Phases of bosons or fermions in confined optical lattices

Phases of Bose or Fermi atoms in optical lattices confined in harmonic traps are studied within the Thomas-Fermi approximation. Critical radii and particle numbers for the onset of Mott insulator states are calculated and phase diagrams are shown in one dimension, and estimated for two and three dimensions. Methods to observe these and phases such as d-wave superconductivity are discussed. Specifically, the collective modes are calculated.