Two-tone spectroscopy of a SQUID metamaterial in the nonlinear regime

Compact microwave resonantors made of superconducting rings containing Josephson junctions (SQUIDs) are attractive candidates for building frequency tunable metamaterials with low losses and pronounced nonlinear properties. We explore the nonlinearity of a SQUID metamaterial by performing a two-tone resonant spectroscopy. The small-amplitude response of the metamaterial under strong driving by a microwave pump tone is investigated experimentally and theoretically. The transmission coefficient $S_{21}$ of a weak probe signal is measured in the presence of the pump tone. Increasing the power of the pump, we observe pronounced oscillations of the SQUID's resonance frequency $f_{\textrm{res}}$. The shape of these oscillations varies significantly with the frequency of the pump tone $f_{\textrm{dr}}$. The response to the probe signal displays instabilities and sidebands. A state with strong second harmonic generation is observed. We provide a theoretical analysis of these observations, which is in good agreement with the experimental results.