Q035 · Timing photons at two detectors

HBT Photon Correlation

How can you tell equally bright light sources apart by the way their photons arrive? Coincidences between two detectors at zero delay tell them apart. Coherent light, such as a laser, arrives at random, and the correlation stays flat. Thermal light bunches, giving extra coincidences. A single emitter antibunches, because it cannot emit two photons at once.

Split the light from one source between two independent detectors. Collect arrival times and build the correlation curve for coherent light, thermal light in temporal modes and a single emitter.

Recorded clicks $N=0$

Sampled central correlation $\text{—}$

01 / The experimental setup

One source feeds two photon counters.

Each detected photon appears in one output. Compare the two clocks.

Drag to rotate · scroll to zoomSchematic optics · flashes replay recorded clicks

Recorded time

Choose an experiment

Mean source flux

\Phi

1

Detection efficiency

\eta

0.65

Detector & model settings

Source statisticsBackground rate per detector

b

0.005

Fraction sent to detector A

R

0.5

Thermal mode duration

t_c

2

Timestamp jitter width

w

0

Detector dead time

t_d

0

Correlation bin width

\Delta\tau

0.25

Duration per independent record

T

2000

Reproducible seed

q

2735

Changing the model clears records. Rebinning and replay preserve timestamps. Time is in reference units; these are simulated experiments.

Before you measure

Can equal brightness hide different arrival patterns?

The beam splitter is ideal, with vacuum entering its unused port, and photodetection is a stationary point process. Thermal light is built from independent flat temporal modes, which give a triangular correlation peak. The emitter is pumped incoherently. The model includes detector loss, background, uniform timing jitter and a separate nonparalyzable dead time for each detector.

02 / Follow the evidence

From two clocks to one correlation curve

How a pair of clicks enters the curve

Inspect delay bin

\tau=0

This bin, pooled over every recorded A click:

\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.

Compare with the same detector A–A

Numerical results & model checks

Central correlation / one standard error

\text{—}

Central pairs / estimated accidentals

\text{—}

Measured A / B count rates

\text{—}

Detector model central reference

\text{—}

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.

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

03 / Equally bright sources with different arrival patterns

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

\text{—}

Coherent light

Independent arrivals

\text{—}

Thermal modes

Fluctuating intensity groups arrivals

\text{—}

Ready to compare using the current detector settings.

Start with the essentials

Focus question: Do photons prefer arriving together?
One-sentence intuition: Draw conclusions from the recorded clicks first, and only then compare them with the model.

Core mathematical model

Units of time and rate

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

The reference time is arbitrary and is not tied to any atom or real lifetime. All times, rates and CSV values are given in this unit.

Normally ordered correlation

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

A photon is never paired with itself. The splitter sends each photon to exactly one output, so the two detectors register pairs of different photons.

The three light sources

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 light arrives as a Poisson process. Thermal light is a train of independent flat modes that start at a random time; each mode has an exponentially distributed intensity and a geometric photon count. The two-level emitter has equal pump and decay rates, each twice the mean flux. The shape of the thermal peak belongs to this mode model.

Counting pairs from timestamps

C_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 gets its own overlap and its own estimate of accidental pairs from the measured rates. Events from different records are never paired. The central bin covers both negative and positive delays.

Pooled estimate and uncertainty

\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

The jackknife standard error leaves out one record at a time and needs at least three independent records. It is an approximate sampling error and guarantees no confidence level. Estimating rates from finite records adds a bias, and pairs that share a photon are not independent Poisson observations.

Effect of background and timing jitter

g_{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 over the triangular spread of the difference between two independent uniform jitters. Independent background lowers the contrast. With dead time on, the reference is hidden because it ignores the detector's memory of earlier clicks.

Dead time is a detector effect

t_{n+1}^{\rm accepted}-t_n^{\rm accepted}\ge t_d\quad\text{before timestamp jitter}

Each detector recovers on its own, and arrivals it rejects do not extend its dead time. For coherent light, compare A–A with A–B: a dip in a single channel can come from the instrument. The A–A estimate excludes self-pairs and is normalized by the factorial count.

Common difficulties

A dip does not prove a quantum source

Typical misconceptionAny dip in the curve proves the source is nonclassical.

Better mental modelBefore drawing that conclusion, check both channels, the background, the timing resolution and the detector dead time.

Run the experiment

01
Match brightness
Keep the mean flux fixed while switching sources.
What to observe: The average count rate alone cannot tell the photon statistics apart.
02
Collect more records
Collect eight independent records and look at the correlation in the central bin.
What to observe: The error bars come from the spread between independent records.
03
Test the detector
Select the dead-time case and compare same-channel and cross-channel curves.
What to observe: The detector alone can make a single channel look antibunched.