  Improving coherence is a fundamental challenge in quantum simulation and
sensing experiments with trapped ions. Here we discuss, experimentally
demonstrate, and estimate the potential impacts of two different protocols that
enhance, through motional parametric excitation, the coherent spin-motion
coupling of ions obtained with a spin-dependent force. The experiments are
performed on 2D crystal arrays of approximately one hundred $^9$Be$^+$ ions
confined in a Penning trap. By modulating the trapping potential at close to
twice the center-of-mass mode frequency, we squeeze the motional mode and
enhance the spin-motion coupling while maintaining spin coherence. With a
stroboscopic protocol, we measure $5.4 \pm 0.9$ dB of motional squeezing below
the ground-state motion, from which theory predicts a $10$ dB enhancement in
the sensitivity for measuring small displacements using a recently demonstrated
protocol [Science $\textbf{373}$, 673 (2021)]. With a continuous squeezing
protocol, we measure and accurately calibrate the parametric coupling strength.
Theory suggests this protocol can be used to improve quantum spin squeezing,
limited in our system by off-resonant light scatter. We illustrate numerically
the trade-offs between strong parametric amplification and motional dephasing
in the form of center-of-mass frequency fluctuations for improving quantum spin
squeezing in our set-up.
