Cavity-enhanced Brillouin scattering interactions with gigahertz-frequency acoustic phonons offer a promising pathway towards the quantum coherent control of mechanical oscillators. In this thesis, I experimentally investigate single-quantum-level operations applied to thermal me- chanical oscillators by combining optical measurement techniques with Brillouin interactions in crystalline whispering-gallery-mode microresonator devices. These operations are explored for applications in quantum state engineering and optical quantum memories. Generating and characterising non-classical states of mechanical motion currently represents a key challenge in quantum cavity optomechanics, and the realisation of a quantum memory would enable the development of many quantum technologies. The advances reported here contribute to both of these active areas of research.
In a series of three experiments, single- and multi-phonon addition and subtraction opera- tions applied to thermal mechanical states are explored. I present the first experimental inves- tigation of single-phonon addition and subtraction operations using a joint click-dyne detection scheme, where the effect of such operations are verified by observing a characteristic doubling of the mean occupation of the state. These techniques are then extended to multi-phonon subtraction. Here, the 𝑠-parameterised Wigner function of the resulting non-Gaussian states are determined, advancing the state-of-the-art for optics-based mechanical state tomography. Finally, an interferometric detection scheme is employed that implements a superposition of phonon subtractions in two time bins, and the phase coherence between these two operations is demonstrated and studied. In this thesis, I also theoretically investigate the prospects of an optical quantum memory based on Brillouin cavity optomechanics. Using realistic parameters, I show that efficient storage and retrieval of single photons is feasible, and I identify two key applications: temporal multiplexing and temporal mode manipulation. The deleterious effect of thermal noise in such optomechanical quantum light storage is also considered. To conclude, an outlook towards some near-term and long-term experimental goals that can build upon on the achievements reported is presented.