Parity-time Symmetric Potentials Research Articles

Collapse Dynamics of Vector Vortex Beams in Kerr Medium with Parity–Time-Symmetric Lattice Modulation

Based on the two-dimensional (2D) nonlinear Schrödinger equation, we investigate the collapse dynamics of a vector vortex optical field (VVOF) in nonlinear Kerr media with parity–time (PT)-symmetric modulation. The critical power for the collapse of a VVOF in a Kerr-ROLP medium (Kerr medium with a real optical lattice potential) is derived. Numerical simulations indicate that the number, position, propagation distance, and collapse profile of the collapse of a VVOF in sine and cosine parity–time-symmetric potential (SCPT) Kerr media are closely related to the modulation depth, initial powers, and the topological charge number of a VVOF. The VVOF collapses into symmetric shapes during propagation in a Kerr-ROLP medium, and collapse shapes are sensitively related to the density of the PT-symmetric optical lattice potential. In addition, due to gain–loss, the VVOF will be distorted during propagation in the Kerr-SCPT medium, forming an asymmetric shape of collapse. The power evolution of the VVOF in a Kerr-SCPT medium as a function of the transmission distance with different modulating parameters and topological numbers is analyzed in detail. The introduction of PT-symmetric optical lattice potentials into nonlinear Kerr materials may provide a new approach to manipulate the collapse of the VVOF.

Asynchronous quantum Kármán vortex street in two-component Bose-Einstein condensate with PT symmetric potential

The dynamics of a miscible two-component Bose-Einstein condensate (BEC) with PT (parity-time) symmetric potential are investigated numerically. The dynamical behaviors of the system is described by Gross-Pitaevskii (GP) equations under the mean-field theory. Firstly, the ground state of the system is obtained by the imaginary-time propagation method. Then dynamical behaviors are numerically simulated by the time-splitting Fourier pseudo-spectral approach under periodic boundary conditions. By adjusting the width and velocity of the obstacle potential, various patterns such as no vortex, oblique drifting vortex dipole, V-shaped vortex pairs, irregular quantum turbulence and combined modes are studied. It is noted that the shedding vortex pairs in components 1 and 2 are staggered, which is called ``the asynchronous quantum Kármán vortex street''. Here, the ratio of the distance between two vortex pairs in one row to the distance between vortex rows is approximately 0.18, which is less than the stability criterion 0.28 of classical fluid. We calculated the drag force acting on the obstacle potential during generation of the asynchronous quantum Kármán vortex street. It is found that periodical oscillation of the drag force is generated via drifting up or down of the vortex pairs. Meanwhile, we analyzed the influence of the imaginary part of the PT symmetric potential with gain-loss for wake. The trajectory and frequency of the vortex are changed, due to the imaginary part breaks the local symmetry of the system. In addition, the imaginary part affects the stability of the asynchronous quantum Kármán vortex street. Lots of numerical simulations are carried out to determine the parameter regions of various vortex shedding modes. We also proposed an experimental protocol to realize the asynchronous quantum Kármán vortex street in the miscible two-component BEC with PT symmetric potential.

Fundamental, dipole, and vortex solitons in fractional nonlinear Schrödinger equation with a parity-time-symmetric periodic potential

We investigate the existence and stability of two-dimensional (2D) solitons in the framework of the nonlinear fractional Schrödinger equation (NLFSE) with focusing nonlinearity and a parity-time (PT) symmetric periodic potential. Numerical studies reveal that the fundamental, dipole, and localized vortex solitons exist in a semi-infinite band gap. The Lévy index (α) cannot change the phase transition point of the PT-symmetric periodic potential, but it affects the existence and stability of these solitons significantly. With the growth of the Lévy index, the stability regions become markedly wider for the fundamental, dipole solitons and vortex solitons with the topological charge of 2 (S=2). But for a unit topological charge (S=1), the vortex soliton cannot maintain its stability under the usual diffraction effect (α=2). The result shows that the fractional-order diffraction helps to stabilize the vortex solitons with S=1.

