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Journal Club for July 2023: A Space-Time Odyssey - Taming Exceptional Points in Elastodynamics for Sensitivity and Emissivity Enhancement and Asymmetric Wave Steering

Ramathasan Thevamaran's picture

By: Ramathasan Thevamaran (Theva)

Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, WI 53706, USA.

Research Lab: https://thevamaran.engr.wisc.edu

Geometric symmetries—such as translation, rotation, mirror, and fractal symmetries—are commonly found in nature and are utilized often in architecture and engineering as well as for realizing photonic crystals, phononic crystals, and metamaterials. The dynamical (hidden) symmetries—such as time-reversal symmetry and parity-time (PT) symmetry—however, are not apparent in the system geometry and instead require a careful analysis of the equations of motion describing the dynamic behavior of the system to recognize them. Incorporation of such dynamical symmetries and their violations at certain critical points of the parameter space that control the system behavior can lead to intriguing and unusual consequences that have utility in engineering. This perspective article aims to present some of the recent advancements in an emerging field in mechanics—non-Hermitian elastodynamics [1-10]—with their prospective utility in engineering applications that rely on mechanical wave-matter interactions. While an introduction to non-Hermitian systems and exceptional point degeneracies (EPDs) is provided, this article is not intended to be a comprehensive review. Such detailed reviews can be found elsewhere [11-16].

What is an exceptional point degeneracy (EPD)? And what is its uniqueness?

The EPD is a spectral singularity where two or more eigenvalues and the corresponding eigenvectors of a non-Hermitian system coalesce (see Figure 1 for an order two EPD where two eigenmodes coalesce). The non-Hermitian systems can have complex eigenvalues and non-orthogonal eigenvectors—in contrast to the Hermitian Hamiltonians that have real eigenvalues and orthogonal eigenvectors. Because of this, the system does not necessarily have conservative properties, for example, observables such as energy and momentum are not necessarily conserved. 

A special class of these non-Hermitian systems is the Parity-Time (PT)-symmetric system, where the system is symmetric under combined Parity and Time reversal operations. As an example, Figure 1a illustrates a coupled mechanical oscillator (resonator) with one oscillator having energy loss (dissipative mechanism) and the other having an equivalent energy gain (amplification mechanism) respecting the PT-symmetry. When this system is Hermitian, i.e., neither gain nor loss is present, it exhibits two eigenmodes corresponding to an in-phase (low frequency) and an out-of-phase (high frequency) motion. When the gain and loss intensity (non-Hermitian strength) is increased while keeping them balanced, these two modes approach each other to coalesce at a critical non-Hermitian strength (Figure 1b). This coalescence point—EPD—is often interpreted as a phase transition point between an exact PT-phase (where the eigenvalues are real) and a broken PT-phase (where the eigenvalues are complex) (Figure 1b). The eigenvectors that describe the mode shapes are initially orthogonal when the system is Hermitian, and become non-orthogonal when the gain/loss is introduced, and eventually become parallel at the EPD.

Figure 1. (a) A Parity-Time (PT) symmetric elastodynamic system depicted by a coupled oscillators with one (blue) oscillator having loss (damping mechanism) and the other oscillator (red) having equivalent gain (amplification mechanism). The system is neither symmetric under Parity operation alone nor Time operation alone but recovers the symmetry under the combined PT-symmetry operation. (b) The formation of an exceptional point degeneracy (EPD) as the gain/loss intensity is increased; left axis shows the real part of the eigenfrequency and the right axis shows the imaginary part of the eigenfrequency; the regime where the eigenvalues are real (i.e., imaginary part is zero) is referred to as the exact PT-phase and the other as the broken PT-phase. (c) Comparison of an EPD (red) that forms in a non-Hermitian system as a function of the coupling between the two resonant elements to the commonly known diabolic point that forms in a Hermitian system, showing the sublinear bifurcation which enables hypersensitivity.

