# Slow dynamics and thermodynamics of open quantum systems

## TL;DR

This paper introduces a perturbation theory for open quantum systems with slowly changing parameters, enabling analysis of finite-time thermodynamic processes and deriving formulas for efficiency at maximum power.

## Contribution

It develops a perturbation framework for master equations with slowly varying parameters, applicable to finite-time thermodynamics and efficiency calculations.

## Key findings

- Derived a general formula for efficiency at maximum power of finite-time Carnot engines.
- Clarified conditions under which Curzon-Ahlborn efficiency is recovered.
- Provided a method to evaluate work and heat perturbatively in slowly driven systems.

## Abstract

We develop a perturbation theory of quantum (and classical) master equations with slowly varying parameters, applicable to systems which are externally controlled on a time scale much longer than their characteristic relaxation time. We apply this technique to the analysis of finite-time isothermal processes in which, differently from quasi-static transformations, the state of the system is not able to continuously relax to the equilibrium ensemble. Our approach allows to formally evaluate perturbations up to arbitrary order to the work and heat exchange associated to an arbitrary process. Within first order in the perturbation expansion, we identify a general formula for the efficiency at maximum power of a finite-time Carnot engine. We also clarify under which assumptions and in which limit one can recover previous phenomenological results as, for example, the Curzon-Ahlborn efficiency.