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Observation of Critical Phenomena in Parity-Time-Symmetric Quantum Dynamics
Abstract
We experimentally simulate nonunitary quantum dynamics using a single-photon interferometric network and study the information flow between a parity-time- (PT-)symmetric non-Hermitian system and its environment. We observe oscillations of quantum-state distinguishability and complete information retrieval in the PT-symmetry-unbroken regime. We then characterize in detail critical phenomena of the information flow near the exceptional point separating the PT-unbroken and PT-broken regimes, and demonstrate power-law behavior in key quantities such as the distinguishability and the recurrence time. We also reveal how the critical phenomena are affected by symmetry and initial conditions. Finally, introducing an ancilla as an environment and probing quantum entanglement between the system and the environment, we confirm that the observed information retrieval is induced by a finite-dimensional entanglement partner in the environment. Our work constitutes the first experimental characterization of critical phenomena in PT-symmetric nonunitary quantum dynamics.
... From the experimental perspective, several research groups have successfully prepared non-Hermitian quantum simulators using different platforms, including optical systems [28][29][30][31][32][33][34][35][36][37], acoustic systems [38], vacancy centers in solids [39], and cold atom systems [40]. Among these, optical systems are one of the most commonly used experimental platforms. ...
... However, due to the scattering and absorption processes that occur when light propagates * Corresponding author: wujs@sustech.edu.cn in a medium, there is energy dissipation and interference in optical devices, which limits the simulation time of non-Hermitian systems. For example, the quantum random walk, a current experimental hot spot, has only a single-digit evolution time for a non-Hermitian simulation with traditional experimental settings [30][31][32][33][34][35][36][37]. The reported maximum is t max = 7. ...
... To date, several works have realized the simulation of non-Hermitian systems successfully in quantum-walk experiments [30][31][32][33][34]. Unfortunately, the evolution time in traditional experimental settings is highly limited due to the dissipation of light. ...
We present a study on the long-term evolution of a non-Hermitian one-dimensional system with on-site dissipations in quantum-walk dynamics. By investigating various cases, including different evolution operations, lattice sizes, initial positions, and dissipations, we uncover remarkable and unconventional features. They include the emergence of distinct final states after a sufficiently long evolution time, the persistence of metastable states for extended periods, the survival of extended states only within the inner region of a domain-wall structure, and the occurrence of instantaneous ultrafar jumps during the evolution process. To provide insight into the underlying physical mechanisms behind these observations, we undertake a comprehensive investigation accompanied by detailed analysis, clarify the relation between time-evolved (meta)stable states and system eigenstates, and analyze the symmetry of final states. Moreover, our research reveals some phenomena that differ from those seen in previous experimental results, which can be attributed to the limited evolution time in the experiments. These theoretical findings underscore the significance of long-term evolution in non-Hermitian systems based on quantum walks.
... Due to this they lack the dynamical properties that facilitate the decay of the averaged state distinguishability at long times and, consequently, the quasi-Hermitian systems on their own are generally less practical for the reservoir computing tasks of interest. As shown in Appendix D for a two-level system, in this regime measures such as purity, entanglement, and trace distance may exhibit persistent γ-dependent oscillatory behavior, which can be interpreted as a (perpetual) full or partial recovery of dissipated information [99,120,121]. In this view, γ can control the degree of non-Markovianity of a NH model, potentially enhancing memory in specific tasks [122]. ...
... For γ → ∞ we get D(t) → 1 + sinh 2 (2γt) −1/2 → 0. Near the exceptional point γ ≈ h, the distance asymptotically behaves as D(t) ∼ t −1 . The exponent of this power-law decay depends both on the model and the initial state considered [121]. As we have demonstrated, non-Hermiticity endows a quantum reservoir with highly desirable properties for information processing, including controllable dissipation and purification. ...
Non-Hermitian (NH) systems provide a fertile platform for quantum technologies, owing in part to their distinct dynamical phases. These systems can be characterized by the preservation or spontaneous breaking of parity-time reversal symmetry, significantly impacting the dynamical behavior of quantum resources such as entanglement and purity; resources which in turn govern the system's information processing and memory capacity. Here we investigate this relationship using the example of an interacting NH spin system defined on random graphs. We show that the onset of the first exceptional point - marking the real-to-complex spectral transition - also corresponds to an abrupt change in the system's learning capacity. We further demonstrate that this transition is controllable via local disorder and spin interactions strength, thereby defining a tunable learnability threshold. Within the learning phase, the system exhibits the key features required for memory-dependent reservoir computing. This makes explicit a direct link between spectral structure and computational capacity, further establishing non-Hermiticity, and more broadly engineered dissipation, as a dynamic resource for temporal quantum machine learning.
... Furthermore, we were able to access the symmetryprotected entanglement properties of the cluster SPT phase. With these advancements, our work paves the way for physically exploring a broad range of critical phenomena that exist in complicated many-body models [49,[83][84][85][86][87][88][89][90]. ...
... Furthermore, as variational methods are widely employed in many-body simulations, our improved ZNE technique offers an efficient approach for achieving high-fidelity results on current noisy quantum hardware [80,107,108]. Thus, our work paves the way for the quantum simulation of a broad range of phenomena, such as deconfined quantum criticality, entanglement in topologically ordered systems, and critical behaviors in open quantum systems [49,[83][84][85][109][110][111][112][113][114][115]. ACKNOWLEDGMENTS We acknowledge the use of IBM Quantum services for this work. ...
Topology and symmetry play critical roles in characterizing quantum phases of matter. Recent advancements have unveiled symmetry-protected topological (SPT) phases in many-body systems as a unique class of short-range entangled states, notable for their nontrivial edge modes and characteristic ground-state entanglement gap. In this study, we demonstrate the robust simulation of many-body ground states of an Ising-cluster model on a quantum computer. By employing the method of quantum imaginary-time evolution (QITE) combined with enhanced zero-noise extrapolation techniques, we achieve accurate measurements of the transition between trivial and cluster SPT phases. Furthermore, we measured the characteristic edge modes and their associated topological entanglement properties, such as the second R\'enyi entropy, reduced density matrix, and entanglement spectral gap. Our work demonstrates the potential of using QITE in investigating sophisticated quantum phase transitions and critical phenomena on quantum computers.
... Recent advancements in experimental dissipation manipulation have reignited interest in studying open quantum systems, where dissipative processes are fundamental for quantum state preparation [40][41][42]. Specifically, experimental employment of dissipative coupling can achieve Tonks-Girardeau gas of molecules [43] and topological states [44], probing peculiar dynamical behaviors tied to passive parity-time symmetry in dissipative quantum systems [45][46][47][48]. The ability to simulate non-Hermitian systems in open quantum setups has led to a series of discoveries, including non-Hermitian skin effects [49][50][51][52][53] and topological classifications exceeding the standard ten classes [54][55][56][57][58]. Recent studies have unveiled distinct dynamic traits of non-Hermitian skin effects in dissipative systems [4,5,10,59,60], alongside the appearance of chiral damping [61], helical damping [62], and edge bursts [63]. ...
... While F/J decreases from infinity to a finite value, the first EP, denoted by the blue dashed line, always emerges at (F/J)c 2 = 1.577. Figure 5(d) showcases the zero solution of equations (48) and (52), confirming that the theoretical solution of equation (52) in the limit of α → 0, and j − ξ → ∞ is independent of the system size. This independence suggests a scale-free EP at (F/J)c 2 = 1.577 with J assumed to be 1 for simplicity. ...
Exceptional points (EPs) are distinct characteristics of non-Hermitian Hamiltonians that have no counterparts in Hermitian systems. In this study, we focus on EPs in continuous systems rather than discrete non-Hermitian systems, which are commonly investigated in both the experimental and theoretical studies. The non-Hermiticity of the system stems from the local imaginary potential, which can be effectively achieved through particle loss in recent quantum simulation setups. Leveraging the discrete Fourier transform, the dynamics of EPs within the low-energy sector can be well modeled by a Stark ladder system under the influence of a non-Hermitian tilted potential. To illustrate this, we systematically investigate continuous systems with finite imaginary potential wells and demonstrate the distinctive EP dynamics across different orders. Our investigation sheds light on EP behaviors, potentially catalyzing further exploration of EP phenomena across a variety of quantum simulation setups.
... These unique spectral degeneracies, fundamental to quantum mechanics, have been observed in diverse platforms such as photonic crystals, acoustic cavities, and solid-state spins [6][7][8]. EPs catalyze many interesting phenomena, including unconventional transmission or reflection [9-12], enhanced sensing [13][14][15], and unusual quantum criticality [16][17][18][19][20][21]. Notably, the quasistatic and dynamical encirclement around EPs exhibits intriguing band braiding [22][23][24][25][26][27] and chiral mode transfer [28][29][30][31][32][33][34][35][36], respectively, which signify the unique consequences of nontrivial EPs topology and provide broad applications for novel generation of quantum devices [6][7][8]. ...
... These unique spectral degeneracies, fundamental to quantum mechanics, have been observed in diverse platforms such as photonic crystals, acoustic cavities, and solid-state spins [6][7][8]. EPs catalyze many interesting phenomena, including unconventional transmission or reflection [9][10][11][12], enhanced sensing [13][14][15], and unusual quantum criticality [16][17][18][19][20][21]. Notably, the quasistatic and dynamical encirclement around EPs exhibits intriguing band braiding [22][23][24][25][26][27] and chiral mode transfer [28][29][30][31][32][33][34][35][36], respectively, which signify the unique consequences of nontrivial EPs topology and provide broad applications for novel generation of quantum devices [6][7][8]. ...
