Abstract
Our high-precision measurements of the Larmor-to-cyclotron frequency ratios \(\varGamma \) require single trapped, cooled ions in close-to-ideal vacuum. State-of-the-art trapping and high-precision measurement techniques with stable ions will be introduced in this chapter. These techniques allow us to work with the same single ion for the complete measurement period.
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Notes
- 1.
For completion, we also mention the trapping of electrically charged particles in storage rings, which require usually circumferences of at least a few meters up to 27 km.
- 2.
In a Penning trap, different numbers and even different species of charged particles can be confined at the same time. In this thesis, we focus on single trapped ions.
- 3.
Frequencies are quoted sometimes as ordinary frequencies \(\nu \) or angular frequencies \(\omega =2\pi \nu \).
- 4.
In the following, the cyclotron frequency, \(\omega _c,\) is called the free cyclotron frequency in contrast to the modified cyclotron frequency \(\omega _+\). This terminology is widely used in the Penning trap community.
- 5.
The present electrodes have a machining precision of \(\pm 10 \,\upmu \!\, \text {m},\) nowadays \(\pm 5 \,\upmu \!\, \text {m}\) are possible.
- 6.
In dB units the FWHM of a power ratio corresponds to the \(10\cdot \log _{10}(0.5) \approx -3\text {dB}\)-width.
- 7.
For an amplitude ratio: \(1\, \text {dBVrms}=20\log _{10}(1\, \text {Vrms})\).
- 8.
The cooling time constant \(\tau \) is defined in a way that the axial energy (\(\propto z_0^2\)) has a damping of \(\propto \exp {-t/\tau }\).
- 9.
Only thermalized ions of equal species, which oscillate in common mode, contribute to the dip signal.
- 10.
In this context excitation can mean both, real excitation but also deexciation/cooling of an eigenmotion.
- 11.
Creation operator: \(a_ n^\dagger \left| n \right\rangle =\sqrt{n+1} \left| n+1 \right\rangle \) and annihilation operator: \(a_ n \left| n \right\rangle = \sqrt{n} \left| n-1 \right\rangle \).
- 12.
A mode is denoted as cold, when it has been thermalized by the axial resonator.
- 13.
One level: Motional energy is maximal in the axial mode, minimal in the modified cyclotron mode. Other level: Motional energy is minimal in the axial mode, maximal in the modified cyclotron mode.
- 14.
z(t) is modulated with the frequency \(\varOmega _0/2\), since the energy, \(E_z\propto z(t)^2\), oscillates with \(\varOmega _0\).
- 15.
In this thesis, phases will always be denoted in degree and not in radian.
- 16.
Here, I neglect the complete preparation time, which comprises the time of the double-dip measurement, the six \(10\, \text {ms}\)-PnA cycles, the phase unwrapping PnA-cycles (1 s and 2 s) and especially the cooling time of the modes between these cycles.
- 17.
For a coherent spin-flip drive at \(\nu _L\approx 105\, \text {GHz}\) and a Rabi frequency of \(0.1\, \text {Hz}\) the relative width of the Rabi resonance is: \((0.1\, \text {Hz}/105\, \text {GHz})\approx 1\cdot 10^{-12},\) see also Sect. 4.5.
- 18.
\(T_+\approx 3000\, \text {K}\) corresponds to a typical modified cyclotron radius of \(r_+=14\,\upmu \! \, \text {m}\) during the PnA phase evolution time.
- 19.
Calculated by a COMSOL simulation.
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Köhler-Langes, F. (2017). Penning Trap Physics. In: The Electron Mass and Calcium Isotope Shifts. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-50877-1_3
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