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Tunneling-like wave transmission in non-Hermitian chains with mirrored nonreciprocity

We report a peculiar tunneling-like wave phenomenon that occurs in a non-Hermitian chain, which is placed as an interface between two Hermitian chains. In particular, we consider the non-Hermitian chain to consist of two sections governed by nonreciprocal couplings between the sites, where the nonreciprocity is mirrored between the sections in a way that the weaker couplings are directed toward the boundary with the Hermitian chain. As a standalone system, which was widely studied in recent years, each of these sections supports the non-Hermitian skin effect, in which modes are accumulated at the boundary of the weaker coupling. Here, we investigate what happens to a wave that propagates along the Hermitian chain and hits the interface. Due to the skin mode accumulation a total reflection from the interface would be expected, but surprisingly, in certain conditions the wave is fully transmitted to the other side, keeping the interface dark, as if the wave invisibly tunneled through it. Remarkably, the tunneling is independent of the interface length. We derive the phenomenon both for quantum and classical systems, and realize it experimentally in a non-foster electric transmission-line metamaterial. Read more.

Gravitational lensing and tunneling of mechanical waves in synthetic curved spacetime

Black holes are considered among the most fascinating objects that exist in our universe, since in the classical formalism nothing, even no light, can escape from their vicinity due to gravity. The gravitational potential causes the light to bend towards the hole, which is known by gravitational lensing. Here we present a synthetic realization of this phenomenon in a lab-scale 2D network of mechanical circuits, based on analogous condensed matter formalism of Weyl semimetals with inhomogeneous nodal tilt profiles. Some of the underlying network couplings turn out as unstable and non-reciprocal, and are implemented by embedded active feedback interactions in an overall stabilized structure. We demonstrate the lensing by propagating mechanical wavepackets through the network with a programmed funnel-like potential, achieving wave bending towards the circle center. We then demonstrate the versatility of our platform by reprogramming it to mimic quantum tunneling of particles through the event horizon, known by Hawking radiation, achieving an exceptional correspondence to the original mass loss rate within the hole. The network couplings and the potential can be further reprogrammed to realize other curvatures and associated relativistic phenomena. Read more


We study the interplay of two distinct non-Hermitian parameters: directional coupling and onsite gain-loss, together with topology, in coupled 1D non-Hermitian Su-Schrieffer-Heeger (SSH) chains. The SSH model represents one of the simplest two-band models that features boundary localized topological modes. Our study shows how the hybridization between two topological modes can lead to a striking spectral feature of non-Hermitian systems, namely exceptional point (EP). We reveal the existence EP as a singularity in the parameter space of non-Hermitian couplings carrying a half-integer topological charge. We also demonstrate two different localization behaviors observed in the bulk and hybridized modes. While the bulk states and individual topological modes remain localized at the boundaries due to skin effect, the competition between the constituent non-Hermitian parameters can overcome the strength of skin effect and lead to the complete delocalization of these hybridized modes. We obtain explicit analytic solutions for the eigenfunctions and the eigenenergies of the hybridized modes, which exactly match the numerical results and successfully reveal the underlying cause of delocalization and the emergence of EP. Read more


Active control approach to temporal acoustic cloaking

We propose a realization of a transformation-based acoustic temporal cloak using an active closed-loop control approach to an equivalent electromagnetic problem. Unlike the more common spatial cloaks the goal of which is hiding fixed objects from detection, the goal of the temporal cloak is hiding the occurrence of events during a finite period of time. In electromagnetic systems, in which events represent, for example, leakage of signals from transmission lines or optical fibers, temporal cloaking solutions usually rely on nonlinear phenomena related to the fibers properties, or on modulating the properties of the propagation medium itself. In particular, the transformation-based solution requires modulating the constitutive parameters of the medium both in space and time. Our control approach is fully linear, where the required change in the medium parameters is programmed into the controllers and created by external actuators in real-time. This cloaking system keeps the physical medium unchanged, and enables to reprogram the cloaking parameters upon request. We demonstrate our solution in a simulation of a one-dimensional water channel. Read more

Pressure field

Control  signals field

Real-time-controlled artificial quiet channel for acoustic cloaking

We consider the problem of hiding non-stationary objects from acoustic detection in a two-dimensional environment, where both the object's impedance and the properties of the detection signal may vary during operation. The detection signal is assumed to be an acoustic beam created by an array of emitters, which scans the area at different angles and different frequencies. We propose an active control-based solution that creates an effective moving dead zone around the object, and results in an artificial quiet channel for the object to pass through undetected. The control principle is based on mid-domain generation of near uni-directional beams using only monopole actuators. Based on real-time response prediction, these beams open and close the dead zone with a minimal perturbation backwards, which is crucial due to detector observers being located on both sides of the object's route. The back action wave determines the cloak efficiency, and is traded-off with the control effort; the higher is the effort the quieter is the cloaking channel. We validate our control algorithm via numerical experiments in a two-dimensional acoustic waveguide, testing variation in frequency and incidence angle of the detection source. Our cloak successfully intercepts the source by steering the control beams and adjusting their wavelength accordingly. Read more

