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Quantum information dynamics in a high-dimensional parity-time-symmetric system
Abstract
Non-Hermitian systems with parity-time (PT) symmetry give rise to exceptional points (EPs) with exceptional properties that arise due to the coalescence of eigenvectors. Such systems have been extensively explored in the classical domain, where second- or higher-order EPs have been proposed or realized. In contrast, quantum information studies of PT-symmetric systems have been confined to systems with a two-dimensional Hilbert space. Here, by using a single-photon interferometry setup, we simulate the quantum dynamics of a four-dimensional PT-symmetric system across a fourth-order exceptional point. By tracking the coherent, nonunitary evolution of the density matrix of the system in PT-symmetry unbroken and broken regions, we observe the entropy dynamics for both the entire system, and the gain and loss subsystems. Our setup is scalable to the higher-dimensional PT-symmetric systems, and our results point towards the rich dynamics and critical properties.
... In more detail, we see that the conventional metric used in Eq. (5) leads to results completely consistent with experimental observations [49,91,[102][103][104][105][106][107][108][109] provided that it is applied correctly. In fact, in order to physically implement the non-unitary evolution leading to Eq. (5), we can embed the (anti)PT-symmetric system into a larger Hermitian system, realized by adding ancillary qubits, and perform a measuring process [93,103]. ...
... Because of this postselection occurring in the measuring process, the successful implementation of the non-unitary gate is a probabilistic procedure. This experimental limitation [49,102,103,105], is similar to the situation which occurs in Bell inequality tests [113][114][115]. Therefore, we can solve the paradoxes [7,101,116,117] associated with violation of no-go theorems in PT-symmetric theory [101] using normalized density matrix (5), without the need for any modification in the Hilbert space (for more details, see Refs. ...
... Moreover, the information exchange between the system and this entangled partner hidden in the environment may be one of the physical origins of the time oscillations of the distance measures for quantum states of the (anti-)PT-symmetric system. The aforementioned reasons motivate us to define non-Markovianity concept in the (anti-)PT-symmetric systems; however, as also described in Introduction, the dependence of this definition on the postselection process, appearing in all current experimental realizations of (anti-)PT-symmetric dynamics [49,91,[102][103][104][105][106]109], may call into question its validity. Demonstrating the failure of the BLP measure in defining possible non-Markovian effects in (anti-)PT-symmetric systems (see Refs. [105,118]) can support this reasoning. ...
Non-Hermitian systems with parity-time (PT) symmetry and anti-PT symmetry lead to exceptional points (EPs) with intriguing properties related to, e.g., chiral transport and enhanced sensitivity, due to the coalescence of eigenvectors. In this paper, we propose an easily computable tool, based on the Hilbert–Schmidt speed (HSS), not requiring the diagonalization of the evolved density matrix, to detect exactly the EPs, especially in high-dimensional systems. Our theoretical predictions, made without the need for modification of the Hilbert space, are completely consistent with results extracted from recent experiments studying the criticality in (anti-)PT-symmetric systems. Moreover, not modifying the Hilbert space of the non-Hermitian system, we find that the trace distance whose dynamics is known as a faithful witness of non-Markovianity, may be non-contractive under the non-Hermitian evolution of the system. Therefore, it losses one of the most important characteristics which must be met by any standard witness of non-Markovianity. We also address the non-contractivity of quantum Fisher information in non-Hermitian systems.
... In more detail, we see that the conventional metric used in Eq. (5) leads to results completely consistent with experimental observations [49,91,[102][103][104][105][106][107][108][109] provided that it is applied correctly. In fact, in order to physically implement the non-unitary evolution leading to Eq. (5), we can embed the (anti)PT−symmetric system into a larger Hermitian system, realized by adding ancillary qubits, and perform a measuring process [93,103]. ...
... Because of this post-selection occurring in the measuring process, the successful implementation of the non-unitary gate is a probabilistic procedure. This experimental limitation [49,102,103,105], is similar to the situation which occurs in Bell inequality tests [113][114][115]. Therefore, we can solve the paradoxes [7,101,116,117] associated with violation of no-go theorems in PT−symmetric theory [101] using normalized density matrix (5), without the need for any modification in the Hilbert space (for more details, see Refs. ...
... Moreover, the information exchange between the system and this entangled partner hidden in the environment may be one of the physical origins of the time oscillations of the distance measures for quantum states of the (anti-)PT−symmetric system. The aforementioned reasons motivates us to define non-Markovianity concept in the (anti-)PT−symmetric systems, however, as also described in the Introduction, the dependence of this definition on the postselection process, appearing in all current experimental realizations of (anti-)PT−symmetric dynamics [49,91,[102][103][104][105][106]109], may call into question its validity. Demonstrating the failure of the BLP measure in defining possible non-Markovian effects in (anti-)PT−symmetric systems (see Refs. [105,118]) can support this reasoning. ...
Non-Hermitian systems with parity-time () symmetry and anti- symmetry give rise to exceptional points (EPs) with intriguing properties related to, e.g., chiral transport and enhanced sensitivity, due to the coalescence of eigenvectors.
In this paper, we propose a powerful and easily computable tool, based on the Hilbert-Schmidt speed (HSS), which does not require the diagonalization of the evolved density matrix, to detect exactly the EPs and hence the critical behavior of the (anti-)symmetric systems, especially high-dimensional ones. Our theoretical predictions, made without the need for modification of the Hilbert space, which is performed by diagonalizing one of the observables, are completely consistent with results extracted from recent experiments studying the criticality in (anti-)symmetric systems. Nevertheless, not modifying the Hilbert space of the non-Hermitian system, we find that the trace distance, a measure of distinguishability of two arbitrary quantum states, whose dynamics is known as a faithful witness of non-Markovianity in Hermitian systems, may be non-contractive under the non-Hermitian evolution of the system. Therefore, it lacks one of the most important characteristics which must be met by any standard witness of non-Markovianity.
... Since EPs are branch points of Riemann manifolds that represent complex energies, they are responsible for the enhanced sensing and adiabatic mode-switch phenomena [25,26]. Their novel properties have intensified the efforts to engineer EP landscapes [27][28][29] and higher-order EPs in the classical and quantum domains [30][31][32][33][34] . ...
... When higher-dimensional representations of SU(2)-qutrits, qudits-are considered, results in Sec. II remain valid with higher-order EP contours [33,34]. They also remain qualitatively same for other Hermitian couplings K µν σ µ ⊗ σ ν between the unitary and thermal qubits. ...
Minimal, open quantum systems that are governed by non-Hermitian Hamiltonians have been realized across multiple platforms in the past two years. Here we investigate the dynamics of open systems with Hermitian or anti-Hermitian Hamiltonians, both of which can be implemented in such platforms. For a single system subject to unitary and thermal dynamics in a periodic manner, we show that the corresponding Floquet Hamiltonian has a rich phase diagram with numerous exceptional-point (EP) degeneracy contours. This protocol can be used to realize a quantum Hatano-Nelson model that is characterized by asymmetric tunneling. For one unitary and one thermal qubit, we show that the concurrence is maximized at the EP that is controlled by the strength of Hermitian coupling between them. Surprisingly, the entropy of each qubit is also maximized at the EP. Our results point to the multifarious phenomenology of systems undergoing unitary and thermal dynamics.
... Dissipative quantum systems have been widely studied in different contexts [1][2][3][4][5][6]. Among them, physical systems which can be described by non-hermitian Hamiltonians are particularly important. ...
... We see that F (ρ, H 2 ) is almost vanishing for W 3, while it starts to increase and reaches nonzero values for W 3, thus exhibiting a peak around the interval 4 W 6. In addition, F (ρ, H 2 ) decreases and rapidly approaches zero for W 6. This means that, for 4 W 6, the excited state become coherent into the reference eigenbasis, while H 1 and H 2 stand as commuting operators for W 3 and W 6. The solid red bars depict the average standard deviation of F (ρ, H 2 ) over the disorder realizations, thus showing strong fluctuations around the interval 4 W 6, whereas such fluctuations are strongly suppressed for W 3 and W 6. In Figs. 8(b) and 8(c) we set the system sizes N = 50 and N = 100, respectively, and plot the quantity F (ρ, H 2 ) averaged over 3000 realizations, for W ∈ [4,6]. Overall, the larger the system, the higher the fluctuations on F (ρ, H 2 ), which in turn also exhibits higher amplitudes. ...
