---
title: "Neutral Kaons in Flight Interactive Lab | Sandbox Physics"
description: "How can a neutral kaon change flavour while its probability of survival falls? A neutral kaon evolves as a combination of short-lived and long-lived components. Their relative phase changes its flavour as they decay. A small CP-violating amplitude lets the long-lived component decay to two pions."
canonical_url: "https://sandboxphysics.com/labs/neutral-kaon-interferometer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/neutral-kaon-interferometer/"
last_modified: "2026-10-24"
---

P002 · Mixing, decay and CP violation

# Neutral Kaon Interferometer

**How can a neutral kaon change flavour while its probability of survival falls?** A neutral kaon evolves as a combination of short-lived and long-lived components. Their relative phase changes its flavour as they decay. A small CP-violating amplitude lets the long-lived component decay to two pions.

Move the observation gate along a neutral-kaon beam. Follow flavour mixing as particles decay, then prepare a long-lived beam and reveal its rare two-pion channel.

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

Drag the observation gate. The two-pion graph uses a logarithmic scale so the small CP-sensitive tail remains visible.

Calculated probabilities refer to the original beam. The two-pion rate is per short lifetime; its logarithmic axis reveals the small long-lived contribution.

Model and scientific sources

A coherent, monoenergetic vacuum beam with CPT-symmetric mixing. The reference values are from the 2024 PDG: short lifetime 89.54 picoseconds, long lifetime 51.16 nanoseconds, and indirect CP amplitude 0.002228 at 43.51 degrees. The two-pion amplitude ratio is approximated by this indirect CP amplitude; direct CP violation, regeneration in matter, wave-packet separation and detector backgrounds are omitted. The displayed two-pion rate includes the short-lived charged-pion branching fraction. The adjustable CP scale is a teaching comparison. Beam tracks show schematic decay paths, not a simulated detector sample.

$$
g_{\pm}(t)=\frac{e^{-i m_S t-\Gamma_S t/2}\pm e^{-i m_L t-\Gamma_L t/2}}{2}
$$

- [PDG 2024 · CP violation in the quark sector](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-cp-violation.pdf)
- [PDG 2024 · Short-lived neutral kaon](https://pdg.lbl.gov/2024/listings/rpp2024-list-K-zero-S.pdf)
- [PDG 2024 · Long-lived neutral kaon](https://pdg.lbl.gov/2024/listings/rpp2024-list-K-zero-L.pdf)

Physics tutorial

## Mixing takes place while the beam decays

**Background** A neutral kaon and its antiparticle share decay channels. Their superpositions include a short-lived state and a long-lived state.

**Why it matters** The two components accumulate different phases and disappear at very different rates. Their interference changes the flavour probabilities along the beam.

Start with the essentials

- **Focus question**: How do flavour mixing, particle loss and the rare two-pion channel fit together?
- **One-sentence intuition**: An unstable quantum superposition can change flavour while its total surviving probability decreases.

### Core mathematical model

#### Vacuum propagation of the two components

$$
g_\pm(t)=\frac{e^{-i m_S t-\Gamma_S t/2}\pm e^{-i m_L t-\Gamma_L t/2}}{2}
$$

The widths set the disappearance rates. The mass difference sets the relative phase; natural units are used in this expression.

#### Two-pion decay from an initially tagged kaon

$$
\frac{dP_{\pi\pi}}{d(t/\tau_S)}=\frac{B_{S\to\pi^+\pi^-}}{4|p|^2}\left|e_S(t)+\eta_{+-}e_L(t)\right|^2
$$

The short-lived and long-lived decay amplitudes can interfere. In this model the small two-pion amplitude ratio is approximated by the indirect CP-violation parameter.

### Common difficulties

#### Probability relative to the initial beam

**Typical misconception** The two flavour probabilities must always add to one.

**Better mental model** Their sum is the surviving fraction. The missing probability belongs to particles that have decayed.

#### Mixing and CP violation

**Typical misconception** Any flavour oscillation means CP symmetry is broken.

**Better mental model** Mixing also occurs in the CP-conserving comparison. The long-lived two-pion channel is sensitive to the small CP-violating amplitude.

### Run the experiment

1. 01

   #### Follow the beam

   Drag the observation gate through the beam.

   **What to observe:** The flavour balance changes, and the total surviving fraction falls.

2. 02

   #### Prepare the long-lived state

   Choose Rare long-lived two-pion decay.

   **What to observe:** A small nonzero two-pion rate remains after the short-lived contribution has disappeared.

3. 03

   #### Restore the symmetry

   Choose Restore CP symmetry.

   **What to observe:** The two-pion rate of the long-lived preparation becomes zero in this approximation.

Physical conclusion

Flavour change and decay are parts of the same propagation; the rare two-pion channel adds a separate test of CP symmetry.

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)