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The Controlling of Local Quantum Uncertainty of Two Atoms in the Composite Cavity Optomechanical Systems and Phase-Damping Environments
DOI: 10.12677/MP.2022.123004, PDF, HTML, XML, 下载: 75  浏览: 107  国家自然科学基金支持

Abstract: Local quantum uncertainty (LQU) is one of the important measures for characterizing quantum correlations and it plays an integral role in quantum information processing. In this paper, we have studied the effect of dephasing parameters on LQU in the composite cavity optomechanical system and the controlling of various parameters in the system on LQU of two atoms. The results show that as the dephasing parameter increases, the LQU of two atoms decays with a greater decay rate. But, this decay rate is much slower than that of the concurrence, it indicates that LQU is more suitable for characterizing quantum correlations in composite cavity optomechanical systems than concurrence. By increasing the atom-cavityfield (with vibrating mirror) coupling and the vibrating mirror-cavityfield coupling, as well as both increasing the atom-cavityfield (J-C model) coupling and the atom-cavityfield (with vibrating mirror) coupling, the quantum correlation of two atoms can achieve the enhancement in the composite cavity optomechanical system. Furthermore, adjusting the atom-cavityfield (with vibrating mirror) coupling and the mirror-cavityfield coupling can effectively control the period of LQU evolution with time.

1. 引言

2. 系统模型

Figure 1. The schematic diagram of the composite cavity optomechanical system

$\begin{array}{c}{H}_{I}=\hslash {g}_{1}\left({a}_{1}{\sigma }_{+}^{1}+{a}_{1}^{†}{\sigma }_{-}^{1}\right)+\frac{\hslash G{g}_{2}}{{\omega }_{m}}\left({a}_{2}^{†}{b}_{2}{\sigma }_{-}^{2}+{a}_{2}{b}_{2}^{†}{\sigma }_{+}^{2}\right)\\ \text{\hspace{0.17em}}\text{ }\text{ }-\frac{\hslash {G}^{2}}{{\omega }_{m}}{\left({a}_{2}^{†}{a}_{2}\right)}^{2}-\frac{\hslash {g}_{2}^{2}}{{\omega }_{m}}\left({a}_{2}^{†}{a}_{2}{\sigma }_{2}^{z}+{\sigma }_{+}^{2}{\sigma }_{-}^{2}\right)\end{array}$ (1)

${H}_{I}={g}_{1}\left({a}_{1}{\sigma }_{+}^{1}+{a}_{1}^{†}{\sigma }_{-}^{1}\right)+{k}_{1}\left({a}_{2}^{†}{b}_{2}{\sigma }_{-}^{2}+{a}_{2}{b}_{2}^{†}{\sigma }_{+}^{2}\right)-{k}_{2}{\left({a}_{2}^{†}{a}_{2}\right)}^{2}-{k}_{3}\left({a}_{2}^{†}{a}_{2}{\sigma }_{z}^{2}+{\sigma }_{+}^{2}{\sigma }_{-}^{2}\right)$ (2)

${|\Psi \left(0\right)〉}_{{a}_{1}{a}_{2}}=\mathrm{cos}\alpha |{e}_{1}{g}_{2}〉+\mathrm{sin}\alpha |{g}_{1}{e}_{2}〉$ (3)

$|\Psi \left(0\right)〉=\mathrm{cos}\alpha |{e}_{1}{g}_{2}〉|{0}_{m}{0}_{f1}{1}_{f2}〉+\mathrm{sin}\alpha |{g}_{1}{e}_{2}〉|{0}_{m}{0}_{f1}{1}_{f2}〉$ (4)

