Rigorous Quantum Formulation of Parity-Time Symmetric Coupled Resonators

Rigorous quantum formulation of the Parity-Time (PT) symmetry phenomenon in the RF/microwave regime for a pair of coil resonators with lump elements has been presented. The coil resonator is described by the lump-element model that consists of an inductor (L), a resistor (R), and a capacitor (C). Rigorous quantum Hamiltonian for the coupled RLC coil resonators system has been derived through twice basis transforms of the original basis. The first basis transform rotates the original basis such that off-diagonal terms of the governing matrix of the equation system of the coupled coil resonators is reduced to constants. Then a second basis transform obtains the quantum Hamiltonian, including the diagonal effective complex frequencies and off-diagonal coupling terms, together with the transformed basis. With the obtained quantum Hamiltonian, the eigenvalues and eigenvectors of the coupled coil resonators can be obtained as usual as the quantum Hamiltonian. Finally, numerical simulation verifies the correctness of the theory. The quantum formulation of the coupled coil resonators can provide better guideline to design a better PT-symmetric system.