  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.
