Large flux-mediated coupling in hybrid electromechanical system with a transmon qubit

Control over the quantum states of a massive oscillator is important for several technological applications and to test fundamental limits mechanics. Addition an internal degree freedom could be valuable resource such control. Recently, hybrid electromechanical systems using superconducting qubits, based on electric-charge mediated coupling, have been quite successful. Here, we realize device, consisting transmon qubit mechanical resonator coupled magnetic-flux. The coupling stems from quantum-interference phase across tunnel junctions. We demonstrate vacuum rate up 4 kHz by making resonant with readout cavity. Consequently, thermal-motion detected driving hybridized-mode mean-occupancy well below one photon. By tuning away cavity, can enhanced 40 kHz. In this limit, small coherent drive results in splitting spectrum, observe interference signature arising Landau-Zener-St\"uckelberg effect. With improvements coherence, system offers novel platform rich interactions potentially provide full control motional states.