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ID: past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2010/8/2/solution

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Past exam of the laser science and quantum information processing course of the University of Oxford / 2010 / 8 / 2 / Solution by

Codex  0 2026-10-25

Use the two-pair component of polarization-entangled down-conversion to account for identical bosons in the same collected modes. Write

a_{H}^{†}

for a creation operator in mode

a

with horizontal polarization, and similarly for the other modes. For the plus Bell state, the pair-creation operator is

K^{†} = a_{H}^{†} b_{V}^{†} + a_{V}^{†} b_{H}^{†}

. The normalized four-photon state is

∣ Ψ_{2} ⟩ = \frac{( K ^{†} ) ^{2}}{2 3} ∣0 ⟩ = \frac{∣2 H _{a} , 2 V _{b} ⟩ + ∣ H _{a} V _{a} , H _{b} V _{b} ⟩ + ∣2 V _{a} , 2 H _{b} ⟩}{3} .

These are normalized Fock states. In particular,

(a_{H}^{†})^{2} ∣0 ⟩ = 2 ∣2 H_{a} ⟩

; omitting this factor would misnormalize the four-photon input.

The first polarizing beam splitter sends

a_{H}^{†}

to

T_{H}^{†}

and

a_{V}^{†}

to

c_{V}^{†}

. Only the middle term can give exactly one trigger photon and leave another photon in

c

. In the first term both

a

photons reach

T

and none reaches

c

; in the last there is no trigger. Thus the relevant, still unnormalized component is

\frac{1}{3} T_{H}^{†} c_{V}^{†} b_{H}^{†} b_{V}^{†} ∣0 ⟩ .

For the selected paths of the half-wave plate and beam splitter, take

c_{V}^{†} ⟼ \frac{c _{H}^{' †} + c _{V}^{' †}}{2}, b_{σ}^{†} ⟼ \frac{d _{σ}^{†} + e _{σ}^{†}}{2}, σ = H, V .

Here

e

labels the output towards

D_{3}

. An immaterial common reflection phase has been absorbed in the mode definitions. Keeping one photon at

D_{3}

\frac{T _{H}^{†}}{2 6} (c_{H}^{' †} + c_{V}^{' †}) (d_{H}^{†} e_{V}^{†} + d_{V}^{†} e_{H}^{†}) ∣0 ⟩ .

At the final polarizing beam splitter, the

H_{c^{'}} H_{d}

alternative yields one

H

at each of

D_{1}, D_{2}

, while

V_{c^{'}} V_{d}

yields one

V

at each. The two opposite-polarization alternatives send both photons to the same detector and fail fourfold coincidence counting. The accepted component is therefore

\frac{1}{2 6} ∣ H ⟩_{T} (∣ HH V ⟩_{123} + ∣ VV H ⟩_{123}) .

After conditioning and normalization, the three detected output modes have

∣ GHZ ⟩_{123} = \frac{∣ VV H ⟩ + ∣ HH V ⟩}{2} .

The optical mode matching must erase any timing, frequency or other record that distinguishes the two alternatives. Otherwise the same polarization populations could describe an incoherent mixed quantum state, rather than this quantum superposition. The Bouwmeester three-photon GHZ experiment uses this principle. The photonic GHZ preparation by fourfold coincidence is a measurement in the coincidence basis: detecting all four photons projects and consumes them. A click at

T

by itself does not herald a guaranteed freely available three-photon state.

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