= Velocity-averaged Ramsey line
{title2=$\overline P(\delta)=\int_0^\infty f(v)P(\delta;\ell/v,L/v)\,dv$}
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:
$$
\overline P(\delta)=\int_0^\infty f(v)P_{g\to e}(\delta;\ell/v,L/v)\,dv,\qquad
f(v)=\frac{2v^3}{u^4}e^{-(v/u)^2},\quad v>0.
$$
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 $f(v)dv=xe^{-x}dx$, proving normalization. Actual apertures, state selection and readout can change the effective weights.
For an original example choose $\ell=0.10\,\mathrm m$, $L=1.00\,\mathrm m$, $u=v_*=200\,\mathrm{m\,s^{-1}}$ and $\Omega=2\pi\times500\,\mathrm{rad\,s^{-1}}$. The reference particle has $\tau_*=0.500\,\mathrm{ms}$, $T_*=5.00\,\mathrm{ms}$ 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.
\Image[/ramsey-finite-pulse-velocity-average.png]
{title=Finite transit times shape the Ramsey line}
{description=Original calculation of the <finite-pulse Ramsey transition probability> and <velocity-averaged Ramsey line> for the chosen parameters above. The central panel also shows the instantaneous-pulse limit at $v_*$. All curves are model predictions; none reproduce experimental points.}
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.
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