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Andreas Fring - One of the best experts on this subject based on the ideXlab platform.
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metric versus observable operator representation higher spin models
European Physical Journal Plus, 2018Co-Authors: Andreas Fring, Thomas FrithAbstract:We elaborate further on the metric representation that is obtained by transferring the time-dependence from a Hermitian Hamiltonian to the metric operator in a related non-Hermitian system. We provide further insight into the procedure on how to employ the time-dependent Dyson relation and the quasi-Hermiticity relation to solve time-dependent Hermitian Hamiltonian systems. By solving both equations separately we argue here that it is in general easier to solve the former. We solve the mutually related time-dependent Schrodinger equation for a Hermitian and non-Hermitian spin 1/2, 1 and 3/2 model with time-independent and time-dependent metric, respectively. In all models the overdetermined coupled system of equations for the Dyson map can be decoupled algebraic manipulations and reduces to simple linear differential equations and an equation that can be converted into the non-linear Ermakov-Pinney equation.
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exact analytical solutions for time dependent Hermitian Hamiltonian systems from static unobservable non Hermitian Hamiltonians
Physical Review A, 2017Co-Authors: Andreas Fring, Thomas FrithAbstract:We propose a procedure to obtain exact analytical solutions to the time-dependent Schrodinger equations involving explicit time-dependent Hermitian Hamiltonians from solutions to time-independent non-Hermitian Hamiltonian systems and the time-dependent Dyson relation, together with the time-dependent quasi-Hermiticity relation. We illustrate the working of this method for a simple Hermitian Rabi-type model by relating it to a non-Hermitian time-independent system corresponding to the one-site lattice Yang-Lee model.
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metric versus observable operator representation higher spin models
arXiv: Quantum Physics, 2016Co-Authors: Andreas Fring, Thomas FrithAbstract:We elaborate further on the metric representation that is obtained by transferring the time-dependence from a Hermitian Hamiltonian to the metric operator in a related non-Hermitian system. We provide further insight into the procedure on how to employ the time-dependent Dyson relation and the quasi-Hermiticity relation to solve time-dependent Hermitian Hamiltonian systems. By solving both equations separately we argue here that it is in general easier to solve the former. We solve the mutually related time-dependent Schroedinger equation for a Hermitian and non-Hermitian spin 1/2, 1 and 3/2 model with time-independent and time-dependent metric, respectively. In all models the overdetermined coupled system of equations for the Dyson map can be decoupled algebraic manipulations and reduces to simple linear differential equations and an equation that can be converted into the nonlinear Ermakov-Pinney equation.
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unitary quantum evolution for time dependent quasi Hermitian systems with nonobservable Hamiltonians
Physical Review A, 2016Co-Authors: Andreas Fring, M H Y MoussaAbstract:It has been argued that it is incompatible to maintain unitary time evolution for time-dependent non-Hermitian Hamiltonians when the metric operator is explicitly time dependent. We demonstrate here that the time-dependent Dyson equation and the time-dependent quasi-Hermiticity relation can be solved consistently in such a scenario for a time-dependent Dyson map and time-dependent metric operator, respectively. These solutions are obtained at the cost of rendering the non-Hermitian Hamiltonian to be a nonobservable operator as it ceases to be quasi-Hermitian when the metric becomes time dependent.
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minimal length in quantum mechanics and non Hermitian Hamiltonian systems
Physics Letters A, 2009Co-Authors: Bijan Bagchi, Andreas FringAbstract:Deformations of the canonical commutation relations lead to non-Hermitian momentum and position operators and therefore almost inevitably to non-Hermitian Hamiltonians. We demonstrate that such type of deformed quantum mechanical systems may be treated in a similar framework as quasi/pseudo and/or PT-symmetric systems, which have recently attracted much attention. For a newly proposed deformation of exponential type we compute the minimal uncertainty and minimal length, which are essential in almost all approaches to quantum gravity.
Ali Mostafazadeh - One of the best experts on this subject based on the ideXlab platform.
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pseudo Hermitian representation of quantum mechanics
International Journal of Geometric Methods in Modern Physics, 2010Co-Authors: Ali MostafazadehAbstract: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 ${\mathcal{P}\mathcal{T}}$, the true meaning and significance of the so-called charge operators $\mathcal{C}$ and the ${\mathcal{C}\mathcal{P}\mathcal{T}}$-inner products,...
