1. Introduction
Vacuum fluctuations are inevitable according to quantum theory and have been deemed to be the origin of various physical phenomena, such as spontaneous emission [1], Lamb shift [2] and quantum interaction potential between objects [3–5]. It has been shown that the average rate of change of energy of a uniformly moving atom in vacuum, by adopting the symmetric ordering between the operators of atoms and fields in the atom-field interaction Hamiltonian, is attributed to the combined contributions of vacuum fluctuations and the radiation reaction of atoms [6]. This formalism, first proposed by Dalibard, Dupont-Roc and Cohen-Tannoudji, is known as the DDC formalism [7, 8]. With this formalism, it has been shown that when the atom is initially in the higher-level state, the contributions of both vacuum fluctuations and radiation reaction equally lead to a loss of atomic energy and then spontaneous emission occurs. However, for the atom in the ground state, these two contributions exactly cancel out with each other, and as a result, the inertial ground-state atoms are always stable [6].
For the atom in non-inertial motion or in curved spacetime, the radiative behavior increases. When the atom is uniformly accelerated in vacuum, the average rate of change of atomic energy is still contributed by the vacuum fluctuations and the radiation reaction of atoms in the case of the atom coupled linearly to the scalar fields or electromagnetic (EM) fields or gravitational fields [6, 9–12]. However, in contrast to the situation of the inertial ground-state atom, perfect cancellation between the two contributions in the accelerating situation no longer exists. As a result, spontaneous excitation occurs and it is also possible in the case of the atom-field nonlinear interaction, in which an extra contribution involving both the vacuum fluctuations and the radiation reaction appears [13, 14]. When the atom is static outside a Schwarzschild black hole as well as being static or freely falling in de Sitter spacetime, it would also be spontaneously excited in accordance with equivalence principle [15, 16].
The investigations on radiative processes have already been generalized to the generic system composed of multiple particles [17–21]. For example, by studying the identical-particle system with its small dimension compared to the transition wavelength of particles, Dicke first predicted coherence radiation as well as the phenomena of super-radiance and sub-radiance for inter-particle separations much smaller than the transition wavelength [17]. The simplest multi-particle system should be the two-atom system, and much effort has been devoted to studying the collective radiative processes of two entangled atoms in various situations. For example, in [22] the authors studied the spontaneous emission of two identical entangled atoms in unbounded space from the viewpoint of the degree of entanglement and the relative phase, and in [23] the authors investigated the radiative processes of two entangled but not interacting atoms individually coupled to two spatially separated cavities. It is well known that when boundaries are present, the vacuum field modes are modified, and atomic radiative properties as a result of vacuum fluctuations can thus be significantly affected. The results given in [24, 25] show the possibility of controlling the collective transition processes of two entangled atoms by means of the presence of perfect reflecting flat mirrors. Moreover, there is literature exploring the collective radiative processes of two maximally symmetric or antisymmetric entangled atoms, which are accelerated uniformly and synchronously in free space [26–29] or are static near a Schwarzschild black hole [30, 31], in order to disclose the effects caused by the collective non-inertial motion and the curved spacetime background.
In the present paper, we study the collective radiative processes of two accelerated and maximally entangled atoms placed in the vicinity of a reflecting boundary. Note that the same topic has already been studied in the case of two atoms in the monopole interaction with a massless scalar field [32]. However, that is just a toy model. Here we consider a realistic one, i.e. two atoms in the dipole interaction with an EM field, and this model could also help to reveal the effects caused by various orientations of dipole moments of atoms. Moreover, different from the method of calculating response functions used in [32], we here adopt the second-order DDC formalism, which provides a different perspective to understand the underlying physical mechanism for the evolution of energy of a two-atom system. As section 2 shows, this method indicates that the average rate of change of energy of two maximally symmetric or antisymmetric entangled atoms coupled to an EM field is attributed to the contribution of radiation reaction of atoms only but not perturbed by vacuum fluctuations, which is the same as in the case of the atoms coupled to scalar fields [27]. It is worth indicating that the second-order DDC formalism has also been widely used to study the resonant interaction potential of two maximally entangled atoms [33–36]. Our paper is organized as follows. In section 2 , we first derive the second-order DDC formalism for the case of the two-atom system in interaction with an EM field in vacuum, and find that the average rate of change of energy of the two atoms prepared in a maximally entangled state is exclusively the result of the radiation reaction of atoms. In section 3 , we consider two specific cases of the accelerated and entangled atoms placed in free space as well as near a reflecting boundary, and have detailed discussion on their respective average rates of change of the atomic energy. Finally, in section 4 , we give a summary of our study. In this paper, we exploit the units in which ℏ = c = ϵ0 = 1 with ℏ the reduced Plank constant, c the light velocity and ϵ0 the vacuum permittivity.
2. The average rate of change of energy of the two-atom system in interaction with a vacuum EM field
Let us consider two uniformly accelerated atoms in weak interaction with a common fluctuating EM field in vacuum. The two atoms labeled A and B, respectively, are assumed to have the same proper time τ and the accelerating trajectories in terms of τ, as follows:2 ), the expressions of HS(τ), HF(τ) and HI(τ) with regard to τ are,
$\begin{eqnarray}\begin{array}{rcl}{t}_{\xi } & = & \frac{1}{a}{\rm{\sinh }}(a\tau ),\quad {x}_{\xi }=\frac{1}{a}\cosh (a\tau ),\\ {y}_{A} & = & {y}_{B},\quad {z}_{B}-{z}_{A}=L,\end{array}\end{eqnarray}$
with ξ = A, B, a the atomic acceleration and L the interatomic separation, respectively. The atoms are also assumed to have two energy levels, ground states ∣gξ⟩ and excited states ∣eξ⟩, with the energies $-\frac{1}{2}{\omega }_{\xi }$ and $\frac{1}{2}{\omega }_{\xi }$, respectively. The total Hamiltonian of the “atom+field” is expressed as, $\begin{eqnarray}H(\tau )={H}_{S}(\tau )+{H}_{F}(\tau )+{H}_{I}(\tau ),\end{eqnarray}$
where the first two terms on the right are the Hamiltonians of the two atoms and the quantum EM fields, respectively, and the third one describes the atom-field interaction. Specifically in equation ( $\begin{eqnarray}{H}_{S}(\tau )={\omega }_{A}{R}_{3}^{A}(\tau )+{\omega }_{B}{R}_{3}^{B}(\tau ),\end{eqnarray}$
$\begin{eqnarray}{H}_{F}(\tau )=\int {{\rm{d}}}^{3}{\boldsymbol{k}}{\omega }_{{\boldsymbol{k}}}\displaystyle \sum _{\nu =1}^{2}{a}_{{\boldsymbol{k}},\nu }^{\dagger }(t(\tau )){a}_{{\boldsymbol{k}},\nu }(t(\tau ))\frac{{\rm{d}}t}{{\rm{d}}\tau },\end{eqnarray}$
$\begin{eqnarray}{H}_{I}(\tau )=-{{\boldsymbol{D}}}^{A}(\tau )\cdot {\boldsymbol{E}}({{\rm{x}}}_{A}(\tau ))-{{\boldsymbol{D}}}^{B}(\tau )\cdot {\boldsymbol{E}}({{\rm{x}}}_{B}(\tau )).\end{eqnarray}$
Here, ${R}_{3}^{\xi }=\frac{1}{2}\left(| {e}_{\xi }\rangle \langle {e}_{\xi }| -| {g}_{\xi }\rangle \langle {g}_{\xi }| \right)$, ξ = A or B, ${a}_{{\boldsymbol{k}},\nu }^{\dagger }$ and ak,ν are respectively the creation and annihilation operators with the wave vector k, the polarization of field modes ν, and Dξ and E(x) are the electric dipole moment of atom ξ and the electric-field operator, respectively. Furthermore, E(x) in the Coulomb gauge can be expanded by [37], $\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{E}}({\rm{x}}) & = & {\rm{i}}\displaystyle \int {{\rm{d}}}^{3}{\boldsymbol{k}}\sqrt{\frac{{\omega }_{{\boldsymbol{k}}}}{2{(2\pi )}^{3}}}\displaystyle \sum _{\nu =1}^{2}\epsilon ({\boldsymbol{k}},\nu )\\ & & \times \,\left[{a}_{{\boldsymbol{k}},\nu }(t){{\rm{e}}}^{{\rm{i}}{\boldsymbol{k}}\cdot {\boldsymbol{x}}}-{a}_{{\boldsymbol{k}},\nu }^{\dagger }(t){{\rm{e}}}^{-{\rm{i}}{\boldsymbol{k}}\cdot {\boldsymbol{x}}}\right],\end{array}\end{eqnarray}$
with ε(k, ν) the polarization vector which satisfies, $\begin{eqnarray}\displaystyle \sum _{\nu =1}^{2}{\epsilon }_{p}({\boldsymbol{k}},\nu ){\epsilon }_{q}({\boldsymbol{k}},\nu )+\frac{{k}_{p}{k}_{q}}{{{\boldsymbol{k}}}^{2}}={\delta }_{pq},\end{eqnarray}$
and Dξ is expanded by, $\begin{eqnarray}{{\boldsymbol{D}}}^{\xi }=\langle {g}_{\xi }| {{\boldsymbol{D}}}^{\xi }| {e}_{\xi }\rangle {R}_{-}^{\xi }+\langle {e}_{\xi }| {{\boldsymbol{D}}}^{\xi }| {g}_{\xi }\rangle {R}_{+}^{\xi }={{\boldsymbol{d}}}^{\xi }{R}_{-}^{\xi }+{{\boldsymbol{d}}}^{\xi * }{R}_{+}^{\xi },\end{eqnarray}$
