---
title: "Cavity and waveguide quantum electrodynamics"
description: "Follow single excitations into two-photon scattering, finite-band outputs and collected photon modes through eight chapters and three experiments."
canonical_url: "https://sandboxphysics.com/stories/waveguide-qed/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/waveguide-qed/"
---

FR11

# Cavity and waveguide quantum electrodynamics

Sandbox Physics · 2026-10-28

One photon shows how an atom exchanges and releases energy. A second photon exposes saturation and correlations. Follow both into output ports, then ask which temporal mode a distant receiver can collect.

## Start with one excitation; keep following what leaves

Chapters 1–3 build amplitudes, open ports and environmental memory. Chapter 4 adds a second photon. Chapters 5–6 turn the output into a detector or receiver question. Chapters 7–8 read a fixed experiment and build reproducible benchmarks. Each chapter introduces its mathematical tools, develops a worked example and includes exercises with explanations.

Begin directly with Chapter 1. For a broader preparation in photon temporal modes, use FR10 as needed; it is a prerequisite route, not a required completed course. [FR10](https://github.com/yangzichao/sandboxphysics/issues/40) · [Common textbook plan](https://github.com/yangzichao/sandboxphysics/issues/27)

An excitation oscillates between atom and cavity

Excitation probability $P$

Scaled time $gt$

- Atom
- Cavity

Original analytic reference for a resonant, lossless single-mode model. The sum of both curves is one.

1. L1 · 1

   ## [How an excitation moves from an atom into light](https://sandboxphysics.com/stories/waveguide-qed/atom-cavity-exchange/)

   Build probability amplitudes from atom–cavity exchange, then see why a travelling photon needs a spatial wavepacket.

2. L2 · 2

   ## [How to keep track of light after it leaves](https://sandboxphysics.com/stories/waveguide-qed/input-output-markov/)

   Derive an environmental memory kernel, identify the Markov approximation and connect cavity amplitude to output photon flux.

3. L2 · 3

   ## [When an environment sends light back](https://sandboxphysics.com/stories/waveguide-qed/bandwidth-and-return/)

   Separate finite bandwidth, giant-atom propagation and boundary reflections, then explain a localized photon cloud near a band edge.

4. L3 · 4

   ## [Why two photons cannot always be scattered separately](https://sandboxphysics.com/stories/waveguide-qed/two-photon-saturation/)

   Build a two-excitation wavefunction and compare atomic saturation with a linear reference from the same one-photon dynamics.

5. L3 · 5

   ## [From a travelling field to a detector record](https://sandboxphysics.com/stories/waveguide-qed/output-records/)

   Separate port amplitudes, photon flux and detector response, and determine when a spatial pair map can become an arrival-time distribution.

6. L3 · 6

   ## [Receiving a photon also means receiving its mode](https://sandboxphysics.com/stories/waveguide-qed/collected-temporal-modes/)

   Calculate collection probability, conditional mode fidelity and unconditional target weight from a saved complex output envelope.

7. L4 · 7

   ## [Connect a temporal mode to a published experiment](https://sandboxphysics.com/stories/waveguide-qed/kurpiers-photon-interface/)

   Read a fixed Kurpiers photon-interface paper, reproduce its chosen ideal envelope and distinguish squared mean field from photon flux.

8. L4 · 8

   ## [Turn a scattering picture into a reproducible study](https://sandboxphysics.com/stories/waveguide-qed/reproducible-waveguide-study/)

   Use analytic limits, conservation and convergence checks, then independently benchmark weak-drive fluorescence and detector response.

## Three new experiments and four earlier starting points

- [Let two photons meet an atom and compare a linear reference](https://sandboxphysics.com/labs/two-photon-waveguide-scattering/)
- [Follow a finite-band bath into its collected output ports](https://sandboxphysics.com/labs/structured-bath-output-ports/)
- [Match an emitted photon to a receiver’s temporal mode](https://sandboxphysics.com/labs/quantum-node-photon-collection/)

The single-excitation route reuses experiments with explicit models: [cavity exchange](https://sandboxphysics.com/labs/cavity-quantum-exchange/) · [giant-atom return](https://sandboxphysics.com/labs/giant-atom-waveguide-echo/) · [band-gap cloud](https://sandboxphysics.com/labs/bandgap-bound-photon-cloud/) · [photon blockade](https://sandboxphysics.com/labs/single-atom-photon-blockade/).

The new pair map is an equal-time finite-waveguide state. The port experiment keeps finite-band propagation and open losses. The interface experiment evaluates a controlled source and fixed receiver mode. A separate Markov fluorescence benchmark provides a two-time output-correlation check. Their assumptions are stated where each result is used.