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spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies
Physical Review Letters, 2009Co-Authors: Ali MostafazadehAbstract:Spectral singularities are spectral points that spoil the completeness of the eigenfunctions of certain non-Hermitian Hamiltonian operators. We identify spectral singularities of complex scattering potentials with the real energies at which the reflection and transmission coefficients tend to infinity, i.e., they correspond to resonances having a zero width. We show that a waveguide modeled using such a potential operates like a resonator at the frequencies of spectral singularities. As a concrete example, we explore the spectral singularities of an imaginary PT-symmetric barrier potential and demonstrate the above resonance phenomenon for a certain electromagnetic waveguide.
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application of pseudo Hermitian quantum mechanics to a pt symmetric Hamiltonian with a continuum of scattering states
Journal of Mathematical Physics, 2005Co-Authors: Ali MostafazadehAbstract:We extend the application of the techniques developed within the framework of the pseudo-Hermitian quantum mechanics to study a unitary quantum system described by an imaginary PT-symmetric potential v(x) having a continuous real spectrum. For this potential that has recently been used, in the context of optical potentials, for modeling the propagation of electromagnetic waves traveling in a waveguide half and half filled with gain and absorbing media, we give a perturbative construction of the physical Hilbert space, observables, localized states, and the equivalent Hermitian Hamiltonian. Ignoring terms of order three or higher in the non-Hermiticity parameter ζ, we show that the equivalent Hermitian Hamiltonian has the form p2∕2m+(ζ2∕2)∑n=0∞{αn(x),p2n} with αn(x) vanishing outside an interval that is three times larger than the support of v(x), i.e., in 2∕3 of the physical interaction region the potential v(x) vanishes identically. We provide a physical interpretation for this unusual behavior and comme...
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application of pseudo Hermitian quantum mechanics to a pt symmetric Hamiltonian with a continuum of scattering states
arXiv: Quantum Physics, 2005Co-Authors: Ali MostafazadehAbstract:We extend the application of the techniques developed within the framework of the pseudo-Hermitian quantum mechanics to study a unitary quantum system described by an imaginary PT-symmetric potential v(x) having a continuous real spectrum. For this potential that has recently been used, in the context of optical potentials, for modelling the propagation of electromagnetic waves travelling in a wave guide half and half filed with gain and absorbing media, we give a perturbative construction of the physical Hilbert space, observables, localized states, and the equivalent Hermitian Hamiltonian. Ignoring terms of order three or higher in the non-Hermiticity parameter zeta, we show that the equivalent Hermitian Hamiltonian has the form $\frac{p^2}{2m}+\frac{\zeta^2}{2}\sum_{n=0}^\infty\{\alpha_n(x),p^{2n}\}$ with $\alpha_n(x)$ vanishing outside an interval that is three times larger than the support of $v(x)$, i.e., in 2/3 of the physical interaction region the potential $v(x)$ vanishes identically. We provide a physical interpretation for this unusual behavior and comment on the classical limit of the system.
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pseudo hermiticity versus pt symmetry ii a complete characterization of non Hermitian Hamiltonians with a real spectrum
Journal of Mathematical Physics, 2002Co-Authors: Ali MostafazadehAbstract:We give a necessary and sufficient condition for the reality of the spectrum of a non-Hermitian Hamiltonian admitting a complete set of biorthonormal eigenvectors.
M H Y Moussa - One of the best experts on this subject based on the ideXlab platform.
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a time dependent pseudo Hermitian Hamiltonian for a cavity mode with pure imaginary frequency
Physica A-statistical Mechanics and Its Applications, 2021Co-Authors: R A Dourado, M A De Ponte, M H Y MoussaAbstract:Abstract In this work we first present a general treatment for a time-dependent (TD) pseudo-Hermitian quadratic Hamiltonian, considering an equally TD and non-Hermitian Dyson map. Then we particularize our study to a TD pseudo-Hermitian Hamiltonian describing a cavity mode with pure imaginary frequency. Our goal is to produce squeezed states of the radiation field with an infinite degree of squeezing at a finite time interval. This mechanism is prevented by Hermitian Hamiltonians which allow an infinite degree of squeezing only asymptotically in time. Beyond the specifics of the technical treatment of the problem, we also present a discussion of its experimental implementation in the domain of the radiation–matter interaction. Such a discussion, is crucial to put into perspective the relevance of the quantum mechanics of pseudo-Hermitian Hamiltonians, whose effects can go beyond those coming from Hermitian Hamiltonians. Basically, we conclude that pseudo-Hermitian Hamiltonians enable much higher pumping rates than those of Hermitian processes.