The exploration of novel phases and the elucidation of correspondences between topological invariants and their intriguing properties are pivotal in the realm of topological physics. Here, we investigate a complex exceptional structure, termed the composite exceptional ring (CER), composed of a third-order exceptional ring and multiple Weyl exceptional rings. We establish a direct correspondence between Chern numbers and the distinctive behaviors exhibited by these exceptional structures. Notably, we demonstrate that band braiding during quasistatic encircling processes correlates with bands possessing nontrivial Chern numbers, leading to triple (double) periodic spectra for cases with topologically nontrivial (trivial) middle bands. Moreover, the Chern numbers predict mode transfer behaviors during dynamical encircling process. We propose experimental schemes to realize CER in cold atoms, emphasizing the critical role of Chern numbers as both a measurable quantity and a descriptor of the exceptional physics inherent to dissipative systems. The discovery of CER opens significant avenues for expanding the scope of topological classifications in non-Hermitian systems, with promising applications in quantum computing and metrology.
... Thus, the non-conserved norm is essential for describing information dynamics in NH systems. In quantum information, a trace of the density matrix is a central concept in various formulae characterizing information properties, such as von Neumann entropy [50], Rényi entropy [45][46][47]51], and trace distance measuring of the distinguishability of two quantum states [8,18,52,53]. ...
... Our results show that the intertwining of (anti-)PT symmetry leads to new information dynamics patterns: damped oscillation with an overall decrease (increase) and asymptotically stable damped oscillation. The approaches of Hermitian quantum Rényi entropy or distinguishability adopted in [8,18,53,54] not only degenerate the three distinguished patterns to the same one, but they also distort it. The degeneracy is caused by the normalization of the nonunitary evolved density matrix, which leads to the loss of information about the total probability flow between the open system and the environment, while our approach based on the non-normalized density matrix reserves all the information related to the nonunitary time evolution. ...
We reveal the continuous change of information dynamics patterns in anyonic-PT symmetric systems that originates from the continuity of anyonic-PT symmetry. We find there are three information dynamics patterns for anyonic-PT symmetric systems: damped oscillations with an overall decrease (increase) and asymptotically stable damped oscillations, which are three-fold degenerate and are distorted using the Hermitian quantum Rényi entropy or distinguishability. It is the normalization of the non-unitary evolved density matrix that causes the degeneracy and distortion. We give a justification for non-Hermitian quantum Rényi entropy being negative. By exploring the mathematics and physical meaning of the negative entropy in open quantum systems, we connect negative non-Hermitian quantum Rényi entropy and negative quantum conditional entropy, paving the way to rigorously investigate negative entropy in open quantum systems.
... Ever since the pioneering discovery [1] by Bender and Boettcher of NH systems with PT (product of parity and time-reversal symmetries) symmetry exhibiting real energy spectra, these systems have become intriguing subjects for further investigations. Along with the mathematical advances in NH physics [2][3][4][5], recent experimental studies on NH properties in optical systems [6][7][8][9][10], electronic systems [11][12][13], acoustic systems [14][15][16] have prompted the exploration of this field in a significant way. * Author to whom any correspondence should be addressed. ...
... and σ σ σ denote the Pauli matrices. The presence of the d 1z term in the d 1 d 1 d 1 -vector poses a challenge for computing the Berry phase from equation (9). Hence, to make d 1z zero, we shall perform a unitary transformation on h 1 (k). ...
This work comprehensively investigates the non-Hermitian skin effect (NHSE) in a spinless Bernevig-Hughes-Zhang -like model in one dimension. It is generally believed that a system with non-reciprocal hopping amplitudes demonstrates NHSE. However, we show that there are exceptions, and more in-depth analyses are required to decode the presence of NHSE or its variants in a system. The fascinating aspects of our findings, depending on the inclusion of non-reciprocity in the inter-orbital hopping terms, concede the existence of conventional NHSE or NHSE at both edges and even a surprising absence of NHSE. The topological properties and the (bi-orthogonal) bulk-boundary correspondence, enumerated via computation of the (complex) Berry phase and spatial localization of the edge modes, highlight the topological phase transitions occurring therein. Further, to facilitate a structured discussion of the non-Hermitian model, we split the results into PT symmetric and non- PT symmetric cases with a view to comparing the two.
... The 2 × 2 matrix generating the dynamics is non-Hermitian and has imaginary instantaneous eigenvalues whenever k < a /a and real eigenvalues otherwise. Matrices with this structure and unitary transformations of them appear frequently in the study of non-Hermitian quantum systems [70][71][72][73][74][75]. Note that these Bogoliubov equations are simply parametric oscillators in disguise [76]. ...
Inspired by recent advances in observational astrophysics and continued explorations in the field of analog gravity, we discuss the prospect of simulating models of cosmology within the context of synthetic mechanical lattice experiments. We focus on the physics of expanding universe scenarios described by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. Specifically, quantizing scalar fluctuations in a background FLRW spacetime leads to a quadratic bosonic Hamiltonian with temporally varying pair production terms. Here we present a mapping that provides a one-to-one correspondence between these classes of cosmology models and feedback-coupled mechanical oscillators. As proof of principle, we then perform experiments on a synthetic mechanical lattice composed of such oscillators. We simulate two different FLRW expansion scenarios with universes dominated by vacuum energy and matter and discuss our experimental results.
Published by the American Physical Society 2025
... In particular, the implementation of NH dynamics has been achieved using a nitrogen-vacancy center in diamond through selective microwave pulses [44]. In additon, it has been achieved in a single-photon interferometric network [45]. This further facilitates the experimental realization of the present proposal. ...
Enhancing the sensitivity of quantum sensing near an exceptional point represents a significant phenomenon in non-Hermitian (NH) systems. However, the application of this property in time-modulated NH systems remains largely unexplored. In this work, we propose two theoretical schemes to achieve enhanced quantum sensing in time-modulated NH systems by leveraging the coalescence of eigenvalues and eigenstates. We conduct a comprehensive analysis of the full energy spectrum, including both real and imaginary components, the population distribution of eigenstates, and various characteristics associated with optimal conditions for sensitivity enhancement. Numerical simulations confirm that eigenvalue-based quantum sensors exhibit a 9.21-fold improvement compared to conventional Hermitian sensors, aligning with the performance of existing time-independent NH sensors. Conversely, for eigenstate-based quantum sensors, the enhancement reaches up to 50 times that of conventional Hermitian sensors, surpassing the results of existing time-independent NH sensors. Moreover, the eigenstate-based sensor exhibits divergent susceptibility even when not close to an exceptional point. Our findings pave the way for advanced sensing in time-sensitive contexts, thereby complementing existing efforts aimed at harnessing the unique properties of open systems.
... As shown in Fig. 2(d), in the unbroken phase, the single-QE dynamics displays prominent population retrieval on top of fractional decay; in the broken phase, the QE population in the formed bound state always keeps the value of 1/2. Similar phenomena have been observed in PT-symmetric systems [80,81], but here we uncover these phenomena in a fundamentally different setting. We also numerically compute the two-QE dynamics (see Ref. [74]), which is qualitatively different in the two phases. ...
Dissipative light-matter coupling plays a vital role in non-Hermitian physics, but it remains largely unexplored in waveguide QED systems. In this work, we find that by employing pseudo-Hermitian symmetry rather than anti-PT symmetry, the concept of dissipative coupling could be generalized and applied to the field of waveguide QED. This leads to a series of intriguing results, such as spontaneous breaking of pseudo-Hermitian symmetry across the exceptional points (EPs), level attraction between the bound states, and critical transition across the EPs for the population of quantum emitters in the bound state. Thanks to the tunability of photonic bands in crystal waveguides, we also demonstrate that dissipative light-matter coupling leads to the emergence of nonstandard third-order exceptional points with chiral spatial profiles in a topological waveguide QED system. This work provides a promising paradigm for studying non-Hermitian quantum phenomena in waveguide QED systems.
... As shown in Fig. 2(d), in the unbroken phase, the single-QE dynamics displays prominent population retrieval on top of fractional decay; in the broken phase, the QE population in the formed bound state always keeps the value of 1/2. Similar phenomena have been observed in PT -symmetric systems [96,97], but here we uncover these phenomena in a fundamentally different setting. We also numerically compute the two-QE dynamics (see Ref. [74]), which is qualitatively different in the two phases. ...
Dissipative light-matter coupling plays a vital role in non-Hermitian physics, but it remains largely unexplored in waveguide QED systems. In this work, we find that by employing pseudo-Hermitian symmetry rather than anti- PT symmetry, the concept of dissipative coupling could be generalized and applied to the field of waveguide QED. This leads to a series of intriguing results, such as spontaneous breaking of pseudo-Hermitian symmetry across the exceptional points (EPs), level attraction between the bound states, and critical transition across the EPs for the population of quantum emitters in the bound state. Thanks to the tunability of photonic bands in crystal waveguides, we also demonstrate that dissipative light-matter coupling leads to the emergence of nonstandard third-order exceptional points with chiral spatial profiles in a topological waveguide QED system. This work provides a promising paradigm for studying non-Hermitian quantum phenomena in waveguide QED systems.
Published by the American Physical Society 2025
... These unique spectral degeneracies, fundamental to quantum mechanics, have been observed in diverse platforms such as photonic crystals, acoustic cavities, and solid-state spins [6][7][8] . EPs catalyze many interesting phenomena, including unconventional transmission or reflection [9][10][11][12] , enhanced sensing [13][14][15] , and unusual quantum criticality [16][17][18][19][20] . It should be emphasized that the quasistatic and dynamical encirclement around EPs exhibits intriguing band braiding [21][22][23][24][25][26] and chiral mode transfer [27][28][29][30][31][32] , respectively, which signify the unique consequences of nontrivial EPs topology and provide broad applications for novel generation of quantum devices [6][7][8] . ...