with ${R}_{+}^{\xi }=| {e}_{\xi }\rangle \langle {g}_{\xi }| $ and ${R}_{-}^{\xi }=| {g}_{\xi }\rangle \langle {e}_{\xi }| $ the raising and lowering operators, respectively.With the total Hamiltonian equation (2 ), the dynamical variables of the atoms and the fields satisfy the following Heisenberg equation of motion:9 ) can be written as the sum of the free operator ${{ \mathcal O }}^{f}(\tau )$ and the source operator ${{ \mathcal O }}^{s}(\tau )$, of which the respective evolution occurs regardless of the atom-field interaction Hamiltonian and completely results from the atom-field coupling. Then, the free parts of ${R}_{3}^{\xi }$, ${R}_{\pm }^{\xi }$ and ak,ν are obtained as,
$\begin{eqnarray}\frac{{\rm{d}}}{{\rm{d}}\tau }{ \mathcal O }(\tau )={\rm{i}}\left[{H}_{S}(\tau )+{H}_{F}(\tau )+{H}_{I}(\tau ),{ \mathcal O }(\tau )\right],\end{eqnarray}$
where ${ \mathcal O }$ represents ${R}_{3}^{\xi }$, ${R}_{\pm }^{\xi }$, ak,ν and ${a}_{{\boldsymbol{k}},\nu }^{\dagger }$ respectively, and [, ] represents the commutator of two operators. The operator ${ \mathcal O }(\tau )$ on the left in equation ( $\begin{eqnarray}\left\{\begin{array}{l}{R}_{3}^{\xi ,f}(\tau )={R}_{3}^{\xi ,f}({\tau }_{0}),\\ {R}_{\pm }^{\xi ,f}(\tau )={R}_{\pm }^{\xi ,f}({\tau }_{0}){{\rm{e}}}^{\pm {\rm{i}}{\omega }_{\xi }(\tau -{\tau }_{0})},\\ {a}_{{\boldsymbol{k}},\nu }^{f}(t(\tau ))={a}_{{\boldsymbol{k}},\nu }^{f}(t({\tau }_{0})){{\rm{e}}}^{-{\rm{i}}{\omega }_{{\boldsymbol{k}}}(t(\tau )-t({\tau }_{0}))}.\end{array}\right.\end{eqnarray}$
For their source parts, we preserve them to the first order ${{ \mathcal O }}^{s}\approx {{ \mathcal O }}^{(1)}$ with the approximation of the weak atom-field coupling and have the following: $\begin{eqnarray}\left\{\begin{array}{r}{R}_{3}^{\xi ,(1)}(\tau )=-{\rm{i}}\displaystyle \sum _{l}{\displaystyle \int }_{{\tau }_{0}}^{\tau }{\rm{d}}\tau ^{\prime} {E}_{l}^{f}({{\rm{x}}}_{\xi }(\tau ^{\prime} ))\left[{D}_{l}^{\xi ,f}(\tau ^{\prime} ),{R}_{3}^{\xi ,f}(\tau )\right],\\ {R}_{\pm }^{\xi ,(1)}(\tau )=-{\rm{i}}\displaystyle \sum _{l}{\displaystyle \int }_{{\tau }_{0}}^{\tau }{\rm{d}}\tau ^{\prime} {E}_{l}^{f}({{\rm{x}}}_{\xi }(\tau ^{\prime} ))\left[{D}_{l}^{\xi ,f}(\tau ^{\prime} ),{R}_{\pm }^{\xi ,f}(\tau )\right],\\ {a}_{{\boldsymbol{k}},\nu }^{(1)}(t(\tau ))=-{\rm{i}}\displaystyle \sum _{\xi ,l}{\displaystyle \int }_{{\tau }_{0}}^{\tau }{\rm{d}}\tau ^{\prime} {D}_{l}^{\xi ,f}(\tau ^{\prime} )\left[{E}_{l}^{f}({{\rm{x}}}_{\xi }(\tau ^{\prime} )),{a}_{{\boldsymbol{k}},\nu }^{f}(t(\tau ))\right],\end{array}\right.\end{eqnarray}$
with ${E}_{l}^{f}({{\rm{x}}}_{\xi }(\tau ^{\prime} ))$ and ${D}_{l}^{\xi ,f}(\tau ^{\prime} )$ being the lth components of ${{\boldsymbol{E}}}^{{\boldsymbol{f}}}({{\rm{x}}}_{\xi }(\tau ^{\prime} ))$ and ${{\boldsymbol{D}}}^{\xi ,f}(\tau ^{\prime} )$.According to the DDC procedures [7, 8], the change rate of energy of the atom, for example, atom A, is attributed to the combination of the contribution of vacuum fluctuations[vf-contribution] and that of the radiation reaction of atoms [rr-contribution], which result from the free field ${E}_{j}^{f}({{\rm{x}}}_{A}(\tau ))$ and the source field ${E}_{j}^{s}({{\rm{x}}}_{A}(\tau ))$, respectively, i.e., 10 ) and (11 ), the leading [second-order] contributions are shown by,
$\begin{eqnarray}\begin{array}{rcl}\frac{{\rm{d}}{H}_{A}(\tau )}{{\rm{d}}\tau } & = & -{\rm{i}}\displaystyle \sum _{j}{E}_{j}({{\rm{x}}}_{A}(\tau ))\left[{D}_{j}^{A}(\tau ),{\omega }_{A}{R}_{3}^{A}(\tau )\right]\\ & = & {\left(\frac{{\rm{d}}{H}_{A}(\tau )}{{\rm{d}}\tau }\right)}_{vf}+{\left(\frac{{\rm{d}}{H}_{A}(\tau )}{{\rm{d}}\tau }\right)}_{rr},\end{array}\end{eqnarray}$
where, using symmetric operator ordering, $\begin{eqnarray}{\left(\displaystyle \frac{{\rm{d}}{H}_{A}(\tau )}{{\rm{d}}\tau }\right)}_{vf}=-\displaystyle \frac{{\rm{i}}}{2}\displaystyle \sum _{j}\left\{{E}_{j}^{f}({{\rm{x}}}_{A}(\tau )),\left[{D}_{j}^{A}(\tau ),{\omega }_{A}{R}_{3}^{A}(\tau )\right]\right\},\end{eqnarray}$
and $\begin{eqnarray}{\left(\frac{{\rm{d}}{H}_{A}(\tau )}{{\rm{d}}\tau }\right)}_{rr}=-\frac{{\rm{i}}}{2}\displaystyle \sum _{j}\left\{{E}_{j}^{s}({{\rm{x}}}_{A}(\tau )),\left[{D}_{j}^{A}(\tau ),{\omega }_{A}{R}_{3}^{A}(\tau )\right]\right\}.\end{eqnarray}$
Here {, } denotes the anti-commutator of two operators. With equations ( $\begin{eqnarray}\displaystyle \begin{array}{rcl}{\left(\frac{{\rm{d}}{H}_{A}(\tau )}{{\rm{d}}\tau }\right)}_{vf} & = & \frac{{\rm{i}}}{2}\displaystyle \sum _{j,l}{\displaystyle \int }_{{\tau }_{0}}^{\tau }{\rm{d}}\tau ^{\prime} \left\{{E}_{j}^{f}({{\rm{x}}}_{A}(\tau )),{E}_{l}^{f}({{\rm{x}}}_{A}(\tau ^{\prime} ))\right\}\\ & & \times \,\frac{{\rm{d}}}{{\rm{d}}\tau }\left[{D}_{j}^{A,f}(\tau ),{D}_{l}^{A,f}(\tau ^{\prime} )\right],\end{array}\end{eqnarray}$
and $\begin{eqnarray}\displaystyle \begin{array}{rcl}{\left(\frac{{\rm{d}}{H}_{A}(\tau )}{{\rm{d}}\tau }\right)}_{rr} & = & \frac{{\rm{i}}}{2}\displaystyle \sum _{\xi ,j,l}{\displaystyle \int }_{{\tau }_{0}}^{\tau }{\rm{d}}\tau ^{\prime} \left[{E}_{j}^{f}({{\rm{x}}}_{A}(\tau )),{E}_{l}^{f}({{\rm{x}}}_{\xi }(\tau ^{\prime} ))\right]\\ & & \times \,\frac{{\rm{d}}}{{\rm{d}}\tau }\left\{{D}_{j}^{A,f}(\tau ),{D}_{l}^{\xi ,f}(\tau ^{\prime} )\right\}.\end{array}\end{eqnarray}$
Following similar procedures, we can obtain the vf-contribution and the rr-contribution to the change rate of energy of the whole two-atom system, ${\left(\frac{{\rm{d}}{H}_{S}(\tau )}{{\rm{d}}\tau }\right)}_{vf/rr}$. After taking their expectation values over the EM-field vacuum state ∣{0}⟩ and the two-atom state ∣ψS⟩, we then obtain the vf-contribution and rr-contribution to the average rate of change of energy of the two-atom system and they are, respectively, as follows: $\begin{eqnarray}\displaystyle {\left\langle \frac{{\rm{d}}{H}_{S}(\tau )}{{\rm{d}}\tau }\right\rangle }_{vf}=2{\rm{i}}\displaystyle \sum _{\xi ,j,l}{\int }_{{\tau }_{0}}^{\tau }{\rm{d}}\tau ^{\prime} {C}_{jl}^{F}\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi }(\tau ^{\prime} )\right)\frac{{\rm{d}}}{{\rm{d}}\tau }{\chi }_{jl}^{\xi }(\tau ,\tau ^{\prime} ),\end{eqnarray}$
and $\begin{eqnarray}\displaystyle {\left\langle \frac{{\rm{d}}{H}_{S}(\tau )}{{\rm{d}}\tau }\right\rangle }_{rr}=2{\rm{i}}\displaystyle \sum _{\xi ,\xi ^{\prime} ,j,l}{\int }_{{\tau }_{0}}^{\tau }{\rm{d}}\tau ^{\prime} {\chi }_{jl}^{F}\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} )\right)\frac{{\rm{d}}}{{\rm{d}}\tau }{C}_{jl}^{\xi \xi ^{\prime} }(\tau ,\tau ^{\prime} ).\end{eqnarray}$
Here, $\begin{eqnarray}\displaystyle {C}_{jl}^{F}\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi }(\tau ^{\prime} )\right)\equiv \frac{1}{2}\langle \{0\}| \{{E}_{j}^{f}({{\rm{x}}}_{\xi }(\tau )),{E}_{l}^{f}({{\rm{x}}}_{\xi }(\tau ^{\prime} ))\}| \{0\}\rangle ,\end{eqnarray}$
$\begin{eqnarray}\displaystyle {\chi }_{jl}^{F}\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} )\right)\equiv \displaystyle \frac{1}{2}\langle \{0\}| [{E}_{j}^{f}({{\rm{x}}}_{\xi }(\tau )),{E}_{l}^{f}({{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} ))]| \{0\}\rangle ,\end{eqnarray}$
are the symmetric and antisymmetric correlation functions of the EM fields, and, $\begin{eqnarray}\displaystyle \begin{array}{rcl}{\chi }_{jl}^{\xi }(\tau ,\tau ^{\prime} ) & \equiv & \frac{1}{2}\langle {\psi }_{S}| [{D}_{j}^{\xi ,f}(\tau ),{D}_{l}^{\xi ,f}(\tau ^{\prime} )]| {\psi }_{S}\rangle ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\displaystyle \begin{array}{rcl}{C}_{jl}^{\xi \xi ^{\prime} }(\tau ,\tau ^{\prime} ) & \equiv & \displaystyle \frac{1}{2}\langle {\psi }_{S}| \{{D}_{j}^{\xi ,f}(\tau ),{D}_{l}^{\xi ^{\prime} ,f}(\tau ^{\prime} )\}| {\psi }_{S}\rangle ,\end{array}\end{eqnarray}$
are the antisymmetric and symmetric statistical functions of the atoms. The total average rate of change of energy of the two atoms coupled to a fluctuating EM field in vacuum is finally given by, $\begin{eqnarray}\displaystyle \left\langle \frac{{\rm{d}}{H}_{S}(\tau )}{{\rm{d}}\tau }\right\rangle ={\left\langle \frac{{\rm{d}}{H}_{S}(\tau )}{{\rm{d}}\tau }\right\rangle }_{vf}+{\left\langle \frac{{\rm{d}}{H}_{S}(\tau )}{{\rm{d}}\tau }\right\rangle }_{rr}.\end{eqnarray}$
Suppose that the two-atom system is initially prepared in one of the following states: the factorizable states ∣gAgB⟩ and ∣eAeB⟩, and the maximally symmetric or antisymmetric entangled states $| {\psi }_{\pm }\rangle =\frac{1}{\sqrt{2}}\left(| {g}_{A}{e}_{B}\rangle \pm | {e}_{A}{g}_{B}\rangle \right)$. With the help of equations (8 ) and (21 ), we find that when the two-atom system is initially prepared in the maximally entangled states, i.e. ∣ψS⟩ = ∣ψ±⟩, the atomic antisymmetric statistical function ${\chi }_{jl}^{\xi }(\tau ,\tau ^{\prime} )$ is vanishing, and thus equation (17 ) is equal to zero. This suggests that in the case of two atoms interacting with an EM field, the average rate of change of energy of the two-atom system initially prepared in a maximally entangled state is unaffected by vacuum fluctuations and wholly contributed by the radiation reaction of atoms. However, for a factorizable state, ∣gAgB⟩ or ∣eAeB⟩, the average rate of change of energy of the two atoms is attributed to the combined contributions of vacuum fluctuations and the radiation reaction of atoms. The above results in the EM case are shown to be consistent with the case of two atoms in interaction with a massless scalar field [27].
