  Tests of quantum mechanics on a macroscopic scale require extreme control
over mechanical motion and its decoherence. Quantum control of mechanical
motion has been achieved by engineering the radiation-pressure coupling between
a micromechanical oscillator and the electromagnetic field in a resonator.
Furthermore, measurement-based feedback control relying on cavity-enhanced
detection schemes has been used to cool micromechanical oscillators to their
quantum ground states. In contrast to mechanically tethered systems, optically
levitated nanoparticles are particularly promising candidates for matter-wave
experiments with massive objects, since their trapping potential is fully
controllable. In this work, we optically levitate a femto-gram dielectric
particle in cryogenic free space, which suppresses thermal effects sufficiently
to make the measurement backaction the dominant decoherence mechanism. With an
efficient quantum measurement, we exert quantum control over the dynamics of
the particle. We cool its center-of-mass motion by measurement-based feedback
to an average occupancy of 0.65 motional quanta, corresponding to a state
purity of 43%. The absence of an optical resonator and its bandwidth
limitations holds promise to transfer the full quantum control available for
electromagnetic fields to a mechanical system. Together with the fact that the
optical trapping potential is highly controllable, our experimental platform
offers a route to investigating quantum mechanics at macroscopic scales.
