Ideal single-mode factorial moment

Given: The number state $\mathrm{\mid }1⟩$, the ideal single-mode count PVM, the vacuum state in the unused port, a passive two-mode unitary with $0\le \eta \le 1$, and the assigned finite-mode model.

The finite-mode postulate supplies an ideal number-count PVM but makes no claim about a laboratory detector or a measured count record (Finite-mode dipole and detector assignment).

Assume AC and ${\mathrm{A}\mathrm{C}}_{\omega }$ as mathematical prerequisites of the finite-mode passive implementer and Fock suppliers; these are set-theoretic assumptions, not physical postulates (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

For the declared number distribution, $g^{(2)}(0)={\sum }_{n}n(n-1)p_n\mathrm{/}({\sum }_{n}n{p}_n{)}^2$ when the denominator is finite and positive (Ideal single-mode normally ordered second factorial statistic).

The ladder actions give $a\mathrm{\mid }1⟩=\mathrm{\mid }0⟩$, $a\mathrm{\mid }0⟩=0$, $a^{\ast }a\mathrm{\mid }1⟩=\mathrm{\mid }1⟩$, and $\mathrm{\mid }1⟩,\mathrm{\mid }0⟩$ are orthonormal (Fixed-mode number law).

A one-quantum input and vacuum at the second port have joint output state $t\mathrm{\mid }1,0⟩+r\mathrm{\mid }0,1⟩$ and ideal count probabilities $\mathrm{\mid }t{\mathrm{\mid }}^2=\eta$, $\mathrm{\mid }r{\mathrm{\mid }}^2=1-\eta$ (Passive beam-splitter photon statistics).

Ideal number-count effects are PVM projections with probabilities given by the Born rule; they do not imply a physical device or recorded data (Declared ideal measurements and instruments of the framework).