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The field of cavity optomechanics has achieved groundbreaking photoniccontrol and detection of mechanical oscillators, based on their coupling tolinear electromagnetic modes. Lately, however, there is an uprising interest inexploring cavity nonlinearities as a powerful new resource inradiation-pressure interacting systems. Here, we present a flux-mediatedoptomechanical device combining a nonlinear Josephson-based superconductingquantum interference cavity with a mechanical nanobeam. We demonstrate how theintrinsic Kerr nonlinearity of the microwave circuit can be used for acounter-intuitive blue-detuned sideband-cooling scheme based on multi-tonecavity driving and intracavity four-wave-mixing. Based on the largesingle-photon coupling rate of the system of up to $g_0 = 2\pi\cdot 3.6\,$kHzand a high mechanical quality factor $Q_\mathrm{m} \approx 4\cdot 10^{5}$, weachieve an effective four-wave cooperativity of $\mathcal{C}_\mathrm{fw} > 100$and demonstrate four-wave cooling of the mechanical oscillator close to itsquantum groundstate, achieving a final occupancy of $n_\mathrm{m} \sim 1.6$.Our results significantly advance the recently developed platform offlux-mediated optomechanics and demonstrate how cavity Kerr nonlinearities canbe utilized for novel control schemes in cavity optomechanics.

Four-wave-cooling to the single phonon level in Kerr optomechanics