  Propagation and interference of quantum-mechanical particles comprise an
important part of elementary processes in quantum physics, and their essence
can be modeled using a quantum walk, a mathematical concept that describes the
motion of a quantum-mechanical particle among discretized spatial regions. Here
we report the observation of the quantum walks of a phonon, a vibrational
quantum, in a trapped-ion crystal. By employing the capability of preparing and
observing a localized wave packet of a phonon, the propagation of a single
radial local phonon in a four-ion linear crystal is observed with single-site
resolution. The results show an agreement with numerical calculations,
indicating the predictability and reproducibility of the phonon system. These
characteristics may contribute advantageously in advanced experimental studies
of quantum walks with large numbers of nodes, as well as realization of boson
sampling and quantum simulation using phonons as computational resources.
