A quantum optomechanical Mach clock

We introduce a quantum thermodynamic system that can operate as a thermal clock in analogy with the thermal clock first introduced by Ernst Mach in which the temperature difference between cooling bodies in contact can be used as a kind of relational time. Our model is based on an optomechanical system in which photons are transferred irreversibly between two cavities due to the modulation of the cavity cooling rate by a mechanical system coupled to a heat bath. We contrast the ensemble average view with a single system view using the theory of continuous measurement and we show that, by making a weak continuous measurement of the energy difference between the systems, a natural Mach thermal clock arises in an appropriate semiclassical limit. We investigate how quantum fluctuations modify this result.