Ramsey ensemble coherence ($\chi (T)=⟨e^{i\eta T}⟩$)

For an ensemble with static detunings $\delta +\eta$, ideal same-phase quarter-turn pulses give ${\overline{p}}_2(T)=[1+\mathrm{Re}(e^{i\delta T}\chi (T))]/2$. Here $\chi$ is the characteristic function of the offset distribution. A mean-zero Gaussian of RMS width $\sigma$ gives $\chi =e^{-{\sigma }^2{T}^2/2}$; a symmetric uniform interval gives a sinc function. A width alone does not uniquely specify the envelope.