Diabolical Points Research Articles

Long-Lived Magnetization in an Atomic Spin Chain Tuned to a Diabolic Point.

Scaling magnets down to where quantum size effects become prominent triggers quantum tunneling of magnetization (QTM), profoundly influencing magnetization dynamics. Measuring magnetization switching in an Fe atomic chain under a carefully tuned transverse magnetic field, we observe a nonmonotonic variation of magnetization lifetimes around a level crossing, known as the diabolic point (DP). Near DPs, local environment effects causing QTM are efficiently suppressed, enhancing lifetimes by three orders of magnitude. Adjusting interatomic interactions further facilitates multiple DPs. Our Letter provides a deeper understanding of quantum dynamics near DPs and enhances our ability to engineer a quantum magnet.

Electrically tunable topological phase transition in non-Hermitian optical MEMS metasurfaces.

Exceptional points (EPs), unique junctures in non-Hermitian open systems where eigenvalues and eigenstates simultaneously coalesce, have gained notable attention in photonics because of their enthralling physical principles and unique properties. Nonetheless, the experimental observation of EPs, particularly within the optical domain, has proven rather challenging because of the grueling demand for precise and comprehensive control over the parameter space, further compounded by the necessity for dynamic tunability. Here, we demonstrate the occurrence of optical EPs when operating with an electrically tunable non-Hermitian metasurface platform that synergizes chiral metasurfaces with piezoelectric MEMS mirrors. Moreover, we show that, with a carefully constructed metasurface, a voltage-controlled spectral space can be finely tuned to access not only the chiral EP but also the diabolic point characterized by degenerate eigenvalues and orthogonal eigenstates, thereby allowing for dynamic topological phase transition. Our work paves the way for developing cutting-edge optical devices rooted in EP physics and opening uncharted vistas in dynamic topological photonics.

Sensitivity enhancement at higher order exceptional point based on coupled whispering gallery modes in microstructured optical fibers

Sensitivity enhancement at higher order exceptional point (EP) was demonstrated based on coupled whispering gallery modes in microstructured optical fibers. Nonlinear wavelength shift response near the third-order EP (EP(3)) could be obtained at four typical refractive indices (RIs) by exploiting the adjustable RI feature of functional materials. The maximum sensitivity near EP reaches about 1.5 × 104 nm/RIU, and the sensitivity at EP approaches infinity. Besides, the sensing characteristics of a second-order EP system and a diabolic point system are compared with the EP(3) system for the same RI measurement range around 1.45 and the calculation results indicate that EP(3) system experiences an increase in RI sensitivity around EP by 6 and 10 times, respectively. In addition, tuning the real part of the medium RI of gain & loss microcavities helps to achieve tuning of the sample material RI with high sensitivity. Finally, another orthogonal state of the eigenvectors has been found by analyzing the eigenvector evolution from Hermitian regime to EP state for the EP(3) system and a lower limit of detection could be obtained as well. These finds would be of great significance for noise suppression in sensing applications based on interrogation of eigenfrequency splitting near higher order EPs.

Analysis of Hermitian and non-Hermitian diabolic points and exceptional rings in parity-time symmetric ZRC and RLC dimers

Analysis of Hermitian and non-Hermitian diabolic points and exceptional rings in parity-time symmetric ZRC and RLC dimers

Theinfinity-loop microresonator: A new integrated photonic structure working on an exceptional surface

Exceptional points, where eigenvalues and eigenvectors coalesce, impact the behavior of different photonics components that show, e.g., enhanced sensing, coherent perfect absorption, unidirectional lasing, and chirality. However, only a few passive geometries have been developed that work on these points. Here, we introduce a novel non-Hermitian structure based on a microresonator shaped as the infinity symbol twice coupled to a bus waveguide: the infinity-loop microresonator. Unlike other structures working on an exceptional surface, the infinity-loop microresonator can achieve either high- or low-contrast unidirectional reflection with a negligible or identical reflection for counterpropagating light. It allows an easy walking through the Riemann sheet by simply controlling the phase of the light propagating in the bus waveguide, which makes it a tunable component to build more complex topological structures. Furthermore, the infinity-loop microresonator allows sensors that show the features of both an exceptional point device and a diabolic point device simultaneously.

