---
title: "Two-Photon Waveguide Scattering Interactive Lab | Sandbox Physics"
description: "What changes when a second photon reaches an atom before the first has left? A two-level atom can hold one excitation."
canonical_url: "https://sandboxphysics.com/labs/two-photon-waveguide-scattering/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/two-photon-waveguide-scattering/"
---

FR11 · Two photons meet one atom

# Two-Photon Waveguide Scattering

**What changes when a second photon reaches an atom before the first has left?** A two-level atom can hold one excitation. Two overlapping photons therefore scatter together differently from two independent photons. Their joint position probability shows the change even when the average density looks similar.

Send two wavepackets past a single atom. Watch their shared probability spread, and compare the scattered pair with two independently scattered photons.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

A finite nearest-neighbor waveguide and a two-level emitter evolve with exactly two total excitations. The reference uses a harmonic emitter with the same single-excitation response. Maps show equal-time field-position probabilities from the simulated quantum state. Open chain ends can reflect light; the boundary diagnostic reports population near them. This lattice has finite bandwidth and is not a calibrated optical device.

Simulated quantum state

## Where can the pair be together?

Joint field-position probability

Change caused by saturation

Both axes locate a field photon at the same simulation time. Cyan means more probability than the linear reference; coral means less. The difference map uses the joint map’s probability scale.

Normalization and finite-chain checks

$\langle\Psi|\Psi\rangle=1$

The maps integrate to the probability that both excitations are in the waveguide. Values are not renormalized after the atom becomes excited. Pair-probability difference is half the summed absolute difference before binning; it is a field-pair diagnostic. Boundary occupation is the fraction of excitations in the outer four sites at each end.

[Nysteen et al. (2015), two-photon wavefunction dynamics](https://doi.org/10.1088/1367-2630/17/2/023030) · [Continue the waveguide QED course](https://sandboxphysics.com/stories/waveguide-qed/)

Physics tutorial

## When two photons share an atom

**Background** An isolated photon can excite an atom and later leave as light. A two-level atom has only one excited state. A second photon arriving during that encounter sees a system whose response already depends on the first.

**Why it matters** The average photon density can hide this interaction. A joint map asks a sharper question: if both excitations are in the waveguide, where can the two photons be at the same time?

Start with the essentials

- **Focus question**: How does atomic saturation change the outgoing pair?
- **One-sentence intuition**: The linear reference inherits every one-photon reflection and delay. A difference in the pair map therefore reveals the additional effect of restricting the emitter to one excitation.

### Core mathematical model

#### Finite-band waveguide Hamiltonian

$$
\begin{aligned}H/J={}&-\sum_j(a_{j+1}^{\dagger}a_j+\mathrm{h.c.})\\&+(\Delta/J)\sigma^+\sigma^-\\&+(g/J)(a_c^{\dagger}\sigma^-+\sigma^+a_c)\end{aligned}
$$

A field quantum hops between adjacent sites. The emitter at the central site exchanges excitations with the field. Hopping energy fixes the time unit; the lattice spacing fixes distance.

#### Two total excitations

$$
\begin{aligned}|\Psi\rangle={}&\frac{1}{\sqrt2}\sum_{ij}\Phi_{ij}a_i^{\dagger}a_j^{\dagger}|g,0\rangle\\&+\sum_j e_j a_j^{\dagger}|e,0\rangle,\quad\Phi_{ij}=\Phi_{ji}\end{aligned}
$$

The first term has two field photons and a ground-state atom. The second has one field photon and an excited atom. The position map uses the squared magnitude of the symmetric field amplitude, summed into pairs of neighboring sites.

#### A common single-photon reference

$$
\Phi_{ij}^{\rm lin}=\frac{u_i v_j+v_i u_j}{\sqrt{2(1+|\langle u|v\rangle|^2)}}
$$

Each incoming mode evolves in the same one-excitation Hamiltonian. Their overlap stays in the normalization, including when the pulses coincide. The harmonic reference can also store two excitations in its emitter mode.

### Common difficulties

#### A density is an average

**Typical misconception** Two bright bumps trace two individual photon paths.

**Better mental model** The waveguide shows an expected photon number at each site. The joint map gives the probabilities for two positions in repeated measurements.

#### Position and detection time differ

**Typical misconception** The joint map is already a measured coincidence histogram.

**Better mental model** The map samples the simulated field at one time. A collected signal also depends on the selected output port, propagation and detector response.

### Run the experiment

1. 01

   #### Watch the shared encounter

   Use Arrive together and watch the atom brighten. Compare the filled density with the dashed linear reference, then inspect the difference map below.

   **What to observe:** A single-excitation response is insufficient to predict the two-photon probability. Cyan marks an increase relative to the linear reference; coral marks a decrease.

2. 02

   #### Separate the pulses

   Drag the gold pulse handle to the left, or choose Give the atom time. Replay the full encounter.

   **What to observe:** With more time between arrivals, the atom can release an excitation before the other pulse arrives. Finite pulse tails and the chosen coupling set how close the pair comes to independent scattering.

3. 03

   #### Keep the output state

   Pause after the pulses leave the center. Save the run and compare the joint map, norm error and boundary occupation.

   **What to observe:** Every displayed map and density comes from the same state. The saved arrays are equal-time probabilities; a detector timing histogram requires an additional detection model.

Physical conclusion

Compare a saturated emitter with its own linear one-photon response, and follow the whole outgoing pair.

## Related experiments and articles

Cavity and waveguide quantum light · Reading path

### From one excitation to photon pairs and collected modes

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why two photons cannot always be scattered separately**](https://sandboxphysics.com/stories/waveguide-qed/two-photon-saturation/)
2. [Article **Turn a scattering picture into a reproducible study**](https://sandboxphysics.com/stories/waveguide-qed/reproducible-waveguide-study/)
3. [Lab **Single-Atom Photon Blockade** Compare cavity output with atomic side fluorescence](https://sandboxphysics.com/labs/single-atom-photon-blockade/)
4. [Lab **Quantum Node Photon Collection** Improve emission and reception separately](https://sandboxphysics.com/labs/quantum-node-photon-collection/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)