Ideal single-mode normally ordered second factorial statistic

Specialize the finite-mode assignment to $M=1$, identify the coordinate vector $(z)\in {\mathbb{C}}^1$ with its single component $z\in \mathbb{C}$, and write ${\mathcal{F}}_s(\mathbb{C})$ for this one-mode Fock space. Write $\mathrm{\mid }n⟩$ for the occupation vector with $n$ quanta in its unique mode. Let $\rho$ be a density operator (a positive trace-class operator of trace one) on ${\mathcal{F}}_s(\mathbb{C})$. Let ${\mathrm{\Pi }}_n:=\mathrm{\mid }n⟩⟨n\mathrm{\mid }$ be the ideal number-count PVM, with probabilities $p_n:=\mathrm{tr}(\rho {\mathrm{\Pi }}_n)$. Suppose $\bar{n}:={\sum }_{n\ge 0}n{p}_n$ is finite and strictly positive, and $m_2:={\sum }_{n\ge 0}n(n-1)p_n$ is finite. Define the model's equal-time second factorial statistic by $g^{(2)}(0;\rho ):=m_2\mathrm{/}{\bar{n}}^2$. On finite-particle number vectors the one-mode maps satisfy $a\mathrm{\mid }0⟩=0$, $a\mathrm{\mid }n⟩=\sqrt{n}\mathrm{\mid }n-1⟩$ for $n\ge 1$, and $a^{\ast }\mathrm{\mid }n⟩=\sqrt{n+1}\mathrm{\mid }n+1⟩$ for $n\ge 0$. Normal ordering here means placing creation maps to the left of annihilation maps; these actions give $a^{\ast }{a}^{\ast }aa\mathrm{\mid }n⟩=n(n-1)\mathrm{\mid }n⟩$ for $n\ge 0$. Thus $m_2$ is the corresponding number-basis factorial moment, defined here directly by its probability sum rather than by an unqualified expectation of an unbounded operator product. This uses only the ideal counting model; it asserts no time-resolved photocurrent law, empirical detection, or detector realization.

Finite-mode dipole and detector assignment

Fixed-mode number law

Declared ideal measurements and instruments of the framework

The Axiom of Choice

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)