3. The average rate of change of energy of two entangled atoms accelerated in a free or bounded space
In the present study, we mainly focus on the average rate of change of energy of the two-atom system initially prepared in a maximally symmetric or antisymmetric entangled state. According to the above conclusions, the average rate of change of energy is attributed to the contribution of radiation reaction of atoms only, i.e. $\left\langle \frac{{\rm{d}}{H}_{S}(\tau )}{{\rm{d}}\tau }\right\rangle ={\left\langle \frac{{\rm{d}}{H}_{S}(\tau )}{{\rm{d}}\tau }\right\rangle }_{rr}$, and it depends on the antisymmetric correlation function of the EM fields and the symmetric statistical function of the atoms [refer to equation (18 )]. For simplicity, we assume that the two atoms have the same energy-level gap, i.e. ωA = ωB ≡ ω. Thus, according to equations (8 ) and (22 ), we obtain the following equation about the atomic symmetric statistical function ${C}_{jl}^{\xi \xi ^{\prime} }(\tau ,\tau ^{\prime} ):$ 24 ) and (25 ) corresponds to the symmetric state ∣ψ+⟩ and the antisymmetric state ∣ψ−⟩, respectively.
$\begin{eqnarray}\displaystyle \frac{{\rm{d}}}{{\rm{d}}\tau }{C}_{jl}^{\xi \xi ^{\prime} }(\tau ,\tau ^{\prime} )=\left\{\begin{array}{ll}-\omega {d}_{j}^{\xi }{d}_{l}^{\xi ^{\prime} * }\sin [\omega (\tau -\tau ^{\prime} )], & \xi =\xi ^{\prime} \\ \mp \omega {d}_{j}^{\xi }{d}_{l}^{\xi ^{\prime} * }\sin [\omega (\tau -\tau ^{\prime} )], & \xi \ne \xi ^{\prime} ,\end{array}\right.\end{eqnarray}$
and thus the average rate of change of energy of two maximally entangled atoms can be simplified as, $\begin{eqnarray}\displaystyle \begin{array}{rcl}\left\langle \frac{{\rm{d}}{H}_{S}(\tau )}{{\rm{d}}\tau }\right\rangle & = & -2{\rm{i}}\omega \displaystyle \sum _{j,l,\xi =\xi ^{\prime} }{d}_{j}^{\xi }{d}_{l}^{\xi ^{\prime} * }{\displaystyle \int }_{{\tau }_{0}}^{\tau }{\rm{d}}\tau ^{\prime} {\chi }_{jl}^{F}\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} )\right)\sin [\omega (\tau -\tau ^{\prime} )]\\ & & \mp 2{\rm{i}}\omega \displaystyle \sum _{j,l,\xi \ne \xi ^{\prime} }{d}_{j}^{\xi }{d}_{l}^{\xi ^{\prime} * }{\displaystyle \int }_{{\tau }_{0}}^{\tau }{\rm{d}}\tau ^{\prime} {\chi }_{jl}^{F}\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} )\right)\sin [\omega (\tau -\tau ^{\prime} )].\end{array}\end{eqnarray}$
Note that the sign ∓ in the second line on the right of equations (To calculate equation (25 ), we should also obtain the antisymmetric correlation functions of the EM fields to which the two atoms in motion are coupled [equation (20 )], and in the present study we mainly consider the case of two atoms synchronously and uniformly accelerated with the direction of the acceleration perpendicular to that of the interatomic separation. In subsection 3.1 , we first revisit the average rate of change of energy of two accelerated atoms interacting with an EM field in free space using the DDC formalism, and our resultant equation (31 ) is shown to be consistent with the resultant equation (38) given by [29] using the method of calculating response functions. Then, in subsection 3.2 , we further study the situation of two atoms placed in the vicinity of an infinite and perfectly reflecting boundary, which has never previously been discussed.
3.1. The free-space case
We first calculate the antisymmetric correlation functions of the EM fields to which the two accelerated atoms are coupled. In free space, the EM-field two-point correlation functions in the laboratory frame are given by,1 )], we express the two-point correlation functions in the frame of accelerated atoms, with the technique of contour integration and the residue theorem, in the following forms: A.1 )–(A.3 )]. Using equations (27 ) and (28 ) in equation (20 ), the antisymmetric correlation functions of the EM fields are,
$\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl}{\langle \{0\}| {E}_{j}^{f}({{\rm{x}}}_{\xi }){E}_{l}^{f}({\rm{x}}{{\prime} }_{\xi ^{\prime} })| \{0\}\rangle }_{ub} & = & \frac{1}{4{\pi }^{2}}({\partial }_{0}\partial ^{\prime} {\delta }_{jl}-{\partial }_{j}\partial ^{\prime} )\,\displaystyle \frac{1}{{({x}_{\xi }-x{{\prime} }_{\xi ^{\prime} })}^{2}+{({y}_{\xi }-{y}_{\xi ^{\prime} }^{{\prime} })}^{2}+{({z}_{\xi }-z{{\prime} }_{\xi ^{\prime} })}^{2}-{({t}_{\xi }-t{{\prime} }_{\xi ^{\prime} }-{\rm{i}}\epsilon )}^{2}}.\end{array}\end{eqnarray}$
By performing Lorentz transformation on the electric-field operators in the laboratory frame and using the accelerated trajectories of atoms [equation ( $\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl} & & {\langle \{0\}| {E}_{j}^{f}({{\rm{x}}}_{\xi }(\tau )){E}_{l}^{f}({{\rm{x}}}_{\xi }(\tau ^{\prime} ))| \{0\}\rangle }_{ub}=\frac{{\delta }_{jl}}{6\pi }{\displaystyle \int }_{0}^{\infty }{\rm{d}}\omega ({a}^{2}+{\omega }^{2})\omega \left[{\rm{\coth }}\left(\frac{\pi \omega }{a}\right)\cos (\omega {\rm{\Delta }}\tau )-{\rm{i}}\sin (\omega {\rm{\Delta }}\tau )\right],\end{array}\end{eqnarray}$
with ${\rm{\Delta }}\tau =\tau -\tau ^{\prime} $ and δjl being a Kronecker symbol; and for $\xi \ne \xi ^{\prime} $ $\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl} & & {\langle \{0\}| {E}_{j}^{f}({{\rm{x}}}_{\xi }(\tau )){E}_{l}^{f}({{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} ))| \{0\}\rangle }_{ub}\\ & = & {\displaystyle \int }_{0}^{\infty }{\rm{d}}\omega \left[\left(\right.{f}_{jl,ub}^{\xi \xi ^{\prime} }+{\omega }^{2}{g}_{jl,ub}^{\xi \xi ^{\prime} }\left)\right.\sin \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}\frac{aL}{2}\right)+\omega {h}_{jl,ub}^{\xi \xi ^{\prime} }\cos \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}\frac{aL}{2}\right)\right]\\ & & \times \left[{\rm{\coth }}\left(\frac{\pi \omega }{a}\right)\cos (\omega {\rm{\Delta }}\tau )-{\rm{i}}\sin (\omega {\rm{\Delta }}\tau )\right],\end{array}\end{eqnarray}$
where ${f}_{jl,ub}^{\xi \xi ^{\prime} }$, ${h}_{jl,ub}^{\xi \xi ^{\prime} }$ and ${g}_{jl,ub}^{\xi \xi ^{\prime} }$ are respectively the abbreviations of ${f}_{jl,ub}^{\xi \xi ^{\prime} }(a,L)$, ${h}_{jl,ub}^{\xi \xi ^{\prime} }(a,L)$ and ${g}_{jl,ub}^{\xi \xi ^{\prime} }(a,L)$ as the functions of a and L, and the explicit expressions of their nonvanishing components are given in the appendix [see equations ( $\begin{eqnarray}\displaystyle {\chi }_{jl}^{F}{\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi }(\tau ^{\prime} )\right)}_{ub}=-\frac{{\rm{i}}}{6\pi }{\delta }_{jl}[\delta ^{\prime\prime\prime} ({\rm{\Delta }}\tau )-{a}^{2}\delta ^{\prime} ({\rm{\Delta }}\tau )],\end{eqnarray}$
and $\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl}{\chi }_{jl}^{F}{\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} )\right)}_{ub} & = & -\frac{{\rm{i}}}{2}{g}_{jl,ub}^{\xi \xi ^{\prime} }\left[\delta ^{\prime\prime} \left({\rm{\Delta }}\tau +\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{aL}{2}\right)-\delta ^{\prime\prime} \left({\rm{\Delta }}\tau -\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{aL}{2}\right)\right]\\ & & +\frac{{\rm{i}}}{2}{h}_{jl,ub}^{\xi \xi ^{\prime} }\left[\delta ^{\prime} \left({\rm{\Delta }}\tau +\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{aL}{2}\right)+\delta ^{\prime} \left({\rm{\Delta }}\tau -\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{aL}{2}\right)\right]\\ & & +\frac{{\rm{i}}}{2}{f}_{jl,ub}^{\xi \xi ^{\prime} }\left[\delta \left({\rm{\Delta }}\tau +\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{aL}{2}\right)-\delta \left({\rm{\Delta }}\tau -\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{aL}{2}\right)\right],\end{array}\end{eqnarray}$
where δ is the delta function, and $\delta ^{\prime} $, δ″ and $\delta ^{\prime\prime\prime} $ denote the first, second and third derivative with respect to Δτ.Substituting equations (24 ), (29 ) and (30 ) into equation (25 ) and applying equations (A.1 )–(A.3 ), after some straightforward calculations, we finally obtain the average rate of change of energy of two maximally symmetric or antisymmetric entangled atoms that interact with a vacuum EM field and are accelerated in free space:
