Analytic solutions to the Maxwell-London equations and levitation force for a superconducting sphere in a quadrupole field

Recent proposals suggest using magnetically trapped superconducting spheres in the Meissner state to create low-loss mechanical oscillators with long coherence times. In these proposals the derivation of the force on the superconducting sphere and the coupling to the sphere typically relies on a vanishing penetration depth $\lambda$ as well as a specific symmetry (i.e. restricting the position of the sphere to one axis) or heuristic methods (e.g. assigning an equivalent point magnetic dipole moment to the sphere). In this paper we analytically solve the Maxwell-London equations with appropriate boundary conditions for a superconducting sphere in a quadrupole field. The analytic solutions provide the full field distribution for arbitrary $\lambda$ and for an arbitrary sphere position as well as the distribution of shielding currents within the sphere. We furthermore calculate the force acting on the sphere and the maximum field over the volume of the sphere. We show that for a certain range of $\lambda$ the maximum field experienced by the superconducting sphere is actually lower than it is for a non-magnetic sphere.