Robust and High-Capacity Phononic Communications through Topological Edge States by Discrete Degree-of-Freedom Multiplexing

One long-term goal of phononic communications is to achieve the controlled transport of elastic wave signals with enhanced information capacity and improved robustness, which is still challenging given the complexity of elastic waves in terms of vectorial movement and multiple polarizations. Here, we present our theoretical and experimental study of the robust transport of elastic wave signals along interfaces between distinct topological phases. These topological edge states are robust against defects and disorders because they are jointly protected by both pseudospin and valley degrees of freedom (DOFs); thus naturally providing doubled information carriers within a single channel. A topology-based beam splitter is experimentally demonstrated, where the signal's propagation path is uniquely determined and topologically protected. Edge states between distinct topological phases not only offer a route towards new topological phenomena, but also open up an avenue for the design of high-capacity and robust phononic communication devices through discrete DOF multiplexing.

Figure 1

Design of the elastic rod lattice with nontrivial topology. (a) Unit cell of the rod lattice; a = 20 mm is the lattice constant. d = 2 mm and t = 1.2 mm are, respectively, the width and thickness of the elastic rod connecting two neighboring vertices (labeled as p and q) in the hexagonal lattice. (b) Photograph of a sample used in the experiment, where the black region surrounding the sample is covered by an impedance-matched sound absorption adhesive. (c) Experimentally measured transmission spectra through the bulk lattices of QSH-like and QVH-like systems, where the shaded regions represent the corresponding band gaps for u_z (z component of displacement). (d)–(f) Schematic unit-cell configuration (upper panel) and corresponding band structure (lower panel) that exhibits a double-cone dispersion (d), a QSH gap (e), and a QVH gap (f) around the K valley, where QSH and QVH gaps overlap with each other and share the same central-gap frequency of 12.5 kHz. Inset of (f) shows that u_z is negligible for bulk modes 1 and 2, which leads to a wider QVH gap in the experimental measurement.

Figure 2

Topologically nontrivial edge states along interfaces between distinct topological phases. (a) Projected band structures along the x direction for a 24 × 1 VP-SU supercell consisting of 12 cells on each side of the interface. Edge state is characterized by its pseudospin DOF, with blue (red) representing spin up (spin down). (b) Displacement field distributions of the edge state at the green point, where red and blue denote the maximum and zero of the displacement's amplitude. (c),(d), The same as that shown in (a),(b), except that the supercell contains a VQ-SU interface.

Figure 3

Edges states along the VP-SU interface. (a) A straight VP-SU interface is constructed in a sample, which is surrounded by PMLs, as marked by the light-blue region. An excitation source is placed at the red point, and the displacement field inside the red box is scanned in the experiment. (b) Distributions of the u_z energy at 12.8 kHz, with experimental field-scanning data on top of numerical simulation results. (c) Transmission spectrum along the interface is plotted as blue line, which is orders of magnitude larger than those through bulk samples. (d)–(f) The same as those in (a)–(c) except that a zigzag interface is studied. We note that there is an intrinsic loss for the UTR9000 ABS material, which accounts for an approximate 10 dB loss in the transmission spectra in (c),(f).

Figure 4

Transport of elastic wave signals with substantially improved robustness. (a) A straight interface is constructed in three different samples, so that VP-SU, SD-SU, and VP-VQ interfaces are formed. For the VP-SU interface, a random disorder is introduced in a strong way, so that four different types of unit cells (VP, VQ, SU, and SD) are all involved, and they are randomly distributed within the 12 perturbed unit cells, with VP (blue), VQ (red), SU (green), and SD (yellow) unit cells marked by different colors. For the SD-SU and VP-VQ interfaces, the defect is relatively simple and weak, where the 12 perturbed unit cells on different sides of the interface interchange their geometrical configurations, e.g., a VP unit cell is replaced by a VQ one and vice versa. (b) Transmission spectra along the three interfaces are plotted as black (VP-SU), red (SD-SU), and blue (VP-VQ) lines. Green points mark the position of operating frequency at 12.2 kHz, where the distributions of the u field are displayed in (c). Clearly, edge states along the VP-SU interface are more robust against the random lattice disorders than those of VP-VQ interfaces. (c) Distributions of the u field for three different interfaces. WG denotes waveguide.

Figure 5

Effective Hamiltonian. (a)–(c) Schematic unit-cell configuration (upper panel) and corresponding band structure (lower panel) that exhibits a double-cone dispersion (a), a QSH gap (b), and a QVH gap (c), around the K valley. Band structures predicted by corresponding effective Hamiltonians are shown in red, which agree well with the rigorous numerical results denoted in blue.

Figure 6

Topological edge states along VP-SU and VQ-SU interfaces. (a) Modification of geometrical parameters, where the vertices p and q are shifted upwards slightly by a distance of $0.35\mathrm{mm}$ for the QSH domain (upper panel), and the difference between $h_p$ and $h_p$ decreases for the QVH domain, so that $h_p=5.4\mathrm{mm}$ and $h_q=1.8\mathrm{mm}$ (lower panel). (b) Corresponding edge bands after modification (solid lines), as well as those before modification (dotted lines).

Figure 7

A topology-based beam splitter. (a) Schematic layout of the beam splitter composed of four domains, as marked by VP, VQ, SU, and SD. Wave energy is injected at the source position, as indicated by the red point on the VP-SU interface. The whole sample is surrounded by PMLs, as marked by the light-blue region. The region in the black box functions as a filter, so that we obtain an edge mode with specified valley and spin indices at port 1. The $u_z$ energy inside the red box is scanned in the experiment. (b) Distributions of the $u_z$ field at 12.8 kHz, with experimental field-scanning data on top of numerical simulation results. According to the valley-spin locking mechanism, the input signal at port 1 will go to port 2. (c) Transmission spectra at ports 2 and 3, where the signal intensity at port 2 is orders of magnitude larger than that at port 3. (d)–(f), The same as that in (a)–(c), except that the VQ domain is replaced by a VP one, and the signal at port 1 will go to port 3.