= Atomic clock frequency shifts
A narrow fringe makes the center easier to resolve statistically; it does not certify an unbiased <frequency>. For example, with ideal $\pi/2$ pulses and a second-pulse <oscillation phase> offset $\varphi_c$, the <finite-pulse Ramsey transition probability> approaches
$$
P_{g\to e}=\cos^2\!\left(\frac{\delta T-\varphi_c}{2}\right),\qquad
\nu_{\mathrm{center}}-\nu_0=-\frac{\varphi_c}{2\pi T}.
$$
This sign uses exactly the <Rabi detuning> convention and pulse <Hamiltonian operator> above, with the second pulse replacing $\sigma_x$ by $\cos\varphi_c\,\sigma_x+\sin\varphi_c\,\sigma_y$. The factor $2\pi$ converts <angular frequency> to ordinary <frequency>. For the original example, $\varphi_c=0.010$ rad and $T=5\,\mathrm{ms}$ give a shift of about $-0.318\,\mathrm{Hz}$. For finite pulses or distributed flight times, use the full signal and its <atomic clock frequency lock> rather than inserting an unweighted mean $T$.
NIST-7 used <optical pumping> and <fluorescence>. Controls included field-sensitive transitions, independent beam-speed measurements, <microwave> power stabilization, and beam reversal for cavity <quantum phase> errors.<Primary Atomic Frequency Standards at NIST by D. B. Sullivan and others>[§2.2, pp. 50–52: optical readout and frequency-shift controls]{ref}
The connection to <special relativity> is quantitative. A freely moving emitter's <proper time> advances more slowly than laboratory time by $\sqrt{1-v^2/c^2}$. With first-order propagation effects removed in the ideal comparison, its internal laboratory <frequency> is therefore
$$
\frac{\nu_{\mathrm{lab}}-\nu_0}{\nu_0}\simeq-\frac{v^2}{2c^2}.
$$
Correcting an ensemble signal requires the effective <velocity> weights of preparation, interrogation and detection, not simply the oven's mean <speed>. Additional shifts from <Zeeman splitting>, unwanted light, <collisions> and cavity fields likewise require separate evaluation; <feedback control> locks to the observed line, including any uncorrected bias. Comparing clocks at different <gravitational potentials> additionally requires <gravitational time dilation>. The moving-clock experiments in <twin paradox clock experiments> compare calibrated clock readings along different <world lines>; the beam atoms' internal motion is part of the clock mechanism's own correction.
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