Exceptional Degeneracy Research Articles

Realizing Exceptional Points by Floquet Dissipative Couplings in Thermal Atoms.

Exceptional degeneracies and generically complex spectra of non-Hermitian systems are at the heart of numerous phenomena absent in the Hermitian realm. Recently, it was suggested that Floquet dissipative coupling in the space-time domain may provide a novel mechanism to drive intriguing spectral topology with no static analogs, though its experimental investigation in quantum systems remains elusive. We demonstrate such Floquet dissipative coupling in an ensemble of thermal atoms interacting with two spatially separated optical beams, and observe an anomalous anti-parity-time symmetry phase transition at an exception point far from the phase-transition threshold of the static counterpart. Our protocol sets the stage for Floquet engineering of non-Hermitian topological spectra, and for engineering new quantum phases that cannot exist in static systems.

Uncovering exceptional contours in non-Hermitian hyperbolic lattices

Hyperbolic lattices are starting to be explored in search of novel phases of matter. At the same time, non-Hermitian physics has come to the forefront in photonic, optical, phononic, and condensed matter systems. In this work, we introduce non-Hermitian hyperbolic lattices and elucidate its exceptional properties in depth. We use hyperbolic Bloch theory to investigate band structures of hyperbolic lattices in the presence of non-Hermitian on-site gain and loss as well as non-reciprocal hopping. Using various analytical and numerical approaches we demonstrate widely accessible and tunable exceptional points (EPs) and contours in {10,5} tessellations, which we characterize using phase rigidity, energy scaling, and vorticity. We further demonstrate the occurrence of higher-order EPs and contours in the {8,4} tessellations using the method of Newton polygons, supported by vorticity and phase rigidity computations. Finally, we investigate the open boundary spectra and densities of states to compare with results from band theory, along with a demonstration of boundary localisation. Our results unveil an abundance of exceptional degeneracies in hyperbolic non-Hermitian matter.

Multiple exceptional points and phase transitions of a one-dimensional PT-symmetric Lieb photonic lattice

Exceptional points (EPs) in non-Hermitian systems have attracted enormous attention and spawned intriguing prospects for the manipulation of waves. Despite many efforts focusing on the exotic behaviors about EPs, there are only a few studies of phase transitions involving multiple EPs. Here, by employing staggered couplings as well as two pairs of on-site gain/loss, we propose a one-dimensional parity-time (PT)-symmetric Lieb photonic lattice and demonstrate diverse phase transitions of such a multiband structure. Owing to the non-Hermitian chiral symmetry, symmetry-protected higher-order EPs are constructed, and the system exhibits PT symmetry breaking beyond a certain threshold. More importantly, both the relative couplings and the on-site gain/loss can be flexibly reconfigured on demand, which yields the degeneracy of different bands, i.e., the emergence of multiple EPs. We also unveil that the EPs will no longer exist in the presence of a non-Hermitian diagonal disorder. In contrast, the spectrum remains symmetric and the EPs, along with the flatband, are robust against the off diagonal disorder due to the preserved non-Hermitian particle-hole symmetry. Our work not only provides a controllable platform for studying EPs but also sheds light on the exciting non-Hermitian physics based on exceptional degeneracies.

Lasing at a stationary inflection point

The concept of lasers based on the frozen mode regime in active periodic optical waveguides with a 3rd-order exceptional point of degeneracy (EPD) is advanced. The frozen mode regime in a lossless and gainless waveguide is associated with a stationary inflection point (SIP) in the Bloch dispersion relation, where three Bloch eigenmodes coalesce forming the frozen mode. As a practical example, we consider an asymmetric serpentine optical waveguide (ASOW). An ASOW operating near the SIP frequency displays a large group delay of a non-resonant nature that scales as the cube of the waveguide length, leading to a strong gain enhancement when active material is included. Therefore, a laser operating in the close vicinity of an SIP has a gain threshold that scales as a negative cube of the waveguide length. We determine that this scaling law is maintained in the presence of small distributed losses, such as radiation associated with waveguide bends and roughness. In addition, we show that although gain causes a distortion in the modes coalescing at the SIP, the properties of the frozen mode are relatively resistant to such small perturbations and we still observe a large degree of exceptional degeneracy for gain values that bring the system above threshold. Finally, our study also reveals that lasing near an SIP is favored over lasing near a photonic band edge located in close proximity to the SIP. In particular, we observe that an SIP-induced lasing in an ASOW displays lower gain threshold compared to lasing near the photonic regular band edge (RBE), even though the SIP resonance has a lower quality factor than the RBE resonance.

