In recent years Josephson-junction-based parametric amplifiers have been developed for single-photon-level signals in the microwave spectral region, where they found applications in quantum information processing. Typically, Josephson traveling wave parametric amplifiers (JTWPA) have been studied based on classical circuit models. Although this approach shows good agreement with experimental results, it describes the system behavior in the framework of a classical theory, neglecting the quantum nature of the device. A quantum mechanical treatment of the JTWPA was only given in a few recent papers, where [1] derives a Hamiltonian for a chain of unit cells without resonant phase-matching (RPM), and [2] solves the dynamics of the system with and without RPM, also partly considering noise and squeezed state generation. However, our approach differs from [2] in that we give an explicit solution to the resulting nonlinear wave equation for narrow-band signals, starting from discrete chain Hamiltonians with and without RPM.     «

In recent years Josephson-junction-based parametric amplifiers have been developed for single-photon-level signals in the microwave spectral region, where they found applications in quantum information processing. Typically, Josephson traveling wave parametric amplifiers (JTWPA) have been studied based on classical circuit models. Although this approach shows good agreement with experimental results, it describes the system behavior in the framework of a classical theory, neglecting the quantum...     »