Rapidly rotating Bose-Einstein condensates in and near the lowest Landau level.

We create rapidly rotating Bose-Einstein condensates in the lowest Landau level by spinning up the condensates to rotation rates Omega > 99% of the centrifugal limit for a harmonically trapped gas, while reducing the number of atoms. As a consequence, the chemical potential drops below the cyclotron energy 2 variant Planck's over 2pi Omega. While in this mean-field quantum-Hall regime we still observe an ordered vortex lattice, its elastic shear strength is strongly reduced, as evidenced by the observed very low frequency of Tkachenko modes. Furthermore, the gas approaches the quasi-two-dimensional limit. The associated crossover from interacting- to ideal-gas behavior along the rotation axis results in a shift of the axial breathing mode frequency.