  A room-temperature mechanical oscillator undergoes thermal Brownian motion
with an amplitude much larger than the amplitude associated with a single
phonon of excitation. This motion can be read out and manipulated using laser
light using a cavity-optomechanical approach. By performing a strong quantum
measurement, i.e., counting single photons in the sidebands imparted on a
laser, we herald the addition and subtraction of single phonons on the 300K
thermal motional state of a 4GHz mechanical oscillator. To understand the
resulting mechanical state, we implement a tomography scheme and observe highly
non-Gaussian phase-space distributions. Using a maximum likelihood method, we
infer the density matrix of the oscillator and confirm the counter-intuitive
doubling of the mean phonon number resulting from phonon addition and
subtraction.
