Which Otto Engine Is the Fastest?

Abstract

A general thermal machine is characterized by distinct time or energy scales: temperature, coupling to the heat baths, internal free dynamics and interactions, and external driving. We address the question: what parameters determine the speed of a given engine, and how can they be used to reliably compare different types of engine? Specifically, we consider four realizations of the Otto engine, each operating with the same Otto efficiency, and aim to characterize and compare their power outputs. The main insight of this paper is that, irrespective of their very different dynamical implementations, the power of each engine can be factorized into a common characteristic work and an implementation-specific characteristic time. The characteristic work depends only on the internal frequencies of the machine and the temperatures, whereas the characteristic time depends on the coupling strength to the baths and on the implementation-specific driving. This observation allows us to compare the power of different engines through their characteristic times and thereby reduce the relevant parameter space to parameters that capture the dynamical aspects of engine performance. Despite different dynamical implementations, this framework reveals simple common features. In all cases, the maximum operating speed is limited by a common timescale given by the sum of the characteristic thermalization times of the hot and cold baths. Moreover, the four engines fall into two distinct asymptotic classes: implementations in which driving and thermalization occur simultaneously exhibit quadratic scaling, whereas those in which work extraction and thermalization alternate exhibit linear scaling. These results expose general dynamical features that determine the operational speed of quantum Otto engines.