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 number for onset of Mott insulator states are calculated and phase diagrams shown in 1D, and estimated for 2 and 3D. Methods to observe these and novel phases such as d-wave superconductivity is discussed. Specifically the collective modes are calculated.