WaveguideQED.jl: An Efficient Framework for Simulating Non-Markovian Waveguide Quantum Electrodynamics

Matias Bundgaard-Nielsen; Dirk Englund; Mikkel Heuck; Stefan Krastanov

doi:10.22331/q-2025-04-17-1710

WaveguideQED.jl: An Efficient Framework for Simulating Non-Markovian Waveguide Quantum Electrodynamics

Matias Bundgaard-Nielsen^1,2, Dirk Englund³, Mikkel Heuck^1,2, and Stefan Krastanov⁴

¹Department of Electrical and Photonics Engineering, Technical University of Denmark, Building 343, Lyngby, Denmark
²NanoPhoton-Center for Nanophotonics, Technical University of Denmark, Building 343, Lyngby, Denmark
³Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
⁴Manning College of Information and Computer Sciences, University of Massachusetts Amherst, 140 Governors Drive Amherst, Massachusetts 01003, USA

Published:	2025-04-17, volume 9, page 1710
Editor:	Francesco Ciccarello
Eprint:	arXiv:2412.13332v2
Doi:	https://doi.org/10.22331/q-2025-04-17-1710
Citation:	Quantum 9, 1710 (2025).

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Abstract

In this paper, we introduce a numerical framework designed to solve problems within the emerging field of Waveguide Quantum Electrodynamics (WQED). The framework is based on collision quantum optics, where a localized quantum system interacts sequentially with individual time-bin modes. This approach provides a physically intuitive model that allows researchers familiar with tools such as QuTiP in Python, Quantum Optics Toolbox for Matlab, or
in Julia to efficiently set up and execute WQED simulations. Despite its conceptual simplicity, we demonstrate the framework's robust ability to handle complex WQED scenarios. These applications include the scattering of single- or two-photon pulses by quantum emitters or cavities, as well as the exploration of non-Markovian dynamics, where emitted photons are reflected back, thereby introducing feedback mechanisms.

Featured image: (a): Sketch of the system: A two-photon pulse in a right-propagating Waveguide mode is scattered into both left- and right-propagating pulses when interacting with a quantum emitter. The bins represent the granulation made in the simulation software (b): The populations of the emitter and waveguide modes as a function of time. (c)-(e): The three different wavefunctions resulting from the scattering.

Popular summary

Waveguide Quantum Electrodynamics (WQED) is the study of traveling few-photon quantum states interacting with localized quantum systems—like atoms, quantum dots, or superconducting qubits.

Modeling WQED accurately requires keeping track of the full quantum state of the traveling light field, including its temporal structure. Standard tools often treat photons as inputs and outputs without describing their full quantum state, and most methods struggle to handle non-Markovian effects—where emitted photons loop back and influence the system later in time. Simulating such memory effects usually involves significant complexity or computational cost.

WaveguideQED.jl is a new open-source Julia package that offers an efficient, user-friendly framework for simulating WQED. It uses the collision model of quantum optics, where photons are represented as time-bins that sequentially interact with a quantum system. This gives a clear, intuitive picture of light-matter interactions and naturally accommodates feedback and entanglement effects.

A key ingredient in WaveguideQED.jl’s efficiency is its matrix-free implementation of time-dependent operators. Instead of constructing large sparse matrices for each interaction with a time-bin, the framework uses LazyOperators—a feature from QuantumOptics.jl that applies operator actions as efficient kernel functions. This design greatly reduces memory use and computation time, making simulations with fine time resolution feasible and fast.

► BibTeX data

@article{BundgaardNielsen2025waveguideqedjl,
  doi = {10.22331/q-2025-04-17-1710},
  url = {https://doi.org/10.22331/q-2025-04-17-1710},
  title = {Waveguide{QED}.jl: {A}n {E}fficient {F}ramework for {S}imulating {N}on-{M}arkovian {W}aveguide {Q}uantum {E}lectrodynamics},
  author = {Bundgaard-Nielsen, Matias and Englund, Dirk and Heuck, Mikkel and Krastanov, Stefan},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1710},
  month = apr,
  year = {2025}
}

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[3] Nicholas J. Martin, Dominic Hallett, Mateusz Duda, Luke Hallacy, Elena Callus, Luke Brunswick, René Dost, Edmund Clarke, Pallavi K. Patil, Pieter Kok, Maurice S. Skolnick, and Luke R. Wilson, "Purcell-enhanced, directional light–matter interaction in a waveguide-coupled nanocavity", Optica 12 7, 1100 (2025).

[4] Matias Bundgaard-Nielsen, Matthew Kozma, Sofia Arranz Regidor, and Stephen Hughes, "Decoherence in waveguide quantum electrodynamics using matrix product states", Physical Review Research 8 3, 033109 (2026).

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