# Mean-Field Description of Cooperative Scattering by Atomic Clouds

## Abstract

**:**

## 1. Introduction

## 2. General Equations

## 3. Average Quantities

## 4. Specific Radial Distribution

#### 4.1. Uniform Sphere [10,11]

#### 4.2. Parabolic Profile

#### 4.3. Gaussian Profile

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**${\lambda}_{n}/N$ for a uniform sphere with $\sigma =20$. The dashed blue line is the value ${\lambda}_{N}/N=3/2{\sigma}^{2}$.

**Figure 2.**${\omega}_{n}/N$ for a uniform sphere with $\sigma =20$. The dashed blue line is ${\omega}_{N}/N=3/4{\sigma}^{2}tan\sigma $, the dash-dotted red line is the value ${\omega}_{N}/N=-3/4{\sigma}^{2}cot\sigma $.

**Figure 3.**$\langle |\beta \left(t\right){|}^{2}\rangle $ (in units of ${({\Omega}_{0}/\Gamma )}^{2}$) vs. $\Gamma t$ for $\delta =10$ and a uniform sphere with $\sigma =20$ and $N={10}^{3}$, from the analytical MF solution (continuous red line), from the numerical solution of the discrete Equation (1) (dash-dot black line) and from the Timed-Dicke approximated solution, (37) (dash blue line).

**Figure 4.**$\langle |{\beta}^{\left(\mathrm{free}\right)}{\left(t\right)|}^{2}\rangle /\langle |{\beta}^{\left(\mathrm{free}\right)}\left(0\right){|}^{2}\rangle $ vs. $\Gamma t$ for $\delta =10$ and a uniform sphere with $\sigma =20$ and $N={10}^{3}$, from the analytical MF solution (continuous blue line) and from the numerical solution of the discrete Equation (1) (dashed black line). The dashed-dotted red line is the timed-Dicke approximation, $exp(-{\lambda}_{N}\Gamma t)$, and the dotted black line is the single atom decay $exp(-\Gamma t)$.

**Figure 5.**${\lambda}_{n}/N$ for a sphere with parabolic profile, with $\sigma =20$. The dashed blue line is the value ${\lambda}_{N}/N=5/2{\sigma}^{2}$.

**Figure 7.**$\langle |\beta \left(t\right){|}^{2}\rangle $ (in units of ${({\Omega}_{0}/\Gamma )}^{2}$) vs. $\Gamma t$ for $\delta =10$ and a Gaussian sphere with $\sigma =20$ and $N={10}^{3}$, from the analytical MF solution (continuous red line) and from the numerical solution of the discrete Equation (1) (dashed blue line). The dotted black line is the single-atom decay as $exp(-\Gamma t)$.

**Figure 8.**$\langle |\beta \left(t\right){|}^{2}\rangle $ (in units of ${({\Omega}_{0}/\Gamma )}^{2}$) vs. $\Gamma t$ for $\delta =0$ and a Gaussian sphere with $\sigma =20$ and $N={10}^{3}$, from the analytical MF solution (continuous red line) and from the numerical solution of the discrete Equation (1) (dashed blue line). The dotted black line is the single-atom decay, as $exp(-\Gamma t)$.

**Figure 9.**$P/{P}_{1}$ vs. $\Gamma t$ for $\delta =10$ and a Gaussian sphere with $\sigma =20$ and $N={10}^{3}$, from the analytical MF solution (continuous red line) and from the numerical solution of the discrete Equation (22) (dashed blue line).

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**MDPI and ACS Style**

Piovella, N.
Mean-Field Description of Cooperative Scattering by Atomic Clouds. *Atoms* **2023**, *11*, 101.
https://doi.org/10.3390/atoms11070101

**AMA Style**

Piovella N.
Mean-Field Description of Cooperative Scattering by Atomic Clouds. *Atoms*. 2023; 11(7):101.
https://doi.org/10.3390/atoms11070101

**Chicago/Turabian Style**

Piovella, Nicola.
2023. "Mean-Field Description of Cooperative Scattering by Atomic Clouds" *Atoms* 11, no. 7: 101.
https://doi.org/10.3390/atoms11070101