  Magnetically levitated superconductors in the Meissner state can be utilized
as micro-mechanical oscillators with large mass, high quality factors and long
coherence times. In previous works analytical solutions for the magnetic field
distribution around a superconducting sphere in a quadrupole field have been
found and used to derive the trap parameters, while non-spherical geometries
have only been investigated in a few idealized cases. However, superconductors
of almost arbitrary shape can be used as levitators in a magnetic trap and, as
the trap's properties depend strongly on the superconductors shape, allow for a
wider parameter regime to be accessed. Finite element models are suitable to
obtain the field distribution around arbitrarily shaped superconductors in
arbitrary fields, but have not yet been used widely in the context of levitated
superconductors. Here we present a simple numerical model for this purpose and
use it to calculate the field distribution around cylindrical superconductors
in a quadrupole field and to evaluate the trap parameters. We find that the
cylindrical shape, compared to spherical levitators, allows for substantially
higher trap frequencies and coupling strengths. This in turn reduces the
demands on vibration isolation and significantly eases the requirements for
feedback cooling to the ground state. The numerical model is provided as
supplemental material and can easily be adapted to various geometries and trap
fields.
