Commutativity of the adiabatic elimination limit of fast oscillatory components and the instantaneous feedback limit in quantum feedback networks

TL;DR

This paper proves that in quantum feedback networks, the limits of adiabatic elimination of fast oscillatory modes and instantaneous feedback commute, ensuring the consistency of common approximation methods.

Contribution

It demonstrates mathematically that adiabatic elimination and instantaneous feedback limits commute in quantum feedback networks, validating widely used approximation techniques.

Findings

Limits of adiabatic elimination and instantaneous feedback commute.

Both limits involve a Schur complement procedure.

Approximation methods for eliminating strongly coupled cavities are validated.

Abstract

We show that, for arbitrary quantum feedback networks consisting of several quantum mechanical components connected by quantum fields, the limit of adiabatic elimination of fast oscillator modes in the components and the limit of instantaneous transmission along internal quantum field connections commute. The underlying technique is to show that both limits involve a Schur complement procedure. The result shows that the frequently used approximations, for instance to eliminate strongly coupled optical cavities, are mathematically consistent.