Figure 1. (a) A Parity-Time (PT) symmetric elastodynamic system depicted by a coupled oscillators with one (blue) oscillator having loss (damping mechanism) and the other oscillator (red) having equivalent gain (amplification mechanism). The system is neither symmetric under Parity operation alone nor Time operation alone but recovers the symmetry under the combined PT-symmetry operation. (b) The formation of an exceptional point degeneracy (EPD) as the gain/loss intensity (non-Hermitian strength) is increased; left axis shows the real part of the eigenfrequency and the right axis shows the imaginary part of the eigenfrequency; the regime where the eigenvalues are real (i.e., the imaginary part is zero) is referred to as the exact PT-phase and the other as the broken PT-phase. (c) An EPD (red) that forms in a non-Hermitian system as a function of the coupling between the two resonant elements compared to the commonly known diabolic point (black) that forms in a Hermitian system, showing the sublinear bifurcation associated with EPD which enables hypersensitivity.

 

One of the unique features of these EPDs is the sublinear approach of the modes towards the EPD. It can be seen from the Figure 1c that in contrast to a Hermitian degeneracy such as a diabolic point which exhibits a linear bifurcation, the order-two EPD exhibits a square-root bifurcation. Besides this novel nonlinear behavior emerging from a linear system (note that there is no nonlinearity introduced to the system so far), this nature of the coalescence has immediate utility. Compared to diabolic points, the EPD exhibits hyper-sensitivity to small perturbations in the system parameters. Hence, it has been exploited to create hypersensitive electronic, optical, microwave, acoustic, electromechanical, and elastodynamic sensors depending on the framework in which they are implemented [1,16,17].

With this background, let me turn to a more interesting new viewpoint on the non-Hermitian systems where we are just starting to scratch the surface of.

Can we enhance signals using damping?

Damping is commonly used as a design element to attenuate signals, i.e., mechanical waves and vibrations, in engineering systems. But can we use it to enhance signals, particularly without deteriorating the signal quality? The EPD allows us to do so as we have demonstrated experimentally in a recent study [5]. Besides what I have presented above about EPDs in PT-symmetric systems, EPDs can be realized even in the absence of any gain and purely with damping. However, it requires differential damping between the two elements—for example, if we consider the coupled oscillator system in Figure 2, introducing some damping to one and no damping to the other oscillator or introducing higher damping to one and lower damping to the other oscillator can potentially create an EPD. More generally, non-proportional damping where the damping matrix [C]does not commute with mass matrix [M] and stiffness matrix [K]—unlike in proportional damping case where [K]=α[M]+β[K]—can lead to EPDs [10]. This is quite interesting, because certain avoided crossings that we find in the band diagrams commonly studied in phononics and metamaterials could be engineered to form EPDs with the introduction of such non-proportional damping.

Figure 2. Comparison of EPDs in (a) a PT-symmetric system with balanced gain and loss and (b) a passive non-Hermitian system with differential damping (zero gain).

Figure 2. Comparison of EPDs in (a) a PT-symmetric system with balanced gain and loss and (b) a passive non-Hermitian system with differential damping (zero gain). [6]

The consequences are quite remarkable. It has been argued theoretically [18-24] that the eigenvector degeneracy (EVD) associated with the EPDs can lead to an anomalous emissivity enhancement of a source when it is brought at the proximity of an EPD. As we demonstrated experimentally in a non-Hermitian elastodynamic system recently [5], the emitted power from an actuator is enhanced by a universal factor of four (exerted force is enhanced by two-fold) when the system is reconfigured near the EPD compared to its expected resonance-induced enhancement when the system operates far away from the EPD (Figure 3). More importantly, this enhancement is achieved while keeping the quality factor of the signal constant—an intriguing consequence of exact phase of non-Hermitian systems where the imaginary part of the signal remains constant during such reconfiguration (Figure 3d). In other words, incorporation of EPDs into mechanical systems such as MEMS resonators, nano-/AFM-indenters, and robotic actuators provides a new pathway to further boost the actuation power by four-fold while maintaining signal quality—importantly by entirely passive means with no gain/amplification elements. Beyond the technological implications, this finding also provides new insights at a fundamental level to the Purcell Physics [24] and Fermi’s Golden Rule of quantum mechanics [20] which links emission rates to local density of states (LDoS) of a system. Purcell showed that the spontaneous emission rate of a quantum emitter is enhanced by appropriately engineering its surrounding environment, and therefore, the LDoS [24]. These analogies can be understood clearly when the imaginary part of the Green’s function that describes our elastodynamic system is interpreted as the LDoS of the system [5].

