# Squeezed Coherent States in Double Optical Resonance

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

## 3. Results and Discussion

## 4. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SQVS | Squeezed Vacuum State |

SQCS | Squeezed Coherent State |

DOR | Double Optical Resonance |

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**Figure 1.**Schematic presentation of the system of study. A strong radiation field prepared initially in a squeezed coherent state drives the $|g\rangle \leftrightarrow |a\rangle $ transition, resulting in a Stark splitting of both states. This splitting is monitored through the calculation of the population of state $|b\rangle $ as a function of ${\Delta}_{2}$, which is assumed to be weakly coupled to state $|a\rangle $ through a probe field.

**Figure 2.**(

**a**) Photon number distribution of a squeezed coherent state with ${\left|a\right|}^{2}=60$ and $r=0.5$, compared with the respective distribution of a coherent state ($r=0$, black line). (

**b**) Photon number distribution of a squeezed coherent state with ${\left|a\right|}^{2}=60$ and $r=2$. In both panels, the red lines correspond to $\Phi =0$, while the teal lines correspond to $\Phi =\pi /2$.

**Figure 3.**(

**a**) Population of the probe state $|b\rangle $ as a function of the time for various detunings ${\Delta}_{2}$. The parameters of the strong field are: ${\left|a\right|}^{2}=60$, $r=0.5$, ${g}_{1}=0.7\gamma $, and ${\Delta}_{1}=0$. The vertical dashed line corresponds to the time $T=30{\gamma}^{-1}$ where the system is well within its steady-state regime. (

**b**) Population of the probe state $|b\rangle $ as a function of ${\Delta}_{2}$ for various values of ${\left|a\right|}^{2}$. The chosen parameters are: $r=0.5$, ${g}_{1}=0.7\gamma $, and ${\Delta}_{1}=0$.

**Figure 4.**(

**a**) Population of the probe state $|b\rangle $ as a function of ${\Delta}_{2}$ for different degrees of squeezing and $\Phi =0$. Inset: Variance of the SQCS according to Equation (13) for $\Phi =0$. (

**b**) Population of the probe state $|b\rangle $ as a function of ${\Delta}_{2}$ for different degrees of squeezing and $\Phi =\pi /2$. Inset: Variance of the SQCS according to Equation (13) for $\Phi =\pi /2$. In both panels, ${\left|a\right|}^{2}=60$, $T=30{\gamma}^{-1}$, ${g}_{1}=0.7\gamma $, and ${\Delta}_{1}=0$. Black lines: $r=0$, teal lines: $r=1$, orange lines: $r=1.5$, and purple lines: $r=2$.

**Figure 5.**(

**a**) Population of the probe state $|b\rangle $ as a function of ${\Delta}_{2}$ for different degrees of extreme squeezing, $\Phi =\pi /2$ and ${\left|a\right|}^{2}=60$. Black line: $r=0$, teal line: $r=3$, orange line: $r=3.5$, and purple line: $r=4$. In both panels, $T=30{\gamma}^{-1}$, ${g}_{1}=0.7\gamma $, and ${\Delta}_{1}=0$. (

**b**) Population of the probe state $|b\rangle $ as a function of ${\Delta}_{2}$ for different degrees of squeezing and ${\left|a\right|}^{2}=0$, corresponding to a squeezed vacuum state. Teal line: $r=1$, orange line: $r=1.5$, and purple line: $r=2$.

**Figure 6.**(

**a**) Population of the probe state $|b\rangle $ as a function of ${\Delta}_{2}$ for various angles $\Phi $. The squeezed field is detuned from resonance with the $|g\rangle \leftrightarrow |a\rangle $ transition by ${\Delta}_{1}=\gamma $. (

**b**) Population of the probe state $|b\rangle $ as a function of ${\Delta}_{2}$ for various angles $\Phi $. The squeezed field is detuned from resonance with the $|g\rangle \leftrightarrow |a\rangle $ transition by ${\Delta}_{1}=3\gamma $. In both panels, ${\left|a\right|}^{2}=60$, $T=30{\gamma}^{-1}$, ${g}_{1}=0.7\gamma $, and $r=1.5$. Black lines: $\Phi =0$, teal lines: $\Phi =\pi /6$, orange lines: $\Phi =\pi /4$, and purple lines: $\Phi =\pi /2$.

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**MDPI and ACS Style**

Mouloudakis, G.; Lambropoulos, P. Squeezed Coherent States in Double Optical Resonance. *Photonics* **2021**, *8*, 72.
https://doi.org/10.3390/photonics8030072

**AMA Style**

Mouloudakis G, Lambropoulos P. Squeezed Coherent States in Double Optical Resonance. *Photonics*. 2021; 8(3):72.
https://doi.org/10.3390/photonics8030072

**Chicago/Turabian Style**

Mouloudakis, George, and Peter Lambropoulos. 2021. "Squeezed Coherent States in Double Optical Resonance" *Photonics* 8, no. 3: 72.
https://doi.org/10.3390/photonics8030072