Complex frequency detection in a subsystem

Non-Hermitian physics has greatly enriched our understanding of nonequilibrium phenomena and uncovered novel effects such as the non-Hermitian skin effect (NHSE) that has profoundly revolutionized the field. NHSE has been predicted in systems with nonreciprocal couplings which, however, are challenging to realize in experiments. Without nonreciprocal couplings, the NHSE can also emerge in systems with coexisting gauge fields and loss or gain (e.g., in Floquet non-Hermitian systems). However, such Floquet NHSE remains largely unexplored in experiments. Here, we realize the Floquet NHSEs in periodically modulated optical waveguides integrated on a silicon photonic platform. By engineering the artificial gauge fields induced by the periodical modulation, we observe various Floquet NHSE phases and unveil their rich topological transitions. Remarkably, we discover the transitions between the unipolar NHSE phases and an unconventional bipolar NHSE phase, which is accompanied by the directional reversal of the NHSEs. The underlying physics is revealed by the band winding in complex quasienergy space which undergoes a topology change from isolated loops with the same winding to linked loops with opposite windings. Our work unfolds a new route toward Floquet NHSEs originating from the interplay between gauge fields and dissipation effects, and thus offers fundamentally new ways for steering light and other waves.

Parity-time (PT)-symmetric Hamiltonians have widespread significance in non-Hermitian physics. A PT-symmetric Hamiltonian can exhibit distinct phases with either real or complex eigenspectrum, while the transition points in between, the so-called exceptional points, give rise to a host of critical behaviors that holds great promise for applications. For spatially periodic non-Hermitian systems, PT symmetries are commonly characterized and observed in line with the Bloch band theory, with exceptional points dwelling in the Brillouin zone. Here, in nonunitary quantum walks of single photons, we uncover a novel family of exceptional points beyond this common wisdom. These "non-Bloch exceptional points" originate from the accumulation of bulk eigenstates near boundaries, known as the non-Hermitian skin effect, and inhabit a generalized Brillouin zone. Our finding opens the avenue toward a generalized PT-symmetry framework, and reveals the intriguing interplay between PT symmetry and non-Hermitian skin effect.

Non-Hermitian skin effects (NHSEs) enrich the traditional band theories and give rise to unique non-Hermitian topological properties. Here, we reveal a different type of NHSE, termed the V-shaped NHSE, and explore an on-chip optical ring resonator platform to demonstrate the NHSE in telecommunication wavelength. By tuning the hopping and loss modulation, we work out a phase diagram that reveals the phase transition of the NHSE. Besides the conventional NHSE and the frequency-dependent bipolar NHSE, another type of NHSE is observed featuring an interesting V shape in its special skin states distribution. We further design a silicon ring resonator array with loss-assisted antiresonating coupling as the link resonators to demonstrate the nontrivial NHSE. The simulations show that one can tune the number of states piled up at different ends by switching the NHSE from conventional, bipolar to V-shaped types continuously. Our work enriches the NHSE phase transitions and implies potential in on-chip light modulation by the NHSE.

<sec>Quantum simulation serves as a significant tool for studying and understanding novel phenomena in the quantum world. In recent years, it has be realized that apart from quantum platforms, classical systems like photonic crystals, phononic crystals, and mechanical oscillators can also be used to simulate quantum models by analogizing the Schrödinger equation. Among these systems, electrical circuits have emerged as a promising simulation platform owing to their low cost, technological maturity, and ease of scalability, successfully simulating numerous important quantum phenomena. Meanwhile, non-Hermitian physics breaks the Hermiticity of systems’ Hamiltonians in traditional quantum mechanics, providing a fresh perspective for understanding the physics of quantum systems, particularly open quantum systems. Non-Hermitian systems, owing to their manifestation of unique phenomena absent in Hermitian systems, have become emerging research subjects in various fields of physics. However, many non-Hermitian phenomena require specialized configurations that pose relatively high technical thresholds on quantum platforms. For instance, the non-Hermitian skin effect typically requires systems to possess non-reciprocal hopping between lattice sites. Therefore, utilizing flexible electrical circuits to simulate non-Hermitian physics becomes a natural choice.</sec><sec>This paper provides a short review of the current experimental progress in simulating non-Hermitian lattice models by using electrical circuits. It offers a brief introduction to the relevant knowledge of non-Hermitian physics, including mathematical concepts and novel phenomena, as well as the simulation theory of electrical circuits, including the mapping theory of the lattice models, the introduction of non-Hermiticity, and the measurement of physical quantities. The aim is to provide readers with a reference for better understanding or engagement in related researches, thus promoting further development in this field.</sec>

One of the unique features of non-Hermitian Hamiltonians is the non-Hermitian skin effect, namely, that the eigenstates are exponentially localized at the boundary of the system. For open quantum systems, a short-time evolution can often be well described by the effective non-Hermitian Hamiltonians, while long-time dynamics calls for the Lindblad master equations, in which the Liouvillian superoperators generate time evolution. In this Letter, we find that Liouvillian superoperators can exhibit the non-Hermitian skin effect, and uncover its unexpected physical consequences. It is shown that the non-Hermitian skin effect dramatically shapes the long-time dynamics, such that the damping in a class of open quantum systems is algebraic under periodic boundary conditions but exponential under open boundary conditions. Moreover, the non-Hermitian skin effect and non-Bloch bands cause a chiral damping with a sharp wave front. These phenomena are beyond the effective non-Hermitian Hamiltonians; instead, they belong to the non-Hermitian physics of full-fledged open quantum dynamics.

