  We analyze theoretically the single-photon excitation and transmission
spectra of a strong-coupling hybrid optomechanics, where a two-level system
(TLS) is coupled to the mechanical resonator (MR), generating the
Jaynes-Cummings-type polariton doublets. In our model, both the optomichanical
coupling and the TLS-MR coupling are strong. In this parameter region, the
polaron-assisted excitation and reemission processes can strongly affect the
single-photon excitation and output spectra of the cavity. We find that the
fine structure around each sideband can be used to characterize the TLS-MR and
the effective TLS-photon couplings, even at single-quantum level. Thus, the
spectrum structures may make it possible to sensitively probe the quantum
nature of a macroscopic mechanical element. We further provide a possible
approach for tomographic reconstruction of the state of a TLS, utilizing the
single-photon transmission spectra.
