Stimulated dipole-drive transitions of the finite-level model: first-order amplitudes, equal emission and absorption coefficients, and the exact two-level identification

Adopt the prescribed dipole-drive model of The declared prescribed dipole-drive model and its long-wavelength regime with its matter data $W\simeq {\mathbb{C}}^d$, $H_a$, $D_1^W,D_2^W,D_3^W$ and its compact interval $I=[t_0,t_1]$, the driven model of The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models clause (b)(i) and the framework readings of P1, P2 and 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, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions). Assume the Axioms of Countable Choice and of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Let $v_i,v_k$ be orthonormal eigenvectors of $H_a$ with $H_a{v}_i=E_i{v}_i$, $H_a{v}_k=E_k{v}_k$, put $\mathrm{\Delta }:=E_k-E_i$, take $s,t\in I$ with $s\le t$, put $\tau :=t-s$ and let $A_{ki}(t,s)=e^{i{E}_k\tau \mathrm{/}\mathrm{\hslash }}⟨U(t,s)v_i,v_k⟩$ be the model's transition amplitude of clause (e) of the driven model with $U(t,s)$ the propagator of the drive family $H_{\mathrm{d}}(r)=H_a-{\sum }_j{E}_{\mathrm{e}\mathrm{x}\mathrm{t},j}(r)D_j^W$. Then:

(a) (first-order amplitude and remainder) with $A_{ki}^{(1)}(t,s)=\frac{i}{\mathrm{\hslash }}{\sum }_{j=1}^3{\int }_s^t{e}^{i\mathrm{\Delta }(r-s)\mathrm{/}\mathrm{\hslash }}{E}_{\mathrm{e}\mathrm{x}\mathrm{t},j}(r)⟨D_j^W{v}_i,v_k⟩dr,$ one has $A_{ki}(t,s)={\delta }_{ik}+A_{ki}^{(1)}(t,s)+R_{ki}(t,s)$ with $\mathrm{\mid }{R}_{ki}(t,s)\mathrm{\mid }\le e^{K\tau }(K\tau {)}^2\mathrm{/}2$, where $K:={\sup }_{r\in I}\mathrm{\parallel }V(r)\mathrm{\parallel }\mathrm{/}\mathrm{\hslash }$ with $V(r):=-{\sum }_j{E}_{\mathrm{e}\mathrm{x}\mathrm{t},j}(r)D_j^W$; in particular the first-order amplitude is the model's exact amplitude up to the displayed Dyson remainder;

(b) (monochromatic co- and counter-rotating terms) if ${\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(r)={\mathbf{E}}_0\cos (\omega r+\vartheta )$ with ${\mathbf{E}}_0\in {\mathbb{R}}^3$, $\omega ,\vartheta \in \mathbb{R}$ and $W:=-{\sum }_j{E}_{0,j}{D}_j^W$ (a bounded self-adjoint operator on $W$), then $V(r)=W\cos (\omega r+\vartheta )$ and $A_{ki}^{(1)}=a_{+}+a_{-},a_{\pm }=-\frac{i}{2\mathrm{\hslash }}⟨W{v}_i,v_k⟩e^{\mp i{\vartheta }_0}\frac{e^{i(\mathrm{\Delta }\mp \mathrm{\hslash }\omega )\tau \mathrm{/}\mathrm{\hslash }}-1}{i(\mathrm{\Delta }\mp \mathrm{\hslash }\omega )\mathrm{/}\mathrm{\hslash }},{\vartheta }_0:=\omega s+\vartheta ,$ with the value $\tau$ for the quotient when $\mathrm{\Delta }\mp \mathrm{\hslash }\omega =0$; consequently the co-rotating term satisfies $\mathrm{\mid }{a}_{+}\mathrm{\mid }=\mid ⟨W{v}_i,v_k⟩\mid \frac{\mid \sin ((\mathrm{\Delta }-\mathrm{\hslash }\omega )\tau \mathrm{/}(2\mathrm{\hslash }))\mid }{\mathrm{\mid }\mathrm{\Delta }-\mathrm{\hslash }\omega \mathrm{\mid }}(\mathrm{\Delta }\mathrm{\ne }\mathrm{\hslash }\omega ),\mathrm{\mid }{a}_{+}\mathrm{\mid }=\mid ⟨W{v}_i,v_k⟩\mid \frac{\tau }{2\mathrm{\hslash }}(\mathrm{\Delta }=\mathrm{\hslash }\omega ),$ and the counter-rotating term satisfies $\mathrm{\mid }{a}_{-}\mathrm{\mid }\le \mathrm{\mid }⟨W{v}_i,v_k⟩\mathrm{\mid }\mathrm{/}\mathrm{\mid }\mathrm{\Delta }+\mathrm{\hslash }\omega \mathrm{\mid }$ whenever $\mathrm{\Delta }+\mathrm{\hslash }\omega \mathrm{\ne }0$;

