Synthesizing multi-phonon quantum superposition states using three-body interactions with transmon qubits

We propose a scheme for controlling a radio-frequency mechanical resonator at the quantum scale using two superconducting transmon qubits. The qubits are coupled via a capacitor in parallel to a superconducting quantum interference device (SQUID), which has a suspended mechanical beam embedded in one of its arms. Following a theoretical analysis of the quantum system, we find that this configuration, in combination with an in-plane magnetic field, can give rise to a tuneable three-body interaction in the single-photon strong-coupling regime, while enabling suppression of the stray qubit-qubit coupling. Using state-of-the-art parameters and qubit operations at single-excitation levels, we numerically demonstrate the possibility of ground-state cooling as well as high-fidelity preparation of multi-phonon quantum states and qubit-phonon entanglement. Our work significantly extends the quantum control toolbox of radio-frequency mechanical resonators and may serve as a promising architecture for integrating such mechanical elements with transmon-based quantum processors.