Adiabatic pumping via avoided crossings in stiffness-modulated quasiperiodic beams

Emanuele Riva, Vito Casieri, Ferruccio Resta, and Francesco Braghin

Phys. Rev. B 102, 014305 – Published 7 July 2020

Abstract

In this paper we report on adiabatic pumping in quasiperiodic stiffness-modulated beams. We show that distinct topological states populating nontrivial gaps can nucleate avoided crossings characterized by edge-to-edge transitions. Such states are inherently coupled when a smooth variation of the modulation phase is induced along a synthetic dimension, resulting in topological edge-to-edge transport stemming from distinct polarizations of the crossing states. We first present a general framework to estimate the required modulation speed for a given transition probability in time. Then this analysis tool is exploited to tailor topological pumping in a stiffness-modulated beam.

Received 28 March 2020
Accepted 29 June 2020

DOI:https://doi.org/10.1103/PhysRevB.102.014305

Physics Subject Headings (PhySH)

Acoustic metamaterials Acoustic phonons Acoustic wave phenomena Topological insulators

Condensed Matter, Materials & Applied PhysicsGeneral Physics

Authors & Affiliations

Emanuele Riva ^*, Vito Casieri , Ferruccio Resta, and Francesco Braghin

Department of Mechanical Engineering, Politecnico di Milano, 20156, Italy

^*emanuele.riva@polimi.it

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Vol. 102, Iss. 1 — 1 July 2020

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Figure 1
(a)-I Schematic of the spring-mass system. The adopted parameters are: $m = 1$ , $ϕ_{i} = 0.45 π$ , $ϕ_{f} = 0.55 π$ , $k_{0} = 1$ , $α = 0.3 ɛ = 0.005$ . (a)-II Diabatic and adiabatic mode polarizations $x^{D} = {[x_{1}^{D}, x_{2}^{D}]}^{T}$ and $x^{A} = {[x_{1}^{A}, x_{2}^{A}]}^{T}$ associated with the top branch $Ω_{+}$ ; (a)-III and to the bottom branch $Ω_{-}$ . Displacements relative to diabatic states are represented with red lines, while adiabatic states are illustrated with black curves. (b) Adiabatic, diabatic, and approximated diabatic states upon varying the modulation phase parameter $ϕ \in [ϕ_{i}, ϕ_{f}]$ . The yellow band represents the locus of diabatic states $Ω_{1}^{D}$ characterized by the same polarization $x^{D} \approx [1, 0]$ . (c) Estimated coefficients of the stiffness matrix in generalized coordinates. ${\hat{K}}_{1, 1}$ and ${\hat{K}}_{2, 2}$ are the diagonal terms, which are displayed with solid and dashed black lines, respectively. ${\hat{K}}_{1, 2} = {\hat{K}}_{2, 1}$ are the off-diagonal terms (blue line) representing the coupling between states.
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Figure 2
(a)–(c) Spectrograms $| \hat{w} (Ω, t) |_{RMS}$ of the simulated spring-mass system under narrow band spectrum tone burst excitation adopting three different angular velocities $Ω_{m}$ . (a) Fast modulation $Ω_{m} = 1.3 \times 10^{- 3}$ . (b) Intermediate angular velocity $Ω_{m} = 1.9 \times 10^{- 4}$ . (c) Slow modulation $Ω_{m} = 1.6 \times 10^{- 5}$ . (d)–(f) Estimated probability for a state belonging to $Ω_{-}^{A}$ to keep the same polarization and jump to $Ω_{+}^{A}$ in correspondence of $ϕ_{CR} = π / 2$ for distinct angular velocity values. The black curve represents the dynamic probabilities without the approximation introduced in Eq (11). Red curve: Probability time history through full approximation of the equation of motion. The asymptotic solution is illustrated with black dashed lines.
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Figure 3
Schematic of the beam under clamped-clamped boundary conditions and quasiperiodic stiffness modulation. Quasiperiodicity is achieved by means of nonrepeating control signals $k - 1, k, k + 1, ...$ able to locally alter the electrical parameters of the NC shunt. In the schematic, a temporal modulation of $R_{1}$ is assumed. (b) Hofstadter butterfly associated with a commensurate realization of a stiffness-modulated beam upon varying the projection parameter $θ$ . The vertical dashed line corresponds to the configuration adopted to study edge-to-edge transitions. (c) Integrated density of states (IDS) as a function of $θ$ . The IDS is characterized by sharp jumps in correspondence of the band gaps and identified by straight lines $IDS = n + m θ$ . The slope of the lines $m$ determines the labels of the gap $C_{g} = m$ . (d) Spectrum of the system for $θ = 0.075$ and upon varying the modulation phase $ϕ$ . A pair of states is spanning the gap and generate the avoided crossing. (e)I Zoomed view of the avoided crossing. Adiabatic states in black and approximated diabatic states in red. In the figure the schematic of the expected polarization for each branch is illustrated. (e)II and (e)III Estimated modal dependence factor (MDF) between modes 25, 26, and full spectrum.
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Figure 4
(a)–(c) Spectrograms $| \hat{w} (f, t) |_{RMS}$ resulting from the numerical time history obtained through narrow band burst excitation of the quasiperiodic beam. The shape of the force is tailored to favor the excitation of the left-localized mode. The modulation phase is varied with three different angular velocity levels $ω_{m}$ . (a) Fast modulation $ω_{m} = 20 π$ . (b) Intermediate angular velocity $ω_{m} = 3.15 π$ . (c) Slow modulation $ω_{m} = 0.26 π$ . (d)–(f) Corresponding displacement field in space and time. (d) Without edge-to-edge transition. (e) With $50 %$ energy left localized and $50 %$ right localized. (f) Complete edge-to-edge pumping.
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Figure 5
Schematic and adopted notation of stiffness for the patch ( $E_{p}$ ), the sandwich ( $E_{s}$ ), and of relevant modulation parameters.
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