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Exceptional points (EPs) are singularities in the spectra of non-Hermitian operators, where eigenvalues and eigenvectors coalesce. Recently, open quantum systems have been increasingly explored as EP testbeds due to their natural non-Hermitian nature. However, existing works mostly focus on the Markovian limit, leaving a gap in understanding EPs in the non-Markovian regime. In this work, we address this gap by proposing a theoretical framework based on two numerically exact descriptions of non-Markovian dynamics: the pseudomode mapping and the hierarchical equations of motion. The proposed framework enables conventional spectral analysis for EP identification, establishing direct links between EPs and dynamic manifestations in open systems, such as non-exponential decays and enhanced sensitivity to external perturbations. We unveil pure non-Markovian EPs that are unobservable in the Markovian limit. Remarkably, the EP aligns with the Markovian-to-non-Markovian transition, and the EP condition is adjustable by modifying environmental spectral properties. Moreover, we show that structured environments can elevate EP order, thereby enhancing the system's sensitivity. These findings lay a theoretical foundation and open new avenues for non-Markovian reservoir engineering and non-Hermitian physics., Comment: 10+5 pages, 2 figures

We propose how to achieve chiral photon blockade by spinning a nonlinear optical resonator. We show that by driving such a device at a fixed direction, completely different quantum effects can emerge for the counter-propagating optical modes, due to the spinning-induced breaking of time-reversal symmetry, which otherwise is unattainable for the same device in the static regime. Also, we find that in comparison with the static case, robust non-classical correlations against random backscattering losses can be achieved for such a quantum chiral system. Our work, extending previous works on the spontaneous breaking of optical chiral symmetry from the classical to purely quantum regimes, can stimulate more efforts towards making and utilizing various chiral quantum effects, including applications for chiral quantum networks or noise-tolerant quantum sensors.

The fascinating realm of non-Hermitian physics with the interplay of parity (P) and time-reversal (T) symmetry has been witnessing immense attention in exploring unconventional physics at Exceptional Point (EP) singularities. Particularly, the physics of PT-symmetry, anti-PT (APT)-symmetry, and the emergence of EPs have ignited fervor in photonics. Beyond the conventional relation between EP and PT-symmetric phase transitions, this study delves into hosting higher-order EPs in a specially designed APT-symmetric Fabry-P\'erot-type microcavity. We unveil the captivating physics of the parametric encirclement schemes to explore the branch-point behaviors of EPs up to order three in terms of successive state-flipping, while optimizing the designed cavity under APT-symmetric constraints. The insights from our findings are poised to boost research in optical metamaterials, meeting the demands of APT-symmetry and paving the way for a novel class of photonic devices., Comment: 6 pages, 3 figures

We analyze the existence and degeneracies of quantum exceptional, diabolical, and hybrid points of simple bosonic systems, composed of up to six modes with damping and/or amplification and exhibiting nonconventional dynamics. They involve the configurations in which the dynamics typical for PT-symmetric systems is observed only in a subspace of the whole Liouville space of the system states (nonconventional PT-symmetric dynamics) as well as those containing unidirectional coupling. The system dynamics described by quadratic non-Hermitian Hamiltonians is governed by the Heisenberg-Langevin equations. Conditions for the observation of inherited quantum hybrid points with up to sixth-order exceptional and second-order diabolical degeneracies are revealed, though relevant only for short-time dynamics. This raises the question of whether higher-order inherited singularities exist in bosonic systems that exhibit physically meaningful behavior at arbitrary times. On the other hand, for short times, unidirectional coupling of various types enables the concatenation of simple bosonic systems with second- and third-order exceptional degeneracies on demand. This approach allows for the creation of arbitrarily high exceptional degeneracies observed in systems with diverse structures. Methods for numerical identifying the quantum exceptional and hybrid points, and determining their degeneracies are discussed. Rich dynamics of higher-order field-operator moments is analyzed from the point of view of the presence of exceptional and diabolical points with their degeneracies in general., Comment: Partial PT-symmetry like dynamics (nonconventional PT-symmetry) as well as non-Hermitian bosonic systems with unidirectional coupling are investigated. The work is in continuation of the paper entitled "Multiple quantum exceptional, diabolical, and hybrid points in multimode bosonic systems: I. Inherited and genuine singularities"

The existence and degeneracies of quantum exceptional, diabolical, and hybrid (i.e., diabolically degenerated exceptional) singularities of simple bosonic systems composed of up to five modes with damping and/or amplification are analyzed. Their dynamics governed by quadratic non-Hermitian Hamiltonians is followed using the Heisenberg-Langevin equations. Conditions for the observation of inherited quantum hybrid points, observed directly in the dynamics of field operators, having up to third-order exceptional and second-order diabolical degeneracies are revealed. Exceptional and diabolical genuine points and their degeneracies observed in the dynamics of second-order field-operator moments are analyzed. Surprisingly, exceptional degeneracies of only second and third orders are revealed. Nevertheless the analyzed bosonic systems exhibit rich dynamics, also owing to their common second-order diabolical degeneracies., Comment: Inherited EPs with second- and third-order singularities are observed in the bosonic systems with the usual (bidirectional) coupling. In the following paper entitled "Multiple quantum exceptional, diabolical, and hybrid points in multimode bosonic systems: II. Nonconventional PT-symmetric dynamics and unidirectional coupling", the investigation are further extended to other bosonic systems

