Ali Mostafazadeh, (Koç University, Turkey), 07/05, 14:00 GMT
Exactness of the Born approximation, broadband unidirectional invisibility, and implicit regularization of point scatterers in two dimensions
Abstract: Born approximation was proposed by Max Born in his monumental 1926 paper where he laid the foundations of quantum scattering theory. Because of its great theoretical and practical significance, this approximation scheme is covered in most standard textbooks on quantum mechanics. None of these, however, addresses the problem of finding potentials whose scattering problem is exactly solvable by the Born approximation. We give a simple condition under which a potential has this property in two dimensions. This in turn allows us to solve another important open problem of scattering theory, namely constructing potentials that enjoy perfect (non-approximate) unidirectional invisibility in a finite spectral band of arbitrary width. Our analysis provides the first examples of quasi-exactly solvable scattering potentials, i.e., potentials whose scattering problem is exactly solvable for energies not exceeding a given value. These results are particular outcomes of a recent dynamical formulation of time-independent scattering theory. We use this formulation to offer an explicit solution of the scattering problem for a delta-function potential in two dimensions. Our approach involves no divergent terms and reproduces the known expression for the scattering amplitude that is obtained by a regularization of the delta-function potential followed by a coupling constant renormalization. The implicit regularization property of our approach avoids such complications.
References
- F. Loran and A. Mostafazadeh, “Exact Solution of the Two-Dimensional Scattering Problem for a Class of δ-Function Potentials Supported on Subsets of a Line,”J. Phys. A: Math. Theor. 51, 335302 (2018); arXiv: 1708.06003.
- F. Loran and A. Mostafazadeh, “Exactness of the Born Approximation and Broadband Unidirectional Invisibility in Two Dimensions,”Phys. Rev. A 100, 053846 (2019); arXiv: 1904.07737.
Nick Mavromatos, (King’s College, Uni. of London, UK), 14/05, 14:00 GMT
Non-Hermitian Yukawa Interactions: Consistency, Dynamical Mass Generation issues and Physics Motivation
Abstract: I discuss a fermion-(pseudo)scalar field theory model in (3+1)-dimensions, with anti hermitian Yukawa interactions, and demonstrate its consistency as a unitary quantum field theory. The model may be understood within a wider PT-symmetry framework, in the sense of identifying conditions for real energy eigenvalues of the corresponding Hamiltonian operators. Due to energetics, it is not possible to have dynamical mass generation for the fermions unless suitable attractive four fermion interactions are included, with sufficiently strong couplings. In the latter case, we estimate the dynamically generated masses using an appropriate Schwinger-Dyson treatment, for weak Yukawa couplings. Such models, with pseudoscalar-fermion Yukawa interactions, find partial motivation in neutrino physics, as providers of novel scenarios for Majorana neutrino mass generation, which could be of phenomenological interest.
Phillip Mannheim, (University of Connecticut, US), 21/05, 14:00 GMT
Ghost Problems from Pauli-Villars to Fourth-Order Quantum Gravity and their Resolution
Abstract: We review the history of the ghost problem in quantum field theory from the Pauli-Villars regulator theory to currently popular fourth-order derivative quantum gravity theories. While these theories all appear to have unitarity-violating ghost states with negative norm, we show that in fact these ghost states only appear because the theories are being formulated in the wrong Hilbert space. In these theories the Hamiltonians are not Hermitian but instead possess an antilinear symmetry. Consequently, one cannot use inner products that are built out of states and their Hermitian conjugates. Rather, one must use inner products built out of states and their conjugates with respect to the antilinear symmetry, and these latter inner products are positive. In this way one can build quantum theories of gravity in four spacetime dimensions that are unitary.
Fabio Bagarallo, (University of Palermo, Italy), 04/06, 14:00 GMT
Weak pseudo-bosons
Abstract: We show how the notion of pseudo-bosons, originally introduced as operators acting on some Hilbert space, can be extended to a distributional settings. In doing so, we are able to construct a rather general framework to deal with generalized eigenvectors of the multiplication and of the derivation operators. Connections with the quantum damped harmonic oscillator are also briefly discussed.