Non-Hermitian skin effect (NHSE) describes the exponential localization of all eigenstates toward boundaries in non-Hermitian systems, and has attracted intense research interest of late. Here we theoretically propose a scheme in which the NHSE significantly impacts the external motion of a single trapped ion through complex spin-motion dynamics. On the one hand, we show the competition between the NHSE and the coherent Bloch dynamics. On the other hand, since the NHSE manifests as a non-reciprocal flow in occupied phonon modes, we demonstrate that such dynamics can have potential applications in cooling and sensing. Our proposal can be readily implemented using existing experimental techniques, and offers a scalable (in terms of the available ions and phonon modes) simulation platform for relevant non-Hermitian physics.

Non-Hermiticity enables macroscopic accumulation of bulk states, named non-Hermitian skin effects. The non-Hermitian skin effects are well established for single-particle systems, but their proper characterization for general systems is elusive. Here, we propose a general criterion of non-Hermitian skin effects, which works for any finite-dimensional system evolved by a linear operator. The applicable systems include many-body systems and network systems. A system meeting the criterion exhibits enhanced non-normality of the evolution operator, accompanied by exceptional characteristics intrinsic to non-Hermitian systems. Applying the criterion, we discover a new type of non-Hermitian skin effect in many-body systems, which we dub the Fock space skin effect. We also discuss the Fock space skin effect-induced slow dynamics, which gives an experimental signal for the Fock space skin effect.

Non-Hermitian systems have been shown to have a dramatic sensitivity to their boundary conditions. In particular, the non-Hermitian skin effect induces collective boundary localization upon turning off boundary coupling, a feature very distinct from that under periodic boundary conditions. Here we develop a full framework for non-Hermitian impurity physics in a non-reciprocal lattice, with periodic/open boundary conditions and even their interpolations being special cases across a whole range of boundary impurity strengths. We uncover steady states with scale-free localization along or even against the direction of non-reciprocity in various impurity strength regimes. Also present are Bloch-like states that survive albeit broken translational invariance. We further explore the co-existence of non-Hermitian skin effect and scale-free localization, where even qualitative aspects of the system’s spectrum can be extremely sensitive to impurity strength. Specific circuit setups are also proposed for experimentally detecting the scale-free accumulation, with simulation results confirming our main findings.

Topological insulators possess strong topological protection properties and can manipulate the wave propagation to combat disorder and defects. And now they have grown into a large research field in photonic and phononic crystals. However, the conventional topological band theory is used to describe a closed photonic/phononic crystal that is assumed to be a Hermitian system. In fact, actual physical systems often couple with external environment, and generate non-Hermitian Hamiltonians with complex eigenvalues. Recently, many novel topological properties have been induced by the interaction between non-Hermitian phase and topological phase. A prominent example is non-Hermitian skin effect that all eigenstates are localized to the boundary in open system, which is different from the conventional topological edge state. This unique physical phenomenon has inspired various applications, such as wave funneling, enhanced sensing, and topological lasing. In this work, we describe the non-Hermitian skin effect by using winding number domains. The sign of the winding number domain determines the rotation direction of the loops in the complex frequency plane, whose sign can be controlled by the nonreciprocal coupling direction. In this work, we design different topological skin interfaces between different domains with opposite winding numbers to manipulate the energy focusing on middle or two-end of non-Hermitian one-dimensional acoustic cavity chain. In experiment, we use an electroacoustic coupling method, in which a unidirectional coupler composed of microphones, speakers, phase shifters, and amplifiers is used to introduce positive and negative non-reciprocal couplings between the two acoustic cavities, and study the characteristics of these non-reciprocal couplings. Then, the non-reciprocal coupling cavities are extended into a chain structure, and the magnitudes and signs of the non-reciprocal couplings are flexibly controlled by using phase shifters and amplifiers. Through this method, we successfully construct the interfaces between different winding number domains, achieving a one-dimensional non-Hermitian skin effect at various interfaces. The experimental results indicate that the sound can be focused on the middle interface or two-end interfaces for different nonreciprocal coupling distributions, and the skin interface can also be switched from middle to two-end by exchanging the nonreciprocal coupling direction of the domains. Our research results provide greater flexibility for designing acoustic devices and also a new platform for exploring advanced topological acoustic systems for controlling sound propagation.