Non-Hermitian systems have attracted much interest in recent decades, driven partly by the existence of exotic spectral singularities, known as exceptional points (EPs), where the dimensionality of the system evolution operator is reduced. Among various intriguing applications, the discovery of EPs has suggested the potential for implementing a symmetric mode switch, when encircling them in a system parameter space. However, subsequent theoretical and experimental works have revealed that {\it dynamical} encirclement of EPs invariably results in asymmetric mode conversion; namely, the mode switching depends only on the winding direction but not on the initial state. This chirality arises from the failure of adiabaticity due to the complex spectrum of non-Hermitian systems. Although the chirality revealed has undoubtedly made a significant impact in the field, a realization of the originally sought symmetric adiabatic passage in non-Hermitian systems with EPs has since been elusive. In this work, we bridge this gap and theoretically demonstrate that adiabaticity, and therefore a symmetric state transfer, is achievable when dynamically winding around an EP. This becomes feasible by specifically choosing a trajectory in the system parameter space along which the corresponding evolution operator attains a real spectrum. Our findings, thus, offer a promise for advancing various wave manipulation protocols in both quantum and classical domains., Comment: 6 pages, 4 figures

It has been a long-standing goal to improve dispersive qubit readout with squeezed light. However, injected external squeezing (IES) {\it cannot} enable a practically interesting increase in the signal-to-noise ratio (SNR), and simultaneously, the increase of the SNR due to the use of intracavity squeezing (ICS) is even {\it negligible}. Here, we {\it counterintuitively} demonstrate that using IES and ICS together can lead to an {\it exponential} improvement of the SNR for any measurement time, corresponding to a measurement error reduced typically by many orders of magnitude. More remarkably, we find that in a short-time measurement, the SNR is even improved exponentially with {\it twice} the squeezing parameter. As a result, we predict a fast and high-fidelity readout. This work offers a promising path toward exploring squeezed light for dispersive qubit readout, with immediate applications in quantum error correction and fault-tolerant quantum computation., Comment: 6 pages + 3 figures, 12 pages + 5 figures

Entanglement potentials (EPs) enable the characterization and quantification of the nonclassicality of single-mode optical fields by measuring the entanglement generated through beam splitting. We experimentally generated single-photon states and tomographically reconstructed the corresponding two-qubit states to determine EPs defined via popular two-qubit measures of entanglement. These include the potentials for the relative entropy of entanglement (REEP), concurrence, and negativity. Among our experimental states, we found those that are very close (at least for some ranges of parameters) to the theoretical upper and lower bounds on relative EPs (or relative nonclassicality), i.e., when one EP is maximized or minimized for a given value if another EP. We experimentally confirmed the counterintuitive theoretical result of Ref. [Phys. Rev. A 92, 062314 (2015)] that the relative nonclassicality (specifically, the negativity potential for given values of the REEP) of single-photon states can be increased by dissipation., Comment: 11 pages, 4 figures

Hamiltonian exceptional points (HEPs) are spectral degeneracies of non-Hermitian Hamiltonians describing classical and semiclassical open systems with gain and/or loss. However, this definition overlooks the occurrence of quantum jumps in the evolution of open quantum systems. These quantum effects are properly accounted for by considering Liouvillians and their exceptional points (LEPs) [Minganti et al., Phys. Rev. A {\bf 100}, 062131 (2019)]. Here, we explicitly describe how standard quantum process tomography, which reveals the dynamics of a quantum system, can be readily applied to reveal and characterize LEPs of non-Hermitian systems. We conducted experiments on an IBM quantum processor to implement a prototype model simulating the decay of a single qubit through three competing channels. Subsequently, we performed tomographic reconstruction of the corresponding experimental Liouvillians and their LEPs using both single- and two-qubit operations. This example underscores the efficacy of process tomography in tuning and observing LEPs, despite the absence of HEPs in the model., Comment: 9+17 pages, 2+4 figures

The interaction of light and matter at the single-photon level is of central importance in various fields of physics, including, e.g., condensed matter physics, astronomy, quantum optics, and quantum information. Amplification of such quantum light-matter interaction can be highly beneficial to, e.g., improve device performance, explore novel phenomena, and understand fundamental physics, and has therefore been a long-standing goal. Furthermore, simulation of light-matter interaction in the regime of ultrastrong coupling, where the interaction strength is comparable to the bare frequencies of the uncoupled systems, has also become a hot research topic, and considerable progress has been made both theoretically and experimentally in the past decade. In this review, we provide a detailed introduction of recent advances in amplification of quantum light-matter interaction and simulation of ultrastrong light-matter interaction, particularly in cavity and circuit quantum electrodynamics and in cavity optomechanics., Comment: 91 pages+18 figures+1 table

Multi-photon dynamics beyond linear optical materials are of significant fundamental and technological importance in quantum information processing. However, it remains largely unexplored in nonlinear waveguide QED. In this work, we theoretically propose a structured nonlinear waveguide in the presence of staggered photon-photon interactions, which supports two branches of gaped bands for doublons (i.e., spatially bound-photon-pair states). In contrast to linear waveguide QED systems, we identify two important contributions to its dynamical evolution, i.e., single-photon bound states (SPBSs) and doublon bound states (DBSs). Most remarkably, the nonlinear waveguide can mediate the long-range four-body interactions between two emitter pairs, even in the presence of disturbance from SPBS. By appropriately designing system's parameters, we can achieve high-fidelity four-body Rabi oscillations mediated only by virtual doublons in DBSs. Our findings pave the way for applying structured nonlinear waveguide QED in multi-body quantum information processing and quantum simulations among remote sites., Comment: 18 pages; 9 figures

One of the remarkable aspects of quantum steering is its ability to violate local uncertainty complementarity relations. In this vein of study, various steering witnesses employing different uncertainty relations have been developed including Reid's criteria. Here, we introduce a novel complementarity relation between system's quantum and classical uncertainties corresponding to the distillable coherence and the von-Neumann entropy, respectively. We demonstrate a superior steering detection efficiency compared to an entropic uncertainty relation. Notably, our proposed steering witness can detect ``all pure entangled states," while the entropic uncertainty relation cannot. We also experimentally validate such a property through a photonic system. Furthermore, a deeper connection to the uncertainty principle is revealed by showcasing the functionality of our proposed complementarity as a quantifier of measurement incompatibility and quantum steerability under genuine incoherent operations. Our work establishes a clear quantitative and operational link between coherence and steering, which are significant resources of quantum technologies, and underscores our efforts in bridging the uncertainty principle with quantum coherence., Comment: 18 pages, 5 figures

