Performance limits of decay clocks: Fundamental accuracy and resolution limits of quantum clocks with exponential ticking mechanism

Clocks are among the most precise measurement devices ever built, but like anything else, they are bound by the laws of thermodynamics. As a consequence, all clocks are inherently subject to noise and thus cannot be infinitely precise. We examine these limitations and the thermodynamic cost of running a clock. From minimal assumptions, we derive that processes driving a clock must be irreversible. In the simplest memory-less thermodynamic setting, this leads to exponential decay. Under the assumption of a fixed decay rate $\Gamma$, we explore what types of quantum clocks can be built using an exponentially decaying process to generate the ticks of the clock, but with a clockwork that is otherwise unconstrained. Using the average number of ticks $N$ until a clock is off by one tick as the measure for its accuracy, and the inverse average tick time as the measure for its resolution $R$, we show that any increase in accuracy ultimately comes at the cost of resolution, subject to the bound $N\leq \Gamma^2/R^2$. With a periodic process in the clockwork to concentrate the decay event probability to specific times, we can build clocks that approach this optimal accuracy-resolution trade-off. Based on a quantum clock from Schwarzhans et al. [1], that uses as its only resource out of equilibrium heat baths, we design a clock that we conjecture to reach the scaling $N\sim \text{const.}/R^2$ asymptotically. In this example, the entropy production grows linearly with the accuracy, confirming the thermodynamic limitation of the clock performance. [1] Emanuel Schwarzhans, Maximilian P. E. Lock, Paul Erker, Nicolai Friis, and Marcus Huber. Autonomous Temporal Probability Concentration: Clockworks and the Second Law of Thermodynamics. PRX, 11(1):011046, 2021. doi:10.1103/PhysRevX.11.011046. URL https://link.aps.org/doi/10.1103/PhysRevX.11.011046.