Symmetric non-Hermitian Systems Research Articles

Level pinning of anti-PT-symmetric circuits for efficient wireless power transfer.

Wireless power transfer (WPT) technology based on magnetic resonance (a basic physical phenomenon) can directly transfer energy from the source to the load without wires and other physical contacts, and has been successfully applied to implantable medical devices, electric vehicles, robotic arms and other fields. However, due to the frequency splitting of near-field coupling, the resonant WPT system has some unique limitations, such as poor transmission stability and low efficiency. Here, we propose anti-resonance with level pinning for high-performance WPT. By introducing the anti-resonance mode into the basic WPT platform, we uncover the competition between dissipative coupling and coherent coupling to achieve novel level pinning, and construct an effective anti-parity-time (anti-PT)-symmetric non-Hermitian system that is superior to previous PT-symmetric WPT schemes. On the one hand, the eigenvalue of the anti-PT-symmetric system at resonance frequency is always pure real in both strong and weak coupling regions, and can be used to overcome the transmission efficiency decrease caused by weak coupling, as brought about by, for example, a large size ratio of the transmitter to receiver, or a long transmission distance. On the other hand, due to the level pinning effect of the two kinds of coupling mechanisms, the working frequency of the system is guaranteed to be locked, so frequency tracking is not required when the position and size of the receiver change. Even if the system deviates from the matching condition, an efficient WPT can be realized, thereby demonstrating the robustness of the level pinning. The experimental results show that when the size ratio of the transmitter coil to the receiver coil is 4.29 (which is in the weak coupling region), the transfer efficiency of the anti-PT-symmetric system is nearly 4.3 (3.2) times higher than that of the PT-symmetric system when the matching conditions are satisfied (deviated). With the miniaturization and integration of devices in mind, a synthetic anti-PT-symmetric system is used to realize a robust WPT. Anti-PT-symmetric WPT technology based on the synthetic dimension not only provides a good research platform for the study of abundant non-Hermitian physics, but also provides a means of going beyond traditional near-field applications with resonance mechanisms, such as resonance imaging, wireless sensing and photonic routing.

Topological invariant for multiband non-Hermitian systems with chiral symmetry

Topology plays an important role in non-Hermitian systems. How to characterize a non-Hermitian topological system under open-boundary conditions (OBCs) is a challenging problem. A one-dimensional (1D) topological invariant defined on a generalized Brillion zone (GBZ) was recently found to successfully describe the topological property of the two-band Su-Schrieffer-Heeger model. But for a 1D multiband chiral symmetric system under OBCs, it is still controversial how to define the topological invariant. We show in this paper by exact proof and detailed demonstration that to acquire the topological invariant for multiband non-Hermitian models with chiral symmetry, the GBZ as the integral domain should be replaced by a more generalized closed loop. Our work thus establishes the non-Bloch bulk-boundary correspondence for 1D multiband chiral symmetric non-Hermitian systems.

Observation of dark edge states in parity-time-symmetric quantum dynamics.

Topological edge states arise in non-Hermitian parity-time ([Formula: see text])-symmetric systems, and manifest themselves as bright or dark edge states, depending on the imaginary components of their eigenenergies. As the spatial probabilities of dark edge states are suppressed during the non-unitary dynamics, it is a challenge to observe them experimentally. Here we report the experimental detection of dark edge states in photonic quantum walks with spontaneously broken [Formula: see text] symmetry, thus providing a complete description of the topological phenomena therein. We experimentally confirm that the global Berry phase in [Formula: see text]-symmetric quantum-walk dynamics unambiguously defines topological invariants of the system in both the [Formula: see text]-symmetry-unbroken and -broken regimes. Our results establish a unified framework for characterizing topology in [Formula: see text]-symmetric quantum-walk dynamics, and provide a useful method to observe topological phenomena in [Formula: see text]-symmetric non-Hermitian systems in general.

Cooperative-effect-induced one-way steering in open cavity magnonics

We propose to generate and control stationary one-way steering with strong entanglement between photon and magnon modes by the cooperative effect of coherent coupling and dissipative coupling. Due to the combination of two couplings, the system becomes a parity-time-like symmetric non-Hermitian system, and two exceptional points (EPs)-like appear in the real and imaginary parts of the eigenvalues. We demonstrate that the especially obvious quantum entanglement and perfect one-way steering can be obtained around two EPs-like. The continuous variable entanglement and steering produced by this cooperative effect show stronger robustness to environment temperature and system dissipation than that induced by nonlinearity. The one-way steering directivity can be controlled by the relative phase of cooperative dissipation and the frequency detuning of the magnon mode. Our work shows the controllability advantage of the open cavity magnonic system and may open up a platform for the generation of stationary one-way steering.

Transport characteristics of a $${\mathcal {PT}}$$-symmetric non-Hermitian system: effect of environmental interaction

The environmental influence is inevitable but often ignored in the study of electronic transport properties of small-scale systems. Such an environment-mediated interaction can generally be described by a parity-time symmetric non-Hermitian system with a balanced distribution of physical gain and loss. It is quite known in the literature that along with the conventional junction current, another current called bias-driven circular current can be established in a loop geometry depending upon the junction configuration. This current, further, induces a strong magnetic field that can even reach few tesla. What will happen to these quantities when the system interacts with its surrounding environment? Would it exhibit a detrimental response? We address such issues considering a two-terminal ring geometry where the junction setup is described within a tight-binding framework. All the transport quantities are evaluated using the standard Green’s function formalism based on the Landauer–Büttiker approach.

Matter-wave gap solitons and vortices in three-dimensional parity-time-symmetric optical lattices

SummaryPast decades have witnessed the emergence and increasing expansion of parity-time ()-symmetric systems in diverse physical fields and beyond as they manifest entirely all-real spectra, although being non-Hermitian. Nonlinear waves in low-dimensional -symmetric non-Hermitian systems have recently been explored broadly; however, understanding these systems in higher dimensions remains abstruse and has yet to be revealed. We survey, theoretically and numerically, matter-wave nonlinear gap modes of Bose-Einstein condensates with repulsive interparticle interactions in three-dimensional optical lattices with emphasis on multidimensional gap solitons and vortices. Utilizing direct perturbed simulations, we address the stability and instability areas of both localized modes in the underlying linear band gap spectra. Our study provides deep and consistent understandings of the formation, structural property, and dynamics of coherent localized matter waves supported by optical lattices in multidimensional space, thus opening a way for exploring and stabilizing three-dimensional localized gap modes in non-Hermitian systems

Topological properties of the one-dimensional <inline-formula><tex-math id="M2">\begin{document}${\cal {PT}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220796_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220796_M2.png"/></alternatives></inline-formula>-symmetric non-Hermitian spin-orbit-coupled Su-Schrieffer-Heeger model

The topological property and the energy property of one-dimensional non-Hermitian spin-orbit-coupled Su-Schrieffer-Heeger (SSH) model are investigated theoretically, by introducing spin-dependent imaginary potentials with gain and loss effects. It is found that the imaginary potential leads the imaginary energy spectra to appera in the topologically nontrivial region of this system, and the <inline-formula><tex-math id="M5">\begin{document}${\cal {PT}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220796_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220796_M5.png"/></alternatives></inline-formula> phase transition to happen in the topologically trivial region. In addition, the imaginary potential energy and spin-orbit coupling work together to make the topological phase transition occur in the topologically trivial region, and the topological non-trivial region becomes wider. The energy spectrum results show that the imaginary potential energy and the spin-orbit coupling can obviously control the zero-energy states of the system, which mainly lies in the presence of four zero-energy states with four different localities and numbers. This shows the special adjustment effect of imaginary potential energy and spin-orbit coupling on the energy band structure of the system. It is believed that these results are helpful in understanding the topological phase transition behavior of <inline-formula><tex-math id="M6">\begin{document}${\cal {PT}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220796_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220796_M6.png"/></alternatives></inline-formula>-symmetric non-Hermitian system.

Information retrieval and criticality in high-dimensional parity-time-symmetric systems

Recently, impressive progress has been made in the study of non-Hermitian systems with parity-time symmetry, such as observations of topological properties of physical systems and criticality at exceptional points. A crucial aspect of parity-time symmetric nonunitary dynamics is the information flow between the system and the environment. In this paper, we use the physical quantity, distinguishability between quantum states, to uniformly quantify the information flow between low-dimensional and high-dimensional parity-time symmetric non-Hermitian systems and environments. The numerical results show that the oscillation of quantum state distinguishability and complete information retrieval and can be obtained in the parity-time-unbroken phase. However, the information decays exponentially in the parity-time-broken phase. The exceptional point marks the criticality between reversibility and irreversibility of information flow, and the distinguishability between quantum states exhibits the behavior of power-law decay. Understanding these unique phenomena in nonunitary quantum dynamics provides an important perspective for the study of open quantum systems and contributes to their application in quantum information.

Information retrieval and criticality in high-dimensional parity-time-symmetric systems

Recently, impressive progress has been made in the study of non-Hermitian systems with parity-time symmetry, such as observations of topological properties of physical systems and criticality at exceptional points. A crucial aspect of parity-time symmetric nonunitary dynamics is the information flow between the system and the environment. In this paper, we use the physical quantity, distinguishability between quantum states, to uniformly quantify the information flow between low-dimensional and high-dimensional parity-time symmetric non-Hermitian systems and environments. The numerical results show that the oscillation of quantum state distinguishability and complete information retrieval and can be obtained in the parity-time-unbroken phase. However, the information decays exponentially in the paritytime-broken phase. The exceptional point marks the criticality between reversibility and irreversibility of information flow, and the distinguishability between quantum states exhibits the behavior of power-law decay. Understanding these unique phenomena in nonunitary quantum dynamics provides an important perspective for the study of open quantum systems and contributes to their application in quantum information.

Matter-Wave Gap Solitons and Vortices in Three-Dimensional Parity-Time-Symmetric Optical Lattices

Past decades have witnessed the emergence and increasing expansion of parity-time (PT )-symmetric systems in diverse physical fields and beyond as they license entirely all-real spectra, although being non-Hermitian. Nonlinear waves in low-dimensional PT -symmetric non-Hermitian systems have recently been explored broadly, whose understanding in higher dimensions remains abstruse and has yet to be revealed. We survey, theoretically and numerically, matter-wave nonlinear gap modes of Bose-Einstein condensates with repulsive interparticle interactions in three-dimensional PT optical lattices with emphasis on multidimensional gap solitons and vortices. Relying on direct perturbed simulations, we address the stability and instability areas of both localized modes in the underlying linearband gap spectra. Our study provides deep and consistent understandings of the formation, structural property, and dynamics of coherent localized matter waves supported by PT optical lattices in multidimensional space, opening a way for exploring and stabilizing three-dimensional localized gap modes in non-Hermitian systems.

Non-Hermitian higher-order Dirac semimetals

In this Letter, we study three-dimensional non-Hermitian higher-order Dirac semimetals (NHHODSMs). Our focus is on ${C}_{4}$-symmetric non-Hermitian systems where we investigate inversion ($\mathcal{I}$) or time-reversal ($\mathcal{T}$) symmetric models of NHHODSMs having real bulk spectra. We show that they exhibit the striking property that the bulk and surfaces are anti-$\mathcal{P}\mathcal{T}$ and $\mathcal{P}\mathcal{T}$ symmetric, respectively, and so belong to two different topological classes realizing a non-Hermitian topological phase which we call a hybrid-$\mathcal{P}\mathcal{T}$ topological phase. Interestingly, while the bulk spectrum is still fully real, we find that exceptional Fermi rings (EFRs) appear connecting the two Dirac nodes on the surface. This provides a route to probe and utilize both the bulk Dirac physics and exceptional rings/points on equal footing. Moreover, particularly for $\mathcal{T}$-NHHODSMs, we also find real hinge arcs connecting the surface EFRs. We show that this higher-order topology can be characterized using a biorthogonal real-space formula of the quadrupole moment. Furthermore, by applying Hermitian ${C}_{4}$-symmetric perturbations, we discover various phases, particularly (i) an intrinsic $\mathcal{I}$-NHHODSM having hinge arcs and surface exceptional Fermi arcs, and (ii) a $\mathcal{T}$-symmetric skin-topological HODSM which possesses both topological and skin hinge modes. The interplay between non-Hermition and higher-order topology in this Letter paves the way toward uncovering rich phenomena and hybrid functionality that can be readily realized in experiment.

Stable states with nonzero entropy under broken PT symmetry

The $\mathcal{PT}$-symmetric non-Hermitian systems have been widely studied and explored both in theory and in experiment these years due to various interesting features. In this work, we focus on the dynamical features of a triple-qubit system, one of which evolves under local $\mathcal{PT}$-symmetric Hamiltonian. A new kind of abnormal dynamic pattern in the entropy evolution process is identified, which presents a parameter-dependent stable state, determined by the non-Hermiticity of Hamiltonian in the broken phase of $\mathcal{PT}$-symmetry. The entanglement and mutual information of a two-body subsystem can increase beyond the initial values, which do not exist in the Hermitian and two-qubit $\mathcal{PT}$-symmetric systems. Moreover, an experimental demonstration of the stable states in non-Hermitian system with non-zero entropy and entanglement is realized on a four-qubit quantum simulator with nuclear spins. Our work reveals the distinctive dynamic features in the triple-qubit $\mathcal{PT}$-symmetric system and paves the way for practical quantum simulation of multi-party non-Hermitian system on quantum computers.

Experimental Determination of PT-Symmetric Exceptional Points in a Single Trapped Ion.

Exceptional points (EPs) of a non-Hermitian Hamiltonian with parity-time-reversal (PT) symmetry have the potential to drastically enhance the capabilities of metrology and sensing through their power-law growing sensitivity to external perturbation. With the ability of generating and tuning dissipation in a single trapped ion system, we observe rich dynamics and detailed quantum phase transitions from the PT-symmetric phase to the symmetry-breaking phase. In this single qubit full quantum system, we develop a method to precisely determine the location of EP without any fitting parameter, and extract the eigenvalues in a unified way through all parameter regions. We can also obtain the full density matrix by quantum state tomography. Finally, we suggest from theoretical analysis that the periodically driving PT-symmetric non-Hermitian system can be used to measure the magnitude, frequency, and phase of time-dependent perturbation with EP enhancement.

Bulk–edge correspondence and stability of multiple edge states of a $\mathcal{PT}$-symmetric non-Hermitian system by using non-unitary quantum walks

Abstract Topological phases and the associated multiple edge states are studied for parity and time-reversal ($\mathcal{PT}$)-symmetric non-Hermitian open quantum systems by constructing a non-unitary three-step quantum walk retaining $\mathcal{PT}$ symmetry in one dimension. We show that the non-unitary quantum walk has large topological numbers of the $\mathbb{Z}$ topological phase and numerically confirm that multiple edge states appear as expected from the bulk–edge correspondence. Therefore, the bulk–edge correspondence is valid in this case. Moreover, we study the stability of the multiple edge states against a symmetry-breaking perturbation so that the topological phase is reduced to $\mathbb{Z}_2$ from $\mathbb{Z}$. In this case, we find that the number of edge states does not become one unless a pair of edge states coalesce at an exceptional point. Thereby, this is a new kind of breakdown of the bulk–edge correspondence in non-Hermitian systems. The mechanism of the prolongation of edge states against the symmetry-breaking perturbation is unique to non-Hermitian systems with multiple edge states and anti-linear symmetry. Toward experimental verifications, we propose a procedure to determine the number of multiple edge states from the time evolution of the probability distribution.

Observation of Critical Phenomena in Parity-Time-Symmetric Quantum Dynamics.

We experimentally simulate nonunitary quantum dynamics using a single-photon interferometric network and study the information flow between a parity-time- (PT-)symmetric non-Hermitian system and its environment. We observe oscillations of quantum-state distinguishability and complete information retrieval in the PT-symmetry-unbroken regime. We then characterize in detail critical phenomena of the information flow near the exceptional point separating the PT-unbroken and PT-broken regimes, and demonstrate power-law behavior in key quantities such as the distinguishability and the recurrence time. We also reveal how the critical phenomena are affected by symmetry and initial conditions. Finally, introducing an ancilla as an environment and probing quantum entanglement between the system and the environment, we confirm that the observed information retrieval is induced by a finite-dimensional entanglement partner in the environment. Our work constitutes the first experimental characterization of critical phenomena in PT-symmetric nonunitary quantum dynamics.

Non-Hermitian anomalous skin effect

A non-Hermitian topological insulator is fundamentally different from conventional topological insulators. The non-Hermitian skin effect arises in a nonreciprocal tight binding lattice with open edges. In this case, not only topological states but also bulk states are localized around the edges of the nonreciprocal system. We discuss that controllable switching from topological edge states into topological extended states in a chiral symmetric non-Hermitian system is possible. We show that the skin depth decreases with non-reciprocity for bulk states but increases with it for topological zero energy states.

Symmetry-protected localized states at defects in non-Hermitian systems

Understanding how local potentials affect system eigenmodes is crucial for experimental studies of nontrivial bulk topology. Recent studies have discovered many exotic and highly non-trivial topological states in non-Hermitian systems. As such, it would be interesting to see how non-Hermitian systems respond to local perturbations. In this work, we consider chiral and particle-hole -symmetric non-Hermitian systems on a bipartite lattice, including SSH model and photonic graphene, and find that a disordered local potential could induce bound states evolving from the bulk. When the local potential on a single site becomes infinite, which renders a lattice vacancy, chiral-symmetry-protected zero-energy mode and particle-hole symmetry-protected bound states with purely imaginary eigenvalues emerge near the vacancy. These modes are robust against any symmetry-preserved perturbations. Our work generalizes the symmetry-protected localized states to non-Hermitian systems.

On the time evolution at a fluctuating exceptional point

Abstract We theoretically evaluate the impact of drift-free noise on the dynamics of P T ${\cal P}{\cal T}$ -symmetric non-Hermitian systems with an exceptional point, which have recently been proposed for sensors. Such systems are currently considered as promising templates for sensing applications, because of their intrinsically extremely sensitive response to external perturbations. However, this applies equally to the impact of fabrication imperfections and fluctuations in the system parameters. Here we focus on the influence of such fluctuations caused by inevitable (thermal) noise and show that the exceptional-point eigenstate is not stable in its presence. To this end, we derive an effective differential equation for the mean time evolution operator averaged over all realizations of the noise field, and via numerical analysis we find that the presence of noise leads to exponential divergence of any initial state after some characteristic period of time. We therefore show that it is rather demanding to design sensor systems based on continuous operation at an exceptional point.

Theory of superconductivity with non-Hermitian and parity-time reversal symmetric Cooper pairing symmetry

Recently developed parity (P) and time-reversal (T) symmetric non-Hermitian systems govern a rich variety of new and characteristically distinct physical properties, which may or may not have a direct analog in their Hermitian counterparts. We study here a non-Hermitian, PT-symmetric superconducting Hamiltonian that possesses real quasiparticle spectrum in the PT-region of the Brillouin zone. Within a single-band mean-field theory, we find that real quasiparticle energies are possible when the superconducting order parameter itself is either Hermitian or anti-Hermitian. Within the corresponding Bardeen-Cooper-Schrieffer (BCS) theory, we find that several properties are characteristically distinct and novel in the non-Hermitian pairing case than its Hermitian counterpart. One of our significant findings is that while a Hermitian superconductor gives a second order phase transition, the non-Hermitian one produces a robust first order phase transition. The corresponding thermodynamic properties, and the Meissner effect are also modified accordingly. Finally, we discuss how such a PT-symmetric pairings can emerge from an anti-symmetric potential, such as the Dzyloshinskii-Moria interaction, but with an external external bath, or complex potential, among others.

Information Retrieval and Criticality in Parity-Time-Symmetric Systems.

By investigating information flow between a general parity-time (PT-)symmetric non-Hermitian system and an environment, we find that the complete information retrieval from the environment can be achieved in the PT-unbroken phase, whereas no information can be retrieved in the PT-broken phase. The PT-transition point thus marks the reversible-irreversible criticality of information flow, around which many physical quantities such as the recurrence time and the distinguishability between quantum states exhibit power-law behavior. Moreover, by embedding a PT-symmetric system into a larger Hilbert space so that the entire system obeys unitary dynamics, we reveal that behind the information retrieval lies a hidden entangled partner protected by PT symmetry. Possible experimental situations are also discussed.