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