The Hong–Ou–Mandel dip: which alternatives became indistinguishable?

Scan a path, count pairs

Hong, Ou and Mandel’s 1987 apparatus brought two down-converted photons to opposite inputs of a beam splitter and scanned its position. The measured curve counted coincidences between output detectors. The dip revealed a short relative-time scale through interference despite much slower detector timing.

The paper’s plotted coincidences had accidentals subtracted. Its electronics accepted a 7.5-nanosecond coincidence interval, and a displaced interval estimated the accidental contribution. Those acquisition windows are not the approximately hundred-femtosecond coherence scale inferred from the optical scan.

Read the evidence: C. K. Hong, Z. Y. Ou and L. Mandel (1987), Measurement of Subpicosecond Time Intervals between Two Photons by Interference, PRL 59, 2044–2046 — scanned copy

Start with a narrower input model

For our teaching equation, assume exactly one photon in each input, a lossless balanced splitter and two independent pure normalized wave packets. Their inner product includes every internal mode that can distinguish them, after applying the scanned relative delay. Coincidence means one photon in each output, with ideal collection.

At full overlap, the both-transmitted and both-reflected alternatives for that output cancel. The two photons still leave the apparatus, together through one output or the other. Orthogonal polarization or nonoverlapping arrival modes remove the cancellation and restore a coincidence probability of one half.

P_{\mathrm{coin}}(\tau)=\frac{1-\left|\langle f\mid g_{\tau}\rangle\right|^2}{2}

Read the evidence: C. K. Hong, Z. Y. Ou and L. Mandel (1987), Measurement of Subpicosecond Time Intervals between Two Photons by Interference, PRL 59, 2044–2046 — scanned copy A. M. Brańczyk (2017), Hong–Ou–Mandel Interference — author notes, arXiv:1711.00080

A mixed input changes the inference

Equal spectra are not enough to justify that pure-product formula. In an independent mixed-state example with two orthogonal modes, let each photon occupy either mode with equal probability. Half the input pairs then match and half do not; their average zero-delay coincidence probability is one quarter.

This gives a dip only halfway down from the distinguishable reference even with otherwise perfect optics. Spectral correlations between photons require their joint state instead. The historical down-conversion calculation and the independent pure-packet example therefore answer related questions under different input assumptions.

Read the evidence: A. M. Brańczyk (2017), Hong–Ou–Mandel Interference — author notes, arXiv:1711.00080

An unbalanced splitter leaves a floor

Take an invented lossless splitter with intensity reflection 0.60 and transmission 0.40. Perfectly matching photons have a coincidence probability of 0.04; distinguishable photons have 0.52. The corresponding visibility is about 0.923. A nonzero minimum here comes entirely from splitter imbalance.

The result follows by adding the signed double-reflection and double-transmission amplitudes before squaring. Adding their probabilities first gives the distinguishable reference. This provides a useful independent check on a plotted dip: zero is a prediction of balance and matching together, not of photon counting alone.

Report the normalization

In a separate invented count example, a far-delay bin has 110 events and the dip bin has 20, each with an independently estimated constant background of ten. Raw visibility is about 0.818; background-corrected visibility is 0.90. Both describe declared records, and neither can silently stand in for the other.

Keep exposure time, source drift, accidental-window scaling and uncertainty propagation with the curve. Optical displacement must also be converted through the actual path geometry. Moving a reflector can change a round-trip path twice as much; copying a distance axis directly into a delay axis can therefore misstate the inferred width.

Read the evidence: C. K. Hong, Z. Y. Ou and L. Mandel (1987), Measurement of Subpicosecond Time Intervals between Two Photons by Interference, PRL 59, 2044–2046 — scanned copy

The dip is a component test

Good two-photon interference helps a Bell-state analyzer operate as intended. It does not by itself certify polarization entanglement, close detection loopholes or establish a complete quantum communication channel. Those tasks need additional settings and resource records.

The linked Lab collects a coincidence dip in its teaching model. Compare delay, polarization and overlap while keeping the source assumptions visible. Carry that habit into the swapping and dense-coding readings: identify which alternatives the analyzer combines before deciding what a successful detector pattern means.

Read the evidence: A. M. Brańczyk (2017), Hong–Ou–Mandel Interference — author notes, arXiv:1711.00080 J.-W. Pan, D. Bouwmeester, H. Weinfurter and A. Zeilinger (1998), Experimental Entanglement Swapping, PRL 80, 3891–3894