The study of unconventional phases and elucidation of correspondences between topological invariants and their intriguing properties are pivotal in topological physics. Here, we investigate a complex exceptional ring (CER), composed of a third-order exceptional ring and multiple Weyl exceptional rings, and establish a direct correspondence between Chern numbers and the distinctive behaviors of these structures. We show that band braiding during quasistatic encircling processes correlates with nontrivial Chern numbers, resulting in triple (double) periodic spectra for topologically nontrivial (trivial) middle bands. Moreover, Chern numbers predict mode transfer during dynamical encircling. Experimental schemes for realizing CER in cold atoms are proposed, emphasizing the crucial role of Chern numbers as both measurable quantity and descriptor of exceptional physics in dissipative systems. This discovery broadens topological classifications in non-Hermitian systems, with promising applications in quantum computing and metrology.
... To surmount these challenges, we devise an ancilla-based framework for lattice systems, where generic non-unitary operators are implemented by embedding them within unitary operators involving extra "ancilla" qubits. While ancilla-based approaches have previously been demonstrated on small systems [48][49][50] , our approach allows for condensed matter lattice simulations for larger system sizes, and it enables greater programmability in implementing the desired non-Hermitian Hamiltonian evolution. Significantly, the scalability of our framework allows for the implementation of the effectively asymmetric couplings necessary for an extended NHSE lattice, which is a costly endeavor with the existing Pauli string approximation method 38,48 . ...
Lately, the non-Hermitian skin effect (NHSE) has been demonstrated in various classical metamaterials and even ultracold atomic arrays. Yet, its interplay with many-body dynamics have never been experimentally investigated. Here, we report the observation of the NHSE and its many-fermion analog on a universal quantum processor. To implement NHSE accumulation on a quantum computer, the time-evolution circuit not only needs to be non-reciprocal and non-unitary, but must also contain sufficiently many lattice qubits. We demonstrate this by systematically post-selecting ancilla qubits, as demonstrated through two paradigmatic non-reciprocal models on noisy quantum processors, with clear signatures of asymmetric spatial propagation and many-body “Fermi skin” accumulation. To minimize errors from inevitable device noise, time evolution is performed using trainable, variationally optimized quantum circuits. Our demonstration represents an important step in the quantum simulation of non-Hermitian lattices on present-day quantum hardware, and can be readily generalized to more sophisticated many-body models.
... The equatorial rotations R(θ, φ) = exp{−iθ [cos(φ)σ m x + sin(φ)σ m y ]/2} on any S-D transition can be realized by resonantly driving the corresponding transition line using a narrow-linewidth laser at 729 nm, where θ is the rotation angle, φ is the laser phase, while σ m x and σ m y are respectively the Pauli x and y matrices in the representation ofσ z . Using the digital quantum simulation method, which has been widely used in experimental studies of PT -symmetric dynamics [41][42][43], the state preparation and an arbitrary unitary operation can be decomposed into appropriate sequences of S-D equatorial rotations. The states in the S manifold can be identified via state-dependent fluorescence observed using a photomultiplier tube (PMT), while coupling the S 1/2 state to the short-lived state P 1/2 by a laser field at 397 nm. ...
The ability of achieving strong quantum spatial correlations has helped the emergence of quantum information science. In contrast, how to achieve strong quantum temporal correlations (QTCs) has remained as a long-standing challenge, thus hindering their applications in time-domain quantum control. Here we experimentally demonstrate that by using a parity-time ( PT )–symmetric single ion, the conventional QTC limit known as the Lüders bound can be well surpassed within a standard measurement scenario, approaching the predicted maximum QTC value. Our work, as a step toward quantum engineering of PT devices in the time domain, can stimulate more efforts on operating various quantum devices with the aid of strong QTC resources.
Published by the American Physical Society 2025
... The PT -symmetric phase transition is accompanied by the emergence of exceptional points, where two or more eigenvalues and their corresponding eigenvectors coalesce or merge into a single entity [49,50]. In recent decades, there has been a rapid progress in research interest in PT symmetry [51][52][53] due to its unique properties, leading to extensive exploration across various experimental platforms such as optical systems [54,55], electronic circuit systems [56,57], and atomic systems [58,59]. ...
... However, EPs also exist in systems without the parity-time symmetry and are therefore more general [1,10]. Thanks to the singular nature of EPs, a system exhibits many interesting behaviors in their vicinity, including universal criticality [23][24][25][26], non-reciprocal dynamics [27][28][29][30][31][32][33][34][35][36][37][38], enhanced entanglement generation [39,40], and strong sensitivity to external perturbations [41][42][43]. These properties have been confirmed in a wide range of classical and quantum mechanical systems, including optics and photonics [29,30], optomechanics [44,45] acoustics [46][47][48], atomic gases [33,49], and superconducting qubits [40,50,51]. ...
Exceptional points are the branch-point singularities of non-Hermitian Hamiltonians and have rich consequences in open-system dynamics. While the exceptional points and their critical phenomena are widely studied in the non-Hermitian settings without quantum jumps, they also emerge in open quantum systems depicted by the Lindblad master equations, wherein they are identified as the degeneracies in the Liouvillian eigenspectrum. These Liouvillian exceptional points often have distinct properties compared to their counterparts in non-Hermitian Hamiltonians, leading to fundamental modifications of the steady states or the steady-state-approaching dynamics. Since the Liouvillian exceptional points widely exist in quantum systems such as the atomic vapors, superconducting qubits, and ultracold ions and atoms, they have received increasing amount of attention of late. Here, we present a brief review on an important aspect of the dynamic consequence of Liouvillian exceptional points, namely the chiral state transfer induced by the parametric encircling the Liouvillian exceptional points. Our review focuses on the theoretical description and experimental observation of the phenomena in atomic systems that are experimentally accessible. We also discuss the ongoing effort to unveil the collective dynamic phenomena close to the Liouvillian exceptional points, as a consequence of the many-body effects therein. Formally, these phenomena are the quantum-many-body counterparts to those in classical open systems with nonlinearity, but hold intriguing new potentials for quantum applications.
... These applications are inspired by interesting phenomena taking place near EPs, and they are still in the classical regime. Recent studies have affirmed that PT -symmetry can be realized in various quantum systems, such as quantum walk [32], single-photon networks [33][34][35][36][37][38], cold atoms [39,40], superconducting circuits [41,42], and single nitrogen-vacancy centers [43,44]. However, realizing PT -symmetry is subject to certain qualifications, such as the control of gain and loss [5] and the Langevin noise arising therefrom [45,46], the study of quantum properties of light involved in non-Hermiticity still remains at an early stage. ...
Einstein–Podolsky–Rosen (EPR) steering is a type of directional quantum correlation that holds immense significance and broad applications in quantum information processing. While EPR steering has been achieved across various physical systems, research into its implementation in non-Hermitian systems remains in its early stages. In this study, we delve into the realm of non-Hermitian control of EPR steering by leveraging atomic coherence-controlled energy-level cascaded four-wave mixing (ELC-FWM) processes. We derive analytical expressions for the generation of EPR steering within such non-Hermitian nonlinear systems, demonstrating that exceptional points (EPs) and multimode EPR steering can be realized through introducing dressing-control fields. Furthermore, we illustrate that nonlinear coherent channels and the associated steerability distribution of the output modes can be tailored during the EPR steering generation process, which is directly linked to the eigenvalues of non-Hermitian processes. Additionally, we analyze the impact of loss effects on generated multimode EPR steering. Our findings suggest that non-Hermitian control offers a promising all-optical approach for constructing practical quantum networks.
... Recent advancements in experimental dissipation manipulation have reignited interest in studying open quantum systems, where dissipative processes are fundamental for quantum state preparation [23][24][25]. Specifically, experimental employment of dissipative coupling can achieve Tonks-Girardeau gas of molecules [26] and topological states [27], probing peculiar dynamical behaviors tied to passive parity-time symmetry in dissipative quantum systems [28][29][30][31]. The ability to simulate non- * wangr@tjnu.deu.cn ...
Exceptional points (EPs) are distinct characteristics of non-Hermitian Hamiltonians that have no counterparts in Hermitian systems. In this study, we focus on EPs in continuous systems rather than discrete non-Hermitian systems, which are commonly investigated in both the experimental and theoretical studies. The non-Hermiticity of the system stems from the local imaginary potential, which can be effectively achieved through particle loss in recent quantum simulation setups. Leveraging the discrete Fourier transform, the dynamics of EPs within the low-energy sector can be well modeled by a Stark ladder system under the influence of a non-Hermitian tilted potential. To illustrate this, we systematically investigate continuous systems with finite imaginary potential wells and demonstrate the distinctive EP dynamics across different orders. Our investigation sheds light on EP behaviors, potentially catalyzing further exploration of EP phenomena across a variety of quantum simulation setups.
... There has been much recent progress on non-Hermitian quantum systems, such as non-Hermitian topological phenomena [18][19][20][21][22][23][24][25][26][27][28][29][30][31], the non-Hermitian skin effect (NHSE) [30][31][32][33][34], and nontrivial path-integral winding [35,36]. While quantum mechanics dictates Hermitian Hamiltonians, the non-Hermitian models are increasingly proving their relevance in the fields of condensed matter physics [37,38], finding their origins in open or dissipative systems [39][40][41][42][43][44][45][46], optical [47][48][49][50][51][52], acoustic [53][54][55][56][57][58][59][60], electric [61][62][63][64][65][66], mechanical [67][68][69][70][71][72], and solid-state systems [73]. Recently, non-Hermitian physics of non-reciprocal classical fields has laid the foundations for understanding SSB in dynamical and non-equilibrium systems [17]. ...
