We investigate an optomechanical system as a model of an autonomous mechanical pendulum clock in the quantum regime, whose operation relies only on incoherent (thermal) resources. The escapement of the clock, the mechanism that translates oscillatory motion into ticks, is provided by an emitter in the optical cavity and the operation of the clock relies on the existence of a limit cycle. Since the clock is based on an oscillatory degree of freedom, it can overcome the thermodynamic uncertainty relation and is thus more accurate than clocks that rely only on stochastic transitions. Furthermore, by increasing the amount of emitters in the cavity, the clock approaches the behavior expected for a macroscopic pendulum clock, where fluctuations become irrelevant while the clock dynamics becomes completely irreversible. This allows for investigating the quantum-to-classical transition of pendulum clocks.
