A Coupled-Oscillators Model to Analyze the Interaction between a Quartz Resonator and Trapped Ions
The novel application of a piezoelectric quartz resonator for the detection of trapped ions has developed in the observation of the quartz-ions interaction under non-equilibrium conditions, opening new perspectives for high-sensitive motional frequency measurements of radioactive particles. Energized quartz crystals have (long) constant-decay times in the order of milliseconds, permitting the coherent detection of charged particles within short times. In this publication we develop in detail a model governing the interaction between trapped $^{40}$Ca$^+$ ions and a quartz resonator connected to a low-noise amplifier. We apply this model to experimental data and extract relevant information like the coupling constant $g=2\pi \times 1.449(2)$~Hz and the ions' modified-cyclotron frequency in our 7-tesla Penning trap. The study on the latter is specially important for the use of this resonator in precision Penning-trap mass spectrometry. The improvement in sensitivity can be accomplished by increasing the coupling constant through the quality factor of the resonator. This can develop in the use of the hybrid quartz-ion system for other applications.