 Weak-pulse Ramsey spectrum

For two coherent square pulses of equal duration $\tau$, pulse-start separation $T\ge \tau$, resonant Rabi angular frequency $\Omega$, and Rabi detuning $\delta$, the initially lower-state transition probability has the displayed leading weak-excitation form when $|\Omega |\tau \ll 1$. Integrating the first-order probability amplitude over both windows gives a sum of two equal Fourier transforms, with relative quantum phase $\delta T$. The factor four from coherent addition makes the central value $(\Omega \tau {)}^2$. The rotating-wave approximation additionally requires many carrier cycles per pulse and near-resonant weak driving.