Quantum Josephson relations for macroscopic Josephson systems

The quantum version of the Feynman two-state model in describing the Josephson systems is shown to be equivalent to a quantum two-mode coupled oscillator. Such equivalence permits us to treat the Josephson systems in a fully quantum mechanical manner in the two-state model and to reveal secondary macroscopic quantum effects therein. The noncommuting nature of macroscopic variables is crucial in doing so. An interesting ap plication of the present investigation into granular superconductor and high-temperature superconductor is briefly addressed.