Entanglement potentials are popular measures of the nonclassicality of single-mode optical fields. These potentials are defined by the amount of entanglement (measured by, e.g., the negativity or concurrence) of the two-mode field generated by mixing a given single-mode field with the vacuum on a balanced beam splitter. We generalize this concept to define the potentials for Bell nonlocality and quantum steering in specific measurement scenarios, in order to quantify single-mode nonclassicality in a more refined way. Thus, we can study the hierarchy of three types of potentials in close analogy to the well-known hierarchy of the corresponding two-mode quantum correlations. For clarity of our presentation, we focus on the analysis of the nonclassicality potentials for arbitrary vacuum-one-photon superpositions (VOPSs), corresponding to a photon-number qubit. We discuss experimentally feasible implementations for the generation of single-mode VOPS states, their mixing with the vacuum on a balanced beam splitter, and their two-mode Wigner-function reconstruction using homodyne tomography to determine the potentials. We analyze the effects of imperfections, including phase damping and unbalanced beam splitting on the quality of the reconstructed two-mode states and nonclassicality potentials. Although we focus on the analysis of VOPS states, single-mode potentials can also be applied to study the nonclassicality of qudits or continuous-variable systems., Comment: 18 pages, 8 figures

Entanglement potentials are a promising way to quantify the nonclassicality of single-mode states. They are defined by the amount of entanglement (expressed by, e.g., the Wootters concurrence) obtained after mixing the examined single-mode state with a purely classical state; such as the vacuum or a coherent state. We generalize the idea of entanglement potentials to other quantum correlations: the EPR steering and Bell nonlocality, thus enabling us to study mutual hierarchies of these nonclassicality potentials. Instead of the usual vacuum and one-photon superposition states, we experimentally test this concept using specially tailored polarization-encoded single-photon states. One polarization encodes a given nonclassical single-mode state, while the other serves as the vacuum place-holder. This technique proves to be experimentally more convenient in comparison to the vacuum and a one-photon superposition as it does not require the vacuum detection., Comment: 14 pages, 4 figures

The occurrence of exceptional points (EPs) is a fascinating non-Hermitian feature of open systems. A level-repulsion phenomenon between two complex states of an open system can be realized by positioning an EP and its time-reversal (T) conjugate pair in the underlying parameter space. Here, we report the fascinating nonreciprocal response of such two conjugate EPs by using a dual-mode planar waveguide system having two T-symmetric active variants concerning the transverse gain-loss profiles. We specifically reveal a comprehensive all-optical scheme to achieve correlative nonreciprocal light dynamics by using the reverse chirality of two dynamically encircled conjugate EPs in the presence of local nonlinearity. A specific nonreciprocal correlation between two designed T-symmetric waveguide variants is established in terms of their unidirectional transfer of light with a precise selection of modes. Here, the unconventional reverse chiral properties of two conjugate EPs allow the nonreciprocal transmission of two selective modes in the opposite directions of the underlying waveguide variants. An explicit dependence of the nonlinearity level on a significant enhancement of the nonreciprocity in terms of an isolation ratio is explored by investigating the effects of both local Kerr-type and saturable nonlinearities (considered separately). The physical insights and implications of harnessing the properties of conjugate EPs in nonlinear optical systems can enable the growth and development of a versatile platform for building nonreciprocal components and devices., Comment: 10 pages, 7 figures

We demonstrate a novel experimental technique for quantum-state tomography of the collective density matrix. It is based on measurements of the polarization of light, traversing the atomic vapor. To assess the technique's robustness against errors, experimental investigations are supported with numerical simulations. This not only allows to determine the fidelity of the reconstruction, but also to analyze the quality of the reconstruction for specific experimental parameters light tuning and number of measurements). By utilizing the so-called conditional number, we demonstrate that the reconstruction can be optimized for a specific tuning of the system parameters, and further improvement is possible by selective repetition of the measurements. Our results underscore the potential high-fidelity quantum-state reconstruction while optimizing measurement resources.

We propose a heralded nonlocal protocol for implementing an entangling gate on two stationary qubits coupled to spatially separated cavities. By dynamically controlling the evolution of the composite system, the entangling gate can be achieved without real excitations of cavity modes nor atoms. The success of our protocol is conditioned on projecting an auxiliary atom onto a postselected state, which simultaneously removes various detrimental effects of dissipation on the gate fidelity. In principle, the success probability of the gate can approach unity as the single-atom cooperativity becomes sufficiently large. Furthermore, we show its application for implementing single- and two-qubit gates within a decoherence-free subspace that is immune to a collective dephasing noise. This heralded, faithful, and nonlocal entangling gate protocol can, therefore, be useful for distributed quantum computation and scalable quantum networks., Comment: 13 pages, 4figures

Quantum correlations, both spatial and temporal, are the central pillars of quantum mechanics. Over the last two decades, a big breakthrough in quantum physics is its complex extension to the non-Hermitian realm, and dizzying varieties of novel phenomena and applications beyond the Hermitian framework have been uncovered. However, unique features of non-Hermitian quantum correlations, especially in the time domain, still remain to be explored. Here, for the first time, we experimentally achieve this goal by using a parity-time (PT )-symmetric trapped-ion system. The upper limit of temporal quantum correlations, known as the algebraic bound, which has so far not been achieved in the standard measurement scenario, is reached here by approaching the exceptional point (EP), thus showing the unexpected ability of EPs in tuning temporal quantum correlation effects. Our study, unveiling the fundamental interplay of non-Hermiticity, nonlinearity, and temporal quantum correlations, provides the first step towards exploring and utilizing various non-Hermitian temporal quantum effects by operating a wide range of EP devices, which are important for both fundamental studies and applications of quantum EP systems., Comment: 4 figures and 8 pages

