1 SCIENTIFIC REPORTS | (2020) 10:2710 | https://doi.org/10.1038/s41598-020-59603-1 www.nature.com/scientificreports Generation of Schubert polynomial series via nanometre- scale photoisomerization in photochromic single crystal and double-probe optical near-feld measurements Kazuharu Uchiyama 1* , Hirotsugu Suzui 1 , Ryo Nakagomi 1 , Hayato Saigo 2 , Kingo Uchida 3 , Makoto Naruse 4 & Hirokazu Hori 1 Generation of irregular time series based on physical processes is indispensable in computing and artifcial 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- feld processes introduced in a photochromic crystal of diarylethene combined with a simple photon detection protocol. Optical near-feld 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-feld 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 reconfgured in either a soft or hard manner by adjusting the photon detection sensitivity. This is the frst study of Schubert polynomial generation via physical processes or nanophotonics, paving the way for future nano-scale intelligence devices and systems. Irregular time series play critical roles in information and communication technology today, including in secure information transfer 1,2 , Monte Carlo simulations 3 , and machine learning 4,5 . Physical processes in nature are inter- esting resources for providing irregular time series, including deterministic dynamics such as chaos 6 , rather than only truly random sequences such as those caused by single photons 7 . Indeed, chaotic lasers enable interest- ing functionalities ranging from ultrafast random number generation 8 and photonic reservoir computing 9 to decision-making, reinforcement learning 10 , and artifcial data generation 5 . In the case of chaotic time series, a minute initial diference results in signifcantly diferent series, while the series share common attributes specifed by the dynamics therein. It has been found that the high complexity and high correlation of deterministic chaos contribute signifcantly to value alignment and decision-making 10 . In this study, we demonstrated the physical generation of irregular time series from near-feld optical sys- tems using photoisomerization in a photochromic crystal, which was observed using a double-probe scanning near-feld optical microscope (SNOM). Specifcally, we generated Schubert matrices corresponding to Schubert polynomials 11,12 via photon transmission through a photochromic crystal photoisomerized on the nanometre 1 University of Yamanashi, 4-3-11 Takeda, Kofu, Yamanashi, 400-8511, Japan. 2 Nagahama Institute of Bio-Science and Technology, 1266 Tamura, Nagahama, Shiga, 526-0829, Japan. 3 Ryukoku University, 1-5 Yokotani, Oe-cho, Seta, Otsu, Shiga, 520-2194, Japan. 4 Department of Information Physics and Computing, Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Bunkyo-ku, Tokyo, 113-8656, Japan. *email: kuchiyama@ yamanashi.ac.jp OPEN