Soliton structures in the (1+1)-dimensional Ginzburg–Landau equation with a parity-time-symmetric potential in ultrafast optics

In this paper, the （l＋l）-dimensional variable-coefficient complex Ginzburg-Landau （CGL） equation with a parity- time （PT） symmetric potential U（x） is investigated. Although the CGL equations with a PT-symmetric potential are less reported analytically, the analytic solutions for the CGL equation are obtained with the bilinear method in this paper. Via the derived solutions, some soliton structures are presented with corresponding parameters, and the influences of them are analyzed and studied. The single-soliton structure is numerically verified, and its stability is analyzed against additive and multiplicative noises. In particular, we study the soliton dynamics under the impact of the PT-symmetric potential. Results show that the PT-symmetric potential plays an important role for obtaining soliton structures in ultrafast optics, and we can design fiber lasers and all-optical switches depending on the different amplitudes of soliton-like structures.

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.

Kármán vortex street in a spin–orbit-coupled Bose–Einstein condensate with PT symmetry

The dynamics of spin–orbit-coupled Bose–Einstein condensate with parity-time symmetry through a moving obstacle potential is simulated numerically. In the miscible two-component condensate, the formation of the Kármán vortex street is observed in one component, while ‘the half-quantum vortex street' is observed in the other component. Other patterns of vortex shedding, such as oblique vortex dipoles, V-shaped vortex pairs, irregular turbulence, and combined modes of various wakes, can also be found. The ratio of inter-vortex spacing in one row to the distance between vortex rows is approximately0.18, which is less than the stability condition 0.28 of classical fluid. The drag force acting on the obstacle potential is simulated. The parametric regions of Kármán vortex street and other vortex patterns are calculated. The range of Kármán vortex street is surrounded by the region of combined modes. In addition, spin–orbit coupling disrupts the symmetry of the system and the gain-loss affects the local particle distribution of the system, which leads to the local symmetry breaking of the system, and finally influences the stability of the Kármán vortex street. Finally, we propose an experimental protocol to realize the Kármán vortex street in a system.

非厄米PT对称多通道光波导体系的调控及动力学

宇称-时间(PARITY-TIME PT)对称光学波导系统中有关PT对称性的调控主要通过周期性变化的复折射率材料来实现.本文中我们给出了一种对于已耦合的双通道波导体系中额外引入周期调制的中性波导操纵PT对称性的构型,通过高频近似将系统转换为重整化参数后的等效未调制系统后,发现可以通过额外引入单频周期调制的光波导去调控体系的PT对称性,并且讨论了周期调制的非厄米三通道波导体系的能谱特性.最后通过解析和数值的方法给出了光在非厄米三通道光学波导中的动力学演化.

Disorder in parity-time symmetric quantum walks

We experimentally investigate the impact of static disorder and dynamic disorder on the non-unitary dynamics of parity-time(PT)-symmetric quantum walks.Via temporally alternating photon losses in an interferometric network,we realize the passive PT-symmetric quantum dynamics for single photons.Controllable coin operations allow us to simulate different environmental influences,which result in three different behaviors of quantum walkers:a standard ballistic spread,a diffusive behavior,and a localization,respectively,in a PT-symmetric quantum walk architecture.

Parity-time symmetric acoustic system constructed by piezoelectric composite plates with active external circuits

Researches on parity-time(PT)symmetry in acoustic field can provide an efficient platform for controlling the travelling acoustic waves with balanced loss and gain.Here,we report a feasible design of PT-symmetric system constructed by piezoelectric composite plates with two different active external circuits.By judiciously adjusting the resistances and inductances in the external circuits,we obtain the exceptional point due to the spontaneous breaking of PT symmetry at the desired frequencies and can observe the unidirectional invisibility.Moreover,the system can be at PT exact phase or broken phase at the same frequency in the same structure by merely adjusting the external circuits,which represents the active control that makes the acoustic manipulation more convenient.Our study may provide a feasible way for manipulating acoustic waves and inspire the application of piezoelectric composite materials in acoustic structures.

Topology of a parity-time symmetric non-Hermitian rhombic lattice

We investigate the topological properties of a trimerized parity–time(PT)symmetric non-Hermitian rhombic lattice.Although the system is PT-symmetric,the topology is not inherited from the Hermitian lattice;in contrast,the topology can be altered by the non-Hermiticity and depends on the couplings between the sublattices.The bulk–boundary correspondence is valid and the Bloch bulk captures the band topology.Topological edge states present in the two band gaps and are predicted from the global Zak phase obtained through the Wilson loop approach.In addition,the anomalous edge states compactly localize within two diamond plaquettes at the boundaries when all bands are flat at the exceptional point of the lattice.Our findings reveal the topological properties of the𝒫PT-symmetric non-Hermitian rhombic lattice and shed light on the investigation of multi-band non-Hermitian topological phases.

