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

J Hofer,M Aspelmeyer

doi:10.1088/1402-4896/ab0c44

Abstract

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 λ 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 λ 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 λ the maximum field experienced by the superconducting sphere is actually lower than it is for a non-magnetic sphere.

Highlights

The last decade has seen significant progress in achieving quantum control over solid state mechanical devices
Several proposals have suggested that quasi-static magnetic levitation of superconductors in the Meissner state allows to further increase both system size and coherence time in such experiments, thereby improving the system performance and enabling access to a completely new parameter regime of macroscopic quantum physics [8][9][10]
The requirements on the magnetic traps are similar to those of atom traps[11], as in both cases a minimum in the magnetic field norm is necessary for levitation

Summary

Introduction

The last decade has seen significant progress in achieving quantum control over solid state mechanical devices. The vector potential and magnetic field inside the sphere are denoted by Ain and Bin, respectively, while Bout is used for the field outside the sphere. As the field is zero inside the sphere and there is by definition no current outside the sphere, the supercurrent density K vanishes everywhere except on the surface of the sphere and is related to the transverse magnetic field by

Magnetic field for finite λ

Levitation force and maximum field

Summary and discussion