A Werner state, which is the singlet Bell state affected by white noise, is a prototype example of states, which can reveal a hierarchy of quantum entanglement, steering, and Bell nonlocality by controlling the amount of noise. However, experimental demonstrations of this hierarchy in a sufficient and necessary way (i.e., by applying measures or universal witnesses of these quantum correlations) have been mainly based on full quantum state tomography, corresponding to measuring at least 15 real parameters of two-qubit states. Here we report an experimental demonstration of this hierarchy by measuring only six elements of a correlation matrix depending on linear combinations of two-qubit Stokes parameters. We show that our experimental setup can also reveal the hierarchy of these quantum correlations of generalized Werner states, which are any two-qubit pure states affected by white noise., Comment: 12 pages, 4 figures

We propose and analyze a novel approach to implement ensemble qubits. The required anharmonicity is provided by a simultaneous decay of two atoms (i.e., two-atom decay), which is achieved by fully quantum degenerate parametric amplification. For an atomic ensemble, the two-atom decay generates and stabilizes a 2D quantum manifold, which is spanned by the ground and single-excited superradiant states. Moreover, this nonlinear decay process can strongly suppress transitions to higher-excited superradiant states, and convert residual transitions into an effective decay from the single-excitation superradiant state to the ground state. Our method does not require Rydberg dipole blockade and, thus, strong atom-atom interactions, compared to previous work. This indicates that it can be applied to typical atomic or spin ensembles in simple experimental setups. Remarkably, our idea is compatible with the cavity protection mechanism, and therefore spin dephasing due to inhomogeneous broadening can be strongly suppressed. The presented ensemble qubit provides a new platform for quantum information processing, and also extends the range of applications of atomic or spin ensembles., Comment: 18 pages, 6 figures

We show that the loss of nonclassicality (including quantum entanglement) cannot be compensated by the (incoherent) amplification of PT-symmetric systems. We address this problem by manipulating the quantum fluctuating forces in the Heisenberg-Langevin approach. Specifically, we analyze the dynamics of two nonlinearly coupled oscillator modes in a PT-symmetric system. An analytical solution allows us to separate the contribution of reservoir fluctuations from the evolution of quantum statistical properties of the modes. In general, as reservoir fluctuations act constantly, the complete loss of nonclassicality and entanglement is observed for long times. To elucidate the role of reservoir fluctuations in a long-time evolution of nonclassicality and entanglement, we consider and compare the predictions from two alternative models in which no fatal long-time detrimental effects on the nonclassicality and entanglement are observed. This is so as, in the first semiclassical model, no reservoir fluctuations are considered at all. This, however, violates the fluctuation-dissipation theorem. The second, more elaborated, model obeys the fluctuation-dissipation relations as it partly involves reservoir fluctuations. However, to prevent from the above long-time detrimental effects, the reservoir fluctuations have to be endowed with the nonphysical properties of a sink model. In both models, additional incorporation of the omitted reservoir fluctuations results in their physically consistent behavior. This behavior, however, predicts the gradual loss of the nonclassicality and entanglement. Thus the effects of reservoir fluctuations related to damping cannot be compensated by those related to amplification., Comment: In version 2, title of the paper was modified. Also some other minor changes were made. 13 pages, 3 figures

Nontrivial spectral properties of non-Hermitian systems can lead to intriguing effects with no counterparts in Hermitian systems. For instance, in a two-mode photonic system, by dynamically winding around an exceptional point (EP) a controlled asymmetric-symmetric mode switching can be realized. That is, the system can either end up in one of its eigenstates, regardless of the initial eigenmode, or it can switch between the two states on demand, by simply controlling the winding direction. However, for multimode systems with higher-order EPs or multiple low-order EPs, the situation can be more involved, and the ability to control asymmetric-symmetric mode switching can be impeded, due to the breakdown of adiabaticity. Here we demonstrate that this difficulty can be overcome by winding around exceptional curves by additionally crossing diabolic points. We consider a four-mode $\cal PT$-symmetric bosonic system as a platform for experimental realization of such a multimode switch. Our work provides alternative routes for light manipulations in non-Hermitian photonic setups., Comment: 10+5 pages, 5+7 figures

The simultaneous ground-state refrigeration of multiple vibrational modes is a prerequisite for observing significant quantum effects of multiple-vibration systems. Here we propose how to realize a large amplification in the net-refrigeration rates based on cavity optomechanics and to largely improve the cooling performance of multivibration modes beyond the resolved-sideband regime. By employing an auxiliary mechanical coupling (AMC) between two mechanical vibrations, the dark mode, which is induced by the coupling of these vibrational modes to a common optical mode and cuts off cooling channels, can be fully removed. We use fully analytical treatments for the effective mechanical susceptibilities and net-cooling rates and find that when the AMC is turned on, the amplification of the net-refrigeration rates by more than six orders of magnitude can be observed. In particular, we reveal that the simultaneous ground-state cooling beyond the resolved-sideband regime arises from the introduced AMC, without which it vanishes. Our work paves the way for quantum control of multiple vibrational modes in the bad-cavity regime., Comment: 12 pages, 5 figures

The experimental observation of quantum phase transitions predicted by the quantum Rabi model in quantum critical systems is usually challenging due to the lack of signature experimental observables associated with them. Here, we describe a method to identify the dynamical critical phenomenon in the quantum Rabi model consisting of a three-level atom and a cavity at the quantum phase transition. Such a critical phenomenon manifests itself as a sudden change of steady-state output photons in the system driven by two classical fields, when both the atom and the cavity are initially unexcited. The process occurs as the high-frequency pump field is converted into the low-frequency Stokes field and multiple cavity photons in the normal phase, while this conversion cannot occur in the superradiant phase. The sudden change of steady-state output photons is an experimentally accessible measure to probe quantum phase transitions, as it does not require preparing the equilibrium state., Comment: has been published in Communcations Physics as a regular article

