Finite-mode light–matter models and detection statistics

(a) Finite oscillator-mode data. Choose an integer $M\ge 1$, the one-particle mode space $\mathfrak{h}={\mathbb{C}}^M$ with a declared orthonormal mode basis $e_1,\dots ,e_M$, and the bosonic Fock space ${\mathcal{F}}_s(\mathfrak{h})={⨁}_{n\ge 0}{\mathfrak{h}}^{{\otimes }_{s}n}$ with vacuum $\mathrm{\Omega }$. Here ${\mathfrak{h}}^{{\otimes }_{s}n}$ is the range of the orthogonal symmetrizer in the completed $n$-fold Hilbert tensor product, with the zero-particle sector $\mathbb{C}\mathrm{\Omega }$. For $\mathbf{n}=(n_1,\dots ,n_M)\in {\mathbb{N}}_0^M$, let $\mathrm{\mid }\mathbf{n}⟩$ be the normalized occupation vector. Define the finite-particle subspace ${\mathcal{F}}_{\mathrm{f}\mathrm{i}\mathrm{n}}:=\{\psi =({\psi }^{(n)}{)}_{n\ge 0}\in {\mathcal{F}}_s(\mathfrak{h}):{\psi }^{(n)}=0\text{ for all but finitely many }n\}=\mathrm{span}\{\mathrm{\mid }\mathbf{n}⟩:\mathbf{n}\in {\mathbb{N}}_0^M\}$. The mode operators are dimensionless and on ${\mathcal{F}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ satisfy $a_j\mathrm{\mid }\mathbf{n}⟩=\sqrt{n_j}\mathrm{\mid }\mathbf{n}-{\mathbf{e}}_j⟩$ and $a_j^{\ast }\mathrm{\mid }\mathbf{n}⟩=\sqrt{n_j+1}\mathrm{\mid }\mathbf{n}+{\mathbf{e}}_j⟩$, with the first expression zero if $n_j=0$. The Fock space is infinite-dimensional for each $M\ge 1$; no particle-number cutoff or tensor factorization of an individual photon into position and spin is part of this assignment.

Assume the Axiom of Choice and countable choice wherever the finite-mode Fock, quadrature, and implementability suppliers require them. These are mathematical hypotheses of the assignment, not physical postulates.

Mode-frequency data. Use $\mathrm{\hslash }>0$ for the reduced Planck constant in joule-seconds under the framework's SI convention. For each selected mode $j$, choose a positive angular frequency ${\omega }_j>0$ in seconds${}^{-1}$; the assigned one-quantum energy scale is $\mathrm{\hslash }{\omega }_j$ joules. These are parameters of the finite-mode model, not measured frequencies.

(b) Finite atom and dipole data. Choose $d\ge 1$, an atomic Hilbert space ${\mathcal{H}}_A={\mathbb{C}}^d$, a Hermitian atomic energy matrix $H_A$ in joules, Hermitian compressed dipole matrices $D_j$ in coulomb-metres, and real mode field scales ${\mathcal{E}}_j$ in volts per metre. The matrix elements of $G_j:={\mathcal{E}}_j{D}_j\mathrm{/}\mathrm{\hslash }$ have units seconds${}^{-1}$, where $\mathrm{\hslash }>0$ is retained in joule-seconds. The finite matrices and scales are chosen model data. If they are obtained by compressing a larger atomic model, that compression's error is a separate model input and is not bounded here.

