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

Abstract

It is well known that typical \(\mathbb {PT}\)-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 \(\mathbb {PT}\) symmetry to arbitrarily large values of the gain-loss coefficient. First, we recapitulate the analysis which demonstrates a possibility of the restoration of the \(\mathbb {PT}\) 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 \( \mathbb {PT}\)-symmetry-breaking transition is found in a deeply subwavelength region. Next, we review a recently proposed possibility to construct stable one-dimensional (1D) \(\mathbb {PT}\)-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 \(\mathbb {PT}\) symmetry in it is also an essentially nonlinear one. Finally, we report new results for unbreakable \(\mathbb {PT}\)-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.