In circuit quantum acoustodynamics (cQAD), superconducting circuits are combined with acoustic resonators to create and control non-classical states of mechanical motion. Simulating these systems is challenging due to the extreme difference in scale between the microwave and mechanical wavelengths. All existing techniques simulate the electromagnetic and mechanical subsystems separately. However, this approach may not be adequate for all cQAD devices. Here, we demonstrate a single simulation of a superconducting qubit coupled to an acoustic and a microwave resonator and introduce two methods for using this simulation to predict the frequencies, coupling rates, and energy-participation ratios of the electromechanical modes of the hybrid system. We also discuss how these methods can be used to investigate important dissipation channels and quantify the nontrivial effects of mode hybridization in our device. Our methodology is flexible and can be extended to other acoustic resonators and quantum degrees of freedom, providing a valuable new tool for designing hybrid quantum systems.
