Many physics laws and mathematical rules are insensitive to order. For example, the addition of numbers disregards the sequence order, e.g., 1+2+3=3+1+2. However, such a commutative property does not always hold. When the outcomes of a set of operations depend on the execution order, they can become “non-Abelian.” In the 20th century, non-Abelian mathematical frameworks have played profound roles in formulating many fundamental laws of modern physics. Famous examples include the classification of hadrons and the unification of electro-weak interactions. Classical physics, such as mechanics, electromagnetism, and optics, were well established before non-Abelian theories came into play. However, this does not mean non-Abelian effects are absent in the classical world. One prominent example is a Rubik’s Cube—the moves made on the cube do not always commute: two sequential moves done in different orders do not necessarily get the color palettes to the same layout. We then ask: how and when non-Abelian phenomena arise in classical waves? Delving into this question, our recent works leverage Berry-phase matrices, which capture the adiabatic evolution of multiple states, to realize non-Abelian braiding in acoustics [1] and photonics [2]. Here, the braiding operations are implemented using coupled waveguide arrays, which are adiabatically modulated along the guiding direction to enforce a multi-state Berry-phase matrix that swaps the modal dwell sites. The evolution of the guiding modes maps to the generators of braid groups. The non-Abelian characteristics are revealed by switching the order of two distinct braiding operations involving at least three modes. Our results offer new perspectives in exploring novel wave-controlling schemes for future technological applications [3].
[1] Z.-G. Chen, R.-Y. Zhang, C. T. Chan, and G. Ma, Classical Non-Abelian Braiding ofAcoustic Modes, Nat. Phys. 18, 179 (2022).
[2] X.-L. Zhang, F. Yu, Z.-G. Chen, Z.-N. Tian, Q.-D. Chen, H.-B. Sun, and G. Ma, Non-Abelian Braiding on Photonic Chips, Nat. Photon. 16, 390 (2022).
[3] Y. Yang, B. Yang, G. Ma, J. Li, S. Zhang, and C. T. Chan, Non-Abelian Physics in Light and Sound, Science 383, eadf9621 (2024).
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