Qm Light Matter And Spectral Transitions

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Fix $\mathrm{\hslash }>0$ and let $\alpha >0$, let $\mathfrak{h}=\mathrm{ran}{M}_{\mathrm{\Pi }}\subseteq L^2({\mathbb{R}}^3;{\mathbb{C}}^3)$ be the transverse space with $\epsilon (k)=\alpha \mathrm{\mid }k\mathrm{\mid }$ of Transverse continuum free Fock Hamiltonian, let $\mathrm{\Lambda }>0$, let $g_1,\dots ,g_J\in \mathfrak{h}$ ($J\ge 1$) satisfy $\mathrm{supp}{g}_j\subseteq \{k:0<\epsilon (k)\le \mathrm{\Lambda }\},\frac{g_j}{\sqrt{\epsilon }}\in \mathfrak{h},$ let ${\eta }_0\in \mathbb{R}$ and let $c_1,\dots ,c_J\in \mathbb{C}$ with $M:={\sum }_{j=1}^J{c}_j{g}_j\in \mathfrak{h}\mathrm{.}$ Fix $\mathrm{\Delta }\in \mathbb{R}$ and define the finite Borel measure ${\nu }_b(B):={\int }_{{\mathbb{R}}^3}{\mathbf{1}}_B(\epsilon (k))\mathrm{\parallel }M(k){\mathrm{\parallel }}_{{\mathbb{C}}^3}^2{d}^{3}k,K_t(x):=\frac{1}{{\mathrm{\hslash }}^{2}t}\mid {\int }_0^t{e}^{ixs\mathrm{/}\mathrm{\hslash }}ds{\mid }^2(t>0),$ and the vector-valued function $A_b(k,t):=-\frac{i{\eta }_0}{\mathrm{\hslash }}M(k){\int }_0^t{e}^{i(\epsilon (k)-\mathrm{\Delta })s\mathrm{/}\mathrm{\hslash }}ds(t\ge 0)\mathrm{.}$ Clause (a) includes $M=0$ and every real ${\eta }_0$, including zero. Clauses (b)-(e) concern the normalized law when $M\mathrm{\ne }0$; no nonzero assumption on ${\eta }_0$ is required. Then:

(a) (factorisation) for every $t>0$ and almost every $k\in {\mathbb{R}}^3$, $\mathrm{\mid }{A}_b(k,t){\mathrm{\mid }}^2={\eta }_0^{2}t{K}_t(\epsilon (k)-\mathrm{\Delta })\mathrm{\parallel }M(k){\mathrm{\parallel }}_{{\mathbb{C}}^3}^2,$ and consequently $\mathrm{\parallel }{A}_b(\cdot ,t){\mathrm{\parallel }}_{\mathfrak{h}}^2={\eta }_0^{2}t{\int }_{\mathbb{R}}{K}_t(E-\mathrm{\Delta })d{\nu }_b(E)$;

(b) (normalisation) if $M\mathrm{\ne }0$, then for every $t>0$ the denominator ${\int }_{\mathbb{R}}{K}_t(E-\mathrm{\Delta })d{\nu }_b(E)$ is finite and strictly positive, so $P_t(B):=\frac{{\int }_B{K}_t(E-\mathrm{\Delta })d{\nu }_b(E)}{{\int }_{\mathbb{R}}{K}_t(E-\mathrm{\Delta })d{\nu }_b(E)}$ defines a Borel probability measure on $\mathbb{R}$ with $P_t(\mathbb{R}\setminus (0,\mathrm{\Lambda }])=0$;

(c) (concentration) if ${\nu }_b=J_b\lambda$ for some $J_b\in L^1(\lambda )$ with $J_b=0$ $\lambda$-almost everywhere on $\mathbb{R}\setminus (0,\mathrm{\Lambda }]$, if $J_b$ is continuous at some $\mathrm{\Delta }$ with $0<\mathrm{\Delta }<\mathrm{\Lambda }$ and $J_b(\mathrm{\Delta })>0$, and if $f:\mathbb{R}\to \mathbb{C}$ is continuous and bounded, then ${\lim }_{t\to \mathrm{\infty }}{\int }_{\mathbb{R}}fd{P}_t=f(\mathrm{\Delta });$

(d) (positivity) if ${\nu }_b=J_b\lambda$ with $J_b\ge 0$ $\lambda$-almost everywhere and $J_b>0$ $\lambda$-almost everywhere on a Borel set $B\subseteq \mathbb{R}$ of positive Lebesgue measure, then $P_t(B)>0$ for every $t>0$;

(e) (tail concentration) under the hypotheses of (c), for every $\delta >0$, ${\lim }_{t\to \mathrm{\infty }}{P}_t(\{E:\mathrm{\mid }E-\mathrm{\Delta }\mathrm{\mid }\ge \delta \})=0$.

No dynamics, transition probability, rate, decay or physical identification is asserted; the lemma computes an explicit kernel, its normalised push-forward law and that law's limit.

