# Loading a Paul Trap: Densities, Capacities, and Scaling in the Saturation Regime

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Equations of Motion and Pseudopotential

#### 2.2. Trapped Electrons?

#### 2.3. Insertion Heating

#### 2.4. Saturation Curves

#### 2.5. Densities

## 3. Methods

_{j}intervals and displayed in the way of a histogram in Figure 4.

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

3DPT | three-dimensional Paul trap |

LPT | linear Paul trap |

rf | radio frequency |

MOT | magneto-optic trap |

## References

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**Figure 1.**Scaled saturation ion number ${\nu}_{s}\left(\lambda \right)={N}_{s}\left(\lambda \right)/\left({q}^{2}{\widehat{R}}_{\mathrm{cut}}^{3}\right)$ for four different values of the trap control parameter q as a function of loading rate $\lambda $. Red squares: $q=0.1$; blue circles: $q=0.2$; green triangles: $q=0.3$; purple diamonds: $q=0.4$. All four loading curves show at least three of the four dynamical regimes predicted in [13,14] and explicitly marked I, II, III, and IV in the case of the curve corresponding to $q=0.2$ (blue circles). For increasing q we observe that the height of the plateaus decreases and the maxima of the plateaus shift over several orders of magnitude toward larger loading rates.

**Figure 2.**Normalized scaled maxima ${\nu}_{s}^{\mathrm{max}}\left(q\right)={N}_{s}^{\mathrm{max}}/\left({q}^{2}{\widehat{R}}_{\mathrm{cut}}^{3}\right)$ of the saturated ion numbers as a function of trap control parameter q. The black dashed line indicates the static, filled-pseudo-potential expectation of the normalized particle number at saturation. The black stars are the ${\nu}_{s}^{\mathrm{max}}\left(q\right)$ values obtained in the case where the rf-driven trap is replaced with its corresponding static pseudo-potential. The filled plot symbols are the ${\nu}_{s}^{\mathrm{max}}\left(q\right)$ values resulting from full three-dimensional molecular-dynamics simulations of the trapped ions under the condition of constant-rate loading. Blue squares, red circles, green triangles, and purple diamonds correspond to ${\widehat{R}}_{\mathrm{cut}}=15,20,25,30$, respectively. The approximate overlap of the plot symbols indicates near-perfect $1/{\widehat{R}}_{\mathrm{cut}}^{3}$ scaling.

**Figure 3.**${\lambda}^{\mathrm{max}}$ as a function of q, illustrating how the loading rate, ${\lambda}^{\mathrm{max}}$, at which the number of the particles in the trap assumes its maximum, shifts toward higher loading rates as a function of increased q. Indicated by the straight blue line, representing the function ${\lambda}^{\mathrm{max}}=0.01\times {10}^{16(q-0.1)}$, approximately exponential sensitivity is observed in the range $0.1<q<0.3$.

**Figure 4.**Ion densities $\widehat{\rho}$ as a function of distance $\widehat{r}$ from the trap’s center (histograms) for four different trap parameters, that is, $q=0.1$ (

**a**), $q=0.2$ (

**b**), $q=0.3$ (

**c**), and $q=0.4$ (

**d**) at their respective loading rates ${\lambda}^{\mathrm{max}}$ (see Figure 3). The red, horizontal line is the pseudo-potential prediction for $\widehat{\rho}$. We see that the pseudopotential predicts the density approximately only for $q=0.1$, while significant deviations from the pseudo-potential prediction are observed for $q>0.1$.

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Blümel, R.
Loading a Paul Trap: Densities, Capacities, and Scaling in the Saturation Regime. *Atoms* **2021**, *9*, 11.
https://doi.org/10.3390/atoms9010011

**AMA Style**

Blümel R.
Loading a Paul Trap: Densities, Capacities, and Scaling in the Saturation Regime. *Atoms*. 2021; 9(1):11.
https://doi.org/10.3390/atoms9010011

**Chicago/Turabian Style**

Blümel, Reinhold.
2021. "Loading a Paul Trap: Densities, Capacities, and Scaling in the Saturation Regime" *Atoms* 9, no. 1: 11.
https://doi.org/10.3390/atoms9010011