  We study the transmission spectra of a Bose Einstein condensate confined in
an optical lattice interacting with two modes of a cavity via nonlinear
two-photon transition. In particular we show that a nonlinear two-photon
interaction between the superfluid (SF) phase and the Mott insulating (MI)
phase of a Bose-Einstein condensate (BEC) and the cavity field show
qualitatively different transmission spectra compared to the one-photon
interaction. We found that when the BEC is in the Mott state, the usual normal
mode splitting present in the one-photon transition is missing in the
two-photon interaction. When the BEC is in the superfluid state, the
transmission spectra shows the usual multiple lorentzian structure. However the
separation between the lorentzians for the two-photon case is much larger than
that for the one-photon case. This study could form the basis for
non-destructive high resolution Rydberg spectroscopy of ultracold atoms or
two-photon spectroscopy of a gas of ultracold atomic hydrogen.