Spontaneous symmetry breaking generally circumvents one-dimensional systems with local interactions in thermal equilibrium. Here, we analyze a category of one-dimensional Hermitian models via local non-Hermitian constructions. Notably, spontaneous symmetry breaking and long-range order may emerge at finite temperatures in such systems under periodic boundary conditions, in sharp contrast to Hermitian constructions. We demonstrate clear numerical evidence, such as order parameters and specific heat, supporting phase diagrams with robust ordered phases. Non-Hermitian physics plays a vital role in prohibiting domain-wall proliferation and promoting spontaneous symmetry breaking. The fermions exhibit an exotic topological nature in their path-integral windings, which uphold nonzero integers -- commonly a non-Hermitian signature -- in the ordered phases, thus offering a novel and spontaneous origin for both symmetry breaking and non-Hermiticity.
... Recently, phase transitions in information dynamics have been demonstrated in a variety of settings ranging from PT-symmetric non-Hermitian systems [28][29][30][31][32][33][34] to systems monitored by projective measurements [35][36][37][38][39][40][41][42][43][44], to more general open systems involving qubits arranged in a variety of space-time geometries [14,21,32,[45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61]. It is therefore natural to wonder if open system dynamics undergo a phase transition in information flow as they are tuned between Markovian and non-Markovian limits. ...
We study a random unitary circuit model of an impurity moving through a chaotic medium. The exchange of information between the medium and impurity is controlled by varying the velocity of the impurity, v_d v d , relative to the speed of information propagation within the medium, v_B v B . Above supersonic velocities, v_d> v_B v d > v B , information cannot flow back to the impurity after it has moved into the medium, and the resulting dynamics are Markovian. Below supersonic velocities, v_d< v_B v d < v B , the dynamics of the impurity and medium are non-Markovian, and information is able to flow back onto the impurity. We show the two regimes are separated by a continuous phase transition with exponents directly related to the diffusive spreading of operators in the medium. This is demonstrated by monitoring an out-of-time-order correlator (OTOC) in a scenario where the impurity is substituted at an intermediate time. During the Markovian phase, information from the medium cannot transfer onto the replaced impurity, manifesting in no significant operator development. Conversely, in the non-Markovian phase, we observe that operators acquire support on the newly introduced impurity. We also characterize the dynamics using the coherent information and provide two decoders which can efficiently probe the transition between Markovian and non-Markovian information flow. Our work demonstrates that Markovian and non-Markovian dynamics can be separated by a phase transition, and we propose an efficient protocol for observing this transition.
... Recent studies on effective non-Hermitian systems, including open quantum systems [34][35][36][37][38][39][40][41], optical systems (non-unitary quantum walk) [42][43][44][45][46][47], electric circuits [48][49][50][51][52][53], among others, have significantly broadened our understanding of condensed matter physics [54][55][56][57]. The non-Hermitian skin effect (NHSE), which supports an extensive number of single-particle eigenstates localized at the boundaries and characterized through the non-Bloch band theory with generalized Brillouin zones [58][59][60], reveals an additional localization formalism in 1D non-Hermitian systems under open-boundary conditions (OBCs) [58][59][60][61][62] and generalizable to higher dimensions [63][64][65][66][67][68][69][70][71][72]. ...
Localization and delocalization are historic topics central to quantum and condensed matter physics. We discover a new delocalization mechanism attributed to a residue imaginary (part of) velocity , feasible for ground states or low-temperature states of non-Hermitian quantum systems under periodic boundary conditions. Interestingly, a disorder field contributing to may allow strong-disorder-limit delocalization when prevails over the Anderson localization. We demonstrate such delocalization with correlation and entanglement behaviors, as well as its many-body nature and generalizability to finite temperatures and interactions. Thus, the nontrivial physics of significantly enriches our understanding of delocalization and breeds useful applications, e.g., in quantum adiabatic processes.
... Recently, Floquet driving has been used to engineer exceptional points [30,[33][34][35][36][37][38][39][40][41][42][43][44][45], demonstrated its control on the spectrum, including the number of exceptional points and the position in parameter space. Both theory [35,36] and experiment [37,40,44,45] have shown exceptional points can be arrived by choosing a proper driving even with a vanishingly small gain and loss. ...
We report a new kind of exceptional points in periodically driven system, called Floquet exceptional points, whose eigenvectors rotate on Bloch sphere and accumulate geometric phase in one time period. The merging of two such kind exceptional points are constrained by their dynamical structure, meaning two order-1/2 exceptional points with same dynamical structure can merge to one order-1 one while those with opposite dynamical structure can not. We show they exist in Floquet bipartite lattices, and the order-1 Floquet exceptional points appear at the phase transition point between quasimomentum gap phases and quasienergy gap phases. The scattering properties around the order-1 Floquet exceptional points is quite novel, which is perfect transparency but detectable in reflection for one of two sides.
... Biorthogonal decomposition was utilized to derive the population in various non-Hermitian dynamics [6,20], but this derivation relies on acquiring the density matrix of states, meanwhile, the direct non-Hermitian quantum measurement without state tomography is unattainable. Furthermore, while non-Hermitian quantum dynamics have been realized by diverse methods [4,5,21], but mainly concentrates on the evolution rather than measurement. In contrast, positive operator value measure (POVM) is a typical generalized measurement without orthogonality restriction, but the POVM can be dilated to standard orthogonal measurement by Naimark's theorem and the projected quantum state in POVM is uncertain [1]. ...
Non-Hermitian dynamics in quantum systems have unveiled novel phenomena, yet the implementation of valid non-Hermitian quantum measurement remains a challenge, because a universal quantum projective mechanism on the complete but skewed non-Hermitian eigenstates is not explicit in experiment. This limitation hinders the direct acquisition of non-Hermitian observable statistics (e.g., non-Hermitian population dynamics), also constrains investigations of non-Hermitian quantum measurement properties such as uncertainty relation. Here, we address these challenges by presenting a non-Hermitian projective protocol and investigating the non-Hermitian uncertainty relation. We derive the uncertainty relation for pseudo-Hermitian (PH) observables that is generalized beyond the Hermitian ones. We then investigate the projective properties of general quantum states onto complete non-Hermitian eigenvectors, and present a quantum simulating method to apply the valid non-Hermitian projective measurement on a direct-sum dilated space. Subsequently, we experimentally construct a quantum simulator in the quantum optical circuit and realize the 3-dimensional non-Hermitian quantum measurement on the single-photon qutrit. Employing this platform, we explore the uncertainty relation experimentally with different PH metrics. Our non-Hermitian quantum measurement method is state-independent and outputs directly the non-Hermitian quantum projective statistics, paving the way for studies of extensive non-Hermitian observable in quantum domain.
... which are real numbers when 0 < a < 1 (the PT -symmetric unbroken regime) [58,59], and imaginary numbers when a > 1 (the PT -symmetric broken regime). Note that the eigenvalues of the HamiltonianĤ PT are zero when a = 1 (the exceptional point). ...
Quantum metrology is an important field in quantum information and plays a crucial role in quantum parameter estimation. Quantum Fisher information (QFI) is widely used to characterize the precision of quantum parameter estimation. However, a full understanding of the precision for different parameterization processes and the role of quantum evolutions in quantum parameter estimations, remains a crucial area of research. In this paper, we introduce a new concept, named as quantum Fisher information power (QFI power), to characterize the QFI-creating capability of quantum evolutions. At the same time, we take both unitary and non-unitary evolutions as examples and study QFI power. The results show that: (i) For unitary evolutions described by spin angular momentum operators, the dynamic of QFI power is dependent on the z component of the Bloch vector, and the rotating angle; (ii) For non-unitary evolutions under decoherence channels, QFI cannot be enhanced when the initial probe state belongs to zero QFI set, and the QFI power is negative when the initial probe state has maximal QFI; (iii) For the PT - and APT -symmetric non-unitary evolutions, the dynamics of QFI power exhibit oscillations in symmetry unbroken regime, while the phenomenon of stable value (SV) occurs in broken regimes with the SV = 1.
... The deformation of the contour specific to a topological invariant is demonstrated to accommodate the non-Hermiticity of the underlying noninteracting Hamiltonian in question [99]. In addition, many studies have focused on the novel topological nature induced by non-Hermiticity [100][101][102][103][104][105][106][107][108][109][110][111][112][113][114][115][116][117][118]. ...
A three-dimensional non-Hermitian Hamiltonian with parity-time symmetry can exhibit a closed exceptional surface (EP surface) in momentum space, which is a non-Hermitian deformation of the degeneracy line. Since the degeneracy line lacks an internal space, the distributions of Berry curvature inside the EP surface become particularly intriguing. This paper studies the distributions taking a toruslike EP surface as an example. In a meridian cross section, the Berry connection exhibits a vortexlike field with only angular components, while the Berry curvature is perpendicular to this cross section; in a equatorial cross section, the Berry curvature forms a closed curve surrounding the central genus. Both Berry connection and curvature converge along the coplanar axis and diverge at the surface. We find the Berry flux depends on the radius of the integration region and is not quantized inside the EP torus. Approaching the surface, the Berry flux tends to infinity and the dynamical phase oscillates violently. We point out that the streamlines of Berry curvature can be used to estimate the zero or nonzero Berry flux. We generalize the above patterns to the case of EP surfaces with complex shapes and present a proposal of realizing the EP surface in an electrical circuit. Our research outcomes enhance the comprehension of EP surfaces and the topological characteristics of non-Hermitian systems with parity-time ( PT ) symmetry.
Published by the American Physical Society 2024
This perspective presents advances in non-Hermitian physics within quantum systems, covering experimental realizations across platforms and applications, along with proposed future research directions.
We identify Floquet exceptional points without a static counterpart, called Floquet π exceptional points (FPEPs), whose eigenvectors rotate on the Bloch sphere and accumulate π Aharonov-Anandan phase in one time period. The term π of FPEPs refers to the Aharonov-Anandan phase of π rather than the quasienergy of π, and an exceptional point with a quasienergy of zero may also be an FPEP. We first show their quasienergy spectral properties and the Aharonov-Anandan phase in a non-Hermitian Floquet two-resonators system. Then we show that they exist in non-Hermitian Floquet bipartite lattices, where there can be an order-1/2 FPEP or an order-1 FPEP. In the lattice model, two order-1/2 FPEPs with same dynamical structure can merge into one order-1 FPEP and the order-1 FPEP appears at the phase transition point between quasimomentum gap phases and quasienergy gap phases. The scattering properties around the order-1 FPEP exhibit perfect transparency but remain detectable in reflection from one of two sides.
