  We study a multi-resonator optomechanical system, consisting of two SiN
membranes coupled to a single optical cavity mode. Correlations in the noise of
the optomechanically coupled thermal baths lead to destructive interference,
which we observe as a reduction in the mechanical noise power spectrum by up to
20 dB, close to the mechanical resonance frequencies. We show that this effect
can be controlled by adjusting the optomechanical interaction strength between
the resonators, and that it originates from the frequency shift of the
optomechanically scattered photons that provide the effective
mechanics-mechanics coupling. Based on this effective coupling, we also
propose, derive and measure a collective effect, cooperativity competition on
mechanical dissipation, which causes the optomechanically broadened linewidth
of one resonator to depend on the coupling efficiency (cooperativity) of the
other resonator.
