The continuous-time differential Lyapunov equation is widely used in linear optimal control theory, a branch of mathematics and engineering. In quantum physics, it is known to appear in Markovian descriptions of linear (quadratic Hamiltonian, linear equations of motion) open quantum systems, typically from quantum master equations. Despite this, the Lyapunov equation is seldom considered a fundamental formalism for linear open quantum systems. In this work we aim to change that. We establish the Lyapunov equation as a fundamental and efficient formalism for linear open quantum systems that can go beyond the limitations of various standard quantum master equation descriptions, while remaining of much less complexity than general exact formalisms. This also provides valuable insights for non-Hermitian quantum physics. In particular, we derive the Lyapunov equation for the most general number conserving linear system in a lattice of arbitrary dimension and geometry, connected to an arbitrary number of baths at different temperatures and chemical potentials. Three slightly different forms of the Lyapunov equation are derived via an equation of motion approach, by making increasing levels of controlled approximations, without reference to any quantum master equation. Then we discuss their relation with quantum master equations, positivity, accuracy and additivity issues, the possibility of describing dark states, general perturbative solutions in terms of single-particle eigenvectors and eigenvalues of the system, and quantum regression formulas. Our derivation gives a clear understanding of the origin of the non-Hermitian Hamiltonian describing the dynamics and separates it from the effects of quantum and thermal fluctuations. Many of these results would have been hard to obtain via standard quantum master equation approaches.
We point out that the Lyapunov exponent of the eigenstate places restrictions on the eigenvalue. Consequently, with regard to non-Hermitian systems, even without any symmetry, the non-conservative Hamiltonians can exhibit real spectra as long as Lyapunov exponents of eigenstates inhibit imaginary parts of eigenvalues. Our findings open up a new route to study non-Hermitian physics.
By means of expressing volumes in phase space in terms of traces of quantum operators, a relationship between the Hamiltonian poles and the Lyapunov exponents in a non Hermitian quantum dynamics, is presented. We illustrate the formalism by characterizing the behavior of the Gamow model whose dissipative decay time, measured by its decoherence time, is found to be inversely proportional to the Lyapunov exponents of the unstable periodic orbits. The results are in agreement with those obtained by means of the semiclassical periodic-orbit approach in quantum resonances theory but using a simpler…
Noise-Enhanced Self-Healing Dynamics in Non-Hermitian Systems
School of Physics, Peking University（北京大学物理学院）；Collaborative Innovation Center of Quantum Matter（量子物质协同创新中心）；Center for High Energy Physics, Peking University（北京大学高能物理中心）
Disorder and non-Hermiticity dramatically impact the topological and localization properties of a quantum system, giving rise to intriguing quantum states of matter. The rich interplay of disorder, non-Hermiticity, and topology is epitomized by the recently proposed non-Hermitian topological Anderson insulator that hosts a plethora of exotic phenomena. Here we experimentally simulate the non-Hermitian topological Anderson insulator using disordered photonic quantum walks, and characterize its localization and topological properties. In particular, we focus on the competition between Anderson l…
Time-periodic driving fields could endow a system with peculiar topological and transport features. In this work, we find dynamically controlled localization transitions and mobility edges in non-Hermitian quasicrystals via shaking the lattice periodically. The driving force dresses the hopping amplitudes between lattice sites, yielding alternate transitions between localized, mobility edge and extended non-Hermitian quasicrystalline phases. We apply our Floquet engineering approach to five representative models of non-Hermitian quasicrystals, obtain the conditions of photon-assisted localizat…
In non-Hermitian crystals showing the non-Hermitian skin effect, ordinary Bloch band theory and Bloch topological invariants fail to correctly predict energy spectra, topological boundary states, and symmetry breaking phase transitions in systems with open boundaries. Recently, it has been shown that a correct description requires to extend Bloch band theory into complex plane. A still open question is whether non-Hermitian skin effect and non-Bloch symmetry-breaking phase transitions can be probed by real-space wave dynamics far from edges, which is entirely governed by ordinary Bloch bands. …
