Start with the two passages through the cavity
Open Weyers and colleagues’ author manuscript, arXiv:1110.2590v1, alongside this guide. Begin with its introduction and cavity description, then compare Figures 1–3. The aim is to follow how a spatial variation of microwave phase becomes a frequency correction for the PTB-CSF2 fountain.
Atoms pass upward through a microwave cavity, turn around under gravity, then pass through it again. The two interactions compare an atomic rhythm with the microwave rhythm. Phase means the position within a cycle: two fields can oscillate at exactly the same frequency while one is slightly ahead of the other.
A fountain reuses one cavity for both encounters, which removes an important mismatch between separate beam-clock zones. But the ascending and descending atoms can cross different positions. If the microwave rhythm is slightly ahead in one part of the cavity, the second passage adds an extra phase to the comparison.
Read the evidence: S. Weyers et al. (2011), Distributed cavity phase frequency shifts of the caesium fountain PTB-CSF2 — author manuscript
Work through one small phase error
Use an ideal Ramsey fringe with instantaneous pulses and a central maximum. The expression below gives the excited-state probability in terms of fringe contrast, the applied frequency minus the unperturbed atomic frequency, and the time between pulses. Frequencies are in hertz and time in seconds. The extra phase is in radians and enters with a plus sign.
A frequency lock follows the central maximum. To keep the cosine at its maximum after adding a positive phase, the applied frequency must move downward. That gives the second line. Reversing the definition of the extra phase reverses this sign.
For a teaching example, one milliradian added over half a second gives about minus 0.318 millihertz. Repeating the measurement reduces random scatter around that displaced centre. If the added phase stays fixed, doubling the waiting time halves its frequency contribution. Changing the launch can change the sampled phase as well, so that condition needs to be checked.
Read the evidence: S. Weyers et al. (2011), Distributed cavity phase frequency shifts of the caesium fountain PTB-CSF2 — author manuscript
Read the points and curves as different objects
Figure 1 plots transition probability against microwave amplitude. Its measured points and model curve help locate the operating amplitude. Figure 2 tests the amplitude-dependent phase effect: the filled circles are measurements, the solid line a calculation, and the grey band shows the effect of varying the assumed cloud size. The statistical error bars and that model band answer different questions.
A cloud samples a range of trajectories. Apertures remove some atoms and the detector weights those that remain. To predict the observed fringe, add the contributions to the signal with those weights. Averaging the angles alone can give the wrong effective phase.
A small arithmetic example makes the distinction visible. Give four contributions equal contrast: three have zero phase and one has a quarter-turn phase. Adding their vectors gives an angle about 0.322 radians; averaging their four angles gives about 0.393 radians. This constructed example illustrates the averaging rule. A real cavity calculation also includes how finite-duration pulses weight the field along each path.
Read the evidence: S. Weyers et al. (2011), Distributed cavity phase frequency shifts of the caesium fountain PTB-CSF2 — author manuscriptG. W. Hoth et al. (2022), Initial Study of the Distributed Cavity Phase Shift for the New Microwave Cavities of Cs Fountains at NIST
Check what a reversal can remove
Figure 3 changes the launch tilt along two directions. Its horizontal coordinate is a controlled change in the fountain trajectory, and its fitted slopes quantify the frequency response. In the uncertainty table, the separate spatial contributions are carried through to a correction and an uncertainty.
For a simple check of the logic, imagine a contribution that reverses from plus ten to minus ten microhertz, sitting on a shared plus two microhertz offset. Averaging the two readings cancels the changing part and leaves the shared offset. A successful reversal constrains the part that changes sign; additional field and trajectory information supplies the rest.
When reading another cavity-phase analysis, note four things: the two atomic paths, the measured response to a controlled change, the field calculation and detection weighting, and the uncertainty attached to the final correction. Keep that correction separate from how precisely the fringe centre was counted.
Read the evidence: S. Weyers et al. (2011), Distributed cavity phase frequency shifts of the caesium fountain PTB-CSF2 — author manuscriptG. W. Hoth et al. (2022), Initial Study of the Distributed Cavity Phase Shift for the New Microwave Cavities of Cs Fountains at NIST