First-order emission and absorption amplitudes of the cutoff dipole model, and the exact model probability of the one-photon level-b projection up to the Dyson tail

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.

• Why the exact probability is not identified with the coefficient. The projection $Q_b$ annihilates the vacuum and every sector of photon number different from one, but the Dyson terms of odd order $n\ge 3$ can return to the one-photon sector; clause (b) therefore controls the exact probability by the tail rather than asserting equality. This is the explicit sense in which the single-vertex coefficient is a first-order model quantity.
• The mirror is an algebraic identity, not a detailed-balance theorem. Clause (c) relates the reverse first-order amplitudes of the same level pair and the same one-photon vector by ${\alpha }_{\mathrm{a}\mathrm{b}\mathrm{s}}=-\overline{{\alpha }_{\mathrm{e}\mathrm{m}}}$. It assumes nothing about populations, temperatures or equilibrium; equality of moduli follows from the Hermitian coupling and the linear-first amplitude convention.