$\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl}{\left\langle \frac{{\rm{d}}{H}_{S}(\tau )}{{\rm{d}}\tau }\right\rangle }_{ub}^{acc} & = & \mp 2| {{\boldsymbol{d}}}^{A}| | {{\boldsymbol{d}}}^{B}| \left[{{ \mathcal Z }}_{xx}^{AB}{\hat{d}}_{x}^{A}{\hat{d}}_{x}^{B}+{{ \mathcal Z }}_{yy}^{AB}{\hat{d}}_{y}^{A}{\hat{d}}_{y}^{B}+{{ \mathcal Z }}_{zz}^{AB}{\hat{d}}_{z}^{A}{\hat{d}}_{z}^{B}+{{ \mathcal Z }}_{xz}^{AB}({\hat{d}}_{z}^{A}{\hat{d}}_{x}^{B}-{\hat{d}}_{x}^{A}{\hat{d}}_{z}^{B})\right]-| {{\boldsymbol{d}}}^{A}{| }^{2}{{ \mathcal Z }}^{AA}-| {{\boldsymbol{d}}}^{B}{| }^{2}{{ \mathcal Z }}^{BB},\end{array}\end{eqnarray}$
where ∣dξ∣ is the magnitude of the electric dipole of the atom ξ, ${\hat{d}}_{i}^{\xi }$ is a unit vector defined as ${\hat{d}}_{i}^{\xi }={d}_{i}^{\xi }/| {{\boldsymbol{d}}}^{\xi }| $, and, $\begin{eqnarray}\displaystyle \displaystyle \left\{\begin{array}{l}{{ \mathcal Z }}^{AA}={{ \mathcal Z }}^{BB}=\frac{{\omega }^{4}}{6\pi }\left(1+\frac{{a}^{2}}{{\omega }^{2}}\right),\\ {{ \mathcal Z }}_{xx}^{AB}=\frac{4{\omega }^{2}L\sqrt{4+{a}^{2}{L}^{2}}(1+{a}^{2}{L}^{2})\cos \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}\frac{aL}{2}\right)-2\omega (4-4{\omega }^{2}{L}^{2}+2{a}^{2}{L}^{2}-{a}^{2}{\omega }^{2}{L}^{4}+{a}^{4}{L}^{4})\sin \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}\frac{aL}{2}\right)}{\pi {L}^{3}{(4+{a}^{2}{L}^{2})}^{5/2}},\\ {{ \mathcal Z }}_{yy}^{AB}=\frac{{\omega }^{2}L(8+6{a}^{2}{L}^{2}+{a}^{4}{L}^{4})\cos \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}\frac{aL}{2}\right)+\omega \sqrt{4+{a}^{2}{L}^{2}}(-4+4{\omega }^{2}{L}^{2}+{a}^{2}{\omega }^{2}{L}^{4})\sin \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}\frac{aL}{2}\right)}{2\pi {L}^{3}{(4+{a}^{2}{L}^{2})}^{2}}\,,\\ {{ \mathcal Z }}_{zz}^{AB}=\frac{-{\omega }^{2}L\sqrt{4+{a}^{2}{L}^{2}}(16+2{a}^{2}{L}^{2}+{a}^{4}{L}^{4})\cos \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}\frac{aL}{2}\right)+\omega (32+20{a}^{2}{L}^{2}-4{a}^{2}{\omega }^{2}{L}^{4}-{a}^{4}{\omega }^{2}{L}^{6})\sin \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}\frac{aL}{2}\right)}{2\pi {L}^{3}{(4+{a}^{2}{L}^{2})}^{5/2}}\,,\\ {{ \mathcal Z }}_{xz}^{AB}=\frac{{\omega }^{2}aL\sqrt{4+{a}^{2}{L}^{2}}(2-{a}^{2}{L}^{2})\cos \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}\frac{aL}{2}\right)-a\omega (4+4{\omega }^{2}{L}^{2}+4{a}^{2}{L}^{2}+{a}^{2}{\omega }^{2}{L}^{4})\sin \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}\frac{aL}{2}\right)}{\pi {L}^{2}{(4+{a}^{2}{L}^{2})}^{5/2}}.\end{array}\right.\end{eqnarray}$
The accelerating result [equation (31 )] is obviously made up of two types of terms. One is proportional to ∣dA∣2 or ∣dB∣2, i.e. related to the dipole moment of one atom only [see the second line on the right of equation (31 )]. We find that no matter whether the two-atom system is initially prepared in the maximally symmetric or antisymmetric entangled state, these interatomic-separation-independent terms are exactly negative, which always leads to the reduction of the two-atom total energy. The appearance of the characteristic factor $\left(1+\frac{{a}^{2}}{{\omega }^{2}}\right)$ in the expression ${{ \mathcal Z }}^{\xi \xi }$ clearly indicates how these terms vary with the atomic acceleration a. Take the system composed of two hydrogen atoms with ω ~ 1015s−1 as an example. It is found that this acceleration-induced effect is remarkable for a ~ 1024m/s2 at least. This situation differs markedly from that of two accelerated atoms coupled to a massless scalar field since the separation-independent terms there have nothing to do with the acceleration and naturally are uncontrolled by atomic non-inertial motion [27]. In fact, the acceleration-dependent factor $\left(1+\frac{{a}^{2}}{{\omega }^{2}}\right)$ is unique for the EM-field case, and it also appears in the contribution of vacuum fluctuations to the energy-level shifts of an accelerated hydrogen atom interacting with the EM field [38]. The other type of terms in the accelerating result (31 ) are related to the dipole moments of both atoms, i.e. those proportional to ∣dA∣∣dB∣ in the first line on the right of equation (31 ). The sign ∓ in these terms corresponds to the two-atom state ∣ψ+⟩ and ∣ψ−⟩, respectively, showing that the symmetric and antisymmetric entangled states play opposite roles in the terms related to both atoms. For example, for some certain interatomic separation, atomic acceleration and polarization of atoms, if these terms help to enhance the total energy of two symmetric entangled atoms, the total energy of the two-atom system in the antisymmetric entangled state must be weakened with the same magnitude of the average variation rate. It can also be seen that these terms have obviously oscillatory behavior, which generally depends on the values of the interatomic separation and atomic acceleration, as well as the orientations of atomic dipole moments. That is to say, for some certain dipole orientations, the values of the interatomic separation and atomic acceleration, the terms related to both atoms are probably vanishing. Notably, when the atoms are in inertial motion, i.e. when a → 0, ${{ \mathcal Z }}_{xz}^{AB}$ in equation (32 ) always vanishes regardless of the values of interatomic separation, and thus the terms proportional to ${\hat{d}}_{z}^{A}{\hat{d}}_{x}^{B}$ or ${\hat{d}}_{x}^{A}{\hat{d}}_{z}^{B}$, which are led by two cross-polarizable atoms, exist in the accelerating result (31 ) but disappear in the inertial-atom case. This suggests that when the dipole moment of one atom is oriented along the interatomic separation L (z-axis) and the other orthogonally to the direction of L (the accelerating direction, x-axis), the average rate of change of energy of two inertial atoms is vanishing, but the accelerated motion of atoms can lead to the variation of energy of these two cross-polarizable atoms. The phenomenon appears since the EM fields, which the accelerated atoms feel in their proper frames, are very different to those the inertial atoms feel, and the cross-components of the EM-field two-point correlation functions ${\langle \{0\}| {E}_{x}^{f}({{\rm{x}}}_{A}(\tau )){E}_{z}^{f}({{\rm{x}}}_{B}(\tau ^{\prime} ))| \{0\}\rangle }_{ub}$ and ${\langle \{0\}| {E}_{z}^{f}({{\rm{x}}}_{A}(\tau )){E}_{x}^{f}({{\rm{x}}}_{B}(\tau ^{\prime} ))| \{0\}\rangle }_{ub}$ are nonzero in the accelerating case [refer to equations (28 ) and (A.1 )–(A.3 )] but vanish in the inertial case. Then, the average rate of change of energy of two inertial entangled atoms in vacuum, under the limit of a → 0, is given by,
$\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl}{\left\langle \frac{{\rm{d}}{H}_{S}(\tau )}{{\rm{d}}\tau }\right\rangle }_{{\rm{u}}{\rm{b}}}^{{\rm{i}}{\rm{n}}{\rm{e}}{\rm{r}}} & = & \mp \frac{| {{\boldsymbol{d}}}^{A}| | {{\boldsymbol{d}}}^{B}| \omega }{2\pi {L}^{3}}\left\{\left({\hat{d}}_{x}^{A}{\hat{d}}_{x}^{B}+{\hat{d}}_{y}^{A}{\hat{d}}_{y}^{B}\right)\left[\omega L\cos \left(\omega L\right)+({\omega }^{2}{L}^{2}-1)\sin \left(\omega L\right)\right]\right.\\ & & \left.-2{\hat{d}}_{z}^{A}{\hat{d}}_{z}^{B}\left[\omega L\cos \left(\omega L\right)-\sin \left(\omega L\right)\right]\right\}-\frac{{\omega }^{4}}{6\pi }\left(| {{\boldsymbol{d}}}^{A}{| }^{2}+| {{\boldsymbol{d}}}^{B}{| }^{2}\right).\end{array}\end{eqnarray}$
In the above equation, when the interatomic separation L is very small compared to the atomic characteristic transition wavelength ∝ ω−1, the average rate of change of energy of the identical-atom system in the symmetric entangled state is the sum of that for two individual atoms, but for the antisymmetric entangled system it is vanishing, showing the well-known phenomena of super-radiance and sub-radiance.3.2. The cases in the presence of a boundary
We next pay attention to the case of two entangled atoms accelerated near an infinite and completely reflecting boundary. Assume that the two atoms are moving parallel to the boundary plane so that the distance to the boundary plane from the atom system is constant. There are two different but basic configurations of the atoms with respect to the boundary. Correspondingly, the interatomic separation could be vertical to or parallel to the boundary plane, and we refer to it as the vertical-alignment and parallel-alignment cases, respectively.
