Delta-kick collimation / Phase optics

Matter-Wave Lens

How can one short pulse slow or refocus a spreading matter wave? The pulse kicks each part of the packet toward the center, harder the farther out it is. After free expansion, the outer parts are moving outward fastest. At the right strength the kick cancels that outward motion and leaves only a small momentum spread, so the packet widens far more slowly. At double strength it converges, regains its original width at twice the pulse time, then spreads again.

Let an atom's wave packet spread out, then apply a short pulse that imprints a quadratic phase. The pulse can stop the spreading, focus the packet back down or drive it through a focus. The momentum distribution changes first, before the density in space has time to move.

The model uses an ideal, isolated Gaussian state of one particle and an instantaneous harmonic kick. Collisions, trap aberrations, the finite pulse duration and thermal ensembles are left out. A narrower momentum spread does not mean entropy was removed, and the Lab does not predict a laboratory temperature.

The wave’s history in 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 shows up in the momentum right away.

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 detections

One detection at a time.

0 / 8000

Each shot is a fresh, independent preparation evolved to the selected time. Positions are drawn from the Born probabilities on the position grid; every shot is kept, and the seed is fixed so runs repeat. The interval below covers shot noise only. Positions do not measure momentum or phase.

Recorded fraction on the right

\text{—}

95% Wilson interval

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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: How can one short pulse slow or refocus a spreading matter wave?
One-sentence intuition: A quadratic phase kick changes the momentum first; the density follows later.

Core mathematical model

Free evolution

\widetilde\psi(k,\tau)=e^{-ik^2\tau/2}\widetilde\psi(k,0)

Between pulses the probability of each momentum stays fixed; only the relative phases change.

A thin matter-wave lens

\psi_+(q)=e^{-i\kappa q^2/2}\psi_-(q)

An instantaneous phase change leaves the position density and the normalization unchanged.

Common difficulties

The surface is not a path

Typical misconceptionThe particle travels across the raised surface.

Better mental modelDepth stands for time and height for probability amplitude. Each horizontal slice is the one-dimensional wavefunction at one moment.

The model is not the experiment

Typical misconceptionNarrower momentum means entropy has been removed.

Better mental modelThe model uses an ideal, isolated Gaussian state of one particle and an instantaneous harmonic kick. Collisions, trap aberrations, the finite pulse duration and thermal ensembles are left out. A narrower momentum spread does not mean entropy was removed, and the Lab 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: Without a pulse the packet keeps spreading. Collimation slows the spreading, and a stronger pulse squeezes the packet down to a waist.
02
Find the cause
Scrub the timeline and compare position and momentum distributions.
What to observe: At the moment of the pulse, the momentum changes but the density does not jump.
03
Collect and compare
Record 400 detections, repeat, pin the run and export the data.
What to observe: The histogram is built from these positions. More shots mean less sampling noise.