--- title: "A Quantum Newton’s Cradle Interactive Lab | Sandbox Physics" description: "Two clouds meet. The rhythm comes back. Release two prepared clouds of impenetrable one-dimensional bosons into a common trap. Scrub repeated collisions, distort the trap and sample correlated atom positions." canonical_url: "https://sandboxphysics.com/labs/quantum-newtons-cradle/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/quantum-newtons-cradle/" last_modified: "2026-10-20" --- 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. > Interactive simulation: open this page in a web browser to change the parameters and run the model. 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 $\tau=0$ Playback Height shows square-root probability density. Transverse width is illustrative. Cyan flow goes left; coral goes right; pale gold is locally still. No individual atom paths are drawn. SCRUB / COMPARE / CAPTURE ## Every collision leaves a pattern. ### Time runs downward 0 0.3 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. - [Yukalov & Girardeau (2005) · Time-dependent Bose–Fermi mapping](https://arxiv.org/html/cond-mat/0507409) - [Kinoshita, Wenger & Weiss (2006) · A quantum Newton’s cradle](https://www.nature.com/articles/nature04693) - [Phys. Rev. A 108, 013324 (2023) · Double-well release into a harmonic trap](https://doi.org/10.1103/PhysRevA.108.013324) Physics tutorial ## A Quantum Newton’s Cradle **Background** Release two prepared clouds of impenetrable one-dimensional bosons into a common trap. Scrub repeated collisions, distort the trap and sample correlated atom positions. **Why it matters** See a collision as a changing density, then ask what an actual position measurement would record. 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 misconception** The dots trace the same atoms through the collision. **Better mental model** Each 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 misconception** This reproduces a calibrated laboratory experiment. **Better mental model** 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. ### Run the experiment 1. 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. 2. 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. 3. 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. Physical conclusion 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. Quantum & fields · Starter path ## Start from a single measurement Predict outcomes and collect counts, building from one qubit to wave packets, bound states and two-photon interference. [Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare it with the frequency of acquired shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/) 1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/) 2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/) 3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/) 4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/) 5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/) 6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/) [View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)