Figure 3. Enhanced emissivity near an EPD. (a) A non-Hermitian elastodynamic system made of coupled resonant elements (a Hookean spring connected to a mass and a KVFD viscoelastic material connected to a mass) that supports the formation of an EPD as a function of the coupling spring stiffness. (b) Experimentally measured response showing the formation of an EPD with actuation force amplification. The (c) real and (d) imaginary parts of the modes showing the exact and broken phases and the EPD. (e) The force amplification factor in the exact phase increasing by a factor of 2 as the system approaches the EPD from far away (right-side of the curve); here, the y-axis shows the force amplification—i.e., output force divided by input force—which is further normalized by the force amplification corresponding to the system operating far away from the EPD (i.e., at higher coupling strength).

Figure 3. Enhanced emissivity near an EPD in an entirely passive non-Hermitian system. (a) A non-Hermitian elastodynamic system made of coupled resonant elements (a Hookean spring connected to a mass and a KVFD viscoelastic material connected to a mass) that supports the formation of an EPD as a function of the coupling spring stiffness. (b) Experimentally measured response showing the formation of an EPD with actuation force amplification. The (c) real and (d) imaginary parts of the modes showing the exact and broken phases and the EPD. (e) The force amplification factor in the exact phase increasing by a (universal) factor of 2 as the system approaches the EPD from far away (right-side of the curve); here, the y-axis shows the force amplification—i.e., output force divided by input force--that is normalized by the force amplification corresponding to the system operating far away from the EPD (i.e., at higher coupling strength). [5]

 

A materials challenge: How do we experimentally realize EPDs in non-Hermitian elastodynamics?

While EPDs can be realized when balanced gain and loss are introduced, realization of EPDs in entirely passive systems with no gain seemed non-trivial. The simplest way to introduce damping in a load bearing elastodynamic system is by incorporating viscoelastic materials as the non-Hermitian elements. However, when we examined various viscoelastic materials that respect different models such as Kelvin-Voigt (KV), Standard Linear Solid (SLS), and Kelvin-Voigt Fractional Derivative (KVFD) viscoelastic models, it turns out only the KVFD viscoelastic material—described by a spring and a springpot in parallel—supports the formation of an EPD (Figure 4) [6]. The KV solids support only an approximate transition with no degeneracy. Even though an SLS material with short relaxation time can still support EPD in theory, such relaxation times are unrealistic. We trace the reasons for this to the loss tangent (tan(δ)=Ed/Es) of the material—given by the ratio of the loss modulus (imaginary part of the dynamic modulus) to the storage modulus (real part of the dynamic modulus)—remaining nearly constant in the frequency regime where the system operates [6]. Only a KVFD viscoelastic material such as polyurethane rubber and PDMS support EPD formation because of their broad spectrum of relaxation times while the natural rubber does not [6].

Figure 4. The viscoelastic material characteristics for the realization of an EPD.

Figure 4. The viscoelastic material characteristics for the realization of an EPD. [6]

 

What are the challenges and potential opportunities?

Last couple of decades have seen the design and realization of phononic crystals and metamaterials that utilize impedance contrast induced wave scatting to control mechanical waves and vibrations as well as to demonstrate exotic wave phenomena that were thought to be impossible, e.g., cloaking, and nonreciprocal transport. In this respect, non-Hermitian symmetries add further elegance to these phononic crystals and metamaterials where their impedance profiles are engineered to be complex rather than just real—i.e., incorporating damping (with/without gain) contrast instead of stiffness/density contrast—to unleash their potential further in unprecedented ways [2, 25-29].

