Complex Photonic Systems with Broken Symmetries: Synergetic Effects of Wave Chaos and Non-Hermiticity

The open quantum system is a part of a bigger closed system; it is typically a small but complex system (often called the system) surrounded by a large but simple system (often called the environment). I here report our new findings about the non-Hermiticity and non-Markovianity of open quantum systems. Since the system exchanges energy and probability with the environment, the effective Hamiltonian of the system is generally non-Hermitian. Complex eigenvalues of all resonant states are given by the non-Hermitian Hamiltonian. For one-body scattering problem, I report our new resolution of unity that comprises all point spectra including all complex eigenvalues. Another interesting feature of open quantum systems is their dynamics. Since the environemnt functions as a memory, the dynamics of the system is generally non-Markovian with the memory kernel coming from the Green's function of the environment. Again for one-body problem, I report our new findings about the power-law decay in the short- and long-term regimes of the dynamics of the system.

We consider a PT-symmetric Fermi gas with an exceptional point, representing the critical point between PT-symmetric and symmetry broken phases. The low energy spectrum remains linear in momentum and is identical to that of a hermitian Fermi gas. The fermionic Green's function decays in a power law fashion for large distances, as expected from gapless excitations, albeit the exponent is reduced from −1 due to the quantum Zeno effect. In spite of the gapless nature of the excitations, the ground state entanglement entropy saturates to a finite value, independent of the subsystem size due to the non-hermitian correlation length intrinsic to the system. Attractive or repulsive interaction drives the system into the PT-symmetry broken regime or opens up a gap and protects PT-symmetry, respectively.

TBC

The one-dimensional Hatano-Nelson model with non-reciprocal hoppings is a prominent example of a relatively simple non-Hermitian quantum-mechanical system, which allows to study various phenomena in open quantum systems without adding extra gain and loss terms. Here, we propose its extension to two dimensions in the form of a flux model with clock-anticlockwise non-reciprocal hopping on each plaquette. Adding the on-site Hubbard type interaction we analyze the formation of the longe-range antiferromagnetic order and its spin excitations. Such a model is non-Hermitian, but $\mathcal{PT}$-symmetric, which leads to the existence of two regions: a region of unbroken $\mathcal{PT}$ symmetry (real-valued spectrum) and a region of broken $\mathcal{PT}$ symmetry with exceptional lines and complex-valued energy spectrum. The transition from one region to another is controlled by the value of the on-site interaction parameter and coincides with the metal-insulator transition. We also analyze the spin wave spectrum, which is characterized by the nearly flat band of zero-energy excitations.

Topological insulators support edge states that are protected against disorder and scattering, with fundamental differences between chiral and helical modes. Chiral edge states, observed in quantum Hall and quantum anomalous Hall systems, emerge in time-reversal symmetry (TRS) broken media, whereas helical edge states, characteristic of the quantum spin Hall effect (QSHE), exist in TRS-preserving systems and support counter-propagating modes with opposite pseudospins. While the spatial properties of these modes have been widely studied, their dynamical transport characteristics remain largely unexplored. We present a microwave experiment that emulates QSHE and enables time-resolved observations of helical edge state propagation. Our measurements reveal robust, pseudospin-polarized transport that is immune to defects and propagates at velocities orders of magnitude lower than free-space waves. These findings provide new insights into topological transport dynamics and highlight implications for photonic and quantum information technologies.

Non-Euclidean optical microcavities are demonstrated as flat photonic devices using transformation optics. Although the devices have a planar structure, they possess the optical characteristics of non-Euclidean microcavity resonators.

We present an optical processing approach for ultrafast image acquisition and real-time image recognition, addressing the limitations of conventional machine vision and imaging methods. This approach is based on the sequential projection of random patterns generated through a scattering medium. It compressively maps an image of a target object into a time-domain data stream. The stream is then processed with an optical reservoir computer, enabling real-time dynamic image recognition and anomaly detection. Beyond image recognition, this method also captures a wide range of temporal phenomena with sub-nanosecond time resolution.

