# Two Photon Processes in an Atom Confined in Gaussian Potential

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Theory

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Variation of two-photon $1s\to 2s$ transition probability amplitude with frequency of incoming photons, ${\omega}_{0}$, for (

**a**) ${V}_{0}=0.2$ a.u. and $\sigma =1.0$ a.u.; (

**b**) ${V}_{0}=0.2$ a.u. and $\sigma =5.0$ a.u.; (

**c**) ${V}_{0}=0.4$ a.u. and $\sigma =1.0$ a.u.; (

**d**) ${V}_{0}=0.4$ a.u. and $\sigma =5.0$ a.u.

**Figure 2.**Variation of two-photon $1s\to 3s$ transition probability amplitude with frequency of incoming photons, ${\omega}_{0}$, for (

**a**) ${V}_{0}=0.2$ a.u. and $\sigma =1.0$ a.u.; (

**b**) ${V}_{0}=0.2$ a.u. and $\sigma =5.0$ a.u.; (

**c**) ${V}_{0}=0.4$ a.u. and $\sigma =1.0$ a.u.; (

**d**) ${V}_{0}=0.4$ a.u. and $\sigma =5.0$ a.u.

**Figure 3.**Variation of two-photon $1s\to 4s$ transition probability amplitude with frequency of incoming photons, ${\omega}_{0}$, for (

**a**) ${V}_{0}=0.2$ a.u. and $\sigma =1.0$ a.u.; (

**b**) ${V}_{0}=0.2$ a.u. and $\sigma =5.0$ a.u.; (

**c**) ${V}_{0}=0.4$ a.u. and $\sigma =1.0$ a.u.; (

**d**) ${V}_{0}=0.4$ a.u. and $\sigma =5.0$ a.u.

**Figure 4.**Variation of two-photon $1s\to 3d$ transition probability amplitude with frequency of incoming photons, ${\omega}_{0}$, for (

**a**) ${V}_{0}=0.2$ a.u. and $\sigma =1.0$ a.u.; (

**b**) ${V}_{0}=0.2$ a.u. and $\sigma =5.0$ a.u.; (

**c**) ${V}_{0}=0.4$ a.u. and $\sigma =1.0$ a.u.; (

**d**) ${V}_{0}=0.4$ a.u. and $\sigma =5.0$ a.u.

**Figure 5.**Variation of two-photon $1s\to 4d$ transition probability amplitude with frequency of incoming photons, ${\omega}_{0}$, for (

**a**) ${V}_{0}=0.2$a.u. and $\sigma =1.0$ a.u.; (

**b**) ${V}_{0}=0.2$ a.u. and $\sigma =5.0$ a.u.; (

**c**) ${V}_{0}=0.4$ a.u. and $\sigma =1.0$ a.u.; (

**d**) ${V}_{0}=0.4$ a.u. and $\sigma =5.0$ a.u.

**Figure 6.**Variation of two-photon $1s\to 3s$ absorption coefficients with frequency of incoming photons, ${\omega}_{0}$, for (

**a**) ${V}_{0}=0.2$ a.u. and $\sigma =1.0$ a.u.; (

**b**) ${V}_{0}=0.2$ a.u. and $\sigma =5.0$ a.u.; (

**c**) ${V}_{0}=0.4$ a.u. and $\sigma =1.0$ a.u.; (

**d**) ${V}_{0}=0.4$ a.u. and $\sigma =5.0$ a.u.

**Figure 7.**Variation of two-photon $1s\to 3d$ absorption coefficients with frequency of incoming photons, ${\omega}_{0}$, for (

**a**) ${V}_{0}=0.2$ a.u. and $\sigma =1.0$ a.u.; (

**b**) ${V}_{0}=0.2$ a.u. and $\sigma =5.0$ a.u.; (

**c**) ${V}_{0}=0.4$ a.u. and $\sigma =1.0$ a.u.; (

**d**) ${V}_{0}=0.4$ a.u. and $\sigma =5.0$ a.u.

**Figure 8.**Variation of two-photon $1s\to 3s$ absorption coefficients with frequency of incoming photons, ${\omega}_{0}$, for ${r}_{0}=40$ a.u. for (

