Image charge shift in high-precision Penning traps

An ion in a Penning trap induces image charges on the surfaces of the trap electrodes. These induced image charges are used to detect the ion's motional frequencies, but they also create an additional electric field, which shifts the free-space cyclotron frequency typically at a relative level of several ${10}^{-11}$. In various high-precision Penning-trap experiments, systematics and their uncertainties are dominated by this so-called image charge shift (ICS). The ICS is investigated in this work by a finite-element simulation and by a dedicated measurement technique. Theoretical and experimental results are in excellent agreement. The measurement is using singly stored ions alternately measured in the same Penning trap. For the determination of the ion's magnetron frequency with relative precision of better than 10 parts per billion, a Ramsey-like technique has been developed. In addition, numerical calculations are carried out for other Penning traps and agree with older ICS measurements.

Figure 1

A hyperbolic Penning trap consisting of a ring and two end-cap electrodes, with the characteristic trap parameters $z_0$ and ${\rho }_0$. The electric quadrupole potential is created by applying the voltage $U_0$ between the ring and the end caps. This electric potential is superimposed with a homogeneous magnetic field $B_0$.

Figure 2

Illustration of the image charge shift (ICS): In the upper part, the Penning trap of the LIONTRAP experiment is shown with a single positively charged ion shifted in the $x$ direction to position $x^{\prime }.$ On the surface of the cylindrical electrodes, the relative change of the induced image charges is shown, which results from the displacement of the ion from the center position. Less induced image charges result in a relative positive change and are sketched in red, and more image charges are sketched in blue. In the lower part, the $x$ component of the electric field simulated by comsol multiphysics at the ion's position is shown, which is generated by the induced image charges when the ion is placed at position $x^{\prime }$. Uncertainties of the individual points are smaller than the point size. The red line indicates the linear approximation of the induced electric field: $E_{\text{image,x}}\left(x^{\prime }\right)=n\times {\mathcal{E}}_x\times x^{\prime },$ where ${\mathcal{E}}_x$ is the linear field gradient in $x$ direction.

Figure 3

Ramsey-like excitation scheme for a high-precision measurement of the magnetron frequency. The pattern contains two identical radial dipolar pulses (gray shaded areas) with pulse lengths of $4.6\text{ms}$ each and excitation frequencies of ${\nu }_{\text{rf}}={\nu }_{-},$ separated by a phase evolution time $T_{\text{evol}}$. The first pulse excites the thermalized magnetron motion of the ion ($r_{-}^{\text{therm}}<10\mu \mathrm{m}$) to $r_{-}^{\text{exc}}\approx 137\mu \mathrm{m}$ and thus imprints a magnetron phase. After the second pulse, the two extreme cases are shown: maximum excitation in red and basically no excitation in blue.

Figure 4

One example of the determination of the magnetron phase by axial frequency shifts. Here, the Ramsey-like excitation scheme has been applied on a single ${}^{12}\mathrm{C}^{6+}$ ion at 11 different phase evolution times ($T_{\text{evol}}=39.9998,...,40\text{s}$). On the $y$ axis, the axial frequency differences $\mathrm{\Delta }{\nu }_z={\nu }_z\left(\text{exc}\right)-{\nu }_z\left(\text{cold}\right)$ are plotted. The error bars are given by the uncertainties of the axial frequencies measured at the deformed voltage setting $[\delta \left(\mathrm{\Delta }{\nu }_z\right)=70\text{mHz}]$ and the intrinsic thermal distribution of the magnetron radius. On the $x$ axis, the final phase evolution times are indicated, which include voltage corrections; see Appendix pp1. The corresponding error bars of the voltage corrected phase evolution time $\delta T_{\text{evol}}^{\text{final}}=2T_{\text{evol}}\delta {\nu }_z^{\text{harm.volt}}/{\nu }_z^{\text{off}}$ are given by the axial frequency uncertainties measured at the harmonic voltage setting ($\delta {\nu }_z^{\text{harm.volt}}=50\text{mHz}$). In this plot, the magnetron phase at $39.9998\mathrm{s}$ and modulo ${360}^{\circ }$ is determined by the fit indicated as a red line with the gray confidence interval: $\varphi (T_{\text{evol}}^{\text{final}}=39.9998\text{s})=249{\left(5\right)}^{\circ }.$

Figure 5

Illustration of the measurement cycle. One complete measurement cycle takes 140 min and reaches a statistical precision of about $480\mu \mathrm{Hz}$ for the magnetron frequency difference. In total, 120 cycles have been recorded over a period of 23 days, where 73 cycles feature a $T_{\text{evol},2}$ of 40 s and the others 20 s.

