## 4 Simulations on the Trapping System
### Electrical Simulations
#### 4.1.1 Paul Trap Stability Diagram  
We used the simulation to extract the geometric coefficients linking the electric field in the center trapping region to the voltages applied to the electrodes. Once the trap geometry has been characterized, the electrical behavior of the electrodes can be predicted, and the stability diagram of the Paul trap can be evaluated. It is important to note that the \"blade\" electrodes are rotated with respect to the endcap electrodes' axes by an angle of \(\theta=45^{\circ}\) (see Figure 6). This rotation implies that the operation of the Paul trap is described by a system of two coupled Mathieu equations [30]. The expression of the dynamic electric potential, with the coordinates rotated from \(\{x^{\prime},y^{\prime}\}\) to the \(\{x,y\}\) frame, can be written as:  
\[V_{RF}(x,y)\,=\,-\frac{e_{0}\phi_{RF}}{2}\frac{\kappa_{RF}}{R_{RF}^{2}}\left( (\cos(\theta)x+\sin(\theta)y)^{2}-(\cos(\theta)y-\sin(\theta)x)^{2}\right) \cos(\omega_{RF}\,t)\]  
where \(\theta\) is the coupling angle, \(\phi_{RF}\) is the amplitude on the potential applied on the blades, \(\omega_{RF}\) is the RF pulsation, \(r_{RF}\) is the distance of the blade electrode from the center of the trap, and \(\kappa_{RF}\) is a geometrical factor smaller than unity that takes into account the electrodes' shapes and shielding effects. Taking into account the contributions of both the dynamical and static electric potentials and the definition of the dimensionless Mathieu parameters \(\vec{q}\) and \(\vec{a}\):  
\[\vec{q}\,=\,q\left(1,\,\,\,\,\,-1,\,\,\,\,\,0\right)\,=\,\frac{2e_{0}\,\phi_ {RF}}{m_{\text{ion}}\,\omega_{RF}^{2}}\frac{\kappa_{RF}}{R_{RF}^{2}}\left(1,\, \,\,\,-1,\,\,\,\,0\right)\]  
\[\vec{a}\,=\,a\,\left(-1,\,\,\,\,\,-1,\,\,\,\,2\right)\,=\,\frac{4e_{0}\,\phi_ {DC}}{m_{\text{ion}}\,\omega_{RF}^{2}}\frac{\kappa_{DC}}{R_{DC}^{2}}\left(-1,\, \,\,\,-1,\,\,\,\,2\right)\]the coupled motion equations for the x and y axes are:  
\[\left\{ \begin{array}{l}\ddot{x}\,=\,2\,q\left(\cos(\theta)x+\sin(\theta)y\right) \,\cos(2\tau)-a\,x\\ \ddot{y}\,=\,2\,q\left(\sin(\theta)x-\cos(\theta)y\right)\,\cos(2\tau)-\alpha\, a\,y\end{array} \right. \tag{4}\]  
where \(\alpha\) is a geometrical factor expressing the ratio between \(a_{x}\) and \(a_{y}\), i.e., \(a_{x}=a_{y}/\alpha=a\). The substitution \(\omega_{\rm RF}t=2\tau\) was also performed in order to make the equations time-dimensionless. The stability condition of this system can be evaluated by applying the Floquet theorem. To do so, Equations (4) must be first expressed as a system of first order differential equations, i.e., in matrix notation \(\dot{\vec{x}}\,=\,B(\tau)\,\vec{x}\), where \(\vec{x}=(x,y,v_{x},v_{y})^{T}\), and the periodic matrix \(B(\tau)=B(\tau+n\pi)\) (\(n\) integer number) is :  
\[B(\tau)\,=\, \begin{pmatrix}0&0&1&0\\ 0&0&0&1\\ 2q\cos(\theta)\cos(2\tau)-a&2q\sin(\theta)\cos(2\tau)&0&0\\ 2q\sin(\theta)\cos(2\tau)&-2q\cos(\theta)\cos(2\tau)-\alpha a&0&0\end{pmatrix} \]  
The Floquet theorem states that the differential equation system has at least one solution of the form [31]:  
\[x^{(j)}(\tau)\,=\,\exp\biggl{(}\,\log(\lambda_{j})\,\frac{\tau}{\pi}\biggr{)} \,p_{j}(\tau)\quad\text{with}\quad p_{j}(\tau+\pi)=p_{j}(\tau) \tag{5}\]  
where \(\lambda_{j}\) are the eigenvalues of the matrix \(Z(\pi)\) defined as:  
\[\dot{Z}(\tau)=B(\tau)\,Z(\tau)\quad\text{with}\quad Z(0)=\mathbb{I}_{4} \tag{6}\]  
The solution Equation (5) can also be reformulated as:  
\[x^{(j)}(\tau)\,=\,\exp\biggl{(}\,\log(|\lambda_{j}|)\,\frac{\tau}{\pi}\biggr{)} \,\exp(i\arg(\lambda_{j}))\,p_{j}(\tau)\]  
Figure 6: Conceptual sketch of the Paul trap electrodes’ position in relation to the electro-optical trap cone-shaped hollow electrodes. The RF electrodes are rotated \(45^{\circ}\) with respect to the endcaps’ frame, and the confinement along the z axis is provided by the negatively charged endcaps. from which it is evident that the absolute value of the eigenvalues \(|\lambda|\) must be less than unity to obtain a stable solution.