  Studying the interplay between multiple coupled mechanical resonators is a
promising new direction in the field of optomechanics. Understanding the
dynamics of the interaction can lead to rich new effects, such as enhanced
coupling and multi-body physics. In particular, multi-resonator optomechanical
systems allow for distinct dynamical effects due to the optical cavity
coherently coupling mechanical resonators. Here, we study the mechanical
response of two SiN membranes and a single optical mode, and find that the
cavity induces a time delay between the local and cavity-transduced thermal
noises experienced by the resonators. This results in an optomechanical phase
lag that causes destructive interference, cancelling the mechanical thermal
noise by up to 20 dB in a controllable fashion, matching our theoretical
expectation. Based on the effective coupling between membranes, we further
propose, derive and measure a collective effect, cooperativity competition on
mechanical dissipation, whereby the linewidth of one resonator depends on the
coupling efficiency (cooperativity) of the other resonator.