Quantum information masking (QIM) allows encoding quantum information in multipartite systems. Complete QIM is of great significance in quantum foundation and application. However, the realization of complete QIM, even for single-qubit encoded information, is still lacking. Here, we propose to demonstrate complete QIM with 4-qubit entangled states. The proposed QIM can be readily extended to multipartite systems with arbitrary number of subsystems, enabling quantum secret sharing (QSS) and quantum teleportation between multiplayers. In experiment, we build up a 4-qubit hyperentangled state to implement complete QIM. The trace distance of 16 encoded single-qubit states falls within the range of 0.12 ± 0.02 to 0.03 ± 0.02. Furthermore, we implement QSS between six players by expanding the 4-qubit state to a 6-qubit state entangled in hybrid manner, in which we observe an average fidelity 0.85 ± 0.03 of the recovered states. Our results open the door towards QIM-enabled quantum information processing and provide applications in quantum communications.
Owing to the presence of exceptional points (EPs), non-Hermitian (NH) systems can display intriguing topological phenomena without Hermitian analogs. However, experimental characterizations of exceptional topological invariants have been restricted to second-order EPs (EP2s) in classical or semiclassical systems. We here propose an NH multi-mode system with higher-order EPs, each of which is underlain by a multifold-degenerate multipartite entangled eigenstate. We implement the NH model by controllably coupling a Josephson-junction-based electronic mode to two microwave resonators. We experimentally quantify the topological invariant for an EP3, by mapping out the complex eigenspectra of the tripartite system along a loop surrounding this EP3 in the parameter space. The nonclassicality of the realized topology is manifested by the observed quantum correlations in the corresponding eigenstates. Our results extend research of exceptional topology to fully quantum-mechanical models with multipartite entangled eigenstates.
Open systems possess unique potentials in high-precision sensing, yet the majority of previous studies rely on the spectral singularities known as "exceptional points." Here, we theoretically propose and experimentally demonstrate universal non-Hermitian sensing in the absence of exceptional points. The scheme makes use of the intrinsic sensitivity of a non-Hermitian probe to weak external fields, which can be understood as the direct consequence of non-Hermiticity. We confirm the basic mechanism by simulating the sensor-field dynamics using photon interferometry, and, as a concrete example, demonstrate the enhanced sensing of signals encoded in the setting angle of a wave plate. While the sensitivity of the probe is ultimately limited by the measurement noise, we find the non-Hermitian sensor showing superior performance under background noises that cannot be suppressed through repetitive measurements. Our experiment opens the avenue of enhanced sensing without exceptional points, complementing existing efforts aimed at harnessing the unique features of open systems.
The need for fast and robust quantum state transfer is an essential element in scalable quantum information processing, leading to widespread interest in shortcuts to adiabaticity for speeding up adiabatic quantum protocols. However, shortcuts to adiabaticity for systems with more than a few levels is occasionally challenging to compute in theory and frequently difficult to implement in experiments. In this paper, we develop a protocol for constructing shortcuts to adiabaticity through the multistate Landau-Zener approach and a stricter adiabatic condition. Importantly, our protocol only requires a few pieces of information about the energy spectrum and just adjusts the evolutionary rate of the system. It means that our protocol has broad applicability to theoretical models and does not require increasing the difficulty of the experiment. As examples, we apply our protocol to state transfer in the two-level Landau-Zener model, the non-Hermitian Su-Schrieffer-Heeger model, and the topological Thouless pump model and find that it can speed up the manipulation speed while remaining robust to Hamiltonian errors. Furthermore, based on the experimental friendliness of our findings, it can potentially be extended to many-body systems, dissipation cases, or Floquet processes. Overall, the proposed shortcut protocol offers a promising avenue for enhancing the efficiency and reliability of quantum state transfer protocols.
Motivated by our recent findings in [Phys. Rev. Lett. 128, 013902, 2022], which introduced a new class of electromagnetic bulk materials whose response is similar to conventional semiconductor transistors, here we propose a one-dimensional (1D) version of such materials based on transmission lines coupled with FET isolators. We demonstrate that the response of this 1D system is nonreciprocal and non-Hermitian, analogous to the idealized transistor-metamaterial, and is also characterized by a broken time-reversal symmetry. We analyze the wave propagation in the system and find that the interaction between the eigenmodes can either lead to gain or loss, depending on the propagation distance. Furthermore, it is also shown that the system may be operated at an exceptional point, wherein the response becomes singular, and the power gain is maximized. Finally, we demonstrate that the exceptional point coincides with the point of operation of typical microwave amplifiers, such as the distributed amplifier.
Open quantum systems display unusual phenomena not seen in closed systems, such as new topological phases and unconventional phase transitions. An interesting example was studied for a quantum spin liquid in the Kitaev model [Phys. Rev. Lett. 126, 077201 (2021)]; an effective non-Hermitian Kitaev model, which incorporates dissipation effects, was shown to give rise to a gapless spin liquid state with exceptional points in the Majorana dispersions. Given that an external magnetic field induces a gapped Majorana topological state in the Hermitian case, the exceptional points may bring about intriguing quantum phenomena under a magnetic field. Here we investigate the non-Hermitian Kitaev model perturbed by a magnetic field. We show that the exceptional points remain gapless up to a nonzero critical magnetic field, in stark contrast with the Hermitian case where an infinitesimal field opens a gap. The gapless state is stable over a wide range of the magnetic field for some particular parameter sets, and, in special cases, undergoes topological transitions to another gapless state with different winding number around the exceptional points without opening a gap. In addition, in the system with edges, we find that the non-Hermitian skin effect is induced by the magnetic field, even for the parameters where the skin effect is absent at zero field. The chirality of edge states is switched through the exceptional points, similarly to the surface Fermi arcs connected by the Weyl points in three-dimensional Weyl semimetals. Our results provide a new possible route to stabilize topological gapless quantum spin liquids under the magnetic field in the presence of dissipation.
We study the one-dimensional non-Hermitian lattices with staggered on-site modulations and nonreciprocal hopping up to the next-nearest-neighboring (NNN) sites. Due to the NNN nonreciprocity, the non-Hermitian skin effect (NHSE) in the system under open boundary conditions (OBC) can be energy dependent and there will be NHSE edges in the eigenenergy spectrum, which separates the eigenstates localized at the opposite ends of the lattice. We find that the interplay between the nonreciprocal hopping and on-site modulations can reverse the direction of the skin effect and modify the position of the NHSE edge. Moreover, by tuning the system parameters, some of the eigenstates under OBC will become fully extended with the corresponding eigenenergies being imaginary under both open and periodic boundary conditions. Thus the extended states can coexist with the NHSE in the same system. The NHSE can even be completely dissolved with all the eigenstates being extended when the modulation is imaginary. Our work unveils the intricate interplay between on-site modulations and nonreciprocal hopping in non-Hermitian systems.
Non-Hermitian matrices are ubiquitous in the description of nature ranging from classical dissipative systems, including optical, electrical, and mechanical metamaterials, to scattering of waves and open quantum many-body systems. Seminal line-gap and point-gap classifications of non-Hermitian systems using K-theory have deepened the understanding of many physical phenomena. However, ample systems remain beyond this description; reference points and lines do not in general distinguish whether multiple non-Hermitian bands exhibit intriguing exceptional points, spectral braids and crossings. To address this we consider two different notions: non-Hermitian band gaps and separation gaps that crucially encompass a broad class of multi-band scenarios, enabling the description of generic band structures with symmetries. With these concepts, we provide a unified and comprehensive classification of both gapped and nodal systems in the presence of physically relevant parity-time ( PT ) and pseudo-Hermitian symmetries using homotopy theory. This uncovers new stable topology stemming from both eigenvalues and wave functions, and remarkably also implies distinct fragile topological phases. In particular, we reveal different Abelian and non-Abelian phases in PT -symmetric systems, described by frame and braid topology. The corresponding invariants are robust to symmetry-preserving perturbations that do not induce (exceptional) degeneracy, and they also predict the deformation rules of nodal phases. We further demonstrate that spontaneous PT symmetry breaking is captured by Chern–Euler and Chern–Stiefel–Whitney descriptions, a fingerprint of unprecedented non-Hermitian topology previously overlooked. These results open the door for theoretical and experimental exploration of a rich variety of novel topological phenomena in a wide range of physical platforms.
We investigate imaginary gap-closed (IGC) points and their associated dynamics in dissipative systems. In a general non-Hermitian model, we derive the equation governing the IGC points of the energy spectrum, establishing that these points are only determined by the Hermitian part of the Hamiltonian. Focusing on a class of one-dimensional dissipative chains, we explore quantum walks across different scenarios and various parameters, showing that IGC points induce a power-law decay scaling in bulk loss probability and trigger a boundary phenomenon referred to as “edge burst.” This observation underscores the crucial role of IGC points under periodic boundary conditions (PBCs) in shaping quantum walk dynamics. Finally, we demonstrate that the damping matrices of these dissipative chains under PBCs possess Liouvillian gapless points, implying an algebraic convergence towards the steady state in long-time dynamics.