Understanding the interplay between quantum coherence and non-Hermitian features would enable the devising of new quantum technologies based on dissipative systems. In turn, quantum coherence can be characterized in terms of the language of multiple quantum coherences (MQCs) originally developed in solid-state nuclear magnetic resonance (NMR), nowadays applied to the detection of quantum chaos, and to the study of criticality in many-body quantum systems. Here we show the usefulness of MQCs for probing equilibrium phase transitions in non-Hermitian systems. To do so, we investigate the connection of quantum coherences and critical points for several paradigmatic non-Hermitian Hamiltonians. For a non-Hermitian two-level system, MQCs witness the parity-symmetry breaking phase transition from the unbroken to the broken phase. Furthermore, for the non-Hermitian transverse field Ising model, MQCs capture the Yang-Lee phase transition in which the ground state energy acquires a nonzero imaginary component. For the disordered Hatano-Nelson model with periodic boundary conditions, MQCs testify the localization of mobility edges in the spectrum of this model. In addition, MQCs signal the topological phase transition exhibited by the complex-energy spectra of the disorder-free Hatano-Nelson model. Finally, we comment on how experimentally probing phase transitions in NMR systems realizing non-Hermitian Hamiltonians. Our results have applications to non-Hermitian quantum sensing, quantum thermodynamics, and in the study of the non-Hermitian skin effect.
... We can determine the value of ⟨U⟩, ⟨V⟩, ⟨U † ⟩ and ⟨U † V⟩ for an input state from the average output photon numbers. As shown in Fig. 1, the experimental setup involves three stages: the specific state preparation, state evolution under either of U, V, U † and U † V, interference-based measurement on the input state and the evolved state [57][58][59][60]. A pair of photons is generated via the spontaneous parametric down conversion in the periodically poled potassium titanyl phosphate crystal (PPKTP), with one serving as a trigger and a signal photon filtered out by an interference filter which restricts the bandwidth of photons to 3nm. ...
... Some wave plates are positioned in order to keep the path-length difference less than the coherence length. We vary the unitary operators pairwise in steps such that all the terms of the left-and right-hand sides of the inequality (3) is sampled and at each setting, where the state of the photons passing each arm is either of |ψ θ ⟩ or |ψ A ⟩. To read out the values of ⟨A⟩, we need to measure the inner product of |ψ θ ⟩ and |ψ A ⟩ [57][58][59][60]. We then apply projective measurements in the basis {|0⟩, |1⟩, ...
Uncertainty relations are one of the most important foundations of quantum physics. In the textbook literatures, uncertainty relations usually refer to the preparation uncertainty. Its original formulation based on variances of two observables limits on the ability to prepare an ensemble of quantum systems for which non-commuting observables will have arbitrary uncertainty. The preparation uncertainty relation has been widely investigated. On the other hand, a unitary operator is a fundamental tenet of quantum theory. Every evolution of a closed quantum system is governed by acting unitary operators on the state of the system and the evolution of an open system can be represented by acting unitary operators on an enlarged system consisting of the quantum system as a subsystem. Therefore, naturally, to understand and quantitatively capture the essence of uncertainty relations for unitary operators is important and timely. Here we report an experimental investigation of a set of uncertainty relations for two unitary operators, which are theoretically derived by using a sequence of fine-grained inequalities. We test these uncertainty relations with single photons and interferometric networks. The unitary uncertainty relation is saturated by any pure qubit state. For higher-dimensional states, it is stronger than the best known bound introduced in the previous literatures. The lower bounds of the unitary uncertainty relations can be even further strengthened by the symmetry of permutation. The experimental findings agree with the predictions of quantum theory and respect the new uncertainty relations.
... We specifically examine asymmetric hopping to build a non-Hermitian form of the Hubbard model [179]. Considering the low-energy effective theory, one can express the effective Hamiltonian describing the low-energy Dirac fermions in the non-Hermitian model on the honeycomb lattice as follows [180]: and sublattice spaces, respectively. Moreover, denotes the Fermi velocity in the absence of non-Hermitian hopping, namely . ...
In our study, we explore high-order exceptional points (EPs), which are crucial for enhancing the sensitivity of open physical systems to external changes. We utilize the Hilbert–Schmidt speed (HSS), a measure of quantum statistical speed, to accurately identify EPs in non-Hermitian systems. These points are characterized by the simultaneous coalescence of eigenvalues and their associated eigenstates. One of the main benefits of using HSS is that it eliminates the need to diagonalize the evolved density matrix, simplifying the identification process. Our method is shown to be effective even in complex, multi-dimensional and interacting Hamiltonian systems. In certain cases, a generalized evolved state may be employed over the conventional normalized state. This necessitates the use of a metric operator to define the inner product between states, thereby introducing additional complexity. Our research confirms that HSS is a reliable and practical tool for detecting EPs, even in these demanding situations.
... 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.
... In recent years, a third model of quantum dynamics obtained by postselecting on quantum trajectories with no quantum jumps has emerged [24,25]. With a non-Hermitian generator H eff (t ) = H (t ) + i (t ) and a nonlinear, tracepreserving equation of motion [26], it maps pure states into pure states but changes the entropy of mixed states [27], thereby commingling salient features of unitary and Lindblad evolution. These non-Hermitian systems occur by considering a subspace of the larger dissipative system which is governed by Lindblad evolution. ...
Thermodynamics constrains changes to the energy of a system, both deliberate and random, via its first and second laws. When the system is not in equilibrium, fluctuation theorems such as the Jarzynski equality further restrict the distributions of deliberate work done. Such fluctuation theorems have been experimentally verified in small, nonequilibrium quantum systems undergoing unitary or decohering dynamics. Yet, their validity in systems governed by a non-Hermitian Hamiltonian has long been contentious due to the false premise of the Hamiltonian's dual and equivalent roles in dynamics and energetics. Here we show that work fluctuations in a non-Hermitian qubit obey the Jarzynski equality even if its Hamiltonian has complex or purely imaginary eigenvalues. With postselection on a dissipative superconducting circuit undergoing a cyclic parameter sweep, we experimentally quantify the work distribution using projective energy measurements and show that the fate of the Jarzynski equality is determined by the parity-time symmetry of, and the energetics that result from, the corresponding non-Hermitian, Floquet Hamiltonian. By distinguishing the energetics from non-Hermitian dynamics, our results provide the recipe for investigating the nonequilibrium quantum thermodynamics of such open systems.
Published by the American Physical Society 2024
... Besides EP2s, non-Hermitian systems can also host higher-order EPs, where more than two eigenmodes coalesce into one [55][56][57][58][59][60][61][62][63]. Very recently, the fourth-order EP was demonstrated using optical elements [64]. It has been shown that higher-order EPs can exhibit greater advantages than EP2s in sensitive detection [65][66][67][68], topological characteristics [60,69,70], and spontaneous emission enhancement [61]. ...
Higher-order exceptional points (EPs), resulting from non-Hermitian degeneracies, have shown greater advantages in sensitive enhancement than second-order EPs (EP2s). Therefore, seeking higher-order EPs in various quantum systems is important for quantum information science. Here we propose a benchmark cavity optomechanical (COM) system consisting of a mechanical resonator coupled to two cavities via radiation pressure for predicting the third-order exceptional point (EP3). We first give the pseudo-Hermitian condition for the non-Hermitian COM system by taking the bath effects into account. Then we consider the mechanical gain effect, and we find that the pseudo-Hermitian COM system without PT symmetry can host both EP3 and EP2 for symmetric and asymmetric cavities. In the symmetric case, only EP3 or EP2 can be predicted in the parameter space, but EP3 and EP2 can be transformed into each other by tuning the optomechanical coupling strength in the asymmetric case. We further consider the case of one cavity with gain. For this case, the pseudo-Hermitian COM system is PT symmetric and can also host EP3 or EP2. The influence of system parameters on them is discussed. Our proposal provides a potential way to realize sensitive detection and study other physical phenomena around higher-order EPs in non-Hermitian COM systems.
... Instead of EP2s, non-Hermitian systems can also host higher-order EPs, where more than two eigenmodes coalesce into one [63][64][65][66][67][68][69][70][71]. It has been shown that higher-order EPs can exhibit greater advantages than EP2s in spontaneous emission enhancement [68], sensitive detection [72][73][74][75], and topological characteristics [76][77][78]. ...
