Wave-front curvature in optical atomic beam clocks

Atomic clocks provide a reproducible basis for our understanding of time and frequency. Recent demonstrations of compact optical clocks, employing thermal atomic beams, have achieved short-term fractional frequency instabilities of order 10-16, competitive with the best international frequency standards available. However, a serious challenge inherent in compact clocks is the necessarily smaller optical beams, which results in rapid variation in interrogating wave fronts. This can cause inhomogeneous excitation of the thermal beam leading to long-term drifts in the output frequency. Here we develop a model for Ramsey-Borde interferometry using optical fields with curved wave fronts and simulate the 40Ca beam clock experiment described by Olson et al. [Phys. Rev. Lett. 123, 073202 (2019)]. The results of Olson et al. showed surprising and unexplained behavior in the response of the atoms in the interrogation. Our model predicts signals consistent with experimental data and can account for the significant sensitivity to laser geometry that was reported. We find the signal-to-noise ratio is maximized when the laser is uncollimated at the interrogation zones to minimize inhomogeneity and also identify an optimal waist size determined by both laser inhomogeneity and the velocity distribution of the atomic beam. We investigate the shifts and stability of the clock frequency, showing that the Gouy phase is the primary source of frequency variations arising from laser geometry.