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Josephson parametric amplifiers (JPA) have become key devices in quantumscience and technology with superconducting circuits. In particular, they canbe utilized as quantum-limited amplifiers or as a source of squeezed microwavefields. Here, we report on the detailed measurements of five flux-driven JPAs,three of them exhibiting a hysteretic dependence of the resonant frequencyversus the applied magnetic flux. We model the measured characteristics bynumerical simulations based on the two-dimensional potential landscape of thedc superconducting quantum interference devices (dc-SQUID), which provide theJPA nonlinearity, for a finite screening parameter $\beta_\mathrm{L}\,{>}\,0$and demonstrate excellent agreement between the numerical results and theexperimental data. Furthermore, we study the nondegenerate response ofdifferent JPAs and accurately describe the experimental results with ourtheory.

Flux-driven Josephson parametric amplifiers: Hysteretic flux response and nondegenerate gain measurements