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.

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 origin0

2\pi

LEFT → RIGHT: POSITIONBACK → FRONT: TIMEHEIGHT: 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)

Myrskog et al. · Magnetic-gradient delta-kick cooling · Physical Review A (2000)

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 misconceptionThe particle travels across the raised surface.

Better mental modelDepth encodes time, height encodes probability amplitude, and each horizontal slice is a one-dimensional wavefunction.

Respect the boundary

Typical misconceptionNarrower momentum means entropy has been removed.

Better mental modelAn 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

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.
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.
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.