Higher-order exceptional points (EPs) in non-Hermitian systems have attracted great interest due to their advantages in sensitive enhancement and distinct topological features. However, realization of such EPs is still a challenge because more fine-tuning parameters are generically required in quantum systems, compared to the second-order EP (EP2). Here, we propose a non-Hermitian three-mode optomechanical system in the blue-sideband regime for predicting the third-order EP (EP3). By deriving the pseudo-Hermitian condition for the proposed system, one cavity with loss and the other with gain must be required. Then we show that EP3 or EP2 can be observed when the mechanical resonator (MR) is neutral, loss, or gain. For the neutral MR, we find that two degenerate or two nondegenerate EP3s can be predicted by tuning system parameters in the parameter space, while four nondegenerate EP2s can be observed when the system parameters deviate from EP3s, which is distinguished from the previous study in the red-detuned optomechanical system. For the gain (loss) MR, we find that only two degenerate EP3s or EP2s can be predicted by tuning enhanced coupling strength. Our proposal provides a potential way to predict higher-order EPs or multiple EP2s and study multimode quantum squeezing around EPs using blue-detuned non-Hermitian optomechanical systems.
... On the other side, later works highlighted the efficient role of non-Hermiticity in the enhancement and protection of entanglement and entropy which is mainly aroused by the local non-Hermitian operations (Chen et al. 2014;Zang et al. 2017;Wang and Fang 2018;Li and Ye 2021). Beyond the entanglement, authors (Roccati et al. 2021) have studied the dynamics of quantum correlations in a PT -symmetric gain-loss system which exhibits stationary quantum correlations and pointed out the usefulness of such stationary quantum correlations in remote-state preparation (Dakic et al. 2012), information encoding (Gu et al. 2012), quantum metrology and sensing (Girolami et al. 2014), etc. Experimental investigations were performed for the quantum information dynamics under non-unitary PT -symmetric evolution (Xiao et al. 2019;Bian et al. 2020). However, the dynamical behaviours of quantum mechanical properties under the non-Hermitian system are addressed, and further investigations on the non-locality under non-Hermitian operations are necessary to understand its physical mechanisms. ...
In this article, we investigate the non-unitary dynamics of a two-qubit system under the action of local non-Hermitian operation from the perspectives of quantum correlations. The correlations are quantified by entanglement, maximal Bell function and different versions of measurement-induced nonlocality (MIN). The analysis is carried out for different initial conditions. We show the advantages of MIN in capturing the nonlocal aspects of the mixed quantum states over the entanglement and Bell function. Interestingly, by proper tuning of the system’s parameter, one can observe the stationary quantum correlations under this non-Hermitian circumstance which have potential applications in quantum information processing and quantum computation.
... We also check the efficiency of the criticality witness W in a high dimensional non-Hermitian system described by a 4 × 4 Hamiltonian [55] H ...
Non-Hermitian systems with exceptional points lead to many intriguing phenomena due to the coalescence of both eigenvalues and corresponding eigenvectors, in comparison to Hermitian systems where only eigenvalues degenerate. In this paper, we propose an alternative proposal based on the entropy uncertainty relation (EUR) to detect the exceptional points and identify different phases of the non-Hermitian systems. We first investigate the EUR in a non-Hermitian system and then reveal a general connection between the EUR and the exceptional points of non-Hermitian system. Compared to the unitary Hermitian dynamics, the behaviors of EUR in the non-Hermitian system are well defined in two different ways depending on whether the system is located in unbroken or broken phase regimes. In unbroken phase regime where EUR undergoes an oscillatory behavior, while in broken phase regime where the oscillation of EUR breaks down. In the dynamical limit, we identify the critical phenomena of non-Hermitian systems in terms of the EUR. It is found that the EUR can exactly detect the critical points of non-Hermitian systems beyond PT-(anti-PT) symmetric systems. Finally, we comment on the possible experimental situation.
... Theoretically, in PT -symmetric systems, Ref. [21] investigated the delayed sudden vanishing of entanglement at exceptional points, [22,23] studied entanglement sudden vanishing, and [24] realized effective entanglement recovery via operators. Entanglement, precision metrology, and sensing enhancement were reported in PT -symmetric systems [25][26][27][28][29][30][31]. Recent experiments have demonstrated topo-indicate the presence of exceptional points and behavior in APT -symmetric systems, especially in high-dimensional qudits [41]. ...
In the past years, many efforts have been made to study various noteworthy phenomena in both parity-time () and anti-parity-time () symmetric systems. However, entanglement dynamics in -symmetric systems has not previously been investigated in both theory and experiments. Here, we investigate the entanglement evolution of two qubits in an -symmetric system. In the -symmetric unbroken regime, our theoretical simulations demonstrate the periodic oscillations of entanglement when each qubit evolves identically, while the nonperiodic oscillations of entanglement when each qubit evolves differently. In particular, when each qubit evolves near the exceptional point in the -symmetric unbroken regime, there exist entanglement sudden vanishing and revival. Moreover, our simulations demonstrate rapid decay and delayed death of entanglement provided one qubit evolves in the -symmetric broken regime. In this work, we also perform an experiment with a linear optical setup. The experimental results agree well with our theoretical simulation results. Our findings reveal novel phenomena of entanglement evolution in the -symmetric system and opens a new direction for future studies on the dynamics of quantum entanglement in multiqubit -symmetric systems or other non-Hermitian quantum systems.
... Theoretically, in PT -symmetric systems, Ref. [21] investigated the delayed sudden vanishing of entanglement at exceptional points, [22,23] studied entanglement sudden vanishing, and [24] realized effective entanglement recovery via operators. Entanglement, precision metrology, and sensing enhancement were reported in PT -symmetric systems [25][26][27][28][29][30][31]. Recent experiments have demonstrated topological edge states based on entanglement in PT -symmetric quantum walks [32], stable states with nonzero entropy under broken PT symmetry [33], and optomechanical dynamics under the PT -and broken-PT -symmetric regimes [34]. ...
In the past years, many efforts have been made to study various noteworthy phenomena in both parity-time (PT) and anti-parity-time (APT) symmetric systems. However, entanglement dynamics in APT-symmetric systems has not previously been investigated in both theory and experiments. Here, we investigate the entanglement evolution of two qubits in an APT-symmetric system. In the APT-symmetric unbroken regime, our theoretical simulations demonstrate the periodic oscillations of entanglement when each qubit evolves identically, while the nonperiodic oscillations of entanglement when each qubit evolves differently. In particular, when each qubit evolves near the exceptional point in the APT-symmetric unbroken regime, there exist entanglement sudden vanishing and revival. Moreover, our simulations demonstrate rapid decay and delayed death of entanglement provided one qubit evolves in the APT-symmetric broken regime. In this work, we also perform an experiment with a linear optical setup. The experimental results agree well with our theoretical simulation results. Our findings reveal novel phenomena of entanglement evolution in the APT-symmetric system and opens a new direction for future studies on the dynamics of quantum entanglement in multiqubit APT-symmetric systems or other non-Hermitian quantum systems.
... 虽然损耗通常被认为会对系统的相干性产生 损坏 [44,45] , 但是满足宇称-时间对称性的经典系统 中的独特的现象和有用的应用说明了平衡增益和 损耗的作用. 从经典系统的研究成果来看, 满足宇 称-时间对称性的量子系统有望显示出对退相干的 鲁棒性, 可能会导致量子信息处理中的长相干时 间. 近年来, 满足宇称-时间对称性的系统的信息论 表征开始被探索 [40][41][42]46] . 除了实际的重要性之外, 这种信息论表征对于更深入地理解宇称-时间对称 系统可以作为开放量子系统是必备的. ...
Recently, impressive progress has been made in the study of non-Hermitian systems with parity-time symmetry, such as observations of topological properties of physical systems and criticality at exceptional points. A crucial aspect of parity-time symmetric nonunitary dynamics is the information flow between the system and the environment. In this paper, we use the physical quantity, distinguishability between quantum states, to uniformly quantify the information flow between low-dimensional and high-dimensional parity-time symmetric non-Hermitian systems and environments. The numerical results show that the oscillation of quantum state distinguishability and complete information retrieval and can be obtained in the parity-time-unbroken phase. However, the information decays exponentially in the parity-time-broken phase. The exceptional point marks the criticality between reversibility and irreversibility of information flow, and the distinguishability between quantum states exhibits the behavior of power-law decay. Understanding these unique phenomena in nonunitary quantum dynamics provides an important perspective for the study of open quantum systems and contributes to their application in quantum information.
... This unusual behaviour, in particular in the adiabatic regime, has been highlighted by recent theoretical [28][29][30][31][32] and experimental [33][34][35][36][37] work associated with encircling exceptional points. Specifically higher order exceptional points have been explored in PT-symmetric systems in [38][39][40][41][42] and in non PT-symmetric systems [43][44][45]. ...
