Distributed cavity phase shift analysis of atomic clocks based on intracavity cooling

Distributed cavity phase shift constitutes a significant source of frequency uncertainty in cold-atom microwave frequency standards and presents challenges in the miniaturization process of high-performance atomic clocks. This paper introduces a method for calculating the distributed cavity phase shift utilizing 3D finite element simulation and Monte Carlo traversal of atoms. The impact of the distributed cavity phase shift on the miniaturized atomic clock is analytically calculated. Results demonstrate that the uncertainties of the distributed cavity phase shift are superior to 2×10-16 at varying atomic temperatures when the Ramsey line width is 10 Hz. Consequently, the long-term stability of this miniaturized atomic frequency standard scheme will not be constrained by the distributed-cavity phase shift, offering a notable advantage in long-term performance among miniaturized atomic clocks. The calculation method can also be extended to rectangular cavities, loop-gap cavities, and in-situ probing microwave cavities with complex structures, proving invaluable for analyzing the distributed cavity phase shift of atomic clocks and optimizing microwave cavity design.