Tailoring reflectionless complex media for non-Abelian braiding of acoustic modes

Multiple scattering of sound and light can be tailored for diverse applications. Despite the great progress enabled by technologies such as time-reversal propagation and wavefront shaping, the full control of the transmission matrix remains a significant challenge. In this work, we propose a multi-scattering-based approach to design reflectionless complex media with an arbitrary unitary transmission matrix. As such, the perfect transmission of waves through such a medium performs a unitary operation. Based on this principle, we experimentally demonstrated braiding of multiple waveguide modes in an acoustic waveguide via multiple scattering and showed non-Abelian characteristics arising from the concatenation of distinct complex media. Furthermore, we show that the principle can be extended for realizing arbitrary unitary operations beyond braiding. Our scheme uses generalized Wigner-Smith operators to design the optimal acoustic complex media with near-arbitrary targeted functionalities. The scheme is generally applicable beyond acoustics, with broad implications to other wave types. Our results demonstrate unprecedented control over multiple-scattering waves and establish complex media as a viable route to modal control and operations on compact platforms, providing a new design paradigm for multimode, reconfigurable wave-based devices with potential applications in multiplexed communication, imaging, and the manipulation of quantum waves.