Short Course on Cold Atomic Gases

Cold atomic gases provide a rich platform for exploring a wide variety of quantum phenomena, such as Bose-Einstein condensation (BEC), superfluidity, entanglement and correlations. In BEC, bosonic atoms condense into a single quantum state and behave as a coherent macroscopic entity. One of the most exciting developments in the study of cold atomic gases has been the use
of optical lattices, periodic potentials created by interfering laser beams. These lattices can be used to simulate the behavior of electrons in solid-state materials, providing a controllable
environment for investigating many-body physics. Within optical lattices, the behavior of atoms can often be described using the tight-binding model, which treats atoms as being localized at lattice sites and hopping to neighboring sites under the influence of quantum tunneling. This model allows for the exploration of phenomena like band gaps, Mott insulators, and topologically protected states.

PART 1: Solitons
LECTURE 1
• Linear versus nonlinear equations
• Superfluids, and the Gross-Pitaevskii equation: Nonlinear Schrödinger versus Euler equations
• Attractively interacting (self-focusing) versus repulsively interacting (defocusing) media.
• Extended, (back-) ground states
• Localized states: Bright versus dark solitons
LECTURE 2
• Bright solitons
– Free versus trapped systems
– One- versus multi-dimensional systems
– Emergence from modulational instability
– Collisions
• Dark solitons
– Domain walls
– Free versus trapped systems
– One- versus multi-dimensional systems
– 2D decay into vortices
PART 2: Tight-binding models for ultracold atoms in optical lattices: general formulation and applications
LECTURE 3
• Tight-binding models for ultracold atoms in optical lattices
• Maximally localized Wannier functions
• Double-well periodic potentials; honeycomb lattices
LECTURE 4
• The Haldane model and its implementation with ultracold atoms
• The Peierls substitution
• Topological phase diagram of the Haldane model
LECTURE 5
• Driven honeycomb lattices and Floquet engineering
• Stroboscopic and non-stroboscopic Floquet Hamiltonians

More information coming soon