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.

  • ANR Project VoLQuan
  • I. Rotating condensates

    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"