Cavity QED characterization of many-body atomic states in double-well potentials -- The role of dissipation

When an incident light beam is scattered off a sample of ultracold atoms trapped in an optical lattice, the statistical properties of the retro-reflected field contain information about the quantum state of the atoms, and permit for example to distinguish between atoms in a superfluid and a Mott insulator state. This paper extends our previous analysis of this problem to include the effects of cavity damping. We use a Monte Carlo wave function method to determine the two-time correlation function and time-dependent physical spectrum of the retro-reflected field in the case of a simple two-well lattice. We also analyze quantitatively the entanglement between the atoms and the light field for atoms in a Mott insulator and a superfluid state.