  Levitated systems are desirable due to reduced clamping losses and reduced
thermal contact. These advantageous properties have been exploited in
optomechanics to achieve ultra-strong coupling between the mechanical mode and
the electromagnetic mode. Such schemes provide an opportunity for the quantum
manipulation of a macroscopic system. In this letter, we report the first
successful experiments with a levitated millimeter-scale neodymium magnet
within a centimeter-scale superconducting aluminum coaxial quarter-wave stub
cavity. The magnet levitated near the top of the stub, where the electric field
is concentrated, perturbs the electric field distribution allowing for small
perturbations in the magnet's position to be detected through shifts in the
resonance frequency. Resonance spectra are collected via a vector network
analyzer (VNA) between temperatures of 5 K and 50 mK revealing movement of the
magnet inside of the cavity. Room temperature measurements and finite element
calculations are done to calculate the shift in frequency for various positions
of the magnet, and an experimentally measured 100 MHz upshift when
transitioning into a superconducting state confirms levitation with remanences
up to 140 times stronger than the critical field of the aluminum. We achieve
levitation heights of 1 - 1.8 mm. We investigate the dependence of levitation
height and levitation temperature on the strength of the magnet and,
surprisingly, we observe that the levitation temperature and height both
increase with permanent magnet strength. Our work describes a novel macroscopic
mechanical system capable of sensing and transducing forces, thus allowing for
the coupling of disparate classical and quantum systems.
