Research

8 pages, 487 KiB

Open AccessArticle

Shannon Entropy and Diffusion Coefficient in Parity-Time Symmetric Quantum Walks

by Zhiyu Tian, Yang Liu and Le Luo

Entropy 2021, 23(9), 1145; https://doi.org/10.3390/e23091145 - 31 Aug 2021

Viewed by 2080

Non-Hermitian topological edge states have many intriguing properties, however, to date, they have mainly been discussed in terms of bulk–boundary correspondence. Here, we propose using a bulk property of diffusion coefficients for probing the topological states and exploring their dynamics. The diffusion coefficient [...] Read more.

Non-Hermitian topological edge states have many intriguing properties, however, to date, they have mainly been discussed in terms of bulk–boundary correspondence. Here, we propose using a bulk property of diffusion coefficients for probing the topological states and exploring their dynamics. The diffusion coefficient was found to show unique features with the topological phase transitions driven by parity–time (PT)-symmetric non-Hermitian discrete-time quantum walks as well as by Hermitian ones, despite the fact that artificial boundaries are not constructed by an inhomogeneous quantum walk. For a Hermitian system, a turning point and abrupt change appears in the diffusion coefficient when the system is approaching the topological phase transition, while it remains stable in the trivial topological state. For a non-Hermitian system, except for the feature associated with the topological transition, the diffusion coefficient in the PT-symmetric-broken phase demonstrates an abrupt change with a peak structure. In addition, the Shannon entropy of the quantum walk is found to exhibit a direct correlation with the diffusion coefficient. The numerical results presented herein may open up a new avenue for studying the topological state in non-Hermitian quantum walk systems. Full article

(This article belongs to the Special Issue Quantum Dynamics with Non-Hermitian Hamiltonians)

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22 pages, 1236 KiB

Open AccessArticle

Evolution of a Non-Hermitian Quantum Single-Molecule Junction at Constant Temperature

by Andrea Grimaldi, Alessandro Sergi and Antonino Messina

Entropy 2021, 23(2), 147; https://doi.org/10.3390/e23020147 - 25 Jan 2021

Cited by 8 | Viewed by 1780

Abstract

This work concerns the theoretical description of the quantum dynamics of molecular junctions with thermal fluctuations and probability losses. To this end, we propose a theory for describing non-Hermitian quantum systems embedded in constant-temperature environments. Along the lines discussed in [A. Sergi et [...] Read more.

This work concerns the theoretical description of the quantum dynamics of molecular junctions with thermal fluctuations and probability losses. To this end, we propose a theory for describing non-Hermitian quantum systems embedded in constant-temperature environments. Along the lines discussed in [A. Sergi et al., Symmetry 10 518 (2018)], we adopt the operator-valued Wigner formulation of quantum mechanics (wherein the density matrix depends on the points of the Wigner phase space associated to the system) and derive a non-linear equation of motion. Moreover, we introduce a model for a non-Hermitian quantum single-molecule junction (nHQSMJ). In this model the leads are mapped to a tunneling two-level system, which is in turn coupled to a harmonic mode (i.e., the molecule). A decay operator acting on the two-level system describes phenomenologically probability losses. Finally, the temperature of the molecule is controlled by means of a Nosé-Hoover chain thermostat. A numerical study of the quantum dynamics of this toy model at different temperatures is reported. We find that the combined action of probability losses and thermal fluctuations assists quantum transport through the molecular junction. The possibility that the formalism here presented can be extended to treat both more quantum states (

\sim 10

) and many more classical modes or atomic particles (

\sim 10^{3} - 10^{5}

) is highlighted. Full article

(This article belongs to the Special Issue Quantum Dynamics with Non-Hermitian Hamiltonians)

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19 pages, 389 KiB

Open AccessArticle

PT -Symmetric Potentials from the Confluent Heun Equation

by Géza Lévai

Entropy 2021, 23(1), 68; https://doi.org/10.3390/e23010068 - 03 Jan 2021

Cited by 2 | Viewed by 1758

Abstract

We derive exactly solvable potentials from the formal solutions of the confluent Heun equation and determine conditions under which the potentials possess

PT

symmetry. We point out that for the implementation of

PT

symmetry, the symmetrical canonical form of the Heun equation is [...] Read more.

