Quantum Theory of LLL Cold-Atom Vortices

In the lowest-Landau-level (LLL) approximation, interacting bosons can be described without further approximation in terms of vortex positions. We report on the quantum theory of LLL vortices in trapped cold-atom systems. For this system, the harmonic theory of vibrating vortices is equivalent to the Bogoliubov theory of weakly interacting cold atoms. We find that a proper description of the vortex system requires the inclusion of invisible vortices located outside the high-atomic density region of the atom cloud. The vortex degrees of freedom have non-local commutation relations, which leads to kinetic vortex-vortex coupling in the semi-classical equations of motion. The excitation spectrum includes collective vortex vibrations and single-particle excitations, and a low-energy ``zero mode'' whose energy vanishes for a trap with perfect circular symmetry. We discuss the effects of small trap asymmetry on this zero mode, and calculate its influence on quantum fluctuations within the Bogoliubov approximation.