Non-Hermitian Hamiltonian of two-level systems in complex quaternionic space: An introduction in electronics

Abstract

An imaginary resistor (Z) based electronic dimer is used to describe the gyroscopic and resistive coupled two-level systems in quaternionic space. We successfully used the quaternionic coefficients to characterize the different classes of non-Hermitian systems: the pseudo-Hermitian and anti-Hermitian, both having exceptional points (EPs) separating the exact and the broken phases. Remarkably, the EPs conical dynamic is fully described to follow a hyperbolic/parabolic shape in the case of parity-time symmetry (PTS)/anti-PTS, respectively. Interestingly, our results demonstrated that the gyroscopic coupling mechanism allows identical non-dissipative but nonoscillatory systems with negative capacitor to exhibit PTS-like behavior.

Experimental observation of real spectra in Parity-Time symmetric ZRC dimers with positive and negative frequencies

“…. The ZRC cell is constituted by a capacitor C , a real resistor R and at least an imaginary resistor Z = jr [30,31]. The value of components may be positive or negative without any restrictions.…”

“…In Optics, the refractive index of the medium must satisfied a relation of the same nature n (x) = n * (−x) [7][8][9][10][11][12][13] and ε (x) = ε * (−x) for dielectric permittivity in Metamaterials [14][15][16][17]. The notion of Parity-Time Symmetry is also extended to many order fields such as Photonics [7][8][9][10][11][12][13], Mechanics [18][19][20], Optomechanics [20][21][22][23], Acoustics [24][25][26] and Electronics [27][28][29][30][31] just to name few. In 2010, Reuter realized the first experimental demonstration of PTsymmetry in Optics by coupling two waveguides having equal amount of gain and loss [7].…”

“…More after it was presented the counterpart of Parity-Time symmetry: The Anti-Parity Time symmetry. In this new Physics, the loss and the gain are exclusively in the dimers [30,[32][33][34][35][36][37][38]. In Quantum Mechanics, the potential follows the relation V (x) = −V * (−x) , the refractive index n(x) = −n * (−x) in Optics and ε (x) = −ε * (−x) for the dielectric permittivity in Metamaterials.…”

“…It also used the Coupled Mode Theory (CMT) and adiabatic elimination in optical waveguides , slowly varying envelope approximation in Nanophotonics to achieve the same reality. Many others works are also presented in these directions [30][31][32][33][34][35][36][37]. Tabeu et al presented in [30] how to achieve Parity-Time symmetry and Anti-Parity time symmetry in electronics using imaginary resistors [31,39,40] to have directly the system of first order ordinary differential equation without the coupling mode theory or the rotating frame.…”

“…Many others works are also presented in these directions [30][31][32][33][34][35][36][37]. Tabeu et al presented in [30] how to achieve Parity-Time symmetry and Anti-Parity time symmetry in electronics using imaginary resistors [31,39,40] to have directly the system of first order ordinary differential equation without the coupling mode theory or the rotating frame. The configurations presented were with Gain-Loss, Gain-Gain and Loss-Loss in the dimer.…”