PT-symmetric Dimer Research Articles

Transmission across a ribbon containing a square [formula omitted] impurity

We study the spectrum and transmission coefficient of plane waves propagating along square ribbons of varying widths, containing a square-shaped, PT-symmetric impurity region. We start with a zero-width ribbon (1D chain) and place a PT symmetric dimer. The spectrum is computed numerically and the instability gain is computed as a function of the gain/loss dimer strength. The transmission coefficient is obtained in closed form and examined as a function of wavevector and the gain/loss parameter. Next, we study a ribbon in a narrow ladder configuration containing a square PT impurity. As before, we compute the instability gain numerically and the transmission coefficient in closed form for the two possible input modes. Finally, we repeat the calculations for a wider ladder ribbon containing a Lieb-like impurity in a PT configuration. For all cases and transmission channels, we obtain transmission divergences in wavevector-gain/loss parameter space, whose number increases with the width of the ribbon.

Experimental study of the nonlinear distortion of non-reciprocal transmission in nonlinear parity-time symmetric LC resonators

At the broken phase of a parity-time (PT)-symmetric dimer where the coupling is weak, the eigenfrequencies are complex conjugate pairs with non-vanishing real parts, leading to oscillation magnitudes with an exponentially growing mode and an exponentially decaying mode. If the large oscillation magnitudes are clamped due to the nonlinear gain of the PT-symmetric dimer, the exponentially growing mode eventually enters the stable oscillations. It was recently demonstrated that such a phenomenon can be utilized for non-reciprocal transmission. The distortion induced by nonlinearity is critical for the non-reciprocal transmission. Here, we experimentally explore the nonlinear distortion in PT-symmetric inductor–capacitor resonators by utilizing discrete components on a printed circuit board. It demonstrates that the IIP3 (the output-referred third-order intercept point) can achieve as high as 38.7 dBm at the frequency of 14.5 MHz corresponding to the maximum forward transmission. The noise figure of the system is measured to be about 11.25 dB.

Analysis of the multi-phenomenal nonlinear system : Testing Integrability and detecting chaos

As a large extension in Hamiltonian form, the system of a PT symmetric dimer of coupled nonlinear oscillators is developed. This system provides an explanation for a number of problems with Hamiltonian dynamics. Integrability is evaluated in the Painlevé sense of the system. The system reported twelve P-cases. First integrals of planar motion are constructed explicitly for each integrable case to show the Liouvillian integrability of the equations of motion. A mixture of numerical approaches is used to test the theoretical conclusions in order to identify the nature of orbits and evaluate the system’s transition from order to chaos. These techniques consist of the Poincaré Section Surface, the maximum Lyapunov Exponent, and the Smaller Alignment Index.

Conserved quantities in non-hermitian systems via vectorization method

Open classical and quantum systems have attracted great interest in the past two decades. These include systems described by non-Hermitian Hamiltonians with parity-time (PT) symmetry that are best understood as systems with balanced, separated gain and loss. Here, we present an alternative way to characterize and derive conserved quantities, or intertwining operators, in such open systems. As a consequence, we also obtain non-Hermitian or Hermitian operators whose expectations values show single exponential time dependence. By using a simple example of a PT-symmetric dimer that arises in two distinct physical realizations, we demonstrate our procedure for static Hamiltonians and generalize it to time-periodic (Floquet) cases where intertwining operators are stroboscopically conserved. Inspired by the Lindblad density matrix equation, our approach provides a useful additionto the well-established methods for characterizing time-invariants in non-Hermitian systems.

‘Approximate’ conservation laws of some nonlinear Schrödinger equation involving a spatially extended system consisting of two coupled elements

In this paper, we construct and analyse the conservation laws of a model of a nonlinear Schrödinger equation involving a spatially extended system consisting of two coupled elements. In the first of two cases, we study the space–time system and, separately, the stationary model arising from a standard transformation. In the second case, we analyse a ‘toy model PT-symmetric dimer’ involving a ‘gain-loss amplitude parameter’ for which only approximate conserved forms exist. In this latter case, Noether's theorem is not applicable as it is not variational and the conserved forms are adapted in an ‘approximate’ and novel way due to the nature of the system.

