Extended Weak Coupling Limit for Pauli-Fierz Operators

TL;DR

This paper establishes a simplified and stronger version of the weak coupling limit for Pauli-Fierz operators, showing that the system-environment interaction converges to quantum white noise in a rigorous mathematical framework.

Contribution

It introduces a more straightforward and robust formulation of the weak coupling limit, connecting the system's evolution to quantum white noise.

Findings

Convergence of the system's evolution to quantum white noise in the weak coupling limit.

Simplification of previous formulations of the quantum Langevin equation.

Rigorous mathematical proof of the limit for Pauli-Fierz operators.

Abstract

We consider the weak coupling limit for a quantum system consisting of a small subsystem and reservoirs. It is known rigorously since [Dav74] that the Heisenberg evolution restricted to the small system converges in an appropriate sense to a Markovian semigroup. In the nineties, Accardi, Frigerio and Lu [AFL90] initiated an investigation of the convergence of the unreduced unitary evolution to a singular unitary evolution generated by a quantum Langevin equation. We present a version of this convergence which is both simpler and stronger than the formulations which we know. Our main result says that in an appropriately understood weak coupling limit the interaction of the small system with environment can be expressed in terms of the so-called quantum white noise.