non-Hermitian Systems Research Articles

Higher Time-Derivative Theories from Space–Time Interchanged Integrable Field Theories

We compare a relativistic and a nonrelativistic version of Ostrogradsky’s method for higher-time derivative theories extended to scalar field theories and consider as an alternative a multi-field variant. We apply the schemes to space–time rotated modified Korteweg–de Vries systems and, exploiting their integrability, to Hamiltonian systems built from space–time rotated inverse Legendre transformed higher-order charges of these systems. We derive the equal-time Poisson bracket structures of these theories, establish the integrability of the latter theories by means of the Painlevé test and construct exact analytical period benign solutions in terms of Jacobi elliptic functions to the classical equations of motion. The classical energies of these partially complex solutions are real when they respect a certain modified CPT-symmetry and complex when this symmetry is broken. The higher-order Cauchy and initial-boundary value problem are addressed analytically and numerically. Finally, we provide the explicit quantization of the simplest mKdV system, exhibiting the usual conundrum of having the choice between having to deal with either a theory that includes non-normalizable states or spectra that are unbounded from below. In our non-Hermitian system, the choice is dictated by the correct sign in the decay width.

Topological phases of many-body non-Hermitian systems

Topological phases of many-body non-Hermitian systems

Incoherent non-Hermitian skin effect in photonic quantum walks

The non-Hermitian skin effect describes the concentration of an extensive number of eigenstates near the boundaries of certain dissipative systems. This phenomenon has raised a huge interest in different areas of physics, including photonics, deeply expanding our understanding of non-Hermitian systems and opening up new avenues in both fundamental and applied aspects of topological phenomena. The skin effect has been associated to a nontrivial point-gap spectral topology and has been experimentally demonstrated in a variety of synthetic matter systems, including photonic lattices. In most of physical models exhibiting the non-Hermitian skin effect full or partial wave coherence is generally assumed. Here we push the concept of skin effect into the fully incoherent regime and show that rather generally (but not universally) the non-Hermitian skin effect persists under dephasing dynamics. The results are illustrated by considering incoherent light dynamics in non-Hermitian photonic quantum walks.

Localization transition in non-Hermitian systems depending on reciprocity and hopping asymmetry.

We studied the single-particle Anderson localization problem for non-Hermitian systems on directed graphs. Random regular graph and various undirected standard random graph models were modified by controlling reciprocity and hopping asymmetry parameters. We found the emergence of left, biorthogonal, and right localized states depending on both parameters and graph structure properties such as node degree d. For directed random graphs, the occurrence of biorthogonal localization near exceptional points is described analytically and numerically. The clustering of localized states near the center of the spectrum and the corresponding mobility edge for left and right states are shown numerically. Structural features responsible for localization, such as topologically invariant nodes or drains and sources, were also described. Considering the diagonal disorder, we observed the disappearance of localization dependence on reciprocity around W∼20 for a random regular graph d=4. With a small diagonal disorder, the average biorthogonal fractal dimension drastically reduces. Around W∼5, localization scars occur within the spectrum, alternating as vertical bands of clustering of left and right localized states.

True exponentially enhanced sensing in the non-Hermitian topological phase

Non-Hermitian systems have been employed to construct a high-sensitivity sensor. To evaluate the performance of the sensors, the quantum Fisher information per photon, or equivalently signal-to-noise ratio per photon, is provided as a “true” sensing criterion, which avoids the trivial contribution from the photon numbers. The specific properties of non-Hermitian systems, e.g., exceptional points and skin effect, have been connected to the true exponentially enhanced sensing performance. To date, the relation between the non-Hermitian topological phase and the true sensing performance has not been reported clearly. Here, we construct the high-sensitivity sensor based on the non-Hermitian Su–Schrieffer–Heeger lattice and establish the relationship between the exponentially enhanced sensing and the non-Hermitian topologically nontrivial phase. The saturation of sensing with the size emerges in the sense of one perturbation. Such a limitation can be surpassed through the change of incident positions of driving fields, and the exponentially enhanced sensing reappears.

Entanglement phase transitions in non-Hermitian Floquet systems

The competition between unitary time evolution and quantum measurements could induce phase transitions in the entanglement characteristics of quantum many-body dynamics. In this work, we reveal such entanglement transitions in the context of non-Hermitian Floquet systems. Focusing on noninteracting fermions in a representative bipartite lattice with balanced gain/loss and under time-periodic quenches, we uncover rich patterns of entanglement transitions due to the interplay between driving and non-Hermitian effects. Specially, we find that the monotonic increase of quenched hopping amplitude could flip the system between volume-law and area-law entangled Floquet phases, yielding alternated entanglement transitions. Meanwhile, the increase of gain/loss strength could trigger area-law to volume-law reentrant transitions in the scaling behavior of steady-state entanglement entropy, which are abnormal and highly unexpected in nondriven systems. Connections between entanglement transitions and parity-time-reversal (PT) transitions in Floquet spectra are further established. Our findings not only build a foundation for exploring entanglement phase transitions in Floquet non-Hermitian setups, but also provide efficient means to engineer and control such transitions by driving fields. Published by the American Physical Society 2024

Amoeba Formulation of Non-Bloch Band Theory in Arbitrary Dimensions

The non-Hermitian skin effect dramatically reshapes the energy bands of non-Hermitian systems, meaning that the usual Bloch band theory is fundamentally inadequate as their characterization. The non-Bloch band theory, in which the concept of Brillouin zone is generalized, has been widely applied to investigate non-Hermitian systems in one spatial dimension. However, its generalization to higher dimensions has been challenging. Here, we develop a formulation of the non-Hermitian skin effect and non-Bloch band theory in arbitrary spatial dimensions, which is based on a natural geometrical object known as the amoeba. Our theory provides a general framework for studying non-Hermitian bands beyond one dimension. Key quantities of non-Hermitian bands, including the energy spectrum, eigenstates profiles, and the generalized Brillouin zone, can be efficiently obtained from this approach. Published by the American Physical Society 2024

Observation of energy localization in nonlinear non-Hermitian systems

Energy localization in Hermitian systems has been utilized to generate ultra-sensitivity. Here, we report the interplay between non-Hermitian parity-time (PT) symmetry breaking and the mode localization transition. In our scheme, a PT-symmetric system consists of two coupled LC (inductor–capacitor) resonators: one has a linear loss and the other has a saturated gain described by a nonlinear model. The nonlinear gain is initially set to be slightly higher than the loss, and the system is operated at the exact PT-symmetric phase close to an exceptional point. The capacitance variation applied on the loss resonator, i.e., perturbation, causes PT-symmetry to break, generating complex frequencies. As a result of nonlinear gain, the resonator will grow to reach its steady state and saturate out the gain. This stable oscillation eliminates the complex frequencies, and the mode is ultimately localized at the gain side. We have observed that the voltage amplitude of the gain resonator due to the perturbation has experienced drastic changes. The amplitude ratio before and after the perturbation is sensitive to the perturbation. Our results provide an approach to study perturbation-driven localization phenomena in a PT-symmetric system and pave the way for sensors with ultrahigh sensitivity.

Experimental Investigation of Lee–Yang Criticality Using Non-Hermitian Quantum System

Lee–Yang theory clearly demonstrates where the phase transition of many-body systems occurs and the asymptotic behavior near the phase transition using the partition function under complex parameters. The complex parameters make the direct investigation of Lee–Yang theory in practical systems challenging. Here we construct a non-Hermitian quantum system that can correspond to the one-dimensional Ising model with imaginary parameters through the equality of partition functions. By adjusting the non-Hermitian parameter, we successfully obtain the partition function under different imaginary magnetic fields and observe the Lee–Yang zeros. We also observe the critical behavior of free energy in vicinity of Lee–Yang zero that is consistent with theoretical prediction. Our work provides a protocol to study Lee–Yang zeros of the one-dimensional Ising model using a single-qubit non-Hermitian system.

Quantum simulation of the bosonic Kitaev chain

Superconducting quantum circuits are a natural platform for quantum simulations of a wide variety of important lattice models describing topological phenomena, spanning condensed matter and high-energy physics. One such model is the bosonic analog of the well-known fermionic Kitaev chain, a 1D tight-binding model with both nearest-neighbor hopping and pairing terms. Despite being fully Hermitian, the bosonic Kitaev chain exhibits a number of striking features associated with non-Hermitian systems, including chiral transport and a dramatic sensitivity to boundary conditions known as the non-Hermitian skin effect. Here, using a multimode superconducting parametric cavity, we implement the bosonic Kitaev chain in synthetic dimensions. The lattice sites are mapped to frequency modes of the cavity, and the in situ tunable complex hopping and pairing terms are created by parametric pumping at the mode-difference and mode-sum frequencies, respectively. We experimentally demonstrate important precursors of nontrivial topology and the non-Hermitian skin effect in the bosonic Kitaev chain, including chiral transport, quadrature wavefunction localization, and sensitivity to boundary conditions. Our experiment is an important first step towards exploring genuine many-body non-Hermitian quantum dynamics.

Mathematical Foundations of the Non-Hermitian Skin Effect

We study the skin effect in a one-dimensional system of finitely many subwavelength resonators with a non-Hermitian imaginary gauge potential. Using Toeplitz matrix theory, we prove the condensation of bulk eigenmodes at one of the edges of the system. By introducing a generalised (complex) Brillouin zone, we can compute spectral bands of the associated infinitely periodic structure and prove that this is the limit of the spectra of the finite structures with arbitrarily large size. Finally, we contrast the non-Hermitian systems with imaginary gauge potentials considered here with systems where the non-Hermiticity arises due to complex material parameters, showing that the two systems are fundamentally distinct.

Exceptional-point-enhanced phase sensing.

Optical sensors, crucial in diverse fields like gravitational wave detection, biomedical imaging, and structural health monitoring, rely on optical phase to convey valuable information. Enhancing sensitivity is important for detecting weak signals. Exceptional points (EPs), identified in non-Hermitian systems, offer great potential for advanced sensors, given their marked response to perturbations. However, strict physical requirements for operating a sensor at EPs limit broader applications. Here, we introduce an EP-enhanced sensing platform featuring plug-in external sensors separated from an EP control unit. EPs are achieved without modifying the sensor, solely through control-unit adjustments. This configuration converts and amplifies optical phase changes into quantifiable spectral features. By separating sensing and control functions, we expand the applicability of EP enhancement to various conventional sensors. As a proof-of-concept, we demonstrate a sixfold reduction in the detection limit of fiber-optic strain sensing using this configuration. This work establishes a universal platform for applying EP enhancement to diverse phase-dependent structures, promising ultrahigh-sensitivity sensing across various applications.

Dynamics of 2 × 2 matrix non-Hermitian quantum systems on Bloch sphere

By casting evolution to the Bloch sphere, the dynamics of 2 × 2 matrix non-Hermitian systems are investigated in detail. This investigation reveals that there are four kinds of dynamical modes for such systems. The different modes are classified by different kinds of fixed points, namely, the elliptic point, spiral point, critical node, and degenerate point. The Hermitian systems and the unbroken PT non-Hermitian cases belong to the category with elliptic points. The degenerate point just corresponds to the systems with exceptional point (EP). The topological properties of the fixed point are also discussed. It is interesting that the topological charge for the degenerate point is two, while the others are one.

Exceptional classifications of non-Hermitian systems

Eigenstate coalescence in non-Hermitian systems is widely observed in diverse scientific domains encompassing optics and open quantum systems. Recent investigations have revealed that adiabatic encircling of exceptional points (EPs) leads to a nontrivial Berry phase in addition to an exchange of eigenstates. Based on these phenomena, we propose in this work an exhaustive classification framework for EPs in non-Hermitian physical systems. In contrast to previous classifications that only incorporate the eigenstate exchange effect, our proposed classification gives rise to finer Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb{Z}}}_{2}$$\\end{document} classifications depending on the presence of a π Berry phase after the encircling of the EPs. Moreover, by mapping arbitrary one-dimensional systems to the adiabatic encircling of EPs, we can classify one-dimensional non-Hermitian systems characterized by topological phase transitions involving EPs. Applying our exceptional classification to various multiband systems, we expect to enhance the understanding of topological phases in non-Hermitian systems.

Bulk-Boundary Correspondence in Point-Gap Topological Phases.

A striking feature of non-Hermitian systems is the presence of two different types of topology. One generalizes Hermitian topological phases, and the other is intrinsic to non-Hermitian systems, which are called line-gap topology and point-gap topology, respectively. Whereas the bulk-boundary correspondence is a fundamental principle in the former topology, its role in the latter has not been clear yet. This Letter establishes the bulk-boundary correspondence in the point-gap topology in non-Hermitian systems. After revealing the requirement for point-gap topology in the open boundary conditions, we clarify that the bulk point-gap topology in open boundary conditions can be different from that in periodic boundary conditions. On the basis of real space topological invariants and the K theory, we give a complete classification of the open boundary point-gap topology with symmetry and show that the nontrivial open boundary topology results in robust and exotic surface states.

Long-term evolution of non-Hermitian systems in quantum-walk dynamics

Long-term evolution of non-Hermitian systems in quantum-walk dynamics

Reconfigurable high-order exceptional points in coupled optical parametric oscillators for enhanced sensing

Non-Hermitian systems around exceptional points (EPs) demonstrate increased sensitivity as the EP order increases. We propose a protocol for constructing a reconfigurable non-Hermitian system consisting of three coupled optical parametric oscillators (OPOs) and demonstrate its enhanced sensitivity around EPs. The coupled OPO system exhibits reconfigurable high-order EPs and corresponding-order enhanced sensitivity (i.e. fourth and sixth order) when it is subjected to pump modulation. Furthermore, the coupled OPO system can synthesize directly a Floquet non-Hermitian system by introducing periodic pump modulation, and can achieve a tunable second-order Floquet EP without periodic modulation of the coupling rates. The convenient flexibility of the coupled OPO system allows it to operate at EPs of different orders and customize a Floquet EP by manipulating the pump parameters. Our protocol can thus contribute to the development of high-performance sensors and precision measurement devices by exploring the unique features of non-Hermitian systems around various EPs.

Entanglement dynamics of two non-Hermitian qubits

The evolution of entanglement in a non-Hermitian quantum system may behave differently compared to its Hermitian counterpart. In this paper, we investigate the entanglement dynamics of two coupled and driven non-Hermitian qubits. Through calculating the concurrence of the system, we find that the evolution of the bipartite entanglement manifests two distinct patterns in the parameter space. In the low non-Hermiticity regime, the concurrence oscillates significantly, while in the opposite regime the same quantity would trend to a stable value. We attribute this phenomenon to parity-time ( PT ) symmetry phase transition. We also study the effect of decoherence on the entanglement dynamics. Our work provides a method to stabilize entanglement by exploiting non-Hermiticity.

Topological extension including quantum jump

Non-Hermitian (NH) systems and the Lindblad form master equation have always been regarded as reliable tools in dissipative modeling. Intriguingly, existing literature often obtains an equivalent NH Hamiltonian by neglecting the quantum jumping terms in the master equation. However, there lacks investigation into the effects of discarded terms as well as the unified connection between these two approaches. In this study, we study the Su–Schrieffer–Heeger model with collective loss and gain from a topological perspective. When the system undergoes no quantum jump events, the corresponding shape matrix exhibits the same topological properties in contrast to the traditional NH theory. Conversely, the occurrence of quantum jumps can result in a shift in the positions of the phase transition. Our study provides a qualitative analysis of the impact of quantum jumping terms and reveals their unique role in quantum systems.

Resonant exceptional points sensing in terahertz metasurfaces

The phenomenon of simultaneous degeneracy of eigenvalues and eigenstates in non-Hermitian systems, known as exceptional points (EPs), renders systems in the vicinity of EP highly sensitive to perturbations in the environment. Recently, research works on terahertz metasurfaces primarily focused on finding the degenerate eigenvalue using the transmission matrix, known as the scattering EP, while neglecting exploration of another form of EP—the resonant EP. In this study, we numerically, theoretically, and experimentally investigate the resonant EP in terahertz metasurfaces, whose unit cell is composed of two unidentical length metal rods in a two-energy level non-Hermitian system. By altering two system parameters, we can manipulate the coupling between the two resonators, thereby directly realizing the singularity of complex frequency through only a single measurement. Furthermore, our simulations and measurements indicate a sensitivity of up to 9046 GHz/(RIU·mm), which underscores the high sensitivity characteristics of EP. Our approach not only contributes a fresh perspective on understanding EP in non-Hermitian terahertz metasurfaces but also proposes alternative methods for high-sensitivity terahertz sensing.