We investigate nonlinear light structuring via total angular momentum (TAM) projection control. Even a thin film of amorphous silicon offers substantial control over light structuring and manipulation. Using third-harmonic generation under tightly focused excitation, we experimentally demonstrate TAM tripling and polarization-sensitive field patterns[1]. The observed behavior is explained by conservation of TAM projection, and is accurately reproduced by vectorial numerical simulations. Beyond this specific case, we developed a general theoretical framework based on symmetry arguments, suitable for both uniform media and nanostructures with discrete rotational symmetries [1,2]. The resulting selection rules describe the TAM content of the nonlinear signal as a function of the pump and material properties, providing a compact and predictive tool for designing structured harmonic generation in a broad range of systems. The resulting framework can be used, for instance, to predict nonlinear circular dichroism, describe geometric phases, or control far-field emission patterns. [1] Menshikov, Evgenii, et al. "Light structuring via nonlinear total angular momentum addition with flat optics." arXiv, 4 Dec. 2024, doi:10.48550/arXiv.2412.03367. [2]Nikitina, Anastasia and Kristina Frizyuk. "Achiral Nanostructures: Perturbative Harmonic Generation and Dichroism Under Vortex and Vector Beams Illumination." Adv. Opt. Mater., vol. 12, no. 25, 1 Sept. 2024, p. 2400732, doi:10.1002/adom.202400732.

Light scattering in random media is usually considered within the framework of the three-dimensional Anderson universality class, with modifications for the vector nature of electromagnetic waves. In this talk, we discuss how the linear dispersiveness of light introduces topological aspects into the picture. The dynamics of electromagnetic waves follow the same differential equations as those of a spin-1 Weyl semimetal. In the presence of disorder, this equivalence leads to a range of phenomena. These include topological protection against localization when helicity hybridization is weak, the emergence of exotic phases in weakly scattering media, and anomalies in optical transparency in the presence of synthetic ‘magnetic fields'. We will reason that some of these effects should be visible already in weakly disordered optical materials.

Topology has been a major research theme in condensed matter physics and is associated with a number of remarkable phenomena such as robust edge states. A prominent example is the quantum Hall effect, in which the topological invariant is directly observable through the Hall resistance. More recently, topology started to be investigated in systems experiencing gain and loss sparking the field of non-Hermitian topology. However, for a long time, a clear observable signature of non-Hermitian topology had been lacking. In this talk, I will show that non-trivial, non-Hermitian topology is in one-to-one correspondence with the phenomenon of directional amplification [1-3] in one-dimensional bosonic systems, e.g., cavity arrays. Directional amplification allows to selectively amplify signals depending on their propagation direction and has attracted much attention as key resource for applications, such as quantum information processing. Remarkably, in non-trivial topological phases, the end-to-end gain grows exponentially with the number of sites. Furthermore, this effect is robust against disorder [2] with the amount of tolerated disorder given by the separation between the complex spectrum and the origin. In collaboration with the group of Ewold Verhagen at AMOLF, Amsterdam, we experimentally demonstrate the connection between non-Hermitian topology and directional amplification in a cavity optomechanical system [4] by realising a bosonic version of the Kitaev-Majorana chain proposed in [5]. Furthermore, we show in the experiment that a similar system proposed in [6] can be utilised as a sensor with a sensitivity that grows exponentially with system size [4]. Our work opens up new routes for the design of multimode robust directional amplifiers and sensors based on non-Hermitian topology that can be integrated in scalable platforms such as superconducting circuits, cavity optomechanical systems, plasmonic waveguides and nanocavity arrays. [1] Wanjura, Brunelli, Nunnenkamp. Nat Commun 11, 3149 (2020).
 [2] Wanjura, Brunelli, Nunnenkamp. Phys. Rev. Lett. 127, 213601 (2021). [3] Brunelli, Wanjura, Nunnenkamp. SciPost Phys 15, 173 (2023). 
[4] Slim, Wanjura, Brunelli, del Pino, Nunnenkamp, Verhagen. Nature 627, 767–771 (2024). [5] McDonald, Pereg-Barnea, Clerk. Phys Rev X 8, 041031 (2018).
 [6] McDonald, Clerk. Nat Commun 11, 5382 (2020).

