# Effective Control of the Optical Bistability of a Three-Level Quantum Emitter near a Nanostructured Plasmonic Metasurface

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## Abstract

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## 1. Introduction

## 2. Model and Equations

#### 2.1. Hamiltonian and Master Equations

#### 2.2. Susceptibility

#### 2.3. Optical Bistability in a Unidirectional Ring Cavity

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The three-level quantum V-system under study (

**a**). A silica sphere coated with a gold nanoshell (

**b**) and a two-dimensional square lattice of such spheres (

**c**). The quantum sample (quantum emitter + array of spheres) is placed inside a unidirectional ring cavity (

**d**).

**Figure 2.**Plots of the input–output field curves for the quantum V-system. The dashed curve corresponds to the case of a medium of sole quantum emitters, i.e., $d=\infty $, in which case, the decay rate is ${\mathsf{\Gamma}}_{0}$. The dot-dashed ($d=0.7c/{\omega}_{c}$) and solid ($d=0.9c/{\omega}_{c}$) curves correspond a quantum-sample medium (emitter + metasurface) with $\overline{\omega}=0.632{\omega}_{p}$. We assumed that ${\omega}_{32}=0$, $r=0$, $C=100$ and $\delta =0$.

**Figure 3.**Plots of the input–output field curves for the quantum sample (emitter + metasurface) for different values of emitter-metasurface distance $\delta $. We take $\overline{\omega}=0.632{\omega}_{p}$, ${\omega}_{32}={\mathsf{\Gamma}}_{0}$, $r=0$, $C=100$ and $d=0.8c/{\omega}_{c}$.

**Figure 4.**Plots of the input–output field curves for the quantum sample (emitter + metasurface) for different distances d of the emitter from the plasmonic metasurface. We take $\overline{\omega}=0.632{\omega}_{p}$, ${\omega}_{32}=0.25{\mathsf{\Gamma}}_{0}$, $r=0$, $C=100$ and $\delta =0.35{\mathsf{\Gamma}}_{0}$.

**Figure 5.**Plots of the input–output field curves for the quantum sample (emitter + metasurface) for various values of the distance d. In (

**a**), the incoherent pumping rate r varies from 0 to $0.28{\mathsf{\Gamma}}_{0}$; in (

**b**), from $0,3{\mathsf{\Gamma}}_{0}$ to $0.37{\mathsf{\Gamma}}_{0}$; and, in (

**c**), from $0,4{\mathsf{\Gamma}}_{0}$ to $1.5{\mathsf{\Gamma}}_{0}$. We take $\overline{\omega}=0.632{\omega}_{p}$, ${\omega}_{32}={\mathsf{\Gamma}}_{0}$, $C=100$ and $\delta =0.2{\mathsf{\Gamma}}_{0}$ and $d=0.8c/{\omega}_{c}$.

**Figure 6.**(

**a**) The population distributions ${\rho}_{11},{\rho}_{22}$ and ${\rho}_{33}$, (

**b**) and the gain spectrum [$Im\left(\chi \right)$] (in units of $\frac{N{\mu}^{2}}{{\epsilon}_{0}\hslash}$) of the quantum V-system as a function of the incoherent pumping r in the presence of a plasmonic nanostructure. We take here ${\omega}_{32}=0.25{\mathsf{\Gamma}}_{0}$, and $\delta =0.2{\mathsf{\Gamma}}_{0}$ and $d=0.8c/{\omega}_{c}$. The horizontal dotted line indicates the zero absorption limit, while the vertical dashed (solid) line indicates the limit for the incoherent pumping to achieve the population inversion ${r}_{T}^{PI}$ (gain ${r}_{T}^{L}$). Absorption takes place in $r<{r}_{T}^{L}$ (the red zone in (

**b**)). Gain without inversion appears in ${r}_{T}^{L}<r<{r}_{T}^{PI}$(the green zone in (

**b**)), while the lasing with inversion takes place when $r>{r}_{T}^{PI}$ (the blue zone in (

**b**)).

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**MDPI and ACS Style**

Hamedi, H.R.; Paspalakis, E.; Yannopapas, V.
Effective Control of the Optical Bistability of a Three-Level Quantum Emitter near a Nanostructured Plasmonic Metasurface. *Photonics* **2021**, *8*, 285.
https://doi.org/10.3390/photonics8070285

**AMA Style**

Hamedi HR, Paspalakis E, Yannopapas V.
Effective Control of the Optical Bistability of a Three-Level Quantum Emitter near a Nanostructured Plasmonic Metasurface. *Photonics*. 2021; 8(7):285.
https://doi.org/10.3390/photonics8070285

**Chicago/Turabian Style**

Hamedi, Hamid R., Emmanuel Paspalakis, and Vassilios Yannopapas.
2021. "Effective Control of the Optical Bistability of a Three-Level Quantum Emitter near a Nanostructured Plasmonic Metasurface" *Photonics* 8, no. 7: 285.
https://doi.org/10.3390/photonics8070285