  Within a cavity quantum electrodynamics description, we characterize the
fluorescent spectrum from ultracold bosons atoms, in the second harmonic
generation (SHG) and resonant cases. Two situations are considered: i) bosons
loaded into an optical lattice and ii) in a trapped two-component dilute
Bose-Einstein Condensate (BEC), in the regime where the Bogoliubov
approximation is often employed. Atom and photon degrees of freedom are treated
on equal footing within an exact time-dependent configuration interaction
scheme, and cavity leakage is included by including classical oscillator baths.
For optical lattices, we consider few bosons in short chains, described via the
Bose-Hubbard model with two levels per site, and we find that the spectral
response grows on increasing the number of atoms at weak interactions, but
diminishes at high interactions (if the number of chain sites does not exceed
the number of atoms), and is shifted to lower frequency. In the BEC regime, the
spectra display at noticeable extent a scaling behavior with the number of
particles and a suitable rescaling of the BEC-cavity and inter-particle
interactions, whilst the SHG spectrum redshifts at large atom-atom
correlations. Overall, our results provide some general trends for the
fluorescence from ultracold bosons in optical cavities, which can be of
reference to experimental studies and further theoretical work.