The notions of chaos and order are central to understanding the statistical physics of many-body systems. Thermalization and the spread of quantum information in chaotic many-body dynamics is presently attracting a lot of attention across various fields, ranging from statistical via cold atom physics to quantum gravity and quantum computing. Vice versa, it is long known how to harness exponential sensitivity to changes in initial conditions for control purposes in classically chaotic systems. We will generalize this concept, using chaos as a resource for efficiently steering many-body quantum dynamics. We will consider quantum chaos control for two prototypical systems of chaotic single- and many-body dynamics: the quantum kicked rotor and the Bose Hubbard model. We will employ semiclassical methods, thereby bridging the classical and quantum chaotic (many-body) world.

We investigate the fate of the non-Hermitian skin effect in one-dimensional systems that conserve the dipole moment and higher moments of a global U(1) charge. The key feature of this effect in m-pole-conserving systems is the generation of an (m+1)-th multipole moment, observable in both the eigenstates and their dynamics. For example, in contrast to the conventional skin effect—where charges are anomalously localized at a single boundary—the dipole-conserving skin effect localizes charges at both boundaries, forming a configuration that maximizes the quadrupole moment. Beyond charge distribution, we show that entanglement entropy serves as a quantum signature of the many-body skin effect. Finally, we study the impact of quenched disorder on the non-Hermitian multipole skin effect. We demonstrate that when only the U(1) charge is conserved, there is a transition between a skin-effect phase, where charges localize at one boundary, and a many-body localized phase, where charges localize at random positions. Under periodic boundary conditions, the non-Hermitian skin effect instead gives way to a delocalized phase with a unidirectional current. If additional multipole moments (such as dipoles) are conserved, the non-Hermitian skin effect remains robust against any amount of disorder. Consequently, under periodic boundary conditions, the system stays delocalized regardless of disorder strength. References: a) "Many-body non-hermitian skin effect for multipoles" (PRL 2024) b) "Non-Hermitian Multipole Skin Effects Challenge Localization" (In Preparation)

We study fluctuation properties in the energy spectra of finite-size honeycomb lattices - graphene billiards - subject to the Haldane-model on-site potential and next-nearest-neighbor tunneling at critical points, referred to as Haldane graphene billiards in the following. The billiards have the shapes of billiards with integrable and chaotic dynamics and in between. It had been shown that the spectral properties of the graphene billiards coincide with those of the nonrelativistic quantum billiard with the corresponding shape, both at the band edges and in the region of low-energy excitations around the Dirac points at zero energy. There, the dispersion relation is linear and, accordingly, the spectrum is described by the same relativistic Dirac equation for massless spin-1/2 particles as relativistic neutrino billiards, whose spectral properties agree with those of nonrelativistic quantum billiards with violated time-reversal invariance. Deviations from the expected behavior are attributed to differing boundary conditions and backscattering at the boundary, which leads to a mixing of valley states corresponding to the two Dirac points, that are mapped into each other through time reversal. We employ a Haldane model to introduce a gap at one of the two Dirac points so that backscattering is suppressed in the energy region of the gap and demonstrate that there the correlations in the spectra comply with those of the neutrino billiard of the corresponding shape. Here, the phase transition from nonrelativistic to relativistic quantum behavior is achieved by adjusting the Haldane parameters.

Information is created when a system is changed. This information can be detected by waves, in our case by electromagnetic waves, created and measured far from the creation point. I will shortly introduce Fisher information and then show how the information transport can be visualized using the Fisher information flow, similar to the Poynting vector showing the energy flow. The main focus of the talk will be on the experimental realization in a microwave waveguide supporting 4 transporting modes.

