## 4 Comparison between Wire and Rod Traps  
In order to describe the confining potential created by an rf field and the physics governing the dynamics in an ion trap, the adiabatic approximation is used. Here, the ion motion is split into a fast oscillatory motion, also called micromotion; and a slow drift motion in the confining ponderomotive potential, called macromotion. Henceforth, the potential created by rf traps can be expressed as an effective conservative potential (averaged over one oscillation period), given by [34]  
\[V_{\text{eff}}(\vec{R}(t))=\frac{e^{2}}{2m\omega^{2}}\langle\vec{E}(\vec{R}(t ),t)^{2}\rangle. \tag{1}\]  
Here, \(\vec{R}(t)\) describes the trajectory of the macromotion, \(\vec{E}\) is the time-dependent electric field, \(e\) is the electronic charge, and \(m\) is the mass of the ion. This results in a pseudo-static effective potential, which can approximately describe the ion trajectories in different multipole traps. This approximation allows the comparison between different trap types via a static potential. The three trap types under consideration are the ideal hyperbolic trap, a trap consisting of cylindrical rods, and a trap using wires. A geometric comparison between the trap types is given in Figure 3a. In order to replicate the curvature of the ideal multipole potential to first and second order, the radius of the cylindrical rods used must fulfill the following relation [34; 35]:  
\[\rho=\frac{r_{0}}{n-1}. \tag{2}\]  
For our wire trap with \(100\,\mathrm{\SIUnitSymbolMicro m}\) diameter wires and a distance from the trap center of \(3\,\mathrm{mm}\), the ideal rod diameter is \(1.5\,\mathrm{mm}\), which is more than an order of magnitude larger than the wire size, and thus, Equation (2) is far from being fulfilled. Due to the vast geometric differences between the three mentioned rod shapes, it is not trivial to predict if and how the potential landscape differs in the different trap types. In an attempt to investigate these differences, we used COMSOL Multiphysics to simulate infinitely long traps without endcaps. For details on the COMSOL simulations, see Appendix A. The geometries are chosen as seen in Figure 3a, with the hyperbolic and cylindrical rod trap having the same inscribed radius of \(2.25\,\mathrm{mm}\), and the wire trap lying concentrically inside the rod. The simulated electric fields are used to calculate an effective potential via Equation (1) assuming OH\({}^{-}\) ions and a trap frequency \(f\) of \(5\,\mathrm{MHz}\). The resultant potential was then averaged over the azimuthal angle and the voltage was varied until the difference in the average effective potential between the hyperbolic trap and the rod/wire trap was minimized. Figure 3: Comparison between different multipole traps. (**a**) True size relations between hyperbolic rods, ideal cylindrical rods, and cylindrical wires are represented. The distance of a hyperbolic rod from the trap center is \(r_{0}=$2.25\,\mathrm{mm}$\), the diameter of an ideal cylindrical rod is \(2\rho=$1.5\,\mathrm{mm}$\) and the wire diameter is \(d=$100\,\mathrm{\SIUnitSymbolMicro m}$\). The wires are placed concentrically with the ideal cylindrical rods and are located \(3\,\mathrm{mm}\) from the trap center. (**b**) The effective potential created by hyperbolic rods is shown along with that obtained from a rod and a wire trap with their voltages matched to the same average potential as in the hyperbolic case. It can be seen that the hyperbolic trap potential exhibits true azimuthal angle independence, while distinct maxima/minima in the potential for traps with cylindrical rods or wires can be observed. Additionally, the modulations in a wire trap have a higher amplitude than those in a rod trap. For instance, the amplitude of fluctuations in a rod trap at \(0.87\cdot r_{0}\) is already attained by a wire trap at \(0.81\cdot r_{0}\).