Dependence of Kerr nonlinearity on junction array size in Josephson parametric amplifiers*

Josephson parametric amplifiers have become indispensable components of superconducting quantum computing devices. However, their gain saturation power, which is limited by Kerr nonlinearity of Josephson junctions, is still too low for large scale readout.One way to suppress the Kerr nonlinearity, therefore improving the gain saturation power, is to replace the single Josephson element with an array of such elements. We investigate the dependence of Kerr nonlinearity on the array size under realistic fabrication constraints. We find the suppression of Kerr nonlinearity is sub-linear in array size for amplifiers with tens to hundreds Josephson element. We believe such a weak dependence accounts for the fact that arraying has not yet brought significant improvement of the dynamic
range in Josephson parametric amplifiers.