  Optomechanical couplings involve both beam-splitter and two-mode-squeezing
types of interactions. While the former underlies the utility of many
applications, the latter creates unwanted excitations and is usually
detrimental. In this work, we propose a simple but powerful method based on
cavity parametric driving to suppress the unwanted excitation that does not
require working with a deeply sideband-resolved cavity. Our approach is based
on a simple observation: as both the optomechanical two-mode-squeezing
interaction and the cavity parametric drive induce squeezing transformations of
the relevant photonic bath modes, they can be made to cancel one another. We
illustrate how our method can cool a mechanical oscillator below the quantum
back-action limit, and significantly suppress the output noise of a
sideband-unresolved optomechanical transducer.