Write ${\mathcal{H}}_{AF}:={\mathcal{H}}_A\widehat{\otimes }{\mathcal{F}}_s(\mathfrak{h})$ for the completed ordered Hilbert tensor product, with the atom factor first, and let $\kappa :{\mathcal{H}}_A\odot {\mathcal{F}}_s(\mathfrak{h})\to {\mathcal{H}}_{AF}$ be the canonical isometric inclusion. Let $I_A$ and $I_F$ denote the identity operators on ${\mathcal{H}}_A$ and ${\mathcal{F}}_s(\mathfrak{h})$. Let ${\mathcal{D}}_0:=\kappa [{\mathcal{H}}_A\odot {\mathcal{F}}_{\mathrm{f}\mathrm{i}\mathrm{n}}]\subseteq {\mathcal{H}}_{AF}$ be the algebraic product domain. If $f_1,\dots ,f_d$ is the standard orthonormal basis of ${\mathcal{H}}_A={\mathbb{C}}^d$, then the vectors $\kappa (f_{\alpha }\otimes \mathrm{\mid }\mathbf{n}⟩)$ form a complete orthonormal basis by the Hilbert tensor product and finite-mode Fock product-basis conclusions for this ordered pair. Hence ${\mathcal{D}}_0$ is dense in ${\mathcal{H}}_{AF}$.

The assigned dipole interaction is the linear operator $V_{\mathrm{d}\mathrm{i}\mathrm{p}}:{\mathcal{D}}_0\to {\mathcal{H}}_{AF}$ defined on elementary tensors by $V_{\mathrm{d}\mathrm{i}\mathrm{p}}(\kappa (\varphi \otimes \psi )):=-{\sum }_{j=1}^M{\mathcal{E}}_j\kappa (D_j\varphi \otimes (a_j+a_j^{\ast })\psi )$ for $\varphi \in {\mathcal{H}}_A$ and $\psi \in {\mathcal{F}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$, and extended linearly to ${\mathcal{D}}_0$. Each map $(\varphi ,\psi )\mapsto \kappa (D_j\varphi \otimes (a_j+a_j^{\ast })\psi )$ is bilinear, so it descends to the algebraic tensor product; $a_j,a_j^{\ast }$ preserve ${\mathcal{F}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$. Each summand has units joules. This is the declared algebraic product domain of the finite-core expression; no graph-core assertion for a closed realization is made. The assignment does not assert that an arbitrary such sum has a self-adjoint realization or generates an evolution; any dynamical use must state and justify its operator domain. The Jaynes–Cummings supplier below gives one specific self-adjoint direct-sum realization.

(c) Mode transformations. A finite mode-coordinate change is assigned only when its finite Bogoliubov matrices satisfy the exact hypotheses of the finite-mode implementability statement: for $U,V\in M_M(\mathbb{C})$, $U{U}^{\ast }-V{V}^{\ast }=I_M$ and $U{V}^T=V{U}^T$. The resulting unitary acts on this finite-mode Fock space. The statement assumes the Axiom of Choice; this is a mathematical assumption, not a physical law. No infinite-mode implementability follows.

