In this chapter , we consider gases of ultracold atoms that are trapped in periodic potentials created by standing waves of laser light known as optical lattices. We start by considering in detail the atom-light interaction and derive from first principles the potential that an alkali atom experiences in an optical field. Then, we turn to many-body physics by showing that gases of ultracold atoms in a sufficiently deep optical lattice are described by the Hubbard models , which are very important in the fields of solid-state and condensed-matter physics. In particular, the high-temperature superconductors are often thought to be described by such a Hub-bard model, so that a balanced Fermi mixture in an optical lattice might shed some new light in this unsolved problem of high-temperature superconductivity.
In this chapter we focus primarily on the Bose-Hubbard model, which applies to a Bose gas in an optical lattice. We show that this model contains a new quantum phase of matter called the Mott-insulator phase, as first discussed by Fisher et al. . Moreover, the Bose gas is predicted to undergo a quantum phase transition from the superfluid state to the Mott-insulator state as a function of the potential depth of the optical lattice, i.e. as a function of the intensity of the lattice laser beams . This quantum phase transition has recently been observed in a beautiful experiment by Greiner et al. , and has attracted much attention. It showed that ultracold atoms in an optical lattice can be used to simulate various lattice models of fundamental importance to condensed-matter physics, which are very difficult, if not impossible, to study in a controlled way in solid-state materials. At present, a large amount of effort from the community working on ultracold atoms is therefore directed on these exciting possiblities. Our recent proposal to create an ultracold superstring in the laboratory is motivated in the same spirit, namely with the aim of exploring high-energy physics problems with ultracold-atom experiments. In that proposal, a one-dimensional optical lattice plays a pivotal role, while the ultracold superstring is described by a supersymmetric version of the Bose-Hubbard model [161, 162].
KeywordsOptical Lattice Quantum Phase Transition Insulator Transition Mott Insulator Wannier Function
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