  Rigorous quantum formulation of the Parity-Time (PT) symmetry phenomenon in
the RF/microwave regime for a coupled 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 LRC 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 reduces to constants.
Then a second basis transform obtains the quantum Hamiltonian, including the
diagonal effective complex frequencies and the off-diagonal coupling terms,
together with the transformed basis. With the obtain 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.
