---
title: "Structured Bath and Output Ports Interactive Lab | Sandbox Physics"
description: "How does the surrounding waveguide change the photon that reaches a port? The waveguide supports only a finite range of frequencies."
canonical_url: "https://sandboxphysics.com/labs/structured-bath-output-ports/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/structured-bath-output-ports/"
---

FR11 · A photon leaves a finite band

# Structured Bath and Output Ports

**How does the surrounding waveguide change the photon that reaches a port?** The waveguide supports only a finite range of frequencies. Moving an emitter toward its band edge changes how the excitation spreads, so the output pulse and the excitation left near the atom change together.

Release one atomic excitation into a finite waveguide. Follow the emitted field to two output ports, then move the atomic frequency toward the band edge and watch the emission slow down.

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

Model assumptions

The rotating-wave model contains one excitation, a nearest-neighbour chain and a central two-level emitter. Time is measured in inverse reference hopping; frequency uses the reference hopping. Chain sites have unit spacing. Endpoint ports and intrinsic emitter loss are Markov reservoirs. Endpoint mismatch can reflect the field; these finite-chain returns are not a prescribed time-delay feedback equation. The scene shows probability density at schematic scale. Exported complex output amplitudes and integrated losses come from the same evolution; no detector click sequence is sampled.

## The pulse that reaches an output port

Cyan: right-port flux. Violet: left-port flux. Amber: right-port intensity after a causal timing response. The response acts on intensity and is separate from a quantum-mode projection.

Output flux and finite timing response $|\xi_R(t)|^2,\ |\xi_L(t)|^2$

> Saved simulated output intensity.

Excitation in the atom, bath and output $P_e,\ P_{\rm bath},\ P_{\rm out}$

> Computed state probabilities or temporal amplitudes.

Right-port receiving mode

The receiver keeps the normalized complex right-port output of the default resonant bath, over the same recording window. Change the atomic frequency or bandwidth to compare collection with temporal-mode matching.

- **Right-port probability $P_R$**: $0.439$
- **Conditional mode fidelity $F_R$**: $1.00$
- **Target-mode probability $W_R=P_RF_R$**: $0.439$

The overlap uses saved complex amplitudes before the detector timing response. Loss and a mismatch to this fixed mode contribute separately to the target-mode probability.

Read the saved record

Output channels are Markov continua attached to the finite chain. The detector response time is 0.6 in reference time units. It broadens a probability-flux record; no stochastic counting data are generated.

Physics tutorial

## The environment shapes the emitted photon

**Background** An atom first transfers its excitation to nearby waveguide sites. Hopping carries that field toward the ports. The chain supports a finite band of frequencies.

**Why it matters** The atom can decay differently even when its coupling is unchanged, because the surrounding modes change with its frequency.

Start with the essentials

- **Focus question**: How does the surrounding waveguide change the photon that reaches a port?
- **One-sentence intuition**: Atomic population and emitted output must be followed together through the finite observation window.

### Core mathematical model

#### A finite propagation band

$$
H/J_{\rm ref}=-j\sum_n(a_{n+1}^\dagger a_n+\mathrm{h.c.})+\Delta\sigma^+\sigma^-+g(a_m^\dagger\sigma^-+\mathrm{h.c.}),\quad\omega(k)=-2j\cos k
$$

The emitter couples at the central site. The band extends over four times the hopping frequency; its group velocity vanishes at either edge.

#### Save what leaves the system

$$
\xi_L=\sqrt{\kappa_{\rm p}}c_1,\quad\xi_R=\sqrt{\kappa_{\rm p}}c_N,\qquad P_e+\sum_n|c_n|^2+P_L+P_R+P_{\rm loss}=1
$$

The port amplitudes, populations and accumulated losses all follow from the same single-excitation state.

### Common difficulties

#### A delayed peak has a cause

**Typical misconception** Every revival proves intrinsic delayed feedback.

**Better mental model** An endpoint can reflect the field back through the chain. Compare port coupling and travel time before attributing a return to the emitter’s environment.

### Run the experiment

1. 01

   #### Move toward the band edge

   Compare Emit into the band and Linger near the edge. Predict whether the atom empties sooner.

   **What to observe:** Near the edge, the output waveform and the remaining population change together.

2. 02

   #### Identify an endpoint return

   Choose An imperfect output port and follow the field back along the chain.

   **What to observe:** The endpoint sends part of the field back. This return is tied to the finite chain geometry.

3. 03

   #### Move the atom closer

   Drag the glowing atom toward the waveguide.

   **What to observe:** The coupling and emission history update continuously. The vertical spacing is a schematic control for coupling.

Physical conclusion

A finite band changes the emission history. A late return from an imperfect endpoint is a boundary reflection with a defined travel time.

## 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 **How to keep track of light after it leaves**](https://sandboxphysics.com/stories/waveguide-qed/input-output-markov/)
2. [Article **From a travelling field to a detector record**](https://sandboxphysics.com/stories/waveguide-qed/output-records/)
3. [Article **Receiving a photon also means receiving its mode**](https://sandboxphysics.com/stories/waveguide-qed/collected-temporal-modes/)
4. [Lab **Quantum Node Photon Collection** Separate loss from mode mismatch](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)