Spin Dephasing as a Probe of Mode Temperature, Motional State Distributions, and Heating Rates in a 2D Ion Crystal

We employ spin-dependent optical dipole forces to characterize the transverse center-of-mass (COM) motional mode of a two-dimensional Wigner crystal of hundreds of $^9$Be$^+$. By comparing the measured spin dephasing produced by the spin-dependent force with the predictions of a semiclassical dephasing model, we obtain absolute mode temperatures in excellent agreement with both the Doppler laser cooling limit and measurements obtained from a previously published technique (B. C. Sawyer et al. Phys. Rev. Lett. \textbf{108}, 213003 (2012)). Furthermore, the structure of the dephasing histograms allows for discrimination between initial thermal and coherent states of motion. We also apply the techniques discussed here to measure, for the first time, the ambient heating rate of the COM mode of a 2D Coulomb crystal in a Penning trap. This measurement places an upper limit on the anomalous single-ion heating rate due to electric field noise from the trap electrode surfaces of $\frac{d\bar{n}}{dt}\sim 5$ s$^{-1}$ for our trap at a frequency of 795 kHz, where $\bar{n}$ is the mean occupation of quantized COM motion in the axial harmonic well.