  Nanomechanical resonators are a key tool for future quantum technologies such
as quantum force sensors and interfaces, and for studies of macroscopic quantum
physics. The ability to prepare room temperature non-classical states is a
major outstanding challenge. Here, we explore the use of measurement-based
state conditioning to achieve this. We demonstrate conditional cooling of a
nanomechanical resonator that has non-Markovian decoherence, and show
theoretically that the non-Markovianity makes quantum squeezing significantly
easier to achieve. We further show that collective measurement of multiple
resonator modes improves the quality of state preparation. This allows us to
achieve collective thermomechanical squeezing, in experiments that go beyond
the validity of the rotating-wave approximation. Our modelling shows that
non-Markovianity and multimode conditioning can both enable room temperature
quantum squeezing with existing technology. Together, our results pave the way
towards realising room temperature quantum nanomechanical devices and towards
their application in quantum technology and fundamental science.
