Illustration of a periodically driven one-dimensional dimerized lattice model (IMAGE)
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Illustration of a periodically driven one-dimensional dimerized lattice model (the Floquet SSH $\alpha$-chain) and its acoustic implementation.}
(a) Schematic of the Floquet SSH $\alpha$-chain composed of two distinct half periods. The two half periods alternately host only next-nearest-neighbor coupling ($J_2$) and nearest-neighbor coupling ($J_1$), respectively.
(b) Acoustic implementation in a waveguide array, where the excitation is injected through the input port, and the propagation direction $x$ is mapped onto an effective time coordinate. The couplings are engineered via connecting tubes: straight tubes implement nearest-neighbor coupling $J_1$, whereas curved tubes implement next-nearest-neighbor coupling $J_2$. \textbf{a}, Illustration of experimental settings and protocols. Left panel: Photograph of a 1D phononic crystal. Inset shows its unit-cell structure for the CTS with $a$=2~cm (lattice constant), $r=8.4$~mm, $r_0=1.6$~mm, $r_1=3.2$~mm, $r_2=3.7$~mm, and $h=36$~mm (see Supplementary Note 4 for details). Right panel illustrates the measurement protocol: First measure the pump-probe response. Then perform SVD on the Fourier transformed response tensor $\chi(f,k)$. The resultant largest singular value and the associated eigenvector give, respectively, the phonon band dispersion and the Bloch wavefunctions. Using these quantities, $k$-space correlation matrix for subsystem $\mathcal{A}$ can be obtained, the Fourier transform of which yields the real-space correlation matrix. Finally, the eigenvalues and eigenvectors of the real-space correlation matrix give the ES and associated entanglement wavefunctions. \textbf{b}, Measured local pump-probe response at a boundary acoustic cavity (marked by the arrow in the photo in \textbf{a}). \textbf{d}, Measured singular values $\sigma_1(f,k)$ (left) and $\sigma_2(f,k)$ (right) for a gTrivial case with $a=2$~cm. While $\sigma_1(f,k)$ gives phonon's band dispersion, $\sigma_2(f,k)$ captures band degeneracies. \textbf{e}, Measured ES for the gTrivial case. \textbf{f}-\textbf{g}, Same as \textbf{d}-\textbf{e}, but for the CTS case. A pair of mid-gap modes at 0.5 in ES in \textbf{g} gives experimental evidence for nontrivial critical topology. Pentagons and other symbols label their positions in the phase diagram. {Sim.}, simulation; {Arb.u.}, arbitrary unit; {Exp.}, experiment.}}}\label{1D}
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