  Levitated particles are a promising platform for precision sensing of
external perturbations and probing the boundary between quantum and classical
worlds. A critical obstacle for these applications is the difficulty of
generating nonclassical states of motion which have not been realized so far.
Here, we show that strong squeezing of the motion of a levitated particle below
the vacuum level is feasible with available experimental parameters. Using
suitable modulation of the trapping potential (which is impossible with clamped
mechanical resonators) and coherent scattering of trapping photons into a
cavity mode, we explore several strategies to achieve strong phase-sensitive
suppression of mechanical fluctuations. We analyze mechanical squeezing in both
transient and steady-state regimes, and discuss conditions for preparing
nonclassical mechanical squeezing. Our results pave the way to full,
deterministic optomechanical control of levitated particles in the quantum
regime.
