  We propose a Ramsey interferometry experiment using an entangled state of N
atoms to reach the Heisenberg limit for the estimation of an atomic phase shift
if the atom number parity is perfectly determined. In a more realistic
situation, due to statistical fluctuations of the atom source and the finite
detection efficiency, the parity is unknown. We then achieve about half the
Heisenberg limit. The scheme involves an ensemble of circular Rydberg atoms
which dispersively interact successively with two initially empty microwave
cavities. The scheme does not require very high-Q cavities. An experimental
realization with about ten entangled Rydberg atoms is achievable with state of
art apparatuses.
