Characterization of unidirectional and bidirectional transparencies in P T-symmetric quantum and classical systems

We studied the scattering state behavior of tight-binding quantum system and classical electrical transmission lines. We distributed on-site energies _n and resistances R _n , respectively, according to a parity-time symmetric distribution. Using the formalism of scattering matrix S and transfer matrix M, we derived analytical expressions for components of transfer matrix M, and through them, we found transmission coefficient T and left (R _L ) and right (R _R ) reflectance. In addition, we found a generalized conservation relation that relates T, R _L and R _R , valid for quantum and classical symmetric systems. We numerically studied the general behavior of T, R _L , R _R and eigenvalues s _± of scattering matrix S in the full range of possible values of the gain and loss parameters. We observed the existence of unidirectional and bidirectional transparency for specific energies E (quantum case) and specific frequencies Ω (transmission line case), for which the following conditions were simultaneously fulfilled: i) T was unitary (T = 1), ii) eigenvalues s _± were degenerate and unimodular s ± = 1 , iii) the product of reflectances tended to zero, R L R R → 0 , and iv) phases ϕ of one or both reflection amplitudes ϕ r L , ϕ r R showed abrupt change of π. This way, we characterized the unidirectional and bidirectional transparency for classical and quantum systems as a function of the gain and loss parameters that describe both models. In addition, the theoretical results showed complete agreement with the numerical calculation. We hope that our work contributes to a better understanding of the influence of the symmetric distribution of gain and loss parameters on the scattering properties of quantum and classical systems.