PT-symmetric Optical Lattices Research Articles

Centrosymmetric multipole solitons with fractional-order diffraction in two-dimensional parity-time-symmetric optical lattices

Multipole solitons in higher-dimensional nonlinear Schrödinger equation with fractional diffraction are of high current interest. This paper studies multipole gap solitons in parity-time (PT)-symmetric lattices with fractional diffraction. The results obtained demonstrate that both on-site and off-site eight-pole solitons with fractional-order diffraction can be stabilized in a two-dimensional (2D) PT-symmetric optical lattice with defocusing Kerr nonlinearity. These solitons are in-phase and centrosymmetric. On-site eight-pole solitons propagate in a square formation, while off-site solitons propagate in a two-by-four formation. Both on-site and off-site solitons are found to be stable within a low-power range in the first band gap. As the Lévy index decreases, the stability regions of both on-site and off-site solitons narrow. Off-site eight-pole solitons can approach the lower edge of the first Bloch band, whereas on-site eight-pole solitons cannot. Additionally, we investigate the transverse power flow vector of these multipole gap solitons, illustrating the transverse energy flow from gain to loss regions.

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.

Two-stage initial-value iterative physics-informed neural networks for simulating solitary waves of nonlinear wave equations

We propose a new two-stage initial-value iterative neural network (IINN) algorithm for solitary wave computations of nonlinear wave equations based on traditional numerical iterative methods and physics-informed neural networks (PINNs). Specifically, the IINN framework consists of two subnetworks, one of which is used to fit a given initial value, and the other incorporates physical information and continues training on the basis of the first subnetwork. Importantly, the IINN method does not require any additional data information including boundary conditions, apart from the given initial value. Corresponding theoretical guarantees are provided to demonstrate the effectiveness of our IINN method. The proposed IINN method is efficiently applied to learn some types of solutions in different nonlinear wave equations, including the one-dimensional (1D) nonlinear Schrödinger equations (NLS) equation (with and without potentials), the 1D saturable NLS equation with PT-symmetric optical lattices, the 1D focusing-defocusing coupled NLS equations, the KdV equation, the two-dimensional (2D) NLS equation with potentials, the 2D amended GP equation with a potential, the (2+1)-dimensional KP equation, and the 3D NLS equation with a potential. These applications serve as evidence for the efficacy of our method. Finally, by comparing with the traditional methods, we demonstrate the advantages of the proposed IINN method.

Bragg scattering of stochastic beams in PT-symmetric photonic lattices.

The propagation of a Gaussian-Schell beam through a PT-symmetric optical lattice, whose index of refraction is represented by a sinusoidal type of function, is theoretically investigated. Within the framework of standard coherence theory, one is able to access and elucidate unexpected consequences of the interplay between the spatial coherence properties of the beam and the non-Hermitian nature of the photonic lattice. We describe how one may use a non-Hermitian periodic medium to enhance the spatial coherence properties of a partially coherent beam.

Localization of the light in two-dimensional PT-symmetric optical lattice

In this paper, the PT-symmetric lattice that its amplitude and period are different in x and y direction. Under different imaginary components and ratio of amplitudes in two directions, the propagation and localization of the light in the PT lattice are simulated. The result shows that both the amplitude ratio of potential between two orthogonal directions and the imaginary part of the potential have vital influences on the localization of light. The PT-symmetry can be broken to achieve localization as the imaginary part of the potential increases. Further more, the relation between the real part of the PT potential and the PT-symmetry breaking threshold is also discussed in this work.

Localization of the Light in 2D PT-Symmetric Optical Lattice

Localization of the Light in 2D PT-Symmetric Optical Lattice

Families of solitons under nonlocal and PT symmetric competing nonlinearity

In this paper, we discover several new soliton families in the PT symmetric optical lattices with the nonlocal competing cubic–quintic nonlinearity, and investigate their existence and stability ranges. We detailedly study the influence of the degree of nonlocality, the quintic nonlinearity and the PT symmetry on these solitons, and obtain the power surfaces of solitons under different degrees of nonlocality and PT symmetry. We demonstrate that solitons can be linearly stabilized in the first Bloch bandgap under the nonlocal focusing cubic and local defocusing quintic nonlinearity, while they can only exist in the semi-infinite Bloch bandgap under the nonlocal defocusing cubic and local focusing quintic nonlinearity.