We describe a novel type of blockade in a hybrid mode generated by linear coupling of photonic and phononic modes. We refer to this effect as hybrid photon-phonon blockade and show how it can be generated and detected in a driven nonlinear optomechanical superconducting system. Thus, we study boson-number correlations in the photon, phonon, and hybrid modes in linearly coupled microwave and mechanical resonators with a superconducting qubit inserted in one of them. We find such system parameters for which we observe eight types of different combinations of either blockade or tunnelling effects (defined via the sub- and super-Poissonian statistics, respectively) for photons, phonons, and hybrid bosons. In particular, we find that the hybrid photon-phonon blockade can be generated by mixing the photonic and phononic modes which do not exhibit blockade., Comment: 26 pages, 14 figures

Superradiant phase transitions (SPTs) are important for understanding light-matter interactions at the quantum level, and play a central role in criticality-enhanced quantum sensing. So far, SPTs have been observed in driven-dissipative systems, but the emergent light fields did not show any nonclassical characteristic due to the presence of strong dissipation. Here we report an experimental demonstration of the SPT featuring the emergence of a highly nonclassical photonic field, realized with a resonator coupled to a superconducting qubit, implementing the quantum Rabi model. We fully characterize the light-matter state by Wigner matrix tomography. The measured matrix elements exhibit quantum interference intrinsic of a photonic mesoscopic superposition, and reveal light-matter entanglement, Comment: 20 pages, 19 figures, 2 tables

Finite simplex lattice models are used in different branches of science, e.g., in condensed matter physics, when studying frustrated magnetic systems and non-Hermitian localization phenomena; or in chemistry, when describing experiments with mixtures. An $n$-simplex represents the simplest possible polytope in $n$ dimensions, e.g., a line segment, a triangle, and a tetrahedron in one, two, and three dimensions, respectively. In this work, we show that various fully solvable, in general non-Hermitian, $n$-simplex lattice models {with open boundaries} can be constructed from the high-order field-moments space of quadratic bosonic systems. Namely, we demonstrate that such $n$-simplex lattices can be formed by a dimensional reduction of highly-degenerate iterated polytope chains in $(k>n)$-dimensions, which naturally emerge in the field-moments space. Our findings indicate that the field-moments space of bosonic systems provides a versatile platform for simulating real-space $n$-simplex lattices exhibiting non-Hermitian phenomena, and yield valuable insights into the structure of many-body systems exhibiting similar complexity. Amongst a variety of practical applications, these simplex structures can offer a physical setting for implementing the discrete fractional Fourier transform, an indispensable tool for both quantum and classical signal processing., Comment: 16 pages

Equivalent approaches to determine eigenfrequencies of the Liouvillians of open quantum systems are discussed using the solution of the Heisenberg-Langevin equations and the corresponding equations for operator moments. A simple damped two-level atom is analyzed to demonstrate the equivalence of both approaches. The suggested method is used to reveal the structure as well as eigenfrequencies of the dynamics matrices of the corresponding equations of motion and their degeneracies for interacting bosonic modes described by general quadratic Hamiltonians. Quantum Liouvillian exceptional and diabolical points and their degeneracies are explicitly discussed for the case of two modes. Quantum hybrid diabolical exceptional points (inherited, genuine, and induced) and hidden exceptional points, which are not recognized directly in amplitude spectra, are observed. The presented approach via the Heisenberg-Langevin equations paves the general way to a detailed analysis of quantum exceptional and diabolical points in infinitely dimensional open quantum systems., Comment: 25 pages, 1 figure, 3 tables

Quantum steering is an important correlation in quantum information theory. A recent work [Nat. Commun. 12, 2410 (2021)] showed that quantum steering is also useful for quantum metrology. Here, we extend the exploration of steering-enhanced quantum metrology from single noiseless phase shifts to superpositions of noisy phase shifts. As concrete examples, we consider a control system that manipulates a target system to pass through a superposition of either dephased or depolarized phase shifts channels. We show that using such superpositions of noisy phase shifts can suppress the effects of noise and improve metrology. Furthermore, we also implemented proof-of-principle experiments for a superposition of dephased phase shifts on the IBM Quantum Experience, demonstrating a clear improvement on metrology., Comment: 12 pages, 7 figures

Pure dephasing originates from the non-dissipative information exchange between quantum systems and environments, and plays a key-role in both spectroscopy and quantum information technology. Often pure dephasing constitutes the main mechanism of decay of quantum correlations. Here we investigate how pure dephasing of one of the components of a hybrid quantum system affects the dephasing rate of the system transitions. We find that, in turn, the interaction, in the case of a light-matter system, can significantly affect the form of the stochastic perturbation describing the dephasing of a subsystem, depending on the adopted gauge. Neglecting this issue can lead to wrong and unphysical results when the interaction becomes comparable to the bare resonance frequencies of subsystems, which correspond to the ultrastrong and deep-strong coupling regimes. We present results for two prototypical models of cavity quantun electrodynamics: the quantum Rabi and the Hopfield model.

Studies have shown that the Hilbert spaces of non-Hermitian systems require nontrivial metrics. Here, we demonstrate how evolution dimensions, in addition to time, can emerge naturally from a geometric formalism. Specifically, in this formalism, Hamiltonians can be interpreted as a Christoffel symbol-like operators, and the Schroedinger equation as a parallel transport in this formalism. We then derive the evolution equations for the states and metrics along the emergent dimensions and find that the curvature of the Hilbert space bundle for any given closed system is locally flat. Finally, we show that the fidelity susceptibilities and the Berry curvatures of states are related to these emergent parallel transports., Comment: 20 pages, 1 figure

