Frequency Selectivity and Enhancement of Efficiency in Wireless Power Transfer-Based Parity-Time Symmetric Trimer

PT-symmetric electronic dimer without gain material

“…The  -symmetry becomes evident in the system as interchanging subindex 1 by 2, j by its complex conjugate −j and time t by −t in the above equations keep them invariant. According to these equations, the role of the gain or loss can be assigned to the imaginary resistor, whose impedance factor multiplies the first derivative terms, in comparison to conventional  circuit equations [11][12][13][14]. As usual, the two second-order equation (1) can be written as a system of four first-order equations using, for instance, the following change of variables:…”

“…Among non-Hermitian Hamiltonians, it belongs to those that are invariant under the joint transformations of spatial reflection (parity ) and temporal inversion (time-reversal  ) which may admit real eigenvalues. This particular class of Hamiltonians, called  -symmetric Hamiltonians, was first introduced in Quantum Mechanics in 1998 by Carl Bender and Stephan Boettcher [1], before rapidly spreading to other branches of physics as optics [2][3][4][5], photonics [4,6], mechanics [7][8][9][10] and electronics [11][12][13][14][15], to mention a few.  -symmetric Hamiltonians generally require that gain and loss into the system should be balanced, so that, as the degree of non-Hermiticity (also called gain/loss parameter) is increased, the system exhibits a transition from  exact phase with real frequencies to  broken phase, where the frequencies become complex.…”