The linearly driven two-level model: rotating-wave approximation with an explicit finite-time error contract and the averaged Rabi populations

Adopt the framework data and 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: $\mathfrak{H}={\mathbb{C}}^2$, the orthonormal basis $(e_{\uparrow },e_{\downarrow })$ and Pauli matrices of Finite spin matrices clause (e), real parameters ${\omega }_0,{\mathrm{\Omega }}_1,\omega$ with $\omega \mathrm{\ne }0$, and $H_{\mathrm{l}}(t):=\frac{\mathrm{\hslash }}{2}({\omega }_0{\sigma }_z+{\mathrm{\Omega }}_1\cos (\omega t){\sigma }_x),t\in I=[a,b],$ read as the model's energy family in the bounded norm-continuous class with unique propagator $U_{\mathrm{l}}(t,s)$ (The bounded norm-continuous driven model: unique unitary propagator, Dyson truncation budget and outcome-law error clause (a)). Put $\mathrm{\Delta }:={\omega }_0-\omega$, let $R(t):=\cos (\omega t\mathrm{/}2)I+i\sin (\omega t\mathrm{/}2){\sigma }_z$ be the rotating frame of The two-level rotating frame: conjugation identities, the transformed families and invariance of the level populations, let $U_{\mathrm{l}}^R(t,s):=R(t)U_{\mathrm{l}}(t,s)R(s{)}^{\ast }$, and define the averaged matrix and the counter-rotating amplitude $A:=\frac{\mathrm{\hslash }}{2}(\mathrm{\Delta }{\sigma }_z+\frac{{\mathrm{\Omega }}_1}{2}{\sigma }_x),B:=\frac{\mathrm{\hslash }{\mathrm{\Omega }}_1}{8}({\sigma }_x+i{\sigma }_y),\epsilon (\tau ):=\frac{\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }}{2\mathrm{\mid }\omega \mathrm{\mid }}[1+\tau (\frac{\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }}{2}+\frac{3\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }}{4}\left)\right]\mathrm{.}$ 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). Then:

(a) Exact rotating-frame family. For every $t\in I$, $R(t)H_{\mathrm{l}}(t)R(t{)}^{\ast }+i\mathrm{\hslash }{R}^{\mathrm{\prime }}(t)R(t{)}^{\ast }=A+e^{2i\omega t}B+e^{-2i\omega t}{B}^{\ast },$ with $A=A^{\ast }$, $\mathrm{\parallel }A\mathrm{/}\mathrm{\hslash }\mathrm{\parallel }\le \frac{1}{2}(\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }+\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }\mathrm{/}2)$ and $\mathrm{\parallel }B\mathrm{/}\mathrm{\hslash }\mathrm{\parallel }=\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }\mathrm{/}4$; the second and third terms are the counter-rotating terms of the drive in the rotating frame.

(b) Rotating-wave error contract. For all $s,t\in I$, with $\tau :=\mathrm{\mid }t-s\mathrm{\mid }$, $\parallel U_{\mathrm{l}}^R(t,s)-e^{-i(t-s)A\mathrm{/}\mathrm{\hslash }}\parallel \le \epsilon (\tau )=\frac{\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }}{2\mathrm{\mid }\omega \mathrm{\mid }}[1+\tau (\frac{\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }}{2}+\frac{3\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }}{4}\left)\right];$ the bound is uniform in $s$ for fixed $\tau$ and grows linearly in $\tau$.

(c) Population comparison. Let $\psi (s)=e_{\uparrow }$ and let $p_{\downarrow }(t):=⟨P_{\downarrow }\psi (t),\psi (t)⟩$ be the exact down-population at time $t$, where $\psi (t)=U_{\mathrm{l}}(t,s)e_{\uparrow }$. Let $p_{\downarrow }^{\mathrm{a}\mathrm{v}\mathrm{g}}(\tau ):=\frac{({\mathrm{\Omega }}_1\mathrm{/}2{)}^2}{{\mathrm{\Delta }}^2+({\mathrm{\Omega }}_1\mathrm{/}2{)}^2}{\sin }^2\text{\hspace{0.17em}⁣}\left(\frac{\tau }{2}\sqrt{{\mathrm{\Delta }}^2+\frac{{\mathrm{\Omega }}_1^2}{4}}\right)({\mathrm{\Omega }}_1\mathrm{\ne }0),p_{\downarrow }^{\mathrm{a}\mathrm{v}\mathrm{g}}\equiv 0({\mathrm{\Omega }}_1=0),$ be the down-population of the initial up-state under the propagator of the averaged family $t\mapsto A$. Then for every $s,t\in I$ $\mid p_{\downarrow }(t)-p_{\downarrow }^{\mathrm{a}\mathrm{v}\mathrm{g}}(t-s)\mid \le 2\epsilon (\mathrm{\mid }t-s\mathrm{\mid })\mathrm{.}$ At resonance $\mathrm{\Delta }=0$ one has $p_{\downarrow }^{\mathrm{a}\mathrm{v}\mathrm{g}}(\tau )={\sin }^2(\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }\tau \mathrm{/}4)$.

(d) Scope. The averaged family is not asserted to be the model's exact dynamics: clause (b) is an error contract bounding the exact rotating-frame propagator against it, and its linear growth in $\tau$ gives no uniform-in-time approximation. The theorem asserts no resonance width, no decay rate, no long-time limit, no measurement record, no laboratory identification and no empirical result.

• Where the error comes from. The two counter-rotating terms $e^{\pm 2i\omega t}B$ are exactly the pieces dropped by the rotating-wave approximation; the supplier's integration by parts converts their size $\mathrm{\parallel }B\mathrm{\parallel }$ and frequency $2\mathrm{\mid }\omega \mathrm{\mid }$ into the prefactor $\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }\mathrm{/}(2\mathrm{\mid }\omega \mathrm{\mid })$ and their time-dependence into the linear-in-$\tau$ factor. For a specified operator-norm tolerance $\eta >0$, the sufficient condition supplied by the contract is $\epsilon (\tau )\le \eta$, which depends on both detuning and coupling. When ${\mathrm{\Omega }}_1=0$ the comparison is exact for every elapsed time; a large value of this upper bound alone does not establish failure of the approximation.
• Circular versus linear. For the circular family the corresponding counter-rotating term is absent and the reduction of the companion theorem is exact; the linear family retains it, and the difference between the two theorems is a property of the declared drive, not of the frame.