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

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.