= Ramsey spectroscopy
{c}
<Ramsey spectroscopy> measures a transition <frequency> by comparing the relative <quantum phase> acquired between two coherent interactions with the reference <oscillator>. In a beam, a state selector and apertures admit particular internal <quantum states> and trajectories; an analyzer and detector measure state-dependent transmission. In an optically prepared <atomic clock>, <optical pumping> and <fluorescence> can perform those functions. The preparation and readout determine which part of the beam contributes; they are not themselves the two coherent interactions.
For an ideal accepted particle, the first resonant half-duration <Rabi pi pulse> splits the <probability amplitude> between $|g\rangle$ and $|e\rangle$. The second combines two indistinguishable alternatives: a transition in the first region followed by no transition in the second, or the reverse. Their relative <quantum phase> depends on the dark flight and the drive <oscillation phases>. Adding <probability amplitudes> gives <quantum interference>; adding only the two alternatives' <probabilities> would erase the fringe. “Dark” means that the transverse drive is absent: a static <magnetic field> and the internal <Hamiltonian operator> can still act.
\Image[/ramsey-beam-interactions-and-dark-drift.png]
{title=Two spatial interaction regions become two timed pulses for each accepted particle}
{description=Original functional <Ramsey spectroscopy> model, not a reconstruction of NIST-7 or a historical apparatus. Equal zone lengths $\ell$ and edge-to-edge gap $L$ give $\tau=\ell/v$ and $T=L/v$ for constant <speed> $v$. Preparation and state-sensitive readout are abstract functions; the diagram does not claim perfectly pure preparation.}
The useful separation is between pulse duration and phase-accumulation time. Short regions can share one coherent <oscillator> while a long drive-free interval resolves small <frequency> differences. A scan over reference <frequency> locates a fringe pattern; controlled changes of the second pulse's <oscillation phase> distinguish that coherent signal from phase-insensitive background. A reference scan must also identify the central fringe before <atomic clock frequency lock>, because neighboring fringes provide competing lock points.
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