Solitons that interfere and survive the collision

When Matter-Wave Solitons Collide

What decides whether two colliding matter-wave solitons make a bright peak or a dark gap where they meet? Their relative phase. In phase, the two waves add where they overlap and build a tall central peak; in opposite phase they cancel there, and a dark gap stays open for the whole encounter. Attraction keeps each packet from spreading, so both come out still compact; without it they simply disperse. The dark gap is interference, not proof that the atoms bounced back.

Send two localized matter waves into each other. Change their relative phase, switch off the attraction and compare whole collisions on a map of position against time.

The model is the dimensionless one-dimensional focusing Gross–Pitaevskii equation. The two separated sech pulses are shaped for an attraction of two, so changing the attraction quenches that same starting shape. Transverse collapse, finite-temperature loss, many-body fragmentation and calibrated experimental imaging are not simulated.

Two attracting coherent matter waves in a waveguide

Density where the clouds meet

0

EVOLUTION TIME

\tau=0

Scrub through time, compare and capture positions

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

The cloud is the prediction; each dot is one sampled position.

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. The raw CSV stores the preparation, random seed, selected time and every position index, followed by the matching model density. Nothing is reconstructed from the dots.

Start with the essentials

Focus question: What decides whether two colliding matter-wave solitons make a bright peak or a dark gap where they meet?
One-sentence intuition: Attraction can balance dispersion. The relative phase changes the density while the waves overlap, but a dark seam in the middle does not tell you which way individual atoms went.

Core mathematical model

Evolution equation

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

A positive coupling means attraction. The field is normalized to two; that is a choice of scale and does not mean there are two atoms.

What the plots show

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

The profile and the time map show probability per unit of dimensionless length. The relief plots the square root of the density as height. It is a picture of the density, not the transverse shape of the cloud.

Common difficulties

An animation is not a measurement

Typical misconceptionThe dots trace the same atoms through the collision.

Better mental modelEach sample comes from a fresh preparation measured at one fixed time. Samples taken earlier stay unchanged when you move the time cursor.

What the model leaves out

Typical misconceptionThis reproduces a calibrated laboratory experiment.

Better mental modelThe model is the dimensionless one-dimensional focusing Gross–Pitaevskii equation. The two separated sech pulses are shaped for an attraction of two, so changing the attraction quenches that same starting shape. Transverse collapse, finite-temperature loss, many-body fragmentation and calibrated experimental imaging are not 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 collision
Press play or scrub the time map. Compare the presets at the same time.
What to observe: With opposite phases, a node stays at the center. Switching off the attraction lets the packets spread.
03
Measure
Capture sixteen independently prepared clouds at the selected time, pin them and export their raw positions.
What to observe: Each sample holds sixty-four independent position detections from a product state. They are simulated measurements, not trajectories followed in time.