Simultaneous Observation of a Topological Edge State and Exceptional Point in an Open and Non-Hermitian Acoustic System

This Letter reports on the experimental observation of a topologically protected edge state and exceptional point in an open and non-Hermitian (lossy) acoustic system. Although the theoretical underpinning is generic to wave physics, the simulations and experiments are performed for an acoustic system. It has nontrivial topological properties that can be characterized by the Chern number provided that a synthetic dimension is introduced. Unidirectional reflectionless propagation, a hallmark of exceptional points, is unambiguously observed in both simulations and experiments.

Figure 1

The schematic of a comblike compound unit cell composed of a main tube (radius $r_0$) with $Q$ closed-end, side-branch tubes (radius $r_0$ and length $l_0$). $d_n$ refers to the distance between the $n$th and ($n+1$)th side-branch tubes. $a_n$ represents the amplitude of the resonant mode of the $n$th side-branch tube. The dimensions are $r_0=14.5\mathrm{mm}$, $d=100\mathrm{mm}$, and $l_0=58\mathrm{mm}$.

Figure 2

(a) Projected acoustic band structure of the compound unit cell with $b=1/3$ as a function of $\varphi$. The gray regions represent three pass bands separated by two band gaps. The Chern number for each pass band is marked. The eigenfrequencies of the edge modes under left (right) incidence is illustrated with blue (red dashed) line. White circles indicate the points where ${\mathrm{\Gamma }}_0=0$. (b) The critical loss ${\mathrm{\Gamma }}_0$ of the edge modes for left (blue line) and right (red dashed line) incidence. The left edge state and right edge state at $\varphi =-2.1$ ($\varphi /\pi =-2/3$) are marked by star and square, respectively. The parameters for the calculation are ${\omega }_0/2\pi =1.75\mathrm{kHz}$ and $\gamma /2\pi =0.48\mathrm{kHz}$, which are extracted from a full wave simulation.

Figure 3

The calculated reflectivity and reflection phase of the finite structure with three unit cells as a function of $\mathrm{\Gamma }$ in the second band gap with $\varphi =-2.1$ (a),(b) for the left incidence and (c),(d) for the right incidence. ${\mathrm{\Gamma }}_1/\omega$, ${\mathrm{\Gamma }}_2/\omega$, ${\mathrm{\Gamma }}_3/\omega$, and ${\mathrm{\Gamma }}_4/\omega$ are the loss values used in the simulation and experiment, and they are 0.0051, 0.0063, 0.0074, and 0.0086, respectively. (e) is the trajectories of eigenvalues ${\lambda }_{\pm }$ in the complex plane. The frequency is fixed at ${\omega }_0/2\pi =1913\mathrm{Hz}$, and the variation parameter is $\mathrm{\Gamma }/2\pi$ with value range (0, 0.012). (f) is the field distribution at the point $({\omega }_0/2\pi ,{\mathrm{\Gamma }}_0/{\omega }_0)=(1913\mathrm{Hz},0.0063)$ for left input and right input.

Figure 4

Simulated and measured reflectivity and reflection phase of the finite structure with three unit cells and different losses ${\mathrm{\Gamma }}_1$, ${\mathrm{\Gamma }}_2$, ${\mathrm{\Gamma }}_3$, and ${\mathrm{\Gamma }}_4$. (a)–(d) refer to the reflectivity for ${\mathrm{\Gamma }}_1$, ${\mathrm{\Gamma }}_2$, ${\mathrm{\Gamma }}_3$, and ${\mathrm{\Gamma }}_4$, respectively. The three gray regions represent the pass bands. (e) The experimental setup. The inset figure is one unit cell of the sample and the small side holes are used to measure the field distribution. The measured reflection phases at different losses (f) for left incidence and (g) for right incidence. (h) The simulated and measured sound absorption at frequencies in the second band gap where $\mathrm{\Gamma }={\mathrm{\Gamma }}_2$. The blue solid lines (red dashed lines) are the simulated results for left incidence (right incidence). The blue open circles (red open triangles) are the experimental results for left incidence (right incidence).