Bayesian Phase Stabilization at the Shot-Noise Limit for Scalable Quantum Networks
High-precision optical phase stabilization in quantum networks is fundamentally constrained by the strict photon-flux and duty-cycle limits required to avoid disturbing fragile quantum states. This challenge becomes especially critical when coordinating multiple independent light sources for multi-step quantum protocols. Here, we develop an integrated phase-stabilization framework that incorporates a Bayesian phase estimator to optimally extract information from sparse single-photon detection events. This approach outperforms conventional maximum-likelihood estimation and achieves the shot-noise limit under minimal photon flux. The framework enables real-time correction of combined phase noise from both nodal lasers and transmission fibers, facilitating a two-step excitation protocol for heralded entanglement generation between separate trapped-ion nodes via single-photon interference. Operating with a detected photon rate of approximately 1 MHz and a duty cycle less than 6.5%, the system maintains interferometric visibility greater than 97% over fiber links of 10 km and 100 km. This phase control yields long-lived ion-ion entanglement with parity contrast exceeding 85%, enabling device-independent quantum key distribution with finite-size security analysis over 10 km and a positive asymptotic key-rate over 100 km. Moreover, the resulting memory-memory entanglement at 10 km survives beyond the average time required to establish it, which is a fundamental requirement for quantum repeaters. This work establishes a robust and scalable foundation for practical long-distance quantum networks.