# Open quantum systems with delayed coherent feedback

## TL;DR

This paper introduces a generalized method for simulating open quantum systems with delayed feedback, including multiple delays and cascaded systems, and extends the quantum regression formula to such systems.

## Contribution

It provides a new simulation technique for delayed feedback in quantum systems and generalizes the quantum regression formula for these scenarios.

## Key findings

- Simulated systems with multiple delays and cascaded feedback.
- Derived a generalized quantum regression formula for delayed feedback.
- Proposed a quantum teleportation protocol for simulating delayed feedback.

## Abstract

We present an elementary derivation and generalisation of a recently reported method of simulating feedback in open quantum systems. We use our generalised method to simulate systems with multiple delays, as well as cascaded systems with delayed backscatter. In addition, we derive a generalisation of the quantum regression formula that applies to systems with delayed feedback, and show how to use the formula to compute two-time correlation functions of the system as well as output field properties. Finally, we show that delayed coherent feedback can be simulated as a quantum teleportation protocol that requires only Markovian resources, pre-shared entanglement, and time travel. The requirement for time travel can be avoided by using a probabilistic protocol.