  We address the out-of-equilibrium thermodynamics of an isolated quantum
system consisting of a cavity optomechanical device. We explore the dynamical
response of the system when driven out of equilibrium by a sudden quench of the
coupling parameter and compute analytically the full distribution of the work
generated by the process. We consider linear and quadratic optomechanical
coupling, where the cavity field is parametrically coupled to either the
position or the square of the position of a mechanical oscillator,
respectively. In the former case we find that the average work generated by the
quench is zero, whilst the latter leads to a non-zero average value. Through
fluctuations theorems we access the most relevant thermodynamical figures of
merit, such as the free energy difference and the amount of irreversible work
generated. We thus provide a full characterization of the out-of-equilibrium
thermodynamics in the quantum regime for nonlinearly coupled bosonic modes. Our
study is the first due step towards the construction and full quantum analysis
of an optomechanical machine working fully out of equilibrium.
