= Ramsey fringe envelope
{c}
{title2=$0\leq P_{g\to e}\leq4p(1-p)$}
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> $\theta$ by
$$
p=\frac{\Omega^2}{q^2}s_p^2,\qquad b=\frac{\delta}{q}s_p,\qquad
\theta=\operatorname{atan2}(b,c_p).
$$
Since $c_p^2+b^2=1-p$, factorizing the real bracket in the original formula gives
$$
P_{g\to e}=4p(1-p)\cos^2\!\left(\frac{\delta T}{2}+\theta\right).
$$
Where $c_p=b=0$, 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.
\Image[/ramsey-fringes-and-single-zone-response.png]
{title=Finite pulses set the envelope while dark flight sets the fine fringes}
{description=Original exact <two-level system> predictions for $\tau=0.500\,\mathrm{ms}$, $T=5.00\,\mathrm{ms}$ and $\Omega=2\pi\times500\,\mathrm{rad\,s^{-1}}$, matching the existing <velocity-averaged Ramsey line> reference particle. The envelope is a bound, not a measured curve or the single-zone line. The central panel changes only the second pulse's <oscillation phase>. No velocity average or experimental data are plotted.}
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