  Mechanical resonators are emerging as an important new platform for quantum
science and technologies. A large number of proposals for using them to store,
process, and transduce quantum information motivates the development of
increasingly sophisticated techniques for controlling mechanical motion in the
quantum regime. By interfacing mechanical resonators with superconducting
circuits, circuit quantum acoustodynamics (cQAD) can make a variety of
important tools available for manipulating and measuring motional quantum
states. Here we demonstrate direct measurements of the phonon number
distribution and parity of nonclassical mechanical states. We do this by
operating our system in the strong dispersive regime, where a superconducting
qubit can be used to spectroscopically resolve phonon Fock states. These
measurements are some of the basic building blocks for constructing acoustic
quantum memories and processors. Furthermore, our results open the door to
performing even more complex quantum algorithms using mechanical systems, such
as quantum error correction and multi-mode operations.