We derive exactly solvable potentials from the formal solutions of the confluent Heun equation and determine conditions under which the potentials possess

PT

symmetry. We point out that for the implementation of

PT

symmetry, the symmetrical canonical form of the Heun equation is more suitable than its non-symmetrical canonical form. The potentials identified in this construction depend on twelve parameters, of which three contribute to scaling and shifting the energy and the coordinate. Five parameters control the

z (x)

function that detemines the variable transformation taking the Heun equation into the one-dimensional Schrödinger equation, while four parameters play the role of the coupling coefficients of four independently tunable potential terms. The potentials obtained this way contain Natanzon-class potentials as special cases. Comparison with the results of an earlier study based on potentials obtained from the non-symmetrical canonical form of the confluent Heun equation is also presented. While the explicit general solutions of the confluent Heun equation are not available, the results are instructive in identifying which potentials can be obtained from this equation and under which conditions they exhibit

PT

symmetry, either unbroken or broken. Full article

(This article belongs to the Special Issue Quantum Dynamics with Non-Hermitian Hamiltonians)

18 pages, 575 KiB

Open AccessEditor’s ChoiceArticle

Two-Qubit Entanglement Generation through Non-Hermitian Hamiltonians Induced by Repeated Measurements on an Ancilla

by Roberto Grimaudo, Antonino Messina, Alessandro Sergi, Nikolay V. Vitanov and Sergey N. Filippov

Entropy 2020, 22(10), 1184; https://doi.org/10.3390/e22101184 - 20 Oct 2020

Cited by 18 | Viewed by 3356

Abstract

In contrast to classical systems, actual implementation of non-Hermitian Hamiltonian dynamics for quantum systems is a challenge because the processes of energy gain and dissipation are based on the underlying Hermitian system–environment dynamics, which are trace preserving. Recently, a scheme for engineering non-Hermitian [...] Read more.

In contrast to classical systems, actual implementation of non-Hermitian Hamiltonian dynamics for quantum systems is a challenge because the processes of energy gain and dissipation are based on the underlying Hermitian system–environment dynamics, which are trace preserving. Recently, a scheme for engineering non-Hermitian Hamiltonians as a result of repetitive measurements on an ancillary qubit has been proposed. The induced conditional dynamics of the main system is described by the effective non-Hermitian Hamiltonian arising from the procedure. In this paper, we demonstrate the effectiveness of such a protocol by applying it to physically relevant multi-spin models, showing that the effective non-Hermitian Hamiltonian drives the system to a maximally entangled stationary state. In addition, we report a new recipe to construct a physical scenario where the quantum dynamics of a physical system represented by a given non-Hermitian Hamiltonian model may be simulated. The physical implications and the broad scope potential applications of such a scheme are highlighted. Full article

(This article belongs to the Special Issue Quantum Dynamics with Non-Hermitian Hamiltonians)

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13 pages, 2138 KiB

Open AccessArticle

Entanglement of Pseudo-Hermitian Random States

by Cleverson Andrade Goulart and Mauricio Porto Pato

Entropy 2020, 22(10), 1109; https://doi.org/10.3390/e22101109 - 30 Sep 2020

Viewed by 1938

Abstract

In a recent paper (A. Fring and T. Frith, Phys. Rev A 100, 101102 (2019)), a Dyson scheme to deal with density matrix of non-Hermitian Hamiltonians has been used to investigate the entanglement of states of a PT-symmetric bosonic system. They [...] Read more.

In a recent paper (A. Fring and T. Frith, Phys. Rev A 100, 101102 (2019)), a Dyson scheme to deal with density matrix of non-Hermitian Hamiltonians has been used to investigate the entanglement of states of a PT-symmetric bosonic system. They found that von Neumann entropy can show a different behavior in the broken and unbroken regime. We show that their results can be recast in terms of an abstract model of pseudo-Hermitian random matrices. It is found however that although the formalism is practically the same, the entanglement is not of Fock states but of Bell states. Full article

(This article belongs to the Special Issue Quantum Dynamics with Non-Hermitian Hamiltonians)

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11 pages, 2028 KiB

Open AccessEditor’s ChoiceArticle

Relative Entropy as a Measure of Difference between Hermitian and Non-Hermitian Systems

by Kabgyun Jeong, Kyu-Won Park and Jaewan Kim

Entropy 2020, 22(8), 809; https://doi.org/10.3390/e22080809 - 23 Jul 2020

Cited by 3 | Viewed by 3384

Abstract

We employ the relative entropy as a measure to quantify the difference of eigenmodes between Hermitian and non-Hermitian systems in elliptic optical microcavities. We have found that the average value of the relative entropy in the range of the collective Lamb shift is [...] Read more.

