Finite-pulse Ramsey phase response ($P=4p(1-p){\cos }^2[(\delta T-\phi )/2+\theta ]$)

Keep the first pulse's oscillation phase zero and give the second phase $\phi$ in the Driven two-level Rabi model. Its upper off-diagonal Hamiltonian operator entry is $\hslash \Omega e^{-i\phi }/2$. Multiplying the two pulse unitary operators and the free-flight unitary operator givesThe Born rule gives the displayed phase-dependent form using the $p,b,\theta$ of the Ramsey fringe envelope. Thus $\phi =0$ reproduces the existing expression, while $\phi =\pi$ 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 $\phi$ 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 ${\theta }^{\prime }(0)=1/\Omega$. Consequently the phase near resonance isAt $\phi =\pi /2$ the exact resonant slope is $\partial P/\partial \nu =-\pi T_{\mathrm{eff}}$. For the plotted parameters $T_{\mathrm{eff}}\simeq 5.637\mathrm{ms}$; the near-resonant estimate $1/(2T_{\mathrm{eff}})\simeq 88.71\mathrm{Hz}$ 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.