Gap solitons supported by two-dimensional parity-time-symmetric optical lattices in saturable media with fourth-order diffraction

The existence and the stability of in-phase dipole gap solitons in two-dimensional (2D) parity-time (PT)-symmetric optical lattices with defocusing saturable nonlinearity and a fourth-order diffraction property are investigated. When different values are used for the coupling constant of the fourth-order diffraction, the critical threshold of the 2D PT-symmetric optical lattices remains unchanged. The in-phase dipole solitons exist in the first finite gap and are shown to be stable in the moderate power region. The existence and stability of dipole solitons with different fourth-order diffraction coupling constants and different saturation parameters are therefore studied. We also investigate the transverse power-flow vector of these gap solitons.

Multipole gap solitons in fractional Schrödinger equation with parity-time-symmetric optical lattices.

We investigate the existence and stability of in-phase three-pole and four-pole gap solitons in the fractional Schrödinger equation supported by one-dimensional parity-time-symmetric periodic potentials (optical lattices) with defocusing Kerr nonlinearity. These solitons exist in the first finite gap and are stable in the moderate power region. When the Lévy index decreases, the stable regions of these in-phase multipole gap solitons shrink. Below a Lévy index threshold, the effective multipole soliton widths decrease as the Lévy index increases. Above the threshold, these solitons become less localized as the Lévy index increases. The Lévy index cannot change the phase transition point of the PT-symmetric optical lattices. We also study transverse power flow in these multipole gap solitons.

Two-dimensional solitons in parity-time-symmetric optical lattices with nonlocal defocusing nonlinearity.

We investigate the evolution of two-dimensional solitons in PT-symmetric optical lattices with defocusing nonlocal nonlinearity. We discuss the existence of 2D fundamental solitons, and we numerically stimulate the propagation of these solitons in the optical lattice. We also discuss how different degrees of nonlocality can affect the evolution of solitons, and discover that nonlocality can have significant impacts on the formation and propagation of the solitons. In addition, we find that PT-symmetry can lead to fission and shifting of solitons, and this phenomenon is also affected by the degree of nonlocality.

Three-dimensional topological solitons in PT-symmetric optical lattices

We address the properties of fully three-dimensional solitons in complex parity-time (PT)-symmetric periodic lattices with focusing Kerr nonlinearity, and uncover that such lattices can stabilize both fundamental and vortex-carrying soliton states. The imaginary part of the lattice induces internal currents in the solitons that strongly affect their domains of existence and stability. The domain of stability for fundamental solitons can extend nearly up to the PT-symmetry breaking point, where the linear lattice spectrum becomes complex. Vortex solitons feature spatially asymmetric profiles in the PT-symmetric lattices, but they are found to still exist as stable states within narrow regions. Our results provide the first example of continuous families of stable three-dimensional propagating solitons supported by complex potentials.

Observation of Parity-Time Symmetry in Optically Induced Atomic Lattices

We experimentally demonstrate PT-symmetric optical lattices with periodical gain and loss profiles in a coherently prepared four-level N-type atomic system. By appropriately tuning the pertinent atomic parameters, the onset of PT-symmetry breaking is observed through measuring an abrupt phase-shift jump between adjacent gain and loss waveguides. The experimental realization of such a readily reconfigurable and effectively controllable PT-symmetric waveguide array structure sets a new stage for further exploiting and better understanding the peculiar physical properties of these non-Hermitian systems in atomic settings.

Multi-peak solitons in PT-symmetric Bessel optical lattices with defects

This paper presents a theoretical analysis of the existence and stability of multi-peak solitons in parity–time-symmetric Bessel optical lattices with defects in nonlinear media. The results demonstrate that there always exists a critical propagation constant μc for the existence of multi-peak solitons regardless of whether the nonlinearity is self-focusing or self-defocusing. In self-focusing media, multi-peak solitons exist when the propagation constant μ > μc. In the self-defocusing case, solitons exist only when μ < μc. Only low-power solitons can propagate stably when random noise perturbations are present. Positive defects help stabilize the propagation of multi-peak solitons when the nonlinearity is self-focusing. When the nonlinearity is self-defocusing, however, multi-peak solitons in negative defects have wider stable regions than those in positive defects.

Soliton dynamics in a PT-symmetric optical lattice with a longitudinal potential barrier.

We present dynamics of spatial solitons propagating through a PT symmetric optical lattice with a longitudinal potential barrier. We find that a spatial soliton evolves a transverse drift motion after transmitting through the lattice barrier. The gain/loss coefficient of the PT symmetric potential barrier plays an essential role on such soliton dynamics. The bending angle of solitons depends on the lattice parameters including the modulation frequency, incident position, potential depth and the barrier length. Besides, solitons tend to gain a certain amount of energy from the barrier, which can also be tuned by barrier parameters.

