  The coherent states that describe the classical motion of a mechanical
oscillator do not have well-defined energy, but are rather quantum
superpositions of equally-spaced energy eigenstates. Revealing this quantized
structure is only possible with an apparatus that measures the mechanical
energy with a precision greater than the energy of a single phonon,
$\hbar\omega_\text{m}$. One way to achieve this sensitivity is by engineering a
strong but nonresonant interaction between the oscillator and an atom. In a
system with sufficient quantum coherence, this interaction allows one to
distinguish different phonon number states by resolvable differences in the
atom's transition frequency. Such dispersive measurements have been studied in
cavity and circuit quantum electrodynamics where experiments using real and
artificial atoms have resolved the photon number states of cavities. Here, we
report an experiment where an artificial atom senses the motional energy of a
driven nanomechanical oscillator with sufficient sensitivity to resolve the
quantization of its energy. To realize this, we build a hybrid platform that
integrates nanomechanical piezoelectric resonators with a microwave
superconducting qubit on the same chip. We excite phonons with resonant pulses
of varying amplitude and probe the resulting excitation spectrum of the qubit
to observe phonon-number-dependent frequency shifts $\approx 5$ times larger
than the qubit linewidth. Our result demonstrates a fully integrated platform
for quantum acoustics that combines large couplings, considerable coherence
times, and excellent control over the mechanical mode structure. With modest
experimental improvements, we expect our approach will make quantum
nondemolition measurements of phonons an experimental reality, leading the way
to new quantum sensors and information processing approaches that use
chip-scale nanomechanical devices.