This talk is based on the following papers:
- F. Bagarello, Weak pseudo-bosons, J. Phys. A, 53, 135201 (2020)
- F. Bagarello, F. Gargano, F. Roccati, A no-go result for the quantum damped harmonic oscillator, Phys. Lett. A, 383, 2836-2838 (2019)
- F. Bagarello, F. Gargano, F. Roccati, Some remarks on few recent results on the damped quantum harmonic oscillator, Ann. of Phys., 414, 168091, (2020)
Dorje Brody, (University of Surrey, UK), 18/06, 14:00 GMT
Evolution Speed of Open Quantum Dynamics
Abstract: Understanding the speed of the evolution of a quantum state is of interest for a variety of reasons in quantum information science. As well as being of interest in its own right, in implementing quantum algorithms for establishing communication and performing computation, the evolution speed determines how fast a given task can be processed. The speed also determines the sensitivity of quantum states against time evolution, and this information can be used to determine error bounds on quantum state estimation. In this talk I will explain how to derive a formula for the speed of evolution of a general open-system quantum dynamics. Of particular interest lies in PT-symmetric open quantum systems that exhibit PT phase transitions. It will be shown that the time dependence of the evolution speed (or the acceleration of the state) exhibits radically different behaviours in the two phases.
The talk is based on joint work with B. Longstaff.
Savannah Garmon, (Osaka Prefecture University, Japan), 02/07, 14:00 GMT
Anomalous exceptional point and non-Markovian Purcell effect at threshold in 1-D continuum systems
Abstract: For a system consisting of a quantum emitter coupled near threshold to a 1-D continuum with a van Hove singularity in the density of states, we demonstrate general conditions such that a characteristic triple level convergence occurs directly on the threshold as the coupling g is shut off. For small g values the eigenvalue and norm of each of these states can be expanded in a Puiseux expansion in terms of powers of g^{2/3}, which suggests the influence of a third order exceptional point. However, in the actual g→0 limit, only two discrete states in fact coalesce as the system can be reduced to a 2 x 2 Jordan block; the third state instead merges with the continuum. The decay width of the resonance state involved in this convergence is proportional to g^{4/3}, significantly enhanced compared to the usual Fermi golden rule. However, non-Markovian dynamics due to the branch point effect are significantly enhanced near the threshold as well. Applying a perturbative analysis in terms of the Puiseux expansion that also takes into account the threshold influence, we show that the relaxation process of the quantum emitter actually results in an unusual 1 – C t^{3/2} decay on the key timescale during which most of the decay occurs.
Relevant background:
1. S. Garmon, T. Petrosky, L. Simine, and D. Segal, Fortschr. Phys. 61, 261 (2013). https://doi.org/10.1002/prop.201200077
2. S. Garmon, M. Gianfreda and N. Hatano, Phys. Rev. A 92, 022125 (2015). http://dx.doi.org/10.1103/PhysRevA.92.022125
3. S. Garmon and G. Ordonez, J. Math. Phys. 58, 062101 (2017).
https://doi.org/10.1063/1.4983809
Yogesh Joglekar, (Indiana Uni. Purdue University, US), 16/07, 14:00 GMT
Conserved quantities and their consequences in PT symmetric systems: theory and experiments [1]
Abstract: 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 in general, and parity-time (PT) systems in particular. Here, we present a complete set of conserved observables for a broad class of PT -symmetric Hamiltonians and experimentally demonstrate their properties and consequences by using a single-photon linear optical circuit. One surprising consequence is the generation of coherence between different eigenstates of a conserved observable — something that is forbidden in Hermitian quantum theories. 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.
[1] Work done in collaboration with Xue Peng group, and Frantisek Ruzicka (Phys. Rev. Research 2, 022039(R) (2020)).
Andrei Smilga, (University of Nantes, France), 23/07, 14:00 GMT
Classical and quantum dynamics of higher-derivative systems
Abstract: A brief review of the physics of systems including higher derivatives in the Lagrangian is given. We outline the Ostrogradsky formalism and prove simple theorems saying that all such systems involve ghosts. At the classical level, that means that the energy is not bounded neither from below, nor from above. At the quantum level, that means that the spectrum of the Hamiltonian is never bounded from below and the vacuum ground state is absent.
If the Hamiltonian includes nontrivial interactions, this usually
leads to collapse and loss of unitarity, but in certain special cases,
this does not happen, however: ghosts are benign.