Singularities of the scattering matrix underlie many of the exotic and exciting phenomena exhibited by non-Hermitian systems. These singularities include coherent perfect absorption (CPA) where a specific signal injected into a system is completely absorbed, exceptional points (EP) where the eigenvalues and associated eigenvectors coalesce and become degenerate, and others. Many previous studies of scattering singularities have carefully tuned and designed their systems to exhibit a single singularity at a particular point in parameter space. These systems also tend to be static, in the sense that there are no parameters that can be easily varied in situ to perturb the system. These works have revealed a great deal about the phenomena and properties of isolated singularities, but the interactions and dynamics of multiple singularities have not been thoroughly explored. We demonstrate that it is possible to study multiple scattering singularities and their dynamics simultaneously by means of generic microwave cavities with complex boundaries and randomly-distributed scattering centers that break all symmetries. One of the key components enabling this exploration is the implementation of tunable boundary conditions and scattering sites within the system. We find that for two-channel systems in one-, two-, and three-dimensional wave propagation systems, scattering singularities are remarkably abundant, topologically protected, and are created/annihilated obeying a set of strict topological constraints. The singularities are easily manipulated in a smooth and continuous manner in systems either with or without reciprocity. Going further, using multiple tunable parameters we can manipulate systems to combine two different singularities at the same point in parameter space. Specifically, for the combination of both CPA and an EP at the same point, we experimentally demonstrate a new application, which is a 50:50 In-phase/Quadrature (I/Q) power splitter that is robust to any variation in injected signal magnitude or phase.

Combining non-Hermiticity and interactions in open quantum correlated systems gives rise to a plethora of intriguing physical phenomena. The networks of non-Hermitian $PT$-symmetric superconducting qubits directly embedded in a low-dissipative resonator [1] are a promising experimental platform to realize quantum simulations for such systems, to identify novel collective quantum phases and excitations. Motivated by recent experiments on single non-Hermitian superconducting [2] and spin-qubits [3] I present here a theoretical analysis of an exemplary $PT$-symmetric superconducting qubits chains described by spin interacting models. A non-Hermitian part of the Hamiltonian can be implemented by two methods: via imaginary staggered longitudinal magnetic field, which corresponds to a local staggered gain and loss terms [1-4], or non-reciprocal exchange interaction between adjacent qubits (Hatano-Nelson model). By making use of a hard-core bosons approach we develop a quasi-classical mean field theory for such PT-quantum systems and obtain the dependence of ground states and low-lying excitations (magnonic modes) on the adjacent spins interaction strength $J$ and the parameter $\gamma$ characterizing the non-Hermiticity of a system. We identify the $PT$ -symmetry broken and unbroken regimes of collective paramagnetic ($P$) and compressible superfluid ($CS$) states, and obtain the spectrum of magnonic modes in these phases. References [1] G. A. Starkov, M. V. Fistul, and I. M. Eremin, Phys. Rev. B 108, 235417 (2023). [2] W. Chen, M. Abbasi, Y. N. Joglekar, and K. W. Murch, Phys. Rev. Lett. 127, 140504 (2021). [3] Y. Wu, et. al., Science 364, 878 (2019). [4] L. Tetling, M. V. Fistul, and I. M. Eremin, Phys. Rev. B 106, 134511 (2022).

I will introduce various kinds of self-organized optimization of stationary lasing states in non-Hermitian photonic systems: uniform pumping of fully chaotic billiard lasers and asymmetric billiard lasers, and selective pumping of fully chaotic billiard lasers. Interesting phenomena such as universal single-mode lasing, locking of a chiral pair near an exceptional point and deep nonlinear scars will be shown.

We investigate the eigenmodes of a superlattice photonic crystal composed of coupled microcavities, focusing on their spectral and spatial properties. By solving the wave equation with appropriate boundary conditions, we analyze how inter-cavity coupling influences mode structure and localization. Our findings reveal nontrivial topological features and distinctive wave dynamics, offering insights into applications in photonic devices and controlled light transport.

