  Quantum magnonics investigates the quantum-mechanical properties of magnons
such as quantum coherence or entanglement for solid-state quantum information
technologies at the nanoscale. The most promising material for quantum
magnonics is the ferrimagnetic yttrium iron garnet (YIG), which hosts magnons
with the longest lifetimes. YIG films of the highest quality are grown on a
paramagnetic gadolinium gallium garnet (GGG) substrate. The literature has
reported that ferromagnetic resonance (FMR) frequencies of YIG/GGG decrease at
temperatures below 50 K despite the increase in YIG magnetization. We
investigated a 97 nm-thick YIG film grown on 500 $\mathrm{\mu}$m-thick GGG
substrate through a series of experiments conducted at temperatures as low as
30 mK, and using both analytical and numerical methods. Our findings suggest
that the primary factor contributing to the FMR frequency shift is the stray
magnetic field created by the partially magnetized GGG substrate. This stray
field is antiparallel to the applied external field and is highly
inhomogeneous, reaching up to 40 mT in the center of the sample. At
temperatures below 500 mK, the GGG field exhibits a saturation that cannot be
described by the standard Brillouin function for a paramagnet. Including the
calculated GGG field in the analysis of the FMR frequency versus temperature
dependence allowed the determination of the cubic and uniaxial anisotropies. We
find that the total anisotropy increases more than three times with the
decrease in temperature down to 2 K. Our findings enable accurate predictions
of the YIG/GGG magnetic systems behavior at low and ultra-low millikelvin
temperatures, crucial for developing quantum magnonic devices.
