---
title: "Quantum Kicks & Dynamical Localization Interactive Lab | Sandbox Physics"
description: "If you keep kicking a quantum rotor, can its momentum stop spreading? It can."
canonical_url: "https://sandboxphysics.com/labs/quantum-kicked-rotor/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/quantum-kicked-rotor/"
last_modified: "2026-10-27"
---

Quantum motion / Interference stops the spreading

# Quantum Kicks & Dynamical Localization

**If you keep kicking a quantum rotor, can its momentum stop spreading?** It can. Over many kicks the phases interfere destructively, so the quantum momentum spread stops growing, while a classical ensemble kicked the same way keeps diffusing. At special resonant kick intervals the phases line up instead, and the spread grows in proportion to the number of kicks.

Hit a rotor with a train of instantaneous cosine kicks and compare its quantum momentum distribution with a classical ensemble. Tune the interval between kicks from suppressed diffusion to resonance, or make every second kick undo the one before.

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

The model is an isolated delta-kicked rotor in one fixed quasimomentum sector, computed with 2048 Fourier modes over 80 kicks. A separate classical ensemble uses 8192 points. Pulse duration, thermal momentum spread, spontaneous emission and interactions are left out. Suppressed spreading over a finite time does not prove localization at long times.

QUANTUM CLASSICAL

KICK 0

TUNE INTERVAL

VIEW RANGE $-64$ $p/(2\hbar k_L)=m+\beta$ to $64$

Probability landscape · height is the square root of probability · both rows share one scale

Preparing the sequence…

Many independent readouts of one preparation

### How much momentum remains?

Collecting pauses the sequence. Each outcome comes from a fresh identical preparation.

Recorded outcomes $0$

Second moment · quantum $0$

Second moment · classical $0$

Second moment · records $\text{—}$

Probability, linear scale

$m+\beta$

Teal: quantum model. Amber: classical reference. White-centered dots: recorded frequency. Purple dashes: pinned quantum model.

Spread after every step

$\langle(m+\beta)^2\rangle\quad\text{vs.}\quad N$

The second moment is taken about zero, not about the mean, so it includes drift as well as width. The teal and amber curves are computed independently.

The whole sequence at a glance

Horizontal: position or momentum. Down: steps. Click to revisit a step. This is probability, not a trajectory.

Model, units and approximations

A zero integer-momentum initial state with a fixed quasimomentum offset. The cosine kick precedes free propagation. The classical comparator starts with uniform angle and the same momentum. Its histogram uses unit momentum bins; its moment uses unbinned momenta.

$$
q_{n+1}=q_n+\kappa\sin\theta_n,\quad\theta_{n+1}=\theta_n+\tau q_{n+1}\pmod{2\pi}
$$

The 3D view groups adjacent bins when needed. Both models share the same square-root height gain and a fixed horizontal range per preparation. The distribution below shows probabilities per native bin. Measurement outcomes sample the quantum model only; no phase reconstruction is claimed.

Primary research：[Ullah et al. (2012)](https://arxiv.org/abs/1202.3166)

Physics tutorial

## Quantum Kicks & Dynamical Localization

**Background** Hit a rotor with a train of instantaneous cosine kicks and compare its quantum momentum distribution with a classical ensemble. Tune the interval between kicks from suppressed diffusion to resonance, or make every second kick undo the one before.

**Why it matters** Start two rules of motion from the same initial populations and compare where they lead.

Start with the essentials

- **Focus question**: If you keep kicking a quantum rotor, can its momentum stop spreading?
- **One-sentence intuition**: Over many steps, coherent kicks can cancel one another by interference. At resonance their phases line up, and the momentum spreads ballistically.

### Core mathematical model

#### One complete step

$$
U=e^{-i\tau(\hat m+\beta)^2/2}e^{-i\kappa\cos\hat\theta}
$$

Each step is a kick followed by free motion. The integer momentum labels and the fixed offset are dimensionless, and the kick interval equals four pi times the displayed ratio.

#### A limit you can check exactly

$$
r=1,\ \beta=0:\quad P_m(N)=J_m^2(N\kappa),\quad\langle m^2\rangle=\frac{N^2\kappa^2}{2}
$$

At the primary resonance, the free-evolution operator is the identity. At half this interval with zero offset, two kicks bring back the initial momentum state.

#### What is sampled

$$
\widehat{M_2}=\frac1n\sum_{j=1}^{n}q_j^2
$$

Each final readout comes from a fresh preparation. The reported uncertainty is one standard error of the sample. These records cannot reconstruct the quantum phase.

### Common difficulties

#### A probability landscape

**Typical misconception** Each peak is an individual particle, or the trajectory one particle follows.

**Better mental model** The model is an isolated delta-kicked rotor in one fixed quasimomentum sector, computed with 2048 Fourier modes over 80 kicks. A separate classical ensemble uses 8192 points. Pulse duration, thermal momentum spread, spontaneous emission and interactions are left out. Suppressed spreading over a finite time does not prove localization at long times.

### Run the experiment

1. 01

   #### Watch quantum and classical part ways

   Play the sequence. Compare the teal quantum ridge with the amber classical ridge.

   **What to observe:** Both rows use the same horizontal scale and the same square-root height scale. For a quantitative comparison, use the linear plots below.

2. 02

   #### Change only the phase

   Choose resonance, then nudge the interval or the quasimomentum.

   **What to observe:** An exact return after two kicks needs zero quasimomentum and the half-period setting.

3. 03

   #### Record a distribution

   Collect 500 outcomes and pin the preparation. Change the experiment, restore the pinned sample and export its raw CSV.

   **What to observe:** The histogram and the second moment come only from recorded outcomes; the classical ensemble is a separate reference.

Physical conclusion

The model is an isolated delta-kicked rotor in one fixed quasimomentum sector, computed with 2048 Fourier modes over 80 kicks. A separate classical ensemble uses 8192 points. Pulse duration, thermal momentum spread, spontaneous emission and interactions are left out. Suppressed spreading over a finite time does not prove localization at long times.

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)