We speculate that the Theory of Everything is a higher-derivative
field theory, characterized by the presence of such benign ghosts and
defined in a higher-dimensional bulk. Our Universe represents then a
classical solution in this theory, having the form of a 3-brane embedded
in the bulk.
Eva-Maria Graefe, (Imperial College London, UK), 30/07, 14:00 GMT
A PT-symmetric kicked top
Abstract: A non-Hermitian PT-symmetric version of the kicked top is introduced to study the interplay of quantum chaos with balanced loss and gain. The classical dynamics arising from the quantum dynamics of the angular momentum expectation values are derived. It is demonstrated that the presence of PT-symmetry can lead to ”stable” mixed regular chaotic behaviour without sinks or sources for subcritical values of the gain-loss parameter. For large values of the kicking strength a strange attractor is observed that also persists if PT-symmetry is broken. Classical structures are also identified in the quantum dynamics. Finally, some of the statistics of the eigenvalues of the quantum system are analysed.
Avadh Saxena, (Los Alamos National Laboratory, US), 13/08, 14:00 GMT
Anti-PT-Symmetric Qubit: Decoherence and Entanglement Entropy [1]
Abstract: We consider a two-level spin system based anti-parity-time (anti-PT)-symmetric qubit and study its decoherence as well as entanglement entropy properties. We compare our findings with that of the corresponding PT-symmetric and Hermitian qubits. First we consider the time-dependent Dyson map for the density matrix of a general non-Hermitian system and then specialize it to the case of the anti-PT-symmetric qubit. We find that the decoherence function for the anti-PT-symmetricqubit decays slower than the PT-symmetric and Hermitian qubits. For the von Neumann entropy we find that for the anti-PT-symmetric qubit it grows more slowly compared to the PT-symmetric and Hermitian qubits. Similarly, we find that the corresponding average Fisher information is much higher compared to the PT-symmetric and Hermitian qubits.
These results demonstrate that anti-PT-symmetric qubits may be better suited for quantum computing and quantum information processing applications than conventional Hermitian or even PT-symmetric qubits.
[1] Work done in collaboration with Julia Cen (LANL).
Li Ge, (City University of New York, US), 20/08, 14:00 GMT
Pseudo-chirality and its conserved pseudo norm in non-Hermitian photonic systems
Abstract: Noether’s theorem relates constants of motion to the symmetries of the system. Here we investigate a manifestation of Noether’s theorem in non-Hermitian systems, where the inner product is defined differently from quantum mechanics. In this framework, a generalized symmetry which we term pseudo-chirality emerges naturally as the counterpart of symmetries defined by a commutation relation in quantum mechanics [1]. Using this observation, we reveal previously unidentified constants of motion in non-Hermitian systems with parity-time and chiral symmetries. We further elaborate the disparate implications of pseudo-chirality induced constant of motion: It signals the pair excitation of a generalized “particle” and the corresponding “hole” but vanishes universally when the pseudo-chiral operator is anti-symmetric. This disparity, when manifested in a non-Hermitian topological lattice with the Landau gauge, depends on whether the lattice size is even or odd. We further discuss previously unidentified symmetries of this non-Hermitian topological system, and we reveal how its constant of motion due to pseudo-chirality can be used as an indicator of whether a pure chiral edge state is excited.
[1] J. Rivero and L. Ge, Phys. Rev. Letts., in press (2020).
Carl Bender, (Washington University in St. Louis, US), 27/08, 14:00 GMT
Unmasking classical PT symmetry
Abstract: This talk reports the latest results of an ongoing research program to study the classical trajectories of the family of PT-symmetric Hamiltonians H = p² +x² (ix)^ε (ε≥0). A rich variety of phenomena, heretofore overlooked, have been discovered, such as the existence of infinitely many separatrix trajectories, infinite sequences of critical initial values associated with limiting classical orbits, regions of broken PT-symmetric classical trajectories, and a
topological transition at ε = 2.
Vladimir Konotop, (Universidade de Lisboa, Portugal), 10/09, 14:00 GMT
The universal form of one-dimensional potentials with spectral singularities
Abstract: We establish necessary and sufficient conditions for localized complex potentials in the Schrödinger equation to enable spectral singularities. All such potentials have the universal form. We describe an algorithm of constructing potentials featuring two or three coexisting spectral singularities, as well as potentials with a second order spectral singularity. The analysis is generalized to the discrete Schrödinger equation: all lattice potentials with spectral singularities have the universal form of the gain-and-loss distribution. Discrete potentials characterized by several spectral singularities at prescribed wavelengths, as well as potentials with second-order spectral singularities in their spectra are constructed.
