Ramsey spectroscopy

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 $|g⟩$ and $|e⟩$. 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.

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.

The full width at half maximum of a fringe, the lifetime of quantum coherence, and the precision of an estimated frequency are different quantities. In the ideal short-pulse limit, write the transfer probability with contrast $0<V\le 1$ and second-pulse oscillation phase $\phi$ asAt quadrature, $P=1/2$ and $|\partial P/\partial \nu |=\pi VT$. For $N$ independent, equally prepared particles with ideal binary readout, the binomial distribution gives ${\sigma }_P=\sqrt{P(1-P)/N}=1/(2\sqrt{N})$. Local propagation of uncertainty therefore yields the displayed ${\sigma }_{\nu }$. For chosen $T=5\mathrm{ms}$, $V=0.8$ and $N=10^4$, it is approximately $0.398\mathrm{Hz}$, although the ideal fringe width is $100\mathrm{Hz}$. This is an original counting-noise estimate at a known quadrature, not the accuracy of a historical clock or a guarantee of finding the correct fringe.

For the illustrative dephasing model $V(T)=e^{-T/T_2}$ with fixed $N$, minimizing ${\sigma }_{\nu }$ maximizes $T{e}^{-T/T_2}$ and gives $T=T_2$: its derivative is $e^{-T/T_2}(1-T/T_2)$. This conclusion changes if longer interrogation reduces the number of accepted particles or the repetition rate. Quantum decoherence, a distribution of flight times, reference oscillator noise and readout errors can reduce contrast; a fitted decay time need not identify one microscopic mechanism. More counting can improve precision while a reproducible cavity quantum phase offset still biases the frequency, as in atomic clock frequency shifts.