Degeneracies and symmetry breaking in pseudo-Hermitian matrices

Real eigenvalues of pseudo-Hermitian matrices, such as real matrices and $\mathcal{PT}\ensuremath{-}\mathrm{symmetric}$ matrices, frequently split into complex conjugate pairs. This is accompanied by the breaking of certain symmetries of the eigenvectors and, typically, also a drastic change in the behavior of the system. In this paper, we classify the eigenspace of pseudo-Hermitian matrices and show that such symmetry breaking occurs if and only if eigenvalues of opposite kinds collide on the real axis of the complex eigenvalue plane. This enables a classification of the disconnected regions in parameter space where all eigenvalues are real---which correspond, physically, to the stable phases of the system. These disconnected regions are surrounded by exceptional surfaces, which comprise all the real-valued exceptional points of pseudo-Hermitian matrices. The exceptional surfaces, together with the diabolic points created by their intersections, comprise all points of pseudo-Hermiticity breaking. In particular, this clarifies that the degeneracy involved in symmetry breaking is not necessarily an exceptional point. We also discuss how our study relates to conserved quantities and derive the conditions for when degeneracies caused by external symmetries are susceptible to thresholdless pseudo-Hermiticity breaking. We illustrate our results with examples from photonics, condensed matter physics, and mechanics.

Geometric Phase Effect in Attosecond Stimulated X-ray Raman Spectroscopy

Conical intersections (CIs) are diabolical points in the potential energy surfaces generally caused by point-wise degeneracy of different electronic states, and give rise to the geometric phases (GPs) of molecular wave functions. Here we theoretically propose and demonstrate that the transient redistribution of ultrafast electronic coherence in attosecond Raman signal (TRUECARS) spectroscopy is capable of detecting the GP effect in excited state molecules by applying two probe pulses including an attosecond and a femtosecond X-ray pulse. The mechanism is based on a set of symmetry selection rules in the presence of nontrivial GPs. The model of this work can be realized for probing the geometric phase effect in the excited state dynamics of complex molecules with appropriate symmetries, using attosecond light sources such as free-electron X-ray lasers.

Dynamically crossing diabolic points while encircling exceptional curves: A programmable symmetric-asymmetric multimode switch

Nontrivial spectral properties of non-Hermitian systems can lead to intriguing effects with no counterparts in Hermitian systems. For instance, in a two-mode photonic system, by dynamically winding around an exceptional point (EP) a controlled asymmetric-symmetric mode switching can be realized. That is, the system can either end up in one of its eigenstates, regardless of the initial eigenmode, or it can switch between the two states on demand, by simply controlling the winding direction. However, for multimode systems with higher-order EPs or multiple low-order EPs, the situation can be more involved, and the ability to control asymmetric-symmetric mode switching can be impeded, due to the breakdown of adiabaticity. Here we demonstrate that this difficulty can be overcome by winding around exceptional curves by additionally crossing diabolic points. We consider a four-mode {{{{{{{mathcal{PT}}}}}}}}-symmetric bosonic system as a platform for experimental realization of such a multimode switch. Our work provides alternative routes for light manipulations in non-Hermitian photonic setups.