Enhanced eigenvector sensitivity and algebraic classification of sublattice-symmetric exceptional points

Exceptional points (EPs) are degeneracy of non-Hermitian Hamiltonians, at which the eigenvalues, along with their eigenvectors, coalesce. Their orders are given by the Jordan decomposition. Here, we focus on higher-order EPs arising in fermionic systems with a sublattice symmetry, which restricts the eigenvalues of the Hamitlonian to appear in pairs of $\lbrace E, -E\rbrace $. Thus, a naive prediction might lead to only even-order EPs at zero energy. However, we show that odd-order EPs can exist and exhibit enhanced sensitivity in the behaviour of eigenvector-coalescence in their neighbourhood, depending on how we approach the degenerate point. The odd-order EPs can be understood as a mixture of higher- and lower-valued even-order EPs. Such an anomalous behaviour is related to the irregular topology of the EPs as the subspace of the Hamiltonians in question, which is a unique feature of the Jordan blocks. The enhanced eigenvector sensitivity can be described by observing how the quantum distance to the target eigenvector converges to zero. In order to capture the eigenvector-coalescence, we provide an algebraic method to describe the conditions for the existence of these EPs. This complements previous studies based on resultants and discriminants, and unveils heretofore unexplored structures of higher-order exceptional degeneracy.

Exceptional Non-Abelian Topology in Multiband Non-Hermitian Systems.

Defective spectral degeneracy, known as exceptional point (EP), lies at the heart of various intriguing phenomena in optics, acoustics, and other nonconservative systems. Despite extensive studies in the past two decades, the collective behaviors (e.g., annihilation, coalescence, braiding, etc.) involving multiple exceptional points or lines and their interplay have been rarely understood. Here we put forward a universal non-Abelian conservation rule governing these collective behaviors in generic multiband non-Hermitian systems and uncover several counterintuitive phenomena. We demonstrate that two EPs with opposite charges (even the pairwise created) do not necessarily annihilate, depending on how they approach each other. Furthermore, we unveil that the conservation rule imposes strict constraints on the permissible exceptional-line configurations. It excludes structures like Hopf link yet permits novel staggered rings composed of noncommutative exceptional lines. These intriguing phenomena are illustrated by concrete models which could be readily implemented in platforms like coupled acoustic cavities, optical waveguides, and ring resonators. Our findings lay the cornerstone for a comprehensive understanding of the exceptional non-Abelian topology and shed light on the versatile manipulations and applications based on exceptional degeneracies in nonconservative systems.

Exceptional degeneracies in non-Hermitian Rashba semiconductors

Exceptional points (EPs) are spectral degeneracies of non-Hermitian (NH) systems where eigenvalues and eigenvectors coalesce, inducing unique topological phases that have no counterpart in the Hermitian realm. Here we consider an NH system by coupling a two-dimensional semiconductor with Rashba spin–orbit coupling (SOC) to a ferromagnet lead and show the emergence of highly tunable EPs along rings in momentum space. Interestingly, these exceptional degeneracies are the endpoints of lines formed by the eigenvalue coalescence at finite real energy, resembling the bulk Fermi arcs commonly defined at zero real energy. We then show that an in-plane Zeeman field provides a way to control these exceptional degeneracies although higher values of non-Hermiticity are required in contrast to the zero Zeeman field regime. Furthermore, we find that the spin projections also coalescence at the exceptional degeneracies and can acquire larger values than in the Hermitian regime. Finally, we demonstrate that the exceptional degeneracies induce large spectral weights, which can be used as a signature for their detection. Our results thus reveal the potential of systems with Rashba SOC for realizing NH bulk phenomena.

