  We resolve phonon number states in the spectrum of a superconducting qubit
coupled to a multimode acoustic cavity. Crucial to this resolution is the sharp
frequency dependence in the qubit-phonon interaction engineered by coupling the
qubit to surface acoustic waves in two locations separated by $\sim40$ acoustic
wavelengths. In analogy to double-slit diffraction, the resulting
self-interference generates high-contrast frequency structure in the
qubit-phonon interaction. We observe this frequency structure both in the
coupling rate to multiple cavity modes and in the qubit spontaneous emission
rate into unconfined modes. We use this sharp frequency structure to resolve
single phonons by tuning the qubit to a frequency of destructive interference
where all acoustic interactions are dispersive. By exciting several detuned yet
strongly-coupled phononic modes and measuring the resulting qubit spectrum, we
observe that, for two modes, the device enters the strong dispersive regime
where single phonons are spectrally resolved.