In this paper, we present in-depth transfer matrix analyses of the Anderson transition in three non-Hermitian (NH) systems, NH Anderson, U(1) and Peierls models, that belong to NH class AI$^{\dagger}$ or NH class A. We first argue a general validity of the transfer matrix analysis, and clarify the symmetry properties of the Lyapunov exponents, scattering ($S$) matrix and two-terminal conductance in these NH models. The unitarity of the $S$ matrix is violated in NH systems, where the two-terminal conductance can take arbitrarily large values. Nonetheless, we show that the transposition symmetry…
This article provides a novel continuous-time state feedback control strategy to stabilize an eigenstate of the Hermitian measurement operator of a two-level quantum system. In open loop, such system converges stochastically to one of the eigenstates of the measurement operator. Previous work has proposed state feedback that destabilizes the undesired eigenstates and relies on a probabilistic analysis to prove convergence. In contrast, we here associate the state observer to an adaptive version of so-called Markovian feedback (essentially, proportional control) and we show that this leads to a…
Topological phases are enriched in non-equilibrium open systems effectively described by non-Hermitian Hamiltonians. While several properties unique to non-Hermitian topological systems were uncovered, the fundamental role of symmetry in non-Hermitian physics has yet to be fully understood, and it has remained unclear how symmetry protects non-Hermitian topological phases. Here we show that two fundamental anti-unitary symmetries, time-reversal and particle-hole symmetries, are topologically equivalent in the complex energy plane and hence unified in non-Hermitian physics. A striking consequen…
We develop a complete theory of symmetry and topology in non-Hermitian physics. We demonstrate that non-Hermiticity ramifies the celebrated Altland-Zirnbauer symmetry classification for insulators and superconductors. In particular, charge conjugation is defined in terms of transposition rather than complex conjugation due to the lack of Hermiticity, and hence chiral symmetry becomes distinct from sublattice symmetry. It is also shown that non-Hermiticity enables a Hermitian-conjugate counterpart of the Altland-Zirnbauer symmetry. Taking into account sublattice symmetry or pseudo-Hermiticity a…
Anomalous Localization and Mobility Edges in Non-Hermitian Quasicrystals with Disordered Imaginary Gauge Fields
Sanli Honors College, School of Physics and Electronics Engineering, Shanxi University（山西大学物理与电子工程学院三立荣誉学院）；Institute of Theoretical Physics and State Key Laboratory of Quantum Optics Technologies and Devices, Shanxi University（山西大学理论物理研究所量子光学技术与器件国家重点实验室）；State Key Laboratory of Quantum Optics Technologies and Devices, Institute of Laser Spectroscopy, Shanxi University（山西大学激光光谱研究所量子光学技术与器件国家重点实验室）
The past decades have witnessed an explosion of interest in topological materials, and a lot of mathematical concepts have been introduced in condensed matter physics. Among them, the bulk-boundary correspondence is the central topic in topological physics, which has inspired researchers to focus on boundary physics. Recently, the concepts of topological phases have been extended to non-Hermitian Hamiltonians, whose eigenvalues can be complex. Besides the topology, non-Hermiticity can also cause a boundary phenomenon called the non-Hermitian skin effect, which is an extreme sensitivity of the …
For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a canonical orthonormal basis in which a previously introduced unitary mapping of H to a Hermitian Hamiltonian h takes a simple form. We use this basis to construct the observables O of the quantum mechanics based on H. In particular, we introduce pseudo-Hermitian position and momentum operators and a pseudo-Hermitian quantization scheme that relates the latter to the ordinary classical position and momentum observables. These allow us to address the problem of determining the conserved probability density and the under…
Physically, one tends to think of non-Hermitian systems in terms of gain and loss: the decay or amplification of a mode is given by the imaginary part of its energy. Here, we introduce an alternative avenue to the realm of non-Hermitian physics, which involves neither gain nor loss. Instead, complex eigenvalues emerge from the amplitudes and phase-differences of waves backscattered from the boundary of insulators. We show that for any strong topological insulator in a Wigner-Dyson class, the reflected waves are characterized by a reflection matrix exhibiting the non-Hermitian skin effect. This…
In arXiv:0709.0483 Gunther and Samsonov outline a ``generalization'' of quantum mechanics that involves simultaneous consideration of Hermitian and non-Hermitian operators and promises to be ``capable to produce effects beyond those of standard Hermitian quantum mechanics.'' We give a simple physical interpretation of Hermiticity and discuss in detail the shortcomings of the above-mentioned composite quantum theory. In particular, we show that the corresponding ``generalization of measurement theory'' suffers from a dynamical inconsistency and that it is by no means adequate to replace the sta…
