Order recognition by Schubert polynomials generated by optical near-field statistics via nanometre-scale photochromism.

Irregular spatial distribution of photon transmission through a photochromic crystal photoisomerized by a local optical near-field excitation was previously reported, which manifested complex branching processes via the interplay of material deformation and near-field photon transfer therein. Furthermore, by combining such naturally constructed complex photon transmission with a simple photon detection protocol, Schubert polynomials, the foundation of versatile permutation operations in mathematics, have been generated. In this study, we demonstrated an order recognition algorithm inspired by Schubert calculus using optical near-field statistics via nanometre-scale photochromism. More specifically, by utilizing Schubert polynomials generated via optical near-field patterns, we showed that the order of slot machines with initially unknown reward probability was successfully recognized. We emphasized that, unlike conventional algorithms, the proposed principle does not estimate the reward probabilities but exploits the inversion relations contained in the Schubert polynomials. To quantitatively evaluate the impact of Schubert polynomials generated from an optical near-field pattern, order recognition performances were compared with uniformly distributed and spatially strongly skewed probability distributions, where the optical near-field pattern outperformed the others. We found that the number of singularities contained in Schubert polynomials and that of the given problem or considered environment exhibited a clear correspondence, indicating that superior order recognition is attained when the singularity of the given situations is presupposed. This study paves way for physical computing through the interplay of complex natural processes and mathematical insights gained by Schubert calculus.

Generation of Schubert polynomial series via nanometre-scale photoisomerization in photochromic single crystal and double-probe optical near-field measurements.

Generation of irregular time series based on physical processes is indispensable in computing and artificial intelligence. In this report, we propose and demonstrate the generation of Schubert polynomials, which are the foundation of versatile permutations in mathematics, via optical near-field processes introduced in a photochromic crystal of diarylethene combined with a simple photon detection protocol. Optical near-field excitation on the surface of a photochromic single crystal yields a chain of local photoisomerization, forming a complex pattern on the opposite side of the crystal. The incoming photon travels through the nanostructured photochromic crystal, and the exit position of the photon exhibits a versatile pattern. We emulated trains of photons based on the optical pattern experimentally observed through double-probe optical near-field microscopy, where the detection position was determined based on a simple protocol, leading to Schubert matrices corresponding to Schubert polynomials. The versatility and correlations of the generated Schubert matrices could be reconfigured in either a soft or hard manner by adjusting the photon detection sensitivity. This is the first study of Schubert polynomial generation via physical processes or nanophotonics, paving the way for future nano-scale intelligence devices and systems.

Author Correction: Nanometre-scale pattern formation on the surface of a photochromic crystal by optical near-field induced photoisomerization.

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Nanometre-scale pattern formation on the surface of a photochromic crystal by optical near-field induced photoisomerization.

We observed nanometre-scale optical near-field induced photoisomerization on the surface of a photochromic diarylethene crystal via molecular structural changes using an optical near-field assisted atomic force microscope. A nanometre-scale concavity was formed on the sample surface due to locally induced photoisomerization. By using this optical near-field induced local photoisomerization, we succeeded in generating a pattern of alphabet characters on the surface of the diarylethene crystal below the optical wavelength scale. Further, by exploiting the photochromism of the investigated material, erasure of the generated pattern was also confirmed, where the evolution of the pattern during erasure depended on the local spatial characteristics of the crystal. These experimental findings demonstrate the fundamental abilities of photochromic crystals in dynamic memorization in nanometre-scale light-matter interactions.