= Rabi transit-time and field averaging
{c}
{title2=$\tau=L/v$}
For an ideal single transit, let the oscillatory <magnetic field> occupy a region of length $L$, take the longitudinal <speed> $v$ constant, and neglect edge fields. Each molecule then experiences one square pulse of duration $\tau=L/v$. This is one contiguous interaction region, distinct from the two separated pulses of <Ramsey interferometry>. The static <magnetic field> sets the selected transition gap; the transverse drive sets <Rabi angular frequency>. A change of RF <frequency> or static <magnetic field> changes <Rabi detuning>.
For an original ideal detector model, suppose every successful transition removes an otherwise accepted molecule and the incident beam occupies the lower state of the selected pair. With accepted no-transition signal $S_0$, constant background $S_{\mathrm{bg}}$, and effective loss contrast $0\leq\eta\leq1$, write
$$
S(\delta)=S_{\mathrm{bg}}+S_0[1-\eta\,\overline P(\delta)].
$$
For identical trajectories $\overline P=P_{g\to e}(L/v)$, giving an intensity dip where transfer is largest. For real trajectory variation, a normalized joint <probability density> $w(v,\epsilon,\Omega')$ gives the static-offset model
$$
\overline P(\delta)=\int_0^\infty\!dv\int_{-\infty}^\infty\!d\epsilon\int_0^\infty\!d\Omega'\,
w(v,\epsilon,\Omega')
\frac{\Omega'^2}{\Omega'^2+(\delta+\epsilon)^2}
\sin^2\!\left[\frac{L}{2v}\sqrt{\Omega'^2+(\delta+\epsilon)^2}\right],
\qquad \int w\,dv\,d\epsilon\,d\Omega'=1.
$$
Here $\epsilon$ is the trajectory's transition-frequency offset and $\Omega'$ its drive strength, each in angular-frequency units. The weighting is for molecules accepted by the apparatus, rather than an assumed unfiltered oven distribution. In particular, designing $\Omega L/v_0=\pi$ does not produce a <Rabi pi pulse> for every speed: at $0.8v_0$ and $1.2v_0$ the resonant transfer probabilities are respectively $\sin^2(5\pi/8)\simeq0.8536$ and $\sin^2(5\pi/12)\simeq0.9330$.
Spread in transit times, static <magnetic fields> and drive amplitudes can reduce or smear oscillations without <quantum decoherence> on any individual trajectory. Even the single-transit square pulse has finite spectral width of order $1/\tau$ in <angular frequency>. If the <magnetic field> varies substantially along one path, its <Hamiltonian operator> changes during the passage and the constant-offset average above must be replaced by a <time-ordered exponential>. Actual molecules may also have several coupled <energy levels>; the isolated pair and perfect loss discriminator are explicit idealizations of this model.
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