Finite-mode dipole and detector assignment

(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.