PT-symmetric Lattices Research Articles

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.

Collapse Dynamics of Vortex Beams in a Kerr Medium with Refractive Index Modulation and PT-Symmetric Lattices

Using the two-dimensional nonlinear Schrödinger equation, the collapse dynamics of vortex beams in a Kerr medium with refractive index modulation and parity–time (PT) symmetric lattices are explored. The critical power for the collapse of vortex beams in a Kerr medium with real optical lattices (i.e., refractive index modulation lattices) was obtained and discussed. Numerical calculations showed that the number of self-focusing points, the locations of the collapse, and the propagation distances for collapse are sensitively dependent on the modulation factors, topological charge numbers, and initial powers. When the vortex optical field propagates in a Kerr medium with real optical lattices, the optical field will collapse into a symmetrical shape. However, the shape of the vortex beam will be chaotically distorted and collapse in asymmetric patterns during propagation in a Kerr medium with PT-symmetric lattices because of the presence of the complex refraction index. Introducing PT-symmetric lattices into nonlinear Kerr materials may offer a new approach to controlling the collapse of vortex beams.

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.

Nonorientability-induced PT phase transition in ladder lattices

We study parity-time (PT) phase transitions in the energy spectra of ladder lattices caused by the interplay between non-orientability and non-Hermitian PT symmetry. The energy spectra show level crossings in circular ladder lattices with increasing on-site energy gain-loss because of the orientability of a normal strip. However, the energy levels show PT phase transitions in PT-symmetric Moebius ladder lattices due to the non-orientability of a Moebius strip. In order to understand the level crossings of PT symmetric phases, we generalize the rotational transformation using a complex rotation angle. We also study the modification of resonant tunneling induced by a sharply twisted interface in PT-symmetric ladder lattices. Finally, we find that the perfect transmissions at the zero energy are recovered at the exceptional points of the PT-symmetric system due to the self-orthogonal states.

Emergent localized states at the interface of a twofold PT -symmetric lattice

We consider the role of non-triviality resulting from a non-Hermitian Hamiltonian that conserves twofold PT-symmetry assembled by interconnections between a PT-symmetric lattice and its time reversal partner. Twofold PT-symmetry in the lattice produces additional surface exceptional points that play the role of new critical points, along with the bulk exceptional point. We show that there are two distinct regimes possessing symmetry-protected localized states, of which localization lengths are robust against external gain and loss. The states are demonstrated by numerical calculation of a quasi-1D ladder lattice and a 2D bilayered square lattice.

Controllable high-speed polariton waves in a PT-symmetric lattice

Parity-time (PT) symmetry gives rise to unusual phenomena in many physical systems, presently attracting a lot of attention. One essential and non-trivial task is the fabrication and design of the PT-symmetric lattices in different systems. Here we introduce a method to realize such a lattice in an exciton-polariton condensate in a planar semiconductor microcavity. We theoretically demonstrate that in the regime, where lattice profile is nearly PT-symmetric, a polariton wave can propagate at very high velocity resulting from the beating of a ground state condensate created in the lowest energy band at very small momentum and a condensate simultaneously created in higher energy states with large momentum. The spontaneous excitation of these two states in the nonlinear regime due to competition between multiple eigenmodes becomes possible since the spectrum of nearly PT-symmetric structure reveals practically identical amplification for Bloch waves from the entire Brillouin zone. There exists a wide velocity range for the resulting polariton wave. This velocity can be controlled by an additional coherent pulse carrying a specific momentum. We also discuss the breakup of the PT-symmetry when the polariton lifetime exceeds a certain threshold value.

Low Threshold Optical Bistability in Aperiodic PT-Symmetric Lattices Composited with Fibonacci Sequence Dielectrics and Graphene

We explore the optical bistability in aperiodic parity–time-symmetric (PT-symmetric) photonic lattices that are composed of Fibonacci sequence dielectrics and graphene at terahertz frequencies. Two Fibonacci sequence dielectrics, viz. aperiodic photonic lattices, are utilized for enhancing band-edge resonances and achieving the electric field localization that can enhance the nonlinearity of graphene. Modulating the gain-loss factor of dielectrics in the PT symmetry lattices further strengthens the nonlinearity effect and, consequently, low threshold bistability is realized. The interval between the upper and lower bistability thresholds enlarges as the momentum relaxation time of graphene changes. Moreover, we show that the bistability threshold can also be flexibly tuned by modulating the graphene chemical potential. The study might be applied in photomemories and optical switches.