[*] Work done in collaboration with Dmitry A. Zezyulin
[1] D A Zezyulin and V. V. Konotop New J. Phys. 22, 013057 (2020)
[2] V V Konotop and D A Zezyulin J. Phys. A 53 305202 (2020)
[3] D A Zezyulin and V. V. Konotop, Opt. Lett. 45 3447 (2020)
Joshua Feinberg, (University of Haifa and Technion, Israel), 24/09, 14:00 GMT
Dynamics of Disordered Mechanical Systems with Large Connectivity, Free
Probability Theory, and Quasi-Hermitian Random Matrices
Abstract: Disordered mechanical systems with high connectivity represent a limit opposite to the more familiar case of disordered crystals. Individual
ions in a crystal are subjected to nearest-neighbor interactions. In
contrast, the systems discussed in this talk have all their degrees of
freedom coupled to each other: Each momentum is typically coupled to all
other momenta, and similarly for the coordinates. Thus, the problem of
linearized small oscillations of such systems involves two full
positive-definite and non-commuting matrices, as opposed to the sparse
matrices associated with disordered crystals. Consequently, the familiar
methods for determining the averaged vibrational spectra of disordered
crystals, introduced many years ago by Dyson and Schmidt, are
inapplicable for highly connected disordered systems. In this talk we
apply random matrix theory (RMT) to calculate the averaged vibrational
spectra of such systems, in the limit of infinitely large system size.
At the heart of our analysis lies a calculation of the average spectrum
of the product of two positive definite random matrices by means of free
probability theory techniques. We also show that this problem is
intimately related with quasi-hermitian random matrix theory (QHRMT),
which means that the `Hamiltonian’ matrix is hermitian with respect to a
non-trivial metric. This extends ordinary hermitian matrices, for which
the metric is simply the unit matrix. The analytical results we obtain
for the spectrum agree well with our numerical results. The latter also
exhibit oscillations at the high-frequency band edge, which fit well the
Airy kernel pattern. We also compute inverse participation ratios of the
corresponding amplitude eigenvectors and demonstrate that they are all
extended, in contrast with conventional disordered crystals. Finally, we
compute the thermodynamic properties of the system from its spectrum of
vibrations.
This talk is based on work done jointly with Roman Riser at the
University of Haifa, which will be published in a couple of upcoming
papers.
Bhabani Prasad Mandal, (BHU, Varanasi, India), 08/10, 14:00 GMT
PT phase transitions: QM to QFT
Abstract: In this talk I will summarize some of our previous works on PT phase transition in quantum mechanical (QM) systems and will talk about PT phase transition in gauge field theoretic models. I will start with a non-Hermitian QES model in 1-dimensional QM systems to demonstrate PT phase transition which we realized way back in the year 2000. Next I will discuss a couple of exactly solvable non-Hermitian models in higher dimensional QM systems in this context. Dirac oscillator in the background of an external magnetic field with imaginary spin orbit coupling will be considered in (2+1) dimension next to talk about PT phase transition in the relativistic system. Lastly by considering natural Hermiticity properties of the ghost fields we cast SU(N) QCD in a newly introduced quadratic gauge as non-Hermitian but symmetric under combined Parity and Time Reversal transformation. As the ghost fields condensate PT symmetry is broken spontaneously . This leads to realization of the transition from deconfined phase to confined phase as PT phase transitions in QCD.
Stéphane Boris Tabeu, (University of Yaounde, Cameroon), 15/10, 14:00 GMT
Non-Hermitian two-level systems with imaginary components in electronics
Abstract: The systems having real spectra without being Hermitian are in expansion today in many fields of physics. The most investigated are Parity-Time Symmetric systems. Their Hamiltonians commute with the joint parity operator P and the time reversal T operator such that [PT, H]=0. Anti-Parity-Time symmetry is the counterpart of Parity-Time symmetry and is very emerging. Its Hamiltonians anti-commute with the PT operator. Generally, they are obtained in the form of matrices after a coupled mode theory, adiabatic elimination or rotating wave approximations. Electronics is presented nowadays as a test bed of non-Hermitian systems due the easy manipulation of components such resistors, inductors and capacitors. The imaginary components bring new configurations in the design of two-level systems without any approximation. The most powerful applications are listed as telemetry and sensing, wireless power transfer, topological circuits, new waveguides for asymmetric and non-reciprocal transport of information.
