measuring both of the mechanical sidebands, and find that the observed optical spring and damping quantitatively agree with theory.

To quantify the role of classical laser noise in this system, as well as optomechanical systems more generally, we also present a detailed theoretical model of optomechanical systems that are subject to classical laser noise. This model describes the roles of amplitude noise, phase noise, and amplitude-phase correlations in the multiple beams that are typically used to cool and measure an optomechanical system. Expressions are derived for the heterodyne spectrum expected for optomechanical systems in the presence of correlated noise sources, and we discuss the limits that classical laser noise imposes on cooling and reliably measuring the mean phonon number.

Cryogenic Apparatus
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![Cryogenic optomechanical system. **a** Two Nd-YAG lasers probe and cool the cryogenic optomechanical system. The lasers are frequency-locked $\sim$ 9 GHz apart by feeding back on the beat signal from a fast photodiode (FPD). The majority of the signal laser’s output serves as a heterodyne local oscillator. The rest is shifted 80 MHz with an acousto-optical modulator (AOM) and then phase modulated using an electro-optical modulator (EOM) with 22% of the power in $\pm$15 MHz sidebands. These beams 