We propose a phase-biased non-Hermitian Josephson junction (NHJJ) composed of two superconductors mediated by a short non-Hermitian link. Such a NHJJ is described by an effective non-Hermitian Hamiltonian derived based on the Lindblad formalism in the weak coupling regime. By solving the Bogoliubov–de Gennes equation, we find that its Andreev spectrum as a function of phase difference exhibits Josephson gaps, i.e., finite phase windows with no Andreev (quasi)bound states. The complex Andreev spectrum and the presence of Josephson gaps constitute particular spectral features of the NHJJ. Moreover, we propose complex supercurrents arising from inelastic Cooper pair tunneling to characterize the anomalous transport in the NHJJ. Additional numerical simulations complement our analytical predictions. We demonstrate that the Josephson effect is strongly affected by non-Hermitian physics.
Quantum sensing utilizing unique quantum properties of non-Hermitian systems to realize ultraprecision measurements has been attracting increasing attention. However, the debate on whether non-Hermitian systems are superior to Hermitian counterparts in sensing remains an open question. Here, we investigate the quantum information in PT-symmetric quantum sensing utilizing two experimental schemes based on the trapped-ion platform and explore the relationship between PT-symmetric quantum sensors and quantum resources. It turns out that the existence of advantages of non-Hermitian quantum sensing heavily depends on additional information resources carried by the extra degrees of freedom introduced to construct PT-symmetric quantum sensors. Moreover, the practical application of non-Hermitian quantum sensing with superior performance is primarily restricted by the extra resource consumption accompanied by the postselection, while the superiority of non-Hermitian quantum sensing can be efficiently achieved in the presence of reasonable quantum resources. Our study provides theoretical references for the efficient construction of non-Hermitian quantum sensors with superior performance based on quantum resources and has potential applications in the research field of quantum precision measurement.
The two-body loss associated with inelastic interactions between fermions plays a significant role in realistic many-body systems. Here we study a two-component non-Hermitian Bardeen-Cooper-Schrieffer superfluidity with spin imbalance and two-body loss characterized by a complex-valued s-wave interaction in a square lattice. At the mean-field level, we map out the zero-temperature phase diagram and observe a dissipation-induced transition from superfluid to normal phase as well as a distinctive revival of the superfluid state in the weakly interacting regime applicable to both spin-balanced and spin-imbalanced systems. In the spin-imbalanced case, we find that the effective density of states involving weight functions reduces the regions in k space contributing to pairing, hence leading to a pronounced expansion of the normal phase on the phase diagram. Additionally, we analyze the order parameter, condensate energy, quasiparticle spectra, momentum distribution, and compressibility to characterize the presenting phases and phase transitions.
Non-Hermitian quantum metrology, an emerging field at the intersection of quantum estimation and non-Hermitian physics, holds promise for revolutionizing precision measurement. Here, we present a comprehensive investigation of non-Hermitian quantum parameter estimation in the quantum regime, with a special focus on achieving Heisenberg scaling. We introduce a concise expression for the quantum Fisher information (QFI) that applies to general non-Hermitian Hamiltonians, enabling the analysis of estimation precision in these systems. Our findings unveil the remarkable potential of non-Hermitian systems to attain the Heisenberg scaling of 1/ t , where t represents time. Moreover, we derive optimal measurement conditions based on the proposed QFI expression, demonstrating the attainment of the quantum Cramér-Rao bound. By constructing non-unitary evolutions governed by two non-Hermitian Hamiltonians, one with parity-time symmetry and the other without specific symmetries, we experimentally validate our theoretical analysis. The experimental results affirm the realization of Heisenberg scaling in estimation precision, marking a substantial milestone in non-Hermitian quantum metrology.
The non-Hermitian skin effects are representative phenomena intrinsic to non-Hermitian systems: the energy spectra and eigenstates under the open boundary condition (OBC) drastically differ from those under the periodic boundary condition (PBC). Whereas a nontrivial topology under the PBC characterizes the non-Hermitian skin effects, their proper measure under the OBC has not been clarified yet. This paper reveals that topological enhancement of nonnormality under the OBC accurately quantifies the non-Hermitian skin effects. Corresponding to spectrum and state changes of the skin effects, we introduce two scalar measures of nonnormality and argue that the non-Hermitian skin effects enhance both macroscopically under the OBC. We also show that the enhanced nonnormality correctly describes phase transitions causing the non-Hermitian skin effects and reveals the absence of non-Hermitian skin effects protected by average symmetry. The topological enhancement of nonnormality governs the perturbation sensitivity of the OBC spectra and the anomalous time-evolution dynamics through the Bauer-Fike theorem.
The Yang-Lee edge singularity was originally studied from the standpoint of mathematical foundations of phase transitions. However, direct observation of anomalous scaling with the negative scaling dimension has remained elusive due to an imaginary magnetic field required for the nonunitary criticality. We experimentally implement an imaginary magnetic field with an open quantum system of heralded single photons, directly measure the partition function, and demonstrate the Yang-Lee edge singularity via the quantum-classical correspondence. We also demonstrate unconventional scaling laws for finite-temperature quantum dynamics.
Quantum jumps induced by gain and loss have different quantum behaviors in non-Hermitian photonics. However, the jump effect on global quantum PT properties has not been comprehensively understood yet. Here, with considering quantum jump of loss and gain in a photonic dimer structure, we analytically obtained the PT-phase diagram under the steady-state condition and defined a Hermitian exchange operator to characterize the PT-symmetry or -broken phase. When we input Fock states into a PT-broken bi-waveguide splitting system, most photons will concentrate in the dominant waveguide with some state distributions. Especially in Hong-Ou-Mandel interferences, if gain is added, whether loss exists or not, g(2) is always larger than zero because loss cannot compensate gain as their different quantum jump effects. The quantum PT-phase diagram paves the way to the quantum state engineering, quantum interferences, and logic operations in non-Hermitian photonic systems.
We study the one-dimensional non-Hermitian lattices with linearly varying nonreciprocal hopping, where the non-Hermitian skin effect (NHSE) is found to be dissolved gradually as the strength of nonreciprocity increases. The energy spectrum under the open boundary condition is composed of real and imaginary eigenenergies when the nonreciprocal hopping is weak. Interestingly, the real eigenenergies form an equally spaced ladder, and the corresponding eigenstates are localized at the boundary with a Gaussian distribution due to NHSE. By increasing the nonreciprocity, the number of real eigenenergies will decrease while more and more eigenenergies become imaginary. Accompanied by the real-imaginary transition in the spectrum, the eigenstates are shifted from the boundary into the bulk of the lattice. When the nonreciprocity gets strong enough, the whole spectrum will be imaginary and the NHSE disappears completely in the system; i.e., all the eigenstates become Gaussian bound states localized inside the bulk. Our work unveils the exotic properties of non-Hermitian systems with spatially varying nonreciprocal hopping.
Exceptional points are interesting physical phenomena in non-Hermitian physics at which the eigenvalues are degenerate and the eigenvectors coalesce. In this paper, we find that in projected non-Hermitian two-level systems (sub-systems under projecting partial Hilbert space) the singularities of exceptional points (EPs) is due to basis defectiveness rather than energy degeneracy or state coalescence. This leads to the discovery of extended exceptional points (EEPs). For EEPs, more subtle structures (e.g., the so-called Bloch peach), additional classification, and “hidden” quantum phase transitions are explored. By using the topologically protected sub-space from two edge states in the non-Hermitian Su–Schrieffer–Heeger model as an example, we illustrate the physical properties of different types of EEPs.
Topology in quench dynamics gives rise to intriguing dynamic topological phenomena, which are intimately connected to the topology of static Hamiltonians yet challenging to probe experimentally. Here we theoretically characterize and experimentally detect momentum-time skyrmions in parity-time (PT)-symmetric non-unitary quench dynamics in single-photon discrete-time quantum walks. The emergent skyrmion structures are protected by dynamic Chern numbers defined for the emergent two-dimensional momentum-time submanifolds, and are revealed through our experimental scheme enabling the construction of time-dependent non-Hermitian density matrices via direct measurements in position space. Our work experimentally reveals the interplay of PT symmetry and quench dynamics in inducing emergent topological structures, and highlights the application of discrete-time quantum walks for the study of dynamic topological phenomena.
Non-Hermitian physical systems have attracted considerable attention lately for their unconventional behaviour around exceptional points (EPs)—spectral singularities at which eigenvalues and eigenvectors coalesce. In particular, many new EP-related concepts such as unidirectional lasing and invisibility, as well as chiral transmission, have been realized. Given the progress in understanding the physics of EPs in various photonic structures, it is surprising that one of the oldest theoretical predictions associated with them, a remarkable broadening of the laser linewidth at an EP, has been probed only indirectly so far. Here, we fill this gap by steering a phonon laser through an EP in a compound optomechanical system formed by two coupled resonators. We observe a pronounced linewidth broadening of the mechanical lasing mode generated in one of the resonators when the system approaches the EP.
Open physical systems with balanced loss and gain exhibit a transition, absent in their solitary counterparts, which engenders modes that exponentially decay or grow with time and thus spontaneously breaks the parity-time PT symmetry. This PT-symmetry breaking is induced by modulating the strength or the temporal profile of the loss and gain, but also occurs in a pure dissipative system without gain. It has been observed that, in classical systems with mechanical, electrical, and electromagnetic setups with static loss and gain, the PT-symmetry breaking transition leads to extraordinary behavior and functionalities. However, its observation in a quantum system is yet to be realized. Here we report on the first quantum simulation of PT-symmetry breaking transitions using ultracold Li-6 atoms. We simulate static and Floquet dissipative Hamiltonians by generating state-dependent atom loss in a noninteracting Fermi gas, and observe the PT-symmetry breaking transitions by tracking the atom number for each state. We find that while the two-state system undergoes a single transition in the static case, its Floquet counterpart, with a periodic loss, undergoes PT-symmetry breaking and restoring transitions at vanishingly small dissipation strength. Our results demonstrate that Floquet dissipation offers a versatile tool for navigating phases where the PT-symmetry is either broken or conserved. The dissipative ultracold Fermi gas provides a starting point for exploring the interplay among dissipation, decoherence, and interactions in open quantum systems.
