---
title: "Turn a scattering picture into a reproducible study"
description: "Use analytic limits, conservation and convergence checks, then independently benchmark weak-drive fluorescence and detector response."
canonical_url: "https://sandboxphysics.com/stories/waveguide-qed/reproducible-waveguide-study/"
language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/waveguide-qed/reproducible-waveguide-study/"
last_modified: "2026-10-28"
---

[Cavity and waveguide quantum electrodynamics](https://sandboxphysics.com/stories/waveguide-qed/)

L4 · Chapter 8

# Turn a scattering picture into a reproducible study

Sandbox Physics · 2026-10-28

A correlation image becomes useful evidence when changes of step size, boundary or receiver leave a result that can still be explained. Start with a prediction that could fail, then distinguish physical changes from numerical error. A small few-photon model can support that entire process.

## Before you begin

Chapters 1–7, numerical step size, total probability and mode integration. Convergence, conditional evolution and quantum regression are introduced through a concrete calculation. Large baths, long delays and more photons need additional methods.

[Review the preceding chapter](https://sandboxphysics.com/stories/waveguide-qed/kurpiers-photon-interface/)

- Design independent model checks and distinguish size effects from discretization error.
- Derive a weak-drive side-fluorescence correlation from the post-detection conditional state.

## Begin with a prediction that can fail

Ask whether widely separated photons approach independent scattering, or whether a broader bath improves exponential decay. State an observable: the field-pair branch probability, integrated absolute linear difference, or target-mode weight. The question then has a numerical comparison that could contradict it.

Save model version, initial state, units, parameters, integration step, sampling interval and end time. Deterministic Schrödinger evolution has no random seed. A later click or noise sampler needs its algorithm and seed recorded separately.

Reading: Sandbox Physics FR11 · Sandbox Physics FR11

## Conservation passes; which limits remain?

A wrong Hermitian Hamiltonian can still conserve norm. Decoupling the emitter must recover known chain propagation. Keeping one resonator permits direct comparison with a two-state analytic coupling matrix. Single- and two-excitation normalizations require separate checks.

For a two-level atom and one resonator, the states $|g,2\rangle$ and $|e,1\rangle$ couple with $\sqrt2g$. Starting in the former gives atomic excitation probability $\sin^2(\sqrt2gt)$. This follows from creation and annihilation operators on number states, independently of the chain solver’s loops.

Halve time step and compare observables at equal physical times. Lengthen the chain while keeping the pulse width, initial distance from the emitter and physical parameters fixed. Changing the initial pulse too would confuse a boundary effect with a new preparation.

$$
a|2\rangle=\sqrt2|1\rangle,\qquad P_e(t)=\sin^2(\sqrt2gt)
$$

This resonant, lossless single-mode limit is a check on the two-excitation machinery, distinct from full waveguide scattering.

Reading: Sandbox Physics FR11

## Build an independent output-correlation reference

Choose a complementary problem with an analytic answer: stationary side fluorescence from a resonantly driven two-level atom. The detection port contains emitter radiation and excludes the directly transmitted coherent drive. A detection resets the atom to its ground state, so it must be excited again before emitting.

The density matrix $\rho$ stores ground- and excited-state probabilities on its diagonal and their coherence off the diagonal. The operator $\sigma_-=|g\rangle\langle e|$ lowers the atom; $\sigma_x=|e\rangle\langle g|+|g\rangle\langle e|$ exchanges the levels. Brackets mean $[A,B]=AB-BA$ and braces mean $\{A,B\}=AB+BA$.

Set $\hbar=1$, drive Hamiltonian $H=\Omega\sigma_x/2$ and population decay rate $\Gamma$. The dissipator is $\mathcal D[\sigma_-]\rho=\sigma_-\rho\sigma_+-\{\sigma_+\sigma_-,\rho\}/2$. Evolve from the ground state after detection and divide its excitation probability by steady-state excitation.

The weak-drive result is $g^{(2)}(\tau)=(1-e^{-\Gamma|\tau|/2})^2$: zero at zero delay and one at long delay. A finite-drive closed form separately identifies errors caused by a drive that is small but not zero.

$$
\dot\rho=-i[H,\rho]+\Gamma\mathcal D[\sigma_-]\rho,\qquad g^{(2)}(\tau)=\frac{P_e(\tau\mid g)}{P_e^{\rm ss}}
$$

This conditional evolution and regression relation belongs to a Markov master equation. It is not applied directly to the finite-memory lattice.

$$
P_e^{\rm ss}=\frac{\Omega^2}{\Gamma^2+2\Omega^2},\qquad g^{(2)}(\tau)\underset{\Omega/\Gamma\to0}{\longrightarrow}(1-e^{-\Gamma|\tau|/2})^2
$$

The numerical route integrates coupled population and coherence equations before comparing with the closed form and limit.

