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Acknowledgements
We thank X. Cao for helping us with sample preparation, and useful discussions with Y. Pan at the early stage of this work. This work is supported by National Key Research and Development Program of China (grant no. 2022YFA1205101), National Science Foundation of China (grant nos. 12274296 and 12192252), Shanghai International Cooperation Program for Science and Technology (grant no. 22520714300) and Shanghai Jiao Tong University 2030 Initiative. B.W. is sponsored by Yangyang Development Fund. E.H. acknowledges financial support from the Israel Science Foundation (grant no. 1170/20).
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Extended data
Extended Data Fig. 1 Observations for different laser beam waists and polarizations.
(a) and (c): plane-wave illumination (Gaussian beam waist ~2 mm); (a) is x- and y-polarized, and (c) is left- and right-handed circularly polarized. (b) and (d): focused-beam illumination (focused beam waist < 0.5 mm); (b) is x- and y-polarized, and (d) is left- and right- handed circularly polarized. When the incident light is linearly polarized (along x or y), the spin distributions under both plane wave and focused beam illumination are very similar. Under circular polarizations, the spin values are stronger in the illumination region but weaker in the diffusion region.
Extended Data Fig. 2 Experimentally observed Brownian spin-locking effect for different Fe3O4 nanoparticle concentrations.
Normalized light intensity distributions (a) and sx distributions (b) for various concentrations of Fe3O4 nanoparticles in water. ➀ and ➁ (yellow boxes) denote the analysis regions used to extract the light intensity and sx values in (c) and (d). The two upper boxes represent regions symmetric to ➀ and ➁ with respect to the laser beam. The laser is incident from the right side of the images. (c) Variation of normalized light intensity with particle concentration. <I1 > ( < I2 > ) represents the average light intensity in region ➀ (➁). (d) Variation of <sx> with particle concentration. <sx1 > (<sx2 > ) represents the average absolute value of sx in region ➀ (➁). As the concentration increases, the spin effect becomes weaker, although its spatial distribution remains similar. While the intensity evolution in both regions (➀ and ➁) is comparable across the concentration range, the spin distributions differ significantly. Specifically, when the concentration of nanoparticles is approximately 2.28\(\times\)108 cm−3, the statistically averaged spin at region ➀, <sx1 > , approaches 0, while <sx2> is about 0.09. Notably, the sx distribution near region ➁ persists over a wide range of concentrations, indicating the robustness of the Brownian spin-locking effect across single and multiple scattering regimes. Data are presented as mean ± s.d. (technical replicates, n = 10).
Extended Data Fig. 3 Spin angular momentum properties of the scattering field for different combinations of multipoles.
(a) Schematic of Mie scattering. The location of the analytical point (black dot) in the figure is \((5\sqrt{2}\lambda ,\,5\sqrt{2}\lambda ,\,0)\). (b) Phase diagram of sρ at the black dot in (a) with respect to Mie scattering coefficients. sρ represents the radial component of the spin angular momentum density in the xy plane. \({\Phi }_{{a}_{1}},\,{\Phi }_{{b}_{1}},\,{\Phi }_{{a}_{2}}\) represent the phases of \({a}_{1},\,{b}_{1},\,{a}_{2}\), respectively. The spin is strong if there is a ±π/2 phase difference between the two coupled modes, and disappears if the phase difference approaches 0 or π. (c) Spin angular momentum distribution of the Mie scattering field under different combinations of Mie coefficients. The short arrows (orange arrows) represent the spin angular momentum. These scattering cases are divided into three different types. One, for instance, the electric dipole, represents topological-insulator-like spin textures with two orbital spin distributions perpendicular to the radiation cones (a1, b1, a2 = 1, 0, 0). This is a typical transverse spin. For the Janus dipole, the spins are parallel to the radiation cones (a1, b1, a2 = 1, i, 0), that is, longitudinal spin. In general, the spin from scattering is neither perpendicular nor parallel to the kinetic momentum.
Extended Data Fig. 4 Statistical properties of the Brownian spin-locking effect and incoherent scattering theory.
(a) The histograms are experimentally observed statistical distributions of the spatial intensity (upper panel) and spin (lower panel), which are obtained from the diffusion regions of Figs. 1d and 1e, respectively. The solid curves are fitted using a Burr distribution for the intensity and a Beta distribution for the spin. (b) The calculated spatial distributions of the normalized intensity and spin from the incoherent scattering theory. (c) The calculated spatial distributions of the normalized intensity and spin from the coherent scattering theory. (d) The theoretical evolution of the intensity and spin distributions by changing m/N from 100% (incoherent) to 0.01% (coherent). As the degree of coherence increases, the spin distributions become wider with enhanced skewness, corresponding to increased spin fluctuations that reduce the spin-locking phenomenon.
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Zhang, X., Chen, P., Li, M. et al. Brownian spin-locking effect. Nat. Mater. (2025). https://doi.org/10.1038/s41563-025-02413-5
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DOI: https://doi.org/10.1038/s41563-025-02413-5