  We analyze an efficient frequency estimation scheme that is applied to
measure the unknown frequency of an atomic state in Ramsey spectroscopy. The
scheme is employing appropriate combinations of uncorrelated probe atoms and
Greenburgur-Horne-Zeilinger (GHZ) type correlated probe atoms to estimate its
frequency. The estimation value of frequency is obtained through the Bayesian
analysis of the final measurement outcomes. The proposed scheme allows us to
obtain better precision than the scheme without quantum correlation and it also
prevents us from ambiguity in the frequency estimation procedure with GHZ
correlations only. We show that the scheme can beat the shot-noise limit and,
in addition, it is found that there is the trade-off relation between the
precision of the frequency estimation and the decoherence rate in the atomic
states.
