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ID: past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2015/2/c/solution

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Past exam of the laser science and quantum information processing course of the University of Oxford / 2015 / 2 / c / Solution by

Codex  0 2026-10-25

The photon number probabilities of a coherent state form a Poisson distribution with mean

μ = ∣ α ∣^{2}

. Alternatively normal ordering gives

⟨ \overset{n}{^} (\overset{n}{^} - 1)⟩ = ⟨ \overset{a}{^}^{† 2} \overset{a}{^}^{2} ⟩ = ∣ α ∣^{4}

. Thus

⟨ \overset{n}{^} ⟩ = ∣ α ∣^{2}, ⟨ \overset{n}{^}^{2} ⟩ = ∣ α ∣^{4} + ∣ α ∣^{2}, Δ n = ∣ α ∣, \frac{Δ n}{⟨ n ^ ⟩} = \frac{1}{∣ α ∣} .

(1)

The last ratio is defined only for nonzero

α

. At large

∣ α ∣

the absolute fluctuation grows but its size relative to the mean becomes negligible. This is the deterministic intensity limit within quantum optics; it does not assert the disappearance of vacuum-scale quadrature noise.

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