Franson Interferometer: Energy-Time Entangled Photon Interference

Interferometer Phases

Alice phase φ_A 0°

Bob phase φ_B 45°

Pump coherence (visibility V) 92%

Pair emission rate 3.0/s

Coincidence Statistics

Bell test: V too low to violate CHSH

Predicted P(center)

50.0%

Correlation E(φA,φB)

0.00

Total coincidences

Pairs emitted

How it works

An SPDC crystal emits energy-time entangled photon pairs. Each photon enters its own unbalanced Mach–Zehnder interferometer with a short arm (S) and a long arm (L, path difference ΔL). Every photon independently picks S or L, so a pair lands in one of four combinations, each ¼ likely: SS, SL, LS, LL.

P(coincidence at delay 0) = ¼[1 + V·cos(φ_A + φ_B)]
P(early, S_A L_B) = P(late, L_A S_B) = ¼   (always, no interference)
Bell/CHSH violation requires  V > 1/√2 ≈ 70.7%

SL and LS arrive at clearly different total times — the "which-path" information is available from timing alone, so these events never interfere and always land in the early/late bins at a flat ¼ each.
SS and LL arrive in the very same time bin (both photons took the matching arm), so no timing information distinguishes them. Because the pump that created the pair was itself coherent, the two indistinguishable two-photon amplitudes interfere, and the center-bin rate oscillates as cos(φ_A + φ_B) — even though φ_A and φ_B live on opposite sides of the lab and no single photon ever showed interference on its own.
Visibility V models how much pump/channel decoherence survives into the two-photon amplitude; only above V ≈ 70.7% can the center-bin correlation violate a CHSH-type Bell inequality, certifying genuine energy-time entanglement rather than a classical timing correlation.

This is a different Bell-test mechanism from polarization-based protocols (BB84, E91, CHSH with polarizers): here the "basis" is a relative phase inside a time-domain interferometer, and the non-local signature shows up only in the joint (coincidence) statistics, never in either detector's singles rate.

Alice's photon Bob's photon

Early 0Center 0Late 0

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