In recent years, several proposals that leverage principles from condensed matter and high-energy physics for engineering laser arrays have been put forward. The most important among these concepts are topology, which enables the construction of robust zero-mode laser devices, and supersymmetry (SUSY), which holds the potential for achieving phase locking in laser arrays. We demonstrate that the relation between supersymmetric coupled bosonic and fermionic oscillators on one side, and bipartite networks (and hence chiral symmetry) on another side can be exploited together with non-Hermitian engineering for building one- and two-dimensional topological laser arrays with in-phase synchronization. To elucidate our strategy, a concrete design is presented starting from the celebrated Su-Schrieffer-Heeger (SSH) model to arrive at a SUSY laser structure that enjoys two key advantages over those reported in previous works. Firstly, the design presented here features a near-uniform geometry for both the laser array and supersymmetric reservoir (i.e., the widths and distances between the cavity arrays are almost the same). Secondly, the uniform field distribution in the presented structure leads to a far-field intensity that scales as N^2 where N is the number of lasing elements. Taken together, these two features can enable the implementation of higher-power laser arrays that are easy to fabricate, and hence provide a roadmap for pushing the frontier of SUSY laser arrays beyond the proof-of-concept phase. Ref.: A topological route to engineering robust and bright supersymmetric laser arrays - S. Datta, M. Alizadeh, R. El-Ganainy, and K. Roychowdhury, Nature Communications Physics, 7, 403 (2024).

The properties of non-Hermitian quantum systems are often less intuitive than those of conventional Hermitian systems. Here we make use of the Husimi representation in phase space to analyse dynamical and spectral features. We consider the flow of the Husimi phase-space distribution in a semiclassical limit, leading to a first order partial differential equation, that helps illuminate the foundations of the full quantum evolution. Further, we demonstrate how ingredients of the dynamics can be used to construct approximate Husimi distributions of characteristic quantum states.

environment. When the coupling is weak and the environment evolves on a much slower timescale than the system, the dynamics can be described by a Markovian master equation, also known as the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation [1,2]. However, this description breaks down in both the short-time [3] and long-time regimes [4], often requiring one to resort to numerical methods. To analyze these regimes analytically, we introduce a simple toy model [5], where a two-level system is coupled to a one-dimensional particle bath characterized by a Dirac dispersion relation. We show that the short- and long-time dynamics are governed by the high- and low-energy features of the spectral density: a high-energy cutoff controls the short-time behavior, while low-energy structure—such as the spectral Dirac gap—determines the long-time decay. By examining the wave function of the system, we find that a bound state emerges when a spectral Dirac gap opens, while the scattering state exhibits exponential growth up to a finite spatial range before rapidly decaying to zero in the large-space limit. Finally, we propose an experimental realization of our toy model using an optical waveguide array in the Su–Schrieffer–Heeger (SSH) configuration. We show that the relevant features of the model—such as the Dirac gap—can be engineered using current photonic technologies, enabling direct observation of the predicted Markovian-to-non-Markovian crossover in realistic parameter regimes. [1] Gorini, V., Kossakowski, A., & Sudarshan, E. C. G. (1976). J. Math. Phys., 17(5), 821–825. [2] Lindblad, G. (1976). Commun. Math. Phys., 48, 119–130. [3] Chiu, C. B., Sudarshan, E. C. G., & Misra, B. (1977). Phys. Rev. D, 16(2), 520. [4] Khalfin, L. A. (1958). Sov. Phys. JETP, 6(6), 1053–1063. [5] Taira, T., Hatano, N., & Nishino, A. (2024). arXiv preprint arXiv:2406.17436.