Our prior studies [4] examining PT-symmetric quasiperiodic, aperiodic, and fractal metastructures, which exhibit scale-free emergence of EPDs in their unfolding (fractal) spectra, suggest that there are underlying universal rules that govern the emergence of EPDs in such non-Hermitian elastodynamic systems which could facilitate their design and experimental realization (Figure 5). For example, the non-Hermitian strength (gain/loss intensity) required to create an EPD turns out to be directly proportional and of the same order as the initial split between the two coalescing modes in the corresponding Hermitian spectrum (Figure 5e) [4]. The density of EPDs that emerge in the spectra could also be predicted by the fractal dimension of the initial Hermitian spectra (Figure 5f) [4]. Beyond these generalized scaling laws, such systems with fractal spectra themselves could potentially be exploited in interesting ways. For example, could the scale-free nature of EPDs in systems exhibiting fractal spectra enable multi-point and multi-scale sensing? Can the scale-free nature of EPDs in fractal systems be exploited for multi-scale emissivity enhancement? 

Figure 5. The universal rules governing the formation of EPDs. Finite Element Models of (a) a PT-symmetric Aubrey-Harper quasi-periodic elastodynamic metastructure, (b) a PT-symmetric H-tree geometric fractal elastodynamic metastructure, (c) a PT-symmetric aperiodic (following Fibonacci sequence) elastodynamic metastructure. (d) A coupled-mode theory model that follows a Fibonacci substitution rule. All these systems exhibit an unfolding (fractal) spectrum as shown in the center and when the gain/loss intensity (non-Hermitian strength) is increased, numerous EPDs form at different regions of the spectra in a scale-free fashion enabling us to investigate the underling relations between initial modes in the Hermitian spectrum and the EPDs that form in the non-Hermitian spectra once the gain/loss is introduced. (e) The relationship between the initial split between the coalescing modes in the corresponding Hermitian spectrum (Δ_0) and the critical non-Hermitian strength (γ_EP) required to coalesce those modes. (f) The relationship between the density of EPDs in the spectrum and the fractal dimension of the spectrum.

Figure 5. The universal rules governing the formation of EPDs. Finite element models (FEM) of (a) a PT-symmetric Aubrey-Harper quasi-periodic metastructure, (b) a PT-symmetric H-tree geometric fractal metastructure, (c) a PT-symmetric aperiodic (following Fibonacci sequence) metastructure. (d) A coupled-mode theory model that follows a Fibonacci substitution rule. All these systems exhibit an unfolding (fractal) spectrum as shown in the center and when the gain/loss intensity (non-Hermitian strength) is increased, numerous EPDs form at different regions of the spectra in a scale-free fashion enabling us to investigate the underlying relations between the modes in the initial Hermitian spectrum and the EPDs that form in the non-Hermitian spectra once the gain/loss is introduced. (e) The relationship between the initial split between the coalescing modes in the corresponding Hermitian spectrum (Δ0) and the critical non-Hermitian strength (γEP) required to coalesce those modes. (f) The relationship between the density of EPDs in the spectrum and the fractal dimension D of the spectrum. [4]

 

Can a PT-symmetric defect in a periodic system yield anything useful? We recently showed that a PT-symmetric defect embedded in a periodic metastructure itself enable the formation of multiple EPDs with their formation tailored by the defect position—i.e., the distance between the Parity-symmetric components of the defect—in addition to the non-Hermitian strength [7]. Besides giving additional experimental controls for tailoring the underlying relations between the initial spectra and the critical non-Hermitian strength required for the EPD formation, the ability to use merely a PT-symmetric defect embedded in a periodic media to create multiple EPDs is certainly beneficial for experimental realization compared to a more complex PT-symmetric system requiring gain and loss control in every unit cell [7]. What are the other consequences of such PT-symmetric defects to the band topology?

