Wavepacket photodetection and normally ordered correlations

Assume the Axiom of Countable Choice. Let $H$ be a complex Hilbert space with the second-variable-linear pairing used for bosonic Fock space, and let $u\in H$ satisfy $\mathrm{\parallel }u\mathrm{\parallel }=1$. Put $K=u^{\perp }$. Let $J_u:{\mathcal{F}}_s(\mathbb{C}u)\widehat{\otimes }{\mathcal{F}}_s(K)\to {\mathcal{F}}_s(H)$ be the one-mode Fock unitary, and for $n\in {\mathbb{N}}_0$ let $e_n=u^{\otimes n}\in (\mathbb{C}u{)}_s^{(n)}$, with $e_0=1$. On ${\mathcal{F}}_s(\mathbb{C}u)$ define $Q_{n}z=⟨e_n,z{⟩}_2{e}_n$, and on ${\mathcal{F}}_s(H)$ put $P_n:=J_u(Q_n\otimes I)J_u^{-1}\mathrm{.}$ For every Borel set $B\subseteq \mathbb{R}$, define the strong sum $E_u(B):={\sum }_{n\in B\cap {\mathbb{N}}_0}{P}_n\mathrm{.}$ Define the mode number operator $N_u:={\sum }_{n=0}^{\mathrm{\infty }}n{P}_n,D(N_u):=\{\mathrm{\Psi }\in {\mathcal{F}}_s(H):{\sum }_{n=0}^{\mathrm{\infty }}{n}^2\mathrm{\parallel }{P}_n\mathrm{\Psi }{\mathrm{\parallel }}^2<\mathrm{\infty }\}\mathrm{.}$ Then $(P_n{)}_{n\ge 0}$ are mutually orthogonal projections with strong sum $I$, $E_u$ is a projection-valued measure supported on ${\mathbb{N}}_0$, and $N_u$ is self-adjoint with these spectral projections. Define $M_u\mathrm{\Psi }:={\sum }_{n=0}^{\mathrm{\infty }}n(n-1)P_n\mathrm{\Psi }$ on $D(M_u):=\{\mathrm{\Psi }\in {\mathcal{F}}_s(H):{\sum }_{n=0}^{\mathrm{\infty }}{n}^2(n-1{)}^2\mathrm{\parallel }{P}_n\mathrm{\Psi }{\mathrm{\parallel }}^2<\mathrm{\infty }\}\mathrm{.}$ Then $M_u=N_u(N_u-I)$ is self-adjoint, and ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ is a graph core for it. On the finite-particle subspace, $N_u=a^{†}(u)a(u),M_u=N_u(N_u-I)=a^{†}(u{)}^{2}a(u{)}^2\mathrm{.}$ Thus the finite-particle restriction of $a^{†}(u{)}^{2}a(u{)}^2$ is essentially self-adjoint with closure $M_u$.

Schwartz/distribution spaces and Hilbert Fock conventions

One-mode factorization of symmetric Fock space

Bosonic and fermionic Fock operators with exact domains

Projection valued measure

Steck, Quantum and Atom Optics

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.

Assume the ideal wavepacket counting postulate for a normalized mode $v$, and let $p_n$ be the resulting count law of a prepared state. Write $\mu :={\sum }_{n=0}^{\mathrm{\infty }}n{p}_n,F_2:={\sum }_{n=0}^{\mathrm{\infty }}n(n-1)p_n,g_v^{(2)}(0):=F_2\mathrm{/}{\mu }^2$ when $0<\mu <\mathrm{\infty }$ and $F_2<\mathrm{\infty }$. Let $J_v:{\mathcal{F}}_s(\mathbb{C}v)\widehat{\otimes }{\mathcal{F}}_s(v^{\perp })\to {\mathcal{F}}_s({\mathcal{H}}_T)$ be the one-mode Fock unitary, let $\mathrm{\mid }n{⟩}_v=v^{\otimes n}$ with $\mathrm{\mid }0{⟩}_v=1$, and let ${\mathrm{\Omega }}_{v^{\perp }}$ be the vacuum in ${\mathcal{F}}_s(v^{\perp })$. Put ${\psi }_n:=J_v(\mathrm{\mid }n{⟩}_v\otimes {\mathrm{\Omega }}_{v^{\perp }})$. For a unit vector $\psi$, $g_v^{(2)}(0;\psi )$ denotes the postulate's value for the pure density operator $\mathrm{\mid }\psi ⟩⟨\psi \mathrm{\mid }$. For $\alpha \in \mathbb{C}\setminus \{0\}$ put $\lambda =\mathrm{\mid }\alpha {\mathrm{\mid }}^2$, $Z_{\lambda }={\sum }_{n=0}^{\mathrm{\infty }}{\lambda }^n\mathrm{/}n!$, and define the coherent-mode vector ${\psi }_{\alpha }:=Z_{\lambda }^{-1\mathrm{/}2}{\sum }_{n=0}^{\mathrm{\infty }}\frac{{\alpha }^n}{\sqrt{n!}}{\psi }_n\mathrm{.}$ For $m\ge 1$, the number preparation is ${\psi }_m$. For $0<r<1$, the geometric one-mode state is ${\rho }_r:=(1-r){\sum }_{n=0}^{\mathrm{\infty }}{r}^n\mathrm{\mid }{\psi }_n⟩⟨{\psi }_n\mathrm{\mid }\mathrm{.}$ Then the ideal model predictions are $g_v^{(2)}(0;{\psi }_{\alpha })=1,g_v^{(2)}(0;{\psi }_m)=1-\frac{1}{m},g_v^{(2)}(0;{\rho }_r)=2.$ The final state has the specified single-mode geometric occupation law; these conclusions do not classify all radiation as coherent, number, or thermal.

