Quantum control over a physical system requires thermal fluctuations and thermal decoherence to be negligible, which becomes more challenging with decreasing natural frequencies of the target system. For microwave circuits, the quantum regime can be reached simply by cooling them to mK temperatures. Radio-frequency (RF) systems in the MHz regime, however, require further cooling or have to be coupled to an auxiliary quantum system with a coupling rate exceeding their thermal decoherence rate. A powerful tool to cool below the thermodynamic bath temperature is sideband-cooling, a technique that originated from the field of trapped ions and cold atoms and that has been applied in cavity optomechanics for groundstate cooling of mechanical motion. Here, we engineer a system of two superconducting LC circuits coupled by a current-mediated photon-pressure interaction and demonstrate sideband-cooling of a hot RF circuit using a microwave cavity and the regime of quantum-coherent coupling between the circuits. Due to a dramatically increased coupling strength, we obtain a large single-photon quantum cooperativity $mathcal{C}_{mathrm{q}0} sim 1$ and reduce the residual thermal RF occupancy by 75% through sideband-cooling with less than a single pump photon. For larger pump powers, the photon-pressure coupling rate exceeds the RF thermal decoherence rate by a factor of three and the RF circuit is cooled into the quantum groundstate. Our results demonstrate photon-pressure coupling with a hot radio-frequency circuit in the quantum regime and lay the foundation for radio-frequency quantum photonics.
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