# Trichromatic π-Pulse for Ultrafast Total Inversion of a Four-Level Ladder System

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Simulations

**Figure 1.**Time evolution of the populations of the four levels, as indicated, for a “π-pulse” transition produced by a tricolor resonant 10 ps Gaussian pulse of central angular frequencies ${\omega}_{1}$, ${\omega}_{2}$ and ${\omega}_{3}$. The inset on the left shows a scheme of the four-level ladder system with the pulse areas that completely invert the population. Atomic sodium has been considered in our study as an example, with transition energies such as ${\omega}_{12}={\omega}_{1}=16,978$ cm${}^{-1}$ (589 nm for 3s–3p), ${\omega}_{23}={\omega}_{2}=8762$ cm${}^{-1}$ (1141 nm for 3p–4s), and ${\omega}_{34}={\omega}_{3}=12,801$ cm${}^{-1}$ (781 nm for 4s–7p). The inset on the right shows a complete oscillation of a “$2\pi $-pulse”, which is produced with the corresponding double pulse areas $2\sqrt{3}\pi $, $4\pi $, and $2\sqrt{3}\pi $.

**Figure 2.**Study of the effect of field detuning on the population inversion from level $3s$ to level 7p in atomic sodium (see Figure 1) for 100 fs pulses. Note that in

**(a**–

**c**) the most yellow regions correspond to 100% population transfer; while in (

**d**–

**f**), they correspond to 80%, 40% and 12%, respectively.

**Figure 3.**Effect of pulse chirp $\phi \u2033$ on the populations of the four levels after the interaction for different durations of the corresponding TL pulses. Indicated are the final populations of levels 7p (red lines) and 4s (blue lines); levels 3p and 3s become basically unpopulated in all cases (green and black lines). Note that the x-axis has been scaled for the different cases as indicated in the legend of the figure. Clearly, the population inversion produced by the longer pulses is less affected by chirp.

${t}_{p}$ | $\phi \u2033$ | ${t}_{{p}_{c}}$ | % Inversion |
---|---|---|---|

10 ps | 100 ps${}^{2}$ | 29.5 ps | ≈ 92% |

1 ps | 0.94 ps${}^{2}$ | 2.8 ps | ≈ 92% |

100 fs | 9765 fs${}^{2}$ | 289 fs | ≈ 92% |

## 3. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Serrat, C.; Corbera, M.; Afa, J. Trichromatic *π*-Pulse for Ultrafast Total Inversion of a Four-Level Ladder System. *Appl. Sci.* **2015**, *5*, 1484-1493.
https://doi.org/10.3390/app5041484

**AMA Style**

Serrat C, Corbera M, Afa J. Trichromatic *π*-Pulse for Ultrafast Total Inversion of a Four-Level Ladder System. *Applied Sciences*. 2015; 5(4):1484-1493.
https://doi.org/10.3390/app5041484

**Chicago/Turabian Style**

Serrat, Carles, Montserrat Corbera, and John Afa. 2015. "Trichromatic *π*-Pulse for Ultrafast Total Inversion of a Four-Level Ladder System" *Applied Sciences* 5, no. 4: 1484-1493.
https://doi.org/10.3390/app5041484