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arXiv:1410.5278 · Journal of Physics A Mathematical and Theoretical

A unidirectionally invisible $\mathcal{P}\mathcal{T}$-symmetric complex crystal with arbitrary thickness

Stefano Longhi

Published Nov 11, 2014• 15 citations
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Abstract

We introduce a new class of parity-time ( )-symmetric complex crystals which are almost transparent and one-way reflectionless over a broad frequency range around the Bragg frequency, i.e., unidirectionally invisible, regardless of the thickness L of the crystal. The -symmetric complex crystal is synthesized by a supersymmetric (SUSY) transformation of a Hermitian square well potential, and exact analytical expressions of transmission and reflection coefficients are given. As L is increased, the transmittance and reflectance from one side remain close to one and zero, respectively, whereas the reflectance from the other side secularly grows like owing to unidirectional Bragg scattering. This is a distinctive feature as compared to the previously studied case of the complex sinusoidal -symmetric potential at the symmetry breaking point, where transparency breaks down as .

DOI
10.1088/1751-8113/47/48/485302

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A unidirectionally invisible PT\mathcal{P}\mathcal{T}PT-symmetric complex crystal with arbitrary thickness
We introduce a new class of parity-time ( )-symmetric complex crystals which are almost transparent and one-way reflectionless over a broad frequency range around the Bragg frequency, i.e., unidirectionally invisible, regardless of the thickness L of the crystal. The -symmetric complex crystal is synthesized by a supersymmetric (SUSY) transformation of a Hermitian square well potential, and exact analytical expressions of transmission and reflection coefficients are given. As L is increased, the transmittance and reflectance from one side remain close to one and zero, respectively, whereas the reflectance from the other side secularly grows like owing to unidirectional Bragg scattering. This is a distinctive feature as compared to the previously studied case of the complex sinusoidal -symmetric potential at the symmetry breaking point, where transparency breaks down as .
We propose an efficient scheme in helium or alkaline earth atomic vapor to achieve a parity-time symmetric Bragg structure using coherent lights. Unidirectional invisibility can be realized in this scheme, i.e., the atomic vapor shows total transparency for probe light incident from one particular direction, but exhibits enhanced Bragg reflection for probe from the opposite side. By changing the relative phase between the coherent lights, this direction can easily be manipulated, providing a convenient way for investigating special properties of 𝒫𝒯 -symmetric Bragg structures.
Reflectionless defects in Hermitian tight-binding lattices, synthesized by the intertwining operator technique of supersymmetric quantum mechanics, are generally not invisible and time-of-flight measurements could reveal the existence of the defects. Here it is shown that, in a certain class of non-Hermitian tight-binding lattices with complex hopping amplitudes, defects in the lattice can appear fully invisible to an outside observer. The synthesized non-Hermitian lattices with invisible defects possess a real-valued energy spectrum; however, they lack parity-time (PT\mathcal{PT}PT) symmetry, which does not play any role in the present work.
We investigate a two-parametric family of one-dimensional non-Hermitian complex potentials with parity-time (PT\mathcal{PT}PT) symmetry. We find that there exist two distinct types of phase transitions, from an unbroken phase (characterized by a real spectrum) to a broken phase (where the spectrum becomes complex). The first type involves the emergence of a pair of complex eigenvalues bifurcating from the continuous spectrum. The second type is associated with the collision of such pairs at the bottom of the continuous spectrum. The first transition type is closely related to spectral singularities (SSs), at which point the transmission and reflection coefficients are divergent simultaneously. The second is associated with the emergence of bound states. In particular, under specific parameter conditions, we construct an exact bound state solution. By systematically exploring the parameter space, we establish a universal relationship governing the number of SSs in these potentials. These findings provide a fundamental theoretical framework for manipulating wave scattering in non-Hermitian systems, offering promising implications for designing advanced optical and quantum devices.
Lossy atomic photonic crystals can be suitably tailored so that the real and imaginary parts of the susceptibility are, respectively, an odd and an even function of position. Such a parity-time (PT)(PT)(PT) space antisymmetry in the susceptibility requires neither optical gain nor negative refraction, but is rather attained by a combined control of the spatial modulation of both the atomic density and their dynamic level shift. These passive photonic crystals made of dressed atoms are characterized by a tunable unidirectional reflectionlessness accompanied by an appreciable degree of transmission. Interestingly, such peculiar properties are associated with non-Hermitian degeneracies of the crystal scattering matrix, which can then be directly observed through reflectivity measurements via a straightforward phase modulation of the atomic dynamic level shift and even off resonance.
We study the PT\mathcal{PT}PT-symmetry breaking for the scattering problem in a one-dimensional non-Hermitian tight-binding lattice model with balanced gain and loss distributed on two adjacent sites. In the scattering process the system undergoes a transition from the exact PT\mathcal{PT}PT-symmetric phase to the phase with spontaneously breaking PT\mathcal{PT}PT symmetry as the amplitude of complex potentials increases. Using the S-matrix method, we derive an exact discriminant, which can be used to distinguish different symmetry phases, and determine the exceptional point for the symmetry breaking analytically. In the PT\mathcal{PT}PT-symmetry-breaking region, we also confirm the appearance of the unique feature, i.e., the coherent perfect absorption laser, in this simple non-Hermitian lattice model. The study of the scattering problem of such a simple model provides an additional way to unveil the physical effect of non-Hermitian PT\mathcal{PT}PT-symmetric potentials.
PT\mathcal{PT}PT-like phase transitions from square roots of supersymmetric Hamiltonians
We introduce a general framework for realizing PT\mathcal{PT}PT-like phase transitions in non-Hermitian systems without imposing explicit parity--time (PT\mathcal{PT}PT) symmetry. The approach is based on constructing a Hamiltonian as the square root of a supersymmetric partner energy-shifted by a constant. This formulation naturally leads to bipartite dynamics with balanced gain and loss and can incorporate non-reciprocal couplings. The resulting systems exhibit entirely real spectra over a finite parameter range precisely when the corresponding passive Hamiltonian lacks a zero mode. As the non-Hermitian parameter representing gain and loss increases, the spectrum undergoes controlled real-to-complex transitions at second-order exceptional points. We demonstrate the versatility of this framework through several examples, including well-known models such as the Hatano--Nelson (HN) and complex Su--Schrieffer--Heeger (cSSH) lattices. Extending the formalism to qqq-commuting matrices further enables the systematic realization of higher-order exceptional points in systems with unidirectional couplings. Overall, this work uncovers new links between non-Hermitian physics and supersymmetry, offering a practical route to engineer photonic arrays with tunable spectral properties beyond what is achievable with explicit PT\mathcal{PT}PT-symmetry.