  Electric noise from metallic surfaces is a major obstacle towards quantum
applications with trapped ions due to motional heating of the ions. Here, we
discuss how the same noise source can also lead to pure dephasing of motional
quantum states. The mechanism is particularly relevant at small ion-surface
distances, thus imposing a new constraint on trap miniaturization. By means of
a free induction decay experiment, we measure the dephasing time of the motion
of a single ion trapped 50~$\mu$m above a Cu-Al surface. From the dephasing
times we extract the integrated noise below the secular frequency of the ion.
We find that none of the most commonly discussed surface noise models for ion
traps describes both, the observed heating as well as the measured dephasing,
satisfactorily. Thus, our measurements provide a benchmark for future models
for the electric noise emitted by metallic surfaces.
