# Maximum power and corresponding efficiency for two-level heat engines and refrigerators: optimality of fast cycles

## TL;DR

This paper demonstrates that the maximum power in a two-level quantum heat engine or refrigerator is universally achieved by a fast cycle with rapid control switching, and the efficiency at maximum power can approach Carnot's limit.

## Contribution

It establishes a universal optimal control protocol for maximum power in two-level thermal machines, regardless of microscopic details or operation mode.

## Key findings

- Maximum power is achieved by a fast Otto-cycle like protocol.
- Derived a closed-form formula for maximum power.
- Efficiency at maximum power can approach Carnot's efficiency.

## Abstract

We study how to achieve the ultimate power in the simplest, yet non trivial, model of a thermal machine, namely a two-level quantum system coupled to two thermal baths. Without making any prior assumption on the protocol, via optimal control we show that, regardless of the microscopic details and of the operating mode of the thermal machine, the maximum power is universally achieved by a fast Otto-cycle like structure in which the controls are rapidly switched between two extremal values. A closed formula for the maximum power is derived, and finite-speed effects are discussed. We also analyse the associated efficiency at maximum power (EMP) showing that, contrary to universal results derived in the slow-driving regime, it can approach Carnot's efficiency, no other universal bounds being allowed.