A non-commuting measurement transfers, via the apparatus, information encoded in a system's state to the external "observer". Classical measurements determine properties of physical objects. In the quantum realm, the very same notion restricts the recording process to orthogonal states as only those are distinguishable by measurements. Therefore, even a possibility to describe physical reality by means of non-hermitian operators should \emph{volens nolens} be excluded as their eigenstates are not orthogonal. Here, we show that non-hermitian operators with real spectrum can be treated within th…
We review the methodology to theoretically treat parity-time- ($\mathcal{PT}$-) symmetric, non-Hermitian quantum many-body systems... (For the full abstract see paper)
We explain why the main conclusion of Bender et al, hep-th/0511229 [J. Phys. A 39 (2006) 1657] regarding the practical superiority of the non-Hermitian description of PT-symmetric quantum systems over their Hermitian description is not valid. Recalling the essential role played by the Hermitian description in the characterization and interpretation of the physical observables, we maintain that as far as the physical aspects of the theory are concerned the Hermitian description is not only unavoidable but also indispensable.
Non-Hermiticity from non-reciprocal hoppings has been shown recently to demonstrate the non-Hermitian skin effect (NHSE) under open boundary conditions (OBCs). Here we study the interplay of this effect and the Anderson localization in a \textit{non-reciprocal} quasiperiodic lattice, dubbed non-reciprocal Aubry-André model, and a \textit{rescaled} transition point is exactly proved. The non-reciprocity can induce not only the NHSE, but also the asymmetry in localized states with two Lyapunov exponents for both sides. Meanwhile, this transition is also topological, characterized by a winding nu…
A non-Hermitian $N-$level quantum model with two free real parameters is proposed in which the bound-state energies are given as roots of an elementary trigonometric expression and in which they are, in a physical domain of parameters, all real. The wave function components are expressed as closed-form superpositions of two Chebyshev polynomials. In any eligible physical Hilbert space of finite dimension $N < \infty$ our model is constructed as unitary with respect to an underlying Hilbert-space metric $Θ\neq I$. The simplest version of the latter metric is finally constructed, at any dimensio…
Information on quantum systems can be obtained only when they are open (or opened) in relation to a certain environment. As a matter of fact, realistic open quantum systems appear in very different shape. We sketch the theoretical description of open quantum systems by means of a projection operator formalism elaborated many years ago, and applied by now to the description of different open quantum systems. The Hamiltonian describing the open quantum system is non-Hermitian. Most studied are the eigenvalues of the non-Hermitian Hamiltonian of many-particle systems embedded in one environment. …
In this paper, based on a one-dimensional non-Hermitian spin model with $\mathcal{RT}$-invariant term, we study the non-Hermitian physics for the two (nearly) degenerate ground states. By using the high-order perturbation method, an effective pseudo-spin model is obtained to describe non-Hermitian physics for the two (nearly) degenerate ground states, which are precisely consistent with the numerical calculations. We found that there may exist effective (anti) $\mathcal{PT}$ symmetry for the effective pseudo-spin model of the two (nearly) degenerate ground states. In particular, there exists s…
We explore a way of finding the link between a non-Hermitian Hamiltonian and a Hermitian one. Based on the analysis of Bethe Ansatz solutions for a class of non-Hermitian Hamiltonians and the scattering problems for the corresponding Hermitian Hamiltonians. It is shown that a scattering state of an arbitrary Hermitian lattice embedded in a chain as the scattering center shares the same wave function with the corresponding non-Hermitian tight binding lattice, which consists of the Hermitian lattice with two additional on-site complex potentials, no matter the non-Hermitian is broken PT symmetry…
We discuss non-Hermitian field theories where the spectrum of the Hamiltonian involves only real energies. We make three observations. (i) The theories obtained from supersymmetric theories by nonanticommutative deformations belong in many cases to this class. (ii) When the deformation parameter is small, the deformed theory enjoys the same supersymmetry algebra as the undeformed one. Half of the supersymmetries are manifest and the existence of another half can be deduced from the structure of the spectrum. (iii) Generically, the conventionally defined S--matrix is not unitary for such theori…