Temporal negative refraction

Negative refraction is a peculiar wave propagation phenomenon that occurs when a wave crosses a boundary between a regular medium and a medium with both constitutive parameters negative at the given frequency. The phase and group velocities of the transmitted wave then turn anti-parallel. Here we propose a temporal analogue of the negative refraction phenomenon using time-dependent media. Instead of transmitting the wave through a spatial boundary we transmit it through an artificial temporal boundary, created by switching both parameters from constant to dispersive with frequency. We show that the resulting dynamics is sharply different from the spatial case, featuring both reflection and refraction in positive and negative regimes simultaneously. We demonstrate our results analytically and numerically using electromagnetic medium. In addition, we show that by a targeted dispersion tuning the temporal boundary can be made nonreflecting, while preserving both positive and negative refraction. Read more

Acoustic analogue of quantum tunneling

Klein tunneling is a counterintuitive quantum-mechanical phenomenon, predicting perfect transmission of relativistic particles through higher energy barriers. This phenomenon was shown to be supported at normal incidence in graphene due to pseudospin conservation. We show that Klein tunneling analogue can occur in classical systems, and remarkably, not relying on mimicking graphene's spinor wavefunction structure. Instead, the mechanism requires a particular form of constitutive parameters of the penetrated medium, yielding transmission properties identical to the quantum tunneling in graphene. We demonstrate this result by simulating tunneling of sound in a two-dimensional acoustic metamaterial. More strikingly, we show that by introducing a certain form of anisotropy, the tunneling can be made unimpeded for any incidence angle, while keeping most of its original Klein dispersion properties. This phenomenon may be denoted by the omnidirectional Klein-like tunneling. The new tunneling mechanism and its omnidirectional variant may be useful for applications requiring lossless and direction-independent transmission of classical waves. 

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Feedback-based topological mechanical metamaterials

Non-Newtonian metamaterials emulating the quantum Hall effect

We introduced a method to design topological mechanical metamaterials that are not constrained by Newtonian dynamics. The unit cells in a mechanical lattice are subjected to active feedback forces that are processed through autonomous controllers, pre-programmed to generate the desired local response in real-time. As an example, we focused on the quantum Haldane model, a two-band topological system supporting the anomalous quantum Hall effect. This model breaks time-reversal symmetry via nonreciprocal coupling terms, the implementation of which in mechanical systems required violating Newton’s third law. We demonstrated that the required topological phase, characterized by chiral edge modes, can be achieved in an analogous mechanical system only with closed-loop control. We demonstrated that the resulting system has all the properties of the quantum model, supporting unidirectional, topologically-protected wave propagation along the metamaterial edges. Read more



Mimicking other quantum topological phenomena on the same platform

Pseudo-spin multipole topological insulator
We derived a closed-loop control strategy to turn the same mass-spring lattice into a topological insulator, emulating the QSHE with no spinning elements. The underlying pseudospin-orbit coupling was obtained by breaking spatial symmetry in real-time. The feedback forces created effective unit cells with different inter and intra site couplings.

The modified Haldane model

We derived a control program to realize a non-Newtonian mechanical topological system with anti-chiral edge sates, on top of the same mass-spring lattice. The complex-valued couplings were polarized in a way that modes on opposite lattice edges propagate in the same direction, balanced by counter-propagating bulk modes. 





Two-dimensional feedback-based topological acoustic waveguides

We realized active autonomous guiding of topological sound beams along arbitrary curved paths in free two-dimensional space. Acoustic transducers, embedded in a slab waveguide, generated desired dispersion profiles in closed-loop by processing real-time pressure field measurements through pre-programmed controllers.

We mimicked the quantum valley Hall effect by actively creating an alternating acoustic impedance pattern across the waveguide. The pattern was traversed by artificial reconfigurable trajectories of different shapes. By topological protection, sound waves between the plates remained localized on the trajectories. 