Understanding open quantum systems using information encoded in its complex eigenvalues has been a subject of growing interest. The higher-order gap ratio of the singular values of an open quantum system can be connected to the nearest-neighbor spacing ratio of positions of classical particles of a harmonically confined log gas with inverse temperature $\beta'(k)$, where $\beta'(k)$ is an analytical function that depends on $k$ and the Dyson’s index $\beta$ = 1, 2, and 4 that characterizes the properties of the associated Hermitized matrix. The results are crucial not only for understanding long-range correlations between the eigenvalues but also provide an excellent way of distinguishing different symmetry classes in an open quantum system. The universality of the higher-order gap ratios are demonstrated using different platforms such as non-Hermitian random matrices, random dissipative Liouvillians, Hamiltonians coupled to a Markovian bath, and Hamiltonians with built-in non-Hermiticity.

n order to describe realistic situations in which a quantum system interacts with external systems, a theory beyond the conventional framework of quantum mechanics for isolated systems is required. Open quantum systems constitute a theoretical framework that describes the realistic situations. It is well known that the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) describes the Markovian time evolution of open quantum systems. However the dynamics of open quantum systems is generally the non-Markovian one, which depends on states of the system of interest in the past, and there is no genaral framework to discuss the non-Markovian time evolution of open quantum systems. Keeping the extension to the non-Markovian region in our mind, we develop the effective Liouvillian method for open quantum systems. In this method, we mainly discuss the dynamics not in the time domain but in the Fourier-Laplace domain, and we use a new pair of the complement projection super-operators in the method. Applying the effective Liouvillian method to the Friedrichs model, we have two results. The first result is that the time evolution of the system of interest is decomposed into five contributions: (i) the dominant term for the rescaled time under the interaction between the system and the environment; (ii) the perturbation terms under the rescaled time; (iii) the contribution from the contour integral around a branch point; (iv) the effects of thermal fluctuation of the environment; (v) the perturbation terms under the original time scale without the interaction. Then we estimate the orders of each term. The second result is that the term (i) obeys the GKSL equation which is equivalent to the one derived by the conventional method. These results mean that I find what terms are neglected by the approximations, which was not obvious in the conventional method. This would be a toehold for studies of the non-Markovian region by picking up contributions from the neglected terms under the Markovian approximation.

Non-Hermitian systems exhibit remarkable response effects tied to distinct spectral scenarios, most notably exceptional points of different order. Existing frameworks treat these cases separately based on their specific spectral decomposition, which complicates the analysis in their vicinity, and limits the practical applicability due to the ill-conditioned nature of these decompositions. Furthermore, much less attention has been paid to the more complicated case of degeneracies with higher geometric multiplicity. This talk presents a general response theory that applies uniformly across all these spectral scenarios. By developing a universal expansion of the spectral quantization condition and Green’s function, both based solely on directly computable Hamiltonian data, we describe how one can capture the qualitative response differences while ensuring smooth parameter dependence. To highlight the applicability of the formalism, we describe how it allows to formulate (1) precise conditions for spectral degeneracies of higher geometric multiplicity, and (2) a continuous hierarchy of spectral response strengths that characterize of all degeneracies quantitatively. Combining both aspects, it follows that scenarios with higher geometric multiplicity give rise to unique super-Lorentzian response, for which the spectral and physical response strength become two separate quantities. Joint work with Subhajyoti Bid References [1] S. Bid and H. Schomerus, Uniform response theory of non-Hermitian systems: Non-Hermitian physics beyond the exceptional point, arXiv:2412.11932

In recent years, non-Hermitian degeneracies, so-called exceptional points, have attracted immense attention, in particular in photonics and optics. Among a range of interesting fundamental aspects, exceptional points have great potential for applications, such as topological mode switching, non-reciprocal devices, and ultrasensitive sensors. Although the research on exceptional points has grown enormously in the last years, the construction of high-order exceptional points, where more than two eigenfrequencies and corresponding modes coalesce, is still challenging. In the first part of the talk we discuss the concept of spectral response strength associated with second- and higher-order exceptional points. This measure not only quantifies the spectral response to perturbations, but also the intensity response to excitations and the dynamical response to initial deviations from the exceptional point. Furthermore, we explore the connection between this concept and the well-established Petermann factor related to isolated eigenfrequencies. In the second part of the talk we present a robust implementation of high-order exceptional points in microring cavities. To do so, we combine the known mirror-induced asymmetric backscattering with the unidirectional coupling between microcavities via adjacent waveguides. We investigate the topology of eigenfrequencies around the exceptional points and demonstrate a cavity-selective sensing scheme. Our numerical results are also described by an intuitive Hamiltonian description. Our scheme allows for the robust and scalable construction of high-order exceptional points in integrated semiconductor platforms.

