  Optomechanical systems have been proven to be very useful for precision
sensing of a variety of forces and effects. In this work, we propose an
opto-magno-mechanical setup for spatial displacement sensing where one mirror
of the optical cavity is levitated in vacuum via diamagnetic forces in an
inhomogenous magnetic field produced by two layers of permanent magnets. We
show that the optomechanical system can sense small changes in separation
between the magnet layers, as the mechanical frequency of the levitated mirror
shifts with changing magnet layer separation $d$. We use Quantum Fisher
Information (QFI) as a figure of merit of the displacement sensing precision,
and study the fundamental precision bound that can be reached in our setup.
Nonlinear interaction inherently present in the optomechanical Hamiltonian
improves the precision, and we show that in the case of a pure state of the
optical cavity, one can achieve extremely small displacement sensing precision
of $\Delta d\sim36\times10^{-21}\text{m}$. Further, we incorporate decoherence
into our system to study the effect of leaking photons from the optical cavity
on the QFI.
