  We establish a method for the determination of the single-photon
optomechanical coupling rate, which characterizes the radiation pressure
interaction in an optomechanical system. The estimation of the rate with which
a mechanical oscillator, initially in a thermal state, undergoes a Hopf
bifurcation, and reaches a limit cycle, allows us to determine the
single-photon optomechanical coupling rate in a simple and consistent way. Most
importantly, and in contrast to other methods, our method does not rely on
knowledge of the system's bath temperature and on a calibration of the signal.
We provide the theoretical framework, and experimentally validate this method,
providing a procedure for the full characterization of an optomechanical
system, which could be extended outside cavity optomechanics, whenever a
resonator is driven into a limit cycle by the appropriate interaction with
another degree of freedom.