Reading: Kimble · Sandbox Physics FR11

## Derive the weak-drive answer

Immediately after detection the atom is in its ground state. At weak drive, its excited amplitude is of order $\Omega/\Gamma$, so ground amplitude can be approximated by one. Drive builds the excited amplitude while it decays at rate $\Gamma/2$. A first-order equation gives its recovery.

Square that solution to obtain conditional excitation. Steady excitation in the same approximation is $(\Omega/\Gamma)^2$. Dividing cancels drive strength and leaves the delay dependence. Stronger drive requires the full equations with ground-state depletion and coherence.

$$
\dot c_e=-\frac\Gamma2c_e-\frac{i\Omega}{2},\quad c_e(0)=0,\quad c_e(\tau)=-\frac{i\Omega}{\Gamma}(1-e^{-\Gamma\tau/2}),\quad P_e(\tau\mid g)\simeq\frac{\Omega^2}{\Gamma^2}(1-e^{-\Gamma\tau/2})^2
$$

This derivation uses nonnegative delay. The stationary correlation extends symmetrically with absolute delay.

## Add timing uncertainty to the same correlation

The benchmark uses $\Gamma=1$ and $\Omega=0.02$. Over delays zero to eighteen, regression and the finite-drive closed form differ by less than $5\times10^{-13}$. Their maximum difference from the strict weak-drive limit is about $3.68\times10^{-4}$. This difference mainly records the finite-drive correction.

Convolve with a normalized Gaussian delay response. Its standard deviation $\sigma$ already combines both detectors’ timing errors. Values $\sigma=0.3$ and $\sigma=1$ raise the zero-delay reading to approximately $0.0179$ and $0.1248$. Finite resolution mixes nearby nonzero correlations into the centre without changing the original field.

These are model references. They do not fit the 1977 experiment’s counts, background or atomic transit, and do not convert the non-Markov spatial map into detector times. They check the relation among an independent master equation, analytic limits and response processing.

Reading: Sandbox Physics FR11

## Choose the next physical question

For collected states, let a receiver follow a returning pulse’s phase and compare fixed and adjustable target-mode weights. For two-photon scattering, vary bandwidth and arrival separation while keeping the same single-photon reference. Add one clearly defined physical degree of freedom at a time.

Long delays, large baths and more photons expand the state space rapidly. A next method can represent successive temporal output modes as a matrix product state, retaining restricted entanglement. Analytic small-system limits and saved records remain the first results that a larger implementation should recover.

## Try the prediction

### [Write a prediction that the calculation could reject](https://sandboxphysics.com/labs/two-photon-waveguide-scattering/)

1. Choose pulse separation or detuning as the sole variable and predict the linear difference.
2. Save two records and compare branch weights at the same end time, checking that boundaries do not dominate.
3. Reproduce the repository step-size and chain-length benchmarks. Report an unresolved effect if its size is comparable to numerical error.

### [Compare cavity output with atomic side fluorescence](https://sandboxphysics.com/labs/single-atom-photon-blockade/)

1. Choose “One photon at a time”, reduce coherent drive and inspect the predicted conditional output correlation. Interpret intracavity number snapshots separately.
2. Choose “Remove the atom” and compare the linear empty cavity with the atom–cavity ladder, then restore the original preset.
3. Explain why cavity output uses a cavity-field jump while this chapter’s side fluorescence uses atomic lowering. They are distinct collection geometries; independently prepared cavity snapshots are not continuous clicks.

### [Improve emission and reception separately](https://sandboxphysics.com/labs/quantum-node-photon-collection/)

1. Keep one emission record fixed and propose a receiver change that increases target weight.
2. Compare conditional fidelity and unconditional target weight, identifying the source of the gain.
3. Change one emission parameter and test whether the receiver remains appropriate.

## Exercises and explanations

1. Does excellent norm conservation establish correct two-photon scattering?

   Read the explanation

   No. A wrong Hermitian model can conserve norm. Add single-mode analytic, linear-reference, single-excitation, step-size and boundary checks. They address different failure modes.

2. Why can timing blur raise the central correlation without instantaneous double emission?

   Read the explanation

   The measured delay averages nearby true delays, including nonzero ones. This fills the observed dip while the original field can retain zero normally ordered coincidence at zero delay.

3. A longer chain changes the result. How do you isolate a finite-size effect from a changed initial state?

   Read the explanation

   Keep pulse width, distance from the emitter, relative coordinates and end time fixed, extending only the remote ends. Compare reflection timing and boundary weight.

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

## Sources and calculation scope

1. [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.

2. [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.

3. [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.

4. [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.