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  • Featured in Physics
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Second-Order Microscopic Nonlinear Optical Susceptibility in a Centrosymmetric Material: Application to Imaging Valence Electron Motion

Chance Ornelas-Skarin1,2,*, Tatiana Bezriadina3,4, Matthias Fuchs5, Shambhu Ghimire1, J. B. Hastings6,1, Quynh L. Nguyen1, Gilberto de la Peña1,7, Takahiro Sato6,1, Sharon Shwartz8 et al.

Mariano Trigo1,7, Diling Zhu6, Daria Popova-Gorelova3,4,9, and David A. Reis1,7,†

  • *Contact author: ccornela@stanford.edu
  • †Contact author: dreis@stanford.edu
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Phys. Rev. X 16, 011006 – Published 7 January, 2026

DOI: https://doi.org/10.1103/7vqw-jbs7

Abstract

We report measurements of phase-matched nonlinear x-ray and optical mixing from single-crystal silicon using subresonant 0.95 eV laser pulses and 9.5 keV hard x-ray pulses from the Linear Coherent Light Source free-electron laser. The mixing signal appears as energy and momentum sidebands to the elastic Bragg peak. It is proportional to the magnitude squared of the relevant temporal and spatial Fourier components of the optically induced microscopic charges and currents. We measure the first- and second-order sideband to the 220 Bragg peak and find that the efficiency is maximized when the applied field is along the reciprocal lattice vector. For an optical intensity of approximately 1012  W/cm2, we measure peak efficiencies of 3×10−7 and 3×10−10 for the first- and second-order sideband, respectively (relative to the elastic Bragg peak). The first-order sideband is consistent with induced microscopic currents along the applied electric field and an isotropic response. The second-order sideband depends nontrivially on the optical field orientation and is consistent with an anisotropic response originating from induced charges along the bonds with C3v site symmetry. The results agree well with first-principles Floquet-Bloch calculations.

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synopsis

X-Ray–Optical Technique Detects Subtle Signal

Published 7 January, 2026

A high-sensitivity measurement reveals the asymmetric shifts of electrons along atomic bonds.

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The detailed valence-electron density of a material plays a central role in its structure and properties. Measuring its atomic-scale structure is challenging because the valence density tends to be delocalized and is a small fraction of the total electron density. We isolated the optically driven valence-electron density in silicon using a nonlinear version of x-ray diffraction. From a single nonlinear Bragg peak corresponding to oscillations at twice the optical frequency, we isolated the valence density along the tetrahedral bonds without requiring phase information. While this nonlinear optical response is forbidden macroscopically, the angstrom-scale resolution provided by a hard x-ray probe allows sensitivity to atomic-scale inversion symmetry breaking, and thus local nonlinear optical responses that are invisible to optical measurements. The measurements agree well with first-principles calculations. The results have implications for studying nontrivial electronic ordering in more complex materials, both near and far from equilibrium.

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Phys. Rev. X 16, 011006– Published 7 January, 2026
  • Received 18 July 2025
  • Accepted 13 November 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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