## 3 Trap Design
### Electro-Optical Trap Design  
Ions are trapped along two directions by a static electric quadrupole potential, whereas along the third direction (x axis), the confinement is provided by a 1D blue-detuned optical lattice. In our electro-optical trap, the static quadrupole potential is created along y and z axes in Figure 3 by two opposite electrodes, while along the third direction (x axis), the confinement is realized by a bow-tie cavity. Since the axis orthogonal to the 2D ion crystals (x axis) coincides with the imaging direction, the static electric quadrupole potential must be produced by a pair of ring-shaped electrodes, ensuring a circular distribution of charges with the x direction as the symmetry axis. These rings must be contained in mechanical supports that do not limit the numerical aperture of the imaging of the ion crystals' plane (y-z in Figure 3). To this end, we chose a conical shape for these electrodes, which we will call \"cone electrodes\" in this article. As an additional constraint, the shape and dimensions of the cone electrodes must be carefully adapted to the electrodes of the Paul trap. The bow-tie cavity is based on a symmetric configuration with four curved mirrors having the same radius of curvature, so that the laser beam traveling in the cavity has a focus in each of its arms, two of which are overlapped in the center of the bow-tie. There, the two running waves create a  
Figure 3: CAD assembly of the electro-optical trap, formed by the cone electrodes and the bow-tie cavity mirror mounts. The two mirror mounts on the left side have been removed for a better view. deep lattice in which the intensity of the two interfering beams is enhanced by a factor proportional to the cavity finesse, thus increasing the trap depth [21]. For facilitating the process of loading the atoms and ions in a single minimum of the optical potential, the lattice constant must be as large as possible, i.e., the two lattice beams must cross at a small angle. In our design, the crossing angle is \(10\,^{\circ}\), the waist of the beams at the crossing is \(40\,\mathrm{\SIUnitSymbolMicro m}\), and the mirrors radius of curvature is \(100\,\mathrm{mm}\). With these parameters, the bow-tie geometry can be entirely defined in the ABCD matrix formalism by imposing the cavity stability [22]. The result, described in the stability diagram by the point \(\{g_{1},g_{2}\}=\{-0.89,-0.9\}\), is a strongly elongated cavity with the crossed arms \(9.48\,\mathrm{cm}\) long and parallel ones \(9.45\,\mathrm{cm}\) long. Regarding the cavity assembly, the mirror mounts are composed of two parts: a ceramic mirror holder and a metallic cover. The mirror, which has a diameter of \(6\,\mathrm{mm}\), is attached to the metallic cover with an ultra-high vacuum compatible glue (e.g., Epoxy Technology 353 ND). In order to tune the cavity resonant frequencies, two of the mirrors are equipped with a circular hollow piezo element (e.g., Noliac NAC2121), which is inserted between the mirror and the cover and held in position with a thin layer of glue. This stack is then connected to the ceramic support by using three small and finely threaded screws (M2x0.25\(\,\mathrm{mm}\)). An elastic element (DuPont Kalrez 8002, an ultra-high vacuum compatible rubber) housed in the mirror compartment offers a contrasting force to the mirror (and the piezo, if present), which keeps the cover stable and separated from the holder even if the screws are not completely tightened. In this way, the mirror can be tilted by acting on the screws. In order to facilitate the decoupling between the vertical and the horizontal degrees of freedom, the three screws are placed at the vertices of a triangle with a vertical side. Moreover, in case one wants to rely on the mechanical precision of the assembly, a metallic hollow spacer can be inserted between the holder and the cover to fix their mutual distance, thus making the whole system \"monolithic\".