QUANTUM FLUID / QUANTIZED CIRCULATION

Quantum Vortex Studio

Imprint two vortex cores into a trapped condensate. Compare opposite and equal circulation, drag a repulsive obstacle, and follow the density holes together with their phase winding. Pause anywhere and collect a position image.

Two-dimensional Gross–Pitaevskii dynamics of a deliberately imprinted, nonstationary state. Markers come from resolved phase windings in sufficiently dense cells. No claim of equilibrium, spontaneous nucleation, finite-temperature lifetime or turbulent scaling.

EVOLVING MODEL FIELD

Dark → bright: low → high density

\tau=0

Scrub timePlayback

Preparing the wave field…

PAUSE. COLLECT. COMPARE.

From a wave to recorded positions.

One ideal position from each fresh, identical preparation. Sampling pauses the film.

Recorded positions

0

Peak model density

0

Integrated density

1

Relative energy drift

0

RECORDED / POSITION COUNTS

No positions acquired. The model field is shown above.

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.

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

Primary sources and model limits：Dalfovo et al. (1999) · Bao et al. (2003) · Neely et al. (2010)

Start with the essentials

Focus question: A hole in density. A full turn of phase.
One-sentence intuition: A vortex combines a depleted core with a phase winding. A density hole alone cannot establish its circulation.

Core mathematical model

The evolving field

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

Length is measured in a fixed unit; energy and time follow from mass and Planck’s constant. The field has unit integrated density.

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 includes the effective atom-number factor. No specific atomic species or trap calibration is assumed.

Count phase turns

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

An integer winding is counted around a resolved cell. Low-density cells are excluded; unresolved cores may be missed.

Common difficulties

Simulation boundaries

Typical misconceptionTwo-dimensional Gross–Pitaevskii dynamics of a deliberately imprinted, nonstationary state. Markers come from resolved phase windings in sufficiently dense cells. No claim of equilibrium, spontaneous nucleation, finite-temperature lifetime or turbulent scaling.

Better mental modelReal-time evolution preserves norm without renormalization. Boundary occupation, high-frequency weight and energy drift warn about numerical limitations.

Run the experiment

01
Watch one controlled contrast
Switch between opposite and same circulation; keep separation unchanged.
What to observe: The same two holes can carry different flows. Compare the phase view.
02
Introduce an obstacle
Raise the obstacle and drag its ring before releasing a new preparation.
What to observe: The potential changes the computed field; dragging restarts the experiment rather than stirring it mid-flight.
03
Collect and compare
Pause, collect positions, and pin the result. Change a setting and compare at the same time.
What to observe: Counts fluctuate. The reference line is the integrated model density, not a fit to the points.