  We investigate the position oscillation of a particle that models the center
of mass quantum state of a trapped Bose-Einstein condensate coupled to the
zero-point fluctuations of the gravitational field. A semiclassical analysis is
performed that allows to calculate the mean square amplitude of the
oscillation. In analogy with the Lamb shift in quantum electrodynamics, this
gives rise to an upshift of the energy of the trapped condensates. We show that
for an elongated trap, the energy shift scales quadratically with the length as
well as cubically with the total number of atoms.
