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 applicability mechanics, has been central topic modern experimental physics. Highly coherent heavy micro-mechanical oscillators controlled by superconducting circuits are promising system for this task. Here, we focus particular on electrostatic coupling motion to weakly anharmonic circuit, namely transmon qubit. In case megahertz mechanical oscillator coupled gigahertz transmon, explain difficulties bridging large electro-mechanical frequency gap. To remedy issue, explore requirements reach phonon-number resolution resonant oscillator.