  One of the main features of superfluids is the presence of topological
defects with quantised circulation. These objects are known as quantum vortices
and exhibit a hydrodynamic behaviour. Nowadays, particles are the main
experimental tool used to visualise quantum vortices and to study their
dynamics. We use a self-consistent model based on the three-dimensional
Gross-Pitaevskii (GP) equation to explore theoretically and numerically the
attractive interaction between particles and quantised vortices at very low
temperature. Particles are described as localised potentials depleting the
superfluid and following Newtonian dynamics. We are able to derive analytically
a reduced central-force model that only depends on the classical degrees of
freedom of the particle. Such model is found to be consistent with the GP
simulations. We then generalised the model to include deformations of the
vortex filament. The resulting long-range mutual interaction qualitatively
reproduces the observed generation of a cusp on the vortex filament during the
particle approach. Moreover, we show that particles can excite Kelvin waves on
the vortex filament through a resonance mechanism even if they are still far
from it.