$\begin{array}{c}|\Psi \left(t\right)〉=A\left(t\right)|{e}_{1}{g}_{2}{0}_{m}{0}_{f1}{1}_{f2}〉+B\left(t\right)|{g}_{1}{e}_{2}{0}_{m}{0}_{f1}{1}_{f2}〉+C\left(t\right)|{g}_{1}{g}_{2}{0}_{m}{1}_{f1}{1}_{f2}〉\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+D\left(t\right)|{g}_{1}{e}_{2}{1}_{m}{0}_{f1}{0}_{f2}〉+E\left(t\right)|{e}_{1}{e}_{2}{1}_{m}{0}_{f1}{0}_{f2}〉\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+F\left(t\right)|{g}_{1}{e}_{2}{1}_{m}{1}_{f1}{0}_{f2}〉+S\left(t\right)|{g}_{1}{g}_{2}{0}_{m}{0}_{f1}{1}_{f2}〉\end{array}$ (5)

$i\hslash \frac{\text{d}}{\text{d}t}|\Psi \left(t\right)〉=H|\Psi \left(t\right)〉$ (6)

$\begin{array}{l}{M}_{1}\left(0\right)=\mathrm{cos}\alpha \\ {M}_{2}\left(0\right)=\mathrm{sin}\alpha \\ {M}_{3}\left(0\right)={M}_{4}\left(0\right)={M}_{5}\left(0\right)={M}_{6}\left(0\right)={M}_{7}\left(0\right)=0\end{array}$ (7)

$\begin{array}{l}A\left(t\right)=\frac{a\cdot \mathrm{cos}\left({g}_{1}t\right)}{R}\left[R\mathrm{cos}\frac{Rt}{2}+i\mathrm{sin}\frac{Rt}{2}\left({k}_{2}-2{k}_{3}\right)\right]\mathrm{exp}\left(\frac{i}{2}{k}_{2}t\right)\\ B\left(t\right)=b\cdot \mathrm{exp}\left[i\left({k}_{2}+2{k}_{3}\right)t\right]\\ C\left(t\right)=-\frac{ia\cdot \mathrm{sin}\left({g}_{1}t\right)}{R}\left[R\mathrm{cos}\frac{Rt}{2}+i\mathrm{sin}\frac{Rt}{2}\left({k}_{2}-2{k}_{3}\right)\right]\mathrm{exp}\left(\frac{i}{2}{k}_{2}t\right)\\ D\left(t\right)=-\frac{2ia\cdot \mathrm{cos}\left({g}_{1}t\right)\mathrm{sin}\frac{Rt}{2}}{R}\mathrm{exp}\left(\frac{i}{2}{k}_{2}t\right)\\ E\left(t\right)=-\frac{2a\cdot \mathrm{sin}\left({g}_{1}t\right)\mathrm{sin}\frac{Rt}{2}}{R}\mathrm{exp}\left(\frac{i}{2}{k}_{2}t\right)\\ F\left(t\right)=S\left(t\right)=0\end{array}$ (8)

$\begin{array}{c}{\rho }_{{a}_{1}{a}_{2}}\left(t\right)=A\left(t\right)A{\left(t\right)}^{*}|{e}_{1}{g}_{2}〉〈{e}_{1}{g}_{2}|+A\left(t\right)B{\left(t\right)}^{*}|{e}_{1}{g}_{2}〉〈{e}_{1}{g}_{2}|+B\left(t\right)A{\left(t\right)}^{*}|{e}_{1}{g}_{2}〉〈{e}_{1}{g}_{2}|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+B\left(t\right)B{\left(t\right)}^{*}|{g}_{1}{e}_{2}〉〈{g}_{1}{e}_{2}|+C\left(t\right)C{\left(t\right)}^{*}|{g}_{1}{g}_{2}〉〈{g}_{1}{g}_{2}|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E\left(t\right)E{\left(t\right)}^{*}|{e}_{1}{e}_{2}〉〈{e}_{1}{e}_{2}|+F\left(t\right)F{\left(t\right)}^{*}|{g}_{1}{e}_{2}〉〈{g}_{1}{e}_{2}|\end{array}$ (9)

