  Single crystal Lithium Ferrite (LiFe) spheres of sub-mm dimension are
examined at mK temperatures, microwave frequencies and variable DC magnetic
field, for use in hybrid quantum systems and condensed matter and fundamental
physics experiments. Strong coupling regimes of the photon-magnon interaction
(cavity magnon polariton quasi-particles) were observed with coupling strength
of up to 250 MHz at 9.5 GHz (2.6\%) with magnon linewidths of order 4 MHz (with
potential improvement to sub-MHz values). We show that the photon-magnon
coupling can be significantly improved and exceed that of the widely used
Yttrium Iron Garnet crystal, due to the small unit cell of LiFe, allowing twice
more spins per unit volume. Magnon mode softening was observed at low DC fields
and combined with the normal Zeeman effect creates magnon spin wave modes that
are insensitive to first order order magnetic field fluctuations. This effect
is observed in the Kittel mode at 5.5 GHz (and another higher order mode at 6.5
GHz) with a DC magnetic field close to 0.19 Tesla. We show that if the cavity
is tuned close to this frequency, the magnon polariton particles exhibit an
enhanced range of strong coupling and insensitivity to magnetic field
fluctuations with both first order and second order insensitivity to magnetic
field as a function of frequency (double magic point clock transition), which
could potentially be exploited in cavity QED experiments.
