The stored ion induces image charges on the surfaces of the trap electrodes. The oscillating ion movement creates an image current. For detecting the axial frequency of the ion, a tuned resonant circuit is used, to which a cryogenic amplifier is connected. A sketch of the detection system can be found in Figure 5.1.
The measurement principle now looks like this:
The ion is driven by a radio frequency (RF) electric field applied at about 4 MHz. The axial frequency νz of the ion is matched to this frequency by choosing appropriate trap voltage (see equation 3.2). The ion is analogous to a driven damped harmonic oscillator. In this case, the damping of the harmonic oscillator is caused by the connected tuned circuit, and the driving force comes from the radio frequency excitation.
For determining ν+ and ν-, a tunable excitation frequency around the assumed frequency range of ν± is shone in the trap. If the resonance is hit, the energy of the excited mode increases. Since the trap is not perfectly harmonic, the modes depend on each other. For the measurement of ν+, the relevant dependence can be described by
δνz/νz = [ B2/(4π2B0mνz2) - 1/(2mc2) - (3C4/qU)⋅(νz/ν+)2 ] E+ ,(5.1)
where B2 and C4 describe the anharmonic terms of the magnetic and electric field and E+ is the cyclotron energy. Thus with increasing E+, the natural axial frequency is changed. To still keep the axial frequency locked, the ring voltage must be adjusted. By observing the ring voltage the frequency at which the radial mode is excited is measured. By using this principle all three modes (νz, ν+, ν-) can be determined. Based on this νc can be calculated using equation 3.5. A sample measurement of the reduced cyclotron frequency is shown in Figure 5.2.