${\rho }_{{a}_{1}{a}_{2}}\left(t\right)=\left(\begin{array}{cccc}{|E\left(t\right)|}^{2}& 0& 0& 0\\ 0& {|A\left(t\right)|}^{2}& A\left(t\right)B{\left(t\right)}^{*}& 0\\ 0& B\left(t\right)A{\left(t\right)}^{*}& {|B\left(t\right)|}^{2}+{|F\left(t\right)|}^{2}& 0\\ 0& 0& 0& {|C\left(t\right)|}^{2}\end{array}\right)$ (10)

$\epsilon \left(\rho \right)=\underset{i,j}{\sum }\left({K}_{i}\otimes {K}_{j}\right)\rho {\left({K}_{i}\otimes {K}_{j}\right)}^{†}$ (11)

${K}_{0}=\left(\begin{array}{cc}1& 0\\ 0& \sqrt{1-\lambda }\end{array}\right),{K}_{1}=\left(\begin{array}{cc}0& 0\\ 0& \sqrt{\lambda }\end{array}\right)$ (12)

3. 局部量子不确定性与量子纠缠

$\mathcal{U}\left({\rho }_{AB}\right)=\underset{{K}_{A}}{\mathrm{min}}I\left({\rho }_{AB},{K}_{A}\otimes 1{\text{l}}_{B}\right)$ (13)

$\mathcal{U}\left(\rho \right)=1-{\lambda }_{\mathrm{max}}\left\{W\right\}$ (14)

${w}_{ij}=Tr\left\{\sqrt{\rho }\left({\sigma }_{i}\otimes 1{\text{l}}_{B}\right)\sqrt{\rho }\left({\sigma }_{j}\otimes 1{\text{l}}_{B}\right)\right\}$ (15)

$\rho =\left(\begin{array}{cc}\begin{array}{cc}{\rho }_{11}& 0\\ 0& {\rho }_{22}\end{array}& \begin{array}{cc}0& {\rho }_{14}\\ {\rho }_{23}& 0\end{array}\\ \begin{array}{cc}0& {\rho }_{32}\\ {\rho }_{41}& 0\end{array}& \begin{array}{cc}{\rho }_{33}& 0\\ 0& {\rho }_{44}\end{array}\end{array}\right)$ (16)

$\rho =\frac{1}{4}\underset{\alpha ,\beta }{\sum }{T}_{\alpha \beta }{\sigma }_{\alpha }\otimes {\sigma }_{\beta }$ (17)

