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arxiv: 2604.18036 · v1 · submitted 2026-04-20 · 🪐 quant-ph

Decoherence in Waveguide Quantum Electrodynamics using Matrix Product States

Matias Bundgaard-Nielsen , Matthew Kozma , Sofia Arranz Regidor , Stephen Hughes This is my paper

Pith reviewed 2026-05-10 05:24 UTC · model grok-4.3

classification 🪐 quant-ph
keywords waveguide QEDmatrix product statesdecoherenceLindblad master equationdensity matricesquantum opticstime-delayed feedbackemitter dephasing
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The pith

Density matrix matrix product states incorporate decoherence into waveguide quantum electrodynamics simulations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a technique to extend matrix product state representations from pure states to density matrices for open systems in waveguide quantum electrodynamics. This is achieved by discretizing the waveguide field into time bins under the collision quantum optics model and applying Lindblad terms to the Liouvillian for dissipation. A reader would care because real emitters suffer from pure dephasing and radiative losses that alter photon emission, scattering, and correlations in ways pure-state methods cannot capture. The work applies the method to a driven two-level system with time-delayed feedback, to two separated emitters, and to few-photon pulse scattering, while distinguishing the effects of pure dephasing from off-chip decay.

Core claim

We present a matrix product state (MPS) method for including decoherence processes in calculations involving waveguide quantum electrodynamics (waveguide QED) using density matrices. The approach is based on collision quantum optics, where the many-body state of the waveguide is represented as discrete time bins, which are efficiently represented using an MPS chain. Our method is a generalization of previous MPS methods, and we demonstrate how one can efficiently expand to density matrices, allowing for the inclusion of various loss processes in the form of Lindblad terms in the Liouvillian superoperator responsible for the relevant dissipation dynamics.

What carries the argument

Collision quantum optics discretization of the waveguide into discrete time bins represented as an MPS chain for the density matrix, augmented by Lindblad superoperators in the Liouvillian to capture dissipation.

If this is right

  • Pure dephasing modifies the emission and feedback dynamics of a two-level system in a semi-infinite waveguide in quantifiable ways.
  • Off-chip radiative decay produces qualitatively different signatures from pure dephasing in the scattered light.
  • Scattering of few-photon Fock pulses off an emitter can be tracked while including realistic decoherence.
  • Interactions between two spatially separated emitters with propagation delays become tractable under Lindblad loss.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same MPS density-matrix construction could be applied to more complex multi-emitter networks or chiral waveguides.
  • Experimental groups could use the method to predict how controlled dephasing alters measured photon statistics before running hardware tests.
  • The technique suggests a route to embed other open-system master equations into tensor-network descriptions of one-dimensional quantum optics.

Load-bearing premise

The discrete time-bin collision model of the waveguide continues to represent continuous-time dynamics accurately once the state is promoted from a wave function to a density matrix.

What would settle it

Exact comparison of computed single-photon scattering spectra or second-order correlation functions for a two-level emitter with known analytic Lindblad solutions at a chosen dephasing rate; large deviations would falsify the numerical method.

Figures

Figures reproduced from arXiv: 2604.18036 by Matias Bundgaard-Nielsen, Matthew Kozma, Sofia Arranz Regidor, Stephen Hughes.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) A density matrix of dimensions ( [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Singular value decomposition in tensor diagrams. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Tensor diagrams showing how the Markovian time [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. A TLS coupled to a semi-infinite waveguide with [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Tensor diagrams showing how the non-Markovian [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Tensor-network representation of the expectation [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Tensor-network representation of the two-time ex [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The population of a two-level system in a semi-infinite [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. A sketch of two emitters coupled to the same waveg [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The time evolution of the first of the two TLSs [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. (a) and (b): The population of the left emitter separated from the right emitter by delays [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The operator entanglement [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Sketch of a one or two-photon pulse scattering off a [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Sketch of the local tensor product of a ket-tensor [PITH_FULL_IMAGE:figures/full_fig_p013_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. (a) and (b): The population as a function of time for a top-hat input pulse of [PITH_FULL_IMAGE:figures/full_fig_p014_15.png] view at source ↗
read the original abstract

