The Lagrangian approach to a Josephson traveling-wave parametric amplifier

Future progress and development in ultrasensitive detection and quantum-limited amplification of microwave signals in a wide frequency band is expected to be achieved by using new type architectures of parametric amplifiers based on chains of coupled Josephson junctions or superconducting quantum interference devices (SQUID). Over the past years the classical theory of the amplifiers in the framework of a nonlinear wave equation governing the node flux evolution along an array of coupled Josephson oscillators has been developed under continuum approximation. Although such amplifiers imply the quantum nature utilizing the Josephson effects there is no quantum approach to take into account quantum phenomena in Josephson traveling-wave parametric amplifiers (JTWPA). Here we formulate the Lagrangian approach to a one-dimensional chain of coupled Josephson junctions and take the first step to develop the quantum theory of JTWPAs by deriving an approximate quantum Hamiltonian of the chain