Observation of the AC Stark shift and idler-resonance in two-tone measurements of a Josephson resonator

We report on the behaviour of a nonlinear Josephson cavity driven by two microwave signals. Using a single tone, the cavity exhibits the expected Duffing response of the device at high powers. For two-tone measurements, we use a strong pump at frequency f_p, and simultaneously measure the cavity with a weak probe. For f_p > f_c (cavity frequency), a Lorentzian response of the cavity is shifted in frequency, corresponding to the AC Stark-shift of the Josephson resonator. For f_p < f_c, we observe an abrupt jump in the shifted cavity frequency and an additional feature in the probe spectrum with net gain, occurring at a frequency mirrored about f_p. We attribute this feature to resonance condition of an idler tone, generated by 4-wave mixing, with the cavity frequency, in good agreement with an analytical theory. Our results provide insight into the physics of the driven nonlinear Josephson cavity, and a starting point for exploring the strongly-driven quantum regime.