Ideal wavepacket number counting

One-mode photon number operator and count PVM

Density operators and scalar spectral probability

The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$

The real exponential function and the number $e$ by a power series

For $\mathrm{\mid }r\mathrm{\mid }<1$, ${\sum }_{k\ge 0}{r}^k=1\mathrm{/}(1-r)$, and for $\mathrm{\mid }r\mathrm{\mid }\ge 1$ the series diverges

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

Glauber (1963)

Steck, Quantum and Atom Optics

Assume the ideal wavepacket-counting postulate for a selected mode $v\in {\mathcal{H}}_T$ with $\mathrm{\parallel }v\mathrm{\parallel }=1$, and let $\rho$ be the incoming density operator on ${\mathcal{F}}_s({\mathcal{H}}_T)$ (a positive self-adjoint trace-class operator with trace one). By that postulate, $P_n(v)$ are the mode-count projections and $p_n=\mathrm{tr}(\rho P_n(v))$ are the count probabilities, so $p_n\ge 0$ and ${\sum }_{n=0}^{\mathrm{\infty }}{p}_n=1$. Define the incoming mean and second factorial moment by $\mu :={\sum }_{n=0}^{\mathrm{\infty }}n{p}_n,F_2:={\sum }_{n=0}^{\mathrm{\infty }}n(n-1)p_n,$ and assume $0<\mu <\mathrm{\infty }$ and $F_2<\mathrm{\infty }$. Define $g_N^{(2)}(0):=F_2\mathrm{/}{\mu }^2$; for this input state $\rho$, the postulate denotes the same statistic by $g_v^{(2)}(0)$.

Let $0<\eta \le 1$ be a dimensionless fixed efficiency and assume no background or dark counts. To specify independent recording, put $b_{\eta }(0)=1-\eta$, $b_{\eta }(1)=\eta$, and $\mathrm{\Omega }:=\{(n,x):n\in {\mathbb{N}}_0,x=(x_1,\dots ,x_n)\in \{0,1{\}}^n\},\mathcal{A}:=\mathcal{P}(\mathrm{\Omega })\mathrm{.}$ This is countable under the explicit encoding $(n,x)\mapsto (n,{\sum }_{j=1}^n{x}_j{2}^{j-1})$. Give the space the point masses $\mathbb{P}(\{(n,x)\}):=p_n{\prod }_{j=1}^n{b}_{\eta }(x_j),$ where the empty product for $n=0$ is $1$. For each fixed $n$, the product factors define the conditional product law $Q_n$ on $\{0,1{\}}^n$; hence its coordinates are independent Bernoulli variables with success probability $\eta$. Since ${\sum }_{x\in \{0,1{\}}^n}{Q}_n(\{x\})=(\eta +(1-\eta ){)}^n=1$, these point masses sum to one and define a probability measure $\mathbb{P}$ on $(\mathrm{\Omega },\mathcal{A})$. Define $N(n,x):=n$ and $K(n,x):={\sum }_{j=1}^n{x}_j$. For every nonnegative random variable $Y$ on this space, use the expectation $\mathbb{E}Y:={\sum }_{(n,x)\in \mathrm{\Omega }}Y(n,x)\mathbb{P}(\{(n,x)\})\mathrm{.}$ The marginal law of $N$ is $(p_n)$, and the conditional count law of $K$ given $N=n$ is the binomial law induced by $Q_n$. If $\mathbb{E}K>0$, define $g_K^{(2)}(0):=\mathbb{E}[K(K-1)]\mathrm{/}(\mathbb{E}K{)}^2$. Then $\mathbb{E}K=\eta \mu ,\mathbb{E}[K(K-1)]={\eta }^2{F}_2\mathrm{.}$ Consequently the normalized zero-delay factorial correlation is unchanged: $g_K^{(2)}(0)=\frac{\mathbb{E}[K(K-1)]}{(\mathbb{E}K{)}^2}=\frac{F_2}{{\mu }^2}=g_N^{(2)}(0)=g_v^{(2)}(0)\mathrm{.}$ This is a model deduction conditional on independent constant-efficiency thinning; it is not a universal property of real detectors.

Ideal one-mode intensity correlations separate model states

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

Steck, Quantum and Atom Optics

Glauber (1963)