The course will start out by introducing the fundamental ideas for cooling and trapping ultracold atoms and review the quantum mechanical framework that underlies the description of interacting matter waves in the ultracold regime. This will introduce the idea of degenerate Bose and Fermi gases, and in particular the concept of Bose-Einstein condensation.

After this the main properties of Bose-Einstein condensates will be discussed, including coherence and superfluidity, and for Fermi gases the physics of the BCS transition will be introduced. Conceptually important developments such as optical lattices, Feshbach resonances, artificial gauge fields and others will be explained in detail as well. New developments in the area of strongly correlated gases will be introduced and applications of cold atoms in quantum information or quantum metrology provide the final part of the course.

The course will mostly focus on the theoretical description of ultracold quantum gases, but regularly discuss experimental developments, which go with these.

1. Ultracold atomic gases: cooling and trapping

2. Bose-Einstein condensation and Fermi degeneracy in ideal gases

3. Interacting Bose-Einstein condensates: Gross-Pitaevskii equation.

4. Dynamics of Bose-Einstein condensates. Expanding and oscillating condensates.

5. Elementary excitations. Bogoliubov-De Gennes equations.

6. Two-dimensional Bose gases. Kosterlitz-Thouless transition.

7. Vortices and Superfluidity

8. One-dimensional systems: quasi-condensates and solitons

9. Strongly interacting 1D Bose gases. Impenetrable bosons.

10. Degenerate Fermi gases: BEC and BCS transitions

11. Optical lattices

12. Artificial Gauge fields

13. Applications in quantum information and metrology

While the fundamental concepts of atomic physics and quantum mechanics that are required will be reviewed in the beginning of the course, basic prior knowledge of quantum mechanics is required (e.g. undergraduate quantum mechanics).

Companion course to A211 Advances in Atomic Physics