原子频率标准在过去七十年中实现了持续增加的精度，部分得益于反馈机制，这些机制稳定其输出。同时，量子系统的计时能力已在最近开发的滴答钟框架中被探索，该框架将钟表建模为动态系统，产生随机的滴答序列。然而，缺乏一个能够统一这些视角并纳入反馈到自主量子钟中的理论描述。我们介绍了一个反馈控制的钟表框架，其中从滴答序列中提取的 classical 信息用于影响钟表的后续动力学。我们证明，这种反馈保留了自定时和钟表独立性等核心结构特征，这些特征定义了自主滴答钟。我们进一步将信号噪声比 S 作为评估反馈控制钟表性能的基本指标。将我们的框架应用于两种代表性架构，我们证明经典钟表无法超越无反馈时可实现的最佳信号噪声比。相反，对于量子钟表，我们提供数值证据表明反馈可以提供真正的性能提升，提高最大可实现的信号噪声比。这些结果确立反馈作为推动量子领域计时基本极限的潜在关键成分。
英文摘要：
Atomic frequency standards have achieved steadily increasing precision over the past seventy years, enabled in part by feedback mechanisms that stabilise their output. In parallel, the timekeeping capabilities of quantum systems have been explored within the recently developed ticking-clock framework, which models clocks as dynamical systems producing a stochastic sequence of ticks. However, a theoretical description that unifies these perspectives and incorporates feedback into autonomous quantum clocks has been lacking. We introduce a framework for feedback-controlled clockworks in which classical information extracted from the tick sequence is used to influence the subsequent dynamics of the clock. We show that such feedback preserves the core structural features of self-timing and clockwork independence that characterise autonomous ticking clocks. We further identify the signal-to-noise ratio $\mathfrak{S}$ as the fundamental figure of merit for assessing the performance of feedback-controlled clocks. Applying our framework to two representative architectures, we prove that classical clockworks cannot surpass the optimal signal-to-noise ratio achievable without feedback. In contrast, for quantum clockworks we present numerical evidence that feedback can provide a genuine and robust performance enhancement, improving the maximal attainable signal-to-noise ratio. We provide additional numerical evidence, suggesting that the magnitude of the improvement strongly depends on the amount and type of the stored, classical information. These results establish feedback as a potentially essential ingredient in pushing the fundamental limits of timekeeping in the quantum regime.
Quantum advantages in timekeeping: dimensional advantage, entropic advantage and how to realise them via Berry phases and ultra-regular spontaneous emission
Feedback-based quantum optimization and its classical counterpart: quantum advantage and the power of classical algorithms
Graduate School of Science and Technology, Keio University（庆应义塾大学理工学研究科）；Basic Research Laboratories, NTT, Inc.（日本电报电话公司基础研究所）；NTT Research Center for Theoretical Quantum Information, NTT, Inc.（日本电报电话公司理论量子信息研究中心）