(d) Detector outcome models. A detector's outcome space $(X,\mathrm{\Sigma })$ and POVM $E:\mathrm{\Sigma }\to \mathcal{B}({\mathcal{H}}_{AF})$ are declared model data: each $E(B)$ is bounded and positive, $E(X)=I$, and $E$ is countably additive in the weak operator topology. For a density operator $\rho$ on ${\mathcal{H}}_{AF}$ (a positive trace-class operator of trace one), the assigned probability is $\mathrm{tr}(\rho E(B))$ under the framework's stated measurement convention. An ideal number-resolving model on the field has outcomes $\mathbf{n}\in {\mathbb{N}}_0^M$ and effects $I_A\otimes \mathrm{\mid }\mathbf{n}⟩⟨\mathbf{n}\mathrm{\mid }$; these are bounded tensor maps and projections on ${\mathcal{H}}_{AF}$ by the tensor-map adjoint and composition rules. For ideal homodyne of mode $j$ at phase $\theta \in \mathbb{R}$, let $R_j(\theta )\mathrm{\mid }\mathbf{n}⟩=e^{i\theta n_j}\mathrm{\mid }\mathbf{n}⟩$ and $X_{j,\theta }:={\mathrm{\hslash }}^{-1\mathrm{/}2}{R}_j(\theta )Q_j{R}_j(\theta {)}^{\ast }$, where $\mathcal{U}:{\mathcal{F}}_s({\mathbb{C}}^M)\to L^2({\mathbb{R}}^M)$ is the Fock-to-Schrödinger unitary and $Q_j:={\mathcal{U}}^{-1}{M}_{\sqrt{\mathrm{\hslash }}{x}_j}\mathcal{U}$. Here $M_{\sqrt{\mathrm{\hslash }}{x}_j}$ multiplies by $\sqrt{\mathrm{\hslash }}{x}_j$ on its maximal domain $\{f\in L^2({\mathbb{R}}^M):x_{j}f\in L^2({\mathbb{R}}^M)\}$. Thus $X_{j,\theta }$ is self-adjoint by unitary conjugation and on ${\mathcal{F}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ equals $(e^{-i\theta }{a}_j+e^{i\theta }{a}_j^{\ast })\mathrm{/}\sqrt{2}$. Its spectral PVM $B\mapsto I_A\otimes {\mathbf{1}}_B(X_{j,\theta })$ is defined by the same bounded tensor maps and is the ideal homodyne POVM on the product space. These are ideal outcome assignments, not claims about a detector's optical local oscillator, efficiency, bandwidth, calibration or state update. If a state update or a finite detector environment is modeled, declare its instrument separately. The finite product-preparation/unitary/partial-trace assignment described by the finite open-system postulate is for $H_S={\mathbb{C}}^d$ and $H_E={\mathbb{C}}^m$ with a fixed environment state and a declared joint unitary; it may be used for a separately selected finite-dimensional subsystem but does not cover the Fock field. A reduced-dynamics model involving the Fock field must separately specify a trace-class joint preparation and joint unitary on ${\mathcal{H}}_{AF}$. The trace-class partial trace ${\mathrm{Tr}}_{{\mathcal{F}}_s(\mathfrak{h})}:{\mathcal{S}}_1({\mathcal{H}}_{AF})\to {\mathcal{S}}_1({\mathcal{H}}_A)$ is defined for arbitrary Hilbert factors, is trace-norm contractive, and preserves positivity and trace by the mathematical partial-trace result listed below. No Fock-field reduced dynamics or reservoir limit is adopted here.

(e) Optional approximation regimes. A long-wavelength identification with a spatial electromagnetic mode requires a separately declared atomic extent $L$ and wave number $k_j$ and the scale regime $k_{j}L\ll 1$; no error bound is supplied. A rotating-wave choice keeps only the excitation-exchange terms. For a two-level transition of frequency ${\omega }_0$ coupled to a mode of frequency $\omega$, its usual low-excitation regime is $\mathrm{\mid }g\mathrm{\mid }\sqrt{n+1}\ll \omega ,{\omega }_0$ on the declared photon-number sectors; it is not uniform over the full Fock space. A Markov/GKSL approximation is another separate model choice and must state its rates, reservoir correlation time ${\tau }_B$, system time scale ${\tau }_S$, and regime ${\tau }_B\ll {\tau }_S$. None of these optional approximations is derived from this assignment or from the finite system-environment model.

Finite-mode quadratures and symplectic data

Finite-mode Fock to Schrödinger identification

Finite Bogoliubov implementability

Hilbert tensor construction: inner product, completion and product bases

Bounded tensor maps: extension, norm estimate, adjoint and commutation

Separable partial trace and its basic properties

Dimensions, SI unit conventions and quantity types of the framework

Declared ideal measurements and instruments of the framework

Finite system-environment and Markov model assignments

The Axiom of Choice

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

The preceding radiation and cutoff finite-level emission models remain separate framework branches. This finite-mode assignment does not replace them or assert an identification between their continuum transverse modes and the selected finite oscillator modes. It makes no photon-position, photon-spin, helicity, gauge/canonical/kinetic angular-momentum, spin-orbital, or paraxial-limit claim. No observed counts, homodyne records, uncertainties or empirical support are asserted.

