Free particles / Curved interference

Airy Matter Waves

How can the brightest crest of a free matter wave curve when no force acts on it? The bending crest is an interference pattern, not the particle. A cubic phase on the momentum components makes them add up in step along a parabola in space and time, so the brightest ridge drifts sideways as if accelerating. The center of mass never moves, as expected with no force. Reversing the phase mirrors the curve; because the packet is finite, the ridge eventually fades and changes shape.

Give a matter wave a cubic phase mask and its brightest ridge bends, while its center of mass stays still. Reverse the mask and the curve reverses too. Then build up the position pattern one detection at a time.

A finite one-dimensional wave packet, tapered by a Gaussian, evolves under the free Schrödinger equation. The picture shows the wavefunction, not a particle path, and it does not model a calibrated electron-beam instrument. Because the aperture is finite, the Airy ridge eventually weakens and changes shape.

The wave’s history in space and time

BRIGHTEST POSITION

0

GOLD LINE · CENTER OF MASS

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 brightest peak and the center of mass move differently.

Cyan: brightest position. Gold: center of mass. The finite-aperture brightest lobe can switch; its curve need not remain a parabola. Drag to inspect a time.

$\tau\ \longrightarrow\quad q=x/\ell$

02 / SPACE AND TIME

The interference draws the curve.

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 momentum weights stay the same; only the phases differ.

Current wave and zero-mask reference have identical momentum probabilities. The cubic phase alone changes the spatial pattern.

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

\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)

\widetilde\psi(k,0)=C\exp(-ak^2+ibk^3/3)

\langle q\rangle=-b/(4a),\quad\langle k\rangle=0

The Gaussian spectral envelope makes the Airy packet normalizable. Its brightest ridge can curve while the center remains constant. A finite packet does not preserve its shape forever; the brightest lobe can change. The ideal fold curve is not used as the computed peak.

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.

Voloch-Bloch et al. · Generation of electron Airy beams · Nature (2013)

Start with the essentials

Focus question: How can the brightest crest of a free matter wave curve when no force acts on it?
One-sentence intuition: The brightest interference crest is not the center of mass.

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 finite Airy packet

\widetilde\psi(k,0)=C e^{-ak^2+ibk^3/3}

A Gaussian envelope in momentum makes the packet finite and normalizable. Reversing the cubic phase gives its mirror image.

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 misconceptionA curved maximum proves a force on the particle.

Better mental modelA finite one-dimensional wave packet, tapered by a Gaussian, evolves under the free Schrödinger equation. The picture shows the wavefunction, not a particle path, and it does not model a calibrated electron-beam instrument. Because the aperture is finite, the Airy ridge eventually weakens and changes shape.

Run the experiment

01
Start with a prediction
Compare all three presets at the indicated times.
What to observe: The bright curve flips when the phase mask is reversed, but the center-of-mass line stays straight.
02
Find the cause
Scrub the timeline and compare position and momentum distributions.
What to observe: Different cubic phases leave the momentum weights the same but produce different interference.
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.