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 surfac
e 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.
We use the strong intrinsic non-linearity of a microwave superconducting qubit with a 4 GHz transition frequency to directly detect and control the energy of a micro-mechanical oscillator vibrating at 25 MHz. The qubit and the oscillator are coupled
electrostatically at a rate of approximately $2pitimes$22 MHz. In this far off-resonant regime, the qubit frequency is shifted by 0.52 MHz per oscillator phonon, or about 14 % of the 3.7 MHz qubit linewidth. The qubit behaves as a vibrational energy detector and from its lineshape we extract the phonon number distribution of the oscillator. We manipulate this distribution by driving number state sensitive sideband transitions and creating profoundly non-thermal states. Finally, by driving the lower frequency sideband transition, we cool the oscillator and increase its ground state population up to 0.48$pm$0.13, close to a factor of 8 above its value at thermal equilibrium. These results demonstrate a new class of electromechanics experiments that are a promising strategy for quantum non-demolition measurements and non-classical state preparation.
