Fig. 2: Entanglement entropy and entanglement spectrum in 1D phononic systems.
From: Measuring entanglement entropy and its topological signature for phononic systems
a Measuring entanglement entropy in a 1D phononic crystal via the following procedure: First, measure the acoustic pump-probe response of the system with a speaker, a microphone, and a vector network analyzer (VNA). Second, extract the phononic dispersion and eigenstates by spectral decomposition of the pump-probe response. Third, construct the correlation matrix for subsystem A using the phononic dispersion and eigenstates. Fourth, determine the entanglement entropy from the correlation matrix. The upper-left inset gives the structure of a unit-cell of the phononic crystal. b Photograph of the 1D phononic crystal (top) and the measurement setup with a speaker as the source and a microphone as the detector (down). The black plugs seal the cavities when they are not used for excitation or detection. c Phononic dispersion extracted from the pump-probe response. Cyan curves give the phononic dispersion from the full-wave simulation. d Entanglement entropy of the 1D phononic crystal versus the radius r1 (representing the intra-unit-cell coupling). The contributions of all valence band states are considered with the fermion-filling analog. e Entanglement entropy versus the subsystem size L. Experimental data are represented by the points, while the lines give the area law or the logarithmic law. f Entanglement spectrum versus the radius r1. Points represent the experimental data, while the dotted lines give the simulation results without dissipation. Orange dashed lines in d and f label the topological transition. Geometry parameters are: a = 40 mm, h = 24 mm, d = 16 mm, and r2 = 4 mm.
