Ultracold Bose Gases in Optical Lattices

In this chapter, we review the basic properties of ultracold Bose gases tightly trapped by optical lattice potentials. In Sect. 2.1, we briefly describe the optical-dipole potentials for trapping neutral atoms and show how one generates optical lattice potentials in laboratories. In Sect. 2.2, we derive the single-band Bose–Hubbard model within the tight-binding approximation. In Sect. 2.3, we review the superfluid-Mott-insulator quantum phase transition of the ground state of the Bose–Hubbard model. In Sect. 2.4, we discuss collective excitation modes in the superfluid phase near the Mott-insulator transition. In Sect. 2.5, we formulate an effective-model description for the strongly-correlated regime by using Hilbert-space truncation techniques. This description will be used in Chap. 4 in order to analyze dynamical properties of strongly-correlated bosons in the vicinity of the superfluid-Mott-insulator transition.