We employ the relative entropy as a measure to quantify the difference of eigenmodes between Hermitian and non-Hermitian systems in elliptic optical microcavities. We have found that the average value of the relative entropy in the range of the collective Lamb shift is large, while that in the range of self-energy is small. Furthermore, the weak and strong interactions in the non-Hermitian system exhibit rather different behaviors in terms of the relative entropy, and thus it displays an obvious exchange of eigenmodes in the elliptic microcavity. Full article

(This article belongs to the Special Issue Quantum Dynamics with Non-Hermitian Hamiltonians)

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18 pages, 2357 KiB

Open AccessArticle

Non-Hermitian Floquet Phases with Even-Integer Topological Invariants in a Periodically Quenched Two-Leg Ladder

by Longwen Zhou

Entropy 2020, 22(7), 746; https://doi.org/10.3390/e22070746 - 07 Jul 2020

Cited by 13 | Viewed by 3115

Abstract

Periodically driven non-Hermitian systems could possess exotic nonequilibrium phases with unique topological, dynamical, and transport properties. In this work, we introduce an experimentally realizable two-leg ladder model subjecting to both time-periodic quenches and non-Hermitian effects, which belongs to an extended CII symmetry class. [...] Read more.

Periodically driven non-Hermitian systems could possess exotic nonequilibrium phases with unique topological, dynamical, and transport properties. In this work, we introduce an experimentally realizable two-leg ladder model subjecting to both time-periodic quenches and non-Hermitian effects, which belongs to an extended CII symmetry class. Due to the interplay between drivings and nonreciprocity, rich non-Hermitian Floquet topological phases emerge in the system, with each of them characterized by a pair of even-integer topological invariants

(w_{0}, w_{π}) \in 2 Z \times 2 Z

. Under the open boundary condition, these invariants further predict the number of zero- and

π

-quasienergy modes localized around the edges of the system. We finally construct a generalized version of the mean chiral displacement, which could be employed as a dynamical probe to the topological invariants of non-Hermitian Floquet phases in the CII symmetry class. Our work thus introduces a new type of non-Hermitian Floquet topological matter, and further reveals the richness of topology and dynamics in driven open systems. Full article

(This article belongs to the Special Issue Quantum Dynamics with Non-Hermitian Hamiltonians)

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24 pages, 459 KiB

Open AccessArticle

Time-Dependent Pseudo-Hermitian Hamiltonians and a Hidden Geometric Aspect of Quantum Mechanics

by Ali Mostafazadeh

Entropy 2020, 22(4), 471; https://doi.org/10.3390/e22040471 - 20 Apr 2020

Cited by 20 | Viewed by 3452

Abstract

A non-Hermitian operator H defined in a Hilbert space with inner product

⟨ \cdot | \cdot ⟩

may serve as the Hamiltonian for a unitary quantum system if it is

η

-pseudo-Hermitian for a metric operator (positive-definite automorphism)

η

. The latter defines [...] Read more.

A non-Hermitian operator H defined in a Hilbert space with inner product

⟨ \cdot | \cdot ⟩

may serve as the Hamiltonian for a unitary quantum system if it is

η

-pseudo-Hermitian for a metric operator (positive-definite automorphism)

η

. The latter defines the inner product

⟨ \cdot | η \cdot ⟩

of the physical Hilbert space

H_{η}

of the system. For situations where some of the eigenstates of H depend on time,

η

becomes time-dependent. Therefore, the system has a non-stationary Hilbert space. Such quantum systems, which are also encountered in the study of quantum mechanics in cosmological backgrounds, suffer from a conflict between the unitarity of time evolution and the unobservability of the Hamiltonian. Their proper treatment requires a geometric framework which clarifies the notion of the energy observable and leads to a geometric extension of quantum mechanics (GEQM). We provide a general introduction to the subject, review some of the recent developments, offer a straightforward description of the Heisenberg-picture formulation of the dynamics for quantum systems having a time-dependent Hilbert space, and outline the Heisenberg-picture formulation of dynamics in GEQM. Full article

(This article belongs to the Special Issue Quantum Dynamics with Non-Hermitian Hamiltonians)

