Investigation of PT-symmetric Hamiltonian Systems from an Alternative Point of View

Two non-Hermitian PT-symmetric Hamiltonian systems are reconsidered by means of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the PT-symmetric ones.Compared with the way converting a non-Hermitian Hamiltonian to its Hermitian counterpart,this method has the merit that keeps the Hilbert space of the non-Hermitian PT-symmetric Hamiltonian unchanged.In order to give the positive definite inner product for the PT-symmetric systems,a new operator V,instead of C,can be introduced.The operator V has the similar function to the operator C adopted normally in the PT-symmetric quantum mechanics,however,it can be constructed,as an advantage,directly in terms of Hamiltonians.The spectra of the two non-Hermitian PT-symmetric systems are obtained,which coincide with that given in literature,and in particular,the Hilbert spaces associated with positive definite inner products are worked out.     

PT-invariant Helmholtz optics and its applications to slab waveguides

In optical Integrated circuits (OIC) and optical device design (passive and active), the index of refraction profile is a coordinate dependent parameter. Specifically in slab waveguides which is a basic element in optical engineering, the guided modes and its behavior with respect to waveguide parameters can be extracted from the exact solution of the Helmholtz equation. In the general case, the numerical approach is applied for determination of the electric and magnetic fields. In this paper a suitable algorithm, using the PT-symmetric quantum mechanical approach has been given for light transmission through one-dimensional inhomogeneous media (PT-symmetric index of refraction).Received: 6 October 2003, Published online: 23 December 2003PACS: 03.65.-w Quantum mechanics - 03.50.De Classical electromagnetism, Maxwell equations - 42.82.Et Waveguides, couplers, and arrays     

Geometry of time-dependent PT-symmetric quantum mechanics

A new type of quantum theory known as time-dependent PT-symmetric quantum mechanics has received much attention recently. It has a conceptually intriguing feature of equipping the Hilbert space of a PT-symmetric system with a time-varying inner product. In this work, we explore the geometry of time-dependent PT-symmetric quantum mechanics. We find that a geometric phase can emerge naturally from the cyclic evolution of a PT-symmetric system,and further formulate a series of related differential-geometry concepts, including connection, curvature, parallel transport,metric tensor, and quantum geometric tensor. These findings constitute a useful, perhaps indispensible, tool to investigate geometric properties of PT-symmetric systems with time-varying system's parameters. To exemplify the application of our findings, we show that the unconventional geometric phase [Phys. Rev. Lett. 91 187902(2003)], which is the sum of a geometric phase and a dynamical phase proportional to the geometric phase, can be expressed as a single geometric phase unveiled in this work.     

Finding Short-Range Parity-Time Phase-Transition Points with a Neural Network

The non-Hermitian PT-symmetric system can live in either unbroken or broken PT-symmetric phase. The separation point of the unbroken and broken PT-symmetric phases is called the PT-phase-transition point.Conventionally, given an arbitrary non-Hermitian PT-symmetric Hamiltonian, one has to solve the corresponding Schrodinger equation explicitly in order to determine which phase it is actually in. Here, we propose to use artificial neural network(ANN) to determine the PT-phase-transition points for non-Hermitian PT-symmetric systems with short-range potentials. The numerical results given by ANN agree well with the literature, which shows the reliability of our new method.     

Tunable negative refraction without absorption via electromagnetically induced chirality

We show that negative refraction with minimal absorption can be obtained by means of quantum interference effects similar to electromagnetically induced transparency (EIT). Coupling a magnetic dipole transition coherently with an electric dipole transition leads to electromagnetically induced chirality, which can provide negative refraction without requiring negative permeability and also suppress absorption. This technique allows negative refraction in the optical regime at densities where the magnetic susceptibility is still small and with refraction/absorption ratios that are orders of magnitude larger than those achievable previously. Furthermore, the refractive index can be fine-tuned, which is essential for practical realization of subdiffraction-limit imaging. As with EIT, electromagnetically induced chirality should be applicable to a wide range of systems.     

Relativistic Confinement of Neutral Fermions with Partially Exactly Solvable and Exactly Solvable PT-Symmetric Potentials in the Presence of Position-Dependent Mass

The relativistic problems of neutral fermions subject to a new partially exactly solvable PT-symmetric potential and an exactly solvable PT-symmetric hyperbolic cosecant potential in 1+1 dimensions are investigated. The Dirac equation with the double-well-like mass distribution in the background of the PT-symmetric vector potential coupling can be mapped into the Schrödinger-like equation with the partially exactly solvable double-well potential. The position-dependent effective mass Dirac equation with the PT-symmetric hyperbolic cosecant potential can be mapped into the Schrödinger-like equation with the exactly solvable modified Pöschl-Teller potential. The real relativistic energy levels and corresponding spinor wavefunctions for the bound states have been given in a closed form.     

