Making the P T $$\mathbb {PT}$$ Symmetry Unbreakable

Abstract

It is well known that typical PT -symmetric systems suffer symmetry breaking when the strength of the gain-loss terms, i.e., the coefficient in front of the non-Hermitian part of the underlying Hamiltonian, exceeds a certain critical value. In this article, we present a summary of recently published and newly produced results which demonstrate various possibilities of extending the PT symmetry to arbitrarily large values of the gain-loss coefficient. First, we recapitulate the analysis which demonstrates a possibility of the restoration of the PT symmetry and, moreover, complete avoidance of the breaking in a photonic waveguiding channel of a subwavelength width. The analysis is necessarily based on the system of Maxwell's equations, instead of the usual paraxial approximation. Full elimination of the PT -symmetry-breaking transition is found in a deeply subwavelength region. Next, we review a recently proposed possibility to construct stable one-dimensional (1D) PT -symmetric solitons in a paraxial model with arbitrarily large values of the gain-loss coefficient, provided that the self-trapping of the solitons is induced by self-defocusing cubic nonlinearity, whose local strength grows sufficiently fast from the center to periphery. The model admits a particular analytical solution for the fundamental soliton, and provides full stability for families of fundamental and dipole solitons. It is relevant to stress that this model is nonlinearizable, hence the concept of the PT symmetry in it is also an essentially nonlinear one. Finally, we report new results for unbreakable PT -symmetric solitons in 2D extensions of the 1D model: one with a quasi-1D modulation profile of the local gain-loss coefficient, and another with the fully-2D modulation. These settings admit particular analytical solutions for 2D solitons, while generic soliton families are found in a numerical form. The quasi-1D modulation profile gives rise to a stable family of single-peak 2D solitons, while their dual-peak counterparts tend to be unstable. The soliton stability in the full 2D model is possible if the local gain-loss term is subject to spatial confinement.2 exciton-polariton condensates [36]- [38], and in other physically relevant contexts. In particular, the transitions from unbroken to broken PT symmetry was observed in many experiments. One of prominent experimentally demonstrated applications of the PT symmetry in optics is unidirectional transmission of light [39].Other classical waveguiding settings also admit emulation of the PT symmetry, as demonstrated in acoustics [40] and predicted in optomechanical systems [41]. Also predicted were realizations of this symmetry in atomic Bose-Einstein condensates [42], magnetism [43], mechanical chains of coupled pendula [44], and electronic circuits [45] (in the latter case, the prediction was also demonstrated experimentally). In terms of the theoretical analysis, PT -symmetric extensions were also elaborated for 47], Burgers [48], and sine-Gordon [49] equations, as well as in a system combi...

Attraction centers and parity-time-symmetric delta-functional dipoles in critical and supercritical self-focusing media

“…(1), above which the PT symmetry breaks down [6,7]. Nevertheless, examples of systems with unbreakable PT symmetry are known too [8]. In fact, the linearized version * Electronic address: zyyan@mmrc.iss.ac.cn of the model introduced in the present work also avoids the breakdown, see Eqs.…”

Robust PT symmetry of two-dimensional fundamental and vortex solitons supported by spatially modulated nonlinearity

“…The stability of the stationary states was identified by numerical computation of eigenvalues of small perturbations, using linearized equations (41) for perturbations around the stationary solitons. Finally, the stability predictions, produced by the eigenvalues, were verified by simulations of the perturbed evolution of the solitons (some technical details are reported elsewhere [63]).…”

“…Solitons are also vulnerable to destabilization via the PT -symmetry breaking at the critical value of the gain-loss coefficient [59]. Nevertheless, it was found that, in some settings, the solitons' PT symmetry can be made unbreakable, extending to arbitrarily large values of the strength of the model's imaginary potential [60]- [62], see also a brief review of the unbreakability concept in [63]. The particular property of these models is that self-trapping of solitons is provided not by the self-focusing sign of the nonlinearity, but by the defocusing sign, with the coefficient in front of the cubic term growing fast enough from the center to periphery.…”

Dynamical control of solitons in a parity-time-symmetric coupler by periodic management

“…Actually, PT -symmetric Hamiltonians may admit transformation into Hermitian ones [12,13]. A well-established fact is that the spectrum of Hamiltonians with complex potentials subject to constraint (1) is real below a critical strength of the imaginary part of the potential, at which the PT symmetry gets broken, making the system unstable [14] (exceptions in the form of models with unbreakable PT symmetry are known too [15]). Thus far, the PT symmetry was not directly realized in quantum systems with complex potentials.…”

“…The PT symmetry in an optical waveguide (as well as its CP counterpart) may naturally combine with the material Kerr nonlinearity, giving rise to propagation models based on cubic nonlinear Schrödinger equations (NLSEs) with the complex potentials subject to condition (1). These models may generate PT -symmetric solitons, which were addressed in many theoretical works [18], [23]- [59], [15] (see also reviews [41,42]), and experimentally demonstrated too [34]. Although the presence of the gain and loss makes PT -symmetric media dissipative, solitons exist in them in continuous families, similar to the commonly known situation in conservative models [43], while usual dissipative solitons exist as isolated solutions (attractors, if they are stable) [44].…”