Coupled-oscillator 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 enabled the observation of the quartz-ions interaction under nonequilibrium conditions, opening new perspectives for high-sensitive motional frequency measurements of radioactive particles. Energized quartz crystals have (long) decay-time constants in the order of milliseconds, permitting the coherent detection of charged particles within short time scales. In this paper we develop a detailed model governing the interaction between trapped 40Ca+ ions and a quartz resonator connected to a low-noise amplifier. We apply this model to experimental data and extract the ions' reduced-cyclotron frequency in our 7-T Penning trap setup. We also obtain an upper limit for the coupling constant g with the present quartz-amplifier-trap (QAT) configuration. The study of the reduced-cyclotron frequency is especially important for the use of this resonator in precision Penning-trap mass spectrometry. The improvement in sensitivity can be accomplished by increasing the quality factor of the QAT configuration, which in turn will improve the performance of the system towards the strong-coupling regime.