Eigenmode-Guided Amplification via Spatiotemporal Active Acoustic Metamaterials

TL;DR

The paper tackles controllable eigenmode steering in acoustic metamaterials by introducing a spatiotemporal gain–loss framework with a cross-coupled coefficient $g$ that conserves total energy $E$ while driving the system toward the eigenmode with the largest imaginary eigenvalue. By formulating a nonlinear Hamiltonian and its linear effective counterpart $H_ ext{eff}$, the authors demonstrate deterministic eigenmode collapse in dimers and extend the approach to trimers for cyclic energy routing, with temporal modulation producing either collapse or Rabi-like oscillations near an exceptional point at $|g_0|=\kappa$. They further show that symmetry-forbidden transitions can be overcome via targeted spatiotemporal perturbations, enabling rapid convergence to desired eigenmodes. Full-wave simulations of coupled Helmholtz resonators validate programmable acoustic energy routing and establish a general framework for reconfigurable, time-varying non-Hermitian control in sound systems, with potential applications in adaptive noise control and analog information processing.

Abstract

We present a spatiotemporal gain-loss framework for eigenmode steering in coupled acoustic resonators. A cross-coupled gain-loss coefficient links the gain of one resonator to the intensity of its partner, creating nonlinear feedback that conserves total energy while driving the system toward the eigenmode associated with the eigenvalue having the largest imaginary part-a deterministic eigenmode collapse. Spatial gain-loss profiles shape the eigenvalue spectrum and attractor landscape, while temporal modulation governs the transition dynamics. When symmetry prevents direct access to a target eigenmode, controlled spatiotemporal perturbations enable otherwise symmetry-forbidden transitions and accelerate convergence. Within this framework, parity-time (PT) symmetry appears as a special case, allowing tunable switching between collapse and Rabi-like oscillations near the exceptional point. Full-wave simulations of coupled Helmholtz resonators confirm precise and programmable acoustic energy routing, establishing spatiotemporal gain-loss engineering as a route to reconfigurable wave control and analog information processing.

Figures (4)

• Figure 1: (a) Schematic of the nonlinear acoustic dimer composed of two coupled Helmholtz resonators, where the cross-coupled gain--loss coefficient modulates wall resistance via $R_{a,b} \propto \mp g|\psi_{b,a}|^2$. (b, c) Simulated squared pressures $p_i^2$ and total energy $E$, and (d, e) squared envelope amplitudes $|\psi_i|^2$, for (b, d) $g = 2\pi \times 218~\mathrm{Hz}$ and (c, e) $g = -2\pi \times 218~\mathrm{Hz}$. Positive $g$ concentrates energy in resonator $a$, while negative $g$ favors resonator $b$, consistent with the attractor eigenmodes predicted by the effective Hamiltonian. Dashed horizontal lines denote steady-state fixed points. (f, g) Projection metric $\mathrm{Im}[\langle \psi | H_\mathrm{eff} | \psi \rangle]$ for the simulated field (solid) and the attractor/repulsor eigenvectors of $H_\mathrm{eff}$ (dashed), confirming convergence to the corresponding eigenmode after the transient.
• Figure 2: (a) Imaginary parts $\mathrm{Im}[\lambda_{1,2}]$ of $H_\mathrm{eff}$ showing $\mathcal{PT}$-symmetric and $\mathcal{PT}$-broken phases separated by the exceptional point $|g_0|=\kappa=2\pi\times163~\mathrm{Hz}$. (b) Temporal modulation of $g$ between $\pm g_0$ for time-dependent eigenmode steering. (c) Simulated $|\psi_i|^2$ for $g_0=2\pi\times218~\mathrm{Hz}$ (solid, broken phase) and $g_0=2\pi\times91~\mathrm{Hz}$ (dashed, symmetric phase). In the broken phase, the system collapses to fixed-point eigenmodes (as in Fig. \ref{['fig:fig1']}); in the symmetric phase it exhibits sustained oscillations.
• Figure 3: (a) Schematic of energy routing in the nonlinear acoustic trimer of Eq. (\ref{['eq:trimer_dynamics']}). Cyclic spatiotemporal modulation applies a relative gain $g_0$ sequentially for 60 ms: $(g_a, g_b, g_c) = (g_0, 0, 0) \rightarrow (0, g_0, 0) \rightarrow (0, 0, g_0) \rightarrow \cdots$, steering modal dominance $a \!\rightarrow\! b \!\rightarrow\! c \!\rightarrow\! \cdots$. (b) Simulated $|\psi_{a,b,c}|^2$ showing periodic convergence to distinct fixed points during each step. Representative pressure-field distributions are shown in inset for each interval.
• Figure 4: (a) Spatiotemporal profile for redistributing total energy to cavities $b$ and $c$. (i) Modulation schemes for $g_a$, (ii) for $g_b, g_c$ with symmetric modulation ($g_b = g_c$). (iii) Simulated $|\psi_{a,b,c}|^2$. Energy concentrates in cavity $a$ during the first 60 ms (matching Fig. \ref{['fig:fig3']}(b)) but fails to redistribute to $b$ and $c$ during $60 < t < 120$ ms, unable to reach the target eigenmode $\psi_1' = \{0, 1, -1\}$. (b) Successful redistribution using a spatiotemporal perturbation applied as a transient modification $g_b'(t)$ in (i). (ii) Simulated $|\psi_{a,b,c}|^2$ show that this perturbation enables the previously symmetry-forbidden transition, with convergence to $\psi_1'$ at $t \simeq 100$ ms. Pressure field patterns are sampled and shown for each interval.