Non-Hermitian quantum systems with explicit time dependence are of ever-increasing importance. There are only a handful of models that have been analytically studied in this context. Here, a PT-symmetric non-Hermitian N-level Landau-Zener type problem with two exceptional points of Nth order is introduced. The system is Hermitian far away from the exceptional points, and has purely imaginary eigenvalues between the exceptional points. The full Landau-Zener transition probabilities are derived, and found to show a characteristic binomial behaviour. In the adiabatic limit the final populations are given by the ratios of binomial coefficients. It is demonstrated how this behaviour can be understood on the basis of adiabatic analysis, despite the breakdown of adiabaticity that is often associated with non-Hermitian systems.
... They also pointed out the usefulness of such stationary quantum correlations in information encoding [31], remote-state preparation [32], quantum metrology and sensing [33], etc. To an extent, the authors of [34] experimentally demonstrated the quantum information dynamics through entropy in a high dimensional PT -symmetric system with its exceptional point degeneracies. ...
In this work, we explore the different measures of quantum correlations and quantum teleportation in the Heisenberg XY model for two different cases, namely without PT-symmetric operation and with PT-symmetric operation. Initially, we inspect the quantum correlation measures of thermally entangled states without PT-symmetric operation. Among the different measures, violation of Bell's inequality and concurrence failed to manifest the non-local correlations of the entangled states. Interestingly, the entangled states which show no correlation in terms of Bell's inequality and concurrence are also useful for the teleportation process. Secondly, we incorporate the role of non-unitary PT-symmetric time evolution to the model and study the measures which show the considered states have correlations. The results reveal that the considered model can act as an efficient resource for teleportation with maximum fidelity under non-unitary time evolution.
... In the lower path, the photons directly travel to the measurement device. For a linear optical system, any 2 × 2 unitary measurement operators can be constructed via certain HWPs and QWPs [42][43][44]. In our measurement setup, all of the mutually unbiased bases can be realized by a quarter-wave plate (QWP), a HWP and a polarization beam splitter (PBS). ...
Quantum entanglement is one of the essential resources in quantum information processing. It is of importance to verify whether a quantum state is entangled. At present, a typical quantum certification focused on the classical correlations has attracted widespread attention. Here, we experimentally investigate the relation between quantum entanglement and the classical complementary correlations based on the mutual information, Pearson correlation coefficient and mutual predictability of two-qubit states. Our experimental results show the classical correlations for complementary properties have strong resolution capability to verify entanglement for two qubit pure states and Werner states. We find that the resolution capability has great performance improvement when the eigenstates of the measurement observables constitute a complete set of mutually unbiased bases. For Werner states in particular, the classical complementary correlations based on the Pearson correlation coefficient and mutual predictability can provide the ultimate bounds to certify entanglement.
... In order to achieve the maximal violation of CFFW inequality, we scan the whole permitted measurement strategies of Alice and Bob since the observables are unknown but keeping B 1 and B 2 mutually unbiased. In the experiment, we only need to add certain QWPs in the original measurement setup, namely, a combination of sandwich type QWP-HWP-QWP sequence [36][37][38], and a following PBS to realize the measurement operators. The coincidence counting of the APDs {D1, D3} represent the result a = b = 0, simultaneously. ...
Quantum steering, as a cornerstone of quantum information, is usually used to witness the quantum correlation of bipartite and multi-partite states. Here, we experimentally demonstrate the quantum steering inequality of two-qubit mixed states based on the fine-grained uncertainty relation. Our experimental results show that the steering inequality has potent sensitivity to Werner states and Bell diagonal states. The steering strategy exhibits a strong ability to identify that Werner states are steerable when the decoherence coefficient a>12. Compared to the steering inequality obtained by another stratagem, the steering witness criteria of mixed states based on the fine-grained uncertainty relation demonstrated in our experiment has better precision and accuracy. Moreover, the detection efficiency in our measurement setup is only required to be 50% to close the detection loophole, which means our approach needs less detector efficiency to certificate the steerability of mixed states.
... Besides EP2s, non-Hermitian systems can also host higher-order EPs, where more than two eigenmodes coalesce into one [55][56][57][58][59][60][61][62][63]. Very recently, the fourth-order EP was demonstrated using optical elements [64]. It has been shown that higher-order EPs can exhibit greater advantages than EP2s in sensitive detection [65][66][67][68], topological characteristics [60,69,70], and spontaneous emission enhancement [61]. ...
Higher-order exceptional points (EPs), resulting from non-Hermitian degeneracies, have shown greater advantages in sensitive enhancement than second-order EPs (EP2s). Therefore, seeking higher-order EPs in various quantum systems is important for quantum information science. Here we propose a benchmark cavity optomechanical (COM) system consisting of a mechanical resonator (MR) coupled to two cavities via radiation pressure for predicting the third-order exceptional point (EP3). We first give the pseudo-Hermitian condition for the non-Hermitian COM system by taking the bath effects into account. Then we consider the mechanical gain effect and we find that the pseudo-Hermitian COM system without PT-symmetry can host both the EP3 and EP2 for symmetric and asymmetric cavities. In the symmetric case, only the EP3 or EP2 can be predicted in the parameter space, but the EP3 and EP2 can be transformed into each other by tuning the COM coupling strength in the asymmetric case. We further consider the case of one cavity with gain. For this case, the pseudo-Hermitian COM system is PT-symmetric and can also host the EP3 or EP2. The influence of system parameters on them are discussed. Our proposal provides a potential way to realize sensitive detection and study other physical phenomena around higher-order EP3 in non-Hermitian COM systems.
... In our experiment, qubits are realized on the one hand by the polarization states of single photons (|0 p = |H and |1 p = |V , where H and V stand for horizontal and vertical polarization, respectively), and on the other hand by their spatial modes (|0 s = |D and |1 s = |U , where U denotes the upper path, D denotes the lower path) [23][24][25]. Single photons are generated via type-I spontaneous parametric down-conversion (SPDC) using a 0.5 mm-thick-β-barium-borate (BBO) crystal, pumped by a CW diode laser with a central wavelength 400.8 nm and 80 mW of power. An interference filter is used to restrict the single-photon bandwidth to 3 nm. ...
Iterated quantum protocols with measurement-based selection lead to deterministic chaos for the evolving pure state representing an ensemble of qubits. Deterministic chaos for the pure quantum state may lead to ergodic evolution in the sense that initial states from any small area on the Bloch sphere will cover the whole sphere after a finite number of iterations. We realize two steps of an ergodic protocol in a photonic experiment, where initial qubit states are encoded in the polarization and path degrees of freedom of down-converted photons stemming from a parametric process. We numerically analyze the effect of noise on the time evolution and show that the protocol, described by a Lattès map, remains quasi-ergodic for any initial state if the initial noise is small. Tomographic reconstruction of the quantum states throughout the evolution is consistent with simulations and thus demonstrates ergodicity of the quantum dynamics.
... The notion of memory or non-Markovian behavior arises across diverse platforms, both classical and quantum. In particular, some non-Markovian aspects of quantum information and its flow between the system and its environment have been studied in dissipative [55] or PT symmetric quantum models with a static Hamiltonian [56][57][58]. Our work, on the other hand, introduces memory into the effective non-Hermitian Hamiltonian and leads to nonlinear and sign-dependent effects that are absent in the aforementioned works. ...
Classical open systems with balanced gain and loss, i.e., parity-time (PT) symmetric systems, have attracted tremendous attention over the past decade. Their exotic properties arise from exceptional point degeneracies of non-Hermitian Hamiltonians that govern their dynamics. In recent years, increasingly sophisticated models of PT symmetric systems with time-periodic (Floquet) driving, time-periodic gain and loss, and time-delayed coupling have been investigated, and such systems have been realized across numerous platforms comprising optics, acoustics, mechanical oscillators, optomechanics, and electrical circuits. Here, we introduce a PT symmetric (balanced gain and loss) system with memory and investigate its dynamics analytically and numerically. Our model consists of two coupled LC oscillators with positive and negative resistance, respectively. We introduce memory by replacing either the resistor with a memristor, or the coupling inductor with a meminductor, and investigate the circuit energy dynamics as characterized by PT symmetric or PT symmetry broken phases. Due to the resulting nonlinearity, we find that energy dynamics depend on the sign and strength of initial voltages and currents, as well as the distribution of initial circuit energy across its different components. Surprisingly, at strong inputs, the system exhibits self-organized Floquet dynamics, including a PT symmetry broken phase at vanishingly small dissipation strength. Our results indicate that PT symmetric systems with memory show a rich landscape.