Non-Hermitian systems exhibit an extraordinary sensitivity to boundary conditions, with a notable example being the non-Hermitian skin effect (NHSE), where a significant number of eigenstates accumulate at their open boundaries. In this work, we reveal a supersensitivity feature associated with the NHSE. In an acoustic system with dissipative couplings that preserve the anomalous time-reversal symmetry, we observe a k -dependent skin effect, in which acoustic waves localize at either the left or right open boundaries depending on the perpendicular wave vector. Remarkably, this supersensitivity is so extreme that adding a single unit to one boundary transforms the skin effect from bidirectional to unidirectional. More intriguingly, introducing another single unit to the opposite boundary can restore the original bidirectional skin effect. We directly observe this supersensitivity by detecting acoustic field distributions, consistent with theoretical predictions. Our findings offer a powerful approach to non-Hermitian manipulation and pave the way for high-sensitivity measurement using non-Hermitian systems.

Non-Hermitian (NH) crystals, quasicrystals, and amorphous network display an accumulation of a macroscopic number of states near one of its specific interfaces with vacuum, such as edge, surface, hinge, or corner. This phenomenon is known as the NH skin effect, which can only be observed with open boundary condition. In this regard self-similar fractals, manifesting inner boundaries in the interior of the system, harbor a novel phenomenon, the inner skin effect (ISE). Then the NH skin effect appears at the inner boundaries of the fractal lattice with periodic boundary condition. We showcase this observation by implementing prominent models for NH insulators and superconductors on representative planar Sierpinski carpet fractal lattices. They accommodate both first-order and second-order ISEs at inner edges and corners, respectively, for charged as well as neutral Majorana fermions. Furthermore, over extended parameter regimes ISEs are tied with nontrivial bulk topological invariants, yielding intrinsic ISEs. With the recent success in engineering NH topological phases on highly tunable metamaterial platforms, such as photonic and phononic lattices, as well as topolectric circuits, the proposed ISEs can be observed experimentally at least on fractal metamaterials with periodic boundary condition.

The non-Hermitian skin effect (NHSE), an exotic phenomenon of localization at the boundary, has recently drawn extensive attention and Floquet engineering offers a powerful approach to expanding its scope. The fundamental role of harmonic response in Floquet systems has nonetheless yet to be elucidated. Here, we introduce the concept of harmonic NHSE and demonstrate it experimentally in an acoustic platform. Within our theoretical framework, we demonstrate that incorporating dynamic couplings into the paradigmatic Hatano-Nelson model causes a single-frequency input to generate multiple harmonics with diverse skin morphologies, each dictated by the underlying spectral winding topology at its respective frequency. Experimentally, we implement the requisite dynamic couplings in an acoustic lattice via programmable unidirectional couplers. By simply reconfiguring the excitation and modulation frequencies, we demonstrate controllable switching of unipolar and bipolar harmonic NHSEs, which exhibit uniform and opposite sound gathering toward the sample boundaries, respectively. By introducing the class of harmonic NHSE, our work sets the stage for further investigating non-Hermitian physics with harmonic effects.

We study a Bose-Einstein condensate of ultracold atoms subject to a non-Hermitian spin-orbit coupling, where the system acquires the non-Hermitian skin effect under the interplay of spin-orbit coupling and laser-induced atom loss. The presence of the non-Hermitian skin effect is confirmed through its key signatures in terms of the spectral winding under the periodic boundary condition, the accumulation of eigen wavefunctions at boundaries under an open boundary condition, and bulk dynamics signaled by a directional flow. We show that bulk dynamics, in particular, serves as a convenient signal for experimental detection. The impact of interaction and trapping potentials is also discussed based on the non-Hermitian Gross-Pitaevskii equations. Our work demonstrates that the non-Hermitian skin effect and its rich implications in topology, dynamics, and beyond are well within the reach of current cold-atom experiments.

Single-mode operation is a desirable but elusive property for lasers operating at high pump powers. Typically, single-mode lasing is attainable close to threshold, but increasing the pump power gives rise to multiple lasing peaks due to inter-modal gain competition. We propose a laser with the opposite behavior: multimode lasing occurs at low output powers, but pumping beyond a certain value produces a single lasing mode, with all other candidate modes experiencing negative effective gain. This phenomenon arises in a lattice of coupled optical resonators with non-fine-tuned asymmetric couplings, and is caused by an interaction between nonlinear gain saturation and the non-Hermitian skin effect. The single-mode lasing is observed in both frequency domain and time domain simulations. It is robust against on-site disorder, and scales up to large lattice sizes. This finding might be useful for implementing high-power laser arrays.

Lately, the non-Hermitian skin effect (NHSE) has been demonstrated in various classical metamaterials and even ultracold atomic arrays. Yet, its interplay with many-body dynamics have never been experimentally investigated. Here, we report the observation of the NHSE and its many-fermion analog on a universal quantum processor. To implement NHSE accumulation on a quantum computer, the time-evolution circuit not only needs to be non-reciprocal and non-unitary, but must also contain sufficiently many lattice qubits. We demonstrate this by systematically post-selecting ancilla qubits, as demonstrated through two paradigmatic non-reciprocal models on noisy quantum processors, with clear signatures of asymmetric spatial propagation and many-body “Fermi skin” accumulation. To minimize errors from inevitable device noise, time evolution is performed using trainable, variationally optimized quantum circuits. Our demonstration represents an important step in the quantum simulation of non-Hermitian lattices on present-day quantum hardware, and can be readily generalized to more sophisticated many-body models.