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Mathematically, topological invariants arise from the parallel transport of eigenstates on the energy bands, which, in physics, correspond to the adiabatic dynamical evolution of transient states. It determines the presence of boundary states, while lacking direct measurements. Here, we develop time-varying programmable coupling circuits between acoustic cavities to mimic the Hamiltonians in the Brillouin zone, with which excitation and adiabatic evolution of bulk states are realized in a unit cell. By extracting the Berry phases of the bulk band, topological invariants, including the Zak phase for the SSH model and the Chern number for the AAH model, are obtained convincingly. The bulk state evolution also provides insight into the topological charges of our newly developed non-Abelian models, which are also verified by observing the adiabatic eigenframe rotation. Our work not only provides a general recipe for telling various topological invariants but also sheds light on transient acoustic wave manipulations., Comment: 16 pages, 4 figures

For photonic crystal slab (PCS) structures, bound states in the continuum (BICs) and circularly polarized states (dubbed C-points) are important topological polarization singularities in momentum-space and have attracted burgeoning attention due to their novel topological and optical properties. In our work, the evolution of polarization singularities from BICs to C-points is achieved by breaking the in-plane C2 symmetry of a PCS structure of a square lattice with C4v symmetry. Correspondingly, a BIC is split into two C-points with opposite chirality, incurring distinct optical transmission responses with the incidence of right or left circular polarization (RCP or LCP). Harnessing such chirality selectivity of the C-points, we propose a multifunctional imaging system by integrating the designed PCS into a conventional 4-f imaging system, to realize both the edge imaging and conventional bright-field imaging, determined by the circular polarization state of the light source. In addition to multifunctional imaging, our system also provides a vivid picture about the evolution of the PCS platforms' singularities., Comment: 11 pages, 4 figures

In quantum logic operations, information is carried by the wavefunction rather than the energy distribution. Therefore, the relative phase is essential. Abelian and non-Abelian phases can be emulated in classical waves using passive coupled waveguides with geometric modulation. However, the dynamic phases interference induced by waveguide structure variation is inevitable.To overcome the challenges, we introduce an electroacoustic coupled system that enables the precise control of phase distribution through dynamic modulation of hopping. Such effective hopping is electronically controlled and is utilized to construct various paths in parameter space. These paths lead to state evolution with matrix-valued geometric phases, which correspond to logic operations. We report experimental realizations of several logic operations, including $Y$ gate, $Z$ gate, Hadamard gate and non-Abelian braiding. Our work introduces a temporal process to manipulate transient modes in a compact structure, providing a versatile experimental testbed for exploring other logic gates and exotic topological phenomena.