Probing quantum phases of ultracold atoms in optical lattices by transmission spectra in cavity quantum electrodynamics

Studies of ultracold gases in optical lattices provide a means for testing fundamental and application-oriented quantum many-body concepts of condensed-matter physics in well controllable atomic systems(1); examples include strongly correlated phases and quantum-information processing. Standard methods to observe quantum properties of Bose - Einstein condensates are based on matter - wave interference between atoms released from traps(2-6), a method that ultimately destroys the system. Here, we propose a new approach on the basis of optical measurements that conserves the number of atoms. We prove that atomic quantum statistics can be mapped on transmission spectra of high-Q cavities, where atoms create a quantum refractive index. This can be useful for studying phase transitions(7) - for example, between Mott insulator and superfluid states - as various phases show qualitatively distinct light scattering. Joining the paradigms of cavity quantum electrodynamics and ultracold gases could enable conceptually new investigations of both light and matter at ultimate quantum levels. We predict effects accessible in experiments that recently became possible(8).