# Analytical Solutions of Temporal Evolution of Populations in Optically-Pumped Atoms with Circularly Polarized Light

## Abstract

**:**

## 1. Introduction

## 2. Theory

## 3. Calculated Results

#### 3.1. Results for the ${F}_{g}=4\to {F}_{e}=5$ Transition

#### 3.2. Results for the ${F}_{g}=3\to {F}_{e}=3$ Transition

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix

## References

- Happer, W. Optical pumping. Rev. Mod. Phys.
**1972**, 44, 169–249. [Google Scholar] [CrossRef] - McClelland, J.J. Optical State Preparation of Atoms. In Atomic, Molecular, and Optical Physics: Atoms and Molecules; Dunning, F.B., Hulet, R.G., Eds.; Academic Press: San Diego, CA, USA, 1995; pp. 145–170. [Google Scholar]
- Smith, D.A.; Hughes, I.G. The role of hyperfine pumping in multilevel systems exhibiting saturated absorption. Am. J. Phys.
**2004**, 72, 631–637. [Google Scholar] [CrossRef] - Magnus, F.; Boatwright, A.L.; Flodin, A.; Shiell, R.C. Optical pumping and electromagnetically induced transparency in a lithium vapour. J. Opt. B: Quantum Semiclass. Opt.
**2005**, 7, 109–118. [Google Scholar] [CrossRef] - Han, H.S.; Jeong, J.E.; Cho, D. Line shape of a transition between two levels in a three-level Λ configuration. Phys. Rev. A
**2011**, 84. [Google Scholar] [CrossRef] - Sydoryk, I.; Bezuglov, N.N.; Beterov, I.I.; Miculis, K.; Saks, E.; Janovs, A.; Spels, P.; Ekers, A. Broadening and intensity redistribution in the Na(3p) hyperfine excitation spectra due to optical pumping in the weak excitation limit. Phys. Rev. A
**2008**, 77. [Google Scholar] [CrossRef] - Porfido, N.; Bezuglov, N.N.; Bruvelis, M.; Shayeganrad, G.; Birindelli, S.; Tantussi, F.; Guerri, I.; Viteau, M.; Fioretti, A.; Ciampini, D.; et al. Nonlinear effects in optical pumping of a cold and slow atomic beam. Phys. Rev. A
**2015**, 92. [Google Scholar] [CrossRef] - McClelland, J.J.; Kelley, M.H. Detailed look at aspects of optical pumping in sodium. Phys. Rev. A
**1985**, 31, 3704–3710. [Google Scholar] [CrossRef] [PubMed] - Farrell, P.M.; MacGillivary, W.R.; Standage, M.C. Quantum-electrodynamic calculation of hyperfine-state populations in atomic sodium. Phys. Rev. A
**1988**, 37, 4240–4251. [Google Scholar] [CrossRef] [PubMed] - Balykin, V.I. Cyclic interaction of Na atoms with circularly polarized laser radiation. Opt. Commun.
**1980**, 33, 31–36. [Google Scholar] [CrossRef] - Liu, S.; Zhang, Y.; Fan, D.; Wu, H.; Yuan, P. Selective optical pumping process in Doppler-broadened atoms. Appl. Opt.
**2011**, 50, 1620–1624. [Google Scholar] [CrossRef] [PubMed] - Moon, G.; Shin, S.R.; Noh, H.R. Analytic solutions for the populations of an optically-pumped multilevel atom. J. Korean Phys. Soc.
**2008**, 53, 552–557. [Google Scholar] [CrossRef] - Moon, G.; Heo, M.S.; Shin, S.R.; Noh, H.R.; Jhe, W. Calculation of analytic populations for a multilevel atom at low laser intensity. Phys. Rev. A
**2008**, 78. [Google Scholar] [CrossRef] - Won, J.Y.; Jeong, T.; Noh, H.R. Analytical solutions of the time-evolution of the populations for D
_{1}transition line of the optically-pumped alkali-metal atoms with I = 3/2. Optik**2013**, 124, 451–455. [Google Scholar] [CrossRef] - Noh, H.R. Analytical Study of Optical Pumping for the D1 Line of
^{85}Rb Atoms. J. Korean Phys. Soc.**2104**, 64, 1630–1635. [Google Scholar] [CrossRef] - Moon, G.; Noh, H.R. Analytic solutions for the saturated absorption spectra. J. Opt. Soc. Am. B
**2008**, 25, 701–711. [Google Scholar] [CrossRef] - Moon, G.; Noh, H.R. Analytic Solutions for the Saturated Absorption Spectrum of the
^{85}Rb Atom with a Linearly Polarized Pump Beam. J. Korean Phys. Soc.**2009**, 54, 13–22. [Google Scholar] [CrossRef] - Do, H.D.; Heo, M.S.; Moon, G.; Noh, H.R.; Jhe, W. Analytic calculation of the lineshapes in polarization spectroscopy of rubidium. Opt. Commun.
**2008**, 281, 4042–4047. [Google Scholar] [CrossRef] - Thornton, T.; Marion, J.B. Classical Dynamics of Particles and Systems, 5th ed.; Brooks/Cole: New York, NY, USA, 2004. [Google Scholar]
- Cohen-Tannoudji, C.; Dupont-Roc, J.; Grynberg, G. Atom–Photon Interactions Basic Processes and Applications; Wiley: New York, NY, USA, 1992. [Google Scholar]
- Meystre, P.; Sargent, M., III. Elements of Quantum Optics; Springer: New York, NY, USA, 2007. [Google Scholar]
- Metcalf, H.J.; van der Straten, P. Laser Cooling and Trapping; Springer: New York, NY, USA, 1999. [Google Scholar]

**Figure 1.**An energy level diagram for an optically pumped atom under the influence of circularly polarized light.

**Figure 2.**(

**a**) Energy level diagram of the Cs-D2 line. (

**b**) Energy level diagrams for the ${F}_{g}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}4\phantom{\rule{3.33333pt}{0ex}}\to \phantom{\rule{3.33333pt}{0ex}}{F}_{e}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}5$ cycling transition line and (

**c**) for the ${F}_{g}=3\to {F}_{e}=3$ transition line illuminated by ${\sigma}^{+}$ polarized laser light.

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Noh, H.-R.
Analytical Solutions of Temporal Evolution of Populations in Optically-Pumped Atoms with Circularly Polarized Light. *Symmetry* **2016**, *8*, 17.
https://doi.org/10.3390/sym8030017

**AMA Style**

Noh H-R.
Analytical Solutions of Temporal Evolution of Populations in Optically-Pumped Atoms with Circularly Polarized Light. *Symmetry*. 2016; 8(3):17.
https://doi.org/10.3390/sym8030017

**Chicago/Turabian Style**

Noh, Heung-Ryoul.
2016. "Analytical Solutions of Temporal Evolution of Populations in Optically-Pumped Atoms with Circularly Polarized Light" *Symmetry* 8, no. 3: 17.
https://doi.org/10.3390/sym8030017