Circular Dichroism in Harmonic Light Reveals a Crystal’s Hidden Quantum Geometry

Physicists have long known that the way light and crystals exchange angular momentum is governed by the symmetry of the atomic lattice. A new study published in Nature Photonics shows that this familiar macroscopic rulebook conceals a direct window into one of the most fundamental quantities in modern condensed matter physics: the Berry curvature, a local property of electronic wavefunctions that underlies phenomena from the valley Hall effect to the nonlinear Hall effect. The research team, led by Giancarlo Soavi of Friedrich Schiller University Jena together with theorists at the University of Regensburg, demonstrates that nonlinear circular dichroism in high-harmonic generation is directly proportional to the local Berry curvature at an optical resonance, and that it can be measured with an ultrafast, all-optical tabletop experiment.

The Berry curvature describes how the quantum phase of an electron’s wavefunction twists as the electron moves through momentum space. Inside a crystal, an electron accelerated by an electric field normally acquires only a longitudinal velocity along the field, with a magnitude set by the energy band dispersion. In the presence of a magnetic field, the Lorentz force adds a transverse Hall velocity. Remarkably, electrons can also acquire such a transverse, anomalous velocity with no magnetic field at all, simply by travelling through regions of reciprocal space where the Berry curvature is non-zero. The electron wave-packet velocity contains a term given by the cross product of the crystal momentum’s time derivative and the Berry curvature vector, producing a circular component of motion. Because the electric field of light drives both the longitudinal and transverse components of the electron’s velocity, a natural question arises: is there a direct link between the Berry curvature and the conservation of angular momentum in light-matter interactions?

The symmetries of a crystal make this question even more compelling. Space inversion symmetry requires the Berry curvature to be identical at opposite points in momentum space, while time-reversal symmetry requires it to be exactly opposite. Breaking time-reversal symmetry therefore lifts the degeneracy between valleys, the distinct energy minima that give two-dimensional semiconductors their valleytronic properties. The Berry curvature, together with the quantum metric and their momentum-space derivatives, forms the quantum geometric tensor of the electronic states, and these quantities govern circular dichroism, valley-contrasted optical selection rules, the valley Hall effect, the nonlinear Hall effect and the bulk photovoltaic effect. Although the momentum-resolved Berry curvature is straightforward to compute from the matrix elements of Bloch wavefunctions, a direct experimental measurement of this fundamental quantity has remained extremely challenging.

Previous studies have offered partial glimpses. Seminal work on photoluminescence gave a microscopic interpretation of valley circular dichroism based on the valley-contrasting Berry curvature, and recent high-harmonic generation experiments explored the interplay between Berry curvature and strong-field nonlinear optics, often using transition metal dichalcogenides as model systems. None of these works, however, provided an analytical expression that quantitatively relates an experimentally accessible all-optical response to the Berry curvature. The only direct measurement proposed so far relied on spin-resolved circular dichroism in angle-resolved photoemission spectroscopy, an approach whose range of validity is still under discussion. What has been missing is a quantitative, broadly accessible tabletop probe with ultrafast time resolution.

The new work fills this gap with a remarkably clean theoretical result. Working in a two-band picture with a single vertical optical transition at a well-defined crystal momentum, the researchers show that the circular dichroism of the mth harmonic, defined as the normalized difference between the intensities of the harmonic polarization generated for the two opposite light helicities, equals the local Berry curvature at the resonant momentum divided by the squared interband dipole matrix element. This equation is general and imposes no symmetry-related restrictions, provided the relevant energy levels are sufficiently well separated to prevent overlapping resonances from multiple momentum points. The requirements are simply that the relevant harmonics exist, even harmonics being forbidden in the electric dipole approximation when inversion symmetry is preserved, and that a single optically accessible resonance can be treated as an effective two-level system. The formalism applies not only to direct-gap semiconductors but to any system where a resonant vertical transition drives a harmonic response, and it extends the known connection between linear circular dichroism and Berry curvature into the nonlinear regime.