Nonadiabatic transitions through exceptional points in the band structure of a PT -symmetric lattice

Exceptional points, at which two or more eigenfunctions of a Hamiltonian coalesce, occur in non-Hermitian systems and lead to surprising physical effects. In particular, the behaviour of a system under parameter variation can differ significantly from the familiar Hermitian case in the presence of exceptional points. Here we analytically derive the probability of a non-adiabatic transition in a two-level system driven through two consecutive exceptional points at finite speed. The system is Hermitian far away from the exceptional points. In the adiabatic limit an equal redistribution between the states coalescing in the exceptional point is observed, which can be interpreted as a loss of information when passing through the exceptional point. For finite parameter variation this gets modified. We demonstrate how the transition through the exceptional points can be experimentally addressed in a PT-symmetric lattice using Bloch oscillations.

Dynamics of Multipole Solitons and Vortex Solitons in PT-Symmetric Triangular Lattices with Nonlocal Nonlinearity

We investigate the evolution dynamics of solitons with complex structures in the PT-symmetric triangular lattices with nonlocal nonlinearity, including dipole solitons, six-pole solitons, and vortex solitons. Dipole solitons can be linearly stable with a small degree of gain/loss, while six-pole solitons can only be stable when both the degree of gain/loss and the degree of nonlocality are small. For unstable solitons, some humps will decay quickly or new hotspots will appear during propagation. According to the existence range of dipole solitons, the multipole solitons tend to exist in PT-symmetric triangular lattices whose nonlocal nonlinearity is intermediate. We also consider the vortex solitons with high topological charges in the same triangular lattices and find that their profiles are codetermined by the propagation constant, degree of nonlocality, and topological charge.

Solitons in the fractional Schrödinger equation with parity-time-symmetric lattice potential

We investigate the properties of spatial solitons in the fractional Schrodinger equation (FSE) with parity-time (PT)-symmetric lattice potential supported by the focusing of Kerr nonlinearity. Both one- and two-dimensional solitons can stably propagate in PT-symmetric lattices under noise perturbations. The domains of stability for both one- and two-dimensional solitons strongly depend on the gain/loss strength of the lattice. In the spatial domain, the solitons are rigidly modulated by the lattice potential for the weak diffraction in FSE systems. In the inverse space, due to the periodicity of lattices, the spectra of solitons experience sharp peaks when the values of wavenumbers are even. The transverse power flows induced by the imaginary part of the lattice are also investigated, which can preserve the internal energy balances within the solitons.

Stability and internal interaction of multipole solitons in nonlocal PT-symmetric lattices.

We investigate the existence, stability and internal interaction of two-dimensional multipole solitons in defocusing PT symmetric nonlocal nonlinear media. Compared with nonlocal fundamental solitons in PT-symmetric lattices, the multipole solitons reveal the novel internal interaction such as the oscillation of the gravity center. We consider the change of the gravity center of dipole solitons during propagation, and find that these dipole solitons have "breather-like" or "competition" behavior in different nonlinearities. We also demonstrate that dipole solitons are easier to be stabilized under the intermediate nonlocality. Moreover, two more complicated multipole solitons are studied and it is found that there also exist the novel internal interaction for the multipole solitons.

Families of fundamental and vortex solitons under competing cubic-quintic nonlinearity with complex potentials

We investigate the properties of two-dimensional(2D) fundamental and vortex solitons propagating in PT symmetric lattices with the competing cubic-quintic nonlinearity. We discuss the influence of the competing nonlinearity and the gain-loss coefficient on the existence and stability of both 2D fundamental solitons and vortex solitons, and obtain the existence and stability ranges of them. The stability of solitons under different competing CQ nonlinearity is analyzed by using the VK criterion or the anti-VK criterion. We demonstrate that the whole nonlinearity is determined by the coefficients of nonlinear terms and the propagation constants of solitons. In particular, the power curve of vortex soliton leap and move back near the Bloch band instead of bifurcating from it under the defocusing cubic and focusing quintic nonlinearity.

Lasing With Resonant Feedback in Weakly Modulated Parity-Time Symmetric Lattices

We report on the lasing action in weakly modulated parity-time (PT) symmetric lattices. At the exceptional point in PT-symmetric lattices, the amplitude and phase of unidirectionally scattered waves are independently and flexibly determined by the PT modulations, thus, providing a distinctive route to construct a lasing mode in finite systems with weakly modulated structures. It is revealed that the resonant feedback mechanism plays a crucial role in building up the lasing action. Our finding would enrich the singularity dynamics in the weakly modulated PT systems.

