# Phase-Controlled Absorption and Dispersion Properties of a Multi-Level Quantum Emitter Interacting with Bismuth-Chalcogenide Microparticles

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## Abstract

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## 1. Introduction

## 2. Theory

## 3. Zero Absorption, Optical Transparency and Inversion without Gain

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The quantum emitter studied throughout this work. It is a double-V-type emitter containing two upper states $|2\rangle $ and $|3\rangle $ which decay spontaneously to the two lower states $|0\rangle $ and $|1\rangle $. The system additionally engages with two weak probe electromagnetic fields, both of which are circularly polarized and share the same angular frequency, denoted as ${\omega}_{a}={\omega}_{b}$. The field denoted by a drives the transition $|0\rangle $ to $|2\rangle $ whilst the field denoted by b drives the transition $|0\rangle $ to state $|3\rangle $.

**Figure 2.**The graph depicting the absorption spectrum, represented by the solid curve of $\mathrm{Im}\left({\chi}^{\left(1\right)}\right)$, and the dispersion spectrum, shown as the dashed curve of $\mathrm{Re}\left({\chi}^{\left(1\right)}\right)$, is presented in normalized units ($N{\mu {}^{\prime}}^{2}/(\hslash {\epsilon}_{0}{\mathsf{\Gamma}}_{0})$), as described in Equation (14). The data correspond to the case where the QE is placed in vacuum. We assume that ${\omega}_{32}=1.5{\mathsf{\Gamma}}_{0}$, $\gamma {}^{\prime}=0.3{\mathsf{\Gamma}}_{0}$, ${E}_{b}/{E}_{a}=1.5$.

**Figure 3.**Computational setup: the double-V-type quantum emitter of Figure 1 is placed in the middle of a dimer of $2\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$ microspheres.

**Figure 4.**The graph depicting the absorption spectrum, represented by the solid curve of $\mathrm{Im}\left({\chi}^{\left(1\right)}\right)$, and the dispersion spectrum, shown as the dashed curve of $\mathrm{Re}\left({\chi}^{\left(1\right)}\right)$, is presented in normalized units ($N{\mu {}^{\prime}}^{2}/(\hslash {\epsilon}_{0}{\mathsf{\Gamma}}_{0})$), as described in Equation (14). The data correspond to the computational configuration illustrated in Figure 3. We assume that ${\omega}_{32}=1.5{\mathsf{\Gamma}}_{0}$, $\gamma {}^{\prime}=0.3{\mathsf{\Gamma}}_{0}$, ${E}_{b}/{E}_{a}=1.5$. We have also chosen $\overline{\omega}=12.43$ THz and the gap between the two microspheres to be 2000 $\mathsf{\mu}$m so that ${\mathsf{\Gamma}}_{\perp}=88.51{\mathsf{\Gamma}}_{0}$ and ${\mathsf{\Gamma}}_{\Vert}=0.851{\mathsf{\Gamma}}_{0}$ which provide a degree of QI $p=0.981$. In (

**a**) $\varphi =0$, in (

**b**) $\varphi =\pi /2$, in (

**c**) $\varphi =\pi $, and in (

**d**) $\varphi =3\pi /2$.

**Figure 5.**The same as Figure 4 but for a microsphere gap of 4000 $\mathsf{\mu}$m and frequency $\overline{\omega}=14.1$ THz which provide ${\mathsf{\Gamma}}_{\perp}=17.95{\mathsf{\Gamma}}_{0}$ and ${\mathsf{\Gamma}}_{\Vert}=0.083{\mathsf{\Gamma}}_{0}$ yielding a degree of QI $p=0.995$.

**Figure 7.**The same as Figure 6 but for a $4\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m gap between the microspheres.

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**MDPI and ACS Style**

Papachronis, T.; Kyvelos, N.; Paspalakis, E.; Yannopapas, V.
Phase-Controlled Absorption and Dispersion Properties of a Multi-Level Quantum Emitter Interacting with Bismuth-Chalcogenide Microparticles. *Photonics* **2023**, *10*, 1296.
https://doi.org/10.3390/photonics10121296

**AMA Style**

Papachronis T, Kyvelos N, Paspalakis E, Yannopapas V.
Phase-Controlled Absorption and Dispersion Properties of a Multi-Level Quantum Emitter Interacting with Bismuth-Chalcogenide Microparticles. *Photonics*. 2023; 10(12):1296.
https://doi.org/10.3390/photonics10121296

**Chicago/Turabian Style**

Papachronis, Theodoros, Nikolaos Kyvelos, Emmanuel Paspalakis, and Vassilios Yannopapas.
2023. "Phase-Controlled Absorption and Dispersion Properties of a Multi-Level Quantum Emitter Interacting with Bismuth-Chalcogenide Microparticles" *Photonics* 10, no. 12: 1296.
https://doi.org/10.3390/photonics10121296