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 (${\mathrm{A}\mathrm{C}}_{\omega }$)

Take the one-mode, two-level specialization of the finite-mode assignment and use its SI convention with $\mathrm{\hslash }>0$ in joule-seconds. Let $\omega >0$ be the selected angular frequency in seconds${}^{-1}$, let ${\mathcal{H}}_A={\mathbb{C}}^2$ have orthonormal states $\mathrm{\mid }g⟩,\mathrm{\mid }e⟩$, and choose ${\omega }_0>0$ in seconds${}^{-1}$, $d\in \mathbb{R}$ in coulomb-metres, and $\mathcal{E}\in \mathbb{R}$ in volts per metre. Set ${\sigma }_{+}=\mathrm{\mid }e⟩⟨g\mathrm{\mid }$, ${\sigma }_{-}=\mathrm{\mid }g⟩⟨e\mathrm{\mid }$, ${\sigma }_z=\mathrm{\mid }e⟩⟨e\mathrm{\mid }-\mathrm{\mid }g⟩⟨g\mathrm{\mid }$, and ${\sigma }_x={\sigma }_{+}+{\sigma }_{-}$. Choose $H_A=(\mathrm{\hslash }{\omega }_0\mathrm{/}2){\sigma }_z$ and $D=d{\sigma }_x$; then the dipole expression $-\mathcal{E}D\otimes (a+a^{\ast })$ has energy units. Define the real coupling $g:=-\mathcal{E}d\mathrm{/}\mathrm{\hslash }$ in seconds${}^{-1}$. Retaining from the dipole expression just the excitation-exchange terms ${\sigma }_{+}\otimes a$ and ${\sigma }_{-}\otimes a^{\ast }$ gives coefficient $-\mathcal{E}d=\mathrm{\hslash }g$. Thus the rotating-wave Hamiltonian below is a specified specialization of the model assignment, not an independently measured interaction.

Let ${\mathcal{F}}_s(\mathbb{C})$ be the infinite-dimensional one-mode bosonic Fock space with occupation basis $\mathrm{\mid }n⟩$, $n\in {\mathbb{N}}_0$, and operators $a\mathrm{\mid }n⟩=\sqrt{n}\mathrm{\mid }n-1⟩$, $a^{\ast }\mathrm{\mid }n⟩=\sqrt{n+1}\mathrm{\mid }n+1⟩$ on the finite-particle core. Write $I_A$ and $I_F$ for the identity operators on the atomic and field factors, respectively. Define $\mathrm{\Delta }={\omega }_0-\omega$. For this rotating-wave model, whose usual scale $\mathrm{\mid }g\mathrm{\mid }\sqrt{n+1}\ll \omega ,{\omega }_0$ applies only to the photon-number sectors under study and is not uniform over the full Fock space, set $H_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}=\mathrm{\hslash }\omega I_A\otimes a^{\ast }a+\frac{\mathrm{\hslash }{\omega }_0}{2}{\sigma }_z\otimes I_F+\mathrm{\hslash }g({\sigma }_{+}\otimes a+{\sigma }_{-}\otimes a^{\ast })$ on the finite-particle core of ${\mathbb{C}}^2\widehat{\otimes }{\mathcal{F}}_s(\mathbb{C})$. The Hamiltonian is the exact operator for the selected rotating-wave model; no error bound relative to the unapproximated dipole interaction is asserted.

