--- title: "Matter-Wave Lens Interactive Lab | Sandbox Physics" description: "One brief pulse. A very different future. Let an atom wavepacket expand. Apply a short quadratic phase pulse: collimate its motion, focus it back, or drive it past the focus. Watch the momentum distribution change before the spatial density can move." canonical_url: "https://sandboxphysics.com/labs/delta-kick-matter-wave-lens/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/delta-kick-matter-wave-lens/" last_modified: "2026-10-18" --- DELTA-KICK COLLIMATION / PHASE OPTICS # Matter-Wave Lens Let an atom wavepacket expand. Apply a short quadratic phase pulse: collimate its motion, focus it back, or drive it past the focus. Watch the momentum distribution change before the spatial density can move. > Interactive simulation: open this page in a web browser to change the parameters and run the model. An ideal isolated Gaussian single-particle state and an instantaneous harmonic impulse. No collisions, trap aberrations, finite pulse duration or thermal ensemble. Reduced momentum spread is not entropy removal; this does not predict a laboratory temperature. A SCULPTURE OF SPACE AND TIME POSITION WIDTH $0$ GOLD SLICE · SELECTED TIME $0$ Phase hue · arbitrary origin 0 $2\pi$ LEFT → RIGHT: POSITION BACK → FRONT: TIME HEIGHT: AMPLITUDE This is a one-dimensional wave, shown across its complete history. Depth is time, not another spatial direction. Fixed amplitude gain; the visible position window is −40 to 40. $\tau=8$ 01 / WHAT REALLY MOVES ### The pulse changes what happens next. Cyan: position width with the pulse. Dashed coral: same initial state, no pulse. Purple: pulse time. Drag the cursor to inspect. $\tau\ \longrightarrow\quad\sigma_x/\ell$ 02 / SPACE AND TIME ### Find the waist of the wave. Horizontal position −40 to 40; time 0 at top, 12 at bottom. Fixed square-root density color. Gold is the center; white is the selected time. 03 / POSITION ### A slice through the wave. Cyan: selected wave. Dashed coral: unshaped reference. Linear probability per bin, including overflow in the outer bins. $q=x/\ell$ 04 / MOMENTUM ### The pulse is already in the momentum. Cyan: selected state. Dashed coral: before the pulse. At the pulse, position density is unchanged; the momentum distribution can change. Vertical: momentum probability density. The displayed window is −4 to 4; moments use the full spectrum. $k=p\ell/\hbar$ FROM AMPLITUDES TO EVIDENCE ## One detection at a time. 0 / 8000 Every shot is a new independent preparation at the selected time. Position-grid Born sampling, all shots retained, fixed reproducible seed. The interval below covers shot noise only. Positions do not measure momentum or phase. Recorded fraction on the right $\text{—}$ 95% Wilson interval $\text{—}$ Model probability on the right $\text{—}$ Model momentum width $\text{—}$ ### Let the positions build the pattern. Gold: recorded fractions. Cyan: model probabilities. Position bins span −40 to 40; the first and last bins include every detection beyond the displayed window. Latest 20 records · export all trials | Trial | $q=x/\ell$ | Right half | | --- | --- | --- | The equation, the scale, the limits Position is measured in a freely chosen length unit; momentum and time follow from it. Positive position points right. No laboratory length, atomic mass or temperature is fitted. $$ q=x/\ell,\quad k=p\ell/\hbar,\quad \tau=\hbar t/(m\ell^2) $$ $$ i\partial_\tau\psi=-\tfrac12\partial_q^2\psi,\quad \widetilde\psi(k,\tau)=e^{-ik^2\tau/2}\widetilde\psi(k,0) $$ $$ \psi(q,0)=C e^{-q^2/(4\sigma_0^2)},\quad\psi_+=e^{-i\kappa q^2/2}\psi_- $$ $$ \kappa=r\,\frac{\tau_L/(4\sigma_0^2)}{\sigma_0^2+\tau_L^2/(4\sigma_0^2)} $$ The pulse shears momentum by an amount proportional to position. Strength 1 cancels the position–momentum covariance and minimizes momentum variance. At strength 2, the momentum distribution matches the pre-pulse distribution but position–momentum correlation reverses, allowing refocusing. The pulse preserves norm and purity; the narrowing is not dissipative cooling. The frame at the pulse time is immediately after the pulse. 4096 Fourier points on a periodic interval of length 1024, spacing 0.25. Free evolution is a unitary spectral phase rotation; the lens is a position-phase multiplication. Normalization is performed only at preparation. The visible 80-unit window is a crop; the model, moments and position records use the full grid. Surface height is seven times the square root of position density. Density color saturates at 0.5; phase color uses an arbitrary global hue origin. Neither display choice changes the wave or the records. $\Delta N=0$ $P_{\mathrm{edge}}=0$ $P_{\mathrm{view}}=1$ Edge probability is measured in the outer 10% of the full numerical interval. Discretization and truncation are numerical errors, distinct from the displayed shot-noise interval. No interactions, environmental noise, detector blur or external forces between pulses. [Ammann & Christensen · Delta Kick Cooling · Physical Review Letters (1997)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.78.2088) [Myrskog et al. · Magnetic-gradient delta-kick cooling · Physical Review A (2000)](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.61.053412) Physics tutorial ## Matter-Wave Lens **Background** Let an atom wavepacket expand. Apply a short quadratic phase pulse: collimate its motion, focus it back, or drive it past the focus. Watch the momentum distribution change before the spatial density can move. **Why it matters** Use a phase change to predict the future of a matter wave. Start with the essentials - **Focus question**: One brief pulse. A very different future. - **One-sentence intuition**: A quadratic phase impulse changes momentum before density moves. ### Core mathematical model #### Free evolution $$ \widetilde\psi(k,\tau)=e^{-ik^2\tau/2}\widetilde\psi(k,0) $$ Momentum probabilities remain fixed between pulses; relative phases evolve. #### A thin matter-wave lens $$ \psi_+(q)=e^{-i\kappa q^2/2}\psi_-(q) $$ An instantaneous phase change preserves position density and normalization. ### Common difficulties #### A wave history is not a trajectory **Typical misconception** The particle travels across the raised surface. **Better mental model** Depth encodes time, height encodes probability amplitude, and each horizontal slice is a one-dimensional wavefunction. #### Respect the boundary **Typical misconception** Narrower momentum means entropy has been removed. **Better mental model** An ideal isolated Gaussian single-particle state and an instantaneous harmonic impulse. No collisions, trap aberrations, finite pulse duration or thermal ensemble. Reduced momentum spread is not entropy removal; this does not predict a laboratory temperature. ### Run the experiment 1. 01 #### Start with a prediction Compare all three presets at the indicated times. **What to observe:** No pulse expands, collimation slows expansion, and a stronger pulse forms a waist. 2. 02 #### Locate the cause Scrub the timeline and compare position and momentum distributions. **What to observe:** At the pulse time, momentum changes without an instantaneous density jump. 3. 03 #### Acquire and compare Record 400 detections, repeat, pin the run and export the ledger. **What to observe:** The histogram is computed from these positions; increasing shots reduces sampling noise. Physical conclusion An ideal isolated Gaussian single-particle state and an instantaneous harmonic impulse. No collisions, trap aberrations, finite pulse duration or thermal ensemble. Reduced momentum spread is not entropy removal; this does not predict a laboratory temperature. 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)