  The creation of well understood structures using spectral hole burning is an
important task in the use of technologies based on rare earth ion doped
crystals. We apply a series of different techniques to model and improve the
frequency dependent population change in the atomic level structure of Thulium
Yttrium Gallium Garnet (Tm:YGG). In particular we demonstrate that at zero
applied magnetic field, numerical solutions to frequency dependent three-level
rate equations show good agreement with spectral hole burning results. This
allows predicting spectral structures given a specific hole burning sequence,
the underpinning spectroscopic material properties, and the relevant laser
parameters. This enables us to largely eliminate power dependent hole
broadening through the use of adiabatic hole-burning pulses. Though this system
of rate equations shows good agreement at zero field, the addition of a
magnetic field results in unexpected spectral diffusion proportional to the
induced Tm ion magnetic dipole moment and average magnetic field strength,
which, through the quadratic Zeeman effect, dominates the optical spectrum over
long time scales. Our results allow optimization of the preparation process for
spectral structures in a large variety of rare earth ion doped materials for
quantum memories and other applications.
