# Fundamental accuracy-resolution trade-off for timekeeping devices

**Authors:** Florian Meier, Emanuel Schwarzhans, Paul Erker, Marcus Huber  
**arXiv:** [2301.05173](https://arxiv.org/abs/2301.05173)  
**Date:** 2023-01-12  
**Summary page:** https://www.summarizepaper.com/en/paper/2301.05173v1/

## Key points

- Clocks are driven by irreversible processes, but oscillatory systems can modulate the thermodynamic flux towards equilibrium
- Clock performance is limited by drifts of the oscillatory system and stochastic nature of irreversible events
- Clock accuracy can be improved with stable atomic or nuclear transitions
- Clock resolution at high accuracy is ultimately limited by the number of stochastic events per reference time unit
- The paper proves a universal bound for all clocks whose elementary thermalization events are memoryless
- Figure 1 shows two plots that qualitatively demonstrate ticking behavior with respect to parameter time t, one with Poisson distributed thermalization events and another with temporal probability concentration (TPC)
- Figure 2 sketches an oversampling regime of an exemplary clock: a pendulum in a weakly lit environment
- Quantum mechanics affects this trade-off between accuracy and resolution in clocks, and it can be used to improve clock performance

## Summary

This paper discusses the fundamental trade-off between accuracy and resolution in timekeeping devices from a thermodynamic perspective. All clocks are driven by irreversible processes, but oscillatory systems can be used to modulate the thermodynamic flux towards equilibrium. This modulation can be thought of as a temporal probability concentration for the most elementary thermalization events. The performance of clocks is limited by two factors: the inevitable drifts of the oscillatory system and the intrinsically stochastic nature of irreversible events upon which clock operation is based. Clock accuracy can be improved by finding stable atomic or nuclear transitions that lead to astounding precision in today's clocks. However, clock resolution at high accuracy is ultimately limited by the number of stochastic events per reference time unit. The authors address this essential trade-off between clock accuracy and resolution, proving a universal bound for all clocks whose elementary thermalization events are memoryless. The paper includes two figures that illustrate these concepts. Figure 1 shows two plots that qualitatively demonstrate ticking behavior with respect to parameter time t. The ticks of such a clock are generated by individual thermalization events at rate Γ, where these thermalization events are Poisson distributed in one plot and suppressed through temporal probability concentration (TPC) in another plot. As a result, the probability density for a tick can be concentrated around a desired average value with tick time uncertainty bounded by the width of the TPC window. Figure 2 sketches an oversampling regime of an exemplary clock: a pendulum in a weakly lit environment. The sources of entropy production for this clock include friction within the clockwork itself and matter-light interaction necessary to track pendulum position. In the oversampling regime, where average time between two ticks is much shorter than TPC period, photon reflection off pendulum when it's close to maximum deflection serves as elementary ticking event. The authors also discuss how quantum mechanics affects this trade-off between accuracy and resolution in clocks, and how it can be used to improve clock performance. They conclude that the universal bound they prove for memoryless clocks provides a benchmark for future research on more complex clocks.