Worked rotating-wave error budget for the linearly driven two-level model: dimensionless instantiation and averaged Rabi check

Adopt P1, P2 and P4 in the readings of the declared driven model (The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models clause (a)) and the linear two-level drive model of its clause (f) with ${\mathbb{C}}^2$, the basis $(e_{\uparrow },e_{\downarrow })$ and the Pauli matrices of Finite spin matrices clause (e). Declare angular-frequency parameters $\omega >0$, $\mathrm{\Delta }={\omega }_0-\omega =0$ (resonance) and ${\mathrm{\Omega }}_1=\omega \mathrm{/}50$, so that the dimensionless frequency ratio is ${\mathrm{\Omega }}_1\mathrm{/}\mathrm{\mid }\omega \mathrm{\mid }=1\mathrm{/}50$; all angular frequencies are declared model parameters read in ${\mathrm{s}}^{-1}$ (Dimensions, SI unit conventions and quantity types of the framework), and no value is attributed to a laboratory system. Fix an initial time $s$ and use a compact interval $I$ containing $s$ and every displayed evaluation time $t=s+\tau$, where $\tau \ge 0$; for example $I=[s,s+200\mathrm{/}\omega ]$ contains them all, since $100\pi \mathrm{/}3<200$. Let $p_{\downarrow }(t)$ be the exact down-population of the initially up state under the model's propagator, $p_{\downarrow }^{\mathrm{a}\mathrm{v}\mathrm{g}}(\tau )={\sin }^2(\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }\tau \mathrm{/}4)$ the averaged-model Rabi population of the page's error contract, and $\epsilon (\tau )=\frac{\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }}{2\mathrm{\mid }\omega \mathrm{\mid }}[1+\tau (\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }\mathrm{/}2+3\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }\mathrm{/}4)]$. Then:

(a) the rotating-wave error contract reads $\epsilon (\tau )=\frac{1}{100}(1+\frac{3\omega \tau }{200})$, so that $\epsilon =1\mathrm{/}100$ at $\omega \tau =0$, $\epsilon =5\mathrm{/}200=1\mathrm{/}40$ at $\omega \tau =100$, and $\epsilon =1\mathrm{/}25$ at $\omega \tau =200$; the corresponding population-difference bounds $2\epsilon$ are $2\mathrm{/}100$, $1\mathrm{/}20$ and $2\mathrm{/}25$;

(b) at the declared elapsed time $\omega \tau =100\pi \mathrm{/}3$ one has ${\mathrm{\Omega }}_1\tau \mathrm{/}4=\pi \mathrm{/}6$ and the averaged-model population is exactly $p_{\downarrow }^{\mathrm{a}\mathrm{v}\mathrm{g}}=1\mathrm{/}4$;

(c) for the circular drive of the companion theorem with the same ${\omega }_0,\omega ,{\mathrm{\Omega }}_1$ the model's exact population obeys the same Rabi-type formula with no rotating-wave error, so the non-zero bound of (a) is specific to the linear polarization;

(d) the numbers in (a)-(b) are outputs of the declared model and its error contract; they are not measured values, and clause (a)'s bound is conservative (at $\tau =0$ the two propagators coincide while the operator-norm bound is $\epsilon (0)=1\mathrm{/}100$ and the population-difference bound is $2\epsilon (0)=1\mathrm{/}50$).

• Reading the budget. The error contract is a uniform bound, not a prediction of the actual difference; the example shows the two faces of that: the bound is exact arithmetic (rational numbers at the declared times), and it is conservative at the initial time, where the true difference vanishes.
• No physical numbers. The example deliberately uses only declared ratios and dimensionless elapsed times; no frequency, field amplitude or relaxation parameter of a laboratory species appears.