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 wave function 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 position fluctuation at all. The only non-zero contribution comes from phonons but it is an order of magnitude less than the simple estimate, based on the magnetic width of the effective gaussian wave packet.