Two clouds collide and return

A Quantum Newton’s Cradle

Why do two colliding atom clouds keep returning to their starting shape in a harmonic trap but not in a distorted one? In a harmonic trap every atom swings with the same period, whatever its energy, so every half period the whole density returns to its starting shape. Hard-core repulsion does not spoil this: impenetrable bosons in one dimension have the same position pattern as noninteracting fermions. In a distorted trap different energies have different periods, and the rhythm smears out with no friction.

Release two clouds of one-dimensional bosons that cannot pass through each other into the same trap. Step through their repeated collisions, distort the trap and sample the atom positions, which come out correlated.

The gas is in the zero-temperature Tonks–Girardeau limit and evolves through the time-dependent Bose–Fermi mapping. The starting state is built from displaced orbitals made orthonormal; it is not a solved double-well ground state. The density returning in a harmonic trap does not show whether a real gas thermalizes or not.

One-dimensional quantum gas with infinitely strong repulsion

How much of the initial density returns

0

EVOLUTION TIME

\tau=0

Scrub through time, compare and capture positions

Every collision leaves a pattern.

Time runs downward

00.3

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.

Hard-core position correlations are sampled from the occupied-orbital determinant. The same grid site cannot be occupied twice in one sample.

The model, units and scientific sources

A finite zero-temperature gas of impenetrable bosons. Each side starts from displaced harmonic orbitals, followed by orthonormalization of the combined occupied subspace. At release, the whole gas evolves in one harmonic-plus-quartic trap. This preparation is chosen for clarity; it does not reproduce the Bragg-pulse sequence of the 2006 experiment.

\begin{gathered}i\partial_\tau\varphi_j=\left(-\tfrac12\partial_\xi^2+V\right)\varphi_j\\V=\tfrac12\xi^2+\lambda\xi^4\end{gathered}

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.

R_\rho(\tau)=\left[\int\sqrt{p(\xi,0)p(\xi,\tau)}\,d\xi\right]^2

The return score compares densities only. It is not many-body fidelity. Position probabilities can be mapped to free fermions; the bosonic momentum distribution cannot be read from the orbital FFTs. Anharmonic dephasing alone is not evidence of thermalization.

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: Why do two colliding atom clouds keep returning to their starting shape in a harmonic trap but not in a distorted one?
One-sentence intuition: Bosons that cannot pass through each other have the same position probabilities as a Slater determinant, but a different momentum distribution.

Core mathematical model

Evolution equation

i\partial_\tau\varphi_j=\left[-\frac12\partial_\xi^2+\frac{\xi^2}{2}+\lambda\xi^4\right]\varphi_j

Everything is in dimensionless oscillator units. Each occupied orbital evolves under the same single-particle Hamiltonian, and the many-body position probability comes from their determinant.

What the plots show

\rho(\xi,\tau)=\sum_{j=1}^{N}|\varphi_j|^2,\qquad p=\rho/N

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 gas is in the zero-temperature Tonks–Girardeau limit and evolves through the time-dependent Bose–Fermi mapping. The starting state is built from displaced orbitals made orthonormal; it is not a solved double-well ground state. The density returning in a harmonic trap does not show whether a real gas thermalizes or not.

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: In a harmonic trap the density keeps coming back. A quartic distortion makes the motion drift out of step, even though no friction is added.
03
Measure
Capture sixteen independently prepared clouds at the selected time, pin them and export their raw positions.
What to observe: Each sample has exactly the chosen number of atoms, and their positions show hard-core correlations.