# Calculation of spontaneous emission from a V-type three-level atom in photonic crystals using fractional calculus

## Abstract

Fractional time derivative, an abstract mathematical operator of fractional calculus, is used to describe the real optical system of a V-type three-level atom embedded in a photonic crystal. A fractional kinetic equation governing the dynamics of the spontaneous emission from this optical system is obtained as a fractional Langevin equation. Solving this fractional kinetic equation by fractional calculus leads to the analytical solutions expressed in terms of fractional exponential functions. The accuracy of the obtained solutions is verified through reducing the system into the special cases whose results are consistent with the experimental observation. With accurate physical results and avoiding the complex integration for solving this optical system, we propose fractional calculus with fractional time derivative as a better mathematical method to study spontaneous emission dynamics from the optical system with non-Markovian dynamics.

- Authors:

- Department of Photonics and Institute of Electro-Optical Engineering, National Chiao Tung University, 1001 Tahsueh Rd., Hsinchu 300, Taiwan (China)
- (China)
- Department of Physics, Chinese Culture University, Yangming Mt., Taipei 111, Taiwan (China)

- Publication Date:

- OSTI Identifier:
- 22058761

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review. A

- Additional Journal Information:
- Journal Volume: 84; Journal Issue: 1; Other Information: (c) 2011 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1050-2947

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; ANALYTICAL SOLUTION; CRYSTALS; KINETIC EQUATIONS; LANGEVIN EQUATION; MARKOV PROCESS; MATHEMATICAL OPERATORS; OPTICAL SYSTEMS; PHOTON EMISSION

### Citation Formats

```
Huang, Chih-Hsien, Hsieh, Wen-Feng, Institute of Electro-Optical Science and Engineering, National Cheng Kung University, 1 Dahsueh Rd., Tainan 701, Taiwan, Wu, Jing-Nuo, Cheng, Szu-Cheng, and Li, Yen-Yin.
```*Calculation of spontaneous emission from a V-type three-level atom in photonic crystals using fractional calculus*. United States: N. p., 2011.
Web. doi:10.1103/PHYSREVA.84.013802.

```
Huang, Chih-Hsien, Hsieh, Wen-Feng, Institute of Electro-Optical Science and Engineering, National Cheng Kung University, 1 Dahsueh Rd., Tainan 701, Taiwan, Wu, Jing-Nuo, Cheng, Szu-Cheng, & Li, Yen-Yin.
```*Calculation of spontaneous emission from a V-type three-level atom in photonic crystals using fractional calculus*. United States. doi:10.1103/PHYSREVA.84.013802.

```
Huang, Chih-Hsien, Hsieh, Wen-Feng, Institute of Electro-Optical Science and Engineering, National Cheng Kung University, 1 Dahsueh Rd., Tainan 701, Taiwan, Wu, Jing-Nuo, Cheng, Szu-Cheng, and Li, Yen-Yin. Fri .
"Calculation of spontaneous emission from a V-type three-level atom in photonic crystals using fractional calculus". United States. doi:10.1103/PHYSREVA.84.013802.
```

```
@article{osti_22058761,
```

title = {Calculation of spontaneous emission from a V-type three-level atom in photonic crystals using fractional calculus},

author = {Huang, Chih-Hsien and Hsieh, Wen-Feng and Institute of Electro-Optical Science and Engineering, National Cheng Kung University, 1 Dahsueh Rd., Tainan 701, Taiwan and Wu, Jing-Nuo and Cheng, Szu-Cheng and Li, Yen-Yin},

abstractNote = {Fractional time derivative, an abstract mathematical operator of fractional calculus, is used to describe the real optical system of a V-type three-level atom embedded in a photonic crystal. A fractional kinetic equation governing the dynamics of the spontaneous emission from this optical system is obtained as a fractional Langevin equation. Solving this fractional kinetic equation by fractional calculus leads to the analytical solutions expressed in terms of fractional exponential functions. The accuracy of the obtained solutions is verified through reducing the system into the special cases whose results are consistent with the experimental observation. With accurate physical results and avoiding the complex integration for solving this optical system, we propose fractional calculus with fractional time derivative as a better mathematical method to study spontaneous emission dynamics from the optical system with non-Markovian dynamics.},

doi = {10.1103/PHYSREVA.84.013802},

journal = {Physical Review. A},

issn = {1050-2947},

number = 1,

volume = 84,

place = {United States},

year = {2011},

month = {7}

}