Spectroscopy and Thermometry of Drumhead Modes in a Mesoscopic 2D Coulomb Crystal of $^9$Be$^+$

We demonstrate spectroscopy and thermometry of individual motional modes in a mesoscopic 2D ion array using entanglement between ion valence electron spins and collective motion. Our system is a $\sim $400 $\mu $m-diameter planar crystal of several hundred $^{9}$Be$^{+}$ ions exhibiting complex drumhead modes in the confining potential of a Penning trap. Exploiting precise control over the $^{9}$Be$^{+}$ valence electron spins, we apply a homogeneous spin-dependent optical dipole force to excite arbitrary transverse modes with wavelengths ranging from the array diameter to the interparticle spacing of $\sim $20 $\mu $m. In addition to temperature measurements, this spin-motion entanglement induced by the spin-dependent optical dipole force allows for extremely sensitive detection of external forces ($\sim$100 yN) acting on the ion crystal. Characterization of mode frequencies and temperatures is critical for quantum simulation experiments that make use of the ion spins.