Gravitational effects on geonium and free electron ${\mathrm{g}}_s$-factor measurements in a Penning trap

We present a theoretical analysis of an electron confined by a Penning trap, also known as geonium, that is affected by gravity. In particular, we investigate the gravitational influence on the electron dynamics and the electromagnetic field of the trap. We consider the special case of a homogeneous gravitational field, which is represented by Rindler spacetime. In this spacetime the Hamiltonian of an electron with anomalous magnetic moment is constructed. Based on this Hamiltonian and the exact solution to Maxwell equations for the field of a Penning trap in Rindler spacetime, we derive the transition energies of geonium up to the relativistic corrections of $1/{\mathrm{c}}^2$. These transition energies are used to obtain an extension of the well known ${\mathrm{g}}_s$-factor formula introduced by L. S. Brown and G. Gabrielse [Rev. Mod. Phys. 58, 233 1986].

Figure 1

From left to right: Change from laboratory frame with coordinates $(x^{\prime },y^{\prime },z^{\prime })$, where $\mathit{g}$ points into $z^{\prime }$-direction, to the frame of trap geometry with $(x,y,z)$, where the $z$-axis and ${\mathit{B}}^{(0)}$ are aligned. The angle $\theta$ is determined by the relation $\mathit{g}·{\mathit{B}}^{(0)}=gB\cos \theta$.

Figure 2

After the rotation, shown in Fig. 1, the coordinate center is shifted into the new equilibrium position of the electron motion.

Figure 3

Level scheme for spin states and the lowest cyclotron oscillator states of an electron in a Penning trap. Here, $n$ is the quantum number of cyclotron oscillation and $s=\pm 1/2$ refers to the two spin states of the electron in the Penning trap. The spin-flip transition ${\overline{\omega }}_{a^{\prime }}$ (56) and the cyclotron transition ${\overline{\omega }}_{c^{\prime }}$ (61), used to determine the free electron ${\mathrm{g}}_s$-factor are marked in red. Due to the $n$,$s$-dependence of the relativistic corrections (52), the energy levels are not equidistant.