---
title: "A Photon Cloud That Cannot Escape Interactive Lab | Sandbox Physics"
description: "Put the atom in a gap. A cloud stays behind. Move an atomic transition through a waveguide band edge. Follow the emitted wave and the localized photon cloud, then compare the surviving atom with independently calculated bound states."
canonical_url: "https://sandboxphysics.com/labs/bandgap-bound-photon-cloud/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/bandgap-bound-photon-cloud/"
last_modified: "2026-10-20"
---

TUNE / LOCALIZE / RELEASE

# A Photon Cloud That Cannot Escape

Move an atomic transition through a waveguide band edge. Follow the emitted wave and the localized photon cloud, then compare the surviving atom with independently calculated bound states.

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

A single two-level atom coupled to one site of an ideal resonator array. The rotating-wave, single-excitation Hamiltonian contains no disorder, absorption or additional decay channels. Infinite-chain bound states are analytic references; a finite-time local cloud is not itself a measurement of the total bound-state weight.

AT THE EDGE OF PROPAGATION

EXCITATION STILL IN THE ATOM

$P_e=0$

TOTAL BOUND-STATE WEIGHT · INFINITE CHAIN

$0$

ATOM $0\%$

NEAR ATOM $0\%$

OUTSIDE REGION $0\%$

Schematic resonator array. Ribbon height: fixed square-root probability gain; atom glow: excited population. It is a diagnostic over the waveguide, not a mean field or particle path.

$Jt=0$

01 / WHERE THE EXCITATION GOES

### Scrub through the whole journey.

Full 401-site model, time increasing downward. Drag vertically or use arrows to select a time. Fixed square-root color scale saturates at probability 0.04 per site; white marks the current snapshot.

$x/a\ \longrightarrow\qquad Jt\ \downarrow$

02 / A BAND WITH TWO EDGES

### Drag the atom across the edge.

Cyan: propagation band. Gold: bare atom frequency; drag vertically or use arrows. Coral: the two dressed bound energies, including the weak branch. All are model references.

$ka/\pi\ \longrightarrow\qquad (\omega-\omega_c)/J$

### What stays in the atom?

Cyan: complete time evolution. Dashed coral: the bound-state contribution alone, including interference between both bound states. The continuum is absent from that reference, so it need not agree at early times.

ONE PREPARATION. ONE OUTCOME.

## Find the excitation.

0 / 10000

Each trial starts again in the excited atom, evolves to the selected time, then measures either the atom or one waveguide site. These are ideal independent destructive snapshots, not repeated monitoring of one photon. Fixed seed; every outcome retained.

Recorded atom fraction

$\text{—}$

95% Wilson interval

$\text{—}$

Largest norm error

$\text{—}$

Largest boundary probability

$\text{—}$

### Count the atom. Keep every position.

Gold: acquired frequencies. Cyan: integrated model bins, with a separate atom outcome. All 401 sites are included. Square-root vertical scale; tick labels remain probabilities.

Latest 20 records · export all

| Trial | Outcome | Site position |
| --- | --- | --- |

The model, units and limits

$$
\frac{H}{\hbar J}=\frac{\Delta}{J}|e\rangle\langle e|-\sum_x(|x+1\rangle\langle x|+\mathrm{h.c.})+\sum_j\frac{g_j}{J}(|e\rangle\langle x_j|+\mathrm{h.c.})
$$

Positive position is rightward along the array. Spacing sets length, nearest-neighbor hopping sets frequency, and its inverse sets time. Frequencies are relative to the array center, not negative absolute photon energies. One initial atomic excitation, rotating-wave approximation, vacuum waveguide. No drive after preparation, counter-rotating terms, loss, disorder or readout backaction before the final snapshot.

401 open sites, fourth-order Runge–Kutta at a time step of 0.01, duration 64. No state clipping or renormalization. The displayed norm and outer-eight-site probability audit numerical loss and boundary contamination. Sampling normalizes only the cumulative measurement probabilities by the displayed norm.

$$
E_\pm-\Delta=\frac{g^2\operatorname{sgn}(E_\pm)}{\sqrt{E_\pm^2-4J^2}},\qquad Z_\pm=\left[1+\frac{g^2|E_\pm|}{(E_\pm^2-4J^2)^{3/2}}\right]^{-1}
$$

Both infinite-chain bound states are retained. The total bound overlap is the sum of their atomic weights; the long-time averaged atomic contribution is the sum of their squares. A neighborhood of twelve sites on either side is a finite observation region, not an exact projector onto bound states.

PRIMARY SOURCES

- [Calajó et al. · Atom-field dressed states in slow-light waveguide QED](https://arxiv.org/html/1512.04946v2)
- [Sundaresan et al. · Interacting qubit-photon bound states](https://arxiv.org/html/1801.10167v1)

The model follows the single-excitation resonator-chain treatment; the cited circuit experiment has additional device geometry and two-qubit physics not simulated here.

Physics tutorial

## A Photon Cloud That Cannot Escape

**Background** Move an atomic transition through a waveguide band edge. Follow the emitted wave and the localized photon cloud, then compare the surviving atom with independently calculated bound states.

**Why it matters** The surroundings can remember an emitted excitation.

Start with the essentials

- **Focus question**: Put the atom in a gap. A cloud stays behind.
- **One-sentence intuition**: Outside the propagation band, atom and localized field form a dressed eigenstate.

### Core mathematical model

#### One excitation, one probability budget

$$
P_e(t)+\sum_x|\psi_x(t)|^2=1
$$

Atom and waveguide are mutually exclusive outcomes of the ideal snapshot readout.

#### A finite propagation band

$$
\omega(k)-\omega_c=-2J\cos(ka)
$$

The band edges are two hopping units above and below its center. The same dispersion governs the movie.

### Common difficulties

#### A cloud is not a trajectory

**Typical misconception** The excitation splits into many independently measurable particles.

**Better mental model** Each preparation has one total excitation. Repeated snapshots reveal its distribution; the ribbon height is a square-root probability display.

#### Do not overclaim the reference

**Typical misconception** The reference curve was reconstructed from saved samples.

**Better mental model** A single two-level atom coupled to one site of an ideal resonator array. The rotating-wave, single-excitation Hamiltonian contains no disorder, absorption or additional decay channels. Infinite-chain bound states are analytic references; a finite-time local cloud is not itself a measurement of the total bound-state weight.

### Run the experiment

1. 01

   #### Start with the movie

   Restart from the excited atom. Pause and drag the time cursor through the space–time map.

   **What to observe:** One part propagates away while a localized component stays near the atom.

2. 02

   #### Break the condition

   Move the atomic frequency into the propagation band and compare with the gap.

   **What to observe:** A band-edge effect is not simply weaker coupling. Bound eigenstates exist at both edges even when their weight is tiny.

3. 03

   #### Read independent preparations

   Select a time, collect 500 snapshots, pin the experiment and export all records.

   **What to observe:** Gold frequencies are acquired samples. Cyan and gray are separately labeled predictions.

Physical conclusion

A single two-level atom coupled to one site of an ideal resonator array. The rotating-wave, single-excitation Hamiltonian contains no disorder, absorption or additional decay channels. Infinite-chain bound states are analytic references; a finite-time local cloud is not itself a measurement of the total bound-state weight.

Quantum & fields · Starter path

## Start from a single measurement

Predict outcomes and collect counts, building from one qubit to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare it with the frequency of acquired 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)