The parity-time (PT) symmetry of a non-Hermitian Hamiltonian leads to a real (complex) energy spectrum when the non-Hermiticity is below (above) a threshold. Recently, it has been demonstrated that the non-Hermitian skin effect generates a new type of PT symmetry, dubbed the non-Bloch PT symmetry, featuring unique properties such as high sensitivity to the boundary condition. Despite its relevance to a wide range of non-Hermitian lattice systems, a general theory is still lacking for this generic phenomenon even in one spatial dimension. Here, we uncover the geometric mechanism of non-Bloch PT symmetry and its breaking. We find that non-Bloch PT symmetry breaking occurs by the formation of cusps in the generalized Brillouin zone (GBZ). Based on this geometric understanding, we propose an exact formula that efficiently determines the breaking threshold. Moreover, we predict a new type of spectral singularities associated with the symmetry breaking, dubbed non-Bloch van Hove singularity, whose physical mechanism fundamentally differs from their Hermitian counterparts. This singularity is experimentally observable in linear responses. References: Yu-Min Hu, Hong-Yi Wang, Zhong Wang, and Fei Song. Geometric origin of non-Bloch PT symmetry breaking. Phys. Rev. Lett. 132, 050402 (2024).

Non-Hermitian systems exhibit richer topological properties compared to their Hermitian counterparts. It iswell known that non-Hermitian systems have been classified based on either the symmetry relations for non-Hermitian Hamiltonians or the symmetry relations for nonunitary time-evolution operators in the context of　Floquet topological phases. In this work, we propose that non-Hermitian systems can always be classified in two　ways; a non-Hermitian system can be classified using the symmetry relations for non-Hermitian Hamiltonians or time-evolution operator regardless of the Floquet topological phases or not. We refer to this as dual symmetry classification. To demonstrate this, we successfully introduce a nonunitary quantum walk that exhibits point gaps with a $Z_2$ point-gap topological phase applying the dual symmetry classification and treating the time-evolution operator of this quantum walk as the non-Hermitian Hamiltonian. We also porpose feasible experimenta setups to realize this quantum walk by using quantum optical devices.

The main aim of the present paper is to define an active particle in a quantum framework as a minimal model of quantum active matter and investigate the differences and similarities of quantum and classical active matter. Although the field of active matter has been expanding, most research has been conducted on classical systems. Here, we propose a truly deterministic quantum active-particle model with a nonunitary quantum walk as the minimal model of quantum active matter. We aim to reproduce results obtained previously with classical active Brownian particles; that is, a Brownian particle, with finite energy take-up, becomes active and climbs up a potential wall. We realize such a system with nonunitary quantum walks. We introduce new internal states, the ground state and the excited state, and a new nonunitary operator $N(g)$ for an asymmetric transition between the two states. The non-Hermiticity parameter $g$ promotes the transition to the excited state; hence, the particle takes up energy from the environment. For our quantum active particle, we successfully observe that the movement of the quantum walker becomes more active in a nontrivial manner as we increase the non-Hermiticity parameter $g$, which is similar to the classical active Brownian particle. We also observe three unique features of quantum walks, namely, ballistic propagation of peaks in one dimension, the walker staying on the constant energy plane in two dimensions, and oscillations originating from the resonant transition between the ground state and the excited state both in one and two dimensions.

Two-dimensional optical microcavities provides a platform to explore light wave mechanics influenced by cavity morphology as a non-Hermitian system. Like sound waves resonating in a chamber, light wave in a circular cavity forms quasi-stable states called resonances at specific frequencies. Because of the energy leakage to the outside of the cavity, these resonance frequencies become complex, with imaginary parts representing decay rates. When the cavity shape is slightly deformed from the circular cavity so that there are no geometrical symmetries, nearly degenerate pairs of resonances with high non-orthogonality called chiral pairs appear. We numerically show that a chiral pair in an asymmetric limaçon microcavity exhibits an Exceptional Point when varying two cavity parameters. Furthermore, in a two-dimensional microcavity laser, the nonlinear interactions between light and the medium cause these states to mutually lock, forming a stable lasing state. We show the dynamics interplayed by the chiral pair near an EP.