Necklaces of PT-symmetric dimers

We study light propagation through cyclic arrays, composed by copies of a given $\mathcal{PT}$-symmetric dimer, using a group theoretical approach and finite element modeling. The theoretical mode-coupling analysis suggest the use of these devices as output port replicators where the dynamics are controlled by the impinging light field. This is confirmed to good agreement with finite element propagation in an experimentally feasible necklace of passive $\mathcal{PT}$-symmetric dimers constructed from lossy and lossless waveguides.

Management of Controllable Two-Kuznetsov–Ma Soliton in 𝒫𝒯-Symmetric Dimers

Abstract We study a (2+1)-dimensional coupled nonlinear Schrödinger equation in 𝒫𝒯-symmetric inhomogeneous nonlinear dimers with different values of diffractions and derive a two-Kuznetsov–Ma soliton solution with 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry based on the Darboux and similarity transformation methods. In a diffraction decreasing system with exponential profile, we investigate the management of controllable two-Kuznetsov–Ma soliton by modulating the relation between the values of the maximal effective propagation distance Z m and periodic peak locations Z ij . If Z m <Z ij , Z m =Z ij and Z m >Z ij , we discuss the initial excitation, peak excitation, and complete excitation of the structure with the i th part and j th peak for two-Kuznetsov–Ma soliton.

Integrable oscillator type and Schrödinger type dimers

A PT-symmetric dimer is a two-site nonlinear oscillator dimer or a two-site nonlinear Schrödinger dimer where one site loses and the other site gains energy at the same rate. We present a wide class of integrable oscillator type dimers whose Hamiltonian is of arbitrary even order. Further, we also present a wide class of integrable nonlinear Schrödinger type dimers where again the Hamiltonian is of arbitrary even order. Finally, we consider a recently discussed complex dimer model and point out a few integrable cases in that model.

Optical Lattice Design Assisted by Non-Hermitian Hamiltonians

A brief introduction to non-Hermitian arrays of coupled waveguides is presented. The PT-symmetric dimer is revisited for the sake of clarity. It belongs to the class of photonic lattices with underlying SO(2,1) symmetry that have been shown to provide all-optical conversion from phase to amplitude.

Optical finite representation of the Lorentz group.

We implement a finite-dimensional representation of the 2+1D Lorentz group with a PT-symmetric waveguide array. Our device can be engineered to behave like a multi-port oscillator or directional coupler with amplification. We show that the two-waveguide coupler with linear losses, the Vernier effect in coupled asymmetric micro-cavity lasers, and the so-called linear PT-symmetric dimer belong to this symmetry class of optical systems.

PT-symmetric dimer of coupled nonlinear oscillators

We provide a systematic analysis of a prototypical nonlinear oscillator system respecting PT-symmetry, i.e., one of them has gain and the other an equal and opposite amount of loss. We first discuss various symmetries of the model. We show that both the linear system as well as a special case of the nonlinear system can be derived from a Hamiltonian, whose structure is similar to the Pais–Uhlenbeck Hamiltonian. Exact solutions are obtained in a few special cases. We show that the system is a superintegrable system within the rotating wave approximation (RWA). We also obtain several exact solutions of these RWA equations. Further, we point out a novel superposition in the context of periodic solutions in terms of Jacobi elliptic functions that we obtain in this problem. Finally, we briefly mention numerical results about the stability of some of the solutions.

Spectral and transport properties of the PT-symmetric dimer model

We study the scattering properties of the PT-symmetric tight-binding model with balanced gain and loss parameters. Our main interest is to establish the link between the spectral properties of scattering states and transport characteristics for the case of non-equal couplings between gain/loss sites. The analytical approach we have used allows one to reveal a quite unexpected role of this set-up in comparison with that of equal couplings. In particular, we demonstrate that for the exceptional points characterized by equal eigenvalues of the transfer matrix, the transmission coefficient can be different from one in contrast with the model with equal couplings. The analytical results are complemented by the numerical data.

Dynamics of generalized PT-symmetric dimers with time-periodic gain–loss

A parity-time (PT)-symmetric system with periodically varying-in-time gain and loss modeled by two coupled Schrödinger equations (dimer) is studied. It is shown that the problem can be reduced to a perturbed pendulum-like equation. This is done by finding two constants of motion. Firstly, a generalized problem using Melnikov-type analysis and topological degree arguments is studied for showing the existence of periodic (libration), shift- periodic (rotation), and chaotic solutions. Then these general results are applied to the PT-symmetric dimer. It is interestingly shown that if a sufficient condition is satisfied, then rotation modes, which do not exist in the dimer with constant gain–loss, will persist. An approximate threshold for PT-broken phase corresponding to the disappearance of bounded solutions is also presented. Numerical study is presented accompanying the analytical results.