Bragg-induced power oscillations in PT -symmetric periodic photonic structures

We study Bragg-induced power oscillations in Fourier space between a pair of optical resonant transverse modes propagating through a periodic PT symmetric lattice, represented by a refractive index that includes gain and loss in a balanced way. Our results imply that the PT-symmetric system shows exceptionally rich phenomena absent in its Hermitian counterpart. It is demonstrated that the resonant modes exhibit unique characteristics such as Bragg power oscillations controlled via the PT symmetry, severe asymmetry in mode dynamics, and trapped enhanced transmission. We have also performed numerical simulations in (1+1) and (2+1) dimensions of propagating Gaussian beams to compare with analytical calculations developed under a two-waves model.

Stable solitons in the 1D and 2D generalized nonlinear Schrödinger equations with the periodic effective mass and [formula omitted]-symmetric potentials

We explore the influence of effective-mass modulation on beam dynamics in the one- and two-dimensional generalized linear and nonlinear Schrödinger equations with parity-time (PT) symmetric periodic potentials. In the linear regime, we successively give the first and second PT threshold curves of optical lattices under the modulation of different effective masses, and analyze the associated band structure and diffraction dynamics of beams. In the Kerr-nonlinear regime, a family of novel optical solitons can exist in the semi-infinite gap. As the effective-mass parameter grows, the existing range of these solitons gradually increases whereas their stable domain gradually diminishes. Two-dimensional (2D) stability analysis reveals that the larger effective-mass parameter, gain-and-loss amplitude, and propagation constant may more easily lead to instable solitons. The 1D and 2D transverse power flows are also examined. Our results may open a new design window for the nonlinear optics in the effective mass and PT-symmetric lattices and other related fields.

Non-Hermiticity-induced flat band

We demonstrate the emergence of an entire flat band embedded in dispersive bands at the exceptional point of a PT symmetric photonic lattice. For this to occur, the gain and loss parameter effectively alters the size of the partial flat band windows and band gap of the photonic lattice simultaneously. The mode associated with the entire flat band is robust against changes in the system size and survives even at the edge of the lattice. Our proposal offers a route for controllable localization of light in non-Hermitian systems and a technique for measuring non-Hermiticity via localization.

Surface solitons supported by 𝒫𝒯-symmetric lattice with spatially modulated nonlinearity

We report on the formation and stability of induced surface solitons in parity–time ([Formula: see text]) symmetric periodic systems with spatially modulated nonlinearity. We discover that the spatially modulation of the nonlinearity can affect the existence and stability of surface solitons. These surface solitons can be formed in the semi-infinite and first bandgap. Stability analysis shows that odd surface solitons belonging to semi-infinite bandgap are linearly stably in low power domain, and the stable domain becomes narrow with increasing the strength of spatially modulated nonlinearity, both even surface solitons in semi-infinite bandgap and surface solitons in first bandgap are unstable in their existence domain.

Topologically protected bound states in photonic parity-time-symmetric crystals.

Parity-time (PT)-symmetric crystals are a class of non-Hermitian systems that allow, for example, the existence of modes with real propagation constants, for self-orthogonality of propagating modes, and for uni-directional invisibility at defects. Photonic PT-symmetric systems that also support topological states could be useful for shaping and routing light waves. However, it is currently debated whether topological interface states can exist at all in PT-symmetric systems. Here, we show theoretically and demonstrate experimentally the existence of such states: states that are localized at the interface between two topologically distinct PT-symmetric photonic lattices. We find analytical closed form solutions of topological PT-symmetric interface states, and observe them through fluorescence microscopy in a passive PT-symmetric dimerized photonic lattice. Our results are relevant towards approaches to localize light on the interface between non-Hermitian crystals.

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.

Multi-Channel Optical Isolator Based on Nonlinear Triangular Parity Time Symmetric Lattice

We report the design procedure for a broadband multi-channel cavity-less optical isolator composed of a triangular perturbed nonlinear parity time (PT) symmetric lattice. The interplay between the nonlinearity and the PT-symmetric lattice that results in nonreciprocal transmission enables us to design the isolator outputs and inputs positions as desired. In the transmitting regime, a number of solitons that are simultaneously launched into individual inputs positioned appropriately, on the waveguides’ gain sides in the periodic lattice, swing along the waveguide until they are self-trapped along the respective waveguides axes, where the isolator outputs are located. In the isolating regime, however, the solitons that are launched into the inputs—i.e., the outputs of the transmitting regime—propagate along the waveguides axes, until they exit from the new outputs. Simulations show that the neighboring soliton trajectories in each regime are completely isolated and so are the forward and reverse trajectories in each waveguide cell. Moreover, operation of this cavity-less PT-symmetric isolator that is designed on an inhomogeneous slab waveguide does not suffer from the narrow frequency band limitation, unlike their cavity-based counterparts.