# Phase-shift in vapor cell and compact cold-atom frequency standards

## TL;DR

This paper provides a theoretical analysis of phase-shifts caused by electromagnetic losses in microwave cavities, affecting the accuracy of compact atomic clocks using cold or thermal atoms.

## Contribution

It introduces a detailed calculation of spatial phase variations in cylindrical and spherical cavities and their impact on atomic clock frequency stability.

## Key findings

- Phase-shift depends on cavity geometry and electromagnetic losses.
- Atomic motion causes different atoms to experience different phases.
- Phase-shift can significantly affect the performance of vapor cell and cold-atom clocks.

## Abstract

We report on a theoretical analysis of the phase-shift in compact atomic clocks working either with cold or thermal atoms. It is well known that in a microwave cavity with electromagnetic losses, a traveling wave adds to the standing wave of a given resonant mode. We calculate the spatial varying phase related to this travelling wave for two geometries of interest in clock applications, the cylindrical cavity and the spherical cavity. Due to their motion, the atoms probe different regions of the cavity and then experience different phases of the interrogating microwave field. We show that this combination of atomic motion and spatial changing phase results in a phase-shift of the clock frequency, well known in primary frequency standards, that can affect the metrological performances also of vapor cell and compact cold-atom clocks. In the latter case, we evaluate the phase-shift for a space clock and a ground clock.