Ramsey fringe envelope ($0\le P_{g\to e}\le 4p(1-p)$)

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 $\theta$ 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.