While the squeezing of a propagating field can, in principle, be made arbitrarily strong, the cavity-field squeezing is subject to the well-known 3 dB limit, and thus has limited applications. Here, we propose the use of a fully quantum degenerate parametric amplifier (DPA) to beat this squeezing limit. Specifically, we show that by {\it simply} applying a two-tone driving to the signal mode, the pump mode can, {\it counterintuitively}, be driven by the photon loss of the signal mode into a squeezed steady state with, in principle, an {\it arbitrarily high} degree of squeezing. Furthermore, we demonstrate that this intracavity squeezing can increase the signal-to-noise ratio of longitudinal qubit readout {\it exponentially} with the degree of squeezing. Correspondingly, an improvement of the measurement error by {\it many orders of magnitude} can be achieved even for modest parameters. In stark contrast, using intracavity squeezing of the semiclassical DPA {\it cannot} practically increase the signal-to-noise ratio and thus improve the measurement error. Our results extend the range of applications of DPAs and open up new opportunities for modern quantum technologies., Comment: 8+16 pages, 2+7 figures

We propose a general approach to implement nonadiabatic geometric single- and two-qubit gates beyond the rotating wave approximation (RWA). This protocol is compatible with most optimal control methods used in previous RWA protocols; thus, it is as robust as (or even more robust than) the RWA protocols. Using counter-rotating effects allows us to apply strong drives. Therefore, we can improve the gate speed by 5--10 times compared to the RWA counterpart for implementing high-fidelity ($\geq 99.99\%$) gates. Such an ultrafast evolution (nanoseconds, even picoseconds) significantly reduces the influence of decoherence (e.g., the qubit dissipation and dephasing). Moreover, because the counter-rotating effects no longer induce gate infidelities (in both the weak and strong driving regimes), we can achieve a higher fidelity compared to the RWA protocols. Therefore, in the presence of decoherence, one can implement ultrafast geometric quantum gates with $\geq 99\%$ fidelities., Comment: 13 pages, 8 figures, 3 tables

Entanglement of light and multiple vibrations is a key resource for multi-channel quantum information processing and memory. However, entanglement generation is generally suppressed, or even fully destroyed, by the dark-mode (DM) effect induced by the coupling of multiple degenerate or near-degenerate vibrational modes to a common optical mode. Here we propose how to generate optomechanical entanglement via \emph{DM breaking} induced by synthetic magnetism. We find that at nonzero temperature, light and vibrations are \emph{separable} in the DM-unbreaking regime but \emph{entangled} in the DM-breaking regime. Remarkably, the threshold thermal phonon number for preserving entanglement in our simulations has been observed to be up to \emph{three} orders of magnitude stronger than that in the DM-unbreaking regime. The application of the DM-breaking mechanism to optomechanical networks can make noise-tolerant entanglement networks feasible. These results are quite general and can initiate advances in quantum resources with immunity against both dark modes and thermal noise., Comment: 7 pages, 4 figures

The paradigm of second-order phase transitions (PTs) induced by spontaneous symmetry breaking (SSB) in thermal and quantum systems is a pillar of modern physics that has been fruitfully applied to out-of-equilibrium open quantum systems. Dissipative phase transitions (DPTs) of second order are often connected with SSB, in close analogy with well-known thermal second-order PTs in closed quantum and classical systems. That is, a second-order DPT should disappear by preventing the occurrence of SSB. Here, we prove this statement to be wrong, showing that, surprisingly, SSB is not a necessary condition for the occurrence of second-order DPTs in out-of-equilibrium open quantum systems. We analytically prove this result using the Liouvillian theory of dissipative phase transitions, and demonstrate this anomalous transition in a paradigmatic laser model, where we can arbitrarily remove SSB while retaining criticality, and on a $Z_2$-symmetric model of a two-photon Kerr resonator. This new type of phase transition cannot be interpreted as a "semiclassical" bifurcation, because, after the DPT, the system steady state remains unique., Comment: 9 pages, 2 figures, corrected minor typos

We propose how to achieve significantly enhanced quantum refrigeration and entanglement by coupling a pumped auxiliary cavity to an optomechanical cavity. We obtain both analytical and numerical results, and find optimal-refrigeration and -entanglement conditions under the auxiliary-cavity-assisted (ACA) mechanism. Our method leads to a giant amplification in the net refrigeration rate, and reveals that the ACA entanglement has a much stronger noise-tolerant ability in comparison with the unassisted case. By appropriately designing the ACA mechanism, an effective mechanical susceptibility can be well adjusted, and a genuine tripartite entanglement of cooling-cavity photons, auxiliary-cavity photons, and phonons could be generated. Specifically, we show that both optomechanical refrigeration and entanglement can be greatly enhanced for the blue-detuned driving of the auxiliary cavity but suppressed for the red-detuned case. Our work paves a way towards further quantum control of macroscopic mechanical systems and the enhancement and protection of fragile quantum resources., Comment: Physical Review A, accepted, 16 pages, 10 figures

We propose a protocol to realize nonadiabatic geometric quantum computation of small-amplitude Schr\"odinger cat qubits via invariant-based reverse engineering. We consider a system with a two-photon driven Kerr nonlinearity, which provides a pair of dressed even and odd coherent states, i.e., Schr\"odinger cat states for fault-tolerant quantum computations. An additional coherent field is applied to linearly drive a cavity mode, to induce oscillations between dressed cat states. By designing this linear drive with invariant-based reverse engineering, nonadiabatic geometric quantum computation with cat qubits can be implemented. The performance of the protocol is estimated by taking into account the influence of systematic errors, additive white Gaussian noise, and decoherence including photon loss and dephasing. Numerical results demonstrate that our protocol is robust against these negative factors. Therefore, this protocol may provide a feasible method for nonadiabatic geometric quantum computation in bosonic systems., Comment: 13 pages, 8 figures, has been published as a Regular Article in Physical Review Research