3.2.1. The vertical-alignment case
For the vertical-alignment ('⊥') case shown in figure 1 (left), the acceleration of atoms is along the x − axis, and the two atoms and the boundary plane are located on the z − axis and the 'xoy' plane, respectively. The distance to the boundary is R for atom A (the closer atom) and for atom B it is (R + L).
Figure 1. Two accelerated atoms are aligned vertically to (left) or parallel to (right) a reflecting boundary, with the interatomic separation being L and the distance to the boundary from atom A being R. |
In order to calculate the antisymmetric correlation functions of the EM fields in the presence of an infinite and completely reflecting boundary, let us recall that the two-point function of the vector potential Aμ(x) can be obtained with the help of the method of images as,26 ), and,
$\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl} & & {\langle \{0\}| {A}^{\mu }({\rm{x}}){A}^{\nu }({\rm{x}}^{\prime} )| \{0\}\rangle }_{\perp }={\langle \{0\}| {A}^{\mu }({\rm{x}}){A}^{\nu }({\rm{x}}^{\prime} )| \{0\}\rangle }_{ub}+{\langle \{0\}| {A}^{\mu }({\rm{x}}){A}^{\nu }({\rm{x}}^{\prime} )| \{0\}\rangle }_{\perp ,b}.\end{array}\end{eqnarray}$
Here, $\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl} & & {\langle \{0\}| {A}^{\mu }({\rm{x}}){A}^{\nu }({\rm{x}}^{\prime} )| \{0\}\rangle }_{ub}=\,-\displaystyle \frac{{\eta }^{\mu \nu }}{4{\pi }^{2}[{(x-x^{\prime} )}^{2}+{(y-y^{\prime} )}^{2}+{(z-z^{\prime} )}^{2}-{(t-t^{\prime} -{\rm{i}}\epsilon )}^{2}]}\end{array},\end{eqnarray}$
is the two-point function in free space, and, $\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl} & & {\langle \{0\}| {A}^{\mu }({\rm{x}}){A}^{\nu }({\rm{x}}^{\prime} )| \{0\}\rangle }_{\perp ,b}=\displaystyle \frac{{\eta }^{\mu \nu }+2{r}_{\perp }^{\mu }{r}_{\perp }^{\nu }}{4{\pi }^{2}[{(x-x^{\prime} )}^{2}+{(y-y^{\prime} )}^{2}+{(z+z^{\prime} )}^{2}-{(t-t^{\prime} -{\rm{i}}\epsilon )}^{2}]}\end{array},\end{eqnarray}$
is the modification caused by the presence of the boundary at a distance z from the two-atom system, with ημν = diag(1, − 1, − 1, − 1) and ${r}_{\perp }^{\mu }=(0,0,0,1)$. Then, the EM-field two-point correlation function takes the following form [11]: $\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl} & & {\langle \{0\}| {E}_{j}^{f}({{\rm{x}}}_{\xi }){E}_{l}^{f}({\rm{x}}{{\prime} }_{\xi ^{\prime} })| \{0\}\rangle }_{\perp }={\langle \{0\}| {E}_{j}^{f}({{\rm{x}}}_{\xi }){E}_{l}^{f}({\rm{x}}{{\prime} }_{\xi ^{\prime} })| \{0\}\rangle }_{ub}+{\langle \{0\}| {E}_{j}^{f}({{\rm{x}}}_{\xi }){E}_{l}^{f}({\rm{x}}{{\prime} }_{\xi ^{\prime} })| \{0\}\rangle }_{\perp ,b},\end{array}\end{eqnarray}$
where ${\langle \{0\}| {E}_{j}^{f}({{\rm{x}}}_{\xi }){E}_{l}^{f}({\rm{x}}{{\prime} }_{\xi ^{\prime} })| \{0\}\rangle }_{ub}$ is the two-point function in free space and given by equation ( $\begin{eqnarray}\displaystyle \begin{array}{rcl}{\langle \{0\}| {E}_{j}^{f}({{\rm{x}}}_{\xi }){E}_{l}^{f}({\rm{x}}{{\prime} }_{\xi ^{\prime} })| \{0\}\rangle }_{\perp ,b} & = & -\frac{1}{4{\pi }^{2}}\left[({\delta }_{jl}-2{r}_{\perp ,j}{r}_{\perp ,l}){\partial }_{0}\partial ^{\prime} -{\partial }_{j}\partial ^{\prime} )\right]\\ & & \times \,\displaystyle \frac{1}{{({x}_{\xi }-x{{\prime} }_{\xi ^{\prime} })}^{2}+{({y}_{\xi }-y{{\prime} }_{\xi ^{\prime} })}^{2}+{({z}_{\xi }+z{{\prime} }_{\xi ^{\prime} })}^{2}-{({t}_{\xi }-t{{\prime} }_{\xi ^{\prime} }-{\rm{i}}\epsilon )}^{2}}.\end{array}\end{eqnarray}$
According to the definition of the field antisymmetric correlation function [equation (20 )], it can also be written as the sum of the part in free space and the boundary-caused part, i.e.29 ) and (30 ), and 40 ), the boundary-dependent two-point functions, after the Lorentz transformation and Fourier transforms, can be written as,40 ), we then obtain the boundary-dependent antisymmetric correlation function of the fields as,
$\begin{eqnarray}\displaystyle {\chi }_{jl}^{F}{\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} )\right)}_{\perp }={\chi }_{jl}^{F}{\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} )\right)}_{ub}+{\chi }_{jl}^{F}{\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} )\right)}_{\perp ,b}.\end{eqnarray}$
For ${\chi }_{jl}^{F}{\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} )\right)}_{ub}$, please refer to equations ( $\begin{eqnarray}\displaystyle {\chi }_{jl}^{F}{\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} )\right)}_{\perp ,b}=\frac{1}{2}{\langle \{0\}| [{E}_{j}^{f}({{\rm{x}}}_{\xi }(\tau )),{E}_{l}^{f}({{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} ))]| \{0\}\rangle }_{\perp ,b}.\end{eqnarray}$
In equation ( $\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl} & & {\langle \{0\}| {E}_{j}^{f}({{\rm{x}}}_{\xi }){E}_{l}^{f}({\rm{x}}{{\prime} }_{\xi ^{\prime} })| \{0\}\rangle }_{\perp ,b}={\displaystyle \int }_{0}^{\infty }{\rm{d}}\omega \left[{\rm{\coth }}\left(\frac{\pi \omega }{a}\right)\cos (\omega {\rm{\Delta }}\tau )-{\rm{i}}\sin (\omega {\rm{\Delta }}\tau )\right]\left[\Space{0ex}{3.87ex}{0ex}\left({f}_{jl,\perp }^{\xi \xi ^{\prime} }+{\omega }^{2}{g}_{jl,\perp }^{\xi \xi ^{\prime} }\right)\right.\\ & & \,\left.\times \,\sin \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}\frac{a{{ \mathcal D }}_{\perp }^{\xi \xi ^{\prime} }}{2}\right)+\omega {h}_{jl,\perp }^{\xi \xi ^{\prime} }\cos \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}\frac{a{{ \mathcal D }}_{\perp }^{\xi \xi ^{\prime} }}{2}\right)\right],\end{array}\end{eqnarray}$
where ${{ \mathcal D }}_{\perp }^{AA}=2R$, ${{ \mathcal D }}_{\perp }^{BB}=2(R+L)$, ${{ \mathcal D }}_{\perp }^{AB}={{ \mathcal D }}_{\perp }^{BA}=L+2R$, and the concrete expressions of ${f}_{jl,\perp }^{\xi \xi ^{\prime} }$, ${h}_{jl,\perp }^{\xi \xi ^{\prime} }$ and ${g}_{jl,\perp }^{\xi \xi ^{\prime} }$ are shown in the appendix. With equation ( $\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl} & & {\chi }_{jl}^{F}{\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} )\right)}_{\perp ,b}\\ & = & \frac{{\rm{i}}}{2}{g}_{jl,\perp }^{\xi \xi ^{\prime} }\left[\delta ^{\prime\prime} \left({\rm{\Delta }}\tau -\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{a{{ \mathcal D }}_{\perp }^{\xi \xi ^{\prime} }}{2}\right)-\delta ^{\prime\prime} \left({\rm{\Delta }}\tau +\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{a{{ \mathcal D }}_{\perp }^{\xi \xi ^{\prime} }}{2}\right)\right]\\ & & +\,\frac{{\rm{i}}}{2}{h}_{jl,\perp }^{\xi \xi ^{\prime} }\left[\delta ^{\prime} \left({\rm{\Delta }}\tau -\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{a{{ \mathcal D }}_{\perp }^{\xi \xi ^{\prime} }}{2}\right)+\delta ^{\prime} \left({\rm{\Delta }}\tau +\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{a{{ \mathcal D }}_{\perp }^{\xi \xi ^{\prime} }}{2}\right)\right]\\ & & -\,\frac{{\rm{i}}}{2}{f}_{jl,\perp }^{\xi \xi ^{\prime} }\left[\delta \left({\rm{\Delta }}\tau -\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{a{{ \mathcal D }}_{\perp }^{\xi \xi ^{\prime} }}{2}\right)-\delta \left({\rm{\Delta }}\tau +\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{a{{ \mathcal D }}_{\perp }^{\xi \xi ^{\prime} }}{2}\right)\right].\end{array}\end{eqnarray}$
Inserting equations (24 ), (29 ), (30 ), (39 ) and (42 ) into equation (25 ), since the average rate of change of energy of the atoms, as equation (25 ) shows, is proportional to the antisymmetric correlation functions of the fields, the average rate of change of energy of the two-atom system placed near a boundary should be expressed as the combination of the free-space result and the boundary-induced modification, i.e.31 ) and for the case of the two atoms aligned vertically to and accelerated parallel to the boundary, the boundary-induced modification reads:32 ), (A.4 ) and (A.5 ), it is clear that the average rate of change of energy of the accelerated entangled two-atom system in the presence of the boundary [equation (43 )] generally depends on the initial two-atom state (the symmetric or antisymmetric entangled state), the orientations of atomic dipole moments reflected by ${\hat{d}}_{j}^{\xi }$, as well as the relative scales among the atom-boundary distance R, the interatomic separation L and the inverse of atomic acceleration a−1. We find that when R is infinite compared with L and a−1, the boundary-dependent modifications (44 ) are vanishing through the use of equations (A.4 ) and (A.5 ), and thus this indicates that the average rate of change of energy of the two-atom system placed very far away from the boundary is exactly equal to the result in the free-space case.