... In presence of a dimensionless, small perturbation δ 1, the mode-splitting that is generated at an EPn scales as δ 1/n δ, and thereby provides the enhanced sensitivity that scales with the order of the EP in its proximity. Second [10], third [11], and higher order EPs have been experimentally realized in (semi) classical systems such as optics [12], acoustics, and photonics [13,14], where sensitivity enhancement has been demonstrated [15,16]. These works have also highlighted the role of excess noise in the proximity of the EP [17,18] that arises from nonorthogonal eigenvectors of non-Hermitian matrices. ...
The dynamics of an isolated quantum system is coherent and unitary. Weak coupling to the environment leads to decoherence, which is traditionally modeled with a Lindblad equation for the system's density matrix. Starting from a pure state, such a system approaches a steady state (mixed or otherwise) in an underdamped or overdamped manner. This transition occurs at an eigenvalue degeneracy of a Lindblad superoperator, called an exceptional point (EP), where corresponding eigenvectors coalesce. Recent years have seen an explosion of interest in creating exceptional points in a truly quantum domain, driven by the enhanced sensitivity and topological features EPs have shown in their classical realizations. Here, we present Floquet analysis of a prototypical qubit whose drive or dissipator strengths are varied periodically. We consider models with a single dissipator that generate global loss (phase damping) or mode-selective loss (spontaneous emission). In all cases, we find that periodic modulations lead to EP lines at small dissipator strengths, and a rich EP structure in the parameter space. Our analytical and numerical results show that extending Lindblad Liouvillians to the Floquet domain is a new, potentially preferred route to accessing exceptional points in the transient dynamics towards the Lindblad steady state.
Over the past decade, classical optical systems with gain or loss, modeled by non-Hermitian parity-time symmetric Hamiltonians, have been deeply investigated. Yet, their applicability to the quantum domain with number-resolved photonic states is fundamentally voided by quantum-limited amplifier noise. Here, we show that second-quantized Hermitian Hamiltonians on the Fock space give rise to non-Hermitian effective Hamiltonians that generate the dynamics of corresponding creation and annihilation operators. Using this equivalence between P T-symmetry and symplectic Bogoliubov transformations, we create a quantum optical scheme comprising squeezing, phase-shifters, and beam-splitters for simulating arbitrary non-unitary processes by way of singular value decomposition. In contrast to the post-selection scheme for non-Hermitian quantum simulation, the success probability in this approach is independent of the system size or simulation time and can be efficiently Trotterised similar to a unitary transformation.
Parity-time (PT) symmetric quantum theory can broaden the scope of quantum dynamics beyond unitary evolution which may lead to numerous counter-intuitive phenomena, including single-shot discrimination of non-orthogonal states, faster evolution of state than the standard quantum speed limit, and violation of no-signaling principle. On the other hand, PT -symmetric evolution can be realized as reduced dynamics of a subsystem in real experiments within the scope of standard QT. In this experimental setup, if one side of a composite system is evolved according to a PT -symmetric way, a non-trivial information transfer can happen, i.e. the operation performed at one side can be gathered by the other side. By considering an arbitrary shared state between two parties situated in two distant locations and arbitrary measurements, we show that the PT -symmetric evolution of the reduced subsystem at one side is not sufficient for this information transfer to occur. Specifically, we prove that the information transfer can only happen when the density matrix and the corresponding measurements contain complex numbers. Moreover, we connect the entanglement content of the shared state with the efficacy of information transfer. We find evidence that the task becomes more efficient with the increase of dimension.
Non-Hermitian systems with parity-time [Formula: see text] symmetry and anti-parity-time [Formula: see text] symmetry have exceptional points (EPs) resulting from eigenvector co-coalescence with exceptional properties. In the quantum and classical domains, higher-order EPs for [Formula: see text] symmetry and [Formula: see text]-symmetry systems have been proposed and realized. Both two-qubits [Formula: see text]-[Formula: see text] and [Formula: see text]-[Formula: see text] symmetric systems have seen an increase in recent years, especially in the dynamics of quantum entanglement. However, to our knowledge, neither theoretical nor experimental investigations have been conducted for the dynamics of two-qubits entanglement in the [Formula: see text]-[Formula: see text] symmetric system. We investigate the [Formula: see text]-[Formula: see text] dynamics for the first time. Moreover, we examine the impact of different initial Bell-state conditions on entanglement dynamics in [Formula: see text]-[Formula: see text], [Formula: see text]-[Formula: see text] and [Formula: see text]-[Formula: see text] symmetric systems. Additionally, we conduct a comparative study of entanglement dynamics in the [Formula: see text]-[Formula: see text] symmetrical system, [Formula: see text]-[Formula: see text] symmetrical system, and [Formula: see text]-[Formula: see text] symmetrical systems in order to learn more about non-Hermitian quantum systems and their environments. Entangled qubits evolve in a [Formula: see text]-[Formula: see text] symmetric unbroken regime, the entanglement oscillates with two different oscillation frequencies, and the entanglement is well preserved for a long period of time for the case when non-Hermitian parts of both qubits are taken quite away from the exceptional points.
Non-Hermitian quantum systems with explicit time dependence are of ever increasing importance. There are only a handful of models that have been analytically studied in this context. Here, a PT-symmetric non-Hermitian N-level Landau-Zener type problem with two exceptional points of Nth order is introduced. The system is Hermitian for asymptotically large times, far away from the exceptional points, and has purely imaginary eigenvalues between the exceptional points. The full Landau-Zener transition probabilities are derived, and found to show a characteristic binomial behavior. In the adiabatic limit the final populations are given by the ratios of binomial coefficients. It is demonstrated how this behavior can be understood on the basis of adiabatic analysis, despite the breakdown of adiabaticity that is often associated with non-Hermitian systems.
We experimentally demonstrate a method for detection of entanglement via construction of entanglement witnesses from a limited fixed set of local measurements ( M ). Such a method does not require a priori knowledge about the form of the entanglement witnesses. It is suitable for a scenario where a full state tomography is not available, but the only resource is a limited set of M . We demonstrate the method on pure two-qubit entangled states and mixed two-qubit entangled states, which emerge from photonic implementation of controllable quantum noisy channels. The states we select are motivated by realistic experimental conditions, and we confirm it works well for both cases. Furthermore, possible generalizations to higher-dimensional bipartite systems have been considered, which can potentially detect both decomposable and indecomposable entanglement witnesses. Our experimental results show perfect validity of the method, which indicates that even a limited set of local measurements can be used for quick entanglement detection and further provide a practical test bed for experiments with entanglement witnesses.
Recently, impressive progress has been made in the study of non-Hermitian systems with parity-time symmetry, such as observations of topological properties of physical systems and criticality at exceptional points. A crucial aspect of parity-time symmetric nonunitary dynamics is the information flow between the system and the environment. In this paper, we use the physical quantity, distinguishability between quantum states, to uniformly quantify the information flow between low-dimensional and high-dimensional parity-time symmetric non-Hermitian systems and environments. The numerical results show that the oscillation of quantum state distinguishability and complete information retrieval and can be obtained in the parity-time-unbroken phase. However, the information decays exponentially in the paritytime-broken phase. The exceptional point marks the criticality between reversibility and irreversibility of information flow, and the distinguishability between quantum states exhibits the behavior of power-law decay. Understanding these unique phenomena in nonunitary quantum dynamics provides an important perspective for the study of open quantum systems and contributes to their application in quantum information.
Minimal, open quantum systems that are governed by non-Hermitian Hamiltonians have been realized across multiple platforms in the past two years. Here we investigate the dynamics of open systems with Hermitian or anti-Hermitian Hamiltonians, both of which can be implemented in such platforms. For a single system, the subject to unitary and thermal dynamics in a periodic manner, we show that the corresponding Floquet Hamiltonian has a rich phase diagram with numerous exceptional-point (EP) degeneracy contours. This protocol can be used to realize a quantum Hatano-Nelson model that is characterized by asymmetric tunneling. For one unitary and one thermal qubit, we show that the concurrence is maximized at the EP that is controlled by the strength of Hermitian coupling between them. Surprisingly, the entropy of each qubit is also maximized at the EP. Our results point to the multifarious phenomenology of systems undergoing unitary and thermal dynamics.
