Past exam of the theoretical physics II course of the University of Cambridge / 2015 / Paper 1 / 5 / c

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- Solution c

Solution

Bosonic creation operators commute, so the cross terms cancel:

a_{1}^{†} a_{2}^{†} ∣0 ⟩ = \frac{1}{2} [(b_{1}^{†})^{2} - (b_{2}^{†})^{2}] ∣0 ⟩ = \frac{∣2 , 0 ⟩ - ∣0 , 2 ⟩}{2} .

Thus

P (2, 0) = P (0, 2) = \frac{1}{2}, P (1, 1) = 0.

The square root of two in

(b^{†})^{2} ∣0 ⟩ = 2 ∣2 ⟩

is essential. Destructive interference removes the one-particle-per-output amplitude; the particles do not independently choose ports.

For photons this is the ideal Hong-Ou-Mandel effect. Its laboratory test combines one photon from each input at a balanced beam splitter and scans relative arrival delay while measuring output coincidences. Spatial, polarization and spectral indistinguishability must be controlled; distinguishable photons restore a nonzero coincidence baseline. The original experiment used down-converted photons, a translated beam splitter and coincidence counting, with an independently estimated accidental-coincidence background.^[ref]

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