Ideal wavepacket number counting

Assume Countable Choice and use the positive transverse free-photon model whose one-particle space is ${\mathcal{H}}_T=L^2({\mathbb{R}}^3,d^3\mathbf{k};{\mathbb{C}}^3)$ with its transverse condition and whose state space is the bosonic Fock space ${\mathcal{F}}_s({\mathcal{H}}_T)$. In the unit convention where $\mathbf{k}$ has unit ${\mathrm{m}}^{-1}$, let $u\in {\mathcal{H}}_T^{\nu }$ be a normalized transverse wavepacket for the invariant measure $d\nu =d^3\mathbf{k}\mathrm{/}(2\mathrm{\mid }\mathbf{k}\mathrm{\mid })$, so that $u(\mathbf{k})$ has unit metre and $\int \mathrm{\mid }u{\mathrm{\mid }}^{2}d\nu =1$. Let $v=Wu$, where $Wf(\mathbf{k})=f(\mathbf{k})\mathrm{/}\sqrt{2\mathrm{\mid }\mathbf{k}\mathrm{\mid }}$; then $v\in {\mathcal{H}}_T$, $v(\mathbf{k})$ has unit ${\mathrm{m}}^{3\mathrm{/}2}$, and $\mathrm{\parallel }v{\mathrm{\parallel }}_2=1$. In the Fock factorization ${\mathcal{F}}_s({\mathcal{H}}_T)\cong {\mathcal{F}}_s(\mathbb{C}v)\widehat{\otimes }{\mathcal{F}}_s(v^{\perp })$, $P_n(v)$ is the orthogonal projection onto mode occupation $n$ tensored with the identity on the orthogonal-mode Fock factor. These projections are mutually orthogonal and have strong sum $I$; their count PVM is $E_v(B)={\sum }_{n\in B\cap {\mathbb{N}}_0}{P}_n(v)$, and $N_v={\sum }_{n}n{P}_n(v)$ on the domain ${\sum }_n{n}^2\mathrm{\parallel }{P}_n(v)\mathrm{\Psi }{\mathrm{\parallel }}^2<\mathrm{\infty }$. A density operator $\rho$ means a positive self-adjoint trace-class operator with $\mathrm{tr}\rho =1$, using the trace pairing obtained from the Fock space's second-variable-linear convention by conjugation. For $n\in {\mathbb{N}}_0$, put $p_n=\mathrm{tr}(\rho P_n(v))$. The mathematical density/PVM result makes $(p_n)$ a countably additive probability law under the stated Countable Choice hypothesis.

Adopt as the ideal detector postulate that this law is the model probability for recording exactly $n$ photons in the selected mode. The detector is number resolving, has unit mode efficiency, and has no mode mismatch, background counts or dark counts. This is an additional physical measurement assumption; it is not deduced from the CCR or from the free-field equations.

When ${\sum }_{n}n{p}_n<\mathrm{\infty }$ and ${\sum }_{n}n(n-1)p_n<\mathrm{\infty }$, define the first number moment and normally ordered second factorial moment by ${\mu }_v(\rho ):={\sum }_{n=0}^{\mathrm{\infty }}n{p}_n,F_v^{(2)}(\rho ):={\sum }_{n=0}^{\mathrm{\infty }}n(n-1)p_n\mathrm{.}$ The PVM sums define the unbounded-state moments. In this ideal mode model, define the normally ordered second-mode correlation to be the spectral factorial expectation $F_v^{(2)}(\rho )$. The self-adjoint closure of the finite-particle operator $a^{†}(v{)}^{2}a(v{)}^2$ is $N_v(N_v-I)$ by One-mode photon number operator and count PVM, so this definition agrees with the normally ordered operator on its finite-particle core and uses the spectral expectation for states outside that core. It does not assert a trace product of $\rho$ with an unbounded operator. If ${\mu }_v(\rho )>0$, define the dimensionless same-mode zero-delay statistic $g_v^{(2)}(0):=\frac{F_v^{(2)}(\rho )}{{\mu }_v(\rho {)}^2}\mathrm{.}$ Here $0$ is the conventional zero-delay label for this same-mode statistic; no temporal detector response or pointwise product of electric-field distributions is defined.

Positive transverse free photon model

Quantum field quantities and compatible SI normalizations

Positive transverse photon one-particle space

Transverse one-photon space and helicity projectors

The mass-shell weighted transverse space matches the Lebesgue transverse space

One-mode photon number operator and count PVM

Density operators and scalar spectral probability

Glauber (1963)

Steck, Quantum and Atom Optics

The density/PVM supplier is used only for its density decomposition and scalar probability clauses, which require Countable Choice; its separate unbounded-moment clause requires the Axiom of Choice and is not used. The new physical postulate supplies the detector reading of this countable PVM. It does not claim that the more general finite-instrument measurement postulate or an experimental measurement has derived this detector model.