---
title: "Bloch Oscillations Interactive Lab | Sandbox Physics"
description: "Why does a steady push make a particle in a lattice swing back and forth instead of speeding up?"
canonical_url: "https://sandboxphysics.com/labs/bloch-oscillation-dynamics/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/bloch-oscillation-dynamics/"
last_modified: "2026-10-27"
---

Q060 · Release, detect and compare

# Bloch Oscillations

**Why does a steady push make a particle in a lattice swing back and forth instead of speeding up?** Because in a periodic lattice, a steady push raises the quasimomentum until it wraps around the band, and there the velocity reverses. A wave packet keeps swinging and never speeds away. A state that starts on a single site breathes in and out instead, and disorder spoils the return.

Tilt a periodic lattice and drag through time to follow the motion. Compare a moving wave packet with a state that starts on one site and breathes in place, then add disorder to disturb the revival.

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

The model is a one-band tight-binding chain of 81 sites with open ends, holding one noninteracting particle. Position records are ideal simulated detections of the site the atom is on. The model has no tunneling between bands and no calibrated optical imaging, and it does not prove localization in an infinite system.

02 / Each click gives one position

## Build the distribution, click by click.

Model reference

Mean position · recorded

$\text{—}$

Width · recorded

$\text{—}$

Read / attempted · this time

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Blue bars: recorded frequencies. Amber: model probability. Thin lines: pointwise 95% Wilson intervals. Position is in lattice spacings; mean error is one empirical standard error. No data at this time means no estimate.

03 / Follow the whole evolution

## Returning is different from standing still.

Model probability · click or drag to choose a time

Time runs downward. Color uses a fixed logarithmic probability scale from one hundred-thousandth to one; dark cells are not proof of zero probability.

Mean position over time · model line and recorded dots

$\text{—}$

Revival fidelity compares the full state with the initial state. Position images alone cannot reconstruct it.

04 / The potential the atom moves through

## Change the slope.

$V_n/E_*$

Drag upward for a force to the right: the site energy falls to the right. A slope does not change the physical positions of the sites.

Quasimomentum · model only $ka/\pi$

An 81-point Fourier diagnostic over the first Brillouin zone, separate from the acquired site records. It is not a simulated time-of-flight image.

$\text{—}$

05 / Save a sample for comparison

## Ask the same question under different conditions.

Up to eight frozen samples. A new acquisition after saving starts another sample.

Scientific model & limits

One particle, 81 sites from −40 to 40, open ends and nearest-neighbor tunneling. All energies use a fixed reference energy; time uses the corresponding reduced-Planck time. The finite Hamiltonian is diagonalized completely and evolved unitarily. No state renormalization hides numerical error.

$$
H=-J\sum_n(|n\rangle\langle n+1|+\mathrm{h.c.})+\sum_n(\epsilon_n-Fan)|n\rangle\langle n|
$$

The initial state is a normalized discrete Gaussian with a phase gradient, or one occupied site at zero width. Disorder starts as independent uniform noise, optionally convolved with a Gaussian kernel normalized by its squared weights. It is not optical speckle or a quasiperiodic potential.

Single band throughout: no band excitation or Landau–Zener leakage is calculated. The Bloch period and clean-chain reference apply without disorder and before boundary effects. A finite-time plateau cannot establish infinite-system Anderson localization or many-body localization.

Every trial freshly prepares the state, evolves to the chosen time, and measures position once. State-independent missed detections remain in the CSV with no hidden site. The microscope is a geometric schematic with enlarged spacing; no fluorescence, resolution or optical calibration is simulated.

[Billy et al. · 2008](https://arxiv.org/abs/0804.1621) · [Korsch & Mossmann · 2003](https://arxiv.org/abs/quant-ph/0307203) · [Ben Dahan et al. · 1996](https://doi.org/10.1103/PhysRevLett.76.4508)

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Physics tutorial

## Why a lattice turns motion around

**Background** The atom evolves coherently on a finite chain with an explicit Hamiltonian, and each measurement finds it on a single site.

**Why it matters** From a single still picture you cannot tell confinement, a recurrence and a reflection from the boundary apart.

Start with the essentials

- **Focus question**: Can a constant force make a particle come back?
- **One-sentence intuition**: The periodic band makes the velocity change as the force drives the momentum; a state started on a single site breathes instead of moving its center.

### Core mathematical model

#### The finite chain

$$
H=-J\sum_n(|n\rangle\langle n+1|+\mathrm{h.c.})+\sum_n(\epsilon_n-Fan)|n\rangle\langle n|
$$

Sites run left to right, and a positive force lowers the site energies toward the right. Changing the tunneling leaves the reference energy fixed.

#### Initial state

$$
\psi_n(0)\propto\exp[-(na)^2/(4\sigma_0^2)+ik_0an]
$$

The discrete Gaussian is normalized on the finite chain. Zero width puts the atom on the central site instead. Every shot starts again from the same initial state.

#### Position statistics

$$
\bar x=a\sum_n n p_n,\quad \sigma_x^2=a^2\sum_n(n-\bar x/a)^2p_n
$$

Model probabilities and measured frequencies are kept apart. The measured mean uses only detected positions, and a standard error is given once at least two positions are available.

#### Motion in one band

$$
E(k)=-2J\cos(ka),\quad \hbar\dot k=F
$$

In a clean infinite chain, the velocity follows the slope of the periodic band. As the momentum continues across the zone, the velocity changes sign.

#### Bloch period

$$
T_B=\frac{2\pi\hbar}{|F|a}
$$

This is the analytic return time for a clean single band. With disorder, zero force or strong edge effects, it no longer predicts when the full model returns.

#### Return of the full state

$$
\mathcal F(t)=|\langle\psi(0)|\psi(t)\rangle|^2
$$

This fidelity comes only from the model, because position detections cannot recover the phases needed to measure it. A single-site state breathes even while its center stays put.

### Common difficulties

#### A finite run cannot show an infinite-system phase

**Typical misconception** A narrow packet proves the phase of an infinite system.

**Better mental model** The run lasts a finite time on a finite chain, and the result depends on the sample. The Lab infers no localization length, mobility edge or many-body phase.

#### Only one band is modeled

**Typical misconception** If a strong force works here, it works in a real optical lattice at any strength.

**Better mental model** This model has no second band. A real instrument may leak atoms into higher bands, and without a specified band gap no numerical force limit is stated.

### Run the experiment

1. 01

   #### Predict first

   Compare the first two presets before looking at the data.

   **What to observe:** Each preset includes 200 detections, all clearly marked as simulated.

2. 02

   #### Follow the motion in time

   Drag time or the probability map, then record 17 times.

   **What to observe:** Playback keeps the data and adds no trials. The scan prepares an independent atom at every time.

3. 03

   #### Test the explanation

   Save, change the landscape or initial width, scan again and export the raw records.

   **What to observe:** The saved sample keeps its settings. In the export, a missed detection has no site attached.

Physical conclusion

Keep three things apart: what the model computes, what the sampled detections show and what holds only for an infinite chain.

## Related experiments and articles

Condensed matter · Reading path

### Materials that control current and light

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

1. [Article **Turning graphene flakes into controllable electronic systems**](https://sandboxphysics.com/stories/condensed-matter/graphene-field-effect/)
2. [Lab **Chern Insulator · Follow the Edge** Chern insulator](https://sandboxphysics.com/labs/chern-insulator/)

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)