Negative refraction in ferromagnet-superconductor superlattices

Negative refraction, which reverses many fundamental aspects of classical optics, can be obtained in systems with negative magnetic permeability and negative dielectric permittivity. This Letter documents an experimental realization of negative refraction at millimeter waves, finite magnetic fields, and cryogenic temperatures utilizing a multilayer stack of ferromagnetic and superconducting thin films. In the present case the superconducting YBa2Cu3O7 layers provide negative permittivity while negative permeability is achieved via ferromagnetic (La:Sr)MnO3 layers for frequencies and magnetic fields close to the ferromagnetic resonance. In these superlattices the refractive index can be switched between positive and negative regions using external magnetic field as tuning parameter.     

Reduction of entropic uncertainty in entangled qubits system by local PT-symmetric operation

We investigate the quantum-memory-assisted entropic uncertainty for an entangled two-qubit system in a local quantum noise channel with PT-symmetric operation performing on one of the two particles. Our results show that the quantum-memory-assisted entropic uncertainty in the qubits system can be reduced effectively by the local PT-symmetric operation. Physical explanations for the behavior of the quantum-memory-assisted entropic uncertainty are given based on the property of entanglement of the qubits system and the non-locality induced by the re-normalization procedure for the non-Hermitian PT-symmetric operation.     

Nonlocality in PT-symmetric waveguide arrays with gain and loss

We demonstrate that light propagation in waveguide arrays that include PT-symmetric structures can exhibit strongly nonlocal sensitivity to topology of the array at fixed other parameters. We consider an array composed of lossless waveguides, that includes a pair of PT-symmetric waveguides with balanced gain and loss, and reveal that PT-symmetry breaking thresholds are different for planar and circular array configurations. These results demonstrate that PT-symmetric structures can offer new regimes for optical beam shaping compared to conservative structures.     

Entropic uncertainty relation of a qubit-qutrit Heisenberg spin model and its steering

We investigate the quantum-memory-assisted entropic uncertainty relation (QMA-EUR) in a Heisenberg XYZ mixed-spin (1/2, 1) model. Coupling strength, Dzyaloshinskii-Moriya (DM) interaction and inhomogeneous magnetic field, respectively, contributing to QMA-EUR by a thermal entanglement in the hybrid-spin model are studied in detail. Furthermore, we compare the uncertainty of the bipartite hybrid model with those of qubit-qubit and qutrit-qutrit systems. Meanwhile, the effects of local PT-symmetric operation and weak measurement on the steering of entropic uncertainty are analyzed. We find that the local PT-symmetric operation can reduce the entropic uncertainty, and the entropic uncertainty can also be decreased by weak measurement reversal.     

缺陷对左手材料负折射的调控行为

采用电路板刻蚀技术制备左手材料样品，由劈尖法分别研究了无缺陷和引入缺陷时左手材料的负折射特性.实验结果表明，引入两种点缺陷材料与无缺陷材料的功率峰值之比分别为1.035和1.256，且负折射率的绝对值分别增大了9.6%和19.6%；引入空位缺陷材料与无缺陷材料的功率峰值之比最大为1.973，最小为0.364，负折射率的绝对值最大增大了68.3%，最小增大了9.6%.缺陷的存在改变了左手材料周期性结构，形成新的电磁谐振条件，使其负折射率和功率峰值发生了变化，实现了对左手材料负折射率的调控. 关键词： 左手材料 负折射 缺陷     

Extension of PT-symmetric quantum mechanics to the Dirac theory with position-dependent mass

We present a new method to construct the exactly solvable PT-symmetric potentials within the framework of the position-dependent effective mass Dirac equation with the vector potential coupling scheme in 1 + 1 dimensions. In order to illustrate the procedure, we produce three PT-symmetric potentials as examples, which are PT-symmetric harmonic oscillator-like potential, PT-symmetric potential with the form of a linear potential plus an inversely linear potential, and PT-symmetric kink-like potential, respectively. The real relativistic energy levels and corresponding spinor components for the bound states are obtained by using the basic concepts of the supersymmetric quantum mechanics formalism and function analysis method.     

Restoration of Coherence by Local PT-Symmetric Operator

In this work, we mainly investigate effect of PT-symmetric operation on the dynamics of the relative entropy of coherence for a two-level system within non-Markovian environments, and put forward a feasible physical scheme to recover coherence by means of optimal PT-symmetric operation. The results show that the damaged quantum coherence can be restored to a large extent. Furthermore, the freezing phenomenon of the coherence can be detected by using the optimal PT-symmetric operation strength within the non-Markovian environments.