$\begin{eqnarray}\displaystyle {\left\langle \frac{{\rm{d}}{H}_{S}(\tau )}{{\rm{d}}\tau }\right\rangle }_{\perp }^{{\rm{a}}{\rm{c}}{\rm{c}}}={\left\langle \frac{{\rm{d}}{H}_{S}(\tau )}{{\rm{d}}\tau }\right\rangle }_{ub}^{{\rm{a}}{\rm{c}}{\rm{c}}}+{\left\langle \frac{{\rm{d}}{H}_{S}(\tau )}{{\rm{d}}\tau }\right\rangle }_{\perp ,b}^{{\rm{a}}{\rm{c}}{\rm{c}}}.\end{eqnarray}$
The free-space result is exactly given by equation ( $\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl}{\left\langle \frac{{\rm{d}}{H}_{S}(\tau )}{{\rm{d}}\tau }\right\rangle }_{\perp ,b}^{{\rm{a}}{\rm{c}}{\rm{c}}} & = & \mp 2| {{\boldsymbol{d}}}^{A}| | {{\boldsymbol{d}}}^{B}| \left[{{ \mathcal P }}_{xx}^{AB}{\hat{d}}_{x}^{A}{\hat{d}}_{x}^{B}+{{ \mathcal P }}_{yy}^{AB}{\hat{d}}_{y}^{A}{\hat{d}}_{y}^{B}+{{ \mathcal P }}_{zz}^{AB}{\hat{d}}_{z}^{A}{\hat{d}}_{z}^{B}+{{ \mathcal P }}_{xz}^{AB}({\hat{d}}_{z}^{A}{\hat{d}}_{x}^{B}+{\hat{d}}_{x}^{A}{\hat{d}}_{z}^{B})\right]\\ & & \mp | {{\boldsymbol{d}}}^{A}{| }^{2}\left[{{ \mathcal P }}_{xx}^{AA}{\hat{d}}_{x}^{A}{\hat{d}}_{x}^{A}+{{ \mathcal P }}_{yy}^{AA}{\hat{d}}_{y}^{A}{\hat{d}}_{y}^{A}+{{ \mathcal P }}_{zz}^{AA}{\hat{d}}_{z}^{A}{\hat{d}}_{z}^{A}+2{{ \mathcal P }}_{xz}^{AA}{\hat{d}}_{x}^{A}{\hat{d}}_{z}^{A}\right]\\ & & \mp | {{\boldsymbol{d}}}^{B}{| }^{2}\left[{{ \mathcal P }}_{xx}^{BB}{\hat{d}}_{x}^{B}{\hat{d}}_{x}^{B}+{{ \mathcal P }}_{yy}^{BB}{\hat{d}}_{y}^{B}{\hat{d}}_{y}^{B}+{{ \mathcal P }}_{zz}^{BB}{\hat{d}}_{z}^{B}{\hat{d}}_{z}^{B}+2{{ \mathcal P }}_{xz}^{BB}{\hat{d}}_{x}^{B}{\hat{d}}_{z}^{B}\right],\end{array}\end{eqnarray}$
with $\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl}{{ \mathcal P }}_{jl}^{\xi \xi ^{\prime} } & = & \omega \left(\right.{f}_{jl,\perp }^{\xi \xi ^{\prime} }+{\omega }^{2}{g}_{jl,\perp }^{\xi \xi ^{\prime} }\left)\right.\sin \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}\frac{a{{ \mathcal D }}_{\perp }^{\xi \xi ^{\prime} }}{2}\right)+{\omega }^{2}{h}_{jl,\perp }^{\xi \xi ^{\prime} }\cos \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}\frac{a{{ \mathcal D }}_{\perp }^{\xi \xi ^{\prime} }}{2}\right).\end{array}\end{eqnarray}$
With the use of equations (For the case of the two-atom system placed very near the boundary, i.e. when the atom-boundary distance R is much smaller than both the interatomic separation L and the inverse of atomic acceleration a−1, one can easily obtain the corresponding result with the limit of R → 0. Here, we do not show the explicit expression since it is too lengthy, but give the detailed comparison, in terms of the terms related to the dipole moment of only one atom and those of both atoms, between the free-space case and the limiting vertical-alignment case. When R → 0, the terms only related to atom A are $-\frac{| {{\boldsymbol{d}}}^{A}{| }^{2}{\omega }^{4}}{3\pi }\left(1+\frac{{a}^{2}}{{\omega }^{2}}\right){\hat{d}}_{z}^{A}{\hat{d}}_{z}^{A}$ for the system in the symmetric entangled state, but are $-\frac{| {{\boldsymbol{d}}}^{A}{| }^{2}{\omega }^{4}}{3\pi }\left(1+\frac{{a}^{2}}{{\omega }^{2}}\right)({\hat{d}}_{x}^{A}{\hat{d}}_{x}^{A}+{\hat{d}}_{y}^{A}{\hat{d}}_{y}^{A})$ for the antisymmetric entangled state. By comparing these with the corresponding terms in the free-space case [refer to the first term in the second line of equation (31 )], we find that the terms in the limiting case are significantly affected by different orientations of the dipole moment of atom A, which also depends on the initial two-atom state. Concretely, when atom A placed extremely close to the boundary is polarizable parallel to the boundary plane (the 'xoy' plane), e.g. ${\hat{{\boldsymbol{d}}}}^{A}=\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0\right)$, the terms only related to atom A are vanishing for two symmetric entangled atoms while for two antisymmetric entangled atoms are double the corresponding terms in the free-space case. However, when atom A is polarizable vertically to the boundary plane, i.e. ${\hat{{\boldsymbol{d}}}}^{A}=(0,0,1)$, they vanish for the antisymmetric state while for the symmetric state are twice that of their free-space counterpart. This phenomenon, which appears as an infinite and completely reflecting boundary plane, is considered here, and the vertical- and parallel-to-boundary components of the EM fields reflected on this boundary plane are zero and double, respectively. In this limiting case, the terms only related to atom B, in sharp contrast to those in the free-space case, which are independent of the interatomic separation [refer to the second term in the second line of equation (31 )], exhibiting remarkably oscillatory behavior in the form of $\sin \left(\frac{2\omega }{a}{\sinh }^{\,-1}aL\right)$ and $\cos \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{\,-1}aL\right)$. Finally, we find in the limiting case that the terms related to both atoms depend more crucially on the orientations of dipole moments of both atoms since they are found to be proportional to ${\hat{d}}_{z}^{A}{\hat{d}}_{z}^{B}$ and ${\hat{d}}_{z}^{A}{\hat{d}}_{x}^{B}$ when R → 0. Specifically in the case of the two atoms vertically aligned very near the boundary, only when the closer atom is polarizable along the direction vertical to the boundary and while the other one is along the direction vertical to the boundary or the acceleration of atoms, the terms related to both atoms are nonvanishing and contribute to the average rate of change of energy of the two-atom system.
Some figures are given to illustrate the situations for general values of the boundary-atom distance R. We take the two atoms prepared in the symmetric entangled state ∣ψ+⟩ as an example and assume the interatomic separation satisfying L ~ 10ω−1. Based on the result (43 ), there are five different but basic orientations of the dipole moments of two atoms contributing to the average rate of change of energy of the accelerated atoms in the vertical-alignment case. The two atoms could be identically polarizable, with their dipole moments oriented along the acceleration or interatomic-separation direction, or the direction vertical to both the former, i.e. correspondingly ${\hat{{\boldsymbol{d}}}}^{A}={\hat{{\boldsymbol{d}}}}^{B}=(1,0,0)$ or (0, 0, 1) or (0, 1, 0) in equation (43 ). The atoms could also be cross-polarizable, with the dipole moments oriented along the acceleration and interatomic-separation directions, respectively, i.e. ${\hat{{\boldsymbol{d}}}}^{A}=(1,0,0)$, ${\hat{{\boldsymbol{d}}}}^{B}=(0,0,1)$ or ${\hat{{\boldsymbol{d}}}}^{A}=(0,0,1)$, ${\hat{{\boldsymbol{d}}}}^{B}=(1,0,0)$. In figure 2, we compare how the average rates of change of energy of two accelerated atoms change with the boundary-atom distance in the five different polarization cases. From this figure, it can be seen that the average rates of change of energy are always negative. When the atom closer to the boundary [atom A] is polarizable vertically to the boundary, the absolute values of the average rates of change of energy first decrease remarkably as R grows from zero, while they significantly increase when the dipole moment of the closer atom is oriented parallel to the boundary. With the increase of R, the average rates of change of energy in all five polarization cases then go through several oscillations and finally remain stable. Figure 3 shows, for the fixed orientations of atomic dipole moments, how the average rates of change of energy of the atoms are affected by the atomic acceleration, and it is apparently shown that the atomic acceleration can significantly enhance the absolute values of the average rate of change of energy of the two-atom system.
