Chapter 1 Quantum gases in optical lattices

The experimental realization of correlated quantum phases with ultra-cold gases in optical lattices and their theoretical understanding has witnessed remarkable progress during the last decade. In this review we introduce basic concepts and tools to describe the many-body physics of quantum gases in optical lattices. This includes the derivation of effective lattice Hamiltonians from first principles and an overview of the emerging quantum phases. Additionally, state-of-the-art numerical tools to quantitatively treat bosons or fermions on different lattices are introduced. With the realization of Bose-Einstein condensates (BEC)1–3 and quantum degenerate Fermi gases4–6 a new field of research opened up, in which quan-tum phenomena driven by the interplay of a macroscopic number of atoms are the focus. Early experiments explored mainly coherent effects in dilut