The two-level inversion model in an oscillating field: resonance, averaged Rabi inversion and its error contract

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 basis $(\mathrm{\mid }\mathrm{E}⟩,\mathrm{\mid }\mathrm{G}⟩)$ of a declared two-level inversion model: two orthogonal unit vectors with the Pauli matrices of Finite spin matrices clause (e) acting on their span, so that ${\sigma }_z=\mathrm{diag}(1,-1)$ distinguishes the two levels. Declare real parameters ${\mathrm{\Delta }}_0>0$ (half the inversion splitting, joules), $\mu >0$ (a dipole-like coupling, in $\mathrm{C}\mathrm{m}$), $E_0>0$ (a field amplitude, in $\mathrm{V}{\mathrm{m}}^{-1}$) and a drive frequency $\omega >0$ in ${\mathrm{s}}^{-1}$, and declare the model's energy family $H(t):={\mathrm{\Delta }}_0{\sigma }_z+2\mu E_0\cos (\omega t){\sigma }_x,t\in \mathbb{R},$ read as the energy of the declared model, with $\mathrm{\hslash }$ the reduced Planck constant of Dimensions, SI unit conventions and quantity types of the framework; populations are the diagonal entries of the two projections onto $\mathbb{C}\mathrm{\mid }\mathrm{E}⟩$ and $\mathbb{C}\mathrm{\mid }\mathrm{G}⟩$ in the evolved preparation. Fix the coupling ratio $x:=\mu E_0\mathrm{/}{\mathrm{\Delta }}_0$ (dimensionless). Then:

(a) Identification with the declared linear drive. $H(t)=\frac{\mathrm{\hslash }}{2}({\omega }_0{\sigma }_z+{\mathrm{\Omega }}_1\cos (\omega t){\sigma }_x)$ with ${\omega }_0=\frac{2{\mathrm{\Delta }}_0}{\mathrm{\hslash }},{\mathrm{\Omega }}_1=\frac{4\mu E_0}{\mathrm{\hslash }},{\omega }_0\in {\mathrm{s}}^{-1},{\mathrm{\Omega }}_1\in {\mathrm{s}}^{-1},$ so the model is an instance of the declared linear two-level drive model of clause (f) with $\mathrm{\Delta }={\omega }_0-\omega$; the level splitting is $2{\mathrm{\Delta }}_0$.

(b) Resonance and averaged inversion. At resonance $\omega ={\omega }_0$, i.e. $\mathrm{\hslash }\omega =2{\mathrm{\Delta }}_0$, the averaged model of the page's error contract predicts that the initial $\mathrm{\mid }\mathrm{E}⟩$ population is transferred to $\mathrm{\mid }\mathrm{G}⟩$ with probability $p_{\mathrm{G}}^{\mathrm{a}\mathrm{v}\mathrm{g}}(\tau )={\sin }^2\text{\hspace{0.17em}⁣}\left(\frac{\mu E_0\tau }{\mathrm{\hslash }}\right),$ which equals $1$ at the inversion time $\tau =\pi \mathrm{\hslash }\mathrm{/}(2\mu E_0)$; equivalently ${\mathrm{\Omega }}_1\tau \mathrm{/}4=\pi \mathrm{/}2$ at that time.

(c) Error contract at inversion. At the inversion time $\tau =\pi \mathrm{\hslash }\mathrm{/}(2\mu E_0)$ the exact $\mathrm{\mid }\mathrm{G}⟩$ population $p_{\mathrm{G}}$ satisfies $\mid p_{\mathrm{G}}-1\mid \le 2x(1+\frac{3\pi }{2}),$ so with the declared ratio $x=1\mathrm{/}100$ the contract gives $p_{\mathrm{G}}\ge 1-\frac{2+3\pi }{100}$;

(d) status. The model, its parameters and the time $\tau$ are declared model data; no measured transition frequency, no laboratory molecule, cavity or field, no damping or relaxation and no measurement record appears, and the averaged formula is accompanied by the explicit contract of clause (c), not asserted as exact.

• What is and is not asserted. The example asserts a conditional model statement: the declared two-level family has a resonance at $\mathrm{\hslash }\omega =2{\mathrm{\Delta }}_0$ at which the averaged model inverts the population at the time $\pi \mathrm{\hslash }\mathrm{/}(2\mu E_0)$, with an explicit error contract. It does not assert that any molecule, cavity or field realizes these parameters, and it records no measured frequency; the source's reported numerical frequency is not used as a premise.
• Why the contract is needed. The source treatment drops the counter-rotating term; the contract of clause (c) is what makes that step quantitative, and the ratio $x=\mu E_0\mathrm{/}{\mathrm{\Delta }}_0$ is exactly the small parameter controlling it.