# Limit cycles in periodically driven open quantum systems

## TL;DR

This paper derives a general algebraic condition for periodically driven open quantum systems to approach a unique limit cycle, extending Spohn's condition for equilibrium to driven systems.

## Contribution

It introduces a novel algebraic criterion ensuring convergence to a limit cycle in periodically driven open quantum systems.

## Key findings

- Provides a necessary and sufficient condition for limit cycle convergence.
- Extends Spohn's algebraic condition to non-equilibrium driven systems.
- Offers a framework for analyzing long-time behavior of quantum systems under periodic driving.

## Abstract

We investigate the long-time behavior of quantum N-level systems that are coupled to a Markovian environment and subject to periodic driving. As our main result, we obtain a general algebraic condition ensuring that all solutions of a periodic quantum master equation with Lindblad form approach a unique limit cycle. Quite intuitively, this criterion requires that the dissipative terms of the master equation connect all subspaces of the system Hilbert space during an arbitrarily small fraction of the cycle time. Our results provide a natural extension of Spohn's algebraic condition for the approach to equilibrium to systems with external driving.