• What concentrates. Clause (c) says that the normalised energy law of the explicit single-vertex vector converges to the unit mass at the parameter $\mathrm{\Delta }$ whenever the photon density has a continuous positive value there; clause (d) shows that at every finite time a window on which the density is positive almost everywhere still carries strictly positive probability. The two are compatible: the window probability tends to zero only for windows separated from $\mathrm{\Delta }$, and the convergence is of the law, not of the density.

• No physical spike. The limit in (c) is a limit of an ideal model law as the parameter $t$ of the explicit vector grows; it is not an assertion that a physical line is sharp or that a measurement at time $t$ has a sharp outcome.

• Zero coupling and zero vector. When ${\eta }_0=0$, the explicit amplitude vanishes, but the formal kernel law $P_t$ remains well-defined for $M\mathrm{\ne }0$. When $M=0$, only the unnormalized factorization is asserted and all amplitudes vanish; no division by $\mathrm{\parallel }{A}_b\mathrm{\parallel }$ or zero denominator is made.

(a) Adopted readings. Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4 and the identical-particle and spin postulate P6 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations), the radiation extension postulate Radiation extension of the framework: transverse mode and photon assignments and the static finite-level dipole postulate Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field, in the state and outcome readings of The state space of the framework: preparations, pure rays and density operators, The state space: density operators, pure and mixed states, and purity and Assigned sharp observables and the outcome law of the framework, in the radiation sector vocabulary of The radiation sector of the framework: modes and the number, energy and momentum assignments and Declared species sectors and their preparations: the framework's reading of P6, and in the units and dimensions of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axiom of Countable Choice and the Axiom of Choice, the choice hypotheses of the cited Fock space mathematics (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(b) Finite-level matter factor. Take the declared charged pair, the relative-coordinate factor $L^2({\mathbb{R}}^3;\mathbb{C})$, the coordinate multipliers $Q_j$ and the dipole operator components $D_j=-q{Q}_j$ of Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field clauses (i)-(ii), with the Coulomb energy operator $H_{\kappa }$ of clause (iv) of that postulate. Fix, as model data, a declared finite-dimensional subspace $W\subseteq D(H_{\kappa })\cap D(Q_1)\cap D(Q_2)\cap D(Q_3),\dim W=d<\mathrm{\infty },$ with a declared orthonormal basis $(w_1,\dots ,w_d)$ of $W$ (Orthonormal families, complete orthonormal systems and Hilbert bases), and read $W$ as ${\mathbb{C}}^d$ in that basis. The model's compressed matter energy is the Hermitian $d\times d$ matrix $H_a:={\mathrm{\Pi }}_W{H}_{\kappa }{\mid }_W,$ read in joules, where ${\mathrm{\Pi }}_W$ is the orthogonal projection onto $W$ (The Hilbert orthogonal projection onto a closed subspace); the model's compressed dipole matrices are the Hermitian $d\times d$ matrices $D_j^W:={\mathrm{\Pi }}_W{D}_j{\mid }_W(j=1,2,3),$ read in $\mathrm{C}\mathrm{m}$; and the model's levels are the pairs $(E,v)$ with $E\in \mathbb{R}$ (joules) an eigenvalue of $H_a$ and $v\in {\mathbb{C}}^d$ a nonzero eigenvector for $E$. A declared level pair is an ordered pair $(a,b)$ of orthonormal eigenvectors $v_a,v_b$ of $H_a$; its transition energy is $E_a-E_b$. The subspace, its basis and its dimension are model data: they are not derived from the Coulomb model and they replace the unbounded operators $H_{\kappa },D_j$ by their compressions throughout this pair.

(c) Radiation factor and cutoff dipole data. Take the transverse one-particle space $\mathfrak{h}$, the bosonic Fock space $F(\mathfrak{h})$, the vacuum $\mathrm{\Omega }$, the sector projections ${\mathrm{\Pi }}_n$ and the free-field energy observable $H_f={⨁}_{n\ge 0}{T}_n$ with $\epsilon (k)=\mathrm{\hslash }c\mathrm{\mid }k\mathrm{\mid }$ of The radiation sector of the framework: modes and the number, energy and momentum assignments clauses (b),(d), with the domains and self-adjointness of Transverse continuum free Fock Hamiltonian and Hilbert direct-sum construction, read in joules. Fix, as further model data, a cutoff $\mathrm{\Lambda }>0$ (joules), an integer $J\ge 1$ and vectors $g_1,\dots ,g_J\in \mathfrak{h}$ with $\mathrm{supp}{g}_j\subseteq \{k:0<\epsilon (k)\le \mathrm{\Lambda }\},\frac{g_j}{\sqrt{\epsilon }}\in \mathfrak{h},$ For each radiation channel $j=1,\dots ,J$, declare real dimensionless weights $e_{j,i}$ ($i=1,2,3$) and define the Hermitian channel dipole matrix $D_j^{\mathrm{c}\mathrm{h}}:={\sum }_{i=1}^3{e}_{j,i}{D}_i^W\mathrm{.}$ These are compressions of the actual real Cartesian dipole combination ${\sum }_i{e}_{j,i}{D}_i$ on $W$, so they have dipole units. This explicitly defines every channel even when $J>3$; the Cartesian case $J=3$, $e_{j,i}={\delta }_{ji}$ is included. The weights are declared model data, with no assumed normalization or measured direction. Fix also a real coupling number ${\eta }_0$ declared so that the interaction term of clause (d) is read in joules. The cutoff and the mode vectors are model data; no laboratory mode content, no Maxwell field, no mode density and no value of ${\eta }_0$ is asserted.

