= Rabi pi pulse
{c}
{title2=$\Omega\tau_\pi=\pi$}
For a constant-amplitude square pulse on resonance, a <Rabi pi pulse> lasts $\tau_\pi=\pi/\Omega$ and gives $|g\rangle\mapsto-ie^{-i\phi}|e\rangle$ in the rotating frame. A half-duration pulse prepares $(|g\rangle-ie^{-i\phi}|e\rangle)/\sqrt2$. The <quantum pulse area> $\Omega t$ determines these rotations only under the stated resonant, constant-axis assumptions; a detuned drive rotates about a different axis.
As an original numerical illustration, choose $\Omega=2\pi\times1000\ \mathrm{rad\,s^{-1}}$ and hence $\tau_\pi=0.500\ \mathrm{ms}$. These are chosen model parameters, not historical apparatus settings. Holding that duration fixed gives
$$
x=\frac{\delta}{\Omega},\qquad
P_\pi(x)=\frac{\sin^2[(\pi/2)\sqrt{1+x^2}]}{1+x^2}.
$$
Thus $P_\pi(0)=1$, $P_\pi(1)=\tfrac12\sin^2(\pi/\sqrt2)\simeq0.3166$, and $P_\pi(\sqrt3)=0$. The last zero comes from completing a full rotation about the tilted axis, rather than switching the drive off. At zero <Rabi detuning>, doubling the pulse duration restores the ground-state population; continuous driving does not keep transferring population irreversibly.
\Image[/rabi-square-pulse-detuning.png]
{title=Ideal Rabi oscillations and the detuning response of a fixed pi pulse}
{description=Original <Driven two-level Rabi model>, with no measured points. The left panel varies pulse duration; the right fixes $\Omega\tau_\pi=\pi$ and varies <Rabi detuning>. The dotted curve is the population-transfer upper bound, not a fitted spectral line.}
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