Self-defocusing nonlinear coupled system with PT-symmetric super-Gaussian potential.

The stationary solutions of the coupled nonlinear Schrödinger equation with self-defocusing nonlinearity and super-Gaussian form of parity-time (PT) symmetric potential in an optical system have been analyzed. The stationary eigenmodes of the ground and excited states and the influence of the gain/loss coefficient on the eigenvalue spectra are discussed. The threshold condition of the PT-symmetric phase transition of the high and low-frequency modes has been studied. Also, the variation of the threshold values with the coupling constant and the effect of the nonlinearity on the eigenmodes are analyzed. The stability of the solution is verified using the linear-stability analysis. In addition, the power distribution of the fundamental solutions with the propagation, in the two channels of the system, is analyzed in the PT and broken PT regimes.

Amplitude-phase modulated composite waves in coupled nonlocal system with self-induced PT symmetric potential.

We consider a coupled nonlocal nonlinear Schrödinger equation (nNLSE) with self-induced parity-time (PT) symmetric potential and investigate abundant amplitude-phase modulated composite waves manifesting diverse evolution patterns. It is found that the coupled nonlocal model can be decoupled into nNLSEs with self-induced PT symmetric potential under certain constraints through a general linear transformation with amplitude and phase modulation. Based on the exact solutions of the nNLSEs with self-induced PT potential, we study various composite waves superposed by bright and/or dark soliton solutions, rogue waves, bright/dark soliton and periodic soliton, and present the abundant evolution patterns under amplitude-phase modulation. The results here only demonstrate the characteristics of limited superposed composite waves. In fact, there exist infinite possible evolution patterns of composite waves due to the arbitrary amplitude-phase modulation in coupled nonlocal nonlinear system with self-induced PT symmetric potential.

Physics-informed Neural Network method for predicting soliton dynamics supported by complex PT-symmetric potentials

Abstract We examine the deep learning technique referred to as the physics-informed neural network method for approximating non-linear Schrödinger equation under considered parity time symmetric potentials and obtaining multifarious soliton solutions. For the first time, neural networks founded principally physical information are adopted to figure out the solution the examined non-linear partial differential equation and generate six different types of soliton solutions, which are basic, dipole, tripole, quadruple, pentapole and sextupole solitons we consider. We make comparisons between the predicted and actual soliton solutions to see whether deep learning is capable of seeking the solution the partial differential equation described before. We may assess whether physics-informed neural network is capable of effectively providing approximate soliton solutions through the evaluation of squared error between the predicted and numerical results. Besides, we also scrutinize how different activation mechanisms and network architectures impact the capability of selected deep learning technique works. Through the findings we can prove that the neural networks model we established can be utilized to accurately and effectively approximate non-linear Schrödinger equation under consideration and predict the dynamics of soliton solution.

Physics-Informed Neural Network Method for Predicting Soliton Dynamics Supported by Complex Parity-Time Symmetric Potentials

We examine the deep learning technique referred to as the physics-informed neural network method for approximating the nonlinear Schrödinger equation under considered parity-time symmetric potentials and for obtaining multifarious soliton solutions. Neural networks to found principally physical information are adopted to figure out the solution to the examined nonlinear partial differential equation and to generate six different types of soliton solutions, which are basic, dipole, tripole, quadruple, pentapole, and sextupole solitons we consider. We make comparisons between the predicted and actual soliton solutions to see whether deep learning is capable of seeking the solution to the partial differential equation described before. We may assess whether physics-informed neural network is capable of effectively providing approximate soliton solutions through the evaluation of squared error between the predicted and numerical results. Moreover, we scrutinize how different activation mechanisms and network architectures impact the capability of selected deep learning technique works. Through the findings we can prove that the neural networks model we established can be utilized to accurately and effectively approximate the nonlinear Schrödinger equation under consideration and to predict the dynamics of soliton solution.

