  We report on the use of an ultracold ensemble of $^{87}$Rb atoms trapped in a
vertical lattice as a source for a quantum force sensor based on a Ramsey-Raman
type interferometer. We reach spatial resolution in the low micrometer range in
the vertical direction thanks to evaporative cooling down to ultracold
temperatures in a crossed optical dipole trap. In this configuration, the
coherence time of the atomic ensemble is degraded by inhomogeneous dephasing
arising from atomic interactions. By weakening the confinement in the
transverse direction only, we dilute the cloud and drastically reduce the
strength of these interactions, without affecting the vertical resolution. This
allows to maintain an excellent relative sensitivity on the Bloch frequency,
which is related to the local gravitational force, of $5\times10^{-6}$ at 1\,s
which integrates down to $8\times10^{-8}$ after one hour averaging time.
