Photon Scattering from a System of Multi-Level Quantum Emitters. II. Application to Emitters Coupled to a 1D Waveguide

TL;DR

This paper presents a formalism for analyzing photon scattering from multi-level quantum emitters coupled to a 1D waveguide, enabling direct solutions for low-intensity field interactions in waveguide QED systems.

Contribution

It introduces a novel photon-scattering relation tailored for 1D waveguides, extending previous 3D formalism to provide explicit solutions for single photons and weak coherent states.

Findings

Derived a direct solution for photon scattering in 1D waveguides.

Analyzed transmitted and reflected fields for multi-level emitters.

Demonstrated applications to state generation via photo-detection.

Abstract

In a preceding paper we introduced a formalism to study the scattering of low intensity fields from a system of multi-level emitters embedded in a $3$D dielectric medium. Here we show how this photon-scattering relation can be used to analyze the scattering of single photons and weak coherent states from any generic multi-level quantum emitter coupled to a $1$D waveguide. The reduction of the photon-scattering relation to $1$D waveguides provides for the first time a direct solution of the scattering problem involving low intensity fields in the waveguide QED regime. To show how our formalism works, we consider examples of multi-level emitters and evaluate the transmitted and reflected field amplitude. Furthermore, we extend our study to include the dynamical response of the emitters for scattering of a weak coherent photon pulse. As our photon-scattering relation is based on the Heisenberg picture, it is quite useful for problems involving photo-detection in the waveguide architecture. We show this by considering a specific problem of state generation by photo-detection in a multi-level emitter, where our formalism exhibits its full potential. Since the considered emitters are generic, the $1$D results apply to a plethora of physical systems like atoms, ions, quantum dots, superconducting qubits, and nitrogen-vacancy centers coupled to a $1$D waveguide or transmission line.