  We investigate a scheme for observing the third-order exceptional point (EP3)
in an ion-cavity setting. In the lambda-type level configuration, the ion is
driven by a pump field, and the resonator is probed with another weak laser
field. We exploit the highly asymmetric branching ratio of an ion's excited
state to satisfy the weak-excitation limit, which allows us to construct the
non-Hermitian Hamiltonian $(H_{\textrm{nH}})$. Via fitting the
cavity-transmission spectrum, the eigenvalues of $H_{\textrm{nH}}$ are
obtained. The EP3 appears at a point where the Rabi frequency of the pump laser
and the atom-cavity coupling constant balance the loss rates of the system.
Feasible experimental parameters are provided.