The non-Hermitian formalism is used at present in many papers for the description of open quantum systems. A special language developed in this field of physics which makes it difficult for many physicists to follow and to understand the corresponding papers. We show that the characteristic features of the non-Hermitian formalism are nothing but nonlinearities that may appear in the equations when the Hamiltonian is non-Hermitian. They are related directly to singular points (called mostly exceptional points, EPs). At low level density, they may cause counterintuitive physical results which al…
A review is given on the foundations and applications of non-Hermitian classical and quantum physics. First, key theorems and central concepts in non-Hermitian linear algebra, including Jordan normal form, biorthogonality, exceptional points, pseudo-Hermiticity and parity-time symmetry, are delineated in a pedagogical and mathematically coherent manner. Building on these, we provide an overview of how diverse classical systems, ranging from photonics, mechanics, electrical circuits, acoustics to active matter, can be used to simulate non-Hermitian wave physics. In particular, we discuss rich a…
We present a brief review of physical problems leading to indefinite Hilbert spaces and non-hermitian Hamiltonians. With the exception of pseudo-Riemannian manifolds in GR, the problem of a consistent physical interpretation of these structures still waits to be faced. In print in Rev. Mex. Fis.
We explore an alternative way of finding the link between a PT non-Hermitian Hamiltonian and a Hermitian one. Based on the analysis of the scattering problem for an imaginary potential and its time reversal process, it is shown that any real-energy eigenstate of a PT tight-binding lattice with on-site imaginary potentials shares the same wave function with a resonant transmission state of the corresponding Hermitian lattice embedded in a chain. It indicates that the PT eigenstate of a PT non-Hermitian Hamiltonian has connection to the resonance transmission state of the extended Hermitian Hami…
We present an evaluation of some recent attempts at understanding the role of pseudo-Hermitian and PT-symmetric Hamiltonians in modeling unitary quantum systems and elaborate on a particular physical phenomenon whose discovery originated in the study of complex scattering potentials.
We relate topological properties of non-Hermitian systems and observables of quantum open systems by using the Keldysh path-integral method. We express Keldysh Green's functions in terms of effective non-Hermitian Hamiltonians that contain all the relevant topological information. We arrive at a frequency dependent topological index that is linked to the response of the system to perturbations at a given frequency. We show how to detect a transition between different topological phases by measuring the response to local perturbations. Our formalism is exemplified in a 1D Hatano-Nelson model, h…
Non-Hermitian physics in the dissipative many-body system of Rydberg atoms
University of Science and Technology of China（中国科学技术大学）；Key Laboratory of Quantum Information, University of Science and Technology of China（中国科学技术大学量子信息重点实验室）；Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China（中国科学技术大学量子信息与量子科技创新研究院）
We investigate a non-Hermitian Aharonov-Bohm (AB) ring system with a quantum dot (QD) embedded in each of its two arms. The energy levels of the QDs are complex in order to take into account the physical gain or loss of the ring system during its interacting processes with the environment. When there is magnetic flux threading through the ring, by allocating the flux phase factor into the tunneling amplitudes between the QDs and the leads in different ways, the Hamiltonian of the system can be written into different formalisms. We calculate the transmission through the ring by using these diff…
The metric associated with a quasi-Hermitian Hamiltonian and its physical implications are scrutinised. Consequences of the non-uniqueness such as the question of the probability interpretation and the possible and forbidden choices of additional observables are investigated and exemplified by specific illustrative examples. In particular it is argued that the improper identification of observables lies at the origin of the claimed violation of the brachistchrone transition time between orthogonal states. The need for further physical input to remove ambiguities is pointed out.
Quantum phase transitions also occur in non-Hermitian systems. In this work we show that density functional theory, for the first time, uncovers universal behaviors for phase transitions in non-Hermitian many-body systems. To be specific, we first prove that the non-degenerate steady state of a non-Hermitian quantum many-body system is a universal function of the first derivative of the steady state energy with respect to the control parameter. This finding has far-reaching consequences for non-Hermitian systems: (i) It bridges the nonanalytic behavior in physical observable and nonanalytic be…
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 and elaborate on a number of rel…
