Anomalous Thermodynamic Cost of Clock Synchronization

Clock synchronization is critically important in positioning, navigation and timing systems. While its performance has been intensively studied in a wide range of disciplines, much less is known for the fundamental thermodynamics of clock synchronization, what limits the precision and how to optimize the energy cost for clock synchronization. Here, we report the first experimental investigation of two stochastic clocks synchronization, unveiling the thermodynamic relation between the entropy cost and clock synchronization in an open cavity optomechanical system. Two autonomous clocks are synchronized spontaneously by engineering the controllable photon-mediated dissipative optomechanical coupling and the disparate decay rates of hybrid modes. The measured dependence of the degree of synchronization on entropy cost exhibits an unexpected non-monotonic characteristic, indicating that the perfect clock synchronization does not cost the maximum entropy and there exists an optimum. The investigation of transient dynamics of clock synchronization exposes a trade-off between energy and time consumption. Our results reveal the fundamental relation between clock synchronization and thermodynamics, and have a great potential for precision measurements, distributed quantum networks, and biological science.