Text from the interactive explanation

Subcycle videography of Landau–Zener–Majorana transitions in graphene

Eggers, V., Inzani, G., Meierhofer, M., Münster, L., Helml, J., Wallauer, R., Zajusch, S., Ito, S., Machtl, L., Yin, H., Kumpf, C., Bocquet, F. C., Bao, C., Güdde, J., Tautz, F. S., Huber, R., Höfer, U.

Filming electrons as they tunnel between graphene's energy cones, inside a single wave of light.

Physicists want to build electronics that switch at the pace of light itself, steering electrons with the oscillating electric field of a light wave within a few femtoseconds (a femtosecond is a millionth of a billionth of a second). Graphene has already hosted the first such “lightwave” logic gates. To engineer them, you need to see what electrons actually do during one oscillation of light, everywhere in the crystal at once.

This experiment films exactly that, and catches a quantum effect predicted in 1932: electrons leaping a gap they classically should not cross.

Try it: drag the light down to about 0.1 MV/cm, then back up. Switch tunnelling off. Watch the glowing patch on the floor, which is the electrons' momentum map.

Electrons in graphene live on a cone

In a crystal, an electron's energy is tied to its momentum, meaning which way and how fast it travels. For graphene, a sheet of carbon one atom thick, plotting energy against the two directions of momentum gives two cones touching tip to tip, at the Dirac point.

Every allowed state in the lower cone is filled with electrons. The graphene used here carries a few extra electrons, so a small puddle also sits in the upper cone, filled up to 0.2 electronvolts above the tip (the orange ring). Electrons in this puddle are the ones the light can push around.

The experiment “photographs” this landscape. Ultraviolet flashes knock electrons out of the sheet, and an instrument records the energy and momentum of each one. Look at one energy and you see a horizontal slice through the cones: a ring.

The map is zoomed onto one corner of graphene's momentum map, a “K point”, where a cone tip sits.

Momentum on these maps is measured in Å⁻¹ (“per ångström”). An ångström is a ten-billionth of a metre, about the spacing of atoms. The cone's slope is set by graphene's electron speed, vF ≈ 10.7 Å per femtosecond.

Push the puddle

A light wave's electric field shoves every electron the same way at the same instant. On the momentum map, that means the whole puddle slides, back and forth once per light cycle. Here the light oscillates 30 trillion times a second, one cycle every 33 femtoseconds.

When the puddle sits off-centre, more electrons travel one way than the other: that is an electric current. The experiment measured this current directly from the lopsided maps, peaking at 38.2 amperes per centimetre of graphene.

Predict: when is the puddle pushed furthest?

Find the friction

Collisions act like friction: they keep resetting the electrons' momentum, so the puddle can't “coast” as far and its swing shifts earlier. Slide the average time between collisions until your simple model (blue line) lines up with the measured current (blue dots).

Without any friction, momentum piles up while the field pushes, so the swing peaks a quarter cycle (8.3 fs) after the push. The measured current lags by only 4.9 ± 0.2 fs. The authors' full quantum simulations (the semiconductor Bloch equations) reproduce that delay with a mean scattering time of 9.0 ± 0.7 fs. The toy model on this page treats the puddle as one sliding object, so its best match lands a little lower. Even with so little time between collisions, graphene's electrons are so fast that they travel almost 100 Å (10 nanometres) before scattering. That sets the time and length budget for lightwave electronics in graphene.

The leap at the pinch point

Something else happens near the cones' tips. As the field drags electrons of the full lower cone past the tip, the gap separating them from the empty upper cone shrinks almost to nothing. Some electrons don't stay below. They leap.

Cut the cones along the push direction, but a little to one side of the tip, at an offset q. The two cones become two curves separated by a gap of 2ħvF·q, which is tiny near the tip.

In 1932, Landau, Zener, Majorana and Stückelberg worked out what a quantum system does when it's driven through such a near-crossing. Driven slowly, it adapts and stays on its own lower level. Driven fast, it can't adapt and jumps straight across. The chance of jumping is

A bigger gap (larger q) makes leaping exponentially rarer. A stronger field E sweeps electrons through faster and makes it likelier.

Top: electrons (dots) swept along the lower curve. Leapers continue straight onto the upper cone (yellow). Bottom: leap probability against offset. The marker is your lane, and the dashed line is where P falls to 1/e², at 0.04 Å⁻¹ for the experiment's field of 1.57 MV/cm.

Which shape will the camera see?

Light can lift electrons into the upper cone in two very different ways, and they leave different fingerprints on the momentum map.

Gentle light lifts electrons by absorbing photons. Graphene's electrons carry an internal label called pseudospin, and its rules send photo-excited electrons sideways, perpendicular to the field.

Strong light makes them tunnel along the strip you just found, which runs parallel to the field.

The experiment used a strong field. A few femtoseconds into the pulse, which shape did the momentum microscope record?

Field direction is horizontal (ky). Grey lines are the crystal-map edges meeting at the K point. Fig. 3a of the paper, recoloured.

From gentle to strong light

The authors simulated undoped graphene driven at 30 THz without collisions. A ratio above 1 means the distribution is broader across the field (absorption). Below 1 means it is stretched along the field (tunnelling). The Keldysh parameter γ compares how fast the light oscillates with how fast electrons can tunnel. γ ≪ 1 is the strong-field, tunnelling regime, and the experiment sits at γ ≈ 0.075.