Multi-peak gap solitons in PT-symmetric optical lattice in the form of superlattice and dual superlattice

We report the existence and stability of multi-peak gap solitons in PT-symmetric optical lattice with the real part of the refractive index in the form of superlattice and the imaginary part having two different profiles; one with odd periodic lattice and the second with odd superlattice supported by a defocusing nonlinearity. Linear-stability spectra analysis and simulation results show that they can be stable in the scope of the first gap. The profile of the multi-peak solitons and the power-flow vector is found to be different from those of one cycle lattice.

Coherent Perfect Absorption in 𝓟𝓣 Symmetric Optical Lattice

In this work we study scattering in a 𝓟𝓣 symmetric periodic optical lattice, when two coherent beams are incident on it from two opposite directions. We focus our attention on both 𝓟𝓣 broken as well as 𝓟𝓣 unbroken phase, with special emphasis on spectral singularity. This is another example of an analytically solvable model, apart from the 𝓟𝓣 symmetric Scarf II potential discussed in ref. Ahmed (Phys. Rev. A 81, 022102 (2014)), which exhibits the phenomenon of Coherent Perfect Absorption (CPA) with lasing, in the domain of broken 𝓟𝓣 symmetry. At the same time, CPA is neither possible for unbroken 𝓟𝓣 symmetry, nor at the transition point.

Gap solitons in photorefractive medium with PT-symmetric optical lattices

We find the existence and stability conditions of semi-infinite gap solitons in photorefractive media with parity and time (PT) reversal symmetric complex optical lattices. When the strength of the gain/loss term of complex potential is below the critical value, where the PT-symmetry of the system is unbroken, both the finite and semi-infinite bandgap solitons can exist. The dynamical propagation, Vakhitov–Kolokolov (VK) stability criterion, and linear perturbation stability analysis confirm the stability of the semi-infinite bandgap solitons. All gap solitons in the finite bandgap are found to be dynamically unstable even though they satisfy the VK criterion for stability. We reveal that for the complex potential strength above the critical value only unstable semi-infinite bandgap solitons exist due to the presence of complex propagation constant as a result of the broken PT-symmetry. • Gap solitons in photorefractive media with parity and time reversal symmetric optical lattices. • The existence condition for semi-infinite bandgap solitons is found. • The dynamical and linear stabilities of the semi-infinite gap solitons are investigated. • The gap solitons are unstable above a certain value of the model parameter.

Gap solitons in PT-symmetric optical lattices with higher-order diffraction.

The existence and stability of gap solitons are investigated in the semi-infinite gap of a parity-time (PT)-symmetric periodic potential (optical lattice) with a higher-order diffraction. The Bloch bands and band gaps of this PT-symmetric optical lattice depend crucially on the coupling constant of the fourth-order diffraction, whereas the phase transition point of this PT optical lattice remains unchangeable. The fourth-order diffraction plays a significant role in destabilizing the propagation of dipole solitons. Specifically, when the fourth-order diffraction coupling constant increases, the stable region of the dipole solitons shrinks as new regions of instability appear. However, fundamental solitons are found to be always linearly stable with arbitrary positive value of the coupling constant. We also investigate nonlinear evolution of the PT solitons under perturbation.

Mixed-gap vector solitons in parity-time-symmetric mixed linear–nonlinear optical lattices

We report on the existence and stability of mixed-gap vector solitons in parity-time (PT)-symmetric mixed linear&#x2013;nonlinear optical lattices. The first component is single-peaked, and the propagation constant is in the semi-infinite gap. The second component is the out-of-phase dipole mode; its propagation constant belongs to the first finite gap. The imaginary part and the depth of the PT-symmetric nonlinear optical lattice will significantly affect the existence and stability domains of these vector solitons. The propagation constant of the first component can also influence the existence and stability of the vector solitons. Finally, we also study the effect of the PT-symmetric linear optical lattice on the vector solitons&#x2019; stability.

EXCEPTIONAL POINTS IN OPEN AND PT-SYMMETRIC SYSTEMS

Exceptional points (EPs) determine the dynamics of open quantum systems and cause also PT symmetry breaking in PT symmetric systems. From a mathematical point of view, this is caused by the fact that the phases of the wavefunctions (eigenfunctions of a non-Hermitian Hamiltonian) relative to one another are not rigid when an EP is approached. The system is therefore able to align with the environment to which it is coupled and, consequently, rigorous changes of the system properties may occur. We compare analytically as well as numerically the eigenvalues and eigenfunctions of a 2 × 2 matrix that is characteristic either of open quantum systems at high level density or of PT symmetric optical lattices. In both cases, the results show clearly the influence of the environment on the system in the neighborhood of EPs. Although the systems are very different from one another, the eigenvalues and eigenfunctions indicate the same characteristic features.