(d) The coupled model. Write $\varphi (g):=A_{-}(g)+A_{+}(g)$ for the maximal creation-annihilation sum of Closed adjoint sector domains on $F(\mathfrak{h})$. The declared finite-level cutoff dipole model is the pair of the Hilbert space ${\mathbb{C}}^d\otimes F(\mathfrak{h})$ (Hilbert space) and the operator $H_{\mathrm{L}\mathrm{M}}:=H_a\otimes I+I\otimes H_f+{\eta }_0{\sum }_{j=1}^J{D}_j^{\mathrm{c}\mathrm{h}}\otimes \varphi (g_j),D(H_{\mathrm{L}\mathrm{M}}):={\mathbb{C}}^d\otimes D(H_f)\mathrm{.}$ For $d\ge 1$, by Cutoff dipole relative bound clause (c), read with the Hermitian matrices $H_a$, $D_1^{\mathrm{c}\mathrm{h}},\dots ,D_J^{\mathrm{c}\mathrm{h}}$ and the coupling ${\eta }_0$ (its $d$ is the present $d$), the operator $H_{\mathrm{L}\mathrm{M}}$ is self-adjoint on the displayed domain and bounded below (Symmetric, self-adjoint and essentially self-adjoint operators, A bounded linear operator between normed spaces). If $d=0$, both the tensor-product Hilbert space and the displayed domain are $\{0\}$; $H_{\mathrm{L}\mathrm{M}}$ is the unique operator on that space. Its adjoint is the same operator, and $⟨H_{\mathrm{L}\mathrm{M}}0,0⟩=0$, so it is self-adjoint and bounded below without invoking the positive-dimensional supplier. This degenerate case has no levels, level pairs or normalized preparations. For $d\ge 1$, the model declares $H_{\mathrm{L}\mathrm{M}}$ its energy observable, in the reading of Assigned sharp observables and the outcome law of the framework, with energies, mode energies and the coupling read in joules.

(e) The model's evolution and states. The model's evolution is the autonomous unitary family $U(t):=e^{-it{H}_{\mathrm{L}\mathrm{M}}\mathrm{/}\mathrm{\hslash }}$, $t\in \mathbb{R}$, in the reading of clause (ii) of P4 and of Assigned energy observable, unitary flow and the two pictures of autonomous evolution; for $d\ge 1$, a preparation $\rho$ of the model is a state of ${\mathbb{C}}^d\otimes F(\mathfrak{h})$ in the sense of The state space of the framework: preparations, pure rays and density operators, with $\rho (t):=U(t)\rho U(t{)}^{\ast }$, and the outcome law of an assigned sharp observable $(A,u_A)$ is the reading of P2 (Assigned sharp observables and the outcome law of the framework). No measurement, no detector, no environment, no external field, no rate and no record is part of this definition.

(f) Scope. Clauses (b)-(e) are adopted model data and readings. They assert nothing about a laboratory atom, sample, mode content, cutoff or coupling; no Maxwell equation, source model, back-reaction or propagation is assumed; the truncation, the cutoff and the mode vectors are not derived; and no identification of ${\eta }_0$ or of the $g_j$ with laboratory quantities is made. The identification of a physical preparation with a state of this model is a tested identification with uncertainty under P7. The model contains no photon detector, no sampling model, no environment beyond the declared transverse modes, and no rate, decay or linewidth statement.

(a) Adopted readings. Adopt the state and composition postulate P1, the observable and probability postulate P2 and the autonomous-dynamics postulate P4 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution) and the static finite-level dipole postulate Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field clauses (i)-(iv), in the state and outcome readings of The state space of the framework: preparations, pure rays and density operators, The state space: density operators, pure and mixed states, and purity and Assigned sharp observables and the outcome law of the framework and in the units of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Read the finite-level matter data of The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector clause (b): the subspace $W\simeq {\mathbb{C}}^d$, the compressed matter energy $H_a$ (joules) and the compressed dipole matrices $D_1^W,D_2^W,D_3^W$ ($\mathrm{C}\mathrm{m}$).

