# Two-Photon Collective Atomic Recoil Lasing

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. The Three-Level Atom

#### 2.2. The Two-Photon Approximation

#### 2.3. Atomic Motion

#### 2.4. Cavity Field Evolution

## 3. Two-Photon CARL with Weak and Strong Excitation

**Figure 3.**Evolution of probe photon number, $|{\alpha}_{1}{|}^{2}$, bunching parameter, $\left|b\right|$, and mean population difference, $<D>$, for a case of weak excitation. The parameters used are ${U}_{0}/{\omega}_{r}=5\times {10}^{-5}$, ${\Delta}_{eg}=10$, ${\alpha}_{2}=100$, $N=1000$.

**Figure 4.**Snapshots of momentum distribution $({\theta}_{j},{p}_{j})$ for each atom $j=1..1000$ when (

**a**) $t=0{\omega}_{r}^{-1}$, (

**b**) $t=20{\omega}_{r}^{-1}$ and (

**c**) $t=26{\omega}_{r}^{-1}$ in the case of weak excitation. The parameters used are as in Figure 3.

**Figure 5.**Snapshots of population difference distribution $({\theta}_{j},{D}_{j})$ for each atom $j=1..1000$ when (

**a**) $t=0{\omega}_{r}^{-1}$, (

**b**) $t=20{\omega}_{r}^{-1}$ and (

**c**) $t=26{\omega}_{r}^{-1}$ in the case of weak excitation. The parameters used are as in Figure 3.

## 4. Strong Excitation

**Figure 6.**Evolution of probe photon number, $|{\alpha}_{1}{|}^{2}$, bunching parameter, $\left|b\right|$, and mean population difference, $<D>$, for a case of strong excitation. The parameters used are ${U}_{0}/{\omega}_{r}=5\times {10}^{-5}$, ${\Delta}_{eg}=1$, ${\alpha}_{2}=100$, $N=1000$.

**Figure 7.**Snapshots of momentum distribution $({\theta}_{j},{p}_{j})$ for each atom $j=1..1000$ when (

**a**) $t=0{\omega}_{r}^{-1}$, (

**b**) $t=20{\omega}_{r}^{-1}$ and (

**c**) $t=30{\omega}_{r}^{-1}$ in the case of strong excitation. The parameters used are as in Figure 6.

**Figure 8.**Snapshots of population difference distribution $({\theta}_{j},{D}_{j})$ for each atom $j=1..1000$ when (

**a**) $t=0{\omega}_{r}^{-1}$, (

**b**) $t=20{\omega}_{r}^{-1}$ and (

**c**) $t=30{\omega}_{r}^{-1}$ in the case of strong excitation. The parameters used are as in Figure 6.

## 5. Conclusions

## Acknowledgements

## Author Contributions

## Conflicts of Interest

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McKelvie, J.A.; Robb, G.R.M.
Two-Photon Collective Atomic Recoil Lasing. *Atoms* **2015**, *3*, 495-508.
https://doi.org/10.3390/atoms3040495

**AMA Style**

McKelvie JA, Robb GRM.
Two-Photon Collective Atomic Recoil Lasing. *Atoms*. 2015; 3(4):495-508.
https://doi.org/10.3390/atoms3040495

**Chicago/Turabian Style**

McKelvie, James A., and Gordon R.M. Robb.
2015. "Two-Photon Collective Atomic Recoil Lasing" *Atoms* 3, no. 4: 495-508.
https://doi.org/10.3390/atoms3040495