# THe-TRAP Project

## Detection method

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.

_{LC}= 1/√LC, the tuned circuit behaves like a resistor R. The amplified signal is measured and compared with the driving signal.

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 ν_{z} 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

δν_{z}/ν_{z} = [ B_{2}/(4π^{2}B_{0}mν_{z}^{2}) -
1/(2mc^{2}) - (3C_{4}/qU)⋅(ν_{z}/ν_{+})^{2} ] E_{+} ,(5.1)

where B_{2} and C_{4} 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 (ν_{z}, ν_{+}, ν_{-}) can be determined. Based
on this ν_{c} can be calculated using equation 3.5. A sample measurement of the reduced
cyclotron frequency is shown in Figure 5.2.

_{+}. The excitation frequency is once swept from a higher (downsweep) and once from a lower (upsweep) start frequency over the expected resonance frequency. In parallel, the change in the ring voltage which is necessary to lock the axial frequency is measured. The intersection of the two measurements determines the sought for frequency.