     

Bidirectional Partial Generalized Synchronization in Chaotic and Hyperchaotic Systems via a New Scheme

In this paper, a bidirectional partial generalized (lag, complete, and anticipated) synchronization of a class of continuous-time systems is defined. Then based on the active control idea, a new systematic and concrete scheme is developed to achieve bidirectional partial generalized (lag, complete, and anticipated) synchronization between two chaotic systems or between chaotic and hyperchaotic systems. With the help of symbolic-numerical computation, we choose the modified Chua system, Lorenz system, and the hyperchaotic Tamasevicius-Namajunas-Cenys system to illustrate the proposed scheme. Numerical simulations are used to verify the effectiveness of the proposed scheme. It is interesting that partial chaos synchronization not only can take place between two chaotic systems, but also can take place between chaotic and hyperchaotic systems. The proposed scheme can also be extended to research bidirectional partial generalized (lag, complete, and anticipated) synchronization between other dynamical systems.     

Effect of PT-Symmetric Operator on Coherence Under the Non-Markovian Environments

In this work, we mainly investigate effect of PT-symmetric operation on the dynamic behavior of the relative entropy of coherence for a two-level system within non-Markovian environments and put forward a feasible physical scheme to recover coherence by utilizing optimal PT-symmetric operation. The results show that the damaged quantum coherence can be effectively restored under influence of the non-Markovian regimes. Furthermore, the freezing phenomenon of the coherence can be detected by using the optimal PT-symmetric operation strength within the non-Markovian environments.

     

用惠更斯作图法讨论单轴晶体中双波长双负折射现象

近年的研究表明,在右手性的各向异性材料(如普通晶体)中一定条件下也会形成负折射,用惠更斯作图法可以清楚地说明普通单轴晶体中出现的这种情况.本文用惠更斯作图法说明在适当条件下,一束含有两种波长的光入射到单轴晶体中会出现双正折射、一正一负折射、双负折射等物理现象.以钒酸钇晶体为例,计算了发生负折射现象的临界入射角.本文的工...     

Visualization of branch points in PT-symmetric waveguides

The visualization of an exceptional point in a PT-symmetric directional coupler (DC) is demonstrated. In such a system the exceptional point can be probed by varying only a single parameter. Using the Rayleigh-Schr?dinger perturbation theory we prove that the spectrum of a PT-symmetric Hamiltonian is real as long as the radius of convergence has not been reached. We also show how one can use a PT-symmetric directional coupler to measure the radius of convergence for non-PT-symmetric structures. For such systems the physical meaning of the rather mathematical term radius of convergence is exemplified.     

Bifurcated overtones of one-way localized Fabry-Pérot resonances in parity-time symmetric optical lattices

Since the first observation of parity-time(PT) symmetry in optics, varied interesting phenomena have been discovered in both theories and experiments, such as PT phase transition and unidirectional invisibility, which turns PT-symmetric optics into a hotspot in research. Here, we report on the one-way localized Fabry-Pérot(FP) resonance, where a welldesigned PT optical resonator may operate at exceptional points with bidirectional transparency but unidirectional field localization. Overtones of such one-way localized FP resonance can be classified into a blue shifted branch and a red shifted branch. Therefore, the fundamental resonant frequency is not the lowest one. We find that the spatial field distributions of the overtones at the same absolute order are almost the same, even though their frequencies are quite different.     

Asymmetry in Negative Refraction of Nonlinear Waves

Recently, inwardly propagating waves (called antiwaves, AWs) in nonlinear oscillatory systems have attracted much attention. An interesting negative refraction phenomenon has been observed in a bidomain system where one medium supports forwardly propagating waves (normal waves, NWs) and the other AWs. In this paper we find that negative refraction (NR) in nonlinear media has an asymmetric property, i.e., NR can be observed only by applying wave source withproper frequency to one medium, but not the other. Moreover, NR appears always when the incident waves are dense and the refractional waves are sparse. This asymmetry is a particular feature for nonlinear NR, which can neither be observed in linear refraction processes (both positive and negative refractions) nor in nonlinear positive refraction. The mechanism underlying the asymmetry of nonlinear NR are fully understood based on the competition of nonlinear waves.     

Oscillatory stability of quantum droplets in PT-symmetric optical lattice

We study stability and collisions of quantum droplets(QDs) forming in a binary bosonic condensate trapped in parity-time (PT)-symmetric optical lattices. It is found that the stability of QDs in the PT-symmetric system depends strongly on the values of the imaginary part W_0 of the PT-symmetric optical lattices, self-repulsion strength g, and the condensate norm N. As expected,the PT-symmetric QDs are entirely unstable in the broken PT-symmetric phase. However, the PT-symmetric QDs exhibit oscillatory stability with the increase of N and g in the unbroken PT-symmetric phase. Finally, collisions between PT-symmetric QDs are considered. The collisions of droplets with unequal norms are completely different from that in free space. Besides, a stable PT-symmetric QDs collides with an unstable ones tend to merge into breathers after the collision.