Understanding the interplay between quantum coherence and non-Hermitian features would enable the devising of quantum technologies based on dissipative systems. In turn, quantum coherence can be characterized in terms of the language of multiple quantum coherences (MQCs) originally developed in solid-state nuclear magnetic resonance (NMR), nowadays applied to the detection of quantum chaos and to the study of criticality in many-body quantum systems. Here, we show the usefulness of MQCs for probing equilibrium phase transitions in non-Hermitian systems. To do so, we investigate the connection of quantum coherences and critical points for several paradigmatic non-Hermitian Hamiltonians. For a non-Hermitian two-level system, MQCs witness the parity-symmetry-breaking phase transition from the unbroken to the broken phase. Furthermore, for the non-Hermitian transverse field Ising model, MQCs capture the Yang-Lee phase transition in which the ground state energy acquires a nonzero imaginary component. For the disordered Hatano-Nelson (HN) model with periodic boundary conditions, MQCs testify the emergence of mobility edges in the spectrum of this model. In addition, MQCs signal the topological phase transition exhibited by the complex energy spectra of the disorder-free HN model. Finally, we comment on experimentally probing phase transitions in NMR systems, realizing non-Hermitian Hamiltonians. Our results have applications to non-Hermitian quantum sensing, quantum thermodynamics, and in the study of the non-Hermitian skin effect.
The dynamics of an isolated quantum system is coherent and unitary. Weak coupling to the environment leads to decoherence, which is traditionally modeled with a Lindblad equation for the system's density matrix. Starting from a pure state, such a system approaches a steady state (mixed or otherwise) in an underdamped or overdamped manner. This transition occurs at an eigenvalue degeneracy of a Lindblad superoperator, called an exceptional point (EP), where corresponding eigenvectors coalesce. Recent years have seen an explosion of interest in creating exceptional points in a truly quantum domain, driven by the enhanced sensitivity and topological features EPs have shown in their classical realizations. Here, we present Floquet analysis of a prototypical qubit whose drive or dissipator strengths are varied periodically. We consider models with a single dissipator that generate global loss (phase damping) or mode-selective loss (spontaneous emission). In all cases, we find that periodic modulations lead to EP lines at small dissipator strengths and a rich EP structure in the parameter space. Our analytical and numerical results show that extending Lindblad Liouvillians to the Floquet domain is a potentially preferred route to accessing exceptional points in the transient dynamics towards the Lindblad steady state.
Conserved quantities such as energy or the electric charge of a closed system, or the Runge-Lenz vector in Kepler dynamics, are determined by its global, local, or accidental symmetries. They were instrumental in advances such as the prediction of neutrinos in the (inverse) beta decay process and the development of self-consistent approximate methods for isolated or thermal many-body systems. In contrast, little is known about conservation laws and their consequences in open systems. Recently, a special class of these systems, called parity-time (PT) symmetric systems, has been intensely explored for their remarkable properties that are absent in their closed counterparts. A complete characterization and observation of conserved quantities in these systems and their consequences is still lacking. Here, we present a complete set of conserved observables for a broad class of PT-symmetric Hamiltonians and experimentally demonstrate their properties using a single-photon linear optical circuit. By simulating the dynamics of a four-site system across a fourth-order exceptional point, we measure its four conserved quantities and demonstrate their consequences. Our results spell out nonlocal conservation laws in nonunitary dynamics and provide key elements that will underpin the self-consistent analyses of non-Hermitian quantum many-body systems that are forthcoming.
Bulk–boundary correspondence, a guiding principle in topological matter, relates robust edge states to bulk topological invariants. Its validity, however, has so far been established only in closed systems. Recent theoretical studies indicate that this principle requires fundamental revisions for a wide range of open systems with effective non-Hermitian Hamiltonians. Therein, the intriguing localization of nominal bulk states at boundaries, known as the non-Hermitian skin effect, suggests a non-Bloch band theory in which non-Bloch topological invariants are defined in generalized Brillouin zones, leading to a general bulk–boundary correspondence beyond the conventional framework. Here, we experimentally observe this fundamental non-Hermitian bulk–boundary correspondence in discrete-time non-unitary quantum-walk dynamics of single photons. We demonstrate pronounced photon localizations near boundaries even in the absence of topological edge states, thus confirming the non-Hermitian skin effect. Facilitated by our experimental scheme of edge-state reconstruction, we directly measure topological edge states, which are in excellent agreement with the non-Bloch topological invariants. Our work unequivocally establishes the non-Hermitian bulk–boundary correspondence as a general principle underlying non-Hermitian topological systems and paves the way for a complete understanding of topological matter in open systems. Measurements of non-Hermitian photon dynamics show boundary-localized bulk eigenstates given by the non-Hermitian skin effect. A fundamental revision of the bulk–boundary correspondence in open systems is required to understand the underlying physics.
Open physical systems can be described by effective non-Hermitian Hamiltonians that characterize the gain or loss of energy or particle numbers from the system. Experimental realization of optical1–7 and mechanical8–13 non-Hermitian systems has been reported, demonstrating functionalities such as lasing14–16, topological features7,17–19, optimal energy transfer20,21 and enhanced sensing22,23. Such realizations have been limited to classical (wave) systems in which only the amplitude information, not the phase, is measured. Thus, the effects of a systems’s proximity to an exceptional point—a degeneracy of such non-Hermitian Hamiltonians where the eigenvalues and corresponding eigenmodes coalesce24–29—on its quantum evolution remain unexplored. Here, we use post-selection on a three-level superconducting transmon circuit to carry out quantum state tomography of a single dissipative qubit in the vicinity of its exceptional point. We observe the spacetime reflection symmetry-breaking transition30,31 at zero detuning, decoherence enhancement at finite detuning and a quantum signature of the exceptional point in the qubit relaxation state. Our experiments show phenomena associated with non-Hermitian physics such as non-orthogonality of eigenstates in a fully quantum regime, which could provide a route to the exploration and harnessing of exceptional point degeneracies for quantum information processing.
A common wisdom in quantum mechanics is that the Hamiltonian has to be Hermitian in order to ensure a real eigenvalue spectrum. Yet, parity–time (PT)-symmetric Hamiltonians are sufficient for real eigenvalues and therefore constitute a complex extension of quantum mechanics beyond the constraints of Hermiticity. However, as only single-particle or classical wave physics has been exploited so far, an experimental demonstration of the true quantum nature of PT symmetry has been elusive. In our work, we demonstrate two-particle quantum interference in a PT-symmetric system. We employ integrated photonic waveguides to reveal that the quantum dynamics of indistinguishable photons shows strongly counterintuitive features. To substantiate our experimental data, we analytically solve the quantum master equation using Lie algebra methods. The ideas and results presented here pave the way for non-local PT-symmetric quantum mechanics as a novel building block for future quantum devices.
Exceptional points (EPs) are degeneracies of non-Hermitian operators where, in addition to the eigenvalues, the corresponding eigenmodes become degenerate. Classical and quantum photonic systems with EPs have attracted tremendous attention due to their unusual properties, topological features, and an enhanced sensitivity that depends on the order of the EP, i.e., the number of degenerate eigenmodes. Yet, experimentally engineering higher-order EPs in classical or quantum domains remain an open challenge due to the stringent symmetry constraints that are required for the coalescence of multiple eigenmodes. Here, we analytically show that the number-resolved dynamics of a single, lossy waveguide beam splitter, excited by N indistinguishable photons and post-selected to the N-photon subspace, will exhibit an EP of order N+1. By using the well-established mapping between a beam splitter Hamiltonian and the perfect state transfer model in the photon-number space, we analytically obtain the time evolution of a general N-photon state and numerically simulate the system’s evolution in the post-selected manifold. Our results pave the way toward realizing robust, arbitrary-order EPs on demand in a single device.
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.
Physical systems in the time domain may exhibit analogous phenomena in real space, such as time crystals, time-domain Fresnel lenses, and modulational interference in a qubit. Here, we report the experimental realization of time-domain grating using a superconducting qutrit in periodically modulated probe and control fields via two schemes: simultaneous modulation and complementary modulation. Both experimental and numerical results exhibit modulated Autler-Townes (AT) and modulation-induced diffraction (MID) effects. Theoretical results also confirm that the peak positions of the interference fringes of AT and MID effects are determined by the usual two-level relative phases, while the observed diffraction fringes, appearing only in the complementary modulation, are, however, related to the three-level relative phase. Further analysis indicates that such a single-atom time-domain diffraction originates from the correlation effect between the two time-domain gratings. Moreover, we find that the widths of the diffraction fringes are independent of the control-field power. Our results shed light on the experimental exploration of quantum coherence for modulated multilevel systems and may find promising applications in fast all-microwave switches and quantum-gate operations in the strong-driving regime.
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.
We discuss the impact of gain and loss on the evolution of photonic quantum states and find that -symmetric quantum optics in gain/loss systems is not possible. Within the framework of macroscopic quantum electrodynamics we show that gain and loss are associated with non-compact and compact operator transformations, respectively. This implies a fundamentally different way in which quantum correlations between a quantum system and a reservoir are built up and destroyed.
