= Small-detuning error of a Rabi pi pulse
{title2=$1-P_\pi=(\delta/\Omega)^2+O((\delta/\Omega)^4)$}
For the <Driven two-level Rabi model> at fixed resonant pulse duration $T_\pi=\pi/\Omega$, put $x=\delta/\Omega$. Its transfer probability is
$$
P_\pi(x)=\frac{\sin^2[(\pi/2)\sqrt{1+x^2}]}{1+x^2}.
$$
The sine-squared numerator differs from one only at order $x^4$, while the denominator gives $1-x^2+O(x^4)$. Thus the leading population-transfer error is $x^2$. For a target transfer error $\epsilon$, the frequency mismatch in cycles per second should satisfy $|\delta\nu|\lesssim\Omega\sqrt\epsilon/(2\pi)$ in this small-detuning regime. This is not a complete characterization of every coherent gate error.
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