  Levitated nanoparticles and microparticles are excellent candidates for the
realization of extremely isolated mechanical systems, with a huge potential
impact in sensing applications and in quantum physics. Magnetic levitation
based on static fields is a particularly interesting approach, due to the
unique property of being completely passive and compatible with low
temperatures. Here, we show experimentally that micromagnets levitated above
type-I superconductors feature very low damping at low frequency and low
temperature. In our experiment, we detect 5 out of 6 rigid-body mechanical
modes of a levitated ferromagnetic microsphere, using a dc SQUID
(Superconducting Quantum Interference Device) with a single pick-up coil. The
measured frequencies are in agreement with a finite element simulation based on
ideal Meissner effect. For two specific modes we find further substantial
agreement with analytical predictions based on the image method. We measure
damping times $\tau$ exceeding $10^4$ s and quality factors $Q$ beyond $10^7$,
improving by $2-3$ orders of magnitude over previous experiments based on the
same principle. We investigate the possible residual loss mechanisms besides
gas collisions, and argue that much longer damping time can be achieved with
further effort and optimization. Our results open the way towards the
development of ultrasensitive magnetomechanical sensors with potential
applications to magnetometry and gravimetry, as well as to fundamental and
quantum physics.