How do you film 33 femtoseconds?

No camera shutter is that fast. Instead, the team used a stroboscope made of light. An intense infrared pulse (the “pump”) drives the graphene, and an ultraviolet flash lasting less than 10 femtoseconds (the “probe”) arrives a chosen delay later and knocks electrons out to be photographed.

Each laser shot yields only about one photo-electron, but the laser fires 50,000 times a second. A momentum microscope catches every electron and records its direction and energy. Thousands of shots at one delay add up to one frame. Step the delay, with a precision of 230 attoseconds (0.23 fs), and the frames become a movie.

After an electron leaves the sheet, the infrared field is still there and gives it an extra kick. That shifts the whole picture in step with the light wave. The team turned this nuisance into a tool. Tracking the shift frame by frame reconstructs the light's own waveform at the sample with attosecond precision (E = −(ħ/e)·dΔk/dt). They then subtract the shift so that only motion inside the graphene remains. Toggle “raw” to see the uncorrected view.

Each dot is one detected electron, drawn from the illustrative model at this delay and blurred like a real instrument.

The real film

This is the measured movie. On the left, electrons in the upper cone near one K point, seen from above. On the right, a slice through the cones at that point. Each frame is a different delay of the ultraviolet flash.

Video: Eggers et al. (2026), Supplementary Video 1, CC BY 4.0. Re-encoded and resized.

The green curve wobbles at twice the light's frequency. New electrons can only tunnel into the upper cone where there is room. Near the tip, room opens up when the existing puddle has been pushed away from it, which happens twice per cycle (once to each side). The authors describe a “well” that the field drives and that the oscillating electrons open and close. The biggest broadening coincides with the biggest displacement, which rules out simple motion blur. Throughout, the distribution still covers the Dirac point.

Slid along the field (−43.6 fs). Stretched along the field into an ellipse by tunnelling, with the width along the field growing from 0.054 to 0.096 Å⁻¹ while the width across changes by only 0.014 Å⁻¹ (3 fs). Broadened in both directions by collisions (26.3 fs). Relaxed back to a round shape (233 fs).

What breaks the spell

Tunnelling and sliding are coherent: every electron moves in lockstep with the light. Collisions scramble that. The maps let the authors work out which kind of collision matters, by comparing the measurement with simulations that include different scattering channels.

Back to the opening question

You can now read the opening scene like the researchers do. Each light cycle, three things play out in the electrons' momentum map, and the experiment saw all of them directly.

Now happening: —

Slide = current

The field drags the whole puddle. Measured offset up to 0.078 Å⁻¹, lagging the field by 4.9 fs because of collisions every ~9 fs.

Stretch = tunnelling

Landau–Zener–Majorana leaps at the tip feed electrons into a strip along the field, giving the first direct view of this effect in a solid's band structure.

Round-off = phonon collisions

Collisions with lattice vibrations (optical phonons) redistribute electrons fastest, the key limit on coherent light control in graphene.

The opening asked what electrons do during a single oscillation of light, everywhere in a crystal. This experiment answers it for graphene. It gives a panoramic, sub-cycle view of the entire momentum map, and it pins down the budget for lightwave electronics: collisions strike about every 9 fs, yet electrons cover nearly 100 Å between them.

The authors write that the same tool works for “basically any quantum material”. They point to optical band-structure engineering, light-driven phase transitions and the light-field-driven sculpting of molecular orbitals as questions it may soon address.

↑ Back to the light controls. Try the hero again with tunnelling off, and you'll know exactly what's missing.

Source & credits

V. Eggers, G. Inzani, M. Meierhofer, L. Münster, J. Helml, R. Wallauer, S. Zajusch, S. Ito, L. Machtl, H. Yin, C. Kumpf, F. C. Bocquet, C. Bao, J. Güdde, F. S. Tautz, R. Huber & U. Höfer. “Subcycle videography of Landau–Zener–Majorana transitions in graphene.” Nature (2026). doi:10.1038/s41586-026-11214-y. Open access under a Creative Commons Attribution 4.0 licence.

Measured maps, currents, fields and widths (chapters 2, 4, 6) are redrawn from the authors' public source data: Eggers et al., Jülich DATA, doi:10.26165/JUELICH-DATA/HZFKTH (CC BY 4.0). They have been recoloured with a dark version of the paper's colour scale and are otherwise unchanged apart from rounding and per-map normalisation.

Simulated maps and curves (chapters 4 and 7) are the authors' semiconductor Bloch equation results from the same repository (Fig. 4 and Extended Data Figs. 3 and 7), recoloured.

Supplementary Video 1 (chapter 6) from the article, CC BY 4.0. Re-encoded to WebM and resized.

Illustrative models (hero, chapters 1–3, 5, 8) are simplified models made for this page. They use the paper's numbers (Fermi level 0.2 eV, vF = 10.7 Å/fs, 30 THz, 1.57 MV/cm, the Landau–Zener–Majorana formula) but are not the authors' simulations. The “friction” curve in chapter 2 is a one-parameter damped-motion model fitted live against the measured current.