Heralded photons: the missing coincidences need a denominator

A click opens the trial

Grangier, Roger and Aspect used an atomic radiative cascade in their 1986 experiment. Detecting the first photon opened a short gate for two detectors behind a beam splitter receiving the second photon. The source therefore supplied a timing announcement as well as light.

The record needed four counts over the same acquisition: opened gates, clicks in each output and joint clicks within a gate. A small joint count alone says little when almost no light reaches either detector. The singles and gate count supply its reference scale.

Read the evidence: P. Grangier, G. Roger and A. Aspect (1986), Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter, EPL 1, 173–179 — institutional copy

State the classical comparison

Let the gate count, two singles counts and joint count have the subscripts shown below. In the weak-response semiclassical model, each output’s detection probability is proportional to the same nonnegative gate-integrated intensity, with stable efficiencies and conditionally independent detection. Its intensity variance cannot be negative.

That gives a lower bound of one for the normalized anticorrelation parameter. The ideal one-photon input gives zero joint detections instead. This comparison has a detector and sampling contract: gate definitions, backgrounds, saturation and electronic cross-talk cannot be silently changed between numerator and denominator.

\alpha=\frac{N_{\mathrm{gate}}N_{AB}}{N_A N_B}\geq 1\quad\text{(classical, linear response)}

Read the evidence: P. Grangier, G. Roger and A. Aspect (1986), Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter, EPL 1, 173–179 — institutional copy Institut d’Optique, Lab work in photonics, Single-Photon Interferences: Grangier–Roger–Aspect, pp. 71–95

Reconstruct a small count table

For an invented acquisition, take ten thousand gates, one thousand clicks in output A, eight hundred in output B and eight joint clicks. The normalized parameter is one tenth. A consistent event table contains eight joint events, 992 A-only events, 792 B-only events and 8,208 empty gates.

This table prevents double-counting joint clicks as extra trials. It also makes the question reproducible: changing the coincidence window changes which clicks belong together. The example is arithmetic, not a significance claim; a real confidence interval must account for the joint count and the shared denominators.

Read the evidence: P. Grangier, G. Roger and A. Aspect (1986), Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter, EPL 1, 173–179 — institutional copy

Loss and background do different things

Independent thinning of each output cancels from this ratio within the linear-response approximation. Yet low efficiency can leave most gates empty. Even a perfect zero joint probability cannot distinguish a photon in every heralded trial from an ideal mixture containing many vacuum trials and at most one photon.

Dark counts, unrelated emissions and multiple cascade events can add coincidences. Detector dead time or shared electronics can also suppress a joint record artificially. A source assessment therefore needs heralding probability and background checks alongside anticorrelation, with raw and corrected counts kept identifiable.

Attenuation is not a number-state filter

Consider our own Poisson example with a mean of one tenth of a photon per chosen pulse. About nine and a half percent of pulses contain something; among those nonempty pulses, about 4.9 percent still contain two or more photons. Lower brightness increases the empty fraction rather than guaranteeing one photon when a pulse arrives.

In the ideal model, attenuation preserves Poisson statistics. A herald is useful because it changes the ensemble being tested, but it too needs a measured conditional record. Describing either source merely as faint would hide the difference that the triggered experiment was designed to examine.

Read the evidence: Institut d’Optique, Lab work in photonics, Single-Photon Interferences: Grangier–Roger–Aspect, pp. 71–95

Interference asks another question

The same historical source was also used in an interferometer. A fringe probes phase-sensitive alternatives; output anticorrelation probes joint detection statistics. An attenuated classical field can show excellent fringes, so visibility alone does not replace the triggered counting comparison.

The linked quantum eraser and two-photon interference Labs expose related conditional records under their declared source models. They do not acquire the cascade source or measure this heralded parameter. Keep the experimental sequence clear: characterize the input statistics, identify the alternatives and then interpret the chosen coincidence record.