ATTRACT / INTERFERE / SURVIVE

When Matter-Wave Solitons Collide

Send two localized matter waves into each other. Change their relative phase, remove the attraction and compare the entire collision in a time–position map.

Dimensionless one-dimensional focusing Gross–Pitaevskii equation. Two initially separated sech pulses are prepared for attraction two; changing attraction is a quench of the same initial shape. No transverse collapse, finite-temperature loss, many-body fragmentation or calibrated experimental imaging is simulated.

TWO COHERENT MATTER WAVES

DENSITY AT THE MEETING POINT

0

EVOLUTION TIME

\tau=0

SCRUB / COMPARE / CAPTURE

Every collision leaves a pattern.

Time runs downward

01.25

Drag the white line through time. Probability density uses a fixed color scale with square-root gain. The horizontal direction is position, not momentum.

One instant, in detail

Cyan: now. Dashed gray: initial density. Gold: pinned prediction, when available. All curves have unit integral.

RMS WIDTH

0

NORM DRIFT

0

ENERGY DRIFT

0

Max edge fraction

0

High-wave-number fraction

0

SIMULATED POSITION MEASUREMENTS

A cloud is a prediction. These dots are records.

Capture sixteen fresh preparations at the selected time. Every row is a separate destructive position measurement; moving the time cursor does not move old records. The position axis fits all recorded dots.

Each row samples sixty-four independent positions from the same product-state density. This idealized sampling omits depletion, imaging blur and atom loss.

The model, units and scientific sources

A scalar, one-dimensional focusing mean-field equation in an axially free guide. Initial pulses have unit width and unit norm each before their tiny overlap correction; the combined norm is exactly two. Only at attraction two is each isolated prepared pulse a fundamental soliton. Quenches may radiate or breathe. The pair is a separated-pulse approximation, not an exact two-soliton initial solution.

i\partial_\tau\psi=-\tfrac12\partial_\xi^2\psi-g|\psi|^2\psi

Position points right. All controls use dimensionless units. Choose a length unit; the mass and reduced Planck constant set the time unit. For the cradle, the length is the final harmonic oscillator length. No numerical control is a calibrated micrometre, millisecond or magnetic field.

\begin{aligned}\xi&=x/\ell\\\tau&=t\hbar/(m\ell^2)\\p&=\rho\big/\!\int\rho\,d\xi\end{aligned}

The solver uses 512 cradle or 1024 soliton periodic spatial samples and symmetric split-step Fourier evolution. The box is chosen larger than the visible clouds. Edge population, high-wave-number weight and conservation drift are checked over the whole run. Time frames are discrete; drawing between spatial points adds no physical resolution. Color and relief use a fixed square-root display gain.

The gold pinned curve and the recorded dots are separate objects. The raw CSV stores the preparation, random seed, selected time and every position index, followed by the corresponding model density. There is no reconstruction claim.

Start with the essentials

Focus question: A bright peak. A dark seam. The same encounter.
One-sentence intuition: Attraction can balance dispersion. Relative phase changes the density during the encounter, but a central dark seam does not label individual atom paths.

Core mathematical model

Evolution

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

Positive coupling means attraction. The field norm is two; it is a scaling convention, not two literal atoms.

What the display measures

p(\xi,\tau)=\frac{|\psi|^2}{\int |\psi|^2\,d\xi}

The profile and time map show probability per unit dimensionless length. The relief uses square-root height; it is a density drawing, not a transverse cloud shape.

Common difficulties

A movie is not a measurement record

Typical misconceptionThe dots trace the same atoms through the collision.

Better mental modelEach acquired row is a fresh preparation at one fixed time. Earlier records remain frozen when the time cursor moves.

The boundary of the model

Typical misconceptionThis reproduces a calibrated laboratory experiment.

Better mental modelDimensionless one-dimensional focusing Gross–Pitaevskii equation. Two initially separated sech pulses are prepared for attraction two; changing attraction is a quench of the same initial shape. No transverse collapse, finite-temperature loss, many-body fragmentation or calibrated experimental imaging is simulated.

Run the experiment

01
Prepare
Drag either gold handle on the preparation rail to change the starting separation.
What to observe: Both initial clouds move, and the entire evolution is recalculated.
02
Find the encounter
Press play or scrub the time map. Compare the presets at the same time.
What to observe: Opposite phases maintain a central node. Switching attraction off spreads the packets.
03
Measure
Capture sixteen independently prepared clouds at the selected time, pin them and export their raw positions.
What to observe: Each sample contains sixty-four independent product-state position detections. These are simulated measurements, not time trajectories.