The total excitation operator $N_{\mathrm{e}\mathrm{x}\mathrm{c}}:=I_A\otimes a^{\ast }a+{\sigma }_{+}{\sigma }_{-}\otimes I_F$ is conserved on that core. The ground line $\mathbb{C}\mathrm{\mid }g,0⟩$ is invariant. For each $n\in {\mathbb{N}}_0$, the subspace ${\mathcal{K}}_n=\mathrm{span}\{\mathrm{\mid }e,n⟩,\mathrm{\mid }g,n+1⟩\}$ is invariant and the restriction has matrix $H_n=\left(\begin{array}{cc}\mathrm{\hslash }\omega n+\frac{\mathrm{\hslash }{\omega }_0}{2}& \mathrm{\hslash }g\sqrt{n+1}\\ \mathrm{\hslash }g\sqrt{n+1}& \mathrm{\hslash }\omega (n+1)-\frac{\mathrm{\hslash }{\omega }_0}{2}\end{array}\right)\mathrm{.}$ Its eigenvalue splitting is $\mathrm{\hslash }\sqrt{{\mathrm{\Delta }}^2+4g^2(n+1)}$. Define the full Hamiltonian to act on the ground line by $-\mathrm{\hslash }{\omega }_0\mathrm{/}2$ and as the orthogonal direct sum of the $H_n$ on the ${\mathcal{K}}_n$, with squared-block-norm domain $D(H_{\mathrm{J}\mathrm{C}})=\{c\mathrm{\mid }g,0⟩\oplus {⨁}_{n\ge 0}{\psi }_n:c\in \mathbb{C},{\psi }_n\in {\mathcal{K}}_n,{\sum }_{n\ge 0}\mathrm{\parallel }{H}_n{\psi }_n{\mathrm{\parallel }}^2<\mathrm{\infty }\}\mathrm{.}$ This direct-sum operator is self-adjoint and extends $H_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$.

Finite-mode dipole and detector assignment

Dimensions, SI unit conventions and quantity types of the framework

Closed adjoint sector domains

Fixed-mode number law

Hilbert tensor construction: inner product, completion and product bases

The Axiom of Choice

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

The indexed blocks form an orthogonal decomposition by total excitation, not a finite-dimensional approximation of the full Fock space. The RWA inequality above is a model-validity scale for specified sectors, not a uniform bound on omitted counter-rotating terms.

diagram failed to render:
\begin{tikzpicture}[>=stealth,baseline=(current bounding box.center)]
\node[draw,minimum width=2.3cm,minimum height=1.1cm] (g) at (0,0) {$\mathbb C|g,0\rangle$};
\node at (1.8,0) {$\oplus$};
\node[draw,minimum width=3.2cm,minimum height=1.1cm] (k0) at (4.1,0) {$\mathcal K_0=\operatorname{span}\{|e,0\rangle,|g,1\rangle\}$};
\node at (6.2,0) {$\oplus$};
\node[draw,minimum width=3.2cm,minimum height=1.1cm] (k1) at (8.5,0) {$\mathcal K_1=\operatorname{span}\{|e,1\rangle,|g,2\rangle\}$};
\node at (10.5,0) {$\oplus\cdots$};
\end{tikzpicture}

The boxes label mutually orthogonal excitation sectors in $\mathbb{C}\mathrm{\mid }g,0⟩\oplus {⨁}_{n\ge 0}{\mathcal{K}}_n$. Each $\oplus$ denotes the Hilbert direct sum; no transition between blocks is represented. The proof of invariance and self-adjointness is the operator argument above and does not rely on the diagram.

