Laser pulses unlock quantum geometry hidden in crystal electrons

Physicists have long known that electrons inside crystals behave in ways that defy simple intuition. They do not move like tiny balls rolling along predetermined paths. Instead, their motion is governed by quantum geometry—abstract mathematical properties of their quantum states that nonetheless produce measurable, real-world effects. One of these properties, called Berry curvature, can push electrons sideways even without an external magnetic field. It influences how electrons respond to electric fields and how materials interact with light. The problem has always been the same: calculating Berry curvature is straightforward; measuring it in the lab is brutally difficult.

A team from the universities of Jena and Regensburg has now demonstrated a way forward. Using two ultrashort laser pulses and an analytical model, they revealed local Berry curvature in tungsten diselenide, an atomically thin semiconductor. Their results, published in Nature Photonics, matched theoretical predictions with precision—a validation that suggests the method could become a practical tool for studying quantum materials. The experiment worked on a timescale of a few hundred femtoseconds, opening a path toward tracking the fastest electronic changes in real time.

The researchers focused on tungsten diselenide because it belongs to a family of materials that interact strongly with light when thinned to a single atomic layer. This makes it ideal for nonlinear optics, where the light that emerges from the material differs fundamentally from the light that enters it. The team exploited a specific nonlinear effect called second-harmonic generation, in which the material produces light at exactly twice the frequency of the incoming beam. The wavelength becomes half as long. By examining the polarization of this generated light—the pattern in which its electric field oscillates—they could extract information about Berry curvature.

The experiment used two laser pulses, each with a distinct role. The first pulse, called the pump, was circularly polarized and temporarily altered the material's electronic energy levels. It had a wavelength of 1,800 nanometers. The second pulse, the probe, was linearly polarized at 1,460 nanometers and generated the second-harmonic signal that the researchers measured. By varying the timing between the two pulses, they could isolate the brief, light-induced response from the material's ordinary behavior. The key insight was that tungsten diselenide contains two electronic regions called valleys, located at opposite momenta and labeled +K and −K. These are features of the electronic landscape, not physical hollows. Their Berry curvatures have opposite signs. Normally, their optical contributions cancel out. But the circularly polarized pump shifted the valley energy levels differently, creating an imbalance. This imbalance changed the polarization of the generated second-harmonic light in a measurable way.

The team measured the difference between left- and right-circularly polarized components of the generated light—a quantity called second-harmonic circular dichroism. This difference supplied the experimental signal. When they reversed the pump's rotation, the signal's sign reversed as well, exactly as the model predicted. Increasing the pump's intensity also changed the response in the expected way. These checks confirmed that the optical measurement was genuinely connected to the valley-dependent energy shifts, not merely to any random polarization change.

An analytical model then converted the optical measurement into a number. Fitting the wavelength-dependent data yielded a Berry curvature magnitude of 6.2 ± 1.4 square ångströms—a result that agreed with earlier theoretical calculations based on tight-binding and density functional theory. The model had two adjustable parameters: Berry curvature itself and the optical resonance's dephasing time. Everything else was measured directly and held fixed. This means the result is a quantitative extraction supported by theory, not a direct image independent of theoretical assumptions. The time-dependent signal had a measured width of 313 ± 40 femtoseconds, limited by the laser pulses themselves. A femtosecond is one millionth of a billionth of a second.

Lead author Nele Tornow, who works at Jena's Institute of Solid State Physics, framed the achievement plainly: the team wanted to make a quantum property experimentally accessible that had remained difficult to determine in the lab. They succeeded by combining an optical experiment with a model that connected the observed light to the underlying electronic property. Giancarlo Soavi, also at Jena, noted that the optical method allows investigation of such properties on extremely short time scales—a capability that opens the door to tracking how quantum geometry changes during rapid processes.

The work establishes a measurement strategy rather than a universal test for every solid. Applying the method elsewhere will require suitable optical transitions and material-specific analysis. Crystal symmetry determines which harmonic signals can occur. The experiment also measures Berry curvature only near selected resonant states, not across every electronic momentum, and it does not capture the entire quantum geometric tensor, which contains additional information. Future work could use this optical access to investigate valley-based electronics and other fast light-driven phenomena. For now, the advance is a measurable bridge between emitted light and an elusive property that governs how electrons move inside a crystal.