(b) Prescribed field and its regime. Fix a nonempty compact interval $I=[t_0,t_1]\subset \mathbb{R}$ of physical times (seconds) and a prescribed electric field family $t\mapsto {\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(t)=(E_{\mathrm{e}\mathrm{x}\mathrm{t},1}(t),E_{\mathrm{e}\mathrm{x}\mathrm{t},2}(t),E_{\mathrm{e}\mathrm{x}\mathrm{t},3}(t))$, a continuous map $I\to {\mathbb{R}}^3$ whose components are read in $\mathrm{V}{\mathrm{m}}^{-1}$. The field is prescribed: no Maxwell equation, source model, propagation, back-reaction or feedback of the matter on the source is assumed, and no claim is made that a laboratory arrangement realizes it.

(c) The drive Hamiltonian. The declared dipole-drive model is the finite-dimensional family $H_{\mathrm{d}}(t):=H_a-{\sum }_{j=1}^3{E}_{\mathrm{e}\mathrm{x}\mathrm{t},j}(t)D_j^W,t\in I,$ read as the model's energy in joules, on the matter factor $W\simeq {\mathbb{C}}^d$; its control is the bounded self-adjoint family $V(t):=-{\sum }_j{E}_{\mathrm{e}\mathrm{x}\mathrm{t},j}(t)D_j^W$ with $\mathrm{\parallel }V(t)\mathrm{\parallel }\le ({\sum }_{j=1}^3\mathrm{\mid }{E}_{\mathrm{e}\mathrm{x}\mathrm{t},j}(t){\mathrm{\mid }}^2{)}^{1\mathrm{/}2}({\sum }_{j=1}^3\mathrm{\parallel }{D}_j^W{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}<\mathrm{\infty },$ so $t\mapsto H_{\mathrm{d}}(t)$ is norm-continuous on $I$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, A bounded linear operator between normed spaces). The model therefore declares the driven model of The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models clause (b)(i) in its bounded norm-continuous class, with propagator $U(t,s)$ the unique norm-continuous solution of the integral equation $U(t,s)=I-\frac{i}{\mathrm{\hslash }}{\int }_s^t{H}_{\mathrm{d}}(r)U(r,s)dr,U(s,s)=I,$ constructed by the Dyson series of Norm Dyson series and uniqueness. If $t_0<t_1$, it satisfies $i\mathrm{\hslash }{\mathrm{\partial }}_{t}U(t,s)=H_{\mathrm{d}}(t)U(t,s)$ in operator norm, with one-sided derivatives at the endpoints. If $t_0=t_1$, the evolution declaration is only $U(t_0,t_0)=I$ and asserts no time derivative; state and outcome readings are those of clauses (d)-(e) of the driven model, with level populations read through the eigenbasis of $H_a$.

(d) Long-wavelength regime. Declare a characteristic matter length $L>0$ (metres), a reference point $x_0\in {\mathbb{R}}^3$ (metres), a wavevector $k\in {\mathbb{R}}^3$ (inverse metres), and a dimensionless tolerance $0<\delta <1$ as model data. A prescribed spatial phase template for the field of clause (b) is the continuous map ${\mathbf{E}}^{\mathrm{s}\mathrm{p}}:{\mathbb{R}}^3\times I\to {\mathbb{R}}^3,{\mathbf{E}}^{\mathrm{s}\mathrm{p}}(x,t):=\mathrm{Re}({\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(t)e^{ik\cdot (x-x_0)}),$ with components in $\mathrm{V}{\mathrm{m}}^{-1}$, so ${\mathbf{E}}^{\mathrm{s}\mathrm{p}}(x_0,t)={\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(t)$. This template specifies a spatial phase without assuming Maxwell dynamics or a dispersion relation. The long-wavelength (electric-dipole) regime at tolerance $\delta$ means $\mathrm{\mid }k\mathrm{\mid }L\le \delta$: for $\mathrm{\mid }x-x_0\mathrm{\mid }\le L$, the phase obeys $\mathrm{\mid }k\cdot (x-x_0)\mathrm{\mid }\le \delta$. The model adopts the neglect of this spatial phase and uses the reference-point field in the interaction $-{\sum }_j{E}_{\mathrm{e}\mathrm{x}\mathrm{t},j}(t)D_j^W$ of clause (c). The case $k=0$ is the spatially uniform template. The length $L$ is a declared scale, not a compact-support assertion for the matter wavefunctions. The notation $\mathrm{\mid }k\mathrm{\mid }L\ll 1$ denotes choosing a small declared tolerance; no universal tolerance or error estimate for the induced quantum dynamics is asserted. This regime is an adopted model assumption, and identifications of $L$, $k$ and the field with laboratory data are tested identifications with uncertainty under P7.

