Quantum fluid / Interference

Matter-Wave Collision

What happens when two clouds of condensed atoms overlap? Two coherent condensates overlap as waves, and the atom density shows interference fringes. Shift their relative phase and the fringes slide sideways. Repulsion between the atoms changes how the clouds evolve and reshapes the pattern.

Send two coherent condensate clouds toward each other. Shift their phase to move the interference fringes, or release a single cloud and compare its expansion with its momentum distribution. Pause to build a position image from independent ideal samples.

Model assumptions

This is an effective two-dimensional mean-field model of a dilute gas with little depletion, starting from a coherent state chosen in advance. The height shows density; it is not a physical surface. The calculation runs in a finite periodic box and leaves out the thermal gas, imaging calibration and many-body correlations.

EVOLVING MODEL FIELD

Dark → bright: low → high density

\tau=0

Scrub timePlayback

Preparing the wave field…

Momentum from the wavefunction

The cloud spreads. What happens to momentum?

Teal: current statePurple dashes: preparation

Horizontal momentum $p_x/(\hbar/\ell)=k_x\ell$

When does a position image reveal momentum?

After a long free flight, atoms with different velocities separate. Their final positions then approach a scaled image of the release momentum distribution. At a finite flight time, the initial cloud size and its phase still affect the position image.

\mathbf p\simeq\frac{m\mathbf r}{t}

The momentum curves above are calculated directly from the full complex wavefunction on the periodic grid. They are model probabilities. Repulsion during expansion can change them; the sampled position counts do not determine these curves.

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

What is being solved?

The interference presets release two coherent Gaussian packets with opposite momenta and density width 1.2. The flight preset releases one Gaussian at rest with density width 0.8. The initial phase is prescribed; this does not simulate condensation from a thermal gas.

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) · Andrews et al. (1997)

Start with the essentials

Focus question: What happens when two clouds of condensed atoms overlap?
One-sentence intuition: Shifting the phase moves the dark fringes. Turning on repulsion changes how the field evolves.

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.

The interference term

|\psi_1+\psi_2|^2=|\psi_1|^2+|\psi_2|^2+2\operatorname{Re}(\psi_1^*\psi_2)

For coherent fields without interactions, adding them is exact. With repulsion, the combined field has to be evolved as one.

Common difficulties

Limits of the simulation

Typical misconceptionThis is an effective two-dimensional mean-field model of a dilute gas with little depletion, starting from a coherent state chosen in advance. The height shows density; it is not a physical surface. The calculation runs in a finite periodic box and leaves out the thermal gas, imaging calibration and many-body correlations.

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
Read the motion in momentum space
Choose Expansion reveals momentum and scrub the time. Compare the gold flight curve with the purple release curve.
What to observe: The spatial cloud expands, but its Fourier momentum probabilities stay fixed. The finite-flight position conversion approaches them gradually; the initial cloud size still matters.
02
Change one thing
Start with no interaction, then flip the relative phase.
What to observe: The central bright ridge becomes a dark seam.
03
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.
04
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.

Matter-Wave Collision

The cloud spreads. What happens to momentum?

From a wave to recorded positions.