Crossing of the Branch Cut: The Topological Origin of a Universal 2π‐Phase Retardation in Non‐Hermitian Metasurfaces

AbstractFull wavefront control by photonic components requires that the spatial phase modulation on an incoming optical beam ranges from 0 to 2π. Because of their radiative coupling to the environment, all optical components are intrinsically non‐Hermitian systems, often described by reflection and transmission matrices with complex eigenfrequencies. Here, it is shown that parity or time symmetry breaking—either explicit or spontaneous—moves the position of zero singularities of the reflection or transmission matrices from the real axis to the upper part of the complex frequency plane. A universal 0 to 2π‐phase gradient of an output channel as a function of the real frequency excitation is thus realized whenever the discontinuity branch bridging a zero and a pole, that is, a pair of singularities, is crossing the real axis. This basic understanding is applied to engineer electromagnetic fields at interfaces, including, but not limited to, metasurfaces. Non‐Hermitian topological features associated with exceptional degeneracies or branch cut crossing are shown to play a surprisingly pivotal role in the design of resonant photonic systems.

Highly Sensitive Coupled Oscillator Based on an Exceptional Point of Degeneracy and Nonlinearity

We propose a scheme for obtaining highly-sensitive oscillators in a coupled-resonator system with an exceptional point of degeneracy (EPD) and a small instability. The oscillator with the exceptional degeneracy is realized by using two coupled resonators with an almost balanced small-signal gain and loss, that saturates due to nonlinear effects of the active component, resulting in an oscillation frequency that is very sensitive to a perturbation of the circuit. Two cases are investigated, with two parallel LC resonators with balanced small-signal gain and loss that are either coupled wirelessly by mutual inductance or coupled-wired by a capacitor. This paper demonstrates theoretically and experimentally the conditions to obtain a second-order EPD oscillator and analyzes the ultrasensitivity of the oscillation frequency to components' perturbation, including the case of asymmetric perturbation that breaks PT-symmetry. We discuss the effects of nonlinearity on the performance of the oscillator and how the proposed scheme improves the sensing's sensitivity of perturbations. In contrast to previous methods, our proposed degenerate oscillator can sense positive or negative changes of a circuit component. The degenerate oscillator circuit may find applications in various areas such as ultrasensitive sensors, tunable oscillators and modulators.

Third order modal exceptional degeneracy in waveguides with glide-time symmetry

The dispersion of a three-way waveguide is engineered to exhibit exceptional modal characteristics. Two coupled waveguides with Parity-Time (PT) symmetry have been previously demonstrated to exhibit second order exceptional points of degeneracy (EPDs). In this work, we introduce and investigate a particular class of EPDs, applicable from radio frequency to optical wavelengths, whereby three coupled waveguides satisfy Glide-Time (GT) symmetry to exhibit a third order modal degeneracy with a real-valued wavenumber. GT symmetry involves glide symmetry of lossless/gainless components of the waveguide in addition to changing the sign of passive/active elements while applying a glide symmetry operation. This GT-symmetry condition allows three Floquet-Bloch eigenmodes of the structure to coalesce to a real-valued wavenumber at a single frequency, in addition of having one branch of the dispersion diagram with a purely real wavenumber. The proposed scheme may have applications including but not limited to distributed amplifiers, radiating arrays, and sensors, from radio frequency to optics.

Triple Ladder Lumped Circuit With Sixth Order Modal Exceptional Degeneracy

We introduce a circuit topology based on a simple triple-ladder circuit realized with lumped reactive components that provides a sixth order degenerate band-edge (6DBE). The 6DBE is a special kind of sixth-order exceptional point of degeneracy in a lossless and gainless periodic ladder. This degeneracy provides a very flat band edge in the phase-frequency dispersion diagram. The proposed topology exhibits unique structured resonance features associated with a high loaded Q-factor. We investigate the Floquet-Bloch modes in an infinite-length periodic triple-ladder and their dispersion relation using the S parameter formalism. We also provide the approximate analytic expressions of the eigenmodes and dispersion relation around the degenerate point based on the Puiseux series expansion. We investigate the filtering characteristics of a finite-length structure terminated with loads to highlight the special properties of the 6DBE compared to ladders with regular band edge (RBE) and fourth order degenerate band edge (DBE). The circuit framework introduced here with a 6DBE can be exploited in designing novel high Q-factor oscillators, filters, sensors, and pulse shaping networks.