What happens to noise at the proximity to EPDs? It has been pointed out that the hypersensitivity emerging from EPDs may not be beneficial if the noise is also amplified near the EPDs [30,31]. The noise can be of fundamental nature owing to the eigenbasis collapse at EPDs or of technical nature associated with the amplification mechanisms utilized for the realization of EPDs. In a recent study in electromechanical framework [17], we have shown that the signal-to-noise of an accelerometer can be significantly enhanced by overcoming the fundamental noise enhancement if the EPD sensor is weakly coupled to a transmission line where a transmission peak degeneracy (TPD) is realized, and its detuning is utilized as the measure of sensitivity. We have demonstrated a three-fold enhancement in signal-to-noise ratio near the TPD with high dynamic range where this TPD based accelerometer outperforms conventional linear sensors [17]. While this study steers the utilization of EPDs towards beneficial applications, it also raises further questions and opens research opportunities to address the effects of framework/platform specific (technical) noise and strategies for realizing sensors with selectivity, high signal-to-noise ratio, and high dynamic range.

As we have shown in a prior study in acoustic framework [26], incorporation of dissipative (imaginary) nonlinearities—in contrast to stiffness (real) nonlinearities—can lead to novel asymmetric wave steering. For example, we showed that an amplitude-dependent asymmetric wave rectification with high frequency purity—i.e., rectification performed without higher harmonic generation—can be achieved. How could such dissipative nonlinearities be engineered in material systems without the presence of stiffness nonlinearities? (Note that this question has a fundamental limitation in the linear regime; e.g., the Kramer-Kronig relation dictates that the real and imaginary parts of the dynamic modulus of a viscoelastic material are intimately connected in linear viscoelasticity).

Another promising direction is the exploitation of non-resonant EPDs, i.e., the formation of singularities in spectra of operators other than the effective Hamiltonian that describes the dynamics of the resonant system. For example, in the framework of photonics, it is known for a while that EPDs in Bloch modes of a periodic structure can lead to slow light phenomena [32]. A similar approach can be taken in the framework of elastodynamic media for creating slow waves. Such systems do not involve gain or loss elements and instead exploit the non-Hermitian nature of the transfer matrix that describe the wave propagation across the unit cell of the periodic media. These types of EPDs have also been proposed for hypersensitive sensing [33] where the observable now is a sublinear (Wigner) cusp anomaly in the differential cross-section of scattering processes. To some extent, this way of utilizing EPDs can be thought of as an alternative to topological protection since the scattering characteristics remain robust against local perturbations [34] and respond only to global perturbations, e.g. frequency variations of the incident wave.

Last but not least, what are the consequences of incorporating non-Hermitian elements in topological systems to topological phases [35] and boundary effects [36,37]? This direction also has plethora of fundamental questions that could lead to interesting wave physics.

In summary, non-Hermitian symmetries engineered into metamaterials and phononic crystals can lead to unusual properties and functionalities that could benefit mechanical wave and vibration control, highly responsive robotic systems, and precision instrumentation.

Acknowledgements

I would like to thank my present and past research team members Abhishek Gupta, Yanghao Fang, Jizhe Cai, Jiayan Zhang, Kyle Seledic, and Melissa Schmidt-Landin—whose contributions over the years on the non-Hermitian metamaterials research thrust of our lab educated me enough to draft the above article—for their enthusiastic commitment to science and our growth as a team. A special thanks to Prof. Tsampikos Kottos, our theoretical physics collaborator from the Wesleyan University for his unwavering appreciation and commitment towards experimentation and engineering that led to our long-term collaboration resulting in the breadth and the depth of the work presented here. Finally, we acknowledge the financial support for this research thrust from the National Science Foundation, the Army Research Office, and the Wisconsin Alumni Research Foundation.

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