The Poisson scattering ensemble can be defined in analogy to the random Gaussian scattering ensembles, where a closed system described by a Hamiltonian from the GOE, GUE, or GSE is coupled to a finite number of decay channels. This is done via random and uncorrelated coupling amplitudes between the eigenstates of the Hamiltonian and the decay channels. The Poisson scattering ensemble is constructed in exactly the same way, however, the levels are there taken as uncorrelated random numbers. In this work, I plan to derive an analytical expression for the scattering matrix two-point correlation function, the exact analog to the celebrated Verbaarshot-Weidenmueller-Zirnbauer (VWZ) formula for the GOE case. Then I want to discuss two applications: (i) Obtain an analytical result for the fidelity decay in the Poisson case; this is useful for assessing quantum chaos properties by comparing, alongside with the well knwon GO(US)E prediction, to the Poisson case. (ii) Consider the scattering ensemble in the strong coupling limit, where early numerical evidence indicated that the Poisson scattering ensemble would converge to the canonical Gaussian scattering ensemble; this would provide an alternative route to derive or at least approximate the VWZ-formula.

Recently the interest in non-Hermitian disordered models has been revived, due to the claims of instability of a many-body localization to a coupling to a bath. To describe such open quantum systems, one often focuses on an energy leakage to a bath, using effective non-Hermitian Hamiltonians. A well-known Hatano-Nelson model [1], being a 1d Anderson localization (AL) model, with different hopping amplitudes to the right/left, shows AL breakdown, as non-Hermiticity suppresses the interference. Unlike this, we consider models with the complex gain-loss disorder and show that in general these systems tend to the localization due to non-Hermiticity. First, we consider a power-law random banded matrix ensemble (PLRBM) [2], known to show AL transition (ALT) at the power of the power-law hopping decay a=d equal to the dimension d. In [3], we show that a non-Hermitian gain-loss disorder in PLRBM shifts ALT to smaller values d/2

Exceptional points (EPs) are unique phenomena in topological non-Hermitian physics, attracting significant attention across classical and quantum wave systems. EPs exhibit two remarkable properties—degenerate eigenfrequencies and coalescing eigenmodes—that are highly sensitive to perturbations of system parameters. Although most research has focused on linear systems, nonlinear platforms like lasers promise even richer dynamics. Current studies of EP lasers in theory mainly include two approaches: (1) ab initio theories such as steady-state and periodic-inversion ab initio laser theory (SALT and PALT), and (2) spin-flip models (SFMs) for spin vertical-cavity surface-emitting lasers (spin-VCSELs). Both frameworks uncover diverse dynamical behaviors, including cusp bifurcation, frequency comb generation (limit cycles), period doubling leading to chaos, and even hyperchaos. Despite these advances, the specific role of EPs in EP lasers remains to be fully understood. For the ab initio approach, practical limitations of materials and parameters typically prevent operation very close to an EP. In contrast, while SFMs can indeed approach the EP and generate clear pulse trains, a detailed nonlinear dynamical analysis highlighting the fundamental differences between EP lasers and conventional two-mode lasers. In this work, we reveal rich nonlinear dynamical behaviors near an EP using a generalized SFM. Contrary to the conventional progression (single-mode lasing → limit cycles → chaos), we discover an explosive burst of chaotic states compressed between the single-mode lasing state and limit-cycled frequency comb—uniquely captured within pulse train dynamics. This phenomenon enriches the theoretical understanding of the SFM by characterizing its intricate nonlinear dynamics. Practically, since this chaos occurs near the EP, it can be sustained with relatively low power, making it promising for applications such as analog computing and random-number generation.