Identifying insulating states of ultra cold atoms with cavity transmission spectrum

In this paper, we consider the transmission characteristics of an optical cavity loaded with ultracold atoms in a one-dimensional optical lattice at absolute zero temperature. In particular, we consider the situation when the many body quantum state of the ultracold atoms is an insulating state with a fixed number of atoms at each site, which can either be density wave or Mott insulator phase, each showing a different type of discrete lattice translational symmetry. We provide a general framework of understanding the transmission spectrum from a single and two cavities/modes loaded with such insulating phases. Further, we also discuss how such a transmission spectrum changes when these insulating phases make a cross over to the superfluid phase with the changing depth of the optical lattice potential. (paper)