Optimized Design Parameters for a Flux-Driven SNAIL-Based Traveling-Wave Parametric Amplifier

We present an analytic approach for finding optimized design parameters for flux-driven traveling-wave parametric amplifiers with superconducting nonlinear asymmetric inductive elements. The parameter to be optimized, i.e., the gain coefficient, is extracted from a Hamiltonian quantum model for three-wave mixing flux-driven amplifiers. Analytic expressions for the nonlinear coefficients up to the third order in the current-phase relation are derived. The minimum potential energy is determined by finding the roots of a polynomial function. Candidates for optimal operation points are identified by minimizing the residual cross-phase modulation while at the same time maximizing the gain coefficient. Finally, the gain of a traveling-wave parametric amplifier implementation with optimized parameters is evaluated using a quantum model, including substrate losses and chromatic dispersion.