Comparative Numerical Studies of Ion Traps with Integrated Optical Cavities

We study a range of radio-frequency ion-trap geometries and investigate the effect of integrating dielectric cavity mirrors on their trapping potential using numerical modeling. We compare five different ion-trap geometries with the aim to identify ion-trap and cavity configurations that are best suited for achieving small cavity volumes and thus large ion-photon coupling as required for scalable quantum-information networks. In particular, we investigate the trapping-potential distortions caused by the dielectric material of the cavity mirrors in all three dimensions for different mirror orientations with respect to the trapping electrodes. We also analyze the effect of the mirror material properties such as dielectric constants and surface conductivity, and study the effect of surface charges on the mirrors. As well as perfectly symmetric systems, we also consider traps with optical cavities that are not centrally aligned where we find a spatial displacement of the trap center and asymmetry of the resulting trap only at certain cavity orientations. The best trap-cavity configurations with the smallest trapping-potential distortions are those where the cavities are aligned along the major symmetry axis of the electrode geometries. These cavity configurations also appear to be the most stable with respect to any mirror misalignment. Although we consider particular trap sizes in our study, the presented results can be easily generalized and scaled to different trap dimensions.

Figure 1

Geometries of ion traps with integrated optical cavities: (a) linear trap with blade-shaped electrodes; (b) linear trap with wafer electrodes; (c) end-cap trap; (d) stylus trap; (e) surface trap. “rf” and “gnd” mark rf- and grounded electrodes, respectively. For each trap geometry different cavity orientations are considered in this article, e.g., with cavities along the $x$, $y$, or $z$ axis. The insets show the unperturbed trapping potentials for the respective traps.

Figure 2

Three orientations of optical cavities integrated in the blade linear trap. Cavities oriented along the (a) $x$, (b) $y$, and (c) $z$ axis.

Figure 3

Trap depths calculated for different ion traps and different optical-cavity orientations as a function of cavity length: (a) and (b) are the normalized trap depths in the $x$- and $y$-axis directions of the linear traps. The wafer and blade traps behave similarly and are thus represented by a single data set. The inset in (a) shows how the trapping field is deformed by the presence of the cavity in the blade trap at a cavity length of 1 mm. It can be seen that the trap depth is significantly reduced in the $x$ direction due to the intersection of the cavity with the potential. (c) and (d) are the normalized trap depths in the $x$- and $y$-axis directions of the cylindrical traps.

Figure 4

Secular frequencies calculated for different ion traps and different optical-cavity orientations as a function of the cavity length: (a) and (b) are the normalized secular frequencies in the (a) $x$- and (b) $y$-axis directions of the linear traps. The wafer and blade traps behave very similarly and are thus represented by a single data set. (c) and (d) are the normalized secular frequencies in the (c) $x$- and (d) $y$-axis directions of the cylindrical traps.

Figure 5

Configurations of cavity misalignment and the corresponding trapping potentials: (a) translational misalignments along the longitudinal, $x$ axis, and (b) transverse, $y$ axis direction, and (c) skewed misalignment in the transverse, $z$ axis direction.

Figure 6

Anharmonicity of the trap potential due to an axial misalignment of the cavity mirrors.

Figure 7

Electric field in the trap center generated by surface charges on one of the cavity mirrors in the case of (a) blade and wafer trap, (b) end cap and stylus trap, and (c) surface trap design.

Figure 8

(a) Trap depth and (b) secular frequencies calculated for the blade linear trap with optical cavities with various mirror materials. The mirror materials include an additional layer of a high dielectric constant and conductive coatings on the mirror facets or on the entire substrates. The inset in (a) shows the geometry of the ion trap.