Figure 2. Average rate of change of energy of two vertically aligned and accelerated symmetric entangled atoms in the unit of $\frac{| {\boldsymbol{d}}{| }^{2}{\omega }^{4}}{\pi }$ is plotted as a function of the dimensionless atom-boundary distance. Acceleration and interatomic separation are along the x − axis and z − axis, respectively, and the boundary is on the 'xoy' plane. Here, ωL = 10 and aL = 2. Solid red, blue and green lines correspond to the cases of the two atoms identically polarizable along the x − axis, y − axis and z − axis, respectively. Cases of one atom polarizable along the direction of acceleration and the other along the interatomic-separation direction are described by the dashed gray and dotted black lines with atom A (closer atom) polarizable along the x − axis and z − axis, respectively. |
Figure 3. Average rate of change of energy of two vertically aligned and accelerated symmetric entangled atoms in the unit of $\frac{| {\boldsymbol{d}}{| }^{2}{\omega }^{4}}{\pi }$ is plotted as a function of the dimensionless atom-boundary distance. Here, ωL = 10, and the two atoms are polarizable both along the direction of acceleration (left) and interatomic separation (right). Solid line describes the case of two inertial atoms. Dashed red, dotted blue and dot-dashed green lines correspond to aL = 1, aL = 6 and aL = 10, respectively. |
3.2.2. The parallel-alignment case
We then turn to the case of the two atoms aligned parallel to the boundary [the parallel-alignment ('∥') case]. As figure 1 (right) shows, we do not alter the directions of acceleration and separation of the two atoms [still along the x − axis and z − axis, respectively], but rearrange the boundary plane so that it now coincides with the 'xoz' plane, with the distance to each atom being both R. Only under this configuration, with the average rate of change of energy of two accelerated entangled atoms in the parallel-alignment case, as the following equation (49 ) shows, could it also be expressed as the free-space result equation (31 ) plus a modification caused by the boundary, and thus it is convenient to compare the boundary-dependent-only modifications in the vertical- and parallel-alignment cases. Similar to the vertical-alignment case, the two-point correlation function is the sum of the free-space two-point function ${\langle \{0\}| {E}_{j}^{f}({{\rm{x}}}_{\xi }){E}_{l}^{f}({\rm{x}}{{\prime} }_{\xi ^{\prime} })| \{0\}\rangle }_{ub}$ [equation (26 )] and that induced by the boundary, and the boundary-dependent part in this case now reads as follows:29 ) and (30 ), and with the definition ${\chi }_{jl}^{F}{\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} )\right)}_{\parallel ,b}\,=\frac{1}{2}{\langle \{0\}| [{E}_{j}^{f}({{\rm{x}}}_{\xi }(\tau )),{E}_{l}^{f}({{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} ))]| \{0\}\rangle }_{\parallel ,b}$ we can further obtain:A.6 )–(A.9 ). Using equations (25 ), (24 ), (29 ), (30 ), (47 ) and (48 ), we finally obtain the average rate of change of energy of two accelerated entangled atoms placed parallel to the boundary as,31 )] and the boundary-dependent modifications, and the latter reads as follows:44 )], there also exist in the parallel-alignment case [equation (50 )], besides the terms proportional to ${\hat{d}}_{x}^{A}{\hat{d}}_{y}^{B}$ (or ${\hat{d}}_{y}^{A}{\hat{d}}_{x}^{B}$), other types of cross-terms $\unicode{x0007E}{\hat{d}}_{x}^{A}{\hat{d}}_{z}^{B}$ (or ${\hat{d}}_{z}^{A}{\hat{d}}_{x}^{B}$) and $\unicode{x0007E}{\hat{d}}_{y}^{A}{\hat{d}}_{z}^{B}$ (or ${\hat{d}}_{z}^{A}{\hat{d}}_{y}^{B}$). That is to say, in the case of the two atoms placed parallel to the boundary, the average rate of change of energy of the system could generally be nonvanishing when the two atoms are polarizable both parallel to the boundary (along the acceleration and interatomic-separation directions, respectively), or one atom is polarizable parallel to the boundary (along the interatomic-separation direction) and the other vertical to the boundary.
$\begin{eqnarray}\displaystyle \displaystyle \begin{array}{l}{\langle \{0\}| {E}_{j}^{f}({{\rm{x}}}_{\xi }){E}_{l}^{f}({\rm{x}}{{\prime} }_{\xi ^{\prime} })| \{0\}\rangle }_{\parallel ,b}=-\displaystyle \frac{1}{4{\pi }^{2}}\left[({\delta }_{jl}-2{r}_{\parallel ,j}{r}_{\parallel ,l}){\partial }_{0}\partial ^{\prime} -{\partial }_{j}\partial ^{\prime} \right]\\ \,\times \,\displaystyle \frac{1}{{({x}_{\xi }-x{{\prime} }_{\xi ^{\prime} })}^{2}+{({y}_{\xi }+y{{\prime} }_{\xi ^{\prime} })}^{2}+{({z}_{\xi }-z{{\prime} }_{\xi ^{\prime} })}^{2}-{({t}_{\xi }-t{{\prime} }_{\xi ^{\prime} }-i\epsilon )}^{2}},\end{array}\end{eqnarray}$
with r∥ = (0, 1, 0). Correspondingly, the antisymmetric correlation functions of the EM fields are written as, $\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl} & & {\chi }_{jl}^{F}{\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} )\right)}_{\parallel }={\chi }_{jl}^{F}{\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} )\right)}_{ub}+{\chi }_{jl}^{F}{\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} )\right)}_{\parallel ,b},\end{array}\end{eqnarray}$
where ${\chi }_{jl}^{F}{\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} )\right)}_{ub}$ are given by equations ( $\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl} & & {\chi }_{jl}^{F}{\left({{\rm{x}}}_{\xi }(\tau ),{{\rm{x}}}_{\xi ^{\prime} }(\tau ^{\prime} )\right)}_{\parallel ,b}\\ & = & \frac{{\rm{i}}}{2}{g}_{jl,\parallel }^{\xi \xi ^{\prime} }\left[\delta ^{\prime\prime} \left({\rm{\Delta }}\tau -\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{a{{ \mathcal D }}_{\parallel }^{\xi \xi ^{\prime} }}{2}\right)-\delta ^{\prime\prime} \left({\rm{\Delta }}\tau +\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{a{{ \mathcal D }}_{\parallel }^{\xi \xi ^{\prime} }}{2}\right)\right]\\ & & \quad +\frac{{\rm{i}}}{2}{h}_{jl,\parallel }^{\xi \xi ^{\prime} }\left[\delta ^{\prime} \left({\rm{\Delta }}\tau -\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{a{{ \mathcal D }}_{\parallel }^{\xi \xi ^{\prime} }}{2}\right)+\delta ^{\prime} \left({\rm{\Delta }}\tau +\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{a{{ \mathcal D }}_{\parallel }^{\xi \xi ^{\prime} }}{2}\right)\right]\\ & & \quad -\frac{{\rm{i}}}{2}{f}_{jl,\parallel }^{\xi \xi ^{\prime} }\left[\delta \left({\rm{\Delta }}\tau -\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{a{{ \mathcal D }}_{\parallel }^{\xi \xi ^{\prime} }}{2}\right)-\delta \left({\rm{\Delta }}\tau +\frac{2}{a}{{\rm{\sinh }}}^{-1}\frac{a{{ \mathcal D }}_{\parallel }^{\xi \xi ^{\prime} }}{2}\right)\right],\end{array}\end{eqnarray}$
with ${{ \mathcal D }}_{\parallel }^{\xi \xi }=2R$ and ${{ \mathcal D }}_{\parallel }^{\xi \xi ^{\prime} }=\sqrt{{L}^{2}+4{R}^{2}}$ for $\xi \ne \xi ^{\prime} $. In the appendix, we concretely give the expressions of ${f}_{jl,\parallel }^{\xi \xi ^{\prime} }$, ${h}_{jl,\parallel }^{\xi \xi ^{\prime} }$ and ${g}_{jl,\parallel }^{\xi \xi ^{\prime} }$, please refer to equations ( $\begin{eqnarray}\displaystyle {\left\langle \frac{{\rm{d}}}{{\rm{d}}\tau }{H}_{S}(\tau )\right\rangle }_{\parallel }^{{\rm{a}}{\rm{c}}{\rm{c}}}={\left\langle \frac{{\rm{d}}}{{\rm{d}}\tau }{H}_{S}(\tau )\right\rangle }_{ub}^{{\rm{a}}{\rm{c}}{\rm{c}}}+{\left\langle \frac{{\rm{d}}}{{\rm{d}}\tau }{H}_{S}(\tau )\right\rangle }_{\parallel ,b}^{{\rm{a}}{\rm{c}}{\rm{c}}}.\end{eqnarray}$
This result consists of the free-space result [equation ( $\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl}{\left\langle \frac{{\rm{d}}{H}_{S}(\tau )}{{\rm{d}}\tau }\right\rangle }_{\parallel ,b}^{acc} & = & \mp 2| {{\boldsymbol{d}}}^{A}| | {{\boldsymbol{d}}}^{B}| \left[\right.{{ \mathcal Q }}_{xx}^{AB}{\hat{d}}_{x}^{A}{\hat{d}}_{x}^{B}+{{ \mathcal Q }}_{yy}^{AB}{\hat{d}}_{y}^{A}{\hat{d}}_{y}^{B}+{{ \mathcal Q }}_{zz}^{AB}{\hat{d}}_{z}^{A}{\hat{d}}_{z}^{B}+{{ \mathcal Q }}_{xy}^{AB}({\hat{d}}_{x}^{A}{\hat{d}}_{y}^{B}+{\hat{d}}_{y}^{A}{\hat{d}}_{x}^{B})\\ & & +{{ \mathcal Q }}_{xz}^{AB}({\hat{d}}_{x}^{A}{\hat{d}}_{z}^{B}-{\hat{d}}_{z}^{A}{\hat{d}}_{x}^{B})+{{ \mathcal Q }}_{yz}^{AB}({\hat{d}}_{y}^{A}{\hat{d}}_{z}^{B}-{\hat{d}}_{z}^{A}{\hat{d}}_{y}^{B})\left]\right.\\ & & \mp | {{\boldsymbol{d}}}^{A}{| }^{2}\left[{{ \mathcal Q }}_{xx}^{AA}{\hat{d}}_{x}^{A}{\hat{d}}_{x}^{A}+{{ \mathcal Q }}_{yy}^{AA}{\hat{d}}_{y}^{A}{\hat{d}}_{y}^{A}+{{ \mathcal Q }}_{zz}^{AA}{\hat{d}}_{z}^{A}{\hat{d}}_{z}^{A}+2{{ \mathcal Q }}_{xy}^{AA}{\hat{d}}_{x}^{A}{\hat{d}}_{y}^{A}\right]\\ & & \mp | {{\boldsymbol{d}}}^{B}{| }^{2}\left[{{ \mathcal Q }}_{xx}^{BB}{\hat{d}}_{x}^{B}{\hat{d}}_{x}^{B}+{{ \mathcal Q }}_{yy}^{BB}{\hat{d}}_{y}^{B}{\hat{d}}_{y}^{B}+{{ \mathcal Q }}_{zz}^{BB}{\hat{d}}_{z}^{B}{\hat{d}}_{z}^{B}+2{{ \mathcal Q }}_{xy}^{BB}{\hat{d}}_{x}^{B}{\hat{d}}_{y}^{B}\right],\end{array}\end{eqnarray}$
with $\begin{eqnarray}\displaystyle \displaystyle \begin{array}{rcl}{{ \mathcal Q }}_{jl}^{AA} & = & {{ \mathcal Q }}_{jl}^{BB}=\omega \left(\right.{f}_{jl,\parallel }^{\xi \xi }+{\omega }^{2}{g}_{jl,\parallel }^{\xi \xi }\left)\right.\sin \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}aR\right)+{\omega }^{2}{h}_{jl,\parallel }^{\xi \xi }\cos \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}aR\right),\\ {{ \mathcal Q }}_{jl}^{AB} & = & \omega \left(\right.{f}_{jl,\parallel }^{AB}+{\omega }^{2}{g}_{jl,\parallel }^{AB}\left)\right.\sin \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}\frac{a\sqrt{{L}^{2}+4{R}^{2}}}{2}\right)+{\omega }^{2}{h}_{jl,\parallel }^{AB}\cos \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}\frac{a\sqrt{{L}^{2}+4{R}^{2}}}{2}\right).\end{array}\end{eqnarray}$
