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

Two-photon nonlinear spectroscopy of periodically trapped ultracold atoms in a cavity