  We theoretically explore a quantum memory using a single nanoparticle
levitated in an optical dipole trap and subjected to feedback cooling. This
protocol is realized by storing and retrieving a single photon quantum state
from a mechanical mode in levitated cavityless optomechanics. We describe the
effectiveness of the photon-phonon-photon transfer in terms of the fidelity,
the Wigner function, and the zero-delay second-order autocorrelation function.
For experimentally accessible parameters, our numerical results indicate robust
conversion of the quantum states of the input signal photon to those of the
retrieved photon. We also show that high fidelity single-photon wavelength
conversion is possible in the system as long as intense control pulses shorter
than the mechanical damping time are used. Our work opens up the possibility of
using levitated optomechanical systems for applications of quantum information
processing.
