# Point-coupling Hamiltonian for frequency-independent linear optical devices

## TL;DR

This paper introduces the point-coupling Hamiltonian as a versatile model for frequency-independent linear optical devices, enabling formal analysis and simulation of complex optical systems with feedback.

## Contribution

It provides a method to construct the Hamiltonian from classical scattering matrices and demonstrates its application in simulating feedback systems using matrix-product states.

## Key findings

- Derived the quantum scattering matrix from the Hamiltonian.
- Showed equivalence between quantum scattering and classical inverse scattering.
- Applied the model to simulate time-delayed feedback systems.

## Abstract

We present the point-coupling Hamiltonian as a model for frequency-independent linear optical devices acting on propagating optical modes described as a continua of harmonic oscillators. We formally integrate the Heisenberg equations of motion for this Hamiltonian, calculate its quantum scattering matrix, and show that an application of the quantum scattering matrix on an input state is equivalent to applying the inverse of classical scattering matrix on the annihilation operators describing the optical modes. We show how to construct the point-coupling Hamiltonian corresponding to a general linear optical device described by a classical scattering matrix, and provide examples of Hamiltonians for some commonly used linear optical devices. Finally, in order to demonstrate the practical utility of the point-coupling Hamiltonian, we use it to rigorously formulate a matrix-product-state based simulation for time-delayed feedback systems wherein the feedback is provided by a linear optical device described by a scattering matrix as opposed to a hard boundary condition (e.g. a mirror with less than unity reflectivity).