  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 followed by
tomography to determine the mechanical phase-space distribution. Here, we
experimentally perform heralded single- and multi-phonon subtraction via photon
counting to a laser-cooled mechanical thermal state with a Brillouin
optomechanical system at room temperature, and use optical heterodyne detection
to measure the $s$-parameterized Wigner distribution of the non-Gaussian
mechanical states generated. The techniques developed here advance the
state-of-the-art for optics-based tomography of mechanical states and will be
useful for a broad range of applied and fundamental studies that utilize
mechanical quantum-state engineering and tomography.