(e) Scope. Clauses (b)-(d) are adopted model data and readings of a declared finite-dimensional matter model with a prescribed classical field. The model yields stimulated transition calculations through clause (c) and the mathematics of the driven model; it contains no quantized field, no photon number, no vacuum state and no detector, and a prescribed field of this kind does not by itself supply spontaneous emission: the prescribed source is unaffected by the matter's transitions, in contrast to the quantized light-matter model of The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector, where the field's vacuum and the matter's change of level are one coupled model. No rate, lifetime, linewidth, decay or empirical result is asserted.

(a) Adopted readings. Adopt the state and composition postulate P1, the observable and probability postulate P2, the ideal conditional-measurement postulate P3 and the preparation and apparatus discipline P7 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Ideal conditional measurement postulate P3: the projective Lueders update and discarded outcomes, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions), the finite-level cutoff dipole model The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector, the radiation sector definition The radiation sector of the framework: modes and the number, energy and momentum assignments and the declared ideal counting and energy-window detection models The declared ideal counting, energy-window and analyser detection models of the radiation sector, in the readings of The state space of the framework: preparations, pure rays and density operators, The state space: density operators, pure and mixed states, and purity, Assigned sharp observables and the outcome law of the framework and Declared ideal measurements and instruments of the framework, with the units of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axioms of Countable Choice and of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(b) Declared level pair and transition energy. A declared level pair $(a,b)$ of the model is an ordered pair of orthonormal eigenvectors $v_a,v_b\in {\mathbb{C}}^d$ of the compressed matter energy $H_a$ (Orthonormal families, complete orthonormal systems and Hilbert bases, Real and complex inner product spaces, with the inner product linear in the first argument). Its transition energy is $\mathrm{\Delta }:=E_a-E_b$, read in joules; the pair is upward when $\mathrm{\Delta }>0$, meaning level $a$ lies above level $b$ in the model's energy ordering.

(c) Dipole coefficients and single-vertex photon density. For an upward declared level pair $(a,b)$ and the couplings $D_j^{\mathrm{c}\mathrm{h}},g_j,{\eta }_0$ of The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector clauses (c),(d), the model's dipole coefficients are the complex numbers $c_j:=⟨D_j^{\mathrm{c}\mathrm{h}}{v}_a,v_b{⟩}_{{\mathbb{C}}^d}(j=1,\dots ,J),$ the model's single-vertex mode vector is $M:={\sum }_{j=1}^J{c}_j{g}_j\in \mathfrak{h},$ and the model's single-vertex photon density is the finite Borel measure on $\mathbb{R}$ ${\nu }_b(B):={\int }_{{\mathbb{R}}^3}{\mathbf{1}}_B(\epsilon (k))\mathrm{\parallel }M(k){\mathrm{\parallel }}_{{\mathbb{C}}^3}^2{d}^{3}k,B\subseteq \mathbb{R}\text{ Borel},$ read in the model's units and supported in $(0,\mathrm{\Lambda }]$. The pairing is linear in its first argument and conjugate-linear in its second. Thus $c_j$ is the actual coefficient of $v_b$ in $D_j^{\mathrm{c}\mathrm{h}}{v}_a$, and Hermiticity gives $⟨D_j^{\mathrm{c}\mathrm{h}}{v}_b,v_a⟩=\overline{c_j}$. These are exactly the coefficients of Golden-rule coefficient approximate identity clause (c), with no conjugation or change of the transverse mode vectors; its mode vector and measure are the present $M$ and ${\nu }_b$.

(d) Single-vertex emission vector and coefficient. For $t\ge 0$ the model's single-vertex emission vector is the element of $\mathfrak{h}$ $A_b(\cdot ,t):k\mapsto -\frac{i{\eta }_0}{\mathrm{\hslash }}M(k){\int }_0^t{e}^{i(\epsilon (k)-\mathrm{\Delta })s\mathrm{/}\mathrm{\hslash }}ds,$ and the model's single-vertex emission coefficient is ${\gamma }_b(t):=\mathrm{\parallel }{A}_b(\cdot ,t){\mathrm{\parallel }}_{\mathfrak{h}}^2\mathrm{.}$ The vector and the coefficient are defined model quantities: the model assigns them through this clause and through the cutoff dipole mathematics, and they are not asserted to be the exact probability of any measurement of the model. In particular ${\gamma }_b(t)$ is dimensionless and is not a rate; the reading of a rate coefficient requires a separate limit statement, which is not part of this definition.