Topological operations have the merit of achieving certain goals without requiring accurate control over local operational details. To date, topological operations have been used to control geometric phases, and have been proposed as a means for controlling the state of certain systems within their degenerate subspaces[1-8]. More recently, it was predicted that topological operations can be extended to transfer energy between normal modes, provided that the system possesses a specific type of degeneracy known as an exceptional point (EP)[9-11]. Here we demonstrate the transfer of energy between two modes of a cryogenic optomechanical device by topological operations. We show that this transfer arises from the presence of an EP in the device's spectrum. We also show that this transfer is non-reciprocal[12-14]. These results open new directions in system control; they also open the possibility of exploring other dynamical effects related to EPs[15,16], as well as the behavior of thermal and quantum fluctuations in the vicinity of EPs.
Light propagation in periodic environments is often associated with a number of interesting and potentially useful processes. If a crystalline optical potential is also linearly ramped, light can undergo periodic Bloch oscillations, a direct outcome of localized Wannier-Stark states and their equidistant eigenvalue spectrum. Even though these effects have been extensively explored in conservative settings, this is by no means the case in non-Hermitian photonic lattices encompassing both amplification and attenuation. Quite recently, Bloch oscillations have been predicted in parity-time-symmetric structures involving gain and loss in a balanced fashion. While in a complex bulk medium, one intuitively expects that light will typically follow the path of highest amplification, in a periodic system this behavior can be substantially altered by the underlying band structure. Here, we report the first experimental observation of Bloch oscillations in parity-time-symmetric mesh lattices. We show that these revivals exhibit unusual properties like secondary emissions and resonant restoration of PT symmetry. In addition, we present a versatile method for reconstructing the real and imaginary components of the band structure by directly monitoring the light evolution during a cycle of these oscillations.
We propose and analyze a new approach based on parity-time ()
symmetric microcavities with balanced gain and loss to enhance the performance
of cavity-assisted metrology. We identify the conditions under which
-symmetric microcavities allow to improve sensitivity beyond what
is achievable in loss-only systems. We discuss its application to the detection
of mechanical motion, and show that the sensitivity is significantly enhanced
in the vicinity of the transition point from unbroken- to broken-
regimes. We believe that our results open a new direction for
-symmetric physical systems and it may find use in ultra-high
precision metrology and sensing.
Controlling and reversing the effects of loss are major challenges in optical
systems. For lasers losses need to be overcome by a sufficient amount of gain
to reach the lasing threshold. We show how to turn losses into gain by steering
the parameters of a system to the vicinity of an exceptional point (EP), which
occurs when the eigenvalues and the corresponding eigenstates of a system
coalesce. In our system of coupled microresonators, EPs are manifested as the
loss-induced suppression and revival of lasing. Below a critical value, adding
loss annihilates an existing Raman laser. Beyond this critical threshold,
lasing recovers despite the increasing loss, in stark contrast to what would be
expected from conventional laser theory. Our results exemplify the
counterintuitive features of EPs and present an innovative method for reversing
the effect of loss.
An attractive mechanism to induce robust spatially confined states utilizes
interfaces between regions with topologically distinct gapped band structures.
For electromagnetic waves, this mechanism can be realized in two dimensions by
breaking symmetries in analogy to the quantum Hall effect or by employing
analogies to the quantum spin Hall effect, while in one dimension it can be
obtained by geometric lattice modulation. Induced by the presence of the
interface, a topologically protected, exponentially confined state appears in
the middle of the band gap. The intrinsic robustness of such states raises the
question whether their properties can be controlled and modified independently
of the other states in the system. Here, we draw on concepts from passive
non-hermitian parity-time (PT)-symmetry to demonstrate the selective control
and enhancement of a topologically induced state in a one-dimensional microwave
set-up. In particular, we show that the state can be isolated from losses that
affect all other modes in the system, which enhances its visibility in the
temporal evolution of a pulse. The intrinsic robustness of the state to
structural disorder persists in the presence of the losses. The combination of
concepts from topology and non-hermitian symmetry is a promising addition to
the set of design tools for optical structures with novel functionality.
Optical systems combining balanced loss and gain provide a unique platform to implement classical analogues of quantum systems described by non-Hermitian parity–time (PT)-symmetric Hamiltonians. Such systems can be used to create synthetic materials with properties that cannot be attained in materials having only loss or only gain. Here we report PT-symmetry breaking in coupled optical resonators. We observed non-reciprocity in the PT-symmetry-breaking phase due to strong field localization, which significantly enhances nonlinearity. In the linear regime, light transmission is reciprocal regardless of whether the symmetry is broken or unbroken. We show that in one direction there is a complete absence of resonance peaks whereas in the other direction the transmission is resonantly enhanced, a feature directly associated with the use of resonant structures. Our results could lead to a new generation of synthetic optical systems enabling on-chip manipulation and control of light propagation.
When two resonant modes in a system with gain or loss coalesce in both their
resonance position and their width, a so-called "Exceptional Point" occurs
which acts as a source of non-trivial physics in a diverse range of systems.
Lasers provide a natural setting to study such "non-Hermitian degeneracies",
since they feature resonant modes and a gain material as their basic
constituents. Here we show that Exceptional Points can be conveniently induced
in a photonic molecule laser by a suitable variation of the applied pump. Using
a pair of coupled micro-disk quantum cascade lasers, we demonstrate that in the
vicinity of these Exceptional Points the laser shows a characteristic reversal
of its pump-dependence, including a strongly decreasing intensity of the
emitted laser light for increasing pump power. This result establishes photonic
molecule lasers as promising tools for exploring many further fascinating
aspects of Exceptional Points, like a strong line-width enhancement and the
coherent perfect absorption of light in their vicinity as well as non-trivial
mode-switching and the accumulation of a geometric phase when encircling an
Exceptional Point parametrically.
The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of PT symmetry, one obtains new infinite classes of complex Hamiltonians whose spectra are also real and positive. These PT symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space. This paper describes the unusual classical and quantum properties of these theories. {copyright} {ital 1998} {ital The American Physical Society}
One of the fundamental axioms of quantum mechanics is associated with the Hermiticity of physical observables 1 . In the case of the Hamiltonian operator, this requirement not only implies real eigenenergies but also guarantees probability conservation. Interestingly, a wide class of non-Hermitian Hamiltonians can still show entirely real spectra. Among these are Hamiltonians respecting parity-time (PT) symmetry 2-7 . Even though the Hermiticity of quantum observables was never in doubt, such concepts have motivated discussions on several fronts in physics, including quantum field theories 8 , non- Hermitian Anderson models 9 and open quantum systems 10,11 , to mention a few. Although the impact of PT symmetry in these fields is still debated, it has been recently realized that optics can provide a fertile ground where PT-related notions can be implemented and experimentally investigated 12-15 . In this letter we report the first observation of the behaviour of a PT optical coupled system that judiciously involves a complex index potential. We observe both spontaneous PT symmetry breaking and power oscillations violating left-right symmetry. Our results may pave the way towards a new class of PT-synthetic materials with intriguing and unexpected properties that rely on non-reciprocal light propagation and tailored transverse energy flow. Before we introduce the concept of spacetime reflection in optics, we first briefly outline some of the basic aspects of this symmetry within the context of quantum mechanics. In general, a Hamiltonian HD p 2 =2mCV(x
A model is proposed that describes the evolution of a mixed state of a quantum system for which gain and loss of energy or amplitude are present. Properties of the model are worked out in detail. In particular, invariant subspaces of the space of density matrices corresponding to the fixed points of the dynamics are identified, and the existence of a transition between the phase in which gain and loss are balanced and the phase in which this balance is lost is illustrated in terms of the time average of observables. The model is extended to include a noise term that results from a uniform random perturbation generated by white noise. Numerical studies of example systems show the emergence of equilibrium states that suppress the phase transition.
Invisibility by metamaterials is of great interest, where optical properties are manipulated in the real permittivity-permeability plane. However, the most effective approach to achieving invisibility in various military applications is to absorb the electromagnetic waves emitted from radar to minimize the corresponding reflection and scattering, such that no signal gets bounced back. Here, we show the experimental realization of chip-scale unidirectional reflectionless optical metamaterials near the spontaneous parity-time symmetry phase transition point where reflection from one side is significantly suppressed. This is enabled by engineering the corresponding optical properties of the designed parity-time metamaterial in the complex dielectric permittivity plane. Numerical simulations and experimental verification consistently exhibit asymmetric reflection with high contrast ratios around a wavelength of of 1,550 nm. The demonstrated unidirectional phenomenon at the corresponding parity-time exceptional point on-a-chip confirms the feasibility of creating complicated on-chip parity-time metamaterials and optical devices based on their properties.
We demonstrate that the above-threshold behavior of a laser can be strongly affected by exceptional points which are induced by pumping the laser nonuniformly. At these singularities, the eigenstates of the non-Hermitian operator which describes the lasing modes coalesce. In their vicinity, the laser may turn off even when the overall pump power deposited in the system is increased. Such signatures of a pump-induced exceptional point can be experimentally probed with coupled ridge or microdisk lasers.
Using a scattering matrix formalism, we derive the general scattering properties of optical structures that are symmetric under a combination of parity and time reversal (PT). We demonstrate the existence of a transition between PT-symmetric scattering eigenstates, which are norm preserving, and symmetry-broken pairs of eigenstates exhibiting net amplification and loss. The system proposed by Longhi [Phys. Rev. A 82, 031801 (2010).], which can act simultaneously as a laser and coherent perfect absorber, occurs at discrete points in the broken-symmetry phase, when a pole and zero of the S matrix coincide.
We construct a general measure for the degree of non-Markovian behavior in open quantum systems. This measure is based on the trace distance which quantifies the distinguishability of quantum states. It represents a functional of the dynamical map describing the time evolution of physical states, and can be interpreted in terms of the information flow between the open system and its environment. The measure takes on nonzero values whenever there is a flow of information from the environment back to the open system, which is the key feature of non-Markovian dynamics.
