= Solution
A resonant <Rabi pi pulse> satisfies $\Omega T_\pi=\pi$. The supplied $T_\pi=0.256\,\mathrm s$ therefore gives
$$
\boxed{\Omega=\frac\pi{0.256\,\mathrm s}\simeq12.27\,\mathrm{rad\,s^{-1}},\qquad
\frac\Omega{2\pi}=1.953125\,\mathrm{Hz}.}
$$
The ideal rotating-frame operation is $-iX$, a <Pauli X gate> up to <global phase>.
Both logical states belong to the electronic ground level. A fast <electric dipole transition> between them is forbidden, and spontaneous radiative relaxation at a hyperfine radiofrequency is weak. The optical readout field is off during the coherent pulse. Neglecting relaxation is therefore consistent with a population lifetime much longer than $0.256\,\mathrm s$. Magnetic noise, collisions and technical dephasing are additional limitations; the ideal coherent calculation assumes their effects are small over this interval.
At detuning $\delta$, define $\Omega_{\rm eff}=\sqrt{\Omega^2+\delta^2}$. The same two-level <Hamiltonian operator> gives
$$
P_b(t)=\frac{\Omega^2}{\Omega^2+\delta^2}
\sin^2\left(\frac{t\sqrt{\Omega^2+\delta^2}}2\right).
$$
For an accurate fixed-duration $\pi$ rotation, the detuning must be small compared with the <Rabi angular frequency>. Writing $\delta\nu=\delta/(2\pi)$,
$$
\boxed{|\delta\nu|\ll\frac\Omega{2\pi}=1.95\,\mathrm{Hz},}
$$
a fractional mismatch much smaller than $6.1\times10^{-9}$ relative to the $320.7\,\mathrm{MHz}$ carrier. This is an accuracy scale, not an assertion that a particular nonzero detuning gives a perfect gate.
The <small-detuning error of a Rabi pi pulse> quantifies a chosen tolerance. For an initial $|a\rangle$ and the resonant pulse duration,
$$
1-P_b(T_\pi)=\left(\frac\delta\Omega\right)^2+O\!\left((\delta/\Omega)^4\right).
$$
A population-transfer error at most $\epsilon$ consequently requires approximately \b[$|\delta\nu|\lesssim1.95\sqrt\epsilon\,\mathrm{Hz}$]; for example, \b[about $0.20\,\mathrm{Hz}$ for one percent error]. Full gate calibration also includes pulse area and drive phase, beyond this population-transfer criterion.
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