Bose-Einstein condensates in fast rotation

In this short review we present our recent results concerning the rotation of atomic Bose-Einstein condensates confined in quadratic or quartic potentials, and give an overview of the field. We first describe the procedure used to set an atomic gas in rotation and briefly discuss the physics of condensates containing a single vortex line. We then address the regime of fast rotation in harmonic traps, where the rotation frequency is close to the trapping frequency. In this limit the Landau Level formalism is well suited to describe the system. The problem of the condensation temperature of a fast rotating gas is discussed, as well as the equilibrium shape of the cloud and the structure of the vortex lattice. Finally we review results obtained with a quadratic + quartic potential, which allows to study a regime where the rotation frequency is equal to or larger than the harmonic trapping frequency. The possibility to obtain quantum degenerate gases by a combination of laser and evaporative cooling has opened several new lines of research, at the border of atomic, statistical and condensed matter physics (for a review, see e.g. [1, 2, 3, 4]). Among them, the rotation of a Bose-Einstein condensate raises many interesting problems with respect to the case of a classical fluid. Since the condensate is described by a macroscopic wave function ψ(r) = √ ρ(r)e, where ρ and φ are the spatial density and phase of the fluid, there exist strong constraints on the velocity field of the rotating gas. In a place where the spatial density is not zero, this velocity field is given by v = h̄∇φ/M (M is the particle mass), hence ∇×v = 0. The circulation of the velocity field is quantized along any close contour on which ρ 6= 0, and it is a multiple of h/M . The rotation of the fluid is thus only possible through the nucleation of quantized vortices [5, 6], which are singular points (in 2 dimensions) or lines (in 3 dimensions) of vanishing density, and around which the phase φ varies by multiples of 2π. Vortices are universal objects which appear in many macroscopic quantum systems, such as superconductors and superfluid liquid helium. ∗Unité de Recherche de l’Ecole normale supérieure et de l’Université Pierre et Marie Curie, associée au CNRS.