**a**) ${V}_{0}=0.2$ a.u. and $\sigma =1.0$ a.u.; (

**b**) ${V}_{0}=0.2$ a.u. and $\sigma =5.0$ a.u.; (

**c**) ${V}_{0}=0.4$ a.u. and $\sigma =1.0$ a.u.; (

**d**) ${V}_{0}=0.4$ a.u. and $\sigma =5.0$ a.u.

**Figure 9.**Variation of two-photon $1s\to 3d$ absorption coefficients with frequency of incoming photons, ${\omega}_{0}$, for ${r}_{0}=40$ a.u. for (

**a**) ${V}_{0}=0.2$ a.u. and $\sigma =1.0$ a.u.; (

**b**) ${V}_{0}=0.2$ a.u. and $\sigma =5.0$ a.u.; (

**c**) ${V}_{0}=0.4$ a.u. and $\sigma =1.0$ a.u.; (

**d**) ${V}_{0}=0.4$ a.u. and $\sigma =5.0$ a.u.

${\omega}_{0}(\mathrm{Ryd}.)$ | $|{D}_{2}|$ ($1s$ → $2s$) | $|{D}_{2}|$ ($1s$ → $3s$) | ||
---|---|---|---|---|

Present Study | Reference [45] | Present Study | Reference [45] | |

0.3750 | 11.780338 | 11.7805 | 3.235425 | 3.2354 |

0.5250 | 14.731690 | 14.7319 | ||

0.6750 | 41.147800 | 41.1484 | 1.669316 | 1.6693 |

0.6875 | 49.686983 | 49.6878 | 0.696330 | 0.6963 |

0.7000 | 62.658358 | 62.6595 | 0.984604 | 0.9847 |

0.7125 | 84.523402 | 84.5252 | 4.158088 | 4.1583 |

0.7250 | 128.680019 | 128.6835 | 11.215759 | 11.2162 |

0.7375 | 262.153248 | 262.1654 | 34.224314 | 34.2263 |

0.7475 | 1334.059059 | 1334.3261 | 226.765862 | 226.8138 |

0.7650 | 58.200900 | 58.2000 | ||

0.8000 | 38.309623 | 38.3099 | ||

0.8250 | 46.579090 | 46.5797 | ||

0.8500 | 74.418968 | 74.4204 | ||

0.8750 | 219.974861 | 219.9847 | ||

0.8860 | 1117.033823 | 1117.2380 |

**Table 2.**Two-photon transparency frequencies for various potential widths and depths. The data is in atomic units.

σ | ${V}_{0}$ | $1s\to 3s$ | $1s\to 4s$ | $1s\to 4d$ |
---|---|---|---|---|

0.2 | 0.2 | 0.6966575 | 0.6943135 | 0.8417985 |

0.8748915 | ||||

0.4 | 0.6997875 | 0.6974305 | 0.8455535 | |

0.8784775 | ||||

0.6 | 0.7029685 | 0.7005985 | 0.8493665 | |

0.8821205 | ||||

0.8 | 0.7062005 | 0.7038175 | 0.8532385 | |

0.8858195 | ||||

1.0 | 0.7094835 | 0.7070865 | 0.8571695 | |

0.8895765 | ||||

1.0 | 0.2 | 0.7966715 | 0.7946135 | 0.9540155 |

0.9837665 | ||||

0.4 | 0.9129015 | 0.9111615 | 1.0823305 | |

1.1088085 | ||||

0.6 | 1.0413755 | 1.0399605 | 1.2221895 | |

1.2456185 | ||||

0.8 | 1.1810825 | 1.1799705 | 1.3726705 | |

