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