Unambiguous scattering matrix for non-Hermitian systems

Abstract

PT symmetry is a unique platform for light manipulation and versatile use in unidirectional invisibility, lasing, sensing, etc. Broken and unbroken PT -symmetric states in non-Hermitian open systems are described by scattering matrices. A multilayer structure, as a simplest example of the open system, has no certain definition of the scattering matrix, since the output ports can be permuted. The uncertainty in definition of the exceptional points bordering PT -symmetric and PT -symmetry-broken states poses an important problem, because the exceptional points are indispensable in applications as sensing and mode discrimination. Here we derive the proper scattering matrix from the unambiguous relation between the PT -symmetric Hamiltonian and scattering matrix. We reveal that the exceptional points of the scattering matrix with permuted output ports are not related to the PT symmetry breaking. Nevertheless, they can be employed for finding a lasing onset as demonstrated in our time-domain calculations and scattering-matrix pole analysis. Our results are important for various applications of the non-Hermitian systems including encircling exceptional points, coherent perfect absorption, PT -symmetric plasmonics, etc.

Purcell effect in PT-symmetric waveguides

“…The problem of uniqueness of the scattering matrix of the PT-symmetric system has been successfully solved in Ref. [35] using the direct connection of the scattering matrix Ŝ with the PT-symmetric Hamiltonian Ĥ of the one-dimensional multilayer system as Ŝ = exp i Ĥ (𝑡 − 𝑡 0 )/ℏ . Correct positions of the exceptional points then read as 𝑠 1,2 = 𝑡 ± √ 𝑟 𝐿 𝑟 𝑅 , where the scattering matrix defined by the right-hand equation in Eq.…”

Unitary Scattering Protected by Pseudo-Hermiticity

“…The energy conservation from the unitary scattering was reported in several non-Hermitian scat-tering centers [71][72][73][74][75]. Thus, the Hermiticity is unnecessary for a unitary scattering; however, the non-unitary scattering more commonly appears in the non-Hermitian systems because of the lack of energy conservation [76][77][78][79][80][81][82][83][84][85][86][87][88][89][90][91][92]. Then, what is essential for a unitary scattering and the energy conservation in non-Hermitian physics?…”

Quasi-bound states in the continuum induced by $\mathcal{PT}$-symmetry breaking

“…-We start with description of our PT -symmetric system and the origin of phase transition there. The simplest PT -symmetric layered structure is the bilayer one, which is a well-studied system, see, e.g., our recent analysis [59]. The PT -symmetric bilayer consists of just two layers -one with loss (permittivity ε + ) and another with gain (ε − ).…”

“…The PT -symmetry breaking phenomenon is usually described in terms of the scattering matrix eigenvalues and eigenvectors. The scattering matrix of a multilayered structure has the form Ŝ = t r R r L t , where t is the transmission coefficient, r L and r R are the reflection coefficients for the left-and right-incident waves [59]. Eigenvalues s 1,2 of the matrix Ŝ are known to be both unimodular (|s 1,2 | = 1) in the PT -symmetric state and inversely linked (|s 1 | = 1/|s 2 |) in the broken-PT -symmetry state.…”