# Stabilization of All Bell States in a Lossy Coupled-Cavity Array

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Effective Master Equation of the Coupled-Cavity Array System

## 3. Stabilization of Different Bell States via Quantum Feedback Control

**D**. Once the photons out of cavities are detected by the photondetector, a feedback operator ${U}_{\mathrm{fb}}=exp{[-i\lambda \left(\right|g\rangle}_{1}\langle e|+{|e\rangle}_{1}\langle g\left|\right)]=exp(-i\lambda {\sigma}_{x}^{1})$ will be applied to the first atom. In this situation the master equation Equation (8) can be modified to [31]

## 4. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of the Effective Master Equation

## Appendix B. Generation of the Other Two Triplet States

## References

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**Figure 1.**Schematic view of the atom-cavity system. The transition $|e\rangle \leftrightarrow |r\rangle $ is driven by a classical field with a time-independent Rabi frequency $\mathsf{\Omega}$; the transition $|g\rangle \leftrightarrow |r\rangle $ is coupled to the cavity with coupling constant g; $\delta $ and $\Delta $ are corresponding detuning parameters. $\kappa $ is the decay rate of each cavity mode. The transition $|g\rangle \leftrightarrow |e\rangle $ is additionally driven by a microwave field with a Rabi frequency $\omega $ and the hopping rate between neighbouring cavities is G.

**Figure 2.**The concurrence of the system at the time $\mathsf{\Gamma}t=2000$ as a function of $\lambda /\pi $ and $\omega /\mathsf{\Gamma}$ as initial state is $|{\varphi}_{1}{\rangle =|g\rangle}_{1}{|g\rangle}_{2}$, without (

**a**,

**b**) or with (

**c**,

**d**) a large decay rate of atomic spontaneous emission ($\gamma =0.1g$).

**Figure 3.**The concurrence of the system in the case of changing the initial state only ($|{\varphi}_{2}{\rangle =|g\rangle}_{1}{|e\rangle}_{2}$) without (

**a**,

**b**) or with (

**c**,

**d**) a large decay rate of atomic spontaneous emission ($\gamma =0.1g$).

**Figure 4.**The fidelities for the states $|{\phi}_{-}\rangle =\left(\right|ge\rangle -|eg\rangle )/\sqrt{2}$ and $|{\psi}_{+}\rangle =\left(\right|gg\rangle +|ee\rangle )/\sqrt{2}$ under different initial states (

**a**) $|{\varphi}_{1}{\rangle =|g\rangle}_{1}{|g\rangle}_{2}$, (

**b**) $|{\varphi}_{1}{\rangle =|g\rangle}_{1}{|e\rangle}_{2}$ with the full master equation. The selection of related parameters is $\mathsf{\Omega}=g$, $G=g$, $\Delta =200g$, $\omega /\mathsf{\Gamma}=\pm 5$ and the driving frequency is set at $\lambda =0.5\pi $.

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**MDPI and ACS Style**

Liu, B.; Li, D.-X.; Shao, X.-Q.
Stabilization of All Bell States in a Lossy Coupled-Cavity Array. *Entropy* **2019**, *21*, 402.
https://doi.org/10.3390/e21040402

**AMA Style**

Liu B, Li D-X, Shao X-Q.
Stabilization of All Bell States in a Lossy Coupled-Cavity Array. *Entropy*. 2019; 21(4):402.
https://doi.org/10.3390/e21040402

**Chicago/Turabian Style**

Liu, Bing, Dong-Xiao Li, and Xiao-Qiang Shao.
2019. "Stabilization of All Bell States in a Lossy Coupled-Cavity Array" *Entropy* 21, no. 4: 402.
https://doi.org/10.3390/e21040402