Bose Einstein condensates (BEC) owe
their name to the prediction of Bose and Einstein in 1925 that for a
gas of non
interacting particles at very low temperature, a macroscopic fraction
of the
gas is in the state of lower energy, that is condensed. At that time,
this idea
was only theoretical. The first experimental realization of atomic BEC
was
obtained in 1995 by American teams and was awarded the Nobel Prize in
2001.
Since then, a lot of properties of these systems have been studied both
experimentally and theoretically, in particular at the Ecole normale
supérieure
(ENS) in Paris, in the group of Jean Dalibard web
page.

In one of the experiments achieved at ENS, a
laser beam is imposed on the magnetic trap holding the atoms to create
a
harmonic anisotropic rotating potential. For sufficiently large angular
velocities, vortices are detected in the
system. Experimentally, it has been observed that when the vortex is
nucleated,
the vortex line is not straight along the axis of rotation but
bending. The aim of our work was
to justify rigorously some of the experimental observations.

1. Bent vortices

The mathematical
framework is that of the Gross-Pitaevskii energy with a rotating term.
We want to find the properties of the complex valued wave function
minimizing the energy. In a joint work with T.Riviere (PRA 2001),
we define an asymptotic parameter which is
small in the experimental regime and approximate the Gross-Pitaevskii
energy to obtain a simpler form of the
energy which only depends on the number and shape of the vortex lines.
The devices relie on the analysis of vortices developped by
Bethuel-Brezis-Helein and Lin-Riviere for Ginzburg-Landau problems.
This
simpler form of the energy has a term which depends on the vortex
contribution
and one due to rotation. Then we check that our characterization
leads to solutions with a bent vortex for a range of
values of the rotational velocity which are
consistent with the experiments.

In a joint paper with R.Jerrard (PRA 2002 and CRAS 2003), we
analyze the properties of the critical points of this reduced energy.
In particular, we prove that when the trap has a cigar shape, as in the
ENS experiment, the straight vortex is unstable at low rotational
velocities
and stable at larger velocities, so that at the rotational velocity of
nucleation of vortices, the vortex line is indeed bending. On the other
hand, for pancake shape
condensates, the minimizer of the energy is the straight vortex and
there is no
bending. Following recent experiments in the group of J.Dalibard,
we also analyze other critical points such as S vortices.

2.Numerical
simulations

This problem has been at the origin of many
interactions between physics experiments, rigorous mathematical tools
and numerical simulations. In joint works with I.Danaila (PRA 2003 and
2004), we perform numerical simulations of the solutions of the Gross
Pitaevskii equations in imaginary time using finite differences. We
find the different shapes of single vortices: U or S and relate them to
analytical results and derive new open problems. In other settings, we
compute giant vortices.

3. Other
trapping potentials

In a joint work with S.Alama and L.Bronsard (to
appear in ARMA), we analyze a case related to very recent experiments,
where the trapping potential has a quartic term. This created other
patterns for vortices, in particular the appearance of a giant vortex
and a circle of vortices.

4. Fast
rotating regime and vortex lattices

The fast rotation regime presents a strong
analogy with Quantum Hall physics and is attracting a lot of interest
in the condensed matter physics community. In a joint paper with
X.Blanc and J.Dalibard (PRA 2004), for a fast rotating condensate in a
harmonic trap, we investigate the structure of the vortex lattice using
wave functions minimizing the Gross Pitaveskii energy in the Lowest
Landau Level. We find that the minimizer of the energy in the rotating
frame has a distorted vortex lattice for which we plot the typical
distribution. We compute analytically the energy of an infinite regular
lattice and of a class of distorted lattices. We find the optimal
distortion and relate it to the decay of the wave function. Finally,
we generalize our method to other trapping potentials. The
rigorous proofs are given in a joint work with X.Blanc (preprint 2004)
and rely on homogeneization techniques. There are many open problems,
in particular, in order to get a proper lower bound, since our
construction only allows to obtain the upper bound.

II. Flow around an obstacle and
superfluidity

The group of W.Ketterle at MIT has found
experimental
evidence for a critical velocity under which there is no dissipation
when one
moves a detuned laser beam in a Bose-Einstein condensate. In
continuation of a work by Frisch, Pomeau, Rica, we analyze the origin
of this critical
velocity in the low density region close to the boundary layer of the
cloud,
which can be described by a Painleve equation. This is a joint work
with Q.Du and Y.Pomeau (PRL 2003).

In
a joint work with X.Blanc (JMPA 2004), we prove the existence of a
vortex free solution at low velocity for this problem, which is related
to the observed absence of dissipation.

This
project is supportd by a grand from the French Ministry for research,
ACI "Nouvelles Interfaces des Mathématiques"