  Electromagnetically-induced-transparency (EIT) spectroscopy has been used as
a sensitive sensor in quantum metrology applications. The sensitivity of a
sensor strongly depends on the measurement precision of EIT spectrum. In this
work, we present a theoretical study of the spectral lineshape measurement on a
three-level $\Lambda$-type EIT media based on Fisher information (FI) analysis.
Using two kinds of probing source: the single-photon Fock state and the
coherent state, we calculate the FI in an EIT medium and quantify the quantum
advantage and limitations of the single-photon probe. The analysis of FI
structure also provides a clear picture to classify the spectral lineshape into
two different regimes, the EIT and Aulter-Townes splitting (ATS). This work
provides a systematic analysis of the single-photon EIT spectrum, which
provides essential knowledge of quantum sensing based on EIT and deepens our
understanding of spectral characteristics of $\Lambda$-type media.