Extraordinary transmission and reflection in PT-symmetric two-segment-connected triangular optical waveguide networks with perfect and broken integer waveguide length ratios

By adjusting the waveguide length ratio, we study the extraordinary characteristics of electromagnetic waves propagating in one-dimensional(1D) parity-time-symmetric(PT-symmetric) two-segment-connected triangular optical waveguide networks with perfect and broken integer waveguide length ratios respectively. It is found that the number and the corresponding frequencies of the extremum spontaneous PT-symmetric breaking points are dependent on the waveguide length ratio. Near the extremum breaking points, ultrastrong extraordinary transmissions are created and the maximal can arrive at, respectively, 2.4079 × 10^14 and 4.3555 × 10^13 in both kinds of networks. However, bidirectional invisibility can only be produced by the networks with broken integer waveguide length ratio, whose mechanism is explained in detail from the perspective of photonic band structure. The findings of this work can be useful optical characteristic control in the fabrication of PT-symmetric optical waveguide networks, which possesses great potential in designing optical amplifiers,optical energy saver devices, and special optical filters.

Dynamics of Airy beams in parity–time symmetric optical lattices

We investigate the dynamics of airy beams propagating in the parity–time(PT) symmetric optical lattices in linear and nonlinear regimes, respectively. For the linear propagation, the position of the channel guided by the PT lattice can be shifted by tuning the lattice frequency. The underlying physical mechanism of this phenomenon is also discussed. An interesting phenomenon is found in the nonlinear regime in that the airy beam becomes a tilt channel with several Rayleigh lengths. These findings create new opportunities for optical steering and manipulations.

铯原子系统中部分PT对称的研究

本文主要研究铯原子Λ型三能级原子的部分PT对称和相变,利用铯原子基态|6S_(1/2),F=3>、|6S_(1/2),F=4>和激发态|6P_(3/2),F'=4>组成Λ型三能级原子系统,由失谐⊿3=607 MHz的探测光与耦合光形成双光子拉曼吸收,构成损耗通道.增加了共振作用于能级6S_(1/2),F=3>与|6P_(3/2),F'=4>跃迁的泵浦光改变两个基态能级的布居,从而使Λ型三能级系统的吸收减小,在一定条件下形成原子系统的增益通道,从而构成部分PT对称的原子系统.实验中通过改变耦合光和探测光的腰斑比σ,观察到部分PT对称系统中由对称向破缺相的转变.此外研究了探测光束强度分布的不对称程度Dasym,精确地测量了部分PT对称的破缺点,理论计算与实验测量结果相符.本文所报道的部分PT对称性及其相变的结果,为主动操纵非厄米系统中的多维激光束开辟了一条途径,并在设计激光不同部分的光放大和衰减光学器件方面具有潜在的应用价值.

On-chip tunable parity‐time symmetric optoelectronic oscillator

Parity‐time(PT)symmetry breaking offers mode selection capability for facilitating single‐mode oscillation in the optoelectronic oscillator(OEO)loop.However,most OEO implementations depend on discrete devices,which impedes proliferation due to size,weight,power consumption,and cost.In this work,we propose and experimentally demonstrate an on-chip tunable PT‐symmetric OEO.A tunable microwave photonic filter,a PT‐symmetric mode‐selective architecture,and two photodetectors are integrated on a silicon‐on‐insulator chip.By exploiting an on‐chip Mach–Zehnder interferometer to match the gain and loss of two mutually coupled optoelectronic loops,single‐mode oscillation can be obtained.In the experiment,the oscillation frequency of the on-chip tunable PT‐symmetric OEO can be tuned from 0 to 20 GHz.To emulate the integrated case,the OEO loop length is minimized,and no extra-long fiber is used in the experiment.When the oscillation frequency is 13.67 GHz,the single‐sideband phase noise at 10-kHz offset frequency is−80.96 dBc∕Hz and the side mode suppression ratio is 46 dB.The proposed on-chip tunable PT‐symmetric OEO significantly reduces the footprint of the system and enhances mode selection.

Analytic solutions for generalized PT-symmetric Rabi models

We theoretically investigate the exact solutions for generalized parity–time(PT)-reversal-symmetric Rabi models driven by external fields with monochromatic periodic, linear, and parabolic forms, respectively. The corresponding exact solutions are presented in terms of the confluent Heun equations without any approximation. In principle, the analytic solutions derived here are valid in the whole parameter space. Such a kind of study may offer potential coherent control schemes of the PT-symmetric two-level systems.

Multipeaked Solitons in Parity-Time Complex Superlattice with Dual Periods

The existence and stability of multipeaked solitons are investigated in a parity-time symmetric superlattice with dual periods under both self-focusing and self-defocusing nonlinearity. For self-defocusing nonlinearity, dipole solitons with low power and all the odd-peak solitons can exist stably in the first gap, while dipole solitons with high power and even-peak （except two） solitons are unstable. For self-focusing nonlinearity, even-peak out-of-phase solitons can propagate stably in the infinite gap, while odd-peak in-phase solitons are unstable.

