Experimental Observation of Temporal Pumping in Electromechanical Waveguides

Yiwei Xia, Emanuele Riva, Matheus I. N. Rosa, Gabriele Cazzulani, Alper Erturk, Francesco Braghin, and Massimo Ruzzene

Phys. Rev. Lett. 126, 095501 – Published 2 March 2021

Abstract

We experimentally demonstrate temporal pumping of elastic waves in an electromechanical waveguide. Temporal pumping exploits a virtual dimension mapped to time, enabling the generation and control of edge states, typical of two-dimensional systems, in a one-dimensional waveguide. We show experimentally that the temporal modulation of the stiffness drives the transfer of edge states from one boundary of the waveguide to the other. The considered implementation, that consists of an elastic waveguide coupled with tunable electrical impedances, allows the pumping to occur in a controllable manner. The framework presented herein opens new avenues for the manipulation and transport of information through elastic waves, with potential technological applications for digital delay lines and digitally controlled waveguides. This Letter also explores higher-dimensional topological physics using virtual dimensions mapped to time in electromechanical systems.

Received 16 June 2020
Accepted 14 January 2021

DOI:https://doi.org/10.1103/PhysRevLett.126.095501

Physics Subject Headings (PhySH)

Acoustic metamaterials Acoustic wave phenomena

Topological materials

General PhysicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Yiwei Xia ¹, Emanuele Riva ², Matheus I. N. Rosa ³, Gabriele Cazzulani ², Alper Erturk ¹, Francesco Braghin ², and Massimo Ruzzene ³

¹School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
²Department of Mechanical Engineering, Politecnico di Milano, 20156 Milano, Italy
³Department of Mechanical Engineering, University of Colorado Boulder, Boulder, Colorado 80309, USA

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Vol. 126, Iss. 9 — 5 March 2021

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Images

Figure 1
Concept of temporal pumping implemented in an electromechanical elastic beam. The equivalent stiffness $D_{n} (ϕ)$ at the location of the $n$ th piezoelectric patches (red lines) is obtained by sampling the surface $D (x, ϕ) = D_{0} [1 + α \cos (2 π θ x + ϕ)]$ at $x_{n} = n$ [32]. The spatial stiffness modulation with slowly varying temporal phase $ϕ (t)$ induces the transition of the left-localized edge state (input) into a right-localized state (output).
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Figure 2
Dispersion properties and edge states for a beam with equivalent stiffness modulation $D_{n} (ϕ) = D_{0} [1 + α \cos (2 π θ n + ϕ)]$ . (a) Dispersion surfaces as a function of $μ$ and $ϕ$ with information on Chern numbers and gap labels. The inset displays the first five bands for $ϕ = 0$ , highlighting the frequency range considered for the surface plot (shaded blue region). (b) Eigenfrequencies for a finite beam as a function of $ϕ$ (black lines) superimposed to the bulk bands (shaded gray regions), where an edge state (red lines) spans the nontrivial gap with $C_{g} = 1$ . (c) States corresponding to the points marked in (b) showing examples of right-localized mode (I), bulk mode (II), and left-localized mode (III).
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Figure 3
Experimental spectral characterization of modulated electromechanical beam. (a),(b) Measured frequency response as a function of $ϕ$ (contours) for excitation at the (a) left and (b) right boundary, superimposed to the eigenfrequencies of bulk (black) and edge (red) modes. The left excitation reproduces mostly the left-localized portion of the branch of the edge state (dashed lines), while the right excitation identifies the right-localized portion (solid line). (c) Representative experimental response for left- (I) and right- (II) localized modes. Black and red lines represent experimental and numerical data.
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Figure 4
Experimental demonstration of temporal pumping in the electromechanical waveguide. (a) Transient time history illustrating a transition from a left-localized mode to a right-localized mode, induced by a linear temporal phase variation from $ϕ_{1} = 1.6 π \to ϕ_{2} = 0.4 π$ starting at $t = 12 ms$ , with a duration of 2 ms. (b) Spectral content of broadband excitation in quasistatic conditions (top) compared to spectrogram of the temporal pump (bottom), illustrating the adiabatic evolution along the branch of the edge state occurring in the pump with negligible influence of the neighboring bulk mode. (c) Signals at left (blue) and right (red) boundaries of the beam for temporal pumps induced within different modulation windows (shaded gray regions). In the initial 12 ms, steady-state vibrations of the left-localized mode are induced (with duration halved for better visualization), while different phase modulation durations (top, 2 ms; middle, 1.5 ms; bottom, 1 ms) delay the arrival of the signal at the right end of the beam.
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