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.

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 timePlayback

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) · Bao et al. (2003) · Neely et al. (2010)

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 misconceptionThe 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 modelThe 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

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.
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.
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.

Quantum Vortex Studio

From a wave to recorded positions.