Theory of Vortex Resonance in Liquid Helium

A theory is developed which predicts the behavior of a positive ion on a linear vortex in the presence of a dc electric field along the vortex and an ac electric field normal to the vortex. We show that with an appropriate (and accessible) combination of experimental parameters, it should be possible to observe resonances in the ion's velocity along the vortex as a function of dc field and ac frequency. These resonances would arise from the excitation of vortex oscillations and could be utilized to determine experimentally vortex oscillation dispersion relation for small $k$. An experimental verification of the Kelvin result ${\omega }_k=(-\frac{\hslash k^2}{2m})(\ln ka-0.116)$ should thus be possible.