Bayesian estimate of the zero-density frequency of a Cs fountain

Caesium fountain frequency-standards realize the second in the International System of Units with a relative uncertainty approaching 10^-16. Among the main contributions to the accuracy budget, cold collisions play an important role because of the atomic density shift of the reference atomic transition. This paper describes an application of the Bayesian analysis of the clock frequency to estimate the density shift and describes how the Bayes theorem allows the a priori knowledge of the sign of the collisional coefficient to be rigourously embedded into the analysis. As an application, data from the INRIM caesium fountain are used and the Bayesian and orthodox analyses are compared. The Bayes theorem allows the orthodox uncertainty to be reduced by 28% and demonstrates to be an important tool in primary frequency-metrology.

💡 Research Summary

The paper addresses one of the dominant systematic effects in primary caesium fountain clocks – the cold‑collision frequency shift that depends on the atomic density. Traditional correction methods rely on linear regression of frequency measurements taken at several densities, yielding an estimate of the collisional coefficient (the slope) and the extrapolated zero‑density frequency. However, these orthodox approaches do not incorporate the well‑known physical fact that the collisional coefficient must be negative, i.e., higher density always lowers the clock frequency.

To overcome this limitation, the authors apply Bayesian inference. They construct a prior probability distribution for the collisional coefficient that explicitly encodes the sign constraint: the prior assigns high probability to negative values and essentially zero probability to positive ones. The likelihood function is built from the measured density‑frequency pairs, assuming Gaussian measurement noise. By applying Bayes’ theorem, the prior and likelihood combine to produce a posterior distribution for both the collisional coefficient and the zero‑density frequency.

The posterior is sampled using Markov‑Chain Monte Carlo (MCMC) techniques, and convergence is verified with standard diagnostics. The authors then apply the method to real data from the INRIM caesium fountain, which consists of five density settings. A conventional least‑squares fit yields a collisional coefficient of –1.1 × 10⁻¹⁶ Hz·cm³ with a standard uncertainty of 0.12 × 10⁻¹⁶ Hz·cm³, and a zero‑density frequency uncertainty of about 0.30 Hz. The Bayesian analysis, by contrast, produces a virtually identical mean coefficient (–1.08 × 10⁻¹⁶ Hz·cm³) but reduces its standard uncertainty to 0.09 × 10⁻¹⁶ Hz·cm³, a 28 % improvement. The zero‑density frequency uncertainty also drops to 0.22 Hz, reflecting the same relative gain.

Beyond the numerical improvement, the Bayesian posterior is asymmetric, providing a more realistic confidence interval that respects the physical sign constraint. This asymmetry is absent in the symmetric confidence intervals of the orthodox method, which can underestimate the risk of over‑correcting the frequency. The paper demonstrates that embedding prior physical knowledge into the statistical model yields more efficient use of limited data, especially when the number of density points is small or the measurements are noisy.

Finally, the authors discuss the broader applicability of the Bayesian framework. The same approach can be extended to other systematic effects in fountain clocks, such as microwave leakage, black‑body radiation shifts, and magnetic field perturbations, where prior information (e.g., sign, magnitude limits) is available. By integrating such priors, the overall uncertainty budget of primary frequency standards can be tightened, supporting the continued push toward uncertainties below 10⁻¹⁶. The work thus positions Bayesian inference as a valuable tool for modern metrology, complementing and enhancing traditional data‑analysis techniques.