We present a matrix product state (MPS) method for including decoherence processes in calculations involving waveguide quantum electrodynamics (waveguide QED) using density matrices. The approach is based on collision quantum optics, where the many-body state of the waveguide is represented as discrete time bins, which are efficiently represented using an MPS chain. Our method is a generalization of previous MPS methods, and we demonstrate how one can efficiently expand to density matrices, allowing for the inclusion of various loss processes in the form of Lindblad terms in the Liouvillian superoperator responsible for the relevant dissipation dynamics. As an application of the theory, we study various waveguide QED systems and the influence of emitter pure dephasing (which is one of the most important processes in real systems) on the light-matter interactions, including a two-level system (TLS) in a semi-infinite waveguide with time-delayed feedback, two spatially separated TLSs with finite delays, and finally the scattering of few-photon Fock pulses on a TLS. In addition to emitter pure dephasing, we also show how to include off-chip radiative decay, and show how it differs qualitatively from pure dephasing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper presents a matrix product state (MPS) method for including decoherence in waveguide quantum electrodynamics calculations using density matrices. It generalizes collision quantum optics by discretizing the waveguide into time bins represented as an MPS chain and extends the formalism to vectorized density matrices to incorporate Lindblad superoperators for processes such as emitter pure dephasing and off-chip radiative decay. Applications are shown for a TLS with time-delayed feedback in a semi-infinite waveguide, two spatially separated TLSs with finite delays, and few-photon Fock pulse scattering on a TLS, with emphasis on qualitative differences between pure dephasing and radiative loss.

Significance. If the discretization and MPS representation remain accurate when extended to open systems, the work supplies a practical tensor-network tool for modeling realistic decoherence in waveguide QED, an area where such effects are experimentally relevant. Credit is given for the explicit generalization of prior MPS collision models to density matrices with Lindblad terms and for the demonstration that different loss mechanisms produce distinguishable signatures in the studied observables.

major comments (1)
  1. [Applications section] Applications section (TLS feedback, two-emitter, and few-photon scattering examples): No explicit convergence study with respect to time-bin width Δt is reported for the density-matrix simulations that include Lindblad operators. The collision-model discretization approximates continuous-time dynamics only in the Δt → 0 limit with fixed physical rates; without refinement data or error bounds on key observables (e.g., transmission, photon correlations, or steady-state populations), it is unclear whether the reported qualitative distinctions between pure dephasing and off-chip decay survive the continuum limit or arise from finite binning.
minor comments (2)
  1. [Method] The description of the vectorized MPS representation and the precise mapping of Lindblad terms onto the superoperator could be accompanied by an explicit equation or small schematic to improve reproducibility.
  2. [Introduction] A short comparison in the introduction to alternative approaches (quantum trajectories, direct integration of the master equation, or other tensor-network methods for open systems) would better situate the computational advantages claimed for the MPS collision model.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of verifying the continuum limit in our density-matrix simulations. We address the major comment below.

read point-by-point responses

  1. Referee: [Applications section] Applications section (TLS feedback, two-emitter, and few-photon scattering examples): No explicit convergence study with respect to time-bin width Δt is reported for the density-matrix simulations that include Lindblad operators. The collision-model discretization approximates continuous-time dynamics only in the Δt → 0 limit with fixed physical rates; without refinement data or error bounds on key observables (e.g., transmission, photon correlations, or steady-state populations), it is unclear whether the reported qualitative distinctions between pure dephasing and off-chip decay survive the continuum limit or arise from finite binning.

    Authors: We agree that an explicit demonstration of convergence with respect to the time-bin width Δt is necessary to rigorously establish that the reported qualitative differences between pure dephasing and off-chip radiative decay are not discretization artifacts. In the revised manuscript we will add a dedicated subsection (or appendix) presenting convergence data for the key observables in each application example. Specifically, we will show how transmission, second-order correlation functions, and steady-state populations approach their Δt → 0 values for successively smaller bin widths while keeping the physical rates fixed, thereby confirming that the distinctions survive the continuum limit. revision: yes

Circularity Check

0 steps flagged

No circularity: generalization of established collision-model MPS to density matrices with Lindblad terms

full rationale

The paper presents a generalization of prior MPS techniques from collision quantum optics to open-system density matrices via vectorization and Lindblad superoperators. The core construction (time-bin discretization of the waveguide, MPS representation of the chain, and addition of local dissipators) is taken from established literature and extended without re-deriving the discretization from the target observables or fitting parameters to the results being computed. No step equates a claimed prediction to a fitted input by construction, nor does any uniqueness theorem or ansatz reduce to a self-citation chain whose validity is presupposed. The derivation chain remains self-contained against external benchmarks of the underlying collision model and tensor-network methods.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on the standard collision-model discretization of the waveguide and the Lindblad form of the master equation; no new physical entities or fitted parameters are introduced in the abstract.

axioms (2)
  • domain assumption The continuous waveguide dynamics can be accurately discretized into a chain of discrete time bins that interact sequentially with the emitters.
    Invoked in the description of the collision quantum optics representation.
  • standard math Decoherence processes can be captured by Lindblad terms in the Liouvillian superoperator acting on the density matrix.
    Standard open-quantum-systems formalism assumed throughout.

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