1.3932485 | ||||

1.0 | 1.3309895 | 1.3301425 | 1.5328565 | |

1.5507545 | ||||

2.0 | 0.2 | 0.9120575 | 0.9100275 | 1.0821115 |

1.1100905 | ||||

0.4 | 1.1385265 | 1.1359635 | 1.3399855 | |

1.3627115 | ||||

0.6 | 1.3661185 | 1.3623365 | ||

1.6253105 | ||||

0.8 | 1.5894415 | 1.5839235 | 1.8820885 | |

1.8955715 | ||||

1.0 | 1.8046355 | 1.7971595 | 2.1605405 | |

2.1721445 | ||||

5.0 | 0.2 | 0.8393165 | 0.8325805 | 1.0656265 |

1.1942105 | ||||

0.4 | 0.9385475 | 0.9232605 | 1.1849155 | |

1.4908805 | ||||

0.6 | 1.0136965 | 1.7431005 | 1.2573295 | |

0.8 | 1.9660075 | |||

1.0 | 1.4822535 | |||

2.1662075 |

**Table 3.**Two-photon resonance enhancement frequencies for various potential widths and depths. The data is in atomic units.

σ | ${V}_{0}$ | $1s\to 3s$ | $1s\to 4s$ | $1s\to 3d$ | $1s\to 4d$ |
---|---|---|---|---|---|

0.2 | 0.2 | 0.7536975 | 0.7536975 | 0.7536975 | 0.7536975 |

0.8925875 | 0.8925875 | 0.8925875 | |||

0.4 | 0.7574475 | 0.7574475 | 0.7574475 | 0.7574475 | |

0.8963405 | 0.8963405 | 0.8963405 | |||

0.6 | 0.7612565 | 0.7612565 | 0.7612565 | 0.7612565 | |

0.9001505 | 0.9001505 | 0.9001505 | |||

0.8 | 0.7651245 | 0.7651245 | 0.7651245 | 0.7651245 | |

0.9040205 | 0.9040205 | 0.9040205 | |||

1 | 0.7690515 | 0.7690515 | 0.7690515 | 0.7690515 | |

0.9079485 | 0.9079485 | 0.9079485 | |||

1 | 0.2 | 0.8630055 | 0.8630055 | 0.8630055 | 0.8630055 |

1.0038595 | 1.0038595 | 1.0038595 | |||

1.0530765 | |||||

0.4 | 0.9880245 | 0.9880245 | 0.9880245 | 0.9880245 | |

1.1311185 | 1.1311185 | 1.1311185 | |||

1.1809445 | |||||

0.6 | 1.1241155 | 1.1241155 | 1.1241155 | 1.1241155 | |

1.2697875 | 1.2697875 | 1.2697875 | |||

1.3203075 | |||||

0.8 | 1.2702465 | 1.2702465 | 1.2702465 | 1.2702465 | |

1.4189135 | 1.4189135 | 1.4189135 | |||

1.4702285 | |||||

1 | 1.4253685 | 1.4253685 | 1.4253685 | 1.4253685 | |

1.5775475 | 1.5775475 | 1.5775475 | |||

1.6297795 |

**Table 4.**Two-photon resonance enhancement frequencies for various potential widths and depths. The data is in atomic units.

σ | ${V}_{0}$ | $1s\to 2s$ | $1s\to 3s$ | $1s\to 4s$ | $1s\to 3d$ | $1s\to 4d$ |
---|---|---|---|---|---|---|

