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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 ${\phi }_c$, the finite-pulse Ramsey transition probability approachesThis sign uses exactly the Rabi detuning convention and pulse Hamiltonian operator above, with the second pulse replacing ${\sigma }_x$ by $\cos {\phi }_c{\sigma }_x+\sin {\phi }_c{\sigma }_y$. The factor $2\pi$ converts angular frequency to ordinary frequency. For the original example, ${\phi }_c=0.010$ 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.^[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 thereforeCorrecting 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.

The exact finite-pulse Ramsey transition probability separates a slowly varying pulse response from rapid quantum interference. Define the single-zone transfer probability $p$ and a pulse-induced quantum phase $\theta$ bySince $c_p^2+b^2=1-p$, factorizing the real bracket in the original formula givesWhere $c_p=b=0$, the probability is zero and $\theta$ need not be defined. Elsewhere this identity proves the envelope bound without a weak-drive assumption. The envelope $4p(1-p)$ is not the single-zone response $p$: two equal coherent interactions have different transfer weights. A contiguous resonant Rabi pi pulse of duration $2\tau$ provides another useful comparison with the same total drive duration, but no dark interval. Its line is broad because it has no long flight phase.

Let each drive region have length $\ell$ and the drive-free gap have length $L$, measured edge to edge. A particle of constant speed $v>0$ experiences $\tau (v)=\ell /v$ and $T(v)=L/v$. For a normalized detected-beam speed probability density function $f(v)$, sum the transition probabilities, not the probability amplitudes of different particles:The displayed $f$ is an ideal effusion flux distribution, with speed scale $u$, uniform transmission and speed-independent detection efficiency. It is not a claim about any particular clock's selected beam. The substitution $x=(v/u{)}^2$ gives $f(v)dv=x{e}^{-x}dx$, proving normalization. Actual apertures, state selection and readout can change the effective weights.

For an original example choose $\ell =0.10\mathrm{m}$, $L=1.00\mathrm{m}$, $u=v_{\ast }=200\mathrm{m}{\mathrm{s}}^{-1}$ and $\Omega =2\pi \times 500\mathrm{rad}{\mathrm{s}}^{-1}$. The reference particle has ${\tau }_{\ast }=0.500\mathrm{ms}$, $T_{\ast }=5.00\mathrm{ms}$ and resonant $\pi /2$ pulses. The instantaneous-pulse width would be $100\mathrm{Hz}$. The exact finite-pulse width for that single speed is about $88.7\mathrm{Hz}$; the chosen distribution also reduces the on-resonance average to approximately $0.738$ because not every particle receives a $\pi /2$ pulse.

Different velocities smear side fringes because $\delta L/v$ varies across the beam. Nevertheless every term is even in $\delta$, so this symmetric ideal-model average has no odd line asymmetry that could displace the central lock point. Reduced contrast or altered width alone is not evidence of a shifted line center. A speed-dependent transition shift or unequal drive quantum phases changes that conclusion; atomic clock frequency shifts explains why these distinctions matter experimentally.