Gain- or loss-induced localization in one-dimensionalPT-symmetric tight-binding models

Abstract

We investigate the properties of PT-symmetric tight-binding models by considering both bounded and unbounded models. For the bounded case, we obtain closed form expressions for the corresponding energy spectra and we analyze the structure of eigenstates as well as their dependence on the gain/loss contrast parameter. For unbounded PT-lattices, we explore their scattering properties through the development of analytical models. Based on our approach we identify a mechanism that is responsible to the emergence of localized states that are entirely due to the presence of gain and loss. The derived expressions for the transmission and reflection coefficients allow one to better understand the role of PT-symmetry in energy transport problems occurring in such PT-symmetric tight-binding settings. Our analytical results are further exemplified via pertinent examples.

${\mathscr{P}}{\mathscr{T}}$-symmetry breaking in the steady state of microscopic gain–loss systems

“…Finally, to show that  -symmetry breaking in the steady state exists also for extended systems, we generalize the analysis above to coupled resonator arrays with alternating gain and loss [41,42], as depicted in figure 6(a). In this case the amplitudes a n and b n for each unit cell obey where ¢ g is the coupling between the unit cells and  ( ) F t n, are independent thermal noise processes.…”

$\mathcal {PT}$-symmetric optical superlattices

“…Complex crystals show rather unusual scattering and transport properties as compared to ordinary crystals, such as violation of the Friedel's law of Bragg scattering [37,38,44], double refraction and nonreciprocal diffraction [17], unidirectional Bloch oscillations [47], unidirectional invisibility [48,49,50,51,52], and invisible defects [53,54]. Complex crystals described by tight-binding Hamiltonians with complex site energies and/or hopping rates have been investigated in several recent works (see, for instance, [8,9,10,11,27,29,53,55,56,57,58,59,60,61] and references therein). Most of previous studies on non-Hermitian lattices have been limited to consider periodic or bi-periodic crystals, inhomogenous lattices, or lattices in presence of localized defects or disorder.…”

Exceptional points of degeneracy andPTsymmetry in photonic coupled chains of scatterers

“…This approach, which resembles the TB formalism in solid-state physics [52,53,63], was also utilized in [48,50,64,65] to analyze the general properties of discrete interactions in PT-symmetric systems. Following these assumptions, we can recast (1) in a much simpler form:…”

“…Furthermore, such additional radiation losses (at high frequencies) could be manipulated and engineered to produce effects related to EPDs, such as loss-induced transmission [39]. Here, the analysis of lossy chains of infinite length is omitted because it would require the use of an infinitely periodic GF [59][60][61][62], whereas our main scope is to provide a simple but intuitive analysis based on the nearest-neighbor approximation, that resembles the TB approach used in [48,50,64,65]. We consider an example of a chain in vacuum (i.e., These values approximately satisfy (19) because of the lowfrequency choice for this EPD to occur.…”

“…In view of the comparatively simpler (with respect to quantum physics) implementations, optical PTsymmetric structures have elicited a great deal of attention, leading to many interesting observations, including the demonstration of low-threshold lasing and laser absorbers [41,[43][44][45], enhanced nonlinear effects [34,[40][41][42][43], as well as metamaterial-based field manipulations [38,48,49]. In previous works, PT-symmetry has been shown in discrete arrangements of resonators and also using the so-called "tight-binding" (TB) approach [48,50]. Moreover, EPDs have been also observed in 2D and 3D geometries [7,37,51].…”