Vortex Lattices and Quantum Hall Effect – VoLQuan

The aim of this project is to tackle mathematical issues that are relevant for problems at the core of international interest in physics, and for which several Nobel prizes have been awarded: the Quantum Hall effect in Bose Einstein condensates. For that purpose, - our group is made up of two teams, one of mathematicians and one of physicists. - - The physics of quantized vortices is of tremendous importance in the field of quantum fluids and extends beyond condensed matter physics. Indeed, rearrangements of a vortex lattice in the interior of neutron stars have been proposed as an explanation of the so-called glitches in the rotation - frequency of pulsars. The cosmic strings are topological line defects of cosmological extent akin to the vortices of liquid Helium and their role in galaxy formation is the subject of present research in cosmology. Ultracold gaseous Bose-Einstein condensates - allow tests in the laboratory to study various aspects of - macroscopic quantum physics. - In particular, the quest - for vortices has been immediate. - The possibility of tuning many parameters of the system through - atomic physics techniques turns these gaseous systems into extremely - attractive objects. - - The atoms are confined in a potential created by an external - magnetic field and/or a superposition of laser beams, which gives - access to many different geometries. To nucleate vortices, one has - to add a certain amount of angular momentum to the gas. This can be - done by rotation. In - 2000, the ENS team of Jean Dalibard at Laboratoire - Kastler-Brossel succeeded in observing vortices in a single - component condensate, followed by teams at the MIT and Oxford. This - has triggered off many mathematical works on vortices and the - Abrikosov vortex lattice. All these experimental situations are described by a mean-field approach, where the system - is characterized by a macroscopic wave function solving the - Gross-Pitaevskii or nonlinear Schrodinger equation. - The mathematical tools involve PDE's, energy estimates, homogeneisation, - analysis of the spectrum of some operator, and more recently semi-classical analysis - and the introduction of Bargmann spaces. Our interest in this project - is still in the vortex lattice, but in a situation where - the mean field model is no longer valid because the states are - highly correlated, similarly to the fractional quantum Hall states - of electrons in two-dimensional structures. Thus, one has to - consider the quantum $N$ body hamiltonian. - - Current physical interest is now for a rapid rotation regime: the - rotation frequency becomes close to the transverse trapping - frequency. The centrifugal and trapping forces then nearly - compensate each other and the spatial extent of the condensate - becomes very large. This is a most interesting regime since it is - equivalent to the situation leading to the quantum Hall effect in - two-dimensional electron gases. Depending on the ratio between the - number of vortices and the number of particles, the ground state of - the system will be well approximated by a macroscopic wave function - or it will be a strongly correlated state. The former case is the - one - that we have started to study but that still leads to very - interesting open questions, in particular on the study of the - Abrikosov lattice, as well as the behaviour of the lattice - (melting) as a function of temperature. The case of strongly - correlated states is even more fascinating as it is directly - connected to the physics of fractional Quantum Hall phenomena. This - regime has not yet been reached experimentally. It is the core of - our project and requires the analysis of the quantum $N$ body - hamiltonian for bosons. The issues that we want to understand are new and open - mathematically. Indeed, all the mathematical works - on the thermodynamical limit lead to situations where the - asymptotic problem is de-correlated and thus the methods cannot be - used as such. Our problems display links w...