Ergodic Properties of Quantum Birth and Death Chains

TL;DR

This paper investigates the ergodic behavior of quantum Markov processes inspired by classical birth and death chains, providing explicit calculations and characterizing their long-term properties.

Contribution

It introduces a class of quantum birth and death processes, analyzing their geometric, irreducibility, and ergodic properties with explicit methods.

Findings

Established geometric and irreducibility properties for the processes.

Derived explicit formulas for transition operators in homogeneous cases.

Identified diverse ergodic behaviors within the class of quantum chains.

Abstract

We study a class of quantum Markov processes that, on the one hand, is inspired by the micromaser experiment in quantum optics and, on the other hand, by classical birth and death processes. We prove some general geometric properties and irreducibility for non-degenerated parameters. Furthermore, we analyze ergodic properties of the corresponding transition operators. For homogeneous birth and death rates we show how these can be fully determined by explicit calculation. As for classical birth and death chains we obtain a rich yet simple class of quantum Markov chains on an infinite space, which allow only local transitions while having divers ergodic properties.