= Finite-pulse Ramsey phase response
{c}
{title2=$P=4p(1-p)\cos^2[(\delta T-\varphi)/2+\theta]$}
Keep the first pulse's <oscillation phase> zero and give the second phase $\varphi$ in the <Driven two-level Rabi model>. Its upper off-diagonal <Hamiltonian operator> entry is $\hbar\Omega e^{-i\varphi}/2$. Multiplying the two pulse <unitary operators> and the free-flight <unitary operator> gives
$$
\langle e|U_p(\varphi)U_fU_p(0)|g\rangle
=-2i e^{-i\varphi/2}\frac{\Omega}{q}s_p
\left[c_p\cos\frac{\delta T-\varphi}{2}-b\sin\frac{\delta T-\varphi}{2}\right].
$$
The <Born rule> gives the displayed phase-dependent form using the $p,b,\theta$ of the <Ramsey fringe envelope>. Thus $\varphi=0$ reproduces the existing expression, while $\varphi=\pi$ turns a resonant transfer maximum into zero transfer. A single pulse's population is insensitive to its drive phase; the second coherent interaction makes their difference observable. An uncontrolled, uniformly random $\varphi$ instead averages this <probability> to $2p(1-p)$, leaving the broad pulse response but removing the dark-flight fringes.
Finite pulses also contribute to the near-resonant <frequency> slope. For $\Omega\tau=\pi/2$, differentiating $\theta$ at $\delta=0$ gives $\theta'(0)=1/\Omega$. Consequently the phase near resonance is
$$
\frac{\delta T-\varphi}{2}+\theta
=\frac{\delta T_{\mathrm{eff}}-\varphi}{2}+O[(\delta/\Omega)^3],\qquad
T_{\mathrm{eff}}=T+\frac2\Omega=T+\frac{4\tau}{\pi}.
$$
At $\varphi=\pi/2$ the exact resonant slope is $\partial P/\partial\nu=-\pi T_{\mathrm{eff}}$. For the plotted parameters $T_{\mathrm{eff}}\simeq5.637\,\mathrm{ms}$; the near-resonant estimate $1/(2T_{\mathrm{eff}})\simeq88.71\,\mathrm{Hz}$ explains why the exact finite-pulse width below differs from $100\,\mathrm{Hz}$. The slope is exact at resonance under these assumptions, while this width estimate uses the local phase expansion. A <velocity> distribution requires averaging the full response, not substituting one effective flight time.
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