  Helium is recognized as a model system for the study of phase transitions. Of
particular interest is the superfluid phase in two dimensions. We report
measurements on superfluid helium films adsorbed on the surface of a suspended
carbon nanotube. We measure the mechanical vibrations of the nanotube to probe
the adsorbed helium film. We demonstrate the formation of helium layers up to
five atoms thickness. Upon increasing the vapour pressure, we observe
layer-by-layer growth with discontinuities in both the number of adsorbed atoms
and the speed of sound in the adsorbed film. These hitherto unobserved
discontinuities point to a series of first-order layering transitions. Our
results show that helium multilayers adsorbed on a nanotube are of
unprecedented quality compared to previous works. They pave the way to new
studies of quantized superfluid vortex dynamics on cylindrical surfaces, of the
Berezinskii-Kosterlitz-Thouless phase transition in this new geometry, perhaps
also to supersolidity in crystalline single layers as predicted in quantum
Monte Carlo calculations.
