Worked Rabi populations of the circularly driven two-level model: quarter-inversion, pi-pulse and detuned checks

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 circular 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). All angular frequencies below are declared model parameters read in ${\mathrm{s}}^{-1}$ (Dimensions, SI unit conventions and quantity types of the framework); no value is attributed to a laboratory system. For the numerical substitutions below, frequency symbols denote their numerical values in ${\mathrm{s}}^{-1}$ and $t,s$ denote numerical time values in seconds; thus a displayed elapsed time $t-s=\pi \mathrm{/}2$ means a physical duration $(\pi \mathrm{/}2)\mathrm{s}$, and the displayed sine arguments are dimensionless. Define ${\sigma }_{\pm }:=({\sigma }_x\pm i{\sigma }_y)\mathrm{/}2$. Three declared parameter sets are used:

1. Resonant set. ${\omega }_0=\omega =1$, ${\mathrm{\Omega }}_1=2$, so $\mathrm{\Delta }=0$ and $b=2$;
2. Detuned set. ${\omega }_0=1+\sqrt{3}$, $\omega =1$, ${\mathrm{\Omega }}_1=1$, so $\mathrm{\Delta }=\sqrt{3}$ and $b=2$;
3. Undriven set. ${\mathrm{\Omega }}_1=0$ and arbitrary ${\omega }_0,\omega$.

In each set prescribe the displayed circular energy family for all real physical times. For $s,t\in \mathbb{R}$, define $U_{\mathrm{c}}(t,s)$ by the model's unique propagator on any compact interval $J$ containing $s,t$. These restrictions agree: on a larger compact interval containing two choices, uniqueness of the norm Dyson integral equation identifies both restrictions. Thus this definition extends the compact-interval model consistently to all real evaluation times. Let $U_{\mathrm{c}}(t,s)$ be that propagator and let $p_{\downarrow }(t)=⟨P_{\downarrow }{U}_{\mathrm{c}}(t,s)e_{\uparrow },U_{\mathrm{c}}(t,s)e_{\uparrow }⟩$ be the down-population of the initially up state. Then:

(a) for the resonant set, $p_{\downarrow }(t)={\sin }^2(t-s)$, so $p_{\downarrow }=1\mathrm{/}2$ at the quarter-inversion time $t-s=\pi \mathrm{/}4$, $p_{\downarrow }=1$ at the pi-pulse time $t-s=\pi \mathrm{/}2$, and $p_{\downarrow }=0$ again at $t-s=\pi$;

(b) for the detuned set, $p_{\downarrow }(t)=\frac{1}{4}{\sin }^2(t-s)$, so $p_{\downarrow }=1\mathrm{/}4$ at $t-s=\pi \mathrm{/}2$ and the population is exactly $\pi$-periodic in $t-s$;

(c) for the undriven set, $p_{\downarrow }(t)\equiv 0$ and $p_{\uparrow }(t)\equiv 1$ for all $t$, while the transverse coherences $⟨{\sigma }_{\pm }⟩(t):=⟨{\sigma }_{\pm }{U}_{\mathrm{c}}(t,s)e_{\uparrow },U_{\mathrm{c}}(t,s)e_{\uparrow }⟩$ rotate with phase $e^{\pm i{\omega }_0(t-s)}$: the general coherence transforms by this phase at the splitting frequency ${\omega }_0$, while the specified initial up state has both transverse coherences zero;

(d) in every set the population is the outcome probability of the projection $P_{\downarrow }$ in the evolved preparation in the reading of P2; no damping, decoherence, measurement record or laboratory identification is asserted.

• Why the checks are exact. Each population in clauses (a)-(c) is the square of a sine at a rational multiple of $\pi$, so the values $1\mathrm{/}2$, $1$, $0$ and $1\mathrm{/}4$ are exact rational outputs of the model, not numerical approximations or measurements.
• Declared parameters, no measurements. The three sets are declared frequency parameters with the numerical-unit convention above; no frequency, energy or time of any laboratory species is claimed, and no source-reported measured value is used.