Assume the finite-mode assignment with two modes and its ideal number PVM, with AC and ${\mathrm{A}\mathrm{C}}_{\omega }$ as required by the implementer and completed Fock/tensor suppliers. Write ${\mathcal{F}}^{(2)}={\mathcal{F}}_s({\mathbb{C}}^2)$ in the ordered mode basis $(1,0),(0,1)$ and $\mathcal{F}={\mathcal{F}}_s(\mathbb{C})$. The mode-factorization map is the unitary $\mathcal{J}:{\mathcal{F}}^{(2)}\to \mathcal{F}\widehat{\otimes }\mathcal{F}$ determined by $\mathcal{J}\mathrm{\mid }m,n⟩=\mathrm{\mid }m⟩\otimes \mathrm{\mid }n⟩$. Here $\widehat{\otimes }$ is the completed Hilbert tensor product, and the one-mode occupation vectors and their products use the same vacuum and mode order. Its unitary extension is proved below. Let $t,r\in \mathbb{C}$ be dimensionless amplitudes with $\mathrm{\mid }t{\mathrm{\mid }}^2+\mathrm{\mid }r{\mathrm{\mid }}^2=1$, and fix the complete unitary mode matrix $C=\left(\begin{array}{cc}t& -\bar{r}\\ r& \bar{t}\end{array}\right)$. Use the state-action convention that the passive unitary $U_C$ satisfies $U_C\mathrm{\Omega }=\mathrm{\Omega }$, $U_C{a}_1^{\ast }{U}_C^{\ast }=t{a}_1^{\ast }+r{a}_2^{\ast }$, and $U_C{a}_2^{\ast }{U}_C^{\ast }=-\bar{r}{a}_1^{\ast }+\bar{t}{a}_2^{\ast }$ on the finite-particle core. Put $\eta :=\mathrm{\mid }t{\mathrm{\mid }}^2$. For $n\in {\mathbb{N}}_0$ and $0\le k\le n$, write $\left(\genfrac{}{}{0px}{}{n}{k}\right):=n!\mathrm{/}(k!(n-k)!)$. For input $\mathrm{\mid }n,0⟩$ and $0<\eta <1$, the joint output count law is $\mathbb{P}(k,\mathrm{\ell })=\{\begin{array}{ll}\left(\genfrac{}{}{0px}{}{n}{k}\right){\eta }^k(1-\eta {)}^{n-k},& \mathrm{\ell }=n-k,0\le k\le n,\\ 0,& \text{otherwise}\mathrm{.}\end{array}$ At $\eta =0$ and $\eta =1$ the joint laws are respectively point masses at $(0,n)$ and $(n,0)$. Thus the first output count is binomial with parameters $(n,\eta )$. For a dimensionless coherent input $\mathrm{\mid }\alpha ,0⟩$ with $\mathrm{\mid }\alpha ⟩=e^{-\mathrm{\mid }\alpha {\mathrm{\mid }}^2\mathrm{/}2}{\sum }_{n\ge 0}{\alpha }^n\mathrm{\mid }n⟩\mathrm{/}\sqrt{n!}$, define $\mathrm{\mid }\alpha ,0⟩={\mathcal{J}}^{\ast }(\mathrm{\mid }\alpha ⟩\otimes \mathrm{\mid }0⟩)$, equivalently $e^{-\mathrm{\mid }\alpha {\mathrm{\mid }}^2\mathrm{/}2}{\sum }_{n\ge 0}{\alpha }^n\mathrm{\mid }n,0⟩\mathrm{/}\sqrt{n!}$. The output state under this fixed mode factorization is $\mathcal{J}{U}_C\mathrm{\mid }\alpha ,0⟩=\mathrm{\mid }t\alpha ⟩\otimes \mathrm{\mid }r\alpha ⟩$; on ${\mathcal{F}}^{(2)}$ it is ${\mathcal{J}}^{\ast }(\mathrm{\mid }t\alpha ⟩\otimes \mathrm{\mid }r\alpha ⟩)$.

Finite-mode dipole and detector assignment

Finite Bogoliubov implementability

Finite-mode Fock to Schrödinger identification

Fixed-mode number law

Declared ideal measurements and instruments of the framework

The Axiom of Choice

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

Hilbert tensor construction: inner product, completion and product bases

The state-action convention fixes which amplitude is called $t$ and which output is called the first port. The literature often displays the inverse Heisenberg transformation; step 1.1 distinguishes the annihilator matrix $C$ from the state-action creator matrix $C^T$. In the diagram, each line denotes a bosonic mode, and the box denotes the declared two-mode unitary. The number labels are Fock occupations, not photon positions.

The two input labels are the separate mode states whose product is $\mathrm{\mid }n,0⟩$; the outgoing lines label the two modes after the joint unitary. The diagram does not depict a detector, its efficiency, or a physical laboratory layout; the count distribution follows from the proof and the assigned number PVM.

