Levitation of superconducting microrings for quantum magnetomechanics

Levitation of superconductors is becoming an important building block in quantum technologies, particularly, in the rising field of magnetomechanics. In most of the theoretical proposals and experiments, solid geometries, such as spheres are considered for the levitator. Here we demonstrate that replacing them by superconducting rings brings two important advantages: First, the forces acting on the ring remain comparable to those expected for solid objects, whereas the mass of the superconductor is greatly reduced. In turn, this reduction increases the achievable trap frequency. Second, the flux trapped in the ring by in-field cooling yields an additional degree of control for the system. We construct a general theoretical framework with which we obtain analytical formulations for a superconducting ring levitating in an anti-Helmholtz quadrupole field and a dipole field for both zero-field and field cooling. The positions and the trapping frequencies of the levitated rings are analytically found as a function of the parameters of the system and the field applied during the cooling process. Unlike what is commonly observed in bulk superconductors, lateral and rotational stabilities are not granted for this idealized geometry. We, therefore, discuss the requirements for simple superconducting structures to achieve stability in all degrees of freedom.