  We discuss quantum fluctuations of the interface between a superfluid and a
Mott-insulating state of ultracold atoms in a trap. The fluctuations of the
boundary are due to a new type of surface modes, whose spectrum is similar (but
not identical) to classical capillary waves. The corresponding quantum
capillary length sets the scale for the penetration of the superfluid into the
Mott-insulating regime by the proximity effect and may be on the order of
several lattice spacings. It determines the typical magnitude of the interface
width due to quantum fluctuations, which may be inferred from single site
imaging of ultracold atoms in an optical lattice.
