  A closed-loop, lossy optomechanical system consisting of one optical and two
degenerate mechanical resonators is computationally investigated. This system
constitutes an elementary synthetic plaquette derived from the loop phase of
the intercoupling coefficients. In examining a specific quantum attribute, we
delve into the control of quadrature variances within the resonator selected
through the plaquette phase. An amplitude modulation is additionally applied to
the cavity-pumping laser to incorporate mechanical squeezing. Our numerical
analysis relies on the integration-free computation of steady-state covariances
for cooling and the Floquet technique for squeezing. We provide physical
insights into how non-Hermiticity plays a crucial role in enhancing cooling and
squeezing in proximity to exceptional points. This enhancement is associated
with the behavior of complex eigenvalue loci as a function of the
intermechanical coupling rate. Additionally, we demonstrate that the parameter
space embodies an exceptional surface, ensuring the robustness of exceptional
point singularities under experimental parameter variations. However, the pump
laser detuning breaks away from the exceptional surface unless it resides on
the red-sideband by an amount sufficiently close to the mechanical resonance
frequency. Finally, we show that this disparate parametric character entitles
frequency-dependent cooling and squeezing, which is of technological
importance.
