A two-level dipole-drive model: identification, resonant first-order coefficient and its comparison with the exact Rabi probability

Adopt the prescribed dipole-drive model The declared prescribed dipole-drive model and its long-wavelength regime with the following realized two-level matter data. For any declared ${\omega }_0>0$ (${\mathrm{s}}^{-1}$) and $d>0$ ($\mathrm{C}\mathrm{m}$), use the positive framework constant ${\epsilon }_0$ of The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions clause (d) and define $q:={\left(\frac{8\sqrt{2}{\pi }^{3\mathrm{/}2}{\epsilon }_0\mathrm{\hslash }{\omega }_{0}d}{3}\right)}^{1\mathrm{/}3},a:=\frac{\sqrt{2}d}{q},\mu :=\frac{9\mathrm{\hslash }}{10{\omega }_0{a}^2},\kappa :=\frac{q^2}{4\pi {\epsilon }_0}\mathrm{.}$ These have units $\mathrm{C}$, $\mathrm{m}$, $\mathrm{k}\mathrm{g}$ and $\mathrm{J}\mathrm{m}$, respectively. Declare the charged pair of Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field with $m_1=m_2=2\mu$, charges $-q,+q$ and relative-coordinate energy $H_{\kappa }=-({\mathrm{\hslash }}^2\mathrm{/}(2\mu )){\mathrm{\Delta }}_w-\kappa \mathrm{/}\mathrm{\mid }x\mathrm{\mid }$ on $H^2({\mathbb{R}}^3;\mathbb{C})$; ${\mathrm{\Delta }}_w$ is the sum of the three second weak coordinate derivatives. Its dipoles are $D_j=-q{Q}_j$, $Q_{j}f(x)=x_{j}f(x)$ on $D(Q_j)=\{f\in L^2:x_{j}f\in L^2\}$. Here $L^2=L^2({\mathbb{R}}^3;\mathbb{C})$ consists of Lebesgue square-integrable classes, and $H^2$ consists of those classes whose weak coordinate derivatives through order two also belong to $L^2$: $D^{\alpha }f$ is a weak derivative when $\int f{\mathrm{\partial }}^{\alpha }\phi =(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}\int D^{\alpha }f\phi$ for every smooth compactly supported test function $\phi :{\mathbb{R}}^3\to \mathbb{C}$. Here $\alpha =({\alpha }_1,{\alpha }_2,{\alpha }_3)\in {\mathbb{N}}_0^3$, $\mathrm{\mid }\alpha \mathrm{\mid }={\alpha }_1+{\alpha }_2+{\alpha }_3$ and ${\mathrm{\partial }}^{\alpha }$ denotes that sequence of classical coordinate derivatives. In this relative-coordinate space put $g_a(x):=(\pi a^2{)}^{-3\mathrm{/}4}{e}^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(2a^2)},w_{\downarrow }:=g_a,w_{\uparrow }:=-\frac{\sqrt{2}{x}_1}{a}{g}_a,W:=\mathrm{span}\{w_{\uparrow },w_{\downarrow }\}\mathrm{.}$ The verification below proves that this is an orthonormal pair in $D(H_{\kappa })\cap {\bigcap }_{j=1}^{3}D(Q_j)$ and that the actual compressions $H_a={\mathrm{\Pi }}_W{H}_{\kappa }{\mathrm{\mid }}_W$, $D_j^W={\mathrm{\Pi }}_W{D}_j{\mathrm{\mid }}_W$ have the following matrices. Identify $(w_{\uparrow },w_{\downarrow })$ with the coordinate basis $(e_{\uparrow },e_{\downarrow })$ of ${\mathbb{C}}^2$, using the Pauli matrices $({\sigma }_x,{\sigma }_y,{\sigma }_z)$ of Finite spin matrices clause (e): $H_a:=\frac{\mathrm{\hslash }{\omega }_0}{2}{\sigma }_z,D_1^W:=d{\sigma }_x,D_2^W:=D_3^W:=0,$ where ${\omega }_0,d$ are the arbitrary positive parameters fixed above. The two vectors are eigenvectors of the compression, not asserted to be Coulomb eigenfunctions; invariance of $W$ under the uncompressed operators is unnecessary. Declare the field ${\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(r):=E_0\cos (\omega r)e_1,r\in I=[0,\tau ],$ with $E_0>0$ in $\mathrm{V}{\mathrm{m}}^{-1}$ and $\omega >0$ in ${\mathrm{s}}^{-1}$, and $\tau >0$ in seconds; declare $L>0$ in metres, $x_0=0\in {\mathbb{R}}^3$, $k=0\in {\mathbb{R}}^3$ in inverse metres, and a dimensionless phase tolerance ${\delta }_{\mathrm{p}\mathrm{h}}=1\mathrm{/}10$. Use the spatially uniform template ${\mathbf{E}}^{\mathrm{s}\mathrm{p}}(x,r)=E_0\cos (\omega r)e_1$ of the dipole-drive definition (d), so $\mathrm{\mid }k\mathrm{\mid }L=0\le {\delta }_{\mathrm{p}\mathrm{h}}$ for every $L$. These are declared prescribed-field data, with no Maxwell dispersion relation; the temporal frequency $\omega >0$ does not determine $k$ in this model. Let $v_{\uparrow }:=e_{\uparrow }$, $v_{\downarrow }:=e_{\downarrow }$, so that $E_{\uparrow }=\mathrm{\hslash }{\omega }_0\mathrm{/}2$, $E_{\downarrow }=-\mathrm{\hslash }{\omega }_0\mathrm{/}2$ and $\mathrm{\Delta }=E_{\uparrow }-E_{\downarrow }=\mathrm{\hslash }{\omega }_0$. Write $\delta :={\omega }_0-\omega$ for the detuning frequency, distinct from the energy $\mathrm{\Delta }$. Then:

(a) (identification) $H_{\mathrm{d}}(r)=\frac{\mathrm{\hslash }}{2}({\omega }_0{\sigma }_z+{\mathrm{\Omega }}_1\cos (\omega r){\sigma }_x)$ with ${\mathrm{\Omega }}_1=-\frac{2d{E}_0}{\mathrm{\hslash }},$ so the model is exactly the declared linear 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 splitting frequency ${\omega }_0$ and transverse coupling ${\mathrm{\Omega }}_1$;

(b) (resonance and first-order coefficient) at resonance $\omega ={\omega }_0$ the co-rotating first-order amplitude of Stimulated dipole-drive transitions of the finite-level model: first-order amplitudes, equal emission and absorption coefficients, and the exact two-level identification clause (b) for the transition $e_{\downarrow }\to e_{\uparrow }$ from time $0$ to time $\tau$ has modulus $\mathrm{\mid }{a}_{+}\mathrm{\mid }=\mid ⟨V_0{e}_{\downarrow },e_{\uparrow }⟩\mid \frac{\tau }{2\mathrm{\hslash }}=\frac{d{E}_0\tau }{2\mathrm{\hslash }},V_0=-d{E}_0{\sigma }_x,$ and the co-rotating component coefficient is ${\gamma }_{+}=\mathrm{\mid }{a}_{+}{\mathrm{\mid }}^2=(d{E}_0\tau \mathrm{/}(2\mathrm{\hslash }){)}^2$; the counter-rotating term is bounded by $\mathrm{\mid }⟨V_0{e}_{\downarrow },e_{\uparrow }⟩\mathrm{\mid }\mathrm{/}(\mathrm{\Delta }+\mathrm{\hslash }\omega )=d{E}_0\mathrm{/}(2\mathrm{\hslash }\omega )$ at resonance. The full first-order interaction-picture amplitude is $A_1=\frac{id{E}_0}{2\mathrm{\hslash }}[\tau +\frac{e^{2i\omega \tau }-1}{2i\omega }],{\gamma }_1:=\mathrm{\mid }{A}_1{\mathrm{\mid }}^2,$ including interference of both components; ${\gamma }_1$ is an unnormalized first-order approximation, not a normalized-state Born law;

(c) (comparison with the exact averaged Rabi probability) by The linearly driven two-level model: rotating-wave approximation with an explicit finite-time error contract and the averaged Rabi populations clause (c) the averaged down-population of the initial up-state at resonance is $p_{\downarrow }^{\mathrm{a}\mathrm{v}\mathrm{g}}(\tau )={\sin }^2(\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }\tau \mathrm{/}4)={\sin }^2(d{E}_0\tau \mathrm{/}(2\mathrm{\hslash }))$, whose small-$\tau$ expansion begins with $(d{E}_0\tau \mathrm{/}(2\mathrm{\hslash }){)}^2$; the co-rotating coefficient ${\gamma }_{+}$ of clause (b) is therefore the leading term of the exact averaged Rabi probability, and the rotating-wave error contract of the cited item bounds the difference between the exact linear-model population and this averaged probability by $2\epsilon (\tau )$ with $\epsilon (\tau )=\frac{\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }}{2\omega }[1+\tau (\frac{\mathrm{\mid }\delta \mathrm{\mid }}{2}+\frac{3\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }}{4})]$;

(d) (scope) the example is a conditional computation in the declared two-level drive model. The identification of a laboratory two-level atom, molecule or field with the declared data is a tested identification with uncertainty under P7; the model has no quantized field, no vacuum, no spontaneous emission, no damping, no measurement record and no empirical value, and the exact circular and linear families of the driven-dynamics page are models with their own scope.

• First order versus exact. The co-rotating component coefficient of clause (b) is the leading term of the exact averaged Rabi oscillation of clause (c) only at small $\tau$; the full linear-drive first-order coefficient is ${\gamma }_1=\mathrm{\mid }{a}_{+}+a_{-}{\mathrm{\mid }}^2$ and has four times that leading small-time coefficient, so no averaging of the full model is silently assumed; the exact linear model differs from the averaged model by the rotating-wave error, and the exact circular model of the driven-dynamics page is a different declared family, not an approximation proved here.
• No spontaneous emission. The example is stimulated: the field is prescribed and unaffected by the transition, and the model contains no vacuum. Spontaneous emission is treated only in the quantized cutoff dipole model of this pair, under the additional vacuum-preparation assumption of the postulate item.