Assume AC and ${\mathrm{A}\mathrm{C}}_{\omega }$, the assigned finite-mode model, one input mode and a second mode prepared independently in vacuum $\mathrm{\mid }0⟩⟨0\mathrm{\mid }$. Write $\mathcal{F}:={\mathcal{F}}_s(\mathbb{C})$ for each one-mode Fock space and ${\mathcal{F}}^{(2)}:={\mathcal{F}}_s({\mathbb{C}}^2)$ for the two-mode Fock space with ordered coordinate basis $e_1=(1,0),e_2=(0,1)$. In the completed product $\mathcal{F}\widehat{\otimes }\mathcal{F}$, define $\mathcal{I}(\mathrm{\mid }m⟩\otimes \mathrm{\mid }n⟩):=\mathrm{\mid }m,n⟩$ on occupation vectors, where $\mathrm{\mid }m,n⟩$ is the two-mode occupation basis in that ordered mode basis. The product-basis result and finite-mode Fock occupation-basis result listed in Statement references make $\mathcal{I}$ extend uniquely to a unitary $\mathcal{I}:\mathcal{F}\widehat{\otimes }\mathcal{F}\to {\mathcal{F}}^{(2)}$. For $0\le \eta \le 1$, define $C_{\eta }:=\left(\begin{array}{cc}\sqrt{\eta }& -\sqrt{1-\eta }\\ \sqrt{1-\eta }& \sqrt{\eta }\end{array}\right)$ and let $U_{\eta }:=U_{C_{\eta }}$ be the two-mode beam-splitter unitary specified by the beam-splitter result in Statement references on ${\mathcal{F}}^{(2)}$. Transport this unitary to the product space as ${\tilde{U}}_{\eta }:={\mathcal{I}}^{\ast }{U}_{\eta }\mathcal{I}$. Define the vacuum insertion isometry $V_0:\mathcal{F}\to \mathcal{F}\widehat{\otimes }\mathcal{F}$ by $V_0\psi =\psi \otimes \mathrm{\mid }0⟩$. For every trace-class operator $\rho$ on $\mathcal{F}$, the vacuum product operator $\rho \otimes \mathrm{\mid }0⟩⟨0\mathrm{\mid }$ means $V_0\rho V_0^{\ast }$, a trace-class operator by the trace-class ideal and tensor-map suppliers. Define ${\mathrm{\Phi }}_{\eta }(\rho ):={\mathrm{Tr}}_2\text{\hspace{0.17em}⁣}({\tilde{U}}_{\eta }(\rho \otimes \mathrm{\mid }0⟩⟨0\mathrm{\mid }){\tilde{U}}_{\eta }^{\ast }),$ where ${\mathrm{Tr}}_2$ is the trace-class partial trace over the second one-mode factor. Here CPTP means complex-linear, trace-preserving, and positive after every finite-dimensional ancilla amplification on trace-class operators. The map ${\mathrm{\Phi }}_{\eta }$ is CPTP. For $0<\eta <1$ and an input number state $\mathrm{\mid }n⟩$, its output state is ${\mathrm{\Phi }}_{\eta }(\mathrm{\mid }n⟩⟨n\mathrm{\mid })={\sum }_{k=0}^n\left(\genfrac{}{}{0px}{}{n}{k}\right){\eta }^k(1-\eta {)}^{n-k}\mathrm{\mid }k⟩⟨k\mathrm{\mid },$ so ideal number detection has the binomial-thinning law. At $\eta =0$ this number input is mapped to vacuum, and at $\eta =1$ it is unchanged. For a coherent input $\mathrm{\mid }\alpha ⟩$, its output is $\mathrm{\mid }\sqrt{\eta }\alpha ⟩⟨\sqrt{\eta }\alpha \mathrm{\mid }$ for every $0\le \eta \le 1$. On arbitrary input states, ${\mathrm{\Phi }}_0$ maps to vacuum and ${\mathrm{\Phi }}_1$ is the identity.

Finite-mode dipole and detector assignment

Passive beam-splitter photon statistics

Finite-mode Fock to Schrödinger identification

Separable partial trace and its basic properties

Nuclear bounded cyclicity for trace-class operators

Hilbert tensor construction: inner product, completion and product bases

Bounded tensor maps: extension, norm estimate, adjoint and commutation

Fixed-mode number law

Declared ideal measurements and instruments of the framework

The Axiom of Choice

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