# Unraveling Mirror Properties in Time-Delayed Quantum Feedback Scenarios

## TL;DR

This paper derives a microscopic model for quantum feedback involving a mirror and emitter, justifying and generalizing the phenomenological coupling used in quantum optics with delay effects.

## Contribution

It provides a microscopic derivation of the feedback coupling in quantum optical systems, including delay effects, and extends the model to various mirror types.

## Key findings

- Derivation of a microscopic Hamiltonian-based model for mirror-emitter dynamics.
- Delay differential operator equations capturing finite round-trip times.
- Generalization to dissipative and gain mirrors.

## Abstract

We derive in the Heisenberg picture a widely used phenomenological coupling element to treat feedback effects in quantum optical platforms. Our derivation is based on a microscopic Hamiltonian, which describes the mirror-emitter dynamics based on a dielectric, a mediating fully quantized electromagnetic field, and a single two-level system in front of the dielectric. The dielectric is modeled as a a system of identical two-state atoms. The Heisenberg equation yields a system of describing differential operator equations, which we solve in the Weisskopf-Wigner limit. Due to a finite round-trip time between emitter and dielectric, we yield delay differential operator equations. Our derivation motivates and justifies the typical phenomenological assumed coupling element and allows, furthermore, a generalization to a variety of mirrors, such as dissipative mirrors or mirrors with gain dynamics.