Our results open the road for the design of new systems for possible used in the generation and manipulation of quantum bits at room temperature, and for the introduction of electronics in the complex quaternionic space.
Naomichi Hatano, (University of Tokyo, Japan), 22/10, 14:00 GMT
What is the “environment” in open quantum systems?
Abstract: I will tutorially review what I had talked about resonant states with complex eigenvalues in the previous meetings of the PHHQP series, and particularly try to reveal what the “environment” of open quantum systems does mean in the actual physical setups. There are two ways of computing resonant states, namely the Siegert (out-going wave) boundary condition and the Feshbach formalism. The former is a direct consequence of finding zeros of the S matrix. Here the environment is in actual physical setups detectors and other experimental apparatus, which dictates the absorbing boundary condition.The spatial divergence of the resonant wave function means that macroscopic part of the wave amplitudes is absorbed in the detectors. This makes the system non-Hermitian. In the Feshbach formalism, on the other hand, the environmental part of the effective Hamiltonian produces the non-Hermiticity when we define the retarded and advanced Green’s functions of the environment. The choice of the retarded Green’s function corresponds to the out-going wave boundary condition, whereas the choice of the advanced one to the in-coming wave boundary condition. The introduction of ±iε in the denominator of the Green’s function produces the resonant and anti-resonant states. I will finally mention the proper choice of the right-eigenvector in analyzing non-Hermitian systems.
Stefan Rotter, (Vienna University of Technology, Austria), 05/11, 15:00 GMT
Non-Hermitian channels of invisibility across complex media
Abstract: Waves typically propagate very differently through a homogeneous medium like free space than through an inhomogeneous medium like a complex dielectric structure. It has thus been quite surprising to find that one-dimensional scattering systems can be engineered in such a way that by way of the gain and loss added to them, they become not only reflection-less, but entirely scattering-free [1,2], even for pulses propagating through them [3]. Our most recent insight is that a straightforward way to extend these non-Hermitian features to two-dimensional space is to map the wave solutions in free space to those inside a suitably designed non-Hermitian potential landscape such that both solutions share the same spatial distribution of their wave intensity [4]. This mapping turns out to be broadly applicable as a design protocol for a special class of non-Hermitian media across which specific incoming waves form scattering-free propagation channels. This protocol naturally enables the design of structures with a broadband unidirectional invisibility for which outgoing waves are indistinguishable from those of free space. We illustrate this concept through the example of a beam that maintains its Gaussian shape while passing through a randomly assembled distribution of scatterers with gain and loss.
[1] K. G. Makris, A. Brandstötter, P. Ambichl, Z. H. Musslimani, and S. Rotter, Light Sci. Appl. 6, e17035 (2017)
[2] E. Rivet, A. Brandstötter, K. G. Makris, H. Lissek, S. Rotter, and R. Fleury, Nature Physics 14, 942 (2018)
[3] A. Brandstötter, K. G. Makris, and S. Rotter, Phys. Rev. B 99, 115402 (2019)
[4] K. G. Makris, I. Krešić, A. Brandstötter, and S. Rotter, Optica 7, 619 (2020)
Maxim Chernodub, (Institut Denis Poisson, CNRS, Tours, France), 12/11, 15:00 GMT
Spontaneous Non-Hermiticity in Nambu-Jona-Lasinio model and Non-Hermitian Chiral Magnetic Effect
Abstract: This talk discusses two topics. First, we explore the physical consequences of a scenario when the standard Hermitian Nambu–Jona-Lasinio (NJL) model spontaneously develops a non-Hermitian PT-symmetric ground state via dynamical generation of an anti-Hermitian Yukawa coupling. The NJL model is used to model the chiral symmetry breaking in the model of strong interactions, and we argue that its non-Hermitian gound state may potentially be realized in an off-equilibrium environment of quark-gluon plasma created in the heavy-ion collisions. Second, we discuss the emergence of the chiral magnetic effect — anomalous generation of electric current along the axis of the background magnetic field — in the context of a non-Hermitian fermion system.
