Optical Lattices: Theory

TL;DR

This chapter reviews the properties of Bose-Einstein condensates in optical lattices, highlighting their interdisciplinary nature, experimental advancements, and the theoretical models used to study phenomena like quantum phase transitions.

Contribution

It provides an overview of the theoretical understanding of BEC in periodic potentials, emphasizing recent experimental achievements and interdisciplinary connections.

Findings

High-precision experiments enable realization of idealized Hamiltonians.

Interdisciplinary links with solid-state physics and photonics.

Insights into dynamical instabilities and quantum phase transitions.

Abstract

This chapter presents an overview of the properties of a Bose-Einstein condensate (BEC) trapped in a periodic potential. This system has attracted a wide interest in the last years, and a few excellent reviews of the field have already appeared in the literature (see, for instance, [1-3] and references therein). For this reason, and because of the huge amount of published results, we do not pretend here to be comprehensive, but we will be content to provide a flavor of the richness of this subject, together with some useful references. On the other hand, there are good reasons for our effort. Probably, the most significant is that BEC in periodic potentials is a truly interdisciplinary problem, with obvious connections with electrons in crystal lattices, polarons and photons in optical fibers. Moreover, the BEC experimentalists have reached such a high level of accuracy to create in the lab, so to speak, paradigmatic Hamiltonians, which were first introduced as idealized theoretical models to study, among others, dynamical instabilities or quantum phase transitions.