## 4 Simulations on the Trapping System
### Thermal Simulations
#### 4.2.3 Ovens' Simulations  
Thermal simulations with COMSOL Multiphysics on the ovens' system were essential for establishing the behavior of the tubes when high current bursts were applied. The parameters of the current pulses' train were fundamental knobs that had to be experimentally adjusted in order to reach a satisfying trade-off between the number of atoms in the vapor flux and the minimization of the vacuum pollution. For instance, the response of the ovens to a given current could be characterized by detecting the neutrals' flux. The reported trapping system was designed for an experiment that used barium ions, generated through a two-photon resonant photo-ionization process [35]. Neutral barium atoms coming out of the ovens were firstly excited through the intercombination line from the ground state \(6\mathrm{s}^{2}\)\({}^{1}\)S\({}_{0}\) to the level \(5\mathrm{d}6\mathrm{p}\)\({}^{3}\)D\({}_{1}\) by shining light at \(413.2\,\mathrm{nm}\), then they were ionized by absorbing a second photon at the same wavelength. However, barium atoms in the intermediate energy level could decay to the \(6\mathrm{s}5\mathrm{d}\)\({}^{3}\)D\({}_{1}\) or \(6\mathrm{s}5\mathrm{d}\)\({}^{3}\)D\({}_{2}\) metastable levels, emitting a photon at \(659.5\,\mathrm{nm}\) or \(667.5\,\mathrm{nm}\), respectively. Therefore, it was possible to detect the atomic flux out of the ovens by exciting the atoms with blue light and detecting their fluorescence in the red region of the spectrum. The graph displayed in Figure 8 reports the average photon counts for a binning time of \(10\,\mathrm{ms}\), feeding one of the two tubes with a train of 20 current pulses \(400\,\mathrm{ms}\) long with a duty cycle of \(75\%\), linearly modulated in amplitude with a decreasing trend from \(100\,\mathrm{A}\) to \(70\,\mathrm{A}\). The measurement was taken by using the photon counting head Hamamatsu H11870-02 and the photon counting unit Hamamatsu C8855-01. The graph, where \(t=0\) marks the beginning of the current feeding, shows that the first current pulses just heated up the tube without causing the appreciable neutrals' emission and that the atomic flux began in the second half of the pulses' train. When the current was switched off (after \(8\,\mathrm{s}\)), the emission rapidly decreased due to the copper heat-sink draining out the heat. At the end of the sequence, the neutrals' emission spikes caused by the last pulses of the sequence could be recognized.  
Another aspect that played a central role in the optimal functioning of the ovens was the geometry, and in particular the ratio between the internal and external diameters \(\phi_{\text{int}}/\phi_{\text{ext}}\). Increasing this ratio, i.e., reducing the tube section, caused an increment of the dissipated heat via the Joule effect. Nevertheless, the diameters' ratio was limited by the feasibility of its mechanical fabrication (in the presented case, this ratio was 0.8). With lower values, thermal simulations showed a temperature increasing during the current burst much less than the one predicted for \(\phi_{\text{int}}/\phi_{\text{ext}}=0.8\), thus leading to longer current pulses in order to obtain proper atomic vapors.