Interference of Non-Hermiticity with Hermiticity at Exceptional Points

We theoretically study the dissipative Bose-Hubbard model describing the array of tunneling-coupled cavities with non-conservative photon-photon interaction. The bound two-photon states are formed in this system either in the limited range of the center-of-mass wave vectors or in the full Brillouin zone, depending on the strength of the dissipative interaction. Transition between these two regimes is manifested as an exceptional point in the complex energy spectrum. This improves fundamental understanding of the interplay of non-Hermiticity and interactions in the quantum structures and can potentially be used for on-demand nonlinear light generation in photonic lattices.

Exceptional points (EPs) in photonics are associated with non-Hermitian Hamiltonians with energy gain or loss, a notable example is EPs in parity-time symmetric Hamiltonians. We show that, counterintuitively, actual energy gain or loss is not required to generate this type of non-Hermitian degeneracy, since the eigenvalue of the optical Hamiltonian is the mode's propagation constant and not the energy. This is demonstrated by a simple three-layer insulator-metal-insulator (I-M-I) plasmonic waveguide, the eigenmodes of which are known to experience degeneracy at a certain point in the parametric space suggested to be used for rainbow trapping. We identify this point to be, in fact, an EP with the coalescence of the eigenmodes, despite the system having neither parity nor time symmetry. Furthermore, we demonstrate the manifestation of a third-order EP, which is generated by merging two separate EPs in the parametric space of the I-M-I waveguide. The presented results reveal unconventional properties of the Hamiltonian of a simple plasmonic waveguide and provide an insight into the nature of EPs in non-Hermitian plasmonic systems in general, suggesting the possibility of accessing even higher-order EP regimes in simple photonic structures without the need for optical gain or loss.

Exceptional points, resulting from non-Hermitian degeneracies, have the potential to enhance the capabilities of quantum sensing. Thus, finding exceptional points in different quantum systems is vital for developing such future sensing devices. Taking advantage of the enhanced light-matter interactions in a confined volume on a metal nanoparticle surface, here we theoretically demonstrate the existence of exceptional points in a system consisting of quantum emitters coupled to a metal nanoparticle of subwavelength scale. By using an analytical quantum electrodynamics approach, exceptional points are manifested as a result of a strong-coupling effect and observable in a drastic splitting of originally coalescent eigenenergies. Furthermore, we show that exceptional points can also occur when a number of quantum emitters are collectively coupled to the dipole mode of localized surface plasmons. Such a quantum collective effect not only relaxes the strong-coupling requirement for an individual emitter, but also results in a more stable generation of the exceptional points. Furthermore, we point out that the exceptional points can be explicitly revealed in the power spectra. A generalized signal-to-noise ratio, accounting for both the frequency splitting in the power spectrum and the system's dissipation, shows clearly that a collection of quantum emitters coupled to a nanoparticle provides a better performance of detecting exceptional points, compared to that of a single quantum emitter.

Engineered non-Hermitian systems featuring exceptional points (EPs) can lead to a host of extraordinary phenomena in diverse fields ranging from photonics, acoustics, opto-mechanics, and electronics to atomic physics. In optics, non-Hermitian dynamics are typically realized using dissipation and phase-insensitive gain accompanied by unavoidable fluctuations. Here, we introduce non-Hermitian dynamics of coupled optical parametric oscillators (OPOs) arising from phase-sensitive amplification and de-amplification, and show their distinct advantages over conventional non-Hermitian systems relying on laser gain and loss. OPO-based non-Hermitian systems can benefit from the instantaneous nature of the parametric gain, noiseless phase-sensitive amplification, and rich quantum and classical nonlinear dynamics. We show that two coupled OPOs can exhibit spectral anti-parity-time (anti-PT) symmetry and a EP between its degenerate and nondegenerate operation regimes. To demonstrate the distinct potentials of the coupled OPO system compared to conventional non-Hermitian systems, we present higher-order EPs with two OPOs, tunable Floquet EPs in a reconfigurable dynamic non-Hermitian system, and the generation of a squeezed vacuum around EPs, all of which are not easy to realize in other non-Hermitian platforms. We believe our results show that coupled OPOs are an outstanding non-Hermitian setting with unprecedented opportunities to realize nonlinear dynamical systems for enhanced sensing and quantum information processing.

Recently, non-Hermitian Hamiltonians have gained a lot of interest, especially in optics and electronics. In particular, the existence of real eigenvalues of non-Hermitian systems has opened a wide set of possibilities, especially, but not only, for sensing applications, exploiting the physics of exceptional points. In particular, the square root dependence of the eigenvalue splitting on different design parameters, exhibited by 2 × 2 non-Hermitian Hamiltonian matrices at the exceptional point, paved the way to the integration of high-performance sensors. The square root dependence of the eigenfrequencies on the design parameters is the reason for a theoretically infinite sensitivity in the proximity of the exceptional point. Recently, higher-order exceptional points have demonstrated the possibility of achieving the nth root dependence of the eigenfrequency splitting on perturbations. However, the exceptional sensitivity to external parameters is, at the same time, the major drawback of non-Hermitian configurations, leading to the high influence of noise. In this review, the basic principles of PT-symmetric and anti-PT-symmetric Hamiltonians will be shown, both in photonics and in electronics. The influence of noise on non-Hermitian configurations will be investigated and the newest solutions to overcome these problems will be illustrated. Finally, an overview of the newest outstanding results in sensing applications of non-Hermitian photonics and electronics will be provided.

We show how an exceptional point of degeneracy (EPD) is formed in a system composed of an electron beam interacting with an electromagnetic mode guided in a slow wave structure (SWS) with distributed power extraction from the interaction zone. Based on this kind of EPD, a new regime of operation is devised for backward wave oscillators (BWOs) as a synchronous and degenerate regime between a backward electromagnetic mode and the charge wave modulating the electron beam. Degenerate synchronization under this EPD condition means that two complex modes of the interactive system do not share just the wavenumber, but they rather coalesce in both their wavenumbers and eigenvectors (polarization states). In principle this new condition guarantees full synchronization between the electromagnetic wave and the beam's charge wave for any amount of output power extracted from the beam, setting the threshold of this EPD-BWO to any arbitrary, desired, value. Indeed, we show that the presence of distributed radiation in the SWS results in having high-threshold electron-beam current to start oscillations which implies higher power generation. These findings have the potential to lead to highly efficient BWOs with very high output power and excellent spectral purity.

<sec> We theoretically and experimentally demonstrate the anisotropic exceptional points (EPs) in the cavity magnonics system where magnons in a one millimeter-diameter yttrium iron garnet (YIG) sphere are coherently coupled with the microwave photons in a three-dimensional microwave cavity. The damping nature makes the cavity magnonics system inherently non-Hermitian. By solving the eigenvalues and eigenvectors of non-Hermitian Hamiltonian, a series of interesting and essential characteristics of the system can be obtained. Therefore, non-Hermitian physics has received more and more attention in both theory and experiment communities. Among them, exceptional points correspond to the non-Hermitian system’s degenerate states where the eigenvalues of the non-Hermitian system are identical, and the eigenvectors are parallel. The coupled cavity photon-magnon system has high tunability of coupling strength and cavity external damping rate, which is very suitable for studying EPs -related physics.</sec><sec> Exceptional points (EPs) are crucial in all kinds of non-Hermitian physical systems, which have both fundamental and applicational importance. For instance, it can be used for sensitive detection by monitoring spectrum splitting of degenerate modes when a perturbation to be sensed occurs. The EPs can be anisotropic, which means that it has a different function relation when the system approaches the EPs along different parameter paths of the system. In this paper, by carefully designing the parameter space, we observe the anisotropic exceptional point in the coupled cavity photon-magnon system. It shows the linear and square-root behavior when the EPs are approached from different directions in the parameter space. One of the parameters is the position of the YIG sphere in the cavity, which determines the coherent coupling strength between the cavity mode and the magnon mode. Another parameter is the number of the gasket between the cavity signal loading port and the cavity external surface, which determines the external damping rate of the cavity mode. Both of these parameters can be easily and accurately adjusted experimentally.</sec><sec> Our study paves the way for exploring anisotropic EPs based sensing technologies and more non-Hermitian related physics in the cavity magnonics system.</sec>

This study investigates the dynamics of two-qubit systems in open quantum environments, with a focus on the influence of exceptional points (EPs) and phase angle on entanglement dynamics. Two-qubit systems are pivotal in quantum information science, as they are the simplest system that reveals quantum entanglement, a crucial resource for quantum computing and communication. The research explores the effects of coupling, spontaneous emission, and environment on two-qubit states, specifically the $X$-states, within the framework of the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) or Lindblad master equation. EPs are unique to non-Hermitian systems. They affect the system’s dynamics significantly when the system approaches EP. Analytical and numerical analyses reveal that third-order EPs occur at specific parameters for two-qubit system, leading to critical changes in eigenvalues and eigenvectors of the Liouvillian superoperator. The study demonstrates that EPs can enhance or suppress entanglement. The concurrence, a quantitative measure of entanglement, peaks around the EPs under certain initial conditions. Furthermore, the impact of phase angle, $\theta$, that parameterizes the relative phase of maximally entangled states is analyzed. For initial states labeled as $\rho_\Psi(\theta,p)$ in this thesis, the phase angle significantly influences concurrence evolution, enabling tunability of entanglement dynamics. In contrast, for other class of initial states labeled as $\rho_\Phi(\theta,p)$, concurrence exhibits phase invariance, ensuring stability across the variations in $\theta$. Besides this, the analysis shows that the maximum concurrence occurs around EP for some states, the phase angle can also be tuned to increase the concurrence for some initial states. These findings highlight the importance of both EPs and phase angle in optimizing entanglement generation and control. This research offers valuable insights into the interplay between non-Hermitian physics and quantum entanglement, paving the way for advancements in quantum technologies. The outcomes provide possible practical implications in the field of quantum computation.

In this paper, we provide a comprehensive review of unidirectional reflectionless light propagation in photonic devices at exceptional points (EPs). EPs, which are branch point singularities of the spectrum, associated with the coalescence of both eigenvalues and corresponding eigenstates, lead to interesting phenomena, such as level repulsion and crossing, bifurcation, chaos, and phase transitions in open quantum systems described by non‐Hermitian Hamiltonians. Recently, it was shown that judiciously designed photonic synthetic matters could mimic the complex non‐Hermitian Hamiltonians in quantum mechanics and realize unidirectional reflection at optical EPs. Unidirectional reflectionlessness is of great interest for optical invisibility. Achieving unidirectional reflectionless light propagation could also be potentially important for developing optical devices, such as optical network analyzers. Here, we discuss unidirectional reflectionlessness at EPs in both parity‐time ( PT )‐symmetric and non‐ PT ‐symmetric optical systems. We also provide an outlook on possible future directions in this field.

This part is a continuation of the Part I where we built resolutions of identity for certain non-Hermitian Hamiltonians constructed of biorthogonal sets of their eigen- and associated functions for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration are taken with continuous spectrum and the following cases are examined: an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum and an exceptional point situated inside of continuous spectrum. In the present work the rigorous proofs are given for the resolutions of identity in both cases.

The present work addresses the distinction between the topological properties of symmetric and non- symmetric scenarios for the non-Hermitian Su–Schrieffer–Heeger model. The non- symmetric case is represented by non-reciprocity in both the inter- and the intra-cell hopping amplitudes, while the one with symmetry is modeled by a complex on-site staggered potential. In particular, we study the loci of the exceptional points, the winding numbers, band structures, and explore the breakdown of bulk-boundary correspondence (BBC). We further study the interplay of the dimerization strengths on the observables for these cases. The non- symmetric case denotes a more familiar situation, where the winding number abruptly changes by half-integer through tuning of the non-reciprocity parameters, and demonstrates a complete breakdown of BBC, thereby showing non-Hermitian skin effect. The topological nature of the symmetric case appears to follow closely to its Hermitian analogue, except that it shows unbroken (broken) regions with complex (purely real) energy spectra, while another variant of the winding number exhibits a continuous behavior as a function of the strength of the potential, while the conventional BBC is preserved.

Exceptional points (EPs), representing unique non-Hermitian degeneracies in open systems, have been the focus of significant research due to their remarkable phenomena and applications across various fields. While second-order EPs are extensively studied, higher-order EPs—requiring the degeneracy of multiple eigenvalues and corresponding eigenvectors—remain less explored. In this work, we propose a generic approach for realizing, 2Nth-order EP in an N-element photonic molecule (PM). Differing from a common PM consisting of identical microresonators, here a spiral-shaped microresonator is incorporated to form an asymmetric PM, offering unidirectional coupling between eigenmodes and transferring the chirality throughout the whole PM. Moreover, the hierarchical construction of 2Nth-order EPs can be achieved in gain–loss asymmetric PMs, as well as in all-passive ones. In the prototype demonstration based on numerical simulations, the obtained fourth-order EP reveals its superior sensitivity with a three-fold enhancement compared with that of a second-order EP to external perturbations at the single-molecule level. We envisage that the proposed asymmetric PM, circumventing the need for external components or precise control of gain–loss profiles, will pave the way for constructing higher-order EPs in integrated photonic devices and enabling applications in both linear and nonlinear regimes.

Non-Hermitian Hamiltonians may still have real eigenvalues, provided that a combined parity-time (ƤƮ) symmetry exists. The prospect of ƤƮ symmetry has been explored in several physical systems such as photonics, acoustics, and electronics. The eigenvalues in these systems undergo a transition from real to complex at exceptional points (EPs), where the ƤƮ symmetry is broken. Here, we demonstrate the existence of EP in magnonic devices composed of two coupled magnets with different magnon losses. The eigenfrequencies and damping rates change from crossing to anti-crossing at the EP when the coupling strength increases. The magnonic dispersion includes a strong "acoustic-like" mode and a weak "optic-like" mode. Moreover, upon microwave radiation, the ƤƮ magnonic devices act as magnon resonant cavity with unique response compared to conventional magnonic systems.

The states of an open quantum system are coupled via the environment of scattering wave functions. The complex coupling coefficients ω between system and environment arise from the principal value integral and the residuum. At high-level density where the resonance states overlap, the dynamics of the system is determined by exceptional points. At these points, the eigenvalues of two states are equal and the corresponding eigenfunctions are linearly dependent. It is shown in the present paper that Im(ω) and Re(ω) influence the system properties differently in the surrounding of exceptional points. Controlling the system by a parameter, the eigenvalues avoid crossing in energy near an exceptional point under the influence of Re(ω) in a similar manner as it is well known from discrete states. Im(ω), however, leads to width bifurcation and finally (when the system is coupled to one channel, i.e., to one common continuum of scattering wave functions), to a splitting of the system into two parts with different characteristic time scales. The role of observer states is discussed. Physically, the system is stabilized by this splitting since the lifetimes of some states are longer than before, while that of one state is shorter. In the cross section the short-lived state appears as a background term in high-resolution experiments. The wave functions of the long-lived states are mixed in those of the original ones in a comparably large parameter range. Numerical results for the eigenvalues and eigenfunctions are shown for N=2,4, and 10 states coupled mostly to one channel.

The appearance of topological singularities, namely exceptional points (EPs) is an intriguing feature of parameter-dependent open quantum or wave systems. EPs are the special type of non-Hermitian degeneracies where two (or more) eigenstates of the underlying system coalesce. In this paper, we present a three level non-Hermitian Hamiltonian which hosts three interacting eigenstates. The matrix elements are optimized in such a way that the intermediate eigenstate interacts with both the other states and the underlying system hosts at least two different second order EPs. This coupling scheme has proven to be highly effective to encounter higher order EPs. The impact of quasi-static parameter variation along a cyclic contour around the embedded EPs on the dynamics of interacting eigenvalues has been investigated comprehensively in the context of cascaded state conversion. Such dynamics of the eigenvalues show a clear signature of the third order EP. Moreover, we examine the accumulation of phases around the identified EPs and study the hallmark of phase exchange during cascaded state conversions accompanied by the parametric encirclement of the third order EP. A device level implementation of the scheme is possible exploiting multimode anisotropic waveguides or coupled waveguide systems.