In cavity optomechanics, light is used to control mechanical motion. A central goal of the field is achieving single-photon strong coupling, which would enable the creation of quantum superposition states of motion. Reaching this limit requires significant improvements in optomechanical coupling and cavity coherence. Here we introduce an optomechanical architecture consisting of a silicon nitride membrane coupled to a three-dimensional superconducting microwave cavity. Exploiting their large quality factors, we achieve an optomechanical cooperativity of 146,000 and perform sideband cooling of the kilohertz-frequency membrane motion to 34$pm$5 $mu$K, the lowest mechanical mode temperature reported to date. The achieved cooling is limited only by classical noise of the signal generator, and should extend deep into the ground state with superconducting filters. Our results suggest that this realization of optomechanics has the potential to reach the regimes of ultra-large cooperativity and single-photon strong coupling, opening up a new generation of experiments.
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