The term “quantum fluid” refers to any system that exhibits quantum mechanical effects at the macroscopic level and encompasses systems as diverse as superfluids, ultracold atoms, superconductors, etc. Typically, quantum fluids arise in situations where both quantum mechanical effects and quantum statistical effects are significant. From a theoretical standpoint, they significantly differ from classical fluids, not only because they are completely inviscid, but also because angular momentum can enter the systems only in the form of quantized vortices. The remarkable properties of quantum fluids find a wide range of applications, from engineering (where, for instance, liquid Helium is used as a coolant for superconducting magnets) to astrophysics (where they come into play to explain glitches in the rotation of neutron stars) and quantum information (realization of qubits in ring lattices). This course aims at providing a modern introduction to the theoretical description of quantum fluids, with a particular focus on atomic Bose-Einstein condensates. After an introductory part, we shall present the Gross-Pitaevskii equation, which constitutes an accurate description of weakly-interacting Bose gases at low temperatures, some notable stationary and time-dependent solutions, as well as their stability properties. The second part of the course is devoted to more advanced topics (depending also on the students’ interests) such as Waves, Solitons and Vortices. Two “hands- on” sessions are scheduled to let the students practice on the discussed topics: the first one is about the application of the Time-Dependent Variational Lagrangian method to analytically solve the dynamics of simple quantum systems, while the second is about the numerical solution of the time-dependent one-dimensional Gross-Pitaevskii equation.

Students are expected to know the main laws, theorems and methods of first- and second-year courses in Physics (mechanics, thermodynamics, electromagnetism) and of elementary Quantum Physics and Statistical Mechanics

Introduction and historical perspective; The ideal Bose gas, Bose-Einstein condensation, superfluidity; The Gross-Pitaevskii equation and some notable solutions (the homogeneous and the harmonically-trapped condensate); Linear-response theory for the condensate wavefunction: Bogoliubiov solutions, demixing instabilities; Hands-on session with Mathematica: the time-dependent variational Lagrangian method, applications to non-linear breathing oscillations; Waves: Dispersion relation and Landau’s criterion of superfluidity; Solitons: dark and bright; Vortices and Vortex Dynamics: the image-charge method and boundary effects; Hands-on session with Matlab: numerical simulation of the 1D Gross-Pitaevskii equation.

On site

Oral presentation

P.D.2-2 - March