Nearly one century after the birth of quantum mechanics, parity-time symmetry is revolutionizing and extending quantum theories to include a unique family of non-Hermitian Hamiltonians. While conceptually striking, experimental demonstration of parity-time symmetry remains unexplored in quantum electronic systems. The flexibility of photonics allows for creating and superposing non-Hermitian eigenstates with ease using optical gain and loss, which makes it an ideal platform to explore various non-Hermitian quantum symmetry paradigms for novel device functionalities. Such explorations that employ classical photonic platforms not only deepen our understanding of fundamental quantum physics but also facilitate technological breakthroughs for photonic applications. Research into non-Hermitian photonics therefore advances and benefits both fields simultaneously.
We report the experimental detection of bulk topological invariants in nonunitary discrete-time quantum walks with single photons. The nonunitarity of the quantum dynamics is enforced by periodically performing partial measurements on the polarization of the walker photon, which effectively introduces loss to the dynamics. The topological invariant of the nonunitary quantum walk is manifested in the quantized average displacement of the walker, which is probed by monitoring the photon loss. We confirm the topological properties of the system by observing localized edge states at the boundary of regions with different topological invariants. We further demonstrate the robustness of both the topological properties and the measurement scheme of the topological invariants against disorder.
We study mechanical cooling in systems of coupled passive (lossy) and active (with gain) optical resonators. We find that for a driving laser which is red-detuned with respect to the cavity frequency, the supermode structure of the system is radically changed, featuring the emergence of genuine high-order exceptional points. This in turn leads to giant enhancement of both the mechanical damping and the spring stiffness, facilitating low-power mechanical cooling in the vicinity of gain-loss balance. This opens up new avenues of steering micromechanical devices with exceptional points beyond the lowest-order two.
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.
Linear optics underpins fundamental tests of quantum mechanics and quantum technologies. We demonstrate a single reprogrammable optical circuit that is sufficient to implement all possible linear optical protocols up to the size of that circuit. Our six-mode universal system consists of a cascade of 15 Mach-Zehnder interferometers with 30 thermo-optic phase shifters integrated into a single photonic chip that is electrically and optically interfaced for arbitrary setting of all phase shifters, input of up to six photons and their measurement with a 12 single-photon detector system. We programmed this system to implement heralded quantum logic and entangling gates, boson sampling with verification tests, and six-dimensional complex Hadamards. We implemented 100 Haar random unitaries with average fidelity 0.999 ± 0.001. Our system can be rapidly reprogrammed to implement these and any other linear optical protocol, pointing the way to applications across fundamental science and quantum technologies.
Copyright © 2015, American Association for the Advancement of Science.
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.
We discuss methods of quantum state tomography for solid-state systems with a
large nuclear spin I=3/2 in nanometer-scale semiconductors devices based on a
quantum well. Due to quadrupolar interactions, the Zeeman levels of these
nuclear-spin devices become nonequidistant, forming a controllable four-level
quantum system (known as quartit or ququart). The occupation of these levels
can be selectively and coherently manipulated by multiphoton transitions using
the techniques of nuclear magnetic resonance (NMR) [Yusa et al., Nature
(London) 434, 101 (2005)]. These methods are based on an unconventional
approach to NMR, where the longitudinal magnetization is directly
measured. This is in contrast to the standard NMR experiments and tomographic
methods, where the transverse magnetization is detected. The
robustness against errors in the measured data is analyzed by using condition
numbers. We propose several methods with optimized sets of rotations. The
optimization is applied to decrease the number of NMR readouts and to improve
the robustness against errors, as quantified by condition numbers. An example
of state reconstruction, using Monte Carlo methods, is presented. Tomographic
methods for quadrupolar nuclei with higher-spin numbers (including I=7/2) are
also described.
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
Over the last two decades, advances in fabrication have led to significant
progress in creating patterned heterostructures that support either carriers,
such as electrons or holes, with specific band structure or electromagnetic
waves with a given mode structure and dispersion. In this article, we review
the properties of light in coupled optical waveguides that support specific
energy spectra, with or without the effects of disorder, that are
well-described by a Hermitian tight-binding model. We show that with a
judicious choice of the initial wave packet, this system displays the
characteristics of a quantum particle, including transverse photonic transport
and localization, and that of a classical particle. We extend the analysis to
non-Hermitian, parity and time-reversal () symmetric Hamiltonians
which physically represent waveguide arrays with spatially separated, balanced
absorption or amplification. We show that coupled waveguides are an ideal
candidate to simulate -symmetric Hamiltonians and the transition
from a purely real energy spectrum to a spectrum with complex conjugate
eigenvalues that occurs in them.
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.
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler-DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum. Comment: Revised version, to appear in J. Math. Phys
We study a non-Hermitian symmetric generalization of an N-particle, two-mode Bose-Hubbard system, modeling for example a Bose-Einstein condensate in a double well potential coupled to a continuum via a sink in one of the wells and a source in the other. The effect of the interplay between the particle interaction and the non-Hermiticity on characteristic features of the spectrum is analyzed drawing special attention to the occurrence and unfolding of exceptional points (EPs). We find that for vanishing particle interaction there are only two EPs of order N+1 which under perturbation unfold either into [(N+1)/2] eigenvalue pairs (and in case of N+1 odd, into an additional zero-eigenvalue) or into eigenvalue triplets (third-order eigenvalue rings) and single eigenvalues, depending on the direction of the perturbation in parameter space. This behavior is described analytically using perturbational techniques. More general EP unfoldings into eigenvalue rings up to (N+1)th order are indicated.
Exceptional points (EPs) are degeneracies of classical and quantum open systems, which are studied in many areas of physics including optics, optoelectronics, plasmonics, and condensed matter physics. In the semiclassical regime, open systems can be described by phenomenological effective non-Hermitian Hamiltonians (NHHs) capturing the effects of gain and loss in terms of imaginary fields. The EPs that characterize the spectra of such Hamiltonians (HEPs) describe the time evolution of a system without quantum jumps. It is well known that a full quantum treatment describing more generic dynamics must crucially take into account such quantum jumps. In a recent paper [F. Minganti et al., Phys. Rev. A 100, 062131 (2019)], we generalized the notion of EPs to the spectra of Liouvillian superoperators governing open system dynamics described by Lindblad master equations. Intriguingly, we found that in situations where a classical-to-quantum correspondence exists, the two types of dynamics can yield different EPs. In a recent experimental work [M. Naghiloo et al., Nat. Phys. 15, 1232 (2019)], it was shown that one can engineer a non-Hermitian Hamiltonian in the quantum limit by postselecting on certain quantum jump trajectories. This raises an interesting question concerning the relation between Hamiltonian and Lindbladian EPs, and quantum trajectories. We discuss these connections by introducing a hybrid-Liouvillian superoperator, capable of describing the passage from an NHH (when one postselects only those trajectories without quantum jumps) to a true Liouvillian including quantum jumps (without postselection). Beyond its fundamental interest, our approach allows to intuitively relate the effects of postselection and finite-efficiency detectors.
Exceptional points (EPs) correspond to degeneracies of open systems. These are attracting much interest in optics, optoelectronics, plasmonics, and condensed matter physics. In the classical and semiclassical approaches, Hamiltonian EPs (HEPs) are usually defined as degeneracies of non-Hermitian Hamiltonians such that at least two eigenfrequencies are identical and the corresponding eigenstates coalesce. HEPs result from continuous, mostly slow, nonunitary evolution without quantum jumps. Clearly, quantum jumps should be included in a fully quantum approach to make it equivalent to, e.g., the Lindblad master equation approach. Thus, we suggest to define EPs via degeneracies of a Liouvillian superoperator (including the full Lindbladian term, LEPs), and we clarify the relations between HEPs and LEPs. We prove two main theorems: Theorem 1 proves that, in the quantum limit, LEPs and HEPs must have essentially different properties. Theorem 2 dictates a condition under which, in the “semiclassical” limit, LEPs and HEPs recover the same properties. In particular, we show the validity of Theorem 1 studying systems which have (1) an LEP but no HEPs and (2) both LEPs and HEPs but for shifted parameters. As for Theorem 2, (3) we show that these two types of EPs become essentially equivalent in the semiclassical limit. We introduce a series of mathematical techniques to unveil analogies and differences between the HEPs and LEPs. We analytically compare LEPs and HEPs for some quantum and semiclassical prototype models with loss and gain.
