# Free Fermions with a Localized Source

## TL;DR

This paper investigates the dynamics of free fermions on an infinite lattice coupled to a localized source, revealing how the particle growth rate depends on lattice geometry, dimensionality, and input rate, with quantum Zeno effects influencing the behavior.

## Contribution

It provides explicit formulas for particle growth rates in free fermion systems with localized sources, highlighting non-monotonic dependence and quantum Zeno effects.

## Key findings

- Growth rate depends on lattice geometry and dimensionality.
- Non-monotonic relationship between input rate and growth rate.
- Large input rate leads to inverse proportionality due to quantum Zeno effect.

## Abstract

We study an open quantum system of free fermions on an infinite lattice coupled to a localized particle source. In the long time limit, the total number of fermions in the system increases linearly with growth rate dependent on the lattice geometry and dimensionality. We express the growth rate in terms of lattice Green functions and derive explicit formulae in one dimension and for the square lattice. The interplay between the dynamics and the coupling to the environment leads, in contrast to classical systems, to a non-monotonic dependence of the particle growth rate on the input rate. We show that for all lattices the particle growth rate is inversely proportional to the input rate when the latter becomes large. This is a manifestation of the quantum Zeno effect.