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Angle-Dependent Magic Optical Trap for the 6S_{1/2}↔nP_{3/2} Rydberg Transition of Cesium Atoms

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## Abstract

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## 1. Introduction

## 2. Calculation Methods

## 3. Results and Discussion

#### 3.1. Magic Condition for Cs $6{S}_{1/2}$ Ground State and $70{P}_{3/2}$ Rydberg State

#### 3.2. Magic Trapping Condition for Cs $n{P}_{3/2}$ Rydberg States

## 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.**The geometrical parameters of electromagnetic plane waves. $\widehat{k}$, ${\widehat{e}}_{z}$, and $\widehat{\epsilon}$ are the unit wave vector, the quantized axis, and the laser polarization vector, respectively. The parameters ${\theta}_{maj}$, ${\theta}_{min}$, and ${\theta}_{k}$ are the angles between each unit vector and the quantized axis.

**Figure 2.**Trapping of Rydberg atoms. (

**a**) The conventional far-off-resonance red-detuned ODT is attractive for ground states but usually repulsive for highly excited Rydberg states. (

**b**) The direct single-photon transition from $\left|g\right.\u232a$ = $\left|6{S}_{1/2}\right.\u232a$ to $\left|r\right.\u232a$ = $\left|n{P}_{3/2}\right.\u232a$ coupled by a 319-nm ultraviolet laser. The trapping potential of the $\left|g\right.\u232a$ and $\left|r\right.\u232a$ states can be equalized when an ODT laser is tuned to the blue side of the $\left|r\right.\u232a\leftrightarrow \left|a\right.\u232a$ auxiliary transition.

**Figure 3.**Dynamic polarizabilities of Cs $6{S}_{1/2}$ ground state (red dashed lines) and $70{P}_{3/2}$ Rydberg state (blue and black solid lines) between 600 nm and 2000 nm.

**Figure 4.**The dynamic polarizabilities of the ground state and the Rydberg state in the case of linearly polarized light in the range from 1882.220 to 1882.245 nm near the auxiliary transition of $70{P}_{3/2}\leftrightarrow 7{D}_{5/2}$. (

**a**) The quantization axis is perpendicular to the plane of polarization and (

**b**) the quantization axis is parallel to the plane of polarization.

**Figure 5.**The dependence of magic detuning for the $6{S}_{1/2}\leftrightarrow 70{P}_{3/2}$ transition near different auxiliary transitions on ${\left|cos{\theta}_{p}\right|}^{2}$ in the case of linearly polarized light. (

**a**) $70{P}_{3/2}\leftrightarrow 7{D}_{5/2}$ auxiliary transition; (

**b**) $70{P}_{3/2}\leftrightarrow 8{S}_{1/2}$ auxiliary transition; (

**c**) $70{P}_{3/2}\leftrightarrow 6{D}_{5/2}$ auxiliary transition.

**Figure 6.**(

**a**) A magic ODT formed by a strongly focused single-mode Gaussian laser beam. (

**b**) The involved energy level diagram of magic ODTs for the $6{S}_{1/2}\leftrightarrow 70{P}_{3/2}$ single-photon transition coupled by a 318.65798 nm ultra-violet laser. The brown double arrow lines are auxiliary transitions from $70{P}_{3/2}$ to $7{D}_{5/2}$, $8{S}_{1/2}$, and $6{D}_{5/2}$, and the corresponding magic wavelengths are 1882.229946, 1415.535736, and 1142.866537 nm, respectively.

**Figure 7.**In the vicinity of different auxiliary transitions, the magic detuning at magic angle varies with the principal quantum number from 50 to 90. The solid lines are the fitted results.

**Table 1.**At magic angle, magic detunings and magic wavelengths of the $6{S}_{1/2}\leftrightarrow 70{P}_{3/2}$ transition near the $70{P}_{3/2}\leftrightarrow 7{D}_{5/2}$, $70{P}_{3/2}\leftrightarrow 8{S}_{1/2}$, and $70{P}_{3/2}\leftrightarrow 6{D}_{5/2}$ auxiliary transitions.

Auxiliary Transition | Magic Detuning (MHz) | Magic Wavelength (nm) |
---|---|---|

$70{P}_{3/2}\leftrightarrow 7{D}_{5/2}$ | 302.67 | 1882.229946 |

$70{P}_{3/2}\leftrightarrow 8{S}_{1/2}$ | 77.08 | 1415.535736 |

$70{P}_{3/2}\leftrightarrow 6{D}_{5/2}$ | 134.48 | 1142.866537 |

Auxiliary Transition | a (MHz) | b | c (MHz) |
---|---|---|---|

$n{P}_{3/2}\leftrightarrow 7{D}_{5/2}$ | 3.881(6) × 10${}^{8}$ | 3.324(5) | 16.6(5) |

$n{P}_{3/2}\leftrightarrow 8{S}_{1/2}$ | 1.050(2) × 10${}^{8}$ | 3.337(5) | 3.9(1) |

$n{P}_{3/2}\leftrightarrow 6{D}_{5/2}$ | 1.653(5) × 10${}^{8}$ | 3.312(8) | 6.1(4) |

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**MDPI and ACS Style**

Bai, J.; Wang, X.; Hou, X.; Liu, W.; Wang, J. Angle-Dependent Magic Optical Trap for the 6*S*_{1/2}↔*n**P*_{3/2} Rydberg Transition of Cesium Atoms. *Photonics* **2022**, *9*, 303.
https://doi.org/10.3390/photonics9050303

**AMA Style**

Bai J, Wang X, Hou X, Liu W, Wang J. Angle-Dependent Magic Optical Trap for the 6*S*_{1/2}↔*n**P*_{3/2} Rydberg Transition of Cesium Atoms. *Photonics*. 2022; 9(5):303.
https://doi.org/10.3390/photonics9050303

**Chicago/Turabian Style**

Bai, Jiandong, Xin Wang, Xiaokai Hou, Wenyuan Liu, and Junmin Wang. 2022. "Angle-Dependent Magic Optical Trap for the 6*S*_{1/2}↔*n**P*_{3/2} Rydberg Transition of Cesium Atoms" *Photonics* 9, no. 5: 303.
https://doi.org/10.3390/photonics9050303