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unitary quantum evolution for time dependent quasi Hermitian systems with nonobservable Hamiltonians
Physical Review A, 2016Co-Authors: Andreas Fring, M H Y MoussaAbstract:It has been argued that it is incompatible to maintain unitary time evolution for time-dependent non-Hermitian Hamiltonians when the metric operator is explicitly time dependent. We demonstrate here that the time-dependent Dyson equation and the time-dependent quasi-Hermiticity relation can be solved consistently in such a scenario for a time-dependent Dyson map and time-dependent metric operator, respectively. These solutions are obtained at the cost of rendering the non-Hermitian Hamiltonian to be a nonobservable operator as it ceases to be quasi-Hermitian when the metric becomes time dependent.
Hideaki Obuse - One of the best experts on this subject based on the ideXlab platform.
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statistical properties of eigenvalues of the non Hermitian su schrieffer heeger model with random hopping terms
Physical Review E, 2020Co-Authors: Ken Mochizuki, Naomichi Hatano, Joshua Feinberg, Hideaki ObuseAbstract:We explore the eigenvalue statistics of a non-Hermitian version of the Su-Schrieffer-Heeger model, with imaginary on-site potentials and randomly distributed hopping terms. We find that owing to the structure of the Hamiltonian, eigenvalues can be purely real in a certain range of parameters, even in the absence of parity and time-reversal symmetry. As it turns out, in this case of purely real spectrum, the level statistics is that of the Gaussian orthogonal ensemble. This demonstrates a general feature which we clarify that a non-Hermitian Hamiltonian whose eigenvalues are purely real can be mapped to a Hermitian Hamiltonian which inherits the symmetries of the original Hamiltonian. When the spectrum contains imaginary eigenvalues, we show that the density of states (DOS) vanishes at the origin and diverges at the spectral edges on the imaginary axis. We show that the divergence of the DOS originates from the Dyson singularity in chiral-symmetric one-dimensional Hermitian systems and derive analytically the asymptotes of the DOS which is different from that in Hermitian systems.
H B Nielsen - One of the best experts on this subject based on the ideXlab platform.
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reality and hermiticity from maximizing overlap in the future included complex action theory
Progress of Theoretical and Experimental Physics, 2015Co-Authors: Keiichi Nagao, H B NielsenAbstract:In the complex action theory whose path runs over not only past but also future we study a normalized matrix element of an operator O defined in terms of the future state at the latest time TB and the past state at the earliest time TA with a proper inner product which makes a non-normal Hamiltonian at first given normal. We present a theorem which states that provided that the operator O is Q-Hermitian, i.e. Hermitian with regard to the proper inner product the normalized matrix element becomes real and time-develops under a Q-Hermitian Hamiltonian for the past and future states selected such that the absolute value of the transition amplitude from the past state to the future state is maximized. Furthermore, we give a possible procedure to formulate the Q-Hermitian Hamiltonian in terms of Q-Hermitian coordinate and momentum operators, and construct a conserved probability current density. ∗) E-mail: nagao@mx.ibaraki.ac.jp ∗∗) E-mail: hbech@nbi.dk 1 typeset using PTPTEX.cls 〈Ver.0.9〉
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reality and hermiticity from maximizing overlap in the future included complex action theory
arXiv: Quantum Physics, 2015Co-Authors: Keiichi Nagao, H B NielsenAbstract:In the complex action theory whose path runs over not only past but also future we study a normalized matrix element of an operator $\hat{\cal O}$ defined in terms of the future state at the latest time $T_B$ and the past state at the earliest time $T_A$ with a proper inner product that makes normal a given Hamiltonian that is non-normal at first. We present a theorem that states that, provided that the operator $\hat{\cal O}$ is $Q$-Hermitian, i.e., Hermitian with regard to the proper inner product, the normalized matrix element becomes real and time-develops under a $Q$-Hermitian Hamiltonian for the past and future states selected such that the absolute value of the transition amplitude from the past state to the future state is maximized. Furthermore, we give a possible procedure to formulate the $Q$-Hermitian Hamiltonian in terms of $Q$-Hermitian coordinate and momentum operators, and construct a conserved probability current density.