We propose a theoretical protocol to implement multiqubit geometric gates (i.e., the M{\o}lmer-S{\o}rensen gate) using photonic cat-state qubits. These cat-state qubits stored in high-$Q$ resonators are promising for hardware-efficient universal quantum computing. Specifically, in the limit of strong two-photon drivings, phase-flip errors of the cat-state qubits are effectively suppressed, leaving only a bit-flip error to be corrected. Because this dominant error commutes with the evolution operator, our protocol preserves the error bias, and, thus, can lower the code-capacity threshold for error correction. A geometric evolution guarantees the robustness of the protocol against stochastic noise along the evolution path. Moreover, by changing detunings of the cavity-cavity couplings at a proper time, the protocol can be robust against parameter imperfections (e.g., the total evolution time) without introducing extra noises into the system. As a result, the gate can produce multi-mode entangled cat states in a short time with high fidelities., Comment: 16 pages, 8 figures

We theoretically study the dynamics of typical optomechanical systems, consisting of a passive optical mode and an active mechanical mode, in the $\mathcal{PT}$- and broken-$\mathcal{PT}$-symmetric regimes. By fully analytical treatments for the dynamics of the average displacement and particle numbers, we reveal the phase diagram under different conditions and the various regimes of both $\mathcal{PT}$-symmetry and stability of the system. We find that by appropriately tuning either mechanical gain or optomechanical coupling, both phase transitions of the $\mathcal{PT}$-symmetry and stability of the system can be flexibly controlled. As a result, the dynamical behaviors of the average displacement, photons, and phonons are radically changed in different regimes. Our study shows that $\mathcal{PT}$-symmetric optomechanical devices can serve as a powerful tool for the manipulation of mechanical motion, photons, and phonons., Comment: 11 pages, 7 figures

The formalism for non-Hermitian quantum systems sometimes blurs the underlying physics. We present a systematic study of the vielbein-like formalism which transforms the Hilbert space bundles of non-Hermitian systems into the conventional ones, rendering the induced Hamiltonian to be Hermitian. In other words, any non-Hermitian Hamiltonian can be "transformed" into a Hermitian one without altering the physics. Thus we show how to find a reference frame (corresponding to Einstein's quantum elevator) in which a non-Hermitian system, equipped with a non-trivial Hilbert space metric, reduces to a Hermitian system within the standard formalism of quantum mechanics., Comment: 10 pages, 2 figures

The polarizations of optical fields, besides field intensities, provide more degrees of freedom to manipulate coherent light-matter interactions. Here we propose how to achieve a coherent switch of optomechanical entanglement in a polarized-light-driven cavity system. We show that by tuning the polarizations of the driving field, the effective optomechanical coupling can be well controlled and, as a result, quantum entanglement between the mechanical oscillator and the optical transverse electric (TE) mode can be coherently and reversibly switched to that between the same phonon mode and the optical transverse magnetic (TM) mode. This ability of switching optomechanical entanglement with such a vectorial device can be important for building a quantum network being capable of efficient quantum information interchanges between processing nodes and flying photons., Comment: To be published in Nanophotonics

Quantum channels breaking entanglement, incompatibility, or nonlocality are defined as such because they are not useful for entanglement-based, one-sided device-independent, or device-independent quantum information processing, respectively. Here, we show that such breaking channels are related to complementary tests of macrorealism i.e., temporal separability, channel unsteerability, temporal unsteerability, and the temporal Bell inequality. To demonstrate this we first define a steerability-breaking channel, which is conceptually similar to entanglement and nonlocality-breaking channels and prove that it is identical to an incompatibility-breaking channel. A hierarchy of quantum non-breaking channels is derived, akin to the existing hierarchy relations for temporal and spatial quantum correlations. We then introduce the concept of channels that break temporal correlations, explain how they are related to the standard breaking channels, and prove the following results: (1) A robustness-based measure for non-entanglement-breaking channels can be probed by temporal nonseparability. (2) A non-steerability-breaking channel can be quantified by channel steering. (3) Temporal steerability and non-macrorealism can be used for, respectively, distinguishing unital steerability-breaking channels and nonlocality-breaking channels for a maximally entangled state. Finally, a two-dimensional depolarizing channel is experimentally implemented as a proof-of-principle example to demonstrate the hierarchy relation of non-breaking channels using temporal quantum correlations, Comment: Comments welcome

Phase transitions of thermal systems and the laser threshold were first connected more than forty years ago. Despite the nonequilibrium nature of the laser, the Landau theory of thermal phase transitions, applied directly to the Scully-Lamb laser model (SLLM), indicates that the laser threshold is a second-order phase transition, associated with a $U(1)$ spontaneous symmetry breaking (SSB). To capture the genuine nonequilibrium phase transition of the SLLM (i.e., a single-mode laser without a saturable absorber), here we employ a quantum theory of dissipative phase transitions. Our results confirm that the $U(1)$ SSB can occur at the lasing threshold but, in contrast to the Landau theory and semiclassical approximation, they signal that the SLLM "fundamental" transition is a different phenomenon, which we call Liouvillian spectral collapse; that is, the emergence of diabolic points of infinite degeneracy. By considering a generalized SLLM with additional dephasing, we witness a second-order phase transition, with a Liouvillian spectral collapse, but in the absence of symmetry breaking. Most surprisingly, the phase transition corresponds to the emergence of dynamical multistability even without SSB. Normally, bistability is suppressed by quantum fluctuations, while in this case, the very presence of quantum fluctuations enables bistability. This rather anomalous bistability, characterizing the truly dissipative and quantum origin of lasing, can be an experimental signature of our predictions, and we show that it is associated with an emergent dynamical hysteresis., Comment: 20 pages, 11 figures, minor typos corrected