This talk is based on the following papers:
1. Alberto Cortijo and M.C., Non-Hermitian Chiral Magnetic Effect in Equilibrium, Symmetry 12, 761 (2020).
2. Alberto Cortijo, Marco Ruggieri, and M.C., Spontaneous Non-Hermiticity in Nambu-Jona-Lasinio model, ArXiv:2008.11629.
Pijush Ghosh, (Siksha-Bhavana, Visva-Bharati, India), 19/11, 15:00 GMT
Taming Hamiltonian systems with balanced loss and gain via Lorentz interaction
Abstract: We present a Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and gain. The instability of such system, both at the classical as well as quantum level, can be tamed by the introduction of additional Lorentz interaction. The Landau problem with balanced loss and gain is analysed to exemplify the idea. Further, we present non-PT-symmetric Hamiltonian systems with balanced loss and gain which admits periodic solutions. The examples include a mechanical model of
coupled Duffing oscillators with balanced loss and gain exhibiting Hamiltonian chaos, an exactly solvable dimer model with balanced loss and gain that describes the time evolution of the amplitudes of the former model and a vector nonlinear Schrodinger equation with balanced loss and gain. It appears that the criteria for the existence of periodic solutions in classical systems with balanced loss and gain can not be specified only in terms of PT-symmetry. Finally, a method is given to construct exact solutions of a class of cubic vector nonlinear Schrodinger equations with balanced loss and gain by the use of pseudo-unitary transformations.
References:
(1) Pijush K. Ghosh, Taming Hamiltonian systems with balanced loss and gain via Lorentz interaction : General results and a case study with Landau Hamiltonian, J. Phys. A: Math. Theor. 52 (2019) 415202 (24pp) arXiv:1810.04137
(2) Pijush K. Ghosh and Puspendu Roy, On regular and chaotic dynamics of a non-PT-symmetric Hamiltonian system of a coupled Duffing oscillator with balanced loss and gain, J. Phys. A: Math. Theor. 53 (2020) 475202 (27pp) arXiv:2007.07286
(3) Pijush K. Ghosh, Constructing Solvable Models of Vector Non-linear Schrodinger Equation with Balanced Loss and Gain via Non-unitary transformation, arXiv:2008.12252
Mikhail Plyushchay, (Universidad de Santiago, Chile), 03/12, 15:00 GMT
Conformal symmetry plus PT-symmetry for perfect invisibility and related exotics
Abstract: We show how perfectly invisible quantum mechanical zero-gap systems can be obtained by exploiting conformal and PT symmetry. Such systems give rise naturally to the exotic supersymmetric structure and allow ones to generate extreme waves via their intimate relation to the classical KdV equation. We also briefly discuss another application of conformal symmetry, the so-called conformal bridge transformation, by which asymptotically free and harmonically trapped conformal quantum systems can be related, and speculate on a possible relevance of the construction to PT-symmetry.
[1] J. M. Guilarte, M. S. Plyushchay, JHEP 12 (2017) 061.
[2] J. M. Guilarte, M. S. Plyushchay, JHEP 01 (2019) 194.
[3] L. Inzunza, M. S. Plyushchay, A. Wipf, Phys. Rev. D 101 (2020) 10, 105019.
[4] L. Inzunza, M. S. Plyushchay, A. Wipf, JHEP 04 (2020) 028.
Qing-hai Wang, (National University of Singapore, Singapore), 17/12, 15:00 GMT
Reinterpret Non-Hermitian Hamiltonians using time-dependent PT-symetric quantum mechanics
Abstract: In this talk, I will reinterpret an arbitrary finite-dimensional non-Hermitian Hamiltonian using time-dependent PT-symetric quantum mechanics. The conceptual framework is built upon a nontrivial time-dependent metric operator. All the ingredients of our framework, such as
the time-dependent Hilbert space, the observable, and the measurement postulate, can be “realized” by means of dilating and reinterpreting the non-Hermitian system in question as a part of a larger Hermitian system. Aided by our metric operator, we formulate the concepts of stable and unstable phases for generic non-Hermitian systems and argue that they, respectively, generalize the notions of unbroken and broken phases in time-independent
PT -symmetric systems.
XIX<vPHHQP<XX 