Small-detuning error of a Rabi pi pulse ($1-P_{\pi }=(\delta /\Omega {)}^2+O((\delta /\Omega {)}^4)$)

For the Driven two-level Rabi model at fixed resonant pulse duration $T_{\pi }=\pi /\Omega$, put $x=\delta /\Omega$. Its transfer probability isThe 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 $ϵ$, the frequency mismatch in cycles per second should satisfy $|\delta \nu |\lesssim \Omega \sqrt{ϵ}/(2\pi )$ in this small-detuning regime. This is not a complete characterization of every coherent gate error.