Non-Hermitian higher-order Weyl semimetal with surface diabolic points

Higher-order topology in non-Hermitian (NH) systems has recently become one of the most promising and rapidly developing fields in condensed-matter physics. Many distinct phases that were not present in the Hermitian equivalents are revealed in these systems. In this work, we examine how higher-order Weyl semimetals are impacted by NH perturbation. We identify a new type of topological semimetal, i.e., non-Hermitian higher-order Weyl semimetal (NHHOWS) with surface diabolic points. We demonstrate that in such an NHHOWS, new exceptional points inside the bulk can be created and annihilated, therefore allowing us to manipulate their number. At the boundary, these exceptional points are connected through unique surface states with diabolic points and hinge states. For specific system parameters, the surface of NHHOWS behaves as a Dirac phase with linear dispersion or a Luttinger phase with a quadratic dispersion, thus paving a way for Dirac-Luttinger switching. Finally, we employ the biorthogonal technique to reinstate the standard bulk-boundary correspondence for NH systems and compute the topological invariants. The obtained quantized biorthogonal Chern number and quadruple moment topologically protect the unique surface and hinge states, respectively.

Absorption-lasing effects and exceptional points in parity-time symmetric non-Hermitian metaplates

Non-Hermitian systems that possess loss and gain have been receiving a great deal of interest in various wave problems demonstrating diverse unprecedented wave phenomena. Here, we theoretically and numerically investigate the Parity-Time (PT) symmetric metaplate with balanced loss and gain based on shunted piezoelectric layers. We present the coherent perfect absorption and lasing effects (CPAL) for flexural waves occurring in the PT broken phase via designing the circuit parameters and discuss their physical mechanisms. Moreover, we focus on the exceptional points (EP) behaving as thresholds of phase transitions and realize unidirectional reflectionless for incident waves from different directions by adjusting the circuit parameters. We further employ EP to realize unidirectional cloakings. Finally, we contrastively study the origins and sensitivities of the non-Hermitian EP and the Hermitian diabolic point (DP) for elastic waves. Our study explores complex material parameters of elastic plates and offers a reliable platform for designing the non-Hermitian elastic wave phenomena, paving the way for highly sensitive sensors, asymmetric wave control, wave absorption and amplification, and energy harvesting.

Exceptional-point sensing with a quantum interferometer

Recently, multiple studies have suggested that exceptional points (EPs) in lossless nonlinear optical systems can minimize quantum noise arising from the material gain and loss in conventional non-Hermitian systems, offering the possibility of quantum EP sensing. Meanwhile, nonlinear SU(1,1) interferometers have been established as useful in sensing due to their reduced quantum noise. In this work, we demonstrate the existence of EPs in a dual-beam SU(1,1) interferometer with two nonlinear parametric amplifiers. Our analysis of the input-output matrix in terms of joint quadrature amplitudes shows that EPs can be linked to both high signal, through a zero matrix element, and low noise, through noise preservation, in sensing by selecting an appropriate operation gauge of the quadrature amplitudes. Additionally, for a multistage SU(1,1) interferometer, EPs of the overall input-output matrix form multiple bands of high signal-to-noise ratio (SNR) which further separate into two phases indicated by the EPs of the transfer matrix of a repeating unit. Our investigations demonstrate the significance of quantum EPs in quantum interferometer sensing and broaden the operating regimes from diabolical points in some of the conventional SU(1,1) interferometers to EPs while still maintaining a high SNR.

Diabolical points in coupled ridge resonators

In this paper, the resonant optical properties of two coupled ridges on the surface of a single-mode dielectric slab waveguide are theoretically investigated. The diabolical points of a structure can be achieved by near-field coupling and far-field interference of resonance modes in the ridges. The uniform unity transmission and 2 π spectral phase span are demonstrated around the diabolical points. By using the temporal coupled-mode theory, we analyze the generating conditions of diabolical points in coupled ridges and the corresponding group delay around diabolical points. The influences of optical loss and excitation beam width on the optical response at diabolical points of a structure are also discussed. Diabolical points in the waveguide system have potential applications in dispersion compensation and phase-only modulators in integrated optical circuits.

