Spatial Evolution of Quantized Discrete-Mode Operators in a Lossy Nonlinear Josephson-Embedded Transmission Line

We present a description of the spatial evolution of quantized discrete-mode operators along a lossy nonlinear transmission line. The nonlinearity is formed by hundereds or even thousands of Josephson junctions which are placed periodically along a microwave transmission line. Dissipation is added to the system Hamiltonian by coupling the nonlinear transmission line to an Ohmic bath. Using the Hamiltonian of the open quantum system, Heisenberg equations of motion for the discrete mode operators can be derived in terms of quantum Langevin equations. The temporal equations of motion are then translated to the spatial domain to investigate the performance of a nonlinear four-wave-mixing process, while signals propagate along the transmission line.