Behaviors of quantum correlation for accelerated atoms coupled with a fluctuating massless scalar field with a perfectly reflecting boundary

Abstract

In this work, we study the dynamics of quantum correlation for two uniformly accelerated atoms immersed in a bath of fluctuating massless scalar field with a perfectly reflecting plane boundary in the Minkowski vacuum. Firstly, the master equation that governs the system evolution is derived. Then we discuss the generation, revival and decay of quantum correlation for initial zero-correlation state and initial entangled state in various conditions and models. We contrast the behaviors of quantum correlation of accelerated atoms with that of static atoms in a thermal bath at the corresponding Unruh temperature and also make a comparison between behaviors of quantum correlation and entanglement behaviors. We find that behaviors of quantum correlation of uniformly accelerated atoms present some features distinct from that of static atoms immersed in a thermal bath, and behaviors of quantum correlation are more robust than that of entanglement.

Appendix: some relevant results of Ref. [26]

(See Figs. 5, 6, 7 and 8.)

Fig. 5

TMIN as function of $\omega t$ for two atoms in the ground state $|00\rangle$ when two-atom distance $L\rightarrow 0$ for a different temperatures with $\omega z=1$ and b different distances from the boundary with $T/\omega =\frac{1}{2}$. (corresponding to Fig. 2 of Ref. [26])

Fig. 6

TMIN as function of $\omega t$ for two atoms in the ground state $|00\rangle$ when two atoms are placed very far from the boundary ($z\rightarrow \infty$) for a different temperatures with $L\omega =\frac{1}{2}$ and b different two-atom distances with $T/\omega =\frac{3}{10}$. (corresponding to Fig. 3 of Ref. [26])

Fig. 7

TMIN as function of $\omega t$ for two atoms in the ground state $|00\rangle$ for a different temperatures with $L\omega =1$ and $z\omega =2$, b different two-atom distances with $T/\omega =\frac{1}{2}$ and $z\omega =1$ and c different distances from boundary with $T/\omega =\frac{1}{2}$ and $L\omega =\frac{1}{2}$. (corresponding to Fig. 6 of Ref. [26])

Fig. 8

a TMIN as function of $\omega t$ for the initial Werner state $p|\phi _1\rangle \langle \phi _1|+(1-p)\frac{I}{4}$ case when $p=\frac{1}{2}$ for different temperatures with $L\omega =\frac{1}{2}$ and $z\omega =1$. b TMIN as function of $\omega t$ for different two-atom distances with $T/\omega =1$ and $z\omega =1$. c TMIN as function of $\omega t$ for different distances from boundary with $T/\omega =1$ and $\omega L=1$. (corresponding to Fig. 7 of Ref. [26])

In Ref. [26], we study the behaviors of quantum correlation for two atoms immersed in a thermal bath of quantum scalar fields in the presence of a perfectly reflecting plane boundary. In this paper, we obtain the relative coefficients

$$\begin{aligned}&A_1={\omega \over 4\pi }{1+e^{-\beta {\omega }}\over 1-e^{-\beta {\omega }}}\bigg [1-{\sin (2z\omega )\over 2z\omega }\bigg ]\;,\nonumber \\&A_2={\omega \over 4\pi }{1+e^{-\beta {\omega }}\over 1-e^{-\beta {\omega }}}\bigg [{\sin (L\omega )\over {L\omega }} -{\sin (\sqrt{L^2+4z^2}\omega )\over \sqrt{L^2+4z^2}\omega }\bigg ], \nonumber \\&B_1={\omega \over 4\pi }\bigg [1-{\sin (2z\omega )\over 2z\omega }\bigg ]\;,\qquad \qquad \qquad B_2={\omega \over 4\pi }\bigg [{\sin (L\omega )\over {L\omega }}-{\sin (\sqrt{L^2+4z^2}\omega ) \over \sqrt{L^2+4z^2}\omega }\bigg ]. \end{aligned}$$

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In the equilibrium state, we get

$$\begin{aligned} a_{1,1}(\infty )\,=\, a_{2,2}(\infty )\,=\,0,a_{0,3}(\infty )\,=\,a_{3,0}(\infty )\,=\, \frac{1-e^{\omega /T}}{e^{\omega /T}+1},a_{3,3}(\infty )\,=\,\frac{\left( e^{\omega /T}-1\right) ^2}{\left( e^{\omega /T}+1\right) ^2}. \end{aligned}$$

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When two atoms are placed very close to the reflecting boundary ($z\rightarrow 0$), it is found that $A_1=A_2=B_1=B_2=0$. There is no generated quantum correlation.

When temperature $T\rightarrow 0$, we get

$$\begin{aligned} A_1=B_1={\omega \over 4\pi }\bigg [1-{\sin (2z\omega )\over 2z\omega }\bigg ]\;,\quad A_2= B_2={\omega \over 4\pi }\bigg [{\sin (L\omega )\over {L\omega }}-{\sin (\sqrt{L^2+4z^2}\omega )\over \sqrt{L^2+4z^2}\omega }\bigg ]. \end{aligned}$$

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