Recently, apparent nonphysical implications of non-Hermitian quantum mechanics (NHQM) have been discussed in the literature. In particular, the apparent violation of the no-signaling theorem, discrimination of nonorthogonal states, and the increase of quantum entanglement by local operations were reported, and therefore NHQM was not considered as a fundamental theory. Here we show that these and other no-go principles (including the no-cloning and no-deleting theorems) of conventional quantum mechanics still hold in finite-dimensional non-Hermitian quantum systems, including parity-time symmetric and pseudo-Hermitian cases, if its formalism is properly applied. We have developed a modified formulation of NHQM based on the geometry of Hilbert spaces which is consistent with the conventional quantum mechanics for Hermitian systems. Using this formulation the validity of these principles can be shown in a simple and uniform approach.
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.
An exceptional point (EP) is a non-Hermitian degeneracy where both eigenvalues and their corresponding eigenvectors coalesce. It was recently proposed and demonstrated that such spectral singularity can be utilized for enhanced sensing. Potential drawbacks of EP sensing include both fundamental resolution limit and noise effects that might mask the hypersensitive resonant splitting. Here, we address these issues by proposing a parity-time (PT)-symmetric sensing circuit bearing a sixth-order EP. By employing capacitive coupling channel as a sensing platform, we achieve an enhanced resonance shift proportional to the fourth-order root of the perturbation strength and maintain a high resolution for weak perturbation. Due to the low-pass feature of our circuit, thermal noise is mitigated down to a level comparable to its Hermitian counterpart, despite the presence of highly noisy gain/loss elements. Our EP sensing scheme offers combined enhanced sensitivity, improved resolution and nondegraded thermal noise performance, showing an exciting prospect for next-generation sensing technologies.
Breaking symmetry with single spins
The energetics of quantum systems are typically described by Hermitian Hamiltonians. The exploration of non-Hermitian physics in classical parity-time (PT)–symmetric systems has provided fertile theoretical and experimental ground to develop systems exhibiting exotic behavior. Wu et al. now demonstrate that non-Hermitian physics can be found in a solid-state quantum system. They developed a protocol, termed dilation, which transformed a PT-symmetric Hamiltonian into a Hermitian one. This allowed them to investigate PT-symmetric physics with a single nitrogen-vacancy center in diamond. The results provide a starting point for exploiting and understanding the exotic properties of PT-symmetric Hamiltonians in quantum systems.
Science , this issue p. 878
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.
Signaled by nonanalyticities in the time evolution of physical observables, dynamic quantum phase transitions (DQPTs) emerge in quench dynamics of topological systems and possess an interesting geometric origin captured by dynamic topological order parameters (DTOPs). In this Letter, we report the experimental study of DQPTs using discrete-time quantum walks of single photons. We simulate quench dynamics between distinct Floquet topological phases using quantum-walk dynamics and experimentally characterize DQPTs and the underlying DTOPs through interference-based measurements. The versatile photonic quantum-walk platform further allows us to experimentally investigate DQPTs for mixed states and in parity-time-symmetric nonunitary dynamics for the first time. Our experiment directly confirms the relation between DQPTs and DTOPs in quench dynamics of topological systems and opens up the avenue of simulating emergent topological phenomena using discrete-time quantum-walk dynamics.
Sensors play an important part in many aspects of daily life such as infrared sensors in home security systems, particle sensors for environmental monitoring and motion sensors in mobile phones. High-quality optical microcavities are prime candidates for sensing applications because of their ability to enhance light-matter interactions in a very confined volume. Examples of such devices include mechanical transducers, magnetometers, single-particle absorption spectrometers, and microcavity sensors for sizing single particles and detecting nanometre-scale objects such as single nanoparticles and atomic ions. Traditionally, a very small perturbation near an optical microcavity introduces either a change in the linewidth or a frequency shift or splitting of a resonance that is proportional to the strength of the perturbation. Here we demonstrate an alternative sensing scheme, by which the sensitivity of microcavities can be enhanced when operated at non-Hermitian spectral degeneracies known as exceptional points. In our experiments, we use two nanoscale scatterers to tune a whispering-gallery-mode micro-toroid cavity, in which light propagates along a concave surface by continuous total internal reflection, in a precise and controlled manner to exceptional points. A target nanoscale object that subsequently enters the evanescent field of the cavity perturbs the system from its exceptional point, leading to frequency splitting. Owing to the complex-square-root topology near an exceptional point, this frequency splitting scales as the square root of the perturbation strength and is therefore larger (for sufficiently small perturbations) than the splitting observed in traditional non-exceptional-point sensing schemes. Our demonstration of exceptional-point-enhanced sensitivity paves the way for sensors with unprecedented sensitivity. © 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.
Considerable progress in wireless power transfer has been made in the realm of non-radiative transfer, which employs magnetic-field coupling in the near field. A combination of circuit resonance and impedance transformation is often used to help to achieve efficient transfer of power over a predetermined distance of about the size of the resonators. The development of non-radiative wireless power transfer has paved the way towards real-world applications such as wireless powering of implantable medical devices and wireless charging of stationary electric vehicles. However, it remains a fundamental challenge to create a wireless power transfer system in which the transfer efficiency is robust against the variation of operating conditions. Here we propose theoretically and demonstrate experimentally that a parity-time-symmetric circuit incorporating a nonlinear gain saturation element provides robust wireless power transfer. Our results show that the transfer efficiency remains near unity over a distance variation of approximately one metre, without the need for any tuning. This is in contrast with conventional methods where high transfer efficiency can only be maintained by constantly tuning the frequency or the internal coupling parameters as the transfer distance or the relative orientation of the source and receiver units is varied. The use of a nonlinear parity-time-symmetric circuit should enable robust wireless power transfer to moving devices or vehicles. © 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
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.
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.
Compound-photonic structures with gain and loss(1) provide a powerful platform for testing various theoretical proposals on non-Hermitian parity-time-symmetric quantum mechanics(2-5) and initiate new possibilities for shaping optical beams and pulses beyond conservative structures. Such structures can be designed as optical analogues of complex parity-timesymmetric potentials with real spectra. However, the beam dynamics can exhibit unique features distinct from conservative systems due to non-trivial wave interference and phase-transition effects. Here, we experimentally realize parity-time-symmetric optics on a chip at the 1,550 nm wavelength in two directly coupled high-Q silica-microtoroid resonators with balanced effective gain and loss. With this composite system, we further implement switchable optical isolation with a non-reciprocal isolation ratio from -8 dB to +8 dB, by breaking time-reversal symmetry with gain-saturated nonlinearity in a large parameter-tunable space. Of importance, our scheme opens a door towards synthesizing novel microscale photonic structures for potential applications in optical isolators, on-chip light control and optical communications.
Effective manipulation of cavity resonant modes is crucial for emission control in laser physics and applications. Using the concept of parity-time symmetry to exploit the interplay between gain and loss (i.e., light amplification and absorption), we demonstrate a parity-time symmetry-breaking laser with resonant modes that can be controlled at will. In contrast to conventional ring cavity lasers with multiple competing modes, our parity-time microring laser exhibits intrinsic single-mode lasing regardless of the gain spectral bandwidth. Thresholdless parity-time symmetry breaking due to the rotationally symmetric structure leads to stable single-mode operation with the selective whispering-gallery mode order. Exploration of parity-time symmetry in laser physics may open a door to next-generation optoelectronic devices for optical communications and computing.
Copyright © 2014, American Association for the Advancement of Science.
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as , the true meaning and significance of the so-called charge operators and the -inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications and manifestations of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos and biophysics.
How much noise does quantum mechanics require a linear amplifier to add to a signal it processes? An analysis of narrow-band amplifiers (single-mode input and output) yields a fundamental theorem for phase-insensitive linear amplifiers; it requires such an amplifier, in the limit of high gain, to add noise which, referred to the input, is at least as large as the half-quantum of zero-point fluctuations. For phase-sensitive linear amplifiers, which can respond differently to the two quadrature phases ("cosωt" and "sinωt"), the single-mode analysis yields an amplifier uncertainty principle—a lower limit on the product of the noises added to the two phases. A multimode treatment of linear amplifiers generalizes the single-mode analysis to amplifiers with nonzero bandwidth. The results for phase-insensitive amplifiers remain the same, but for phase-sensitive amplifiers there emerge bandwidth-dependent corrections to the single-mode results. Specifically, there is a bandwidth-dependent lower limit on the noise carried by one quadrature phase of a signal and a corresponding lower limit on the noise a high-gain linear amplifier must add to one quadrature phase. Particular attention is focused on developing a multimode description of signals with unequal noise in the two quadrature phases.
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.