We demonstrate a hierarchy of various classes of quantum correlations on experimentally prepared two-qubit Werner-like states with controllable white noise. Werner states, which are white-noise-affected Bell states, are prototypal examples for studying such a hierarchy as a function of the amount of white noise. We experimentally generated Werner states and their generalizations, i.e., partially entangled pure states affected by white noise. These states enabled us to study the hierarchy of the following classes of correlations: separability, entanglement, steering in three- and two-measurement scenarios, and Bell nonlocality. We show that the generalized Werner states (GWSs) reveal fundamentally different aspects of the hierarchy compared to the Werner states. In particular, we find five different parameter regimes of the GWSs, including those steerable in a two-measurement scenario but not violating Bell inequalities. This regime cannot be observed for the usual Werner states. Furthermore, we find threshold curves separating different regimes of the quantum correlations and find the optimal states which allow for the largest amount of white noise, which does not destroy their specific quantum correlations (e.g., unsteerable entanglement). Thus, we could identify the optimal Bell-nondiagonal GWSs, which are, for this specific meaning, more robust against white noise compared to the Bell-diagonal GWSs (i.e., Werner states)., Comment: 23 pages, 9 figures

Recently, there has been intense research in proposing and developing various methods for constructing high-order exceptional points (EPs) in dissipative systems. These EPs can possess a number of intriguing properties related to, e.g., chiral transport and enhanced sensitivity. Previous proposals to realize non-Hermitian Hamiltonians (NHHs) with high-order EPs have been mainly based on either direct construction of spatial networks of coupled modes or utilization of synthetic dimensions, e.g., of mapping spatial lattices to time or photon-number space. Both methods rely on the construction of effective NHHs describing classical or postselected quantum fields, which neglect the effects of quantum jumps, and which, thus, suffer from a scalability problem in the {\it quantum regime}, when the probability of quantum jumps increases with the number of excitations and dissipation rate. Here, by considering the full quantum dynamics of a quadratic Liouvillian superoperator, we introduce a simple and effective method for engineering NHHs with high-order quantum EPs, derived from evolution matrices of system operators moments. That is, by quantizing higher-order moments of system operators, e.g., of a quadratic two-mode system, the resulting evolution matrices can be interpreted as alternative NHHs describing, e.g., a spatial lattice of coupled resonators, where spatial sites are represented by high-order field moments in the synthetic space of field moments. As an example, we consider a $U(1)$-symmetric quadratic Liouvillian describing a {\it bimodal} cavity with incoherent mode coupling, which can also possess anti-$\cal PT$-symmetry, whose field moment dynamics can be mapped to an NHH governing a spatial {\it network} of coupled resonators with high-order EPs., Comment: 14 pages

We propose to create and stabilize long-lived macroscopic quantum superposition states in atomic ensembles. We show that using a fully quantum parametric amplifier can cause the simultaneous decay of two atoms and, in turn, create stabilized atomic Schr\"{o}dinger cat states. Remarkably, even with modest parameters these intracavity atomic cat states can have an extremely long lifetime, up to \emph{4 orders of magnitude} longer than that of intracavity photonic cat states under the same parameter conditions, reaching \emph{tens of milliseconds}. This lifetime of atomic cat states is ultimately limited to \emph{several seconds} by extremely weak spin relaxation and thermal noise. Our work opens up a new way toward the long-standing goal of generating large-size and long-lived cat states, with immediate interests both in fundamental studies and noise-immune quantum technologies., Comment: 7+14 pages, 3+7 figures

We predict the emergence of a time crystal generated by an incoherently driven and dissipative nonlinear optical oscillator, where the nonlinearity also comes from dissipation. We show that a second-order dissipative phase transition of light occurs in the frame rotating at the cavity frequency, while a boundary (dissipative) time crystal emerges in the laboratory frame. We relate these two phenomena by using the Liouvillian superoperator associated with the Lindblad master equation and its symmetries. These results connect the emergence of a second-order dissipative phase transition and a dissipative time crystal in the thermodynamic limit, allowing to interpret them as the same phenomenon in terms of the Liouvillian spectrum, but just in different frames., Comment: 10 pages, 5 figures

We propose a method for the fast generation of nonclassical ground states of the Rabi model in the ultrastrong and deep-strong coupling regimes via the shortcuts-to-adiabatic (STA) dynamics. The time-dependent quantum Rabi model is simulated by applying parametric amplification to the Jaynes-Cummings model. Using experimentally feasible parametric drive, this STA protocol can generate large-size Schr\"{o}dinger cat states, through a process that is 10 times faster compared to adiabatic protocols. Such fast evolution increases the robustness of our protocol against dissipation. Our method enables one to freely design the parametric drive, so that the target state can be generated in the lab frame. A largely detuned light-matter coupling makes the protocol robust against imperfections of the operation times in experiments., Comment: 14 pages, 8 figures

Usually, when investigating exceptional points (EPs) of an open Markovian bosonic system, one deals with spectral degeneracies of a non-Hermitian Hamiltonian (NHH), which can correctly describe the system dynamics only in the semiclassical regime. A recently proposed quantum Liouvillian framework enables to completely determine the dynamical properties of such systems and their EPs (referred to as Liouvillian EPs, or LEPs) in the quantum regime by taking into account the effects of quantum jumps, which are ignored in the NHH formalism. Moreover, the symmetry and eigenfrequency spectrum of the NHH become a part of much larger Liouvillian eigenspace. As such, the EPs of an NHH form a subspace of the LEPs. Here we show that once an NHH of a dissipative linear bosonic system exhibits an EP of a certain finite order $n$, it immediately implies that the corresponding LEP can become of any higher order $m\geq n$, defined in the infinite Hilbert space. Most importantly, these higher-order LEPs can be identified by the coherence and spectral functions at the steady state. The coherence functions can offer a convenient tool to probe extreme system sensitivity to external perturbations in the vicinity of higher-order LEPs. As an example, we study a linear bosonic system of a bimodal cavity with incoherent mode coupling to reveal its higher-order LEPs; particularly, of second and third order via first- and second-order coherence functions, respectively. Accordingly, these LEPs can be additionally revealed by squared and cubic Lorentzian spectral lineshapes in the power and intensity-fluctuation spectra. Moreover, we demonstrate that these EPs can also be associated with spontaneous parity-time (${\cal PT}$) and anti-${\cal PT}$-symmetry breaking in the system studied., Comment: 15 pages, 4 figures