Title: A numerical approach to levitated superconductors and its application to a superconducting cylinder in a quadrupole field

Abstract: 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.