$\begin{array}{l}{w}_{11}=\left(\sqrt{{\lambda }_{1}}+\sqrt{{\lambda }_{4}}\right)\left(\sqrt{{\lambda }_{2}}+\sqrt{{\lambda }_{3}}\right)+\frac{1}{4}\frac{\left({T}_{11}^{2}-{T}_{22}^{2}\right)+\left({T}_{12}^{2}-{T}_{21}^{2}\right)+\left({T}_{03}^{2}-{T}_{30}^{2}\right)}{\left(\sqrt{{\lambda }_{1}}+\sqrt{{\lambda }_{4}}\right)\left(\sqrt{{\lambda }_{2}}+\sqrt{{\lambda }_{3}}\right)}\\ {w}_{22}=\left(\sqrt{{\lambda }_{1}}+\sqrt{{\lambda }_{4}}\right)\left(\sqrt{{\lambda }_{2}}+\sqrt{{\lambda }_{3}}\right)+\frac{1}{4}\frac{\left({T}_{22}^{2}-{T}_{11}^{2}\right)+\left({T}_{21}^{2}-{T}_{12}^{2}\right)+\left({T}_{30}^{2}-{T}_{03}^{2}\right)}{\left(\sqrt{{\lambda }_{1}}+\sqrt{{\lambda }_{4}}\right)\left(\sqrt{{\lambda }_{2}}+\sqrt{{\lambda }_{3}}\right)}\\ {w}_{33}=\frac{1}{2}\left[{\left(\sqrt{{\lambda }_{1}}+\sqrt{{\lambda }_{4}}\right)}^{2}+{\left(\sqrt{{\lambda }_{2}}+\sqrt{{\lambda }_{3}}\right)}^{2}\right]+\frac{1}{8}\left[\frac{{\left({T}_{30}+{T}_{03}\right)}^{2}-{\left({T}_{11}-{T}_{22}\right)}^{2}-{\left({T}_{12}+{T}_{21}\right)}^{2}}{{\left(\sqrt{{\lambda }_{1}}+\sqrt{{\lambda }_{4}}\right)}^{2}}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{8}\left[\frac{{\left({T}_{03}-{T}_{30}\right)}^{2}-{\left({T}_{11}+{T}_{22}\right)}^{2}-{\left({T}_{12}-{T}_{21}\right)}^{2}}{{\left(\sqrt{{\lambda }_{2}}+\sqrt{{\lambda }_{3}}\right)}^{2}}\right]\\ {w}_{12}={w}_{21}=\frac{1}{2}\frac{{T}_{11}{T}_{21}+{T}_{22}{T}_{12}}{\left(\sqrt{{\lambda }_{1}}+\sqrt{{\lambda }_{4}}\right)\left(\sqrt{{\lambda }_{2}}+\sqrt{{\lambda }_{3}}\right)}\\ {w}_{13}={w}_{31}={w}_{23}={w}_{32}=0\end{array}$ (18)

$C\left(t\right)=2\mathrm{max}\left\{0,|{\rho }_{23}|-\sqrt{{\rho }_{11}{\rho }_{44}},|{\rho }_{14}|-\sqrt{{\rho }_{22}{\rho }_{33}}\right\}$ (19)

4. 数值结果与分析

4.1. 复合腔光力学系统的相位阻尼参数对LQU的影响

Figure 2. The effect of dephasing parameters on LQU of two atomsin the composite cavity optomechanical system, where $\alpha =\pi /4$, ${g}_{1}={g}_{2}=1$, and the dephasing parameters $\lambda =0.3,0.6,0.9$ correspond to the solid, dotted and dashed lines respectively

Figure 3. The effect of dephasing parameters on Concurrence of two atoms in the composite cavity optomechanical system, where $\alpha =\pi /4$, ${g}_{1}={g}_{2}=1$, the dephasing parameters $\lambda =0.3,0.6,0.9$ correspond to the solid, dotted and dashed lines respectively

4.2. 相位阻尼噪声环境下复合腔光力学系统的量子关联调控

Figure 4. The controlling of the atom-cavityfield (J-C model) coupling ${g}_{1}$ and the atom-cavityfield (with vibrating mirror) coupling ${g}_{2}$ on the LQU of two atoms in the composite cavity optomechanical system, where $\alpha =\pi /6$, $\lambda =0.3$, ${g}_{1}={g}_{2}=1$ and ${g}_{1}={g}_{2}=2$ correspond to the solid and dotted lines respectively

Figure 5. The controlling of the atom-cavityfield (with vibrating mirror) coupling ${g}_{2}$ on the LQU of two atoms in the composite cavity optomechanical system, where $t=4.1$, $\lambda =0.5$, $\alpha =\pi /12$, ${g}_{1}=1$

Figure 6. The controlling of the vibrating mirror-cavityfield coupling G on the LQU of two atoms in the composite cavity optomechanical system, where $t=4.1$, $\lambda =0.5$, $\alpha =\pi /12$, ${g}_{1}=1$

Figure 7. The controlling of the atom-cavityfield (J-C model) coupling ${g}_{1}$ on the LQU of two atoms in the composite cavity optomechanical system, where $t=4.1$, $\lambda =0.5$, $\alpha =\pi /12$, ${g}_{2}=1$

5. 结论

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