---
title: "Quantum Vortex Studio Interactive Lab | Sandbox Physics"
description: "How can you tell that a hole in a condensate is a quantized vortex?"
canonical_url: "https://sandboxphysics.com/labs/quantized-vortex-dynamics/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/quantized-vortex-dynamics/"
last_modified: "2026-10-27"
---

Quantum fluid / Quantized circulation

# Quantum Vortex Studio

**How can you tell that a hole in a condensate is a quantized vortex?** Look at the phase as well as the density: around a quantized vortex, the phase winds through one full turn. That winding is why the circulation comes only in fixed units. A hole in the density by itself proves nothing.

Imprint two vortices into a trapped condensate. Compare vortices that circulate in opposite directions with ones that circulate the same way, move a repulsive obstacle, and read the local current arrows and the winding around each resolved density hole. Pause at any time to collect an image of atom positions.

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

Model assumptions

The model solves the two-dimensional Gross–Pitaevskii equation for a state that was imprinted on purpose and is not stationary. A vortex is marked only where the grid resolves a phase winding in a cell with enough density. The Lab makes no claim about equilibrium, spontaneous nucleation, lifetimes at finite temperature or turbulent scaling.

EVOLVING MODEL FIELD

Dark → bright: low → high density $\tau=0$

Scrub time Playback

Preparing the wave field…

Pause, then collect positions and compare

### From a wave to recorded positions.

Each fresh, identical preparation gives one ideal position. Sampling pauses the animation.

Peak model density $0$

Resolved vortex cores $2$

Low-density cores may be missed by the winding markers.

What is being solved?

Two imposed phase windings and depleted cores in a Gaussian cloud, confined by a fixed harmonic trap. This initial state is not an equilibrium solution.

Two-dimensional Gross–Pitaevskii mean field, a 64 by 64 Fourier grid on a periodic square of side 24, time step 0.015. No dissipation or real-time renormalization. Brightness uses a fixed square-root display gain; height remains proportional to density. Phase is hidden at very low density.

Integrated density $1$

Relative energy drift $0$

Recorded positions $0$

$$
i\partial_\tau\psi=\left[-\frac12\nabla^2+V+g|\psi|^2\right]\psi
$$

Primary sources and model limits：[Dalfovo et al. (1999)](https://arxiv.org/abs/cond-mat/9806038) · [Bao et al. (2003)](https://arxiv.org/abs/cond-mat/0303239) · [Neely et al. (2010)](https://arxiv.org/abs/0912.3773)

Physics tutorial

## Quantum Vortex Studio

**Background** Imprint two vortices into a trapped condensate. Compare vortices that circulate in opposite directions with ones that circulate the same way, move a repulsive obstacle, and read the local current arrows and the winding around each resolved density hole. Pause at any time to collect an image of atom positions.

**Why it matters** Here the phase of a matter wave becomes something you can see.

Start with the essentials

- **Focus question**: How can you tell that a hole in a condensate is a quantized vortex?
- **One-sentence intuition**: A vortex is a hole in the density with the phase winding around it. The hole alone cannot tell you the circulation.

### Core mathematical model

#### The Gross–Pitaevskii equation

$$
i\partial_\tau\psi=\left[-\frac12\nabla^2+V+g|\psi|^2\right]\psi
$$

Length is measured in a fixed unit, and the units of energy and time follow from the mass and Planck’s constant. The field is normalized so that its density integrates to one.

#### Units and density

$$
E_0=\frac{\hbar^2}{m\ell^2},\quad t_0=\frac{m\ell^2}{\hbar},\quad\int |\psi|^2\,d^2r=1
$$

The interaction strength already includes the effective number of atoms. No particular atomic species or trap calibration is assumed.

#### Counting the phase winding

$$
\oint\nabla\arg\psi\cdot d\mathbf l=2\pi q
$$

The phase is followed around each grid cell and the number of full turns is counted. Cells with too little density are skipped, and cores too small for the grid may be missed.

### Common difficulties

#### Limits of the simulation

**Typical misconception** The model solves the two-dimensional Gross–Pitaevskii equation for a state that was imprinted on purpose and is not stationary. A vortex is marked only where the grid resolves a phase winding in a cell with enough density. The Lab makes no claim about equilibrium, spontaneous nucleation, lifetimes at finite temperature or turbulent scaling.

**Better mental model** The real-time evolution keeps the norm on its own, without renormalizing along the way. Probability near the boundary, weight at high frequencies and drift in the energy are warning signs of numerical limits.

### Run the experiment

1. 01

   #### Change one thing

   Keep the separation fixed and use the Relative circulation menu to switch between opposite and same circulation.

   **What to observe:** The same two holes can carry different flows. Switch to the phase view to tell them apart.

2. 02

   #### Add an obstacle

   Raise the obstacle, drag its ring to a new spot, then release a fresh preparation.

   **What to observe:** The obstacle potential changes the computed field. Dragging it restarts the experiment; it does not stir the cloud while it moves.

3. 03

   #### Collect and compare

   Pause, collect positions and pin the result. Change one setting and compare at the same moment.

   **What to observe:** The counts fluctuate. The reference line comes from integrating the model density; it is not fitted to the points.

Physical conclusion

The model solves the two-dimensional Gross–Pitaevskii equation for a state that was imprinted on purpose and is not stationary. A vortex is marked only where the grid resolves a phase winding in a cell with enough density. The Lab makes no claim about equilibrium, spontaneous nucleation, lifetimes at finite temperature or turbulent scaling.

## Related experiments and articles

Condensed matter · Reading path

### Cold liquids and superconducting order

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

1. [Article **How Landau explained superfluid helium through its excitations**](https://sandboxphysics.com/stories/condensed-matter/landau-superfluid-helium/)
2. [Lab **BEC–BCS Crossover** Pairing across the crossover](https://sandboxphysics.com/labs/bec-bcs-crossover/)
3. [Lab **Quantum Gas Occupation** Quantum gas occupation](https://sandboxphysics.com/labs/quantum-gas-occupation/)

Condensed matter · Reading path

### Magnetism, disorder and topological response

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

1. [Article **How vortex pairs unbind and destroy two-dimensional superfluidity**](https://sandboxphysics.com/stories/condensed-matter/kt-vortex-unbinding/)
2. [Lab **Quantum Gas Occupation** Quantum gas occupation](https://sandboxphysics.com/labs/quantum-gas-occupation/)
3. [Lab **Superconductors in a Magnetic Field** Superconducting critical fields](https://sandboxphysics.com/labs/superconductor-critical-fields/)

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)