Two-dimensional fractional [formula omitted]-symmetric cubic–quintic NLS equation: Double-loop symmetry breaking bifurcations, ghost states and dynamics

In this paper, we address the double-loop symmetry breaking bifurcations of optical vortex solitons in the two-dimensional (2D) fractional cubic–quintic nonlinear Schrödinger (FCQNLS) equation with an imprinted partially parity-time (PPT) symmetric complex potential. The continuous branches of asymmetric solitons bifurcate from the fundamental one as the power exceeds some critical value. It is found that the symmetry breaking point is exactly the point at which the fundamental symmetric soliton is destabilized. Intriguingly, the branches of asymmetry solitons (alias ghost states) are existing with complex conjugate propagation constants, which is solely in fractional media. Moreover, the branch of asymmetric solitons crosses the branch of fundamental symmetric ones when it reaches the second critical value, and eventually merges back into it at the third critical power point. The imaginary part of the propagation constant also exhibits a circular shape. And the stability of the fundamental symmetric soliton is restored. In addition, the stability of two branches of asymmetric solitons differs from each other. The reason generating this phenomenon is analyzed in detail. Besides, the branch of dipole solitons for the first excited state is also discussed numerically, which is always stable in the process of symmetry breaking. The stability and dynamics of symmetric (fundamental and the first excited ones) and asymmetric solitons are explored via the linear stability analysis, direct simulation, modulation as well as adiabatic excitation. These results will provide some theoretical basis for the study of spontaneous symmetry breaking phenomena and related experiments in fractional media with PPT-symmetric potentials.

One-dimensional -symmetric eigenmodes in k-wave number Scarf II potential with defocusing nonlinearity

The stationary states and dynamical evolution of one-dimensional eigenmodes in self-defocusing nonlinear Kerr medium under transverse parity-time () symmetric k-wave number Scarff-II potential have been analysed. We study the effect of the coefficient of the imaginary component of the -symmetric potential on the eigenvalue spectra of the ground and first excited states in linear and nonlinear models. Symmetric eigenmodes with real eigenvalues are obtained in the unbroken regime in both models. The linear stability analysis shows that the solutions are stable in the unbroken regime. The stable propagation of the solutions in linear and nonlinear media supported by -symmetric k-wave number Scarff-II potential is also investigated. The comparison of the solutions in the presence and the absence of nonlinearity reveals that in the self-defocusing Kerr medium nonlinearity suppresses the region of -symmetry. The influence of the width of the complex k-wave number Scarff-II potential and the depth of the real potential function on the -symmetry breaking have also been analysed.

Symmetry breaking of solitons in the PT-symmetric nonlinear Schrödinger equation with the cubic-quintic competing saturable nonlinearity.

The symmetry breaking of solitons in the nonlinear Schrödinger equation with cubic-quintic competing nonlinearity and parity-time symmetric potential is studied. At first, a new asymmetric branch separates from the fundamental symmetric soliton at the first power critical point, and then, the asymmetric branch passes through the branch of the fundamental symmetric soliton and finally merges into the branch of the fundamental symmetric soliton at the second power critical point, while the power of the soliton increases. This leads to the symmetry breaking and double-loop bifurcation of fundamental symmetric solitons. From the power-propagation constant curves of solitons, symmetric fundamental and tripole solitons, asymmetric solitons can also exist. The stability of symmetric fundamental solitons, asymmetric solitons, and symmetric tripole solitons is discussed by the linear stability analysis and direct simulation. Results indicate that symmetric fundamental solitons and symmetric tripole solitons tend to be stable with the increase in the soliton power. Asymmetric solitons are unstable in both high and low power regions. Moreover, with the increase in saturable nonlinearity, the stability region of fundamental symmetric solitons and symmetric tripole solitons becomes wider.

Data driven soliton solution of the nonlinear Schrödinger equation with certain P T-symmetric potentials via deep learning.

