Quantum fluctuations of a vortex in an optical lattice

Using a variational ansatz for the wave function of the Bose-Einstein condensate, we develop a quantum theory of vortices and quadrupole modes in a one-dimensional optical lattice. We study the coupling between the quadrupole modes and Kelvin modes, which turns out to be formally analogous to the theory of parametric processes in quantum optics. This leads to the possibility of squeezing vortices. We solve the quantum multimode problem for the Kelvin modes and quadrupole modes numerically and find properties that cannot be explained with a simple linear-response theory.