  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
micro-mechanical 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
electro-mechanical 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.
