  We present exact analytical solutions to study the coherent interaction
between a single photon and the mechanical motion of a membrane in quadratic
optomechanics. We consider single-photon emission and scattering when the
photon is initially inside the cavity and in the fields outside the cavity,
respectively. Using our solutions, we calculate the single-photon emission and
scattering spectra, and find relations between the spectral features and the
system's inherent parameters, such as: the optomechanical coupling strength,
the mechanical frequency, and the cavity-field decay rate. In particular, we
clarify the conditions for the phonon sidebands to be visible. We also study
the photon-phonon entanglement for the long-time emission and scattering
states. The linear entropy is employed to characterize this entanglement by
treating it as a bipartite one between a single mode of phonons and a single
photon.