Quantum Liouvillian exceptional and diabolical points for bosonic fields with quadratic Hamiltonians: The Heisenberg-Langevin equation approach

Equivalent approaches to determine eigenfrequencies of the Liouvillians of open quantum systems are discussed using the solution of the Heisenberg-Langevin equations and the corresponding equations for operator moments. A simple damped two-level atom is analyzed to demonstrate the equivalence of both approaches. The suggested method is used to reveal the structure as well as eigenfrequencies of the dynamics matrices of the corresponding equations of motion and their degeneracies for interacting bosonic modes described by general quadratic Hamiltonians. Quantum Liouvillian exceptional and diabolical points and their degeneracies are explicitly discussed for the case of two modes. Quantum hybrid diabolical exceptional points (inherited, genuine, and induced) and hidden exceptional points, which are not recognized directly in amplitude spectra, are observed. The presented approach via the Heisenberg-Langevin equations paves the general way to a detailed analysis of quantum exceptional and diabolical points in infinitely dimensional open quantum systems.

Distance between exceptional points and diabolic points and its implication for the response strength of non-Hermitian systems

Exceptional points are non-Hermitian degeneracies in open quantum and wave systems at which not only eigenenergies but also the corresponding eigenstates coalesce. This is in strong contrast to degeneracies known from conservative systems, so-called diabolic points, at which only eigenenergies degenerate. Here we connect these two kinds of degeneracies by introducing the concept of the distance of a given exceptional point in matrix space to the set of diabolic points. We prove that this distance determines an upper bound for the response strength of a non-Hermitian system with this exceptional point. A small distance therefore implies a weak spectral response to perturbations and a weak intensity response to excitations. This finding has profound consequences for physical realizations of exceptional points that rely on perturbing a diabolic point. Moreover, we exploit this concept to analyze the limitations of the spectral response strength in passive systems. A number of optical and photonics systems are investigated to illustrate the theory.

Light-Controlled Exceptional Point Sensor Based on Azo-Functionalized Whispering Gallery Mode Microcavity

Nonlinear topological responses in the vicinity of exceptional points (EPs) significantly enhance mode splitting sensitivity of non-Hermitian systems to external perturbations. In this paper, a new method is proposed to flexibility control EP state of the whispering gallery mode microcavity integrated with photo-sensitive azobenzene coating, which provides a simple, efficient and stabilized scheme to prepare and tune the EP state with no need to delicately control the interaction between the nanoscatters and the microcavity. By introducing one pair of defects infiltrated with liquid samples, refractive index sensing could be realized when the system is operating around EP and the measurement range of 1.33 to 1.4 covers the RI range of most biological analytes. And moreover, performances of the proposed system in nanoparticle detection are also investigated. In comparison with its counterpart operating around diabolic points, our proposed system exhibits more than quadruple increase in sensitivity. Our proposed EP sensor based on light-controlled tuning of the system state makes it a promising candidate for applications in the field of optical sensing and paves a way toward developing tunable EP-based photonic devices.

Diabolical touching point in the magnetic energy levels of topological nodal-line metals

For three-dimensional metals, Landau levels disperse as a function of the magnetic field and the momentum wave number parallel to the field. In this two-dimensional parameter space, it is shown that two conically dispersing Landau levels can touch at a diabolical point---a Landau-Dirac point. The conditions giving rise to Landau-Dirac points are shown to be magnetic breakdown (field-driven quantum tunneling) and certain crystallographic spacetime symmetry. Both conditions are realizable in topological nodal-line metals, as we exemplify with the material candidates $\mathrm{Ca}{X}_{3}$ $(X=\mathrm{As},\mathrm{P})$. The experimental fingerprints of a Landau-Dirac point include (a) anomalous ``batman''-like peaks in the magnetoresistance, (b) circular Landau-Fermi surfaces revealed by angle-dependent ultrasonic attenuation, and (c) the tunability of the frequency onset of optical absorption to zero.

