Ramsey frequency precision (${\sigma }_{\nu }\simeq 1/(2\pi VT\sqrt{N})$)

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 $0<V\le 1$ and second-pulse oscillation phase $\phi$ asAt 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 $T{e}^{-T/T_2}$ 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.