We investigate the physics informed neural network method, a deep learning approach, to approximate soliton solution of the nonlinear Schrödinger equation with parity time symmetric potentials. We consider three different parity time symmetric potentials, namely, Gaussian, periodic, and Rosen-Morse potentials. We use the physics informed neural network to solve the considered nonlinear partial differential equation with the above three potentials. We compare the predicted result with the actual result and analyze the ability of deep learning in solving the considered partial differential equation. We check the ability of deep learning in approximating the soliton solution by taking the squared error between real and predicted values. Further, we examine the factors that affect the performance of the considered deep learning method with different activation functions, namely, ReLU, sigmoid, and tanh. We also use a new activation function, namely, sech, which is not used in the field of deep learning, and analyze whether this new activation function is suitable for the prediction of soliton solution of the nonlinear Schrödinger equation for the aforementioned parity time symmetric potentials. In addition to the above, we present how the network's structure and the size of the training data influence the performance of the physics informed neural network. Our results show that the constructed deep learning model successfully approximates the soliton solution of the considered equation with high accuracy.

Suppression of symmetry breaking of nonlinear modes by defocusing saturable nonlinearity in parity-time symmetric potentials

Symmetry breaking of nonlinear localized modes and suppression of symmetry-breaking bifurcations are reported in the framework of the nonlinear Schrödinger equation with defocusing saturable nonlinearity in parity-time symmetric potentials. We found that, beyond a critical point, one type of the nonlinear modes with asymmetric profiles bifurcates from the branch of the first excited state. We prove that the bifurcation is essentially triggered by instability of the first excited state by linear stability analysis, which implies the symmetry breaking of the nonlinear modes is steerable by changing the stability of the first excited state of the nonlinear mode. A suppressing effect is that the symmetry-breaking bifurcation of the nonlinear modes can be completely suppressed by adjusting the strength of the saturable nonlinearity. This suppressing effect of symmetry-breaking bifurcation is illuminated by analyzing the stability behaviors of the nonlinear modes.

Symmetry-breaking bifurcations and ghost states in the fractional nonlinear Schrödinger equation with a PT-symmetric potential

We report symmetry-breaking and restoring bifurcations of solitons in a fractional Schrödinger equation with cubic or cubic-quintic (CQ) nonlinearity and a parity-time-symmetric potential, which may be realized in optical cavities. Solitons are destabilized at the bifurcation point, and, in the case of CQ nonlinearity, the stability is restored by an inverse bifurcation. Two mutually conjugate branches of ghost states (GSs), with complex propagation constants, are created by the bifurcation, solely in the case of fractional diffraction. While GSs are not true solutions, direct simulations confirm that their shapes and results of their stability analysis provide a "blueprint" for the evolution of genuine localized modes in the system.

Solving forward and inverse problems of the logarithmic nonlinear Schrödinger equation with [formula omitted]-symmetric harmonic potential via deep learning

In this paper, we investigate the logarithmic nonlinear Schrödinger (LNLS) equation with the parity-time (PT)-symmetric harmonic potential, which is an important physical model in many fields such as nuclear physics, quantum optics, magma transport phenomena, and effective quantum gravity. Three types of initial value conditions and periodic boundary conditions are chosen to solve the LNLS equation with PT-symmetric harmonic potential via the physics-informed neural networks (PINNs) deep learning method, and these obtained results are compared with ones deduced from the Fourier spectral method. Moreover, we also investigate the effectiveness of the PINNs deep learning for the LNLS equation with PT symmetric potential by choosing the distinct space widths or distinct optimized steps. Finally, we use the PINNs deep learning method to effectively tackle the data-driven discovery of the LNLS equation with PT-symmetric harmonic potential such that the coefficients of dispersion and nonlinear terms or the amplitudes of PT-symmetric harmonic potential can be approximately found.

Parity-time symmetric plasmonic planar superlens

A superlens which involves plasmonic mode could compensate for the loss of high spatial frequency information and achieve super-resolution. However, their working distance is limited to near field region due to the intrinsic loss of metal and the imaging properties are usually distorted by the experimental errors. We investigate a layered planar superlens structure with parity-time (PT) symmetric potential. By utilizing the transfer matrix method, we observe the eigenspectrum of plasmonic mode depends on PT potential. Numerical simulation results, obtained by using the Finite Element Method, show that the working distance of super-resolution can be increased in the PT symmetric superlens. In addition, the resolution could be improved by adjusting the PT symmetric potential, demonstrating the PT symmetry introduced a great advantage of robustness to experimental errors.