(c) (equal stimulated coefficients) for the two-level case $d=2$ with orthonormal eigenvectors $v_{\uparrow },v_{\downarrow }$, $H_a{v}_{\uparrow }=E_{\uparrow }{v}_{\uparrow }$, $H_a{v}_{\downarrow }=E_{\downarrow }{v}_{\downarrow }$ and $\mathrm{\Delta }=E_{\uparrow }-E_{\downarrow }>0$, the actual upward and downward first-order amplitudes under the same real prescribed field obey $A_{\downarrow \uparrow }^{(1)}(t,s)=-\overline{A_{\uparrow \downarrow }^{(1)}(t,s)}\mathrm{.}$ In particular their squared moduli, the model's first-order stimulated emission and absorption coefficients, are equal. For the monochromatic field of (b), the upward co-rotating component at detuning $\mathrm{\Delta }-\mathrm{\hslash }\omega$ and the reverse downward component at the opposite detuning $\mathrm{\hslash }\omega -\mathrm{\Delta }$ have equal moduli as well; reversing the transition conjugates its matrix element and swaps the two exponential components, with the same overall minus-conjugation relation;

(d) (exact two-level identifications) if $d=2$ with the orthonormal basis $(e_{\uparrow },e_{\downarrow })$ and the Pauli matrices of Finite spin matrices clause (e), if $H_a=\frac{\mathrm{\hslash }{\omega }_0}{2}{\sigma }_z,D_1^W=d{\sigma }_x,D_2^W=d{\sigma }_y,D_3^W=0$ for some real $d$ of unit $\mathrm{C}\mathrm{m}$, and if the prescribed field is ${\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(r)=E_0(\cos (\omega r),\sin (\omega r),0)$ with $E_0\in \mathbb{R}$, then $H_{\mathrm{d}}(r)$ is exactly the declared circular two-level drive model of clause (f) of The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models with transverse coupling ${\mathrm{\Omega }}_1:=-2d{E}_0\mathrm{/}\mathrm{\hslash }$, so the exact Rabi populations of The circularly driven two-level model: exact rotating-frame reduction, Rabi populations, resonance and inversion times clause (b) are the model's level populations; if instead ${\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(r)=E_0(\cos (\omega r),0,0)$ and $D_1^W=d{\sigma }_x$, $D_2^W=D_3^W=0$, then $H_{\mathrm{d}}(r)$ is the declared linear two-level drive model with the same ${\mathrm{\Omega }}_1$ and the rotating-wave error contract of The linearly driven two-level model: rotating-wave approximation with an explicit finite-time error contract and the averaged Rabi populations clauses (b),(c) applies to it when $\omega \mathrm{\ne }0$. For $\omega =0$ the linear model is the constant Hamiltonian $H_{\ast }=(\mathrm{\hslash }\mathrm{/}2)({\omega }_0{\sigma }_z+{\mathrm{\Omega }}_1{\sigma }_x)$ with exact propagator $\cos (q\tau \mathrm{/}2)I-i\sin (q\tau \mathrm{/}2)({\omega }_0{\sigma }_z+{\mathrm{\Omega }}_1{\sigma }_x)\mathrm{/}q$ for $q=\sqrt{{\omega }_0^2+{\mathrm{\Omega }}_1^2}>0$, and $I$ when $q=0$;

(e) (scope) clauses (a)-(d) concern the prescribed classical drive model. The field is unaffected by the matter's transitions; the model contains no quantized field, no photon number and no vacuum, so no spontaneous emission, no emission rate and no photon count is supplied by it. The identifications of $H_a,D_j^W,d,{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ with a laboratory atom and field and of the initial preparation with an eigenvector of $H_a$ are tested identifications with uncertainty under P7, and no value of ${\omega }_0$, $d$, $E_0$, $\omega$ or ${\mathrm{\Omega }}_1$ is asserted.

• Stimulated, not spontaneous. The equal-coefficient statement of clause (c) is the model's first-order version of the equality of stimulated emission and absorption coefficients; the model has no vacuum field and therefore contains no spontaneous emission. The quantized model and its vacuum-initiated emission are separate items of this page, and the relation between the two model classes is an identification, not a derivation.
• What the exact Rabi result adds. Clause (d) shows that for the declared two-level data the model's evolution is exactly the circular or linear two-level family of the driven-dynamics page, The circular family has an exact Rabi population solution; the linear family is compared with its averaged Rabi solution through the stated rotating-wave error bound. Neither comparison asserts an all-time transition rate.