  Quantum optical measurement techniques offer a rich avenue for quantum
control of mechanical oscillators via cavity optomechanics. In particular, a
powerful yet little explored combination utilizes optical measurements to
perform heralded non-Gaussian mechanical state preparation and to determine the
mechanical phase-space distribution. Here, we experimentally perform heralded
single- and multi-phonon subtraction via photon counting to a room temperature
mechanical thermal state with a Brillouin optomechanical system, and use
optical heterodyne detection to measure the $s$-parameterized Wigner
phase-space distribution of the non-Gaussian mechanical states generated. The
techniques developed here will be useful for a broad range of both applied and
fundamental studies that exploit quantum-state engineering and reconstruction
of mechanical motional states.