Figure 6

Magnetron frequency differences of all 120 measurement cycles. The red line indicates the average value and the gray band its $43\mu \mathrm{Hz}$ statistical uncertainty. The fit of the data returns a ${\chi }^2$ of 2. This might be explained by tiny axial frequency fluctuations caused by tilt fluctuations. Since a disentanglement between voltage and tilt fluctuations based on the axial frequency is impossible, the tilt fluctuations are interpreted as voltage fluctuations, which leads to false corrections of the corresponding magnetron frequencies. To account for this, the statistical uncertainty is enlarged by a factor of $\sqrt{2}$ suggested by the reduced ${\chi }^2$. The total statistical uncertainty amounts to $61\mu \mathrm{Hz}$. For details, see text and Appendix pp1.

Figure 7

Illustration of the numerical calculation of the surface charge density $\sigma$ [see Eq. (24)] on the electrodes of a cylindrical Penning trap induced by a single charged particle at $x^{\prime }=-0.5\mathrm{mm}$. A projection of the surfaces of the cylindrical measurement trap of the LIONTRAP experiment into the plane is shown.

Figure 8

Showing the four different simulated trap geometries. The values of the characteristic trap dimensions $z_0$ and ${\rho }_0$ can be found in Table 2. (a) The mass-measurement trap of the THe-Trap experiment consists of a hyperbolically shaped ring electrode, two end-cap electrodes, and two guard electrodes. The splitting of the guard electrodes is not taken into account for the ICS, because this effect is negligible. In general, due to the rotational symmetry along the $z$ axis and the mirror symmetry at $z=0$, the full information of the trap geometry can be obtained from looking at one quarter of a cut [see the hatched area surrounded by the red dashed line in panel (a)] of the whole trap. For the other three traps, therefore, only this quadrant is shown. (b) The precision trap (PT) of the LIONTRAP [31] experiment is a cylindrical Penning trap and consists of 11 electrodes. The inner three electrodes (the Ring and Cor1) are all azimuthally split into two pieces, which allows applying different radio-frequency multipoles. The gaps between the electrodes and the split segments is 140 µm. (c) The FSU-Trap is a hyperbolic Penning-trap mass spectrometer. Similar to THe-Trap, it consists of a ring, two end caps, and two guard electrodes. Like in panel (a) (THe-Trap), the splitting of the guard electrodes into two segments is also not considered. (d) The Porto-Trap is a simplified version of the FSU-Trap geometry with the same characteristic dimensions $z_0$ and ${\rho }_0$. In this approximation, holes in the end-cap electrodes and slits are ignored. Furthermore, the guard electrodes are simplified and the space between the electrodes is closed. These modifications are necessary to make the semianalytical treatment possible.

Figure 9

Relative deviation of the simulated ICS of a pointlike charge inside a hollow cylinder with radius $2.78\mathrm{mm}$ and a total length of 30 mm compared to the analytical calculation as a function of the mesh size. While the linear fit does not show any significant decrease of $\mathrm{\Delta }\mathcal{E}$ for a finer mesh, higher order polynomials do. Finally, the fifth-order polynomial is sufficient to minimize $\mathrm{\Delta }\mathcal{E}$. Higher order polynomials than the fifth order describe the data as well as the fifth-order polynomial but with a larger uncertainty.

Figure 10

The evolution of the coefficients for the Porto-Trap as a function of submatrix size. The values from Table 7 have been subtracted and the difference scaled by ${10}^5$ such that one unit on the vertical axis corresponds to a change of 1 in the least significant digit of the quoted coefficients. The blue crosses and red open diamonds correspond to a weight of $w=0$ and $w=-3$, respectively. The lines connecting the points serve as a guide to the eye in terms of order and highlight data outside the selected range.