Exceptional odd-frequency pairing in non-Hermitian superconducting systems

We first show the realization of exceptional points in a non-Hermitian superconducting system based on a conventional superconductor and then demonstrate that, surprisingly, the system hosts odd-frequency pairing, solely generated by the non-Hermiticity. While there is a coexistence of even- and odd-frequency pairs under general conditions, we find that the even-frequency term vanishes at the exceptional degeneracies, leaving only odd-frequency pairing. This exceptional odd-frequency pairing is directly given by the imaginary part of the eigenvalues at the exceptional points and can be measured from the spectral function. Our results thus put forward non-Hermitian systems as a powerful platform to realize odd-frequency superconducting pairing.

Classification of exceptional nodal topologies protected by PT symmetry

Exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, and parity-time ($\mathcal{PT}$) symmetry, reflecting balanced gain and loss in photonic systems, are paramount concepts in non-Hermitian systems. We here complete the topological classification of exceptional nodal degeneracies protected by $\mathcal{PT}$ symmetry in up to three dimensions and provide simple example models whose exceptional nodal topologies include previously overlooked possibilities such as second-order knotted surfaces of arbitrary genus, third-order knots and fourth-order points.

Dynamically Induced Exceptional Phases in Quenched Interacting Semimetals.

We report on the dynamical formation of exceptional degeneracies in basic correlation functions of nonintegrable one- and two-dimensional systems quenched to the vicinity of a critical point. Remarkably, fine-tuned semimetallic points in the phase diagram of the considered systems are thereby promoted to topologically robust non-Hermitian (NH) nodal phases emerging in the coherent time evolution of a dynamically equilibrating system. Using nonequilibrium Green's function methods within the conserving second Born approximation, we predict observable signatures of these NH nodal phases both in equilibrated spectral functions and in the nonequilibrium dynamics of momentum distribution functions.

Exceptional non-Hermitian phases in disordered quantum wires

We demonstrate the occurrence of nodal non-Hermitian (NH) phases featuring exceptional degeneracies in chiral-symmetric disordered quantum wires, where NH physics naturally arises from the self-energy in a disorder-averaged Green's function description. Notably, we find that at least two nodal points in the clean Hermitian system are required for exceptional points to be effectively stabilized upon adding disorder. We identify and study experimental signatures of our theoretical findings both in the spectral functions and in mesoscopic quantum transport properties of the considered systems. Our results are quantitatively corroborated and exemplified by numerically exact simulations on a microscopic lattice model. The proposed setting provides a conceptually minimal framework for the realization and study of topological NH phases in quantum many-body systems.

Exceptional degeneracies in traveling wave tubes with dispersive slow-wave structure including space-charge effect

The interaction between a linear electron beam and a guided electromagnetic wave is studied in the context of exceptional points of degeneracy (EPD) supported by such an interactive system, focusing on a linear beam traveling wave tube (TWT) with a realistic helix waveguide slow-wave structure (SWS). The interaction is formulated by an analytical model that is a generalization of the Pierce model, assuming a one-dimensional electron flow along a dispersive single-mode guiding SWS and taking into account space-charge effects. The augmented model using phase velocity and characteristic impedance obtained via full-wave simulations is validated by calculating gain vs frequency and comparing it with that from more complex electron beam simulators. This comparison also shows the accuracy of our new model compared with respect to the nondispersive Pierce model. EPDs are then investigated using the augmented model, observing the coalescence of complex-valued wavenumbers and the system's eigenvectors. The point in the complex dispersion diagram at which the TWT-system starts/ceases to exhibit a convection instability, i.e., a mode starts/ceases to grow exponentially along the TWT, is the EPD. We also demonstrate the EPD existence by showing that the Puiseux fractional power series expansion well approximates the bifurcation of the dispersion diagram at the EPD. This latter concept also explains the “exceptional” sensitivity of the TWT-system to changes in the beam's electron velocity when operating near an EPD.

