  A contactless control of fluctuations of phase space variables of a
nanoobject belongs among the key methods needed for ultra-precise
nanotechnology and the upcoming quantum technology of macroscopic systems. Here
we utilize the experimental platform of a single levitating nanoparticle (NP)
to demonstrate essential protocols providing linear amplification of the
mechanical phase space variables together with squeezing of phase space
probability distribution. The protocol combines a controlled fast switching
between the parabolic trapping potential and either weak parabolic or inverted
parabolic amplifying potential leading to amplification of mean value and
variance (fluctuations) along an arbitrary phase space variable and squeezing
along the complementary one. The protocol is completed with cold damping scheme
to control the initial fluctuations of the NP phase space variables. We reached
the amplification gain $|G|>2$, the squeezing coefficient above 4 dB, and the
second-order energy correlation function approaching 3 which corresponds to a
maximum for a stochastic non-equilibrium classical state. These experimental
results will already allow pre-amplification and manipulation of nanomechanical
NP motion for all quantum protocols if the NP cooling towards the ground state
is applied.