To connect this idealized result to a realistic experiment, the team turned to monolayer tungsten diselenide, a prototypical transition metal dichalcogenide with two energetically degenerate valleys at opposite momenta. An off-resonant circularly polarized pump beam breaks time-reversal symmetry through the coherent, valley-exclusive optical Stark and Bloch-Siegert effects, lifting the valley degeneracy by roughly ten millielectronvolts under typical conditions. A linearly polarized probe beam resonant with half the optical band gap then generates a second-harmonic signal. The researchers derived an analytical expression showing that the resulting second-harmonic circular dichroism is proportional to the sum of the pump-dressed Berry curvatures of the two valleys, weighted by valley-resolved resonance denominators. The signal scales quadratically with the pump field amplitude, flips sign when the pump helicity is reversed, and depends on the probe detuning through a complex resonance factor that encodes the dephasing time of the optical transition.

The theory also delivers a striking symmetry insight. Expressed in terms of the Cartesian components of the second-order nonlinear susceptibility, the second-harmonic circular dichroism requires that two susceptibility components are non-zero and, crucially, out of phase with one another, a condition that cannot be deduced from a purely macroscopic description of the nonlinear tensors. When time-reversal symmetry is preserved, the effect can only appear in point groups 1, 3 and bar-6, where circular second-harmonic generation is allowed by momentum conservation. With time-reversal symmetry broken, the effect becomes allowed for all non-centrosymmetric magnetic groups with one- or threefold rotational symmetry, including the symmetry of monolayer transition metal dichalcogenides. The asymmetry between energy levels and Berry curvature at opposite momenta induces the phase mismatch between the susceptibility components, linking macroscopic crystal symmetries directly to the microscopic quantum geometry of the electronic bands.

The experimental demonstration was performed in a two-colour pump-probe geometry on a mechanically exfoliated WSe2 monolayer, using synchronized optical parametric oscillators delivering roughly 100-femtosecond pulses at pump and probe wavelengths of 1,800 and 1,460 nanometres. Helicity-resolved spectra revealed the second harmonic of the pump beam appearing exclusively in the cross-polarized detection channel, as expected from spin angular momentum conservation, while the second harmonic of the probe beam showed a clear difference between left and right circular detection at zero pump-probe delay. The incoherent two-photon photoluminescence, which overlaps spectrally with the probe second harmonic, cancels exactly in the helicity difference, leaving a clean dichroism signal. The response is ultrafast, vanishing on a timescale set only by the pulse durations, with a fitted Gaussian full-width at half-maximum of 313 plus or minus 40 femtoseconds, and it disappears entirely when the pump beam is linearly polarized, confirming that broken time-reversal symmetry is the origin of the effect.

By scanning the pump intensity, the pump-probe delay and the probe wavelength, the team verified every prediction of their analytical model, including the quadratic intensity scaling and a sign reversal of the dichroism as the second-harmonic wavelength was tuned across the light-dressed valley resonances. Because every variable in the model except two was independently measured, fitting the wavelength dependence yielded direct estimates of the dephasing time, 42 plus or minus 11 femtoseconds, and the resonant Berry curvature at the valleys, 6.2 plus or minus 1.4 square angstroms, in good agreement with values from tight-binding and density functional theory calculations. The work establishes second-harmonic circular dichroism as a non-invasive, ultrafast and all-optical probe of the local Berry curvature, and it points toward ultrafast devices that combine spin and valley degrees of freedom with the engineering and readout of quantum geometry, advancing the emerging field of ultrafast all-optical valleytronics.

Subject of Research: Nonlinear circular dichroism as an ultrafast all-optical probe of the local Berry curvature in monolayer WSe2

Article Title: Nonlinear circular dichroism reveals the local Berry curvature

Article References: Nonlinear circular dichroism reveals the local Berry curvature. (n.d.). https://doi.org/10.1038/s41566-026-02025-8

Image Credits: AI Generated

DOI: 10.1038/s41566-026-02025-8

Keywords: Berry curvature, circular dichroism, high-harmonic generation, second-harmonic generation, WSe2, valleytronics, time-reversal symmetry, quantum geometry, nonlinear optics, ultrafast photonics, optical Stark effect, transition metal dichalcogenides