Studies of ultracold gases in optical lattices1 link many disciplines. They allow testing fundamental quantum many-body concepts of condensed-matter physics in well controllable atomic systems 1 , e.g., strongly correlated phases, quantum information processing. Standard methods to observe quantum properties of Bose-Einstein condensates (BEC) are based on matter-wave interference between atoms released from traps 2,3,4,5,6 , destroying the system. Here we propose a new, nondestructive in atom numbers, method based on optical measurements, proving that atomic quantum statistics can be mapped on transmission spectra of high-Q cavities, where atoms create a quantum refractive index. This can be extremely useful for studying phase transitions 7 , e.g. between Mott insulator and superfluid states, since various phases show qualitatively distinct light scattering. Joining the paradigms of cavity quantum electrodynamics (QED) and ultracold gases will enable conceptually new investigations of both light and matter at ultimate quantum levels. We predict effects accessible in experiments, which only recently became possible 8 . All-optical methods to characterize atomic quantum statistics were proposed for homogeneous BEC 9,10,11,12,13 and some modified spectral properties induced by BEC's were attributed to collective emission 9,10 , recoil shifts 12 or local field effects 14 . We show a completely different phenomenon directly reflecting atom quantum statistics due to statedependent dispersion. More precisely, the dispersion shift of a cavity mode depends on the atom number. If the atom number in some lattice region fluctuates from realization to realization, the modes get a fluctuating frequency shift. Thus, in the cavity transmission-spectrum, resonances appear at different frequencies directly reflecting the atom number distribution function. Such a measurement allows then to calculate atomic statistical quantities, e.g., mean value and variance reflected by spectral characteristics such as the central frequency and width.Different phases of a degenerate gas possess similar mean-field densities but different quantum amplitudes. This leads to a superposition of different transmission spectra, which e.g. for a superfluid state (SF) consist of numerous peaks reflecting the discreteness of the matter-field. Analogous discrete spectra reversing the role of atoms and light, thus reflecting the photon structure of electromagnetic fields, were obtained in cavity QED with Rydberg atoms 15 and solid-state superconducting circuits 16 . A quantum phase transition towards a Mott insulator state (MI) is characterized by a reduc- tion of the number of peaks towards a single resonance, because atom number fluctuations are significantly suppressed 17,18 . As our detection scheme is based on nonresonant dispersive interaction independent of a particular level structure, it can be also applied to molecules 19,20 .We consider the quantized motion of N two-level atoms in a deep periodic optical lattice with M sites formed by far off-reso...

We study an ultracold gas of neutral atoms subject to the periodic optical potential generated by a high-Q cavity mode. In the limit of very low temperatures, cavity field and atomic dynamics require a quantum description. Starting from a cavity QED single atom Hamiltonian we use different routes to derive approximative multiparticle Hamiltonians in Bose-Hubbard form with rescaled or even dynamical parameters. In the limit of large enough cavity damping the different models agree. Compared to free space optical lattices, quantum uncertainties of the potential and the possibility of atom-field entanglement lead to modified phase transition characteristics, the appearance of new phases or even quantum superpositions of different phases. Using a corresponding effective master equation, which can be numerically solved for few particles, we can study time evolution including dissipation. As an example we exhibit the microscopic processes behind the transition dynamics from a Mott insulator like state to a self-ordered superradiant state of the atoms, which appears as steady state for transverse atomic pumping.