Repeated quantum interactions Quantum Langevin equation and the low density limit

TL;DR

This paper models a small quantum system interacting repeatedly with a heat bath, establishing a connection between the interaction time scale and the low density limit, leading to a quantum Langevin equation driven by Poisson processes.

Contribution

It introduces a novel relation between the interaction time scale and the low density limit, deriving a quantum Langevin equation from repeated quantum interactions.

Findings

The discrete evolution converges to a quantum Langevin equation as the time scale goes to zero.

A chemical potential is related to the interaction time scale via an exponential relation.

The model bridges discrete quantum interactions with continuous quantum stochastic calculus.

Abstract

We consider a repeated quantum interaction model describing a small system $\Hh_S$ in interaction with each one of the identical copies of the chain $\bigotimes_{\N^*}\C^{n+1}$, modeling a heat bath, one after another during the same short time intervals $[0,h]$. We suppose that the repeated quantum interaction Hamiltonian is split in two parts: a free part and an interaction part with time scale of order $h$. After giving the GNS representation, we establish the relation between the time scale $h$ and the classical low density limit. We introduce a chemical potential $\mu$ related to the time $h$ as follows: $h^2=e^{\beta\mu}$. We further prove that the solution of the associated discrete evolution equation converges strongly, when $h$ tends to 0, to the unitary solution of a quantum Langevin equation directed by Poisson processes.