  We investigate the possibility of ultracold neutron (UCN) storage in quantum
states defined by the combined potentials of the Earth's gravity and the
neutron optical repulsion by a horizontal surface of liquid helium. We analyse
the stability of the lowest quantum state, which is most susceptible to
perturbations due to surface excitations, against scattering by helium atoms in
the vapor and by excitations of the liquid, comprised of ripplons, phonons and
surfons. This is an unusual scattering problem since the kinetic energy of the
neutron parallel to the surface may be much greater than the binding energies
perpendicular. The total scattering time constant of these UCNs at 0.7 K is
found to exceed one hour, and rapidly increasing with decreasing temperature.
Such low scattering rates should enable high-precision measurements of the
scheme of discrete energy levels, thus providing improved access to short-range
gravity. The system might also be useful for neutron beta-decay experiments. We
also sketch new experimental concepts for level population and trapping of UCNs
above a flat horizontal mirror.
