  We show that Rabi oscillations of a degenerate fermionic or bosonic gas
trapped in a double-well potential can be exploited for the interferometric
measurement of external forces at micrometer length scales. The Rabi
interferometer is less sensitive, but easier to implement, than the
Mach-Zehnder since it does not require dynamical beam-splitting/recombination
processes. As an application we propose a measurement of the Casimir-Polder
force acting between the atoms and a dielectric surface. We find that even if
the interferometer is fed with a coherent state of relatively small number of
atoms, and in the presence of realistic experimental noise, the force can be
measured with a sensitivity sufficient to discriminate between thermal and
zero-temperature regimes of the Casimir-Polder potential. Higher sensitivities
can be reached with spin squeezed states.
