Hong-Ou-Mandel Two-Photon Interference

Photon Timing

Arrival delay τ 0 fs

Photon coherence time τ_c 60 fs

Beamsplitter reflectivity R 50%

Interference Result

Indistinguishability g(τ)

1.00

P(coincidence) theory

0.0%

Trial Statistics

Trials

Coincidence

Bunched

Measured P(coinc)

—

How it works

Two identical photons enter the two input ports of a beamsplitter (BS) at nearly the same time. Classically you would expect coincident clicks (one photon in each output) 50% of the time. Quantum mechanically the two ways of getting a coincidence — both reflect, or both transmit — are indistinguishable alternatives that interfere destructively, so the photons always leave together, bunched in the same output port. This is the Hong–Ou–Mandel (HOM) effect (1987), the workhorse two-photon-interference test for photonic quantum computing and quantum networking.

Coincidence probability (BS reflectivity R, transmissivity T=1-R):
P_c(τ) = R² + T² − 2RT·g(τ)²

Photon overlap (Gaussian wavepackets, coherence time τc):
g(τ) = exp( −τ² / (2τc²) )

Perfect 50/50 BS, τ = 0 ⇒ g = 1 ⇒ P_c = 0  (the HOM dip bottoms out)
Large |τ|          ⇒ g → 0 ⇒ P_c → R² + T² = 0.5  (classical limit)

Delay τ — the relative arrival time of the two photons at the beamsplitter. τ = 0 means they arrive perfectly together.
Coherence time τ_c — set by the photons' spectral bandwidth; it controls how narrow the HOM dip is. Shorter coherence time (broader bandwidth) gives a narrower dip.
Reflectivity R — an unbalanced beamsplitter never lets the dip reach zero, because "both reflect" and "both transmit" no longer have equal amplitude to cancel.
Photon stream — each trial samples a real outcome (coincidence vs. bunched) from P_c(τ) via Monte Carlo, so the measured coincidence rate converges to the theory curve — exactly how a real HOM experiment builds up statistics photon pair by photon pair.

Real-world relevance: HOM interference visibility is the standard benchmark for how indistinguishable single-photon sources are (quantum dots, SPDC pairs) — a deep, narrow dip is required for photonic entanglement swapping, linear-optical quantum computing (KLM scheme), and quantum repeaters.

Photon 1 / Photon 2 Coincidence click Bunched click

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