Quantum fluctuations of a vortex in a dilute Bose-Einstein condensate

A vortex in a quasi two-dimensional Bose-Einstein condensate is subject to the Magnus force and can be effectively described as a planar particle in a uniform magnetic field. Quantization of this effective particle leads to the lowest Landau level where the most localized wavefunction is a Gaussian. In this Gaussian state, vortex position seems to fluctuate with an average magnitude set by the magnetic width of the Gaussian. We readdress this problem using the number-conserving version of the Bogoliubov theory. We find that the Bogoliubov mode that might be interpreted as a fluctuation of vortex position actually does not contribute to quantum depletion in the centre of vortex core. Far in the Thomas-Fermi regime the only contribution comes from phonon modes, but it is an order of magnitude less than the simple estimate, based on the magnetic width of the effective Gaussian wave packet. At the same time density of non-condensed atoms inside vortex core is a factor of 0.23 less than far from the core. The non-condensed atoms are expelled from the core by centrifugal force.