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Feedback-based reconfigurable elastic metamaterials

Active boundary and interior wave absorbers

We converted a beam in axial vibration into a tunable and reconfigurable metamaterial using feedback control. Our control algorithm was capable to achieve different regimes of effective dynamic stiffness K and mass M for the same operating frequency, including zero stiffness and negative mass for wave propagation suppression, double negative parameters for backward wave propagation, and double zero parameters for total impedance matching. These properties could be activated, deactivated, or tuned in real-time, given that the design did not require any passive periodic feature embedded in the host structure, unlike traditional metamaterial designs. We considered an external actuation approach, by bonding actuators to an initially homogeneous beam. Read more

Regime 1: M<0, K=0. 

Regime 2: M<0, K<0.

Regime 3: M=0, 1/K=0.

Active boundary and interior wave absorbers

In elastic transmission-line metamaterials with zero/negative parameters

We designed active absorbers for wave propagation in a class of mechanical transmission-line metamaterials. The first absorber achieves a complete elimination of wave reflections from the metamaterial boundaries independently of the frequency regime. The associated controller was implemented in a feedback loop. The second absorber blocks wave propagation beyond a prescribed location at the metamaterial interior with minimized back-scattering, thus generating a sink without any physical boundary present. It was implemented in a feed-forward loop via a unique near uni-directional control wave method, using two concentrated actuators. Both the interior and boundary absorbers were based on an exact fractional order transfer function model that we derived for the metamaterial. The model explicitly exhibits essential wave characteristics, including delays, dispersion, impedance, boundary reflections etc. The resulting controllers were of fractional order as well, and were realized via a dedicated approximation technique. Read more

M<0, K<0, boundary absorber OFF

M<0, K<0, boundary absorber ON

M>0, K>0, interior absorber ON

In acoustic tube waveguides

Traveling waves - extended d'Alembert formula

We created an absorber in the interior of a one-dimensional acoustic waveguide, using active control. The control goal was suppressing wave propagation beyond a prescribed region of the waveguide, but without perturbing the propagation within that region. Unlike boundary control, achieving full absorption in the interior constitutes a challenge of creating both a non-transmitting and a non-reflecting sink. We overcame this challenge by introducing a near uni-directional control wave, created with two concentrated actuators. The residual back-action wave is minimized based on the available control energy. We employed this control wave in two algorithms, a feed-forward and a feedback one, which we denoted by Interior Wave  Suppression.
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In ring waveguides with application to power grids

We studied swing dynamics in electric power grids using a continuous approach. Rather than addressing the problem as oscillations in a discrete system, we modeled the swing dynamics as a propagating electro-mechanical wave. We used a ring geometry with a one-dimensional wave equation to analyze the underlying dynamics. We used the Interior Wave Suppression control method to damp the system dynamics. 

Unlike domains with boundaries such as strings, any concentrated input to the ring generates waves in two directions, thereby preventing total absorption. We showed that the modeling and control methods are implementable in a power grid using Phasor Measurement Units (PMU) as sensors and Flexible AC Transmission System (FACTS) devices, such as Thyristor Controlled Series Compensator (TCSC), as actuators.

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In nonstiff elastic structures vibrating in dispersive medium

We proposed a control algorithm for tracking of nonstiff elastic systems, such as bars in axial or torsional vibration or membranes in transverse vibration, etc., in dispersive medium. The algorithm was based on exact transfer function modeling with fractional order delay-like exponential terms, representing the response evolution due to dispersion. The control scheme involved actuation and measurement only at the structure boundaries, with three main loops.

The first was a velocity control loop that eliminated boundary wave reflections, thus suppressing the system's vibratory modes and actively rigidizing it. The second was position stabilization loop, which compensated the fractional order delay and placed the closed loop poles in any desired location. Finally, a pre-compensator eliminated medium dispersion regardless of the frequency, and produced a rational tracking system with a pure delay. Read more

All control loops OFF

Angular position

Active rigidization loop ON

Angular position


Wave equation with boundary damping - exact solutions

Modal analysis - new orthogonality condition

A common method of solving initial boundary value problems is modal analysis. For systems with conservative boundary conditions the method is well-established, but for systems with boundary damping it does not provide closed form solutions. We derived the exact modal series solution with explicit expressions for the series coefficients for second order systems with damped boundaries. The key derivation element was a new orthogonality condition for the damped eigenfunctions. Read more


Traveling waves - extended d'Alembert formula

We derived a traveling wave form of the exact solution using a single equivalent propagating wave, an extension of the classical d’Alembert formula to finite length structures with boundary damping. We considered nonzero initial displacement at the ends. In the literature, the associated discontinuity is avoided by restricting the initial displacement to be zero at the ends. We found that for nonzero end displacement there exists an additional, piecewise constant term, which we denoted by ‘‘end waves’’, that is essential to keep the response continuous. Read more

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