# Nonlinear Dynamics in Isotropic and Anisotropic Magneto-Optical Traps

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## Abstract

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## 1. Introduction

## 2. Model and Time-Dependent Variational Method

## 3. Results

#### 3.1. Anisotropic MOT

#### 3.2. Isotropic MOT

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Numerical solution of (19) and (20) normalized to ${\alpha}_{0}$. Parameters are indicated in the text. Blue curve: ${\alpha}_{\perp}$; red curve: ${\alpha}_{z}$; green line: equilibrium solution ${\alpha}_{\perp eq}/{\alpha}_{0}=0.47$; black line: ${\alpha}_{zeq}/{\alpha}_{0}=0.33$. Initial conditions: ${\alpha}_{\perp 0}={\alpha}_{z0}={\alpha}_{0}$ and ${\dot{\alpha}}_{\perp 0}={\dot{\alpha}}_{z0}=0$.

**Figure 2.**Numerical solution of (26) normalized to ${\alpha}_{0}$ with ${\omega}_{r}=240\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}/\mathrm{s}$ and $\nu =65\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$. Blue curve: ${\alpha}_{r}$; orange line: equilibrium solution ${\alpha}_{req}/{\alpha}_{0}=0.44$. Initial conditions: ${\alpha}_{r0}={\alpha}_{0}$ and ${\dot{\alpha}}_{r0}=0$, using the same parameters as in Figure 1.

**Figure 3.**Numerical simulation of the anisotropy parameter $\chi $ given by Equation (31) using the simulations of Equations (19) and (20). Blue curve: numerical simulation for $\chi $. The horizontal line shows the equilibrium state ${\chi}_{eq}={\alpha}_{\perp eq}/{\alpha}_{zeq}=1.42$, using the same parameters as in Figure 1.

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**MDPI and ACS Style**

Haas, F.; Soares, L.G.F.
Nonlinear Dynamics in Isotropic and Anisotropic Magneto-Optical Traps. *Atoms* **2022**, *10*, 83.
https://doi.org/10.3390/atoms10030083

**AMA Style**

Haas F, Soares LGF.
Nonlinear Dynamics in Isotropic and Anisotropic Magneto-Optical Traps. *Atoms*. 2022; 10(3):83.
https://doi.org/10.3390/atoms10030083

**Chicago/Turabian Style**

Haas, Fernando, and Luiz Gustavo Ferreira Soares.
2022. "Nonlinear Dynamics in Isotropic and Anisotropic Magneto-Optical Traps" *Atoms* 10, no. 3: 83.
https://doi.org/10.3390/atoms10030083