Experimental Milestones Towards Majorana Braiding with Acoustic Metamaterials

TL;DR

The paper presents a passive acoustic metamaterial implementation of the fully general Kitaev chain with complex order parameter $Δ$ and site-dependent chemical potential $μ$, preserving the model’s symmetries and topological phase diagram. By mapping the complex-coupling problem to a real-valued, experimentally accessible form $\tilde{H}$, the authors realize Majorana-like edge modes in a two-layer acoustic lattice and demonstrate domain-wall control and smooth $Δ$-twists necessary for braiding. They validate the approach through simulations and a 16-unit-cell experiment that shows four mid-gap edge modes localized at domain walls and a clear spectral gap in agreement with theory, supporting the feasibility of adiabatic, topology-protected braiding. The work establishes the fundamental building blocks for scalable Majorana braiding in metamaterials, outlining paths to implement complete braiding protocols and non-abelian information processing with accessible platforms.

Abstract

Here we show the first experimental implementation of the fully general Kitaev chain with complex-valued order parameter $Δ$ and site-varying synthetic chemical potential $μ$, using a passive multilayer acoustic resonator design and fabrication. Our laboratory model faithfully reproduces the key symmetries and the topological phase diagram of the model, and displays robust Majorana-like edge modes spatially localized at smoothly engineered domain walls and energetically localized in the middle of the bulk spectral gap. We demonstrate precise control over mode positioning through smooth spatial variations of $μ$, and validate the stability of the modes and of the spectral gap under continuous and complex variations of $Δ$ -- both critical requirements for topological braiding operations. These results establish and validate the fundamental building blocks for experimental implementation of complete braiding protocols, opening concrete pathways toward accessible non-abelian physics and topologically protected information processing.

Figures (3)

• Figure 1: Acoustic Kitaev chain with complex-valued $\Delta$: Schematic and Simulations. (a) Geometry of H-resonators: $h_1=20$ mm, $w_1 =$ 10 mm, $d$ = 3 mm, and $w_2=$ 5 mm, and simulated acoustic pressure profile at resonant frequency 3.38 kHz. (b) Single unit cell schematic of the amplified Kitaev chain \ref{['Eq:KitaevReal']}, with $t_1=t+\Delta_x$ and $t_2=t-\Delta_x$. Blue (orange) connections represent positive (negative) couplings. (c) Full schematic of the amplified Kitaev chain \ref{['Eq:KitaevReal']}, with the color coding identical to that in (b). The last 8 resonators are labeled in order to specify their positions in the actual laboratory model shown in (d). (d) Section of our laboratory model showing a pair of adjacent acoustic unit cells, with the couplings and resonators labeled according to the schematics (b-c) (some features exaggerated for clarity). (e) Simulated resonant spectrum of the laboratory model as function of the $\mu$-coupling, revealing a topological phase transition exactly at $\mu = t$, as in the theoretical Kitaev chain \ref{['Eq:HamKitaev']}. The couplings $t$, $\Delta_x$, and $\Delta_y$ were fixed at 3 mm, 1 mm, and 1 mm, respectively. Topological edge modes are colored in red and bulk modes in black. (f) Resonant spectrum for the $\mu$-coupling value selected in panel (e), with four mid-gap modes labeled and highlighted in red. (g) Pressure fields of the mid-gap modes from panel (f), confirming that they are topological edge modes. (b-c) are reproductions from emilbraiding.
• Figure 2: Braiding with T-junction Geometry and Validation Simulations.(a) Braiding process of topological interface modes depicting the three steps described in text. In step (2) of the braid, a twisting of the $\Delta$ parameter is required. (b) Simulated resonant spectrum of a laboratory model partially implementing step (1) using the site-dependent $\mu$-profile \ref{['eq:mu']}. The two domain walls were simultaneously shifted by a $\delta_x$ in steps of 0.2, while holding constant $\phi$, $\mu_{\text{min}}$, $\xi$, $x_1$, $x_2$, and $t$ at $\pi/4$, 1 mm, 2 mm, 10, 18, and 2 mm respectively. Topological (bulk) modes are shown in red (black). (c) Pressure fields of the mid-gap resonances highlighted in (b) (note the color coding), corresponding to the initial and final configurations, confirming that the pair of interface modes have been displaced as desired. (d) Simulated resonant spectrum of the laboratory model while twisting the order parameter $\Delta=\Delta_0(\cos \phi +\imath \sin \phi)$ with a variable $\phi$, an operation needed at step (2) of the braiding.
• Figure 3: Experimental Kitaev System with local density of states measurements. (a) STL of the experimental setup showing the two-layer acoustic resonator arrays uncoupled. Each resonator has ports for a microphone, drainage, and a speaker. Coupling ports are placed as dictated by the Hamiltonian. (b) Photograph of the fabricated system with flexible tubing connecting the two layers. (c) Simulated eigenfrequencies showing clear gap with topological modes (red) centered between bulk modes (black). Four topological edge modes are labeled 1-4. (d) Spatial amplitude distribution of the topological edge mode at 3.41 kHz comparing numerical simulations (top panels for each layer) with experimental measurements (bottom panels for each layer). The peak amplitude in the gap is localized at the domain wall. (e) Local density of states (LDOS) measured across all 64 resonators as a function of frequency, confirming spectral gaps closings and openings at the location of the domain walls.