## 2 The Electro-Optical Trap  
The electro-optical potential is created by the superposition of a one-dimensional optical lattice and a static electric quadrupole potential. Specifically, the optical potential is realized by the interference pattern between the two crossed arms of a bow-tie optical cavity, in order to enhance the depth of the optical trapping potential. If the wavelength of the laser injected into the bow-tie cavity is blue-detuned with respect to the nearest atomic transition of the ions, the optical potential will confine the ions only along the lattice direction (x axis in Figure 1), in correspondence with the minima of the interference pattern, thus minimizing the off-resonant scattering rate. For realizing a three-dimensional trapping potential, the confinement along the two remaining axes is ensured by the static electric quadrupole, generated by two electrodes aligned along the lattice axis. Therefore, the approximated potential in the local minimum of the optical lattice intensity pattern can be written as [16]:\[\Phi_{eo}(x,y,z) = \left(\frac{8\,\alpha P}{\pi\,w_{0x}\,w_{0y}}\left(\frac{1}{\bar{w}_ {x}^{2}}-\frac{4\pi^{2}}{d^{2}}\right)\,+\,\frac{\kappa e_{0}\phi}{R^{2}}\right) x^{2}+ \tag{1}\] \[+ \left(\frac{8\,\alpha P}{\pi\,w_{0x}\,w_{0y}}\,\frac{1}{w_{0y}^{2 }}\,-\,\frac{\kappa e_{0}\phi}{2R^{2}}\right)y^{2}\,+\,\left(\frac{8\,\alpha P }{\pi\,w_{0x}\,w_{0y}}\,\frac{1}{\bar{w}_{z}^{2}}\,-\,\frac{\kappa e_{0}\phi} {2R^{2}}\right)z^{2}\]  
where \(\alpha\) is the (negative) state-dependent atomic polarizability, \(w_{0x}\) and \(w_{0y}\) are the beam radii along the x and y axes, \(P\) is the power of the laser beam, \(\kappa\) is a geometrical factor related to the electrodes' shape and mutual position, \(R\) is the distance between the electrodes and the trap center, \(\phi\) is the static electric potential applied on the electrodes, and:  
\[\bar{w}_{x}\,=\,\left(\frac{\cos(\theta_{c})^{2}}{w_{0x}^{2}}+\frac{\sin( \theta_{c})^{2}}{2z_{R}^{2}}\right)^{-1/2}\quad\text{and}\quad\bar{w}_{z}\,=\, \left(\frac{\sin(\theta_{c})^{2}}{w_{0x}^{2}}+\frac{\cos(\theta_{c})^{2}}{2z_{ R}^{2}}\right)^{-1/2}\]  
are the rotated beam radii along the x and z directions, where \(z_{R}\) is the Rayleigh range of the laser beam and \(\theta_{C}\) is the crossing angle between the diagonal arms of the bow-tie cavity. By analogy with the anisotropic three-dimensional harmonic oscillator, the following trapping frequencies can be extracted from Equation (1):  
\[\omega_{\omega_{0,x}}^{2} = \frac{16\,\alpha P}{\pi\,m_{\text{ion}}\,w_{0x}\,w_{0y}}\left( \frac{1}{\bar{w}_{x}^{2}}-\frac{4\pi^{2}}{d^{2}}\right)\,+\,\frac{2\kappa|e| \phi}{m_{\text{ion}}\,R^{2}}\] \[\omega_{\omega_{0,y}}^{2} = \frac{16\,\alpha P}{\pi\,m_{\text{ion}}\,w_{0x}\,w_{0y}}\,\frac{1 }{w_{0y}^{2}}\,-\,\frac{\kappa|e|\phi}{m_{\text{ion}}\,R^{2}} \tag{2}\] \[\omega_{e_{0,z}}^{2} = \frac{16\,\alpha P}{\pi\,m_{\text{ion}}\,w_{0x}\,w_{0y}}\,\frac{1 }{\bar{w}_{z}^{2}}\,-\,\frac{\kappa|e|\phi}{m_{\text{ion}}\,R^{2}}\]  
Once the trapping wavelength, the crossing angle between the two cavity arms (through which the lattice constant can be tuned), and the electrodes' geometry are fixed, the electro-optical potential depends only on the laser power \(P\) and on the applied potential \(\phi\). A \"stability diagram\" of the electro-optical trap can be deduced from Equation (2): in fact, the potential acting on the ions provides a stable confinement in all three directions if the frequencies associated with the harmonic potentials are all non-negative. For instance, the stability plot evaluated for the trap parameters \(\lambda=451.7\,\)nm  
Figure 1: Sketch of the electro-optical trap, in which a static electric quadrupole potential (red arrows) is used for trapping ions along two orthogonal directions (y and z axes), while the confinement along the antitrapping direction (x axis) is provided by the interference pattern of two crossed Gaussian laser beams (drawn in blue).