# Repeated interaction processes in the continuous-time limit, applied to quadratic fermionic systems

## TL;DR

This paper analyzes the continuous-time limit of repeated interactions in fermionic systems using Lindblad equations, establishing conditions for convergence to a unique stationary state and exploring various examples.

## Contribution

It introduces a novel criterion for convergence in fermionic Lindblad dynamics, extending control theory concepts to quantum many-body systems.

## Key findings

- Derived a necessary and sufficient condition for convergence to a stationary state.
- Applied the criterion to various fermionic models, including spin chains.
- Connected the convergence condition to the Kalman criterion from control theory.

## Abstract

We study a class of Lindblad equation on finite-dimensional fermionic systems. The model is obtained as the continuous-time limit of a repeated interaction process between fermionic systems with quadratic Hamiltonians, a setup already used by Platini and Karevski for the one-dimensional XY model. We prove a necessary and sufficient condition for the convergence to a unique stationary state, which is similar to the Kalman criterion in control theory. Several examples are treated, including a spin chain with interactions at both ends.