(e) Normalised single-vertex photon-energy law and ideal detection. For $t>0$ and $M\mathrm{\ne }0$ the model's normalised single-vertex photon-energy law is the Borel probability measure on $\mathbb{R}$ $P_{b,t}(B):=\frac{{\int }_B{K}_t(E-\mathrm{\Delta })d{\nu }_b(E)}{{\int }_{\mathbb{R}}{K}_t(E-\mathrm{\Delta })d{\nu }_b(E)},K_t(x):=\frac{1}{{\mathrm{\hslash }}^{2}t}\mid {\int }_0^t{e}^{ixs\mathrm{/}\mathrm{\hslash }}ds{\mid }^2,$ whose denominator is finite and strictly positive by The first-order emitted-photon energy law: factorised kernel, normalisation, concentration at the transition energy and positivity on positive-measure windows clause (b). This probability law is defined independently of ${\eta }_0$, including ${\eta }_0=0$. Its amplitude-based preparation reading additionally requires $A_b(\cdot ,t)\mathrm{\ne }0$. For $t>0$ and $M\mathrm{\ne }0$, this is equivalent to ${\eta }_0\mathrm{\ne }0$, since $\mathrm{\parallel }{A}_b(\cdot ,t){\mathrm{\parallel }}^2={\eta }_0^{2}t\int K_t(E-\mathrm{\Delta })d{\nu }_b(E)$ and the integral is positive. Under this condition the declared single-vertex one-photon preparation is the unit vector $v_b\otimes (A_b(\cdot ,t)\mathrm{/}\mathrm{\parallel }{A}_b(\cdot ,t){\mathrm{\parallel }}_{\mathfrak{h}})$ of ${\mathbb{C}}^d\otimes \mathfrak{h}\subseteq {\mathbb{C}}^d\otimes F(\mathfrak{h})$, and for a Borel window $B\subseteq [0,\mathrm{\infty })$ the declared ideal energy-window detector of window $B$ of The declared ideal counting, energy-window and analyser detection models of the radiation sector clause (c) has the Born probability $P_{b,t}(B)$ in that preparation, because the energy law of a unit one-photon vector $f$ is the push-forward of $\mathrm{\parallel }f(k){\mathrm{\parallel }}_{{\mathbb{C}}^3}^2{d}^{3}k$ under $k\mapsto \epsilon (k)$ (The radiation sector of the framework: modes and the number, energy and momentum assignments clause (e)). When ${\eta }_0=0$, $A_b(\cdot ,t)=0$ and no preparation obtained by normalizing that emission amplitude is declared; the formal law $P_{b,t}$ remains the probability measure defined above. A detected photon energy is therefore a spectral reading of the assigned free-field energy observable $(H_f,\mathrm{J})$, not a sharp energy of a normalisable state and not a quantum number of the model's mode labels.

(f) Scope. Clauses (b)-(e) fix the model's spectral vocabulary. They assert no sharp spectral line: the energy law of clause (e) is spread over the support of ${\nu }_b$. A point mass at $\mathrm{\Delta }$ (the probability measure assigning ${\mathbf{1}}_B(\mathrm{\Delta })$ to a Borel set $B$) is the limiting idealisation of The first-order emitted-photon energy law: factorised kernel, normalisation, concentration at the transition energy and positivity on positive-measure windows clause (c) only under its hypotheses: ${\nu }_b=J_b\lambda$, where $\lambda$ is Lebesgue measure on $\mathbb{R}$ and $J_b$ is an integrable nonnegative density representative vanishing almost everywhere off $(0,\mathrm{\Lambda }]$, continuous at $0<\mathrm{\Delta }<\mathrm{\Lambda }$ and satisfying $J_b(\mathrm{\Delta })>0$. Under these hypotheses, $\int fd{P}_{b,t}\to f(\mathrm{\Delta })$ as $t\to \mathrm{\infty }$ for every bounded continuous $f:\mathbb{R}\to \mathbb{C}$; no such limit is asserted for other allowed level pairs or densities. In particular, if $\mathrm{\Delta }>\mathrm{\Lambda }$, the finite-time laws assign all mass to $(0,\mathrm{\Lambda }]$ and cannot converge to the point mass at $\mathrm{\Delta }$. They supply no spectrometer, no calibration, no detector response beyond the declared ideal window, no efficiency, no background, no sampling model, no count and no measurement record; the identification of a laboratory spectrometer with a declared window is a tested identification with uncertainty under P7. The single-vertex quantities of clauses (d)-(e) are conditional model data; their relation to the model's exact outcome probabilities is a separate proved statement.

Adopt, as additional model assumptions of this pair, the following clauses. It is adopted together with the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4 and the preparation and apparatus discipline P7 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions), the radiation extension postulate Radiation extension of the framework: transverse mode and photon assignments, the static finite-level dipole postulate Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field and the coupled model The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector, in the readings of The radiation sector of the framework: modes and the number, energy and momentum assignments, The state space of the framework: preparations, pure rays and density operators, Assigned sharp observables and the outcome law of the framework and Dimensions, SI unit conventions and quantity types of the framework. Assume the Axioms of Countable Choice and of Choice, the choice hypotheses of the cited mathematics (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(i) Vacuum preparation. Let $(a,b)$ be a declared upward level pair of Spectral-transition readings of the coupled model: dipole coefficients, the single-vertex photon density and ideal energy-window detection clause (b), with $\mathrm{\Omega }$ the vacuum of the radiation sector. The postulate declares that a preparation described as the declared finite-level matter in level $a$ with no quanta in the declared transverse radiation sector is represented by the ray of the unit vector $v_a\otimes \mathrm{\Omega }$ of ${\mathbb{C}}^d\otimes F(\mathfrak{h})$, and that its evolution is the model's autonomous evolution $t\mapsto e^{-it{H}_{\mathrm{L}\mathrm{M}}\mathrm{/}\mathrm{\hslash }}(v_a\otimes \mathrm{\Omega })$. This representation of a physical preparation is an adopted model assumption; it is not derived from P1-P7, from the radiation extension or from the Coulomb model.