Dynamical Evolution of an Effective Two-Level System with PT Symmetry

We investigate the dynamics of parity-and time-reversal（PT） symmetric two-energy-level atoms in the presence of two optical and one radio-frequency fields. The strength and relative phase of fields can drive the system from the unbroken to the broken PT symmetric regions. Compared with the Hermitian model, Rabi-type oscillation is still observed, and the oscillation characteristics are also adjusted by the strength and relative phase in the region of the unbroken symmetry. At the exception point, the oscillation breaks down. To better understand the underlying properties we study the effective Bloch dynamics and find that the emergence of the PT components of the fixed points is the feature of the PT symmetry breaking and the projections in the x–y plane can be controlled with high flexibility compared with the standard two-level system with the PT symmetry. It helps to study the dynamic behavior of the complex PT symmetric model.

Solvability of a class of PT-symmetric non-Hermitian Hamiltonians：Bethe ansatz method

We use the Bethe ansatz method to investigate the Schrdinger equation for a class of PT-symmetric non-Hermitian Hamiltonians. Elementary exact solutions for the eigenvalues and the corresponding wave functions are obtained in terms of the roots of a set of algebraic equations. Also, it is shown that the problems possess sl（2） hidden symmetry and then the exact solutions of the problems are obtained by employing the representation theory of sl（2） Lie algebra. It is found that the results of the two methods are the same.

PT对称多线圈并联无线电能传输系统参数优化

针对宇称时间(parity-time,PT)对称多线圈并联无线电能传输(wireless power transfer,WPT)系统参数相互关联,系统参数配置困难,提出一种基于PT对称的多线圈并联WPT系统参数优化设计方法。建立基于PT对称多线圈并联WPT系统数学模型,给出系统输出功率、传输效率以及有效传输距离的一般性影响规律表达式。采用一种非线性规划和遗传算法相结合的优化方法,以输出功率为目标函数,以传输效率和有效传输距离为约束条件,对系统发射-接受线圈数及谐振参数进行寻优。根据优化结果搭建实验样机,实验结果表明系统在满足所需传输效率及有效传输距离下,系统功率输出达到设计要求,验证了理论分析和优化方法的有效性。

傅里叶变换求解一维PT对称delta势阱中的束缚态

具有PT对称性的系统是量子力学研究前沿热点.一维PT对称delta势阱中束缚态问题是PT系统最简化模型,对其精确求解是具有科研价值的教学问题.本文通过求解一维PT对称delta势的定态薛定谔方程研究了PT对称性对系统束缚态以及本征能量的影响.首先,采用傅里叶变换对一维PT对称delta势的定态薛定谔方程进行了求解并讨论.其次,通过数值手段确定了一维PT对称delta势阱中束缚态的轮廓以及系统的本征能量.最后,分析和探讨了PT对称势强度与本征能量的关系.本文把量子力学教学中传统的厄米delta势阱本征值问题扩展到非厄米delta势阱问题,是对量子力学教学必做习题的有益且及时地扩展.

SS型PT对称磁耦合无线电能传输系统特性分析

为深入分析PT对称系统的电路特性,给实际系统设计提供理论准则,基于电路理论,从系统的工作原理出发对系统进行建模,证实了PT对称系统的工作机制实质为实本征态机制,并给出了系统结构以及运行模式和系统参数之间的数学关系,为实际系统的设计提供了参数准则。在此基础上,对比分析了不同实本征态的阻抗特性及能效特性,为实际系统的工作模式选取提供了理论依据。

PT对称结构同相和反相放大调制器的仿真研究

在现代微波和光通信系统中,迫切需要研制具有复合功能和特别性能的新型元器件。本文研究设计了一个包含电光介质和缺陷腔的一维PT对称结构模型,用于构造多功能复合器件。研究发现,该结构具有一个增益的缺陷模式。以该模式波长作为入射波载波波长,电信号调制成光信号的同时还被放大。通过设置偏置电场,调制的光信号还可以被同相或反相放大。在此基础上设计了一个消除信号噪声的模型。该研究对设计光集成器件具有一定的参考意义。

共轭线性对称性及其对PT-对称量子理论的应用

传统量子系统的哈密顿是自伴算子,哈密顿的自伴性不仅保证系统遵循酉演化和保持概率守恒,而且也保证了它自身具有实的能量本征值,这类系统称为自伴量子系统.然而,确实存在一些物理系统(如PT-对称量子系统),其哈密顿不是自伴的,这类系统称为非自伴量子系统.为了深入研究PT-对称量子系统,并考虑到算子PT的共轭线性性,首先讨论了共轭线性算子的一些性质,包括它们的矩阵表示和谱结构等;其次,分别研究了具有共轭线性对称性和完整共轭线性对称性的线性算子,通过它们的矩阵表示,给出了共轭线性对称性和完整共轭线性对称性的等价刻画;作为应用,得到了关于PT-对称及完整PT-对称算子的一些有趣性质,并通过一些具体例子,说明了完整PT-对称性对张量积运算不具有封闭性,同时说明了完整PT-对称性既不是哈密顿算子在某个正定内积下自伴的充分条件,也不是必要条件.