Free Bosons with a Localized Source

TL;DR

This paper studies the dynamics of non-interacting bosons injected into a lattice, revealing a critical input rate that determines whether the boson number grows exponentially or quadratically over time, with exponential growth occurring in all dimensions except two.

Contribution

It provides a detailed analysis of the critical input rate for exponential growth in bosonic quantum walks and constructs a microscopic Hamiltonian model explaining the instability.

Findings

Exponential growth occurs when input rate exceeds a critical value.

Critical input rate is positive in all dimensions except two.

In two dimensions, the growth is always exponential.

Abstract

We analyze the time evolution of an open quantum system driven by a localized source of bosons. We consider non-interacting identical bosons that are injected into a single lattice site and and perform a continuous time quantum walks on a lattice. We show that the average number of bosons grows exponentially with time when the input rate exceeds a certain lattice-dependent critical value. Below the threshold, the growth is quadratic in time, which is still much faster than the naive linear in time growth. We compute the critical input rate for hyper-cubic lattices and find that it is positive in all dimensions $d$ except for $d=2$ where the critical input rate vanishes---the growth is always exponential in two dimensions. To understand the exponential growth, we construct an explicit microscopic Hamiltonian model which gives rise to the open system dynamics once the bath is traced out. Exponential growth is identified with a region of dynamic instability of the Hamiltonian system.