Controlled self-similar matter waves in PT-symmetric waveguide

We study the dynamics of Bose-Einstein condensate coupled to a waveguide with parity-time symmetric potential in the presence of quadratic-cubic nonlinearity modelled by Gross-Pitaevskii equation with external source. We employ the self-similar technique to obtain matter wave solutions, such as bright, kink-type, rational dark and Lorentzian-type self-similar waves for this model. The dynamical behavior of self-similar matter waves can be controlled through variation of trapping potential, external source and nature of nonlinearities present in the system.

Parity-time symmetry light bullets in a cold Rydberg atomic gas.

A scheme is proposed to generate stable light bullets (LBs) in a cold Rydberg atomic system with a parity-time (PT) symmetric potential, by utilizing electromagnetically induced transparency (EIT). Using an incoherent population pumping between two low-lying levels and spatial modulations of control and auxiliary laser fields, we obtain a two-dimensional (2D) periodic optical potential with PT symmetry. Based on PT symmetry potential and the long-range Rydberg-Rydberg atomic interaction, the system may support slow LBs with low light intensity. Further, it is found that the local and non-local nonlinear coefficients and PT-symmetric potential can be tuned and used to manipulate the behavior of LBs.

Solitons supported by competing nonlinearity, higher order dispersion and [formula omitted]-symmetric potential

The existence and stability properties of optical solitons in competing cubic-quintic nonlinearity, third order dispersion and parity-time symmetric potentials are investigated. The complex potentials are found as solutions of an inverse problem which predicts explicit expression of the complex potential supporting a required phase gradient structure. It has been found that interplay of competing nonlinearity, third order dispersion, spatial modulation of the refractive index and associated gain/loss profile can give rise to stable solitons in some parameter regimes. Also the solitons are stable even when the parity-time symmetry of underlying linear model is broken. Nonlinear mode excitations have been shown by adiabatic change of the third order dispersion coefficient (which in turn changes the refractive index and gain/loss distribution) whereby a stable nonlinear mode in the parameter domain where the linear PT symmetric phase is unbroken, is excited to another stable nonlinear mode belonging to the parameter domain where linear PT symmetric phase is broken.

Three-dimensional optical solitons formed by the balance between different-order nonlinearities and high-order dispersion/diffraction in parity-time symmetric potentials

Under parity-time symmetric potentials, different-order nonlinearities such as cubic, quintic and septimal nonlinearities, altogether with their combinations and second-order and fourth-order dispersions/diffractions are simultaneously considered to form three-dimensional optical solitons. Based on some high-order nonlinear Schrodinger equations, three-dimensional analytical optical soliton solutions are found. In the defocusing cubic nonlinear case, three-dimensional optical soliton without fourth-order diffraction/dispersion is stable than that with fourth-order diffraction/dispersion. However, in the defocusing cubic and focusing quintic nonlinear case, the stability situation of soliton is just on the contrary. Among all combinations of nonlinearity, the stability of three-dimensional optical soliton in the cubic-quintic nonlinear case is better than that in the cubic nonlinear case, but worse than that in the cubic-quintic-septimal nonlinear case. In the quintic-septimal nonlinear case, three-dimensional optical soliton is unstable and will collapse ultimately.

3D cubic and power-law nonlinear Schrödinger equations with different coefficients of dispersion/diffraction and their optical soliton solutions in parity-time symmetric potentials

The 3D cubic and power-law nonlinear Schrödinger equations with different coefficients of dispersion/diffraction is presented in parity-time symmetric potentials, and their optical soliton solutions are analytically discovered after the accurate and optimal balance among various effects. As special examples, analytical soliton solutions considering cubic–quintic, cubic–septimal and quintic–septimal nonlinearities are revealed. In order to analyze their dynamic properties more clearly, graphical representations of each solution are presented.