Discrete solitons and scattering of lattice waves in guiding arrays with a nonlinear PT-symmetric defect.

Discrete fundamental and dipole solitons are constructed, in an exact analytical form, in an array of linear waveguides with an embedded PT-symmetric dimer, which is composed of two nonlinear waveguides carrying equal gain and loss. Fundamental solitons in tightly knit lattices, as well as all dipole modes, exist above a finite threshold value of the total power. However, the threshold vanishes for fundamental solitons in loosely knit lattices. The stability of the discrete solitons is investigated analytically by means of the Vakhitov-Kolokolov (VK) criterion, and, in the full form, via the computation of eigenvalues for perturbation modes. Fundamental and dipole solitons tend to be stable at smaller and larger values of the total power (norm), respectively. The increase of the strength of the coupling between the two defect-forming sites leads to strong expansion of the stability areas. The scattering problem for linear lattice waves impinging upon the defect is considered too.

PT-symmetric dimer of coupled nonlinear oscillators

We provide a systematic analysis of a prototypical nonlinear oscillator system respecting PT-symmetry i.e., one of them has gain and the other an equal and opposite amount of loss. Starting from the linear limit of the system, we extend considerations to the nonlinear case for both soft and hard cubic nonlinearities identifying symmetric and anti-symmetric breather solutions, as well as symmetry breaking variants thereof. We propose a reduction of the system to a Schr\"odinger type PT-symmetric dimer, whose detailed earlier understanding can explain many of the phenomena observed herein, including the PT phase transition. Nevertheless, there are also significant parametric as well as phenomenological potential differences between the two models and we discuss where these arise and where they are most pronounced. Finally, we also provide examples of the evolution dynamics of the different states in their regimes of instability.

Eigenstates and instabilities of chains with embedded defects

We consider the eigenvalue problem for one-dimensional linear Schrödinger lattices (tight-binding) with an embedded few-sites linear or nonlinear, Hamiltonian or non-conservative defect (an oligomer). Such a problem arises when considering scattering states in the presence of (generally complex) impurities as well as in the stability analysis of nonlinear waves. We describe a general approach based on a matching of solutions of the linear portions of the lattice at the location of the oligomer defect. As specific examples, we discuss both linear and nonlinear, Hamiltonian and PT-symmetric dimers and trimers. In the linear case, this approach provides us a handle for semi-analytically computing the spectrum [this amounts to the solution of a polynomial equation]. In the nonlinear case, it enables the computation of the linearization spectrum around the stationary solutions. The calculations showcase the oscillatory instabilities that strongly nonlinear states typically manifest.

Asymmetric wave propagation through nonlinear PT-symmetric oligomers

In the present paper, we consider nonlinear PT-symmetric dimers and trimers (more generally, oligomers) embedded within a linear Schrödinger lattice. We examine the stationary states of such chains in the form of plane waves, and analytically compute their reflection and transmission coefficients through the nonlinear PT symmetric oligomer, as well as the corresponding rectification factors which clearly illustrate the asymmetry between left and right propagation in such systems. We examine not only the existence but also the dynamical stability of the plane wave states and interestingly find them to be unstable except in the vicinity of the linear limit. Lastly, we generalize our numerical considerations to the more physically relevant case of Gaussian initial wavepackets and confirm that the asymmetry in the transmission properties persists in the case of such wavepackets, as well. The role of potential asymmetries in the nonlinearity or in the gain/loss pattern is also considered.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

NonlinearlyPT-symmetric systems: Spontaneous symmetry breaking and transmission resonances

We introduce a class of PT-symmetric systems which include mutually matched nonlinear loss and gain (inother words, a class of PT-invariant Hamiltonians in which both the harmonic and anharmonic parts are non-Hermitian). For a basic system in the form of a dimer, symmetric and asymmetric eigenstates, including multistable ones, are found analytically. We demonstrate that, if coupled to a linear chain, such a nonlinear PT-symmetric dimer generates new types of nonlinear resonances, with the completely suppressed or greatly amplified transmission, as well as a regime similar to the electromagnetically-induced transparency (EIT). The implementation of the systems is possible in various media admitting controllable linear and nonlinear amplification of waves.