## Sources and calculation scope

1. [Jaynes and Cummings, Comparison of quantum and semiclassical radiation theories, Proc. IEEE 51, 89–109 (1963)](https://doi.org/10.1109/PROC.1963.1664)

   Section II.B, pp. 93–95, Eqs. (35)–(36): single-mode exchange with detuning, checked in the scanned author archive at bayes.wustl.edu. Chapter 1 independently derives its two-amplitude rotating-wave, single-excitation limit.

2. [Gardiner and Collett, Input and output in damped quantum systems, Phys. Rev. A 31, 3761 (1985)](https://doi.org/10.1103/PhysRevA.31.3761)

   Section II specifies rotating-wave coupling to a flat bath and derives input–output boundary relations. The course states its own port phase convention and treats the finite-band chain separately from a white-noise port.

3. [Sheremet et al., Waveguide quantum electrodynamics, Rev. Mod. Phys. 95, 015002 (2023)](https://doi.org/10.1103/RevModPhys.95.015002)

   The theory overview and Appendix A place single-excitation propagation and input–output relations in their model context. Our two-amplitude examples and schematic figures are original teaching calculations.

4. [Shen and Fan, Strongly correlated two-photon transport, Phys. Rev. Lett. 98, 153003 (2007)](https://arxiv.org/abs/quant-ph/0701170v2)

   Fixed author version v2, 13 March 2007. Used for the asymptotic two-photon scattering distinction and correlated component. The finite tight-binding wavepacket model has its own bandwidth and boundaries; it is not the paper’s infinite linear-dispersion S matrix.

5. [Nysteen et al., Scattering of two photons on a quantum emitter, New J. Phys. 17, 023030 (2015)](https://doi.org/10.1088/1367-2630/17/2/023030)

   Section 2 and the symmetric two-excitation wavefunction establish normalization and photon-density conventions. Author preprint arXiv:1409.1256v1 was also read. Published article is CC BY 3.0; no figure or dataset is copied here.

6. [Kurpiers et al., Deterministic quantum state transfer and remote entanglement, Nature 558, 264–267 (2018)](https://arxiv.org/abs/1712.08593v1)

   Frozen preprint v1, 22 December 2017, Figure 2c,f and Appendix C. We recalculate the ideal sech envelope and the vacuum–one-photon mean-field mapping at the declared 10.4 MHz scale. No experimental data or source artwork is redistributed; transfer efficiency and process tomography are outside the reproduced observable.

7. [Kimble, Dagenais and Mandel, Photon antibunching in resonance fluorescence, Phys. Rev. Lett. 39, 691 (1977)](https://doi.org/10.1103/PhysRevLett.39.691)

   Published 12 September 1977, Eq. (5) relates fluorescence correlations to conditional ground-state evolution. The complementary Markov benchmark uses that observable relation; it does not fit the historical counts or experimental apparatus.

8. [Sandbox Physics, single-excitation finite-band waveguide model and bound-state references](https://github.com/yangzichao/sandboxphysics/blob/main/docs/quantum/waveguide-memory.md)

   Nearest-neighbour chain dispersion, infinite-chain bound-state reference and finite-window definitions. These are model calculations with declared units, not measured photonic-crystal data.

9. [Sandbox Physics, existing giant-atom propagation model](https://github.com/yangzichao/sandboxphysics/tree/main/src/scripts/physics-lab/experiments/waveguide-memory/model)

   The existing model uses 401 sites, maximum time 64 and explicit propagation between two connections to the same emitter. Default separation is 32 and the compact-state preset uses 30. Boundary monitoring and analytic references are retained.

10. [Sandbox Physics FR11, finite two-excitation scattering model](https://github.com/yangzichao/sandboxphysics/blob/main/docs/frontier/fr11/two-photon-model.md)

    State basis, ordered pair-map normalization, harmonic comparison, convergence and independent single-mode checks. Exported spatial data remain separate from continuous detector records.

11. [Sandbox Physics FR11, finite-band ports and collected-mode model](https://github.com/yangzichao/sandboxphysics/blob/main/docs/frontier/fr11/ports-and-node-model.md)

    The finite bath keeps its field amplitudes and output losses. Its detector response is a declared synthetic filter. The separate controlled source retains complex output samples for receiver-mode projection.

12. [Sandbox Physics FR11, reproducible ideal temporal-mode benchmark](https://github.com/yangzichao/sandboxphysics/blob/main/scripts/frontier/fr11/reproduce-node-benchmark.mjs)

    Recalculates ideal sech normalization, finite-window survival, overlap and the quarter-versus-half mean-field distinction. Parameters and calculated reference values are saved with the source; no measured points enter the benchmark.

13. [Sandbox Physics FR11, independent side-fluorescence regression benchmark](https://github.com/yangzichao/sandboxphysics/blob/main/scripts/frontier/fr11/fluorescence-benchmark.mjs)

    Ground-reset optical-Bloch integration versus a finite-drive closed form, weak-drive limit and normalized Gaussian delay response. Deterministic samples and errors are retained in docs/frontier/fr11/fluorescence-benchmark.json.