(ii) The emission question. The postulate declares that the model's vacuum-initiated emission question for the pair is read through the declared ideal sharp measurement with outcome projection $Q_b=\mathrm{\mid }{v}_b⟩⟨v_b\mathrm{\mid }\otimes {\mathrm{\Pi }}_1$ (or through any other declared measurement of The declared ideal counting, energy-window and analyser detection models of the radiation sector whose outcome structure is explicitly declared), with its Born probability read as in P2; the model's exact probability of that outcome is the quantity studied by the emission items of this page. The identification of this outcome with the registration of an emitted photon by a laboratory apparatus is a tested identification with uncertainty under P7.

(iii) The entire field and environment content. The declared field content of the model is the quantized transverse sector with its vacuum $\mathrm{\Omega }$, the mode vectors $g_1,\dots ,g_J$ with cutoff $\mathrm{\Lambda }$ and the coupling ${\eta }_0$ of The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector clauses (c),(d). The postulate adopts no thermal radiation state, no additional modes, no reservoir or environment, no measurement apparatus and no noise model. In particular the reading of $\mathrm{\Omega }$ as the field's zero-point (vacuum) state is a model reading, and the term "spontaneous" refers to the emission process initiated by this declared vacuum state, not to a derivation of the vacuum from classical electrodynamics or of the coupling from a current.

(iv) What a rate claim would require. The postulate explicitly does not declare a rate, a lifetime, a linewidth or an exponential decay law. Such a claim would require additional adopted assumptions — for instance a declared weak-coupling window, a declared mode-density or environment model, and a declared measurement and sampling model — together with the corresponding mathematics; the theorems of this page state only the limits of the model's explicit single-vertex coefficient and the exact-model control of one projection probability, and they record those limits as model quantities.

(v) Contrast with the prescribed drive. The prescribed-field dipole-drive model of The declared prescribed dipole-drive model and its long-wavelength regime is a different model class: its field is an external datum unaffected by the matter, it contains no vacuum state and no photon number, and it supplies only stimulated transitions. The present postulate is the additional assumption under which an emission process at all is represented by the model; the two classes are not interchangeable, and no derivation of one from the other is asserted.

(vi) Scope. Clauses (i)-(v) are adopted assumptions and readings. They assert no laboratory source, atom, mode content, cutoff or coupling value; no detector response, efficiency, background or calibration; no observed spectrum, count, rate or lifetime; and no statistical inference. The identification of a laboratory system and apparatus with the model and its outcome structure is a tested identification with uncertainty under P7, and the empirical support of the radiation extension or of the coupling is the nonlogical relation recorded on this page, not a proof of the postulate.

Adopt the model of The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector with its data $d$, $H_a$, $D_j^{\mathrm{c}\mathrm{h}}$, $\mathrm{\Lambda }$, $g_j$, ${\eta }_0$, and a declared upward level pair $(a,b)$ with transition energy $\mathrm{\Delta }=E_a-E_b>0$ and page coefficients $c_j=⟨D_j^{\mathrm{c}\mathrm{h}}{v}_a,v_b⟩$, mode vector $M$ and photon density ${\nu }_b$ of Spectral-transition readings of the coupled model: dipole coefficients, the single-vertex photon density and ideal energy-window detection clauses (b),(c). Write $H_0:=H_a\otimes I+I\otimes H_f$ and $B:={\eta }_0{\sum }_j{D}_j^{\mathrm{c}\mathrm{h}}\otimes \varphi (g_j)$, so that $H_{\mathrm{L}\mathrm{M}}=H_0+B$ on ${\mathbb{C}}^d\otimes D(H_f)$. Assume the Axioms of Countable Choice and of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Fix $T>0$, put $C:=\frac{2}{\mathrm{\hslash }}{\sum }_{j=1}^J\mathrm{\parallel }{D}_j^{\mathrm{c}\mathrm{h}}\mathrm{\parallel }\mathrm{\parallel }{g}_j{\mathrm{\parallel }}_{\mathfrak{h}},{\epsilon }_2(t):={\sum }_{n\ge 2}\frac{(\mathrm{\mid }{\eta }_0\mathrm{\mid }Ct{)}^n}{\sqrt{n!}}(0\le t\le T),$ and let ${\mathrm{\Psi }}_0:=v_a\otimes \mathrm{\Omega }$. For $t\ge 0$ let ${\mathrm{\Psi }}_1(t)$ denote the first interaction-picture Dyson term of the model with initial vector ${\mathrm{\Psi }}_0$, as in Vacuum emission Dyson convergence clause (b). Then:

(a) (single-vertex emission vector) the component of ${\mathrm{\Psi }}_1(t)$ in the one-photon sector at level $b$ is $v_b\otimes A_b(\cdot ,t),$ with $A_b(\cdot ,t)$ the single-vertex emission vector of Spectral-transition readings of the coupled model: dipole coefficients, the single-vertex photon density and ideal energy-window detection clause (d) and The first-order emitted-photon energy law: factorised kernel, normalisation, concentration at the transition energy and positivity on positive-measure windows; consequently the single-vertex emission coefficient of the model is ${\gamma }_b(t)={\eta }_0^{2}t{\int }_{\mathbb{R}}{K}_t(E-\mathrm{\Delta })d{\nu }_b(E)(t>0);$

(b) (exact model probability up to the Dyson tail) let $Q_b:=\mathrm{\mid }{v}_b⟩⟨v_b\mathrm{\mid }\otimes {\mathrm{\Pi }}_1$ and let $p_b(t):=\mathrm{\parallel }{Q}_b{e}^{-it{H}_{\mathrm{L}\mathrm{M}}\mathrm{/}\mathrm{\hslash }}{\mathrm{\Psi }}_0{\mathrm{\parallel }}^2$ be the model's probability of the spectral value $1$ of the assigned sharp observable $(Q_b,1)$ in the evolved preparation of ${\mathrm{\Psi }}_0$ at time $t$ (the reading of P2, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Assigned sharp observables and the outcome law of the framework). Then for $0\le t\le T$, $\mid p_b(t)-{\gamma }_b(t)\mid \le {\epsilon }_2(t)(2\mathrm{\parallel }{A}_b(\cdot ,t){\mathrm{\parallel }}_{\mathfrak{h}}+{\epsilon }_2(t));$ in particular the exact probability equals the single-vertex coefficient up to the explicit Dyson tail, which is not asserted to be small;

(c) (absorption mirror) let $g\in \mathfrak{h}$ be a unit vector, let ${\mathrm{\Psi }}_1^g(t)$ be the first interaction-picture Dyson term with initial vector ${\mathrm{\Phi }}_0=v_b\otimes g$, defined as a norm integral using the finite-photon-sector creation/contraction action, and define the physical transition amplitudes with the output vector in the first slot, ${\alpha }_{\mathrm{e}\mathrm{m}}(t):=⟨{\mathrm{\Psi }}_1(t),v_b\otimes g⟩,{\alpha }_{\mathrm{a}\mathrm{b}\mathrm{s}}(t):=⟨{\mathrm{\Psi }}_1^g(t),v_a\otimes \mathrm{\Omega }⟩\mathrm{.}$ Then for every $t\ge 0$, ${\alpha }_{\mathrm{e}\mathrm{m}}(t)=-\frac{i{\eta }_0}{\mathrm{\hslash }}{\int }_0^t{e}^{-i\mathrm{\Delta }s\mathrm{/}\mathrm{\hslash }}⟨M,e^{-is\epsilon \mathrm{/}\mathrm{\hslash }}g{⟩}_{\mathfrak{h}}ds,{\alpha }_{\mathrm{a}\mathrm{b}\mathrm{s}}(t)=-\frac{i{\eta }_0}{\mathrm{\hslash }}{\int }_0^t{e}^{+i\mathrm{\Delta }s\mathrm{/}\mathrm{\hslash }}\overline{⟨M,e^{-is\epsilon \mathrm{/}\mathrm{\hslash }}g{⟩}_{\mathfrak{h}}}ds,$ and hence ${\alpha }_{\mathrm{a}\mathrm{b}\mathrm{s}}(t)=-\overline{{\alpha }_{\mathrm{e}\mathrm{m}}(t)},\mathrm{\mid }{\alpha }_{\mathrm{a}\mathrm{b}\mathrm{s}}(t)\mathrm{\mid }=\mathrm{\mid }{\alpha }_{\mathrm{e}\mathrm{m}}(t)\mathrm{\mid }\mathrm{.}$ This compares the reverse transitions for the same level pair, the same unit one-photon vector $g$ and the same real coupling ${\eta }_0$, retaining arbitrary complex Hermitian dipole matrices and transverse mode vectors;

(d) (scope) clauses (a)-(c) are statements about the model's first-order Dyson terms and about one projection probability of the exact model evolution. No rate, exponential decay, linewidth, environment, sampling model or measurement record is asserted; the identification of a laboratory preparation with ${\mathrm{\Psi }}_0$ and of a laboratory detector with the observable $Q_b$ is a tested identification with uncertainty under P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions); and no value of ${\eta }_0$, $D_j^{\mathrm{c}\mathrm{h}}$, $g_j$ or $\mathrm{\Lambda }$ is asserted.