Requiring that a Hamiltonian be Hermitian is overly restrictive. A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but satisfies the less restrictive and more physical condition of space-time reflection symmetry (PT symmetry). One might expect a non-Hermitian Hamiltonian to lead to a violation of unitarity. However, if PT symmetry is not spontaneously broken, it is possible to construct a previously unnoticed symmetry C of the Hamiltonian. Using C, an inner product whose associated norm is positive definite can be constructed. The procedure is general and works for any PT-symmetric Hamiltonian. Observables exhibit CPT symmetry, and the dynamics is governed by unitary time evolution. This work is not in conflict with conventional quantum mechanics but is rather a complex generalization of it.
The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose +complex conjugate) is replaced by the physically transparent condition of space–time reflection ( ) symmetry. If H has an unbroken symmetry, then the spectrum is real. Examples of -symmetric non-Hermitian quantum-mechanical Hamiltonians are and . Amazingly, the energy levels of these Hamiltonians are all real and positive!
Does a -symmetric Hamiltonian H specify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answer is that if H has an unbroken symmetry, then it has another symmetry represented by a linear operator . In terms of , one can construct a time-independent inner product with a positive-definite norm. Thus, -symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution.
The Lee model provides an excellent example of a -symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm 'ghost' state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. The correct interpretation of the ghost is simply that the non-Hermitian Lee-model Hamiltonian is -symmetric. The operator for the Lee model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.
Exploiting the interplay between gain, loss and the coupling strength between different optical components creates a variety of new opportunities in photonics to generate, control and transmit light. Inspired by the discovery of real eigenfrequencies for non-Hermitian Hamiltonians obeying parity–time (PT) symmetry, many counterintuitive aspects are being explored, particularly close to the associated degeneracies also known as ‘exceptional points’. This Review explains the underlying physical principles and discusses the progress in the experimental investigation of PT-symmetric photonic systems. We highlight the role of PT symmetry and non-Hermitian dynamics for synthesizing and controlling the flow of light in optical structures and provide a roadmap for future studies and potential applications.
In recent years, notions drawn from non-Hermitian physics and parity-time (PT) symmetry have attracted considerable attention. In particular, the realization that the interplay between gain and loss can lead to entirely new and unexpected features has initiated an intense research effort to explore non-Hermitian systems both theoretically and experimentally. Here we review recent progress in this emerging field, and provide an outlook to future directions and developments. © 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
Non-Hermitian degeneracies, also known as exceptional points, have recently emerged as a new way to engineer the response of open physical systems, that is, those that interact with the environment. They correspond to points in parameter space at which the eigenvalues of the underlying system and the corresponding eigenvectors simultaneously coalesce. In optics, the abrupt nature of the phase transitions that are encountered around exceptional points has been shown to lead to many intriguing phenomena, such as loss-induced transparency, unidirectional invisibility, band merging, topological chirality and laser mode selectivity. Recently, it has been shown that the bifurcation properties of second-order non-Hermitian degeneracies can provide a means of enhancing the sensitivity (frequency shifts) of resonant optical structures to external perturbations. Of particular interest is the use of even higher-order exceptional points (greater than second order), which in principle could further amplify the effect of perturbations, leading to even greater sensitivity. Although a growing number of theoretical studies have been devoted to such higher-order degeneracies, their experimental demonstration in the optical domain has so far remained elusive. Here we report the observation of higher-order exceptional points in a coupled cavity arrangement-specifically, a ternary, parity-time-symmetric photonic laser molecule-with a carefully tailored gain-loss distribution. We study the system in the spectral domain and find that the frequency response associated with this system follows a cube-root dependence on induced perturbations in the refractive index. Our work paves the way for utilizing non-Hermitian degeneracies in fields including photonics, optomechanics, microwaves and atomic physics.
The study of non-Hermitian systems with parity–time (PT) symmetry is a rapidly developing frontier. Realized in recent experiments, PT-symmetric classical optical systems with balanced gain and loss hold great promise for future applications. Here we report the experimental realization of passive PT-symmetric quantum dynamics for single photons by temporally alternating photon losses in the quantum walk interferometers. The ability to impose PT symmetry allows us to realize and investigate Floquet topological phases driven by PT-symmetric quantum walks. We observe topological edge states between regions with different bulk topological properties and confirm the robustness of these edge states with respect to PT-symmetry-preserving perturbations and PT-symmetry-breaking static disorder. Our results contribute towards the realization of quantum mechanical PT-synthetic devices and suggest exciting possibilities for the exploration of the topological properties of non-Hermitian systems using discrete-time quantum walks.
By studying the information flow between a general parity-time (PT) symmetric non-Hermitian system and its environment, we find that the complete information retrieval from the environment can be achieved in the PT-unbroken phase, whereas no retrieval can be made in the PT-broken phase. Thus the PT transition point also marks the reversible/irreversible transition of the information flow. To understand the physics behind this criticality, we embed a PT-symmetric system into a larger Hilbert space in which the total system obeys unitary dynamics, and show that the dimension of the ancilla required for the extension diverges at the exceptional point where the recurrence time of the information retrieval exhibits critical behavior.
Network centrality has important implications well beyond its role in physical and information transport analysis; as such, various quantum walk-based algorithms have been proposed for measuring network vertex centrality. In this work, we propose a continuous-time quantum walk algorithm for determining vertex centrality, and show that it generalizes to arbitrary graphs via a statistical analysis of randomly generated scale-free and Erd\H{o}s-R\'enyi networks. As a proof of concept, the algorithm is detailed on a 4-vertex star graph and physically implemented via linear optics, using spatial and polarization degrees of freedoms of single photons. This paper reports the first successful physical demonstration of a quantum centrality algorithm.
Parity–time (PT)-symmetric crystals are a class of non-Hermitian systems that allow, for example, the existence of modes with real propagation constants, for self-orthogonality of propagating modes, and for uni-directional invisibility at defects. Photonic PT-symmetric systems that also support topological states could be useful for shaping and routing light waves. However, it is currently debated whether topological interface states can exist at all in PT-symmetric systems. Here, we show theoretically and demonstrate experimentally the existence of such states: states that are localized at the interface between two topologically distinct PT-symmetric photonic lattices. We find analytical closed form solutions of topological PT-symmetric interface states, and observe them through fluorescence microscopy in a passive PT-symmetric dimerized photonic lattice. Our results are relevant towards approaches to localize light on the interface between non-Hermitian crystals.
This paper surveys perturbation theory for the pseudo–inverse (Moore–Penrose generalized inverse), for the orthogonal projection onto the column space of a matrix, and for the linear least squares problem.
A short resume is given about the nature of exceptional points (EPs) followed
by discussions about their ubiquitous occurrence in a great variety of physical
problems. EPs feature in classical as well as in quantum mechanical problems.
They are associated with symmetry breaking for -symmetric
Hamiltonians, where a great number of experiments have been performed in
particular in optics, and to an increasing extent in atomic and molecular
physics. EPs are involved in quantum phase transition and quantum chaos, they
produce dramatic effects in multichannel scattering, specific time dependence
and more. In nuclear physics they are associated with instabilities and
continuum problems. Being spectral singularities they also affect approximation
schemes.
The development of new artificial structures and materials is today one of the major research challenges in optics. In most studies so far, the design of such structures has been based on the judicious manipulation of their refractive index properties. Recently, the prospect of simultaneously using gain and loss was suggested as a new way of achieving optical behaviour that is at present unattainable with standard arrangements. What facilitated these quests is the recently developed notion of 'parity-time symmetry' in optical systems, which allows a controlled interplay between gain and loss. Here we report the experimental observation of light transport in large-scale temporal lattices that are parity-time symmetric. In addition, we demonstrate that periodic structures respecting this symmetry can act as unidirectional invisible media when operated near their exceptional points. Our experimental results represent a step in the application of concepts from parity-time symmetry to a new generation of multifunctional optical devices and networks.
This paper describes an algorithm for simultaneously diagonalizing by orthogonal transformations the blocks of a partitioned matrix having orthonormal columns.
It is almost a quarter of a century since Chandler Davis and William Kahan brought together the key ideas of what Stewart later completed and defined to be the CS decomposition (CSD) of a partitioned unitary matrix. This paper outlines some germane points in the history of the CSD, pointing out the contributions of Jordan, of Davis and Kahan, and of Stewart, and the relationship of the CSD to the “direct rotation” of Davis and Kato. The paper provides an easy to memorize, constructive proof of the CSD, reviews one of its important uses, and suggests a motivation for the CSD which emphasizes how generally useful it is. It shows the relation between the CSD and generalized singular value decompositions, and points out some useful nullity properties one form of the CSD trivially reveals. Finally it shows how, via the QR factorization, the CSD can be used to obtain interesting results for partitioned nonsingular matrices. We suggest the CSD be taught in its most general form with no restrictions on the two by two partition, and initially with no mention of angles between subspaces.
In 1998, Bender and Boettcher found that a wide class of Hamiltonians, even though non-Hermitian, can still exhibit entirely real spectra provided that they obey parity-time requirements or PT symmetry. Here we demonstrate experimentally passive PT-symmetry breaking within the realm of optics. This phase transition leads to a loss induced optical transparency in specially designed pseudo-Hermitian guiding potentials.
The quantum mechanical brachistochrone system with a PT-symmetric Hamiltonian is Naimark-dilated and reinterpreted as a subsystem of a Hermitian system in a higher-dimensional Hilbert space. This opens a way to a direct experimental implementation of the recently hypothesized PT-symmetric ultrafast brachistochrone regime of Bender et al. [Phys. Rev. Lett. 98, 040403 (2007)] in an entangled two-spin system.