Phonon-number resolution of voltage-biased mechanical oscillators with weakly anharmonic superconducting circuits

Observing quantum phenomena in macroscopic objects, and the potential discovery of a fundamental limit in the applicability of quantum mechanics, has been a central topic of modern experimental physics. Highly coherent and heavy micromechanical oscillators controlled by superconducting circuits are a promising system for this task. Here we focus in particular on the electrostatic coupling of motion to a weakly anharmonic circuit, namely, the transmon qubit. In the case of a megahertz mechanical oscillator coupled to a gigahertz transmon, we explain the difficulties in bridging the large electromechanical frequency gap. To remedy this issue, we explore the requirements to reach phonon-number resolution in the resonant coupling of a megahertz transmon and a mechanical oscillator.