  Quantum memories facilitate the storage and retrieval of quantum information
for on-chip and long-distance quantum communications. Thus, they play a
critical role in quantum information processing and have diverse applications
ranging from aerospace to medical imaging fields. Bulk acoustic wave (BAW)
phonons are one of the most attractive candidates for quantum memories because
of their long lifetime and high operating frequency. In this work, we establish
a modeling approach that can be broadly used to design hybrid magnonic
high-overtone bulk acoustic wave resonator (HBAR) structures for high-density,
long-lasting quantum memories and efficient quantum transduction devices. We
illustrate the approach by investigating a hybrid magnonic system, where BAW
phonons are excited in a gadolinium iron garnet (GGG) thick film via coupling
with magnons in a patterned yttrium iron garnet (YIG) thin film. We present
theoretical and numerical analyses of the diffraction-limited BAW phonon
lifetimes, modeshapes, and their coupling strengths to magnons in planar and
confocal YIG/GGG HBAR structures. We utilize Fourier beam propagation and
Hankel transform eigenvalue problem methods and discuss the effectiveness of
the two methods to predict the HBAR phonons. We discuss strategies to improve
the phonon lifetimes, since increased lifetimes have direct implications on the
storage times of quantum states for quantum memory applications. We find that
ultra-high, diffraction-limited, cooperativities and phonon lifetimes on the
order of ~10^5 and ~10 milliseconds, respectively, could be achieved using a
CHBAR structure with 10mum lateral YIG dimension. Additionally, the confocal
HBAR structure will offer more than 100-fold improvement of integration
density. A high integration density of on-chip memory or transduction centers
is naturally desired for high-density memory or transduction devices.
