Using Entanglement to Measure Temperatures and Frequencies of Individual Normal Modes in a Strongly Coupled 2D Plasma of Be$^+$

Confined non-neutral plasmas of ions in the regime of strong coupling serve as a platform for studying a diverse range of phenomena including: dense astrophysical matter, quantum computation/simulation, dynamical decoupling, and precision measurements. We describe a method of simultaneously detecting and measuring the temperature of transverse plasma modes in two-dimensional crystals of cold $^9$Be$^+$ confined within a Penning trap. \footnote{B. C. Sawyer et al., Phys. Rev. Lett. {\bf 108}, 213003 (2012).} We employ a spin-dependent optical dipole force (ODF) generated from off-resonant laser beams to directly excite plasma modes transverse to the crystal plane of $\sim 100$ ions. Extremely small mode excitations ($\sim 1$ nm) may be detected through spin-motion entanglement induced by an ODF as small as 10 yN , and even the shortest-wavelength ($\sim 20$ $\mu$m) modes are excited and detected through the spin dependence of the force. This mode-specific thermometry has facilitated characterization and mitigation of ion heating sources in this system. Future work may include sub-yN force detection, spectroscopy/thermometry of the more complex in-plane oscillations, and implementation/confirmation of sub-Doppler cooling.