Atomic clock locking with Bayesian quantum parameter estimation: scheme and experiment

Atomic clocks are crucial for science and technology, but their sensitivity is often restricted by the standard quantum limit. To surpass this limit, correlations between particles or interrogation times must be leveraged. Although the sensitivity can be enhanced to the Heisenberg limit using quantum entanglement, it remains unclear whether the scaling of sensitivity with total interrogation time can achieve the Heisenberg scaling. Here, we design an adaptive Bayesian quantum frequency estimation protocol that approaches the Heisenberg scaling and experimentally demonstrate its validity with a cold-atom coherent-population-trapping (CPT) clock. In further, we achieve robust and high-precision closed-loop locking of the cold-atom CPT clock by utilizing our Bayesian quantum frequency estimation protocol. In comparison to the conventional proportional-integral-differential locking, our Bayesian locking scheme not only yields an improvement of 5.1(4) dB in fractional frequency stability, but also exhibits better robustness against technical noises. Our findings not only provide a robust and high-precision approach to lock atomic clocks, but also hold promising applications in various interferometry-based quantum sensors, such as quantum magnetometers and atomic interferometers.