ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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One-photon wavepacket has zero ideal same-mode factorial correlation

Physical scope

One exactly prepared photon in one normalized transverse mode, counted by the ideal number-resolving model with optional independent fixed-efficiency loss. It asserts no zero coincidence probability for arbitrary multimode detectors or apparatus with background.

Example

Assume the ideal mode-counting and thinning models. Prepare the normalized one-photon state $ψ_{1}$ in the selected transverse mode, with all orthogonal modes in vacuum, and take $0 < η \leq 1$ . The ideal count law has $N = 1$ , so its mean is $1$ and its same-mode zero-delay factorial correlation is $g_{N}^{(2)} (0) = 0$ . After independent loss, the registered count is Bernoulli with $P (K = 1) = η$ and $P (K = 0) = 1 - η$ , with mean $η$ and $g_{K}^{(2)} (0) = 0$ . This example is conditional on the one-mode and no-background assumptions and does not assert zero coincidence probability for arbitrary multimode detectors.

Example references

Ideal one-mode intensity correlations separate model states

Ideal independent loss preserves normalized factorial correlation

Steck, Quantum and Atom Optics

Glauber (1963)

Facts & Assumptions

Given: The selected one-photon number state, the ideal count law, and independent fixed-efficiency recording with $0 < η \leq 1$ .

[F1]

The mode state with $m = 1$ has $μ = 1$ , $F_{2} = 0$ , and $g^{(2)} (0) = 1 - 1 / m = 0$ . Ideal one-mode intensity correlations separate model states

[F2]

Conditional on $N = n$ , each photon is recorded independently with probability $η$ , with no background or dark counts; for one photon this gives a Bernoulli count and the normalized factorial correlation is unchanged. Ideal independent loss preserves normalized factorial correlation

Verification

Given: [F1], [F2], and the one-photon preparation and efficiency in the Example.

1.1F1givenalgebra

The statement of [F1] at $m = 1$ gives $μ = 1$ and $F_{2} = 1 (1 - 1) = 0$ , so the postulate's denominator is positive and $g_{N}^{(2)} (0) = 0$ .

2.1F2step 1.1givenalgebra∎

By [F2], conditional on the certain incoming count $N = 1$ , the only possible registered counts are $K = 0$ and $K = 1$ , with probabilities $1 - η$ and $η$ . Thus $E K = η > 0$ and $E [K (K - 1)] = 0$ , giving $g_{K}^{(2)} (0) = 0$ ; this conclusion concerns the selected ideal mode and the stated apparatus assumptions only.

One-photon wavepacket has zero ideal same-mode factorial correlation

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources