Electrical circuit simulation of non-Hermitian lattice models

<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>

Quantum simulation is one of the main contents of quantum information science, aiming to simulate and investigate poorly controllable or unobtainable quantum systems by using controllable quantum systems. Quantum simulation can be implemented in quantum computers, quantum simulators, and small quantum devices. Non-Hermitian systems have aroused research interest increasingly in recent two decades. On one hand, non-Hermitian quantum theories can be seen as the complex extensions of the conventional quantum mechanics, and are closely related to open systems and dissipative systems. On the other hand, both quantum systems and classical systems can be constructed as non-Hermitian systems with novel properties, which can be used to improve the precision of precise measurements. However, a non-Hermitian system is more difficult to simulate than a Hermitian system in that the time evolution of it is no longer unitary. In this review, we introduce recent research progress of quantum simulations of non-Hermitian systems. We mainly introduce theoretical researches to simulate typical non-Hermitian quantum systems by using the linear combinations of unitaries, briefly showing the advantages and limitations of each proposal, and we briefly mention other theoretical simulation methods, such as quantum random walk, space embedded and dilation. Moreover, we briefly introduce the experimental quantum simulations of non-Hermitian systems and novel phenomena in nuclear magnetic resonance, quantum optics and photonics, classical systems, etc. The recent progress of the combinations of quantum simulation and non-Hermitian physics has promoted the development of the non-Hermitian theories, experiments and applications, and expand the scope of application of quantum simulations and quantum computers.

A longstanding tool to characterize the evolution of open Markovian quantum systems is the GKSL (Gorini-Kossakowski-Sudarshan-Lindblad) master equation. However, in some cases, open quantum systems can be effectively described with non-Hermitian Hamiltonians, which have attracted great interest in the last twenty years due to a number of unconventional properties, such as the appearance of exceptional points. Here, we present a short review of these two different approaches aiming in particular to highlight their relation and illustrate different ways of connecting non-Hermitian Hamiltonian to a GKSL master equation for the full density matrix.

Recently, a Dirac exceptional point (EP) was reported in a non-Hermitian system. Unlike a Dirac point in Hermitian systems, this Dirac EP has coalesced eigenstates in addition to the degenerate energy. Also different from a typical EP, the two energy levels connected at this Dirac EP remain real in its vicinity and display a linear instead of square root dispersion, forming a tilted Dirac cone in the hybrid space consisting of a momentum dimension and a synthetic dimension for the strength of non-Hermiticity. In this report, we first present simple three-band and two-band matrix models with a Dirac EP, where the linear dispersion of the tilted Dirac cone can be expressed analytically. Importantly, our analysis also reveals that there exist Hermitian and non-Hermitian systems that have the same (real-valued) energy spectrum in their entire parameter space, with the exception that one or more degeneracies in the former are replaced by Dirac EPs in the later. Finally, we show the existence of an imaginary Dirac cone with an EP at its center.

Non-Hermitian systems have attracted considerable interest over the last few decades due to their unique spectral and dynamical properties not encountered in Hermitian counterparts. An intensely debated question is whether non-Hermitian systems, described by pseudo-Hermitian Hamiltonians with real spectra, can offer enhanced sensitivity for parameter estimation when they are operated at or close to exceptional points. However, much of the current analysis and conclusions are based on mathematical formalism developed for Hermitian quantum systems, which is questionable when applied to pseudo-Hermitian Hamiltonians, whose Hilbert space is intrinsically nonflat. Here, we develop a covariant formulation of quantum Fisher information (QFI) defined on the deformed Hilbert space of pseudo-Hermitian Hamiltonians. This covariant framework ensures the preservation of the state norm and enables a consistent treatment of parameter sensitivity. We further show that the covariant QFI of pseudo-Hermitian systems is dual to the ordinary QFI of corresponding Hermitian systems. Importantly, this correspondence naturally imposes an upper bound on the covariant QFI and allows one to identify optimal projections which saturate the corresponding classical Fisher information to this ultimate limit. The developed framework also enables to set the criteria under which pseudo-Hermitian sensors can exhibit an advantage over their Hermitian counterparts of the same dimensionality.

This thesis concerns dynamics of wave propagations in non-Hermitian and nonlinear systems. In the first part, I study a non-Hermitian Dirac equation (NHDE) which serves as a novel platform to investigate Dirac points in the non-Hermitian regime. Semi-Hermiticity, which is the combination of pseudo-Hermiticity and anti-PT symmetry, enables the formation of diabolic points in non- Hermitian systems. The semi-Hermitian NHDE exhibits an anomalous Klein tunnelling with non-conserving flux, and boundary states akin to the Jackiw- Rebbi modes. These theoretical predictions were validated experimentally using a synthetic lattice constructed from optical fiber loops. The second part of this thesis extends the study of non-Hermitian systems to the nonlinear regime. First, by integrating Kerr-type nonlinearity into a 1D non-Hermitian lattice effectively described by the 1D NHDE, my study reveals the existence of topological gap solitons, characterized by eigenvalues residing within the topological gap and pseudo-spin textures in the solitons mode profiles. My next study involves a lattice that incorporates the non-Hermitian skin effect and saturable gain to achieve single mode lasing at high operational power. I propose a circuit analogue of the circuit design and demonstrate successful single mode operation at high power through simulations.

Quantum phase transitions occur in non-Hermitian systems. In this work we show that density functional theory, for the first time, uncovers universal critical behaviors for quantum phase transitions and quantum entanglement in non-Hermitian many-body systems. To be specific, we first prove that the non-degenerate steady state of a non-Hermitian quantum many body system is a universal function of the first derivative of the steady state energy with respect to the control parameter. This finding has far-reaching consequences for non-Hermitian systems. First, it bridges the non-analytic behavior of physical observable and no-analytic behavior of steady state energy, which explains why the quantum phase transitions in non-Hermitian systems occur for finite systems. Second, it predicts universal scaling behaviors of any physical observable at non-Hermitian phase transition point with scaling exponent being (1 − 1/p) with p being the number of coalesced states at the exceptional point. Third, it reveals that quantum entanglement in non-Hermitian phase transition point presents universal scaling behaviors with critical exponents being (1 − 1/p). These results uncover universal critical behaviors in non-Hermitian phase transitions and provide profound connections between entanglement and phase transition in non-Hermitian quantum many-body physics.

We analyze the chiral magnetic effect for non-Hermitian fermionic systems using the bi-orthogonal formulation of quantum mechanics. In contrast to the Hermitian counterparts, we show that the chiral magnetic effect takes place in equilibrium when a non-Hermitian system is considered. The key observation is that for non-Hermitian charged systems, there is no strict charge conservation as understood in Hermitian systems, so the Bloch theorem preventing currents in the thermodynamic limit and in equilibrium does not apply.

Non-Hermitian physics, such as the non-Hermitian skin effect (NHSE), is well-established in classical platforms, but its emergence in intrinsically Hermitian or quantum systems remains a key challenge. Bridging this gap is crucial for connecting non-Hermitian concepts with foundational quantum many-body theory. Here, we systematically investigate this by studying a quantum subsystem with an effective non-Hermitian Hamiltonian arising from its exact frequency-dependent self-energy. We further employ complex-frequency detection, including excitation, synthesis, and fingerprint, to probe physical responses induced by complex driving frequencies. Our calculations reveal that both complex frequency excitation and synthesis are incompatible with the non-Hermitian approximation and cannot characterize the presence of the NHSE. In contrast, the complex-frequency fingerprint successfully detects the distinctive responses induced by the NHSE through the introduction of a double-frequency Green’s function. Our work provides a platform for studying non-Hermitian physics and its unconventional response in quantum systems rigorously without relying on any approximations. Non-Hermitian physics, like the non-Hermitian skin effect (NHSE), is well-explored in classical platforms but remains challenging in quantum systems. Here, the authors study a quantum subsystem described by an effective non-Hermitian Hamiltonian derived from its exact frequency-dependent self-energy, and show that only complex-frequency fingerprinting can uniquely detect NHSE-induced responses, providing a rigorous framework for exploring non-Hermitian phenomena without approximation.

Self-propulsion is a quintessential aspect of biological systems, which can induce nonequilibrium phenomena that have no counterparts in passive systems. Motivated by biophysical interest together with recent advances in experimental techniques, active matter has been a rapidly developing field in physics. Meanwhile, over the past few decades, topology has played a crucial role to understand certain robust properties appearing in condensed matter systems. For instance, the nontrivial topology of band structures leads to the notion of topological insulators, where one can find robust gapless edge modes protected by the bulk band topology. We here review recent progress in an interdisciplinary area of research at the intersection of these two fields. Specifically, we give brief introductions to active matter and band topology in Hermitian systems, and then explain how the notion of band topology can be extended to nonequilibrium (and thus non-Hermitian) systems including active matter. We review recent studies that have demonstrated the intimate connections between active matter and topological materials, where exotic topological phenomena that are unfeasible in passive systems have been found. A possible extension of the band topology to nonlinear systems is also briefly discussed. Active matter can thus provide an ideal playground to explore topological phenomena in qualitatively new realms beyond conservative linear systems.

Non-hermitian systems have gained a lot of interest in recent years. However, notions of chaos and localization in such systems have not reached the same level of maturity as in the Hermitian systems. Here, we consider non-hermitian interacting disordered Hamiltonians and attempt to analyze their chaotic behavior or lack of it through the lens of the recently introduced non-hermitian analog of the spectral form factor and the complex spacing ratio. We consider three widely relevant non-hermitian models which are unique in their ways and serve as excellent platforms for such investigations. Two of the models considered are short-ranged and have different symmetries. The third model is long-ranged, whose hermitian counterpart has itself become a subject of growing interest. All these models exhibit a deep connection with the non-hermitian Random Matrix Theory of corresponding symmetry classes at relatively weak disorder. At relatively strong disorder, the models show the absence of complex eigenvalue correlation, thereby, corresponding to Poisson statistics. Our thorough analysis is expected to play a crucial role in understanding disordered open quantum systems in general.

This paper reports result of calculation and experimental realization of an electromechanical system that consists of a high-Q mechanical oscillator parametrically coupled in the manner of a capacitive transducer with a radio frequency (RF) circuit, which is in turn inductively coupled with another RF circuit. The system operates in the resolved sideband regime when the mechanical oscillator's frequency is larger than the electrical circuits' bandwidths. Using two coupled RF circuits allowed one to enhance the interaction between them and the mechanical oscillator which is one of flexural vibrational modes of a free-edge circular silicon wafer. Such a coupled electromechanical system can be used as a high-sensitive capacitive vibration sensor.

The dynamics of Hermitian many-body quantum systems has long been a challenging subject due to the complexity induced by the particle-particle interactions. In contrast, this difficulty may be avoided in a well-designed non-Hermitian system. The exceptional point (EP) in a non-Hermitian system admits a peculiar dynamics: the final state being a particular eigenstate, coalescing state. In this work, we study the dynamic transition from a trivial insulating state to an $\ensuremath{\eta}$-pairing state in a composite non-Hermitian Hubbard system. The system consists of two subsystems, A and B, which are connected by unidirectional hoppings. We show that the dynamic transition from an insulating state to an $\ensuremath{\eta}$-pairing state occurs by the probability flow from A to B: the initial state is prepared as an insulating state of A, while B is left empty. The final state is an $\ensuremath{\eta}$-pairing state in B but empty in A. Analytical analyses and numerical simulations show that the speed of relaxation of the off-diagonal long-range order pair state depends on the order of the EP, which is determined by the number of pairs and the fidelity of the scheme is immune to the irregularity of the lattice.

Metasurfaces are two-dimensional arrays of coupled resonant nanostructures designed for controlling various aspects of light scattering and light emission. In our research, we focus on plasmonic metasurfaces composed of metallic nanoantenna arrays supporting dispersive lattice resonances. We present some theoretical and experimental advancements in combining these lattice resonances with active media, which include breaking parity-time symmetry in arrays with spatially engineered gain and loss. We develop an experimental method based on interferometric autocorrelation, targeting the investigation of resonant properties of such non-Hermitian systems by exploiting the enhancement of two-photon excited luminescence in the vicinity of plasmonic nanoantennas. Our findings reveal novel design strategies for distributed feedback lasing and active light shaping by resonant periodic nanostructures.

Non-Hermitian systems display remarkable response effects that directly reflect a variety of distinct spectral scenarios, such as exceptional points where the eigensystem becomes defective. However, present frameworks treat the different scenarios as separate cases, following the singular mathematical change between different spectral decompositions from one scenario to another. This not only complicates the coherent description near the spectral singularities where the response qualitatively changes, but also impedes the application to practical systems, as the determination of these decompositions is manifestly ill conditioned. Here we develop a general response theory of non-Hermitian systems that uniformly applies across all spectral scenarios. We unravel this response by formulating a uniform expansion of the spectral quantization condition, as well as a uniform expansion of the Green's function, where both expansions exclusively involve directly calculable data from the Hamiltonian. These data smoothly vary with external parameters and energy as spectral singularities are approached and attained, and nevertheless capture the qualitative differences of the response in these scenarios. We furthermore present two direct applications of this framework. First, in the context of the quantization condition, we determine the precise conditions for spectral degeneracies of geometric multiplicity greater than unity, as well as the perturbative behavior around these cases. Second, in the context of the Green's function, we formulate a hierarchy of spectral response strengths that varies continuously across all parameter space, and thereby also reliably determines the response strength of exceptional points. Finally, we join both themes and demonstrate, both generally and in concrete examples, that the previously inaccessible scenarios of higher geometric multiplicity result in unique variants of the super-Lorentzian response. Our approach widens the scope of non-Hermitian response theory to capture all spectral scenarios on an equal and uniform footing, identifies the exact mechanisms that lead to the qualitative changes of physical signatures, and renders non-Hermitian response theory fully applicable to numerical descriptions of practical systems. Published by the American Physical Society 2025