  Multiparameter estimation, which aims to simultaneously determine multiple
parameters in the same measurement procedure, attracts extensive interests in
measurement science and technologies. Here, we propose a multimode many-body
quantum interferometry for simultaneously estimating linear and quadratic
Zeeman coefficients via an ensemble of spinor atoms. Different from the scheme
with individual atoms, by using an $N$-atom multimode GHZ state, the
measurement precisions of the two parameters can simultaneously attain the
Heisenberg limit, and they respectively depend on the hyperfine spin number $F$
in the form of $\Delta p \propto 1/(FN)$ and $\Delta q \propto 1/(F^2N)$.
Moreover, the simultaneous estimation can provide better precision than the
individual estimation. Further, by taking a three-mode interferometry with Bose
condensed spin-1 atoms for an example, we show how to perform the simultaneous
estimation of $p$ and $q$. Our scheme provides a novel paradigm for
implementing multiparameter estimation with multimode quantum correlated
states.