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21 pages, 1102 KiB

Open AccessEditor’s ChoiceArticle

Non-Hermitian Hamiltonians and Quantum Transport in Multi-Terminal Conductors

by Nikolay M. Shubin, Alexander A. Gorbatsevich and Gennadiy Ya. Krasnikov

Entropy 2020, 22(4), 459; https://doi.org/10.3390/e22040459 - 17 Apr 2020

Cited by 3 | Viewed by 3270

Abstract

We study the transport properties of multi-terminal Hermitian structures within the non-equilibrium Green’s function formalism in a tight-binding approximation. We show that non-Hermitian Hamiltonians naturally appear in the description of coherent tunneling and are indispensable for the derivation of a general compact expression [...] Read more.

We study the transport properties of multi-terminal Hermitian structures within the non-equilibrium Green’s function formalism in a tight-binding approximation. We show that non-Hermitian Hamiltonians naturally appear in the description of coherent tunneling and are indispensable for the derivation of a general compact expression for the lead-to-lead transmission coefficients of an arbitrary multi-terminal system. This expression can be easily analyzed, and a robust set of conditions for finding zero and unity transmissions (even in the presence of extra electrodes) can be formulated. Using the proposed formalism, a detailed comparison between three- and two-terminal systems is performed, and it is shown, in particular, that transmission at bound states in the continuum does not change with the third electrode insertion. The main conclusions are illustratively exemplified by some three-terminal toy models. For instance, the influence of the tunneling coupling to the gate electrode is discussed for a model of quantum interference transistor. The results of this paper will be of high interest, in particular, within the field of quantum design of molecular electronic devices. Full article

(This article belongs to the Special Issue Quantum Dynamics with Non-Hermitian Hamiltonians)

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11 pages, 494 KiB

Open AccessArticle

Control of the Geometric Phase in Two Open Qubit–Cavity Systems Linked by a Waveguide

by Abdel-Baset A. Mohamed and Ibtisam Masmali

Entropy 2020, 22(1), 85; https://doi.org/10.3390/e22010085 - 10 Jan 2020

Cited by 3 | Viewed by 2193

Abstract

We explore the geometric phase in a system of two non-interacting qubits embedded in two separated open cavities linked via an optical fiber and leaking photons to the external environment. The dynamical behavior of the generated geometric phase is investigated under the physical [...] Read more.

We explore the geometric phase in a system of two non-interacting qubits embedded in two separated open cavities linked via an optical fiber and leaking photons to the external environment. The dynamical behavior of the generated geometric phase is investigated under the physical parameter effects of the coupling constants of both the qubit–cavity and the fiber–cavity interactions, the resonance/off-resonance qubit–field interactions, and the cavity dissipations. It is found that these the physical parameters lead to generating, disappearing and controlling the number and the shape (instantaneous/rectangular) of the geometric phase oscillations. Full article

(This article belongs to the Special Issue Quantum Dynamics with Non-Hermitian Hamiltonians)

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20 pages, 358 KiB

Open AccessArticle

Theory of Response to Perturbations in Non-Hermitian Systems Using Five-Hilbert-Space Reformulation of Unitary Quantum Mechanics

by Miloslav Znojil

Entropy 2020, 22(1), 80; https://doi.org/10.3390/e22010080 - 09 Jan 2020

Cited by 5 | Viewed by 2505

Abstract

Non-Hermitian quantum-Hamiltonian-candidate combination

H_{λ}

of a non-Hermitian unperturbed operator

H = H_{0}

with an arbitrary “small” non-Hermitian perturbation

λ W

is given a mathematically consistent unitary-evolution interpretation. The formalism generalizes the conventional constructive Rayleigh–Schrödinger perturbation expansion technique. It is sufficiently general [...] Read more.

Non-Hermitian quantum-Hamiltonian-candidate combination

H_{λ}

of a non-Hermitian unperturbed operator

H = H_{0}

with an arbitrary “small” non-Hermitian perturbation

λ W

is given a mathematically consistent unitary-evolution interpretation. The formalism generalizes the conventional constructive Rayleigh–Schrödinger perturbation expansion technique. It is sufficiently general to take into account the well known formal ambiguity of reconstruction of the correct physical Hilbert space of states. The possibility of removal of the ambiguity via a complete, irreducible set of observables is also discussed. Full article

(This article belongs to the Special Issue Quantum Dynamics with Non-Hermitian Hamiltonians)

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