Exceptional degeneracy in a waveguide periodically loaded with discrete gain and radiation loss elements

We demonstrate that a periodic waveguide comprising of uniform lossless segments together with discrete gain and radiating elements supports exceptional points of degeneracy (EPDs). We provide analytical expressions for all possible conditions that guarantee the occurrence of an EPD, i.e., the coalescence of eigenvalues and eigenvectors. We show that EPDs are not only achieved using symmetric gain and radiation periodic loading, but they are also obtained using asymmetric gain and radiation loss conditions. We illustrate the characteristics of the degenerate electromagnetic modes, showing the dispersion diagram and discussing the tunability of the EPD frequency. We show a special condition, and we refer to it as a parity-time-glide symmetry, which leads to a degeneracy that is occurring at all frequencies of operation. The class of EPDs proposed in this work is very promising for many applications that incorporate discrete-distributed coherent sources and radiation loss elements; operating in the vicinity of such special degeneracy conditions leads to a potential performance enhancement in a variety of microwave and optical resonators, antennas, and devices and can be extended to a new class of active integrated antenna arrays and radiating laser arrays.

Exceptional topology of non-Hermitian systems

The current understanding of the role of topology in non-Hermitian (NH) systems and its far-reaching physical consequences observable in a range of dissipative settings are reviewed. In particular, how the paramount and genuinely NH concept of exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, leads to phenomena drastically distinct from the familiar Hermitian realm is discussed. An immediate consequence is the ubiquitous occurrence of nodal NH topological phases with concomitant open Fermi-Seifert surfaces, where conventional band-touching points are replaced by the aforementioned exceptional degeneracies. Furthermore, new notions of gapped phases including topological phases in single-band systems are detailed, and the manner in which a given physical context may affect the symmetry-based topological classification is clarified. A unique property of NH systems with relevance beyond the field of topological phases consists of the anomalous relation between bulk and boundary physics, stemming from the striking sensitivity of NH matrices to boundary conditions. Unifying several complementary insights recently reported in this context, a picture of intriguing phenomena such as the NH bulk-boundary correspondence and the NH skin effect is put together. Finally, applications of NH topology in both classical systems including optical setups with gain and loss, electric circuits,s and mechanical systems and genuine quantum systems such as electronic transport settings at material junctions and dissipative cold-atom setups are reviewed.

Complex Berry curvature pair and quantum Hall admittance in non-Hermitian systems

We propose complex Berry curvatures associated with the non-Hermitian Hamiltonian and its Hermitian adjoint and use these to reveal new physics in non-Hermitian systems. We give the complex Berry curvature and Berry phase for the two-dimensional non-Hermitian Dirac model. The imaginary part of the complex Berry phase induces susceptance so that the quantum Hall conductance is generalized to admittance for non-Hermitian systems. This implies that the non-Hermiticity of physical systems can induce intrinsic capacitive or inductive properties, depending on the non-Hermitian parameters. We analyze the complex energy band structures of the two-dimensional non-Hermitian Dirac model, determine the point and line gaps, and identify the conditions for their closure. We find that closure is associated with the exceptional degeneracy of the energy bands in the parameter space, which, in turn, is associated with topological phase transitions. In the continuum limit, we obtain the complex Berry phase in the parameter space.

Two-level system as topological actuator for nanomechanical modes

We investigate theoretically the dynamics of two quasidegenerate mechanical modes coupled through an open quantum two-level system. A mean-field approach shows that by engineering the retarded response of the two-level system with a coherent drive, the non-Hermitian mechanical spectrum exhibits an exceptional degeneracy point where the two modes coalesce. We show that this degeneracy can be exploited to manipulate the vectorial polarization of the mechanical oscillations. We find that adiabatically varying the detuning and the intensity of the drive induces a rotation of the mechanical polarization, which enables the topological and chiral actuation of one mode from the other. This topological manifestation of the degeneracy is further supported by quantum-jump Monte Carlo simulations to account for the strong quantum fluctuations due to the spontaneous emission of the two-level system. Our presentation focuses on a promising realization based on flexural modes of a carbon-nanotube cantilever coupled to a single-molecule electric dipole irradiated by a laser.