2 | 0.2 | 0.9625345 | 0.9625345 | 0.9625345 | 0.9625345 | 0.9625345 |

1.1264855 | 1.1264855 | 1.1264855 | 1.1264855 | |||

1.1802345 | 1.1802345 | |||||

0.4 | 1.1717755 | 1.1717755 | 1.1717755 | 1.1717755 | 1.1717755 | |

1.3771515 | 1.3771515 | 1.3771515 | 1.3771515 | |||

1.4370505 | 1.4370505 | |||||

0.6 | 1.3696755 | 1.3696755 | 1.3696755 | 1.3696755 | 1.3696755 | |

1.6383765 | 1.6383765 | 1.6383765 | 1.6383765 | |||

1.7046825 | 1.7046825 | |||||

0.8 | 1.5528565 | 1.5528565 | 1.5528565 | 1.5528565 | 1.5528565 | |

1.9086325 | 1.9086325 | 1.9086325 | 1.9086325 | |||

1.9809315 | 1.9809315 | |||||

1 | 1.7214035 | 1.7214035 | 1.7214035 | 1.7214035 | 1.7214035 | |

2.1863955 | 2.1863955 | 2.1863955 | 2.1863955 | |||

2.2641365 | 2.2641365 | |||||

5 | 0.2 | 0.9017875 | 0.9017875 | 0.9017875 | 0.9017875 | 0.9017875 |

1.2076625 | 1.2076625 | 1.2076625 | ||||

1.2833555 | ||||||

0.4 | 1.0133065 | 1.0133065 | 1.0133065 | 1.0133065 | 1.0133065 | |

1.4933425 | 1.4933425 | 1.4933425 | ||||

1.6303775 | ||||||

0.6 | 1.1047275 | 1.1047275 | 1.1047275 | 1.1047275 | 1.1047275 | |

1.7368975 | 1.7368975 | 1.7368975 | ||||

1.9763745 | ||||||

0.8 | 1.1844925 | 1.1844925 | 1.1844925 | 1.1844925 | 1.1844925 | |

1.9540065 | 1.9540065 | 1.9540065 | ||||

2.3162335 | ||||||

1 | 1.2556685 | 1.2556685 | 1.4262795 | 1.2556685 | 1.2749545 | |

2.1478855 | 2.1478855 | 2.1478855 | ||||

2.6377985 |

**Table 5.**First few energy levels of a hydrogen atom under the effect of a Gaussian potential and loose spherical confinement for ${r}_{0}=50$ a.u. and for various values of σ and ${V}_{0}$.

n | l | $\sigma (a.u.)$ | ${V}_{0}=0.0(a.u.)$ | ${V}_{0}=0.2(a.u.)$ | ${V}_{0}=0.4(a.u.)$ |
---|---|---|---|---|---|

1 | 0 | 1.0 | −0.500000 | −0.557966 | −0.622137 |

2 | 0 | −0.125000 | −0.130189 | −0.135136 | |

2 | 1 | −0.125000 | −0.126463 | −0.128125 | |

3 | 0 | −0.055556 | −0.057016 | −0.058382 | |

3 | 1 | −0.055556 | −0.056037 | −0.056578 | |

3 | 2 | −0.055556 | −0.055564 | −0.055573 | |

4 | 0 | −0.031204 | −0.031818 | −0.032386 | |

4 | 1 | −0.031216 | −0.031428 | −0.031665 | |

4 | 2 | −0.031233 | −0.031238 | −0.031244 | |

4 | 3 | −0.031246 | −0.031246 | −0.031246 | |

1 | 0 | 5.0 | −0.679948 | −0.861344 | |

2 | 0 | −0.202437 | −0.301700 | ||

2 | 1 | −0.229055 | −0.354691 | ||

3 | 0 | −0.070459 | −0.094417 | ||

3 | 1 | −0.076117 | −0.114672 | ||

3 | 2 | −0.087940 | −0.155671 | ||

4 | 0 | −0.036800 | −0.043339 | ||

4 | 1 | −0.038271 | −0.046155 | ||

4 | 2 | −0.038904 | −0.047266 | ||

4 | 3 | −0.033498 | −0.044919 |

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Lumb, S.; Lumb, S.; Prasad, V.
Two Photon Processes in an Atom Confined in Gaussian Potential. *Atoms* **2016**, *4*, 6.
https://doi.org/10.3390/atoms4010006

**AMA Style**

Lumb S, Lumb S, Prasad V.
Two Photon Processes in an Atom Confined in Gaussian Potential. *Atoms*. 2016; 4(1):6.
https://doi.org/10.3390/atoms4010006

**Chicago/Turabian Style**

Lumb, Sonia, Shalini Lumb, and Vinod Prasad.
2016. "Two Photon Processes in an Atom Confined in Gaussian Potential" *Atoms* 4, no. 1: 6.
https://doi.org/10.3390/atoms4010006