Nonlinear rf-SQUID metamaterials: two-tone spectroscopy and parametric effects beyond the Duffing regime

An experimental and theoretical study of the response of an rf-SQUID metamaterial to strong driving by means of two-tone resonant spectroscopy is presented. The superconducting metamaterial consists of an array of non-hysteretic rf-SQUIDs. We have observed pronounced oscillations of the intrinsic resonance frequency of the meta-atoms with the power of the driving tone. The shapes of these oscillations vary with the frequencies of the driving and probe signals. The response to the probe signal shows remarkable features such as parametric instabilities and sidebands and allows the direct imaging of a bistability. Our theoretical analysis is based on the classical nonlinear dynamics of rf-SQUIDs and in good agreement with experimental observations. We also present first results on parametric amplification by means of an rf-SQUID metamaterial beyond the Duffing regime.