  We propose a scheme for controlling a radio-frequency mechanical resonator at
the quantum level using a superconducting qubit. The mechanical part of the
circuit consists of a suspended micrometer-long beam that is embedded in the
loop of a superconducting quantum interference device (SQUID) and is connected
in parallel to a transmon qubit. Using realistic parameters from recent
experiments with similar devices, we show that this configuration can enable a
tuneable optomechanical interaction in the single-photon ultrastrong-coupling
regime, where the radiation-pressure coupling strength is larger than both the
transmon decay rate and the mechanical frequency. We investigate the dynamics
of the driven system for a range of coupling strengths and find an optimum
regime for ground-state cooling, consistent with previous theoretical
investigations considering linear cavities. Furthermore, we numerically
demonstrate a protocol for generating hybrid discrete- and continuous-variable
entanglement as well as mechanical Schr\"{o}dinger cat states, which can be
realised within the current state of the art. Our results demonstrate the
possibility of controlling the mechanical motion of massive objects using
superconducting qubits at the single-photon level and could enable applications
in hybrid quantum technologies as well as fundamental tests of quantum
mechanics.
