Ramsey clocks: a stable fringe can carry a phase bias

The oscillator supplies more than a frequency

Ramsey interrogation compares atomic evolution with the phases of two interactions. In a fountain, atoms can cross the same microwave cavity on the way up and down. Using one cavity helps suppress a phase difference between separate interaction zones, but different trajectories can still sample different parts of its field.

Weyers and colleagues evaluated distributed cavity phase in PTB-CSF2 through measurements and field calculations. A later NIST study examined a different cavity design. Their apparatus-specific analyses illustrate why a clean fringe alone is insufficient to certify the field phase seen by the detected atoms.

Read the evidence: S. Weyers et al. (2011), Distributed cavity phase frequency shifts of the caesium fountain PTB-CSF2 — author manuscript G. W. Hoth et al. (2022), Initial Study of the Distributed Cavity Phase Shift for the New Microwave Cavities of Cs Fountains at NIST

Declare the sign before quoting a shift

Use an ideal two-level, instantaneous-pulse convention with a central maximum. Frequency is in cycles per second, the interval is free-evolution time, and the extra phase is defined by its positive sign inside the cosine below. Contrast is between zero and one.

The lock follows the central maximum connected continuously to zero extra phase. Its frequency offset is therefore the negative extra phase divided by the interval and a full turn in radians. Reversing the definition of the added phase reverses that sign; a stated convention is part of the result.

\begin{aligned}P_e&=\frac{1+C\cos[2\pi(\nu-\nu_0)T+\Delta\phi]}{2}\\\nu_{\rm lock}-\nu_0&=-\frac{\Delta\phi}{2\pi T}\end{aligned}

Read the evidence: S. Weyers et al. (2011), Distributed cavity phase frequency shifts of the caesium fountain PTB-CSF2 — author manuscript

A small invented phase has a measurable consequence

As a teaching example, add one milliradian of effective phase over a half-second interrogation. The chosen convention gives a frequency shift of about minus 0.318 millihertz. Repeating the same measurement can make its scatter small while leaving this offset unchanged.

Doubling the interval halves this particular fixed-phase contribution. That is a conditional statement, not a general clock improvement: changing launch conditions or pulse geometry can also change the sampled phase. An interval scan is informative only when its accompanying apparatus changes are tracked.

Read the evidence: S. Weyers et al. (2011), Distributed cavity phase frequency shifts of the caesium fountain PTB-CSF2 — author manuscript

The detected ensemble has weights

A cavity has finite apertures and a cloud has a distribution of positions and velocities. The recorded population averages the atoms that survive preparation, flight and detection. An unweighted field map is therefore not yet the phase correction of a measured Ramsey signal.

For illustration, combine three equal-contrast contributions with zero phase and one with a quarter-turn phase. Their normalized phasor has an angle of about 0.322 radians, whereas the arithmetic mean of the four angles is about 0.393 radians. Phase averaging is generally a vector operation; finite pulses add sensitivity weighting.

A reversal tests the part that changes sign

Tilting a fountain or changing cavity feeds can test particular spatial asymmetries. In an invented pair of measurements, suppose a reversible contribution changes from plus ten to minus ten microhertz while a common two-microhertz offset remains. Averaging cancels the first contribution and retains the second.

The NIST initial study explicitly checked simplified numerical fields against analytic and prior model cases. Such validation has its own accuracy limits; an initial model is not a complete uncertainty evaluation. A null reversal or numerical agreement must be interpreted for the component actually tested.

Separate counting precision from the phase model

The existing Ramsey Lab allows comparison of finite counts, contrast and fringe ambiguity. It does not solve a three-dimensional lossy cavity field or calculate a real fountain’s trajectory-weighted phase shift. Its frequency estimate belongs to its declared interrogation model.

Use a clock record to ask both how uncertain the population-derived frequency is and which systematic phase contributions move its center. The projection-noise article addresses the first question; the collision article adds another mechanism for the second.