Liouvillian spectral collapse in the Scully-Lamb laser model

Phase transitions of thermal systems and the laser threshold were first connected more than forty years ago. Despite the nonequilibrium nature of the laser, the Landau theory of thermal phase transitions, applied directly to the Scully-Lamb laser model (SLLM), indicates that the laser threshold is a second-order phase transition, associated with a $U(1)$ spontaneous symmetry breaking (SSB). To capture the genuine nonequilibrium phase transition of the SLLM (i.e., a single-mode laser without a saturable absorber), here we employ a quantum theory of dissipative phase transitions. Our results confirm that the $U(1)$ SSB can occur at the lasing threshold but, in contrast to the Landau theory and semiclassical approximation, they signal that the SLLM "fundamental" transition is a different phenomenon, which we call Liouvillian spectral collapse; that is, the emergence of diabolic points of infinite degeneracy. By considering a generalized SLLM with additional dephasing, we witness a second-order phase transition, with a Liouvillian spectral collapse, but in the absence of symmetry breaking. Most surprisingly, the phase transition corresponds to the emergence of dynamical multistability even without SSB. Normally, bistability is suppressed by quantum fluctuations, while in this case, the very presence of quantum fluctuations enables bistability. This rather anomalous bistability, characterizing the truly dissipative and quantum origin of lasing, can be an experimental signature of our predictions, and we show that it is associated with an emergent dynamical hysteresis.

Spectra, eigenstates and transport properties of a -symmetric ring

We study, analytically and numerically, a simple -symmetric tight-binding ring with an onsite energy a at the gain and loss sites. We show that if a ≠ 0, the system generically exhibits an unbroken -symmetric phase. We study the nature of the spectrum in terms of the singularities in the complex parameter space as well as the behavior of the eigenstates at large values of the gain and loss strength. We find that in addition to the usual exceptional points (EPs), there are ‘diabolical points’, and inverse EPs at which complex eigenvalues reconvert into real eigenvalues. We also study the transport through the system. We calculate the total flux from the source to the drain, and how it splits along the branches of the ring. We find that while usually the density flows from the source to the drain, for certain eigenstates a stationary ‘backflow’ of density from the drain to the source along one of the branches can occur. We also identify two types of singular eigenstates, i.e. states that do not depend on the strength of the gain and loss, and classify them in terms of their transport properties.

Tuning of the Berry curvature in 2D perovskite polaritons.

The engineering of the energy dispersion of polaritons in microcavities through nanofabrication or through the exploitation of intrinsic material and cavity anisotropies has demonstrated many intriguing effects related to topology and emergent gauge fields such as the anomalous quantum Hall and Rashba effects. Here we show how we can obtain different Berry curvature distributions of polariton bands in a strongly coupled organic-inorganic two-dimensional perovskite single-crystal microcavity. The spatial anisotropy of the perovskite crystal combined with photonic spin-orbit coupling produce two Hamilton diabolical points in the dispersion. An external magnetic field breaks time-reversal symmetry owing to the exciton Zeeman splitting and lifts the degeneracy of the diabolical points. As a result, the bands possess non-zero integral Berry curvatures, which we directly measure by state tomography. In addition to the determination of the different Berry curvatures of the multimode microcavity dispersions, we can also modify the Berry curvature distribution, the so-called band geometry, within each band by tuning external parameters, such as temperature, magnetic field and sample thickness.

Enhanced emission from a single quantum dot in a microdisk at a deterministic diabolical point.

We report on controllable cavity modes by controlling the backscattering by two identical scatterers. Periodic changes of the backscattering coupling between two degenerate cavity modes are observed with the changing angle between two scatterers and elucidated by a theoretical model using two-mode approximation and numerical simulations. The periodically appearing single-peak cavity modes indicate mode degeneracy at diabolical points. Interactions between single quantum dots and cavity modes are then investigated. Enhanced emission of a quantum dot with a six-fold intensity increase is obtained in a microdisk at a diabolical point. This method to control cavity modes allows large-scale integration, high reproducibility and flexible design of the size, the location, the quantity and the shape for scatterers, which can be applied for integrated photonic structures with scatterer-modified light-matter interaction.