The development of laser-cooling technology has advanced cold-atomic-frequency standards, thus enhancing the precision of defining a second to 10^-16. Additionally, the second definition with a frequency uncertainty of 10^-19, which offers even greater accuracy, is anticipated to be established in the optical-frequency standard. The cold-atomic-frequency standard offers a higher frequency accuracy and a lower frequency drift rate compared with the classical atomic-frequency standard, thereby providing higher long-term stability. Among the various physical factors affecting the accuracy of atomic-frequency standards, the distributed-cavity phase shift is one of the most important sources of uncertainty and a key factor to be considered in the miniaturization of cold-atomic-frequency standards. Currently, the analysis and calculation of the distributed-cavity phase shift primarily involves the Fourier decomposition of three-dimensional phase distributions in column coordinates. This method accurately solves the phase distribution of microwave cavities with simple structures and the corresponding frequency shifts. However, for phase distributions in rectangular cavities, annular cavities, and cylindrical cavities with multiple openings in the sidewalls, which are relatively complex, this method requires remodeling and is complicated. Rapid advancements in computer technology have enabled rapid three-dimensional finite-element simulations that accurately simulate the phase distribution in microwave cavities. This facilitates the optimization of microwave-cavity designs, thus reducing the effect of the distributed-cavity phase shift.

This study examines the phase distribution of an intracavity-cooling falling-detection atomic clock based on three-dimensional finite-element simulations. Initially, each atom is assigned a position based on a Gaussian distribution and specified with an initial velocity based on the Maxwell–Boltzmann distribution. The direction of the initial velocity is assumed to be completely random, and atomic collisions are disregarded. The sensitivity function is employed to calculate the change in the transition probability caused by each atom experiencing a phase variation, thus allowing for the determination of the total transition probability change and the distributed-cavity phase shift.

The simulation results (Fig. 3) show the phase distribution in a microwave cavity with and without sidewall opening. Apart from the slight phase-gradient change in the Y-direction caused by the structural asymmetry, the overall phase change is minimal. This suggests that opening the sidewall of the microwave cavity to introduce a laser field is feasible. Figure 4 provides a detailed analysis of the phase distribution inside the cavity. It reveals abrupt phase changes at approximately 20 mm in the radial direction. Additionally, the phase undulation within the cutoff waveguide is substantial, which results in significant distributed-cavity phase shifts when the atoms experience these phase changes. Figure 5 shows the calculated distributed-cavity phase shifts under different temperatures and Ramsey linewidths. When the Ramsey linewidths are constant, the motion range of the atoms within the microwave cavity increases with the atomic temperature. Consequently, the phase changes intensify, thus resulting in greater frequency shifts and uncertainties. However, under wider Ramsey linewidths, the distributed-cavity phase shifts are primarily caused by the position change of the atomic cluster due to gravity. The effect of the atomic-cluster temperature becomes more significant when the Ramsey linewidth is narrower. Under the same atomic temperature, the frequency uncertainty of the distributed-cavity phase shift decreases as the linewidth increases because of the inversely proportional relationship between time and the Ramsey linewidth for free evolution. Considering the structure and short-term stability of the microwave cavity, we observe that when the Ramsey linewidth is 10 Hz, the atomic clouds experience better phase shifts and power symmetry, and the uncertainty of the distributed-cavity phase shift is less than 2×10^-16, thus indicating greater potential for long-term stability.

This study introduces a Monte Carlo method for traversing-atom analysis to calculate the distributed-cavity phase shift of microwave atomic clocks. This method is based on a three-dimensional finite-element simulation and is applied to cavity-cooled falling miniaturized atomic clocks. The results indicate that introducing laser light into the sidewall aperture minimally affects the intracavity phase distribution. Additionally, the intracavity-cooling atomic clocks do not experience significant phase variations near the aperture or inside the cutoff waveguide during operation, which is advantageous for evaluating the distributed-cavity phase shift. The frequency uncertainty degree is less than 2×10^-16, which indicates that the long-term stability of the atomic clock is not limited by the distributed-cavity phase shift. This design for the miniaturization of atomic clocks is highly promising and is expected to offer more competitive time–frequency benchmarks with lower frequency uncertainties and better long-term stability compared with other miniaturized atomic clocks. The proposed calculation method can be extended to the distributed-cavity phase-shift calculation of rectangular cavities, loop-gap cavities, and other microwave cavities with complex structures, thus facilitating the microwave-cavity design of miniaturized atomic clocks.