Official description

Ultracold neutral atoms in optical lattices add a new chapter to a paradigmatic physics problem: the motion of massive particles in a periodic potential. The mathematics behind this problem have been studied for over a century, with foundational work by F. Bloch, G. Floquet, A. Lyapunov, and E. Mathieu, among others. Textbooks on solid state physics cover the eigenvalue problem in the context of naturally occurring crystal lattice geometries, occupied by electrons as ideal fermions. Similar band structure is also realized by atoms in optical lattices, but at the length scale set by the wavelength of light and at nanokelvin energy scales. The cold-atom scenario offers a choice of particle statistics (bosons as well as fermions), dynamically tunable lattice depth, tunability of atom-atom interaction strength, and both in-situ and time-of-flight probes. These twelve lectures will give an introduction to this rich physical system. Topics covered will include how to make an optical lattice; symmetries of the system; the formation of bands through Bragg scattering; Bloch states and quasi-momentum; the tight-binding limit; band mapping through time-of-flight imaging; the Hubbard Model; response to external forces; transport and currents; and the Mott insulator transition. The course will assume fluency in electromagnetism, quantum mechanics, statistical mechanics, and rudimentary field theory (such as raising and lowering operators).