= Ramsey frequency precision
{c}
{title2=$\sigma_\nu\simeq1/(2\pi VT\sqrt N)$}
The <full width at half maximum> of a fringe, the lifetime of <quantum coherence>, and the precision of an estimated <frequency> are different quantities. In the ideal short-pulse limit, write the transfer <probability> with contrast $0<V\leq1$ and second-pulse <oscillation phase> $\varphi$ as
$$
P=\frac{1+V\cos(\delta T-\varphi)}2,\qquad \delta=2\pi(\nu_0-\nu).
$$
At quadrature, $P=1/2$ and $|\partial P/\partial\nu|=\pi VT$. For $N$ independent, equally prepared particles with ideal binary readout, the <binomial distribution> gives $\sigma_P=\sqrt{P(1-P)/N}=1/(2\sqrt N)$. Local <propagation of uncertainty> therefore yields the displayed $\sigma_\nu$. For chosen $T=5\,\mathrm{ms}$, $V=0.8$ and $N=10^4$, it is approximately $0.398\,\mathrm{Hz}$, although the ideal fringe width is $100\,\mathrm{Hz}$. This is an original counting-noise estimate at a known quadrature, not the accuracy of a historical clock or a guarantee of finding the correct fringe.
For the illustrative <dephasing> model $V(T)=e^{-T/T_2}$ with fixed $N$, minimizing $\sigma_\nu$ maximizes $Te^{-T/T_2}$ and gives $T=T_2$: its <derivative> is $e^{-T/T_2}(1-T/T_2)$. This conclusion changes if longer interrogation reduces the number of accepted particles or the repetition rate. <Quantum decoherence>, a distribution of flight times, reference <oscillator> noise and readout errors can reduce contrast; a fitted decay time need not identify one microscopic mechanism. More counting can improve precision while a reproducible cavity <quantum phase> offset still biases the <frequency>, as in <atomic clock frequency shifts>.
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