RELEASE / COLLIDE / RETURN

A Quantum Newton’s Cradle

Release two prepared clouds of impenetrable one-dimensional bosons into a common trap. Scrub repeated collisions, distort the trap and sample correlated atom positions.

Zero-temperature Tonks–Girardeau limit, evolved through the time-dependent Bose–Fermi mapping. Prepared displaced orbitals are orthonormalized; they are not a solved double-well ground state. Density recurrence in a harmonic trap does not establish thermalization or its absence in a real gas.

A ONE-DIMENSIONAL QUANTUM GAS

RETURN OF THE DENSITY

0

EVOLUTION TIME

\tau=0

SCRUB / COMPARE / CAPTURE

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

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.

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 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: Two clouds meet. The rhythm comes back.
One-sentence intuition: Impenetrable bosons share the position probabilities of a Slater determinant. Their momentum distribution is different.

Core mathematical model

Evolution

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

Dimensionless oscillator units. Each occupied orbital follows the same single-particle Hamiltonian; the many-body position probability follows from their determinant.

What the display measures

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

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 modelZero-temperature Tonks–Girardeau limit, evolved through the time-dependent Bose–Fermi mapping. Prepared displaced orbitals are orthonormalized; they are not a solved double-well ground state. Density recurrence in a harmonic trap does not establish thermalization or its absence in a real gas.

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: A harmonic trap returns the density. A quartic distortion dephases the motion without adding friction.
03
Measure
Capture sixteen independently prepared clouds at the selected time, pin them and export their raw positions.
What to observe: Each sample contains exactly the selected atom number, with hard-core position correlations.