Quartz-superconductor quantum electromechanical system

Quartz bulk acoustic wave oscillators support mechanical modes with very high resonance frequencies and extremely high quality factors. As such, they provide an appealing platform for quantum optics experiments with phonons, gravitational wave detection, and tests of quantum mechanics. We propose to cool and measure the motion of a quartz oscillator using a transmon, with the coupling mediated by a tuneable superconducting LC circuit. The mechanical motion ($\sim$250MHz) is resonantly coupled to the LC circuit ($\sim$250MHz) by a piezoelectric interaction, the LC circuit is coupled to the transmon ($\sim$8GHz) via sideband transitions, and there is a smaller direct coupling between the quartz oscillator and the transmon. By driving the transmon on its red sideband, the mechanical and electrical oscillators may be cooled close to their quantum ground state. By observing the fluorescence of the qubit, the occupations of the oscillators may be determined via the motional sidebands they induce. A minimal model of this system consists of a qubit coupled to two oscillators, which are themselves mutually coupled. The steady-state of the system and the qubit fluorescence spectrum are evaluated analytically using a perturbative projection operator technique, and verified numerically.