ηΦg_{AB}^{(2)}-1=\frac{S_AS_B}{(S_A+b)(S_B+b)}(g_{\rm source}^{(2)}-1),\quad S_A=R\eta\Phi,\ S_B=(1-R)\eta\Phi

The detector reference averages this excess correlation over the delay bin and the triangular difference of two independent uniform timestamp jitters. Independent background reduces contrast. With dead time on, this reference is withheld because it omits detector memory.

Dead time is a detector effect

tn+1accepted−tnaccepted≥tdbefore timestamp jittert_{n+1}^{\rm accepted}-t_n^{\rm accepted}\ge t_d\quad\text{before timestamp jitter}

Each detector recovers independently. Rejected arrivals do not extend its dead time. Compare A–A with A–B for coherent light: the single-channel dip can be instrumental. The A–A estimator excludes self-pairs and uses the factorial count normalization.

Common difficulties

Model and evidence

Typical misconceptionAny dip proves a nonclassical source.

Better mental modelInspect both channels, backgrounds, timing resolution and the detector dead time.

Run the experiment

  1. 01

    Match brightness

    Keep the mean flux fixed while switching sources.

    What to observe: Mean counts alone do not distinguish photon statistics.
  2. 02

    Accumulate

    Collect eight independent records and inspect the central correlation.

    What to observe: Uncertainty comes from independent records, not invented smooth data.
  3. 03

    Challenge the detector

    Select the dead-time case and compare same-channel and cross-channel curves.

    What to observe: The instrument can make one channel look antibunched.