Note that the boundary plane in the parallel-alignment case is located on the 'xoz' plane, and thus the unit vector ${\hat{d}}_{y}^{\xi }$ is vertical to but ${\hat{d}}_{z}^{\xi }$ is parallel to the boundary plane, which in the vertical-alignment case [the boundary is on the 'xoy' plane] equates to ${\hat{d}}_{z}^{\xi }$ and ${\hat{d}}_{y}^{\xi }$, respectively. Thus, it can be seen that compared to the boundary-induced modifications in the vertical-alignment case [equation (When the two atoms are placed very far away from the boundary (R → ∞), the boundary-dependent modification [equation (50 )], with the use of equations (51 ) and (A.6 )–(A.9 ), is found to be vanishing. When the two atoms are located parallel to and extremely near the boundary, i.e. when R → 0, remarkable distinctions in comparison with the free-space and limiting vertical-alignment cases are found as follows. In the present case, the terms only related to the atom ξ are $-\frac{| {{\boldsymbol{d}}}^{\xi }{| }^{2}{\omega }^{4}}{3\pi }\left(1+\frac{{a}^{2}}{{\omega }^{2}}\right){\hat{d}}_{y}^{\xi }{\hat{d}}_{y}^{\xi }$ for the symmetric entangled state, but are $-\frac{| {{\boldsymbol{d}}}^{\xi }{| }^{2}{\omega }^{4}}{3\pi }\left(1+\frac{{a}^{2}}{{\omega }^{2}}\right)({\hat{d}}_{x}^{\xi }{\hat{d}}_{x}^{\xi }+{\hat{d}}_{z}^{\xi }{\hat{d}}_{z}^{\xi })$ for the antisymmetric entangled state. Recall again that here the boundary plane is located on the 'xoz' plane, and the unit vector ${\hat{d}}_{y}^{\xi }$ is vertical to but ${\hat{d}}_{x}^{\xi }$ and ${\hat{d}}_{z}^{\xi }$ are parallel to the boundary plane. Thus, it can be seen that here the terms related to atom A only are consistent with those in the vertical-alignment case. The terms related to atom B are interatomic-separation-independent and influenced by the orientations of dipole moments of atoms, obviously different from the vertical-alignment case in which the corresponding terms significantly depend on the interatomic separation and also from the free-space case in which the corresponding terms are the polarization-independent constant $-\frac{| {{\boldsymbol{d}}}^{B}{| }^{2}{\omega }^{4}}{6\pi }\left(1+\frac{{a}^{2}}{{\omega }^{2}}\right)$. For the terms related to both atoms, we find that they are proportional to ${\hat{d}}_{y}^{A}{\hat{d}}_{y}^{B}$ and oscillatory in the form of $\sin \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{\,-1}\frac{aL}{2}\right)$ and $\cos \left(\frac{2\omega }{a}{{\rm{\sinh }}}^{-1}\frac{aL}{2}\right)$. Thus, this suggests that these terms contribute only when the dipole moments of the two atoms both have the components vertical to the boundary plane ('xoz' plane), which is consistent with the vertical-alignment situation. As a result, in the case of the two atoms placed parallel to and extremely close to the boundary, we find that when the two atoms are both polarizable along the direction parallel to the boundary, e.g. ${\hat{{\boldsymbol{d}}}}^{A}=(1,0,0)$ and ${\hat{{\boldsymbol{d}}}}^{B}=(0,0,1)$, the average rate of change of energy of two symmetric entangled atoms vanishes as if the two atoms were a closed system. However, for the antisymmetric entangled state, the average rate of change of energy is nonzero $\left[-\frac{(| {{\boldsymbol{d}}}^{A}{| }^{2}+| {{\boldsymbol{d}}}^{B}{| }^{2}){\omega }^{4}}{3\pi }\left(1+\frac{{a}^{2}}{{\omega }^{2}}\right)\right]$ and double the terms only related to one atom in the free-space case $\left[-\frac{(| {{\boldsymbol{d}}}^{A}{| }^{2}+| {{\boldsymbol{d}}}^{B}{| }^{2}){\omega }^{4}}{6\pi }\left(1+\frac{{a}^{2}}{{\omega }^{2}}\right)\right]$. Finally, we use figure 4 to show, in the various cases of the orientations of dipole moments of atoms, the variations of average rate of change of energy of the accelerated and maximally symmetric entangled atoms as a function of the atom-boundary distance R. It is found that when R grows from zero, the absolute values of the average rate of change of energy rapidly become smaller for the two atoms polarizable both vertically to the boundary but become larger in the other polarization cases.
Figure 4. Average rate of change of energy of two parallel-aligned and accelerated symmetric entangled atoms in the unit of $\frac{| {\boldsymbol{d}}{| }^{2}{\omega }^{4}}{\pi }$ is plotted as a function of the dimensionless atom-boundary distance. Acceleration and interatomic separation are along the x − axis and z − axis, respectively, and the boundary is on the 'xoz' plane. Here, ωL = 10 and aL = 2. Orientations of dipole moments of two atoms are assumed to be identical (left) and different (right), respectively. |
4. Summary
In this paper, we study the average rate of change of energy of two maximally entangled atoms in dipole interaction with a fluctuating vacuum EM field and synchronously accelerated in the vicinity of an infinite and perfectly reflecting boundary. With the second-order DDC formalism, we find that for the two-atom system prepared in the maximally symmetric or antisymmetric entangled state, the average rate of change of energy is only caused by the contribution of radiation reaction of atoms. However, for the factorizable ground or excited state, the vacuum fluctuations of free fields and the radiation reaction both contribute. These results are found to be consistent with those obtained in the case of the two atoms in monopole interaction with a vacuum massless scalar field [27].
Using this method, we first revisit the average rate of change of energy of two uniformly accelerated and entangled atoms in free space, and our result is consistent with that obtained in [29] using the method of calculating response functions. We note that the result in free space contains two types of terms, i.e. related to the dipole moment of one atom only and those of both atoms. The terms related to one atom only are independent of the interatomic separation, and always cause the reduction of the two-atom energy no matter that the two atoms are initially prepared in the symmetric or antisymmetric state. Compared to the separation-independent terms in the scalar-field case, which are independent of atomic acceleration, these terms in the EM-field case can be remarkably controlled by atomic acceleration. The terms related to the dipole moments of two atoms, for the two-atom symmetric or antisymmetric entangled state, have the same absolute value but opposite signs. Comparing these terms with their counterparts in the inertial-atom case, the former also includes the contributions led by two cross-polarizable atoms, which indicates that when the dipole moment of one atom is oriented along the interatomic separation and the other along the acceleration (orthogonal to the direction of L), the accelerated motion of atoms can lead to the variation of energy of the two cross-polarizable atoms, which does not occur for two inertial atoms.
When the boundary is present, we concretely consider two configurations of atoms with respect to the boundary, i.e. the two atoms are aligned parallel to and vertically to the boundary. In these cases, the average rates of change of energy of the two atoms are expressed as the sum of the result in the free space and the boundary-dependent modifications, and they generally depend on the initial symmetric or antisymmetric state, the orientations of atomic dipole moments, as well as the relative scales among the atom-boundary distance R, the interatomic separation L and the inverse of atomic acceleration a−1. When R is very large compared to L and a−1, i.e. when the two atoms are placed very far away from the boundary, the average rates of change of energy are exactly equal to the free-space result. When the two atoms are located vertically to and very near the boundary (R → 0), we find that the terms only related to the closer atom (atom A) are significantly affected by the orientations of atomic dipole moment, which also depends on the initial two-atom state. Specifically, these terms are zero for the symmetric entangled state but are double their free-space counterpart for the antisymmetric entangled state, when the atom is polarizable parallel to the boundary. However, for the atom polarizable vertically to the boundary, these terms vanish for the antisymmetric entangled state but are twice their free-space counterpart for the symmetric entangled state. In contrast to the counterparts in the free-space case, the terms related to the farther atom (atom B) are obviously oscillating with the interatomic separation. For the terms related to both atoms, we find that they are always vanishing unless the dipole moments of the two atoms both have the components vertical to the boundary. When the two atoms are placed parallel to and very near the boundary, it is found that the terms related to atom A are the same as those in the vertical-alignment case; the terms related to atom B are interatomic-separation-independent and only influenced by the orientations of atomic dipole moments. Moreover, we find that the terms related to both atoms are nonvanishing only when the dipole moments of two atoms both have the components vertical to the boundary plane, which is consistent with the vertical-alignment situation. The result in the limiting parallel-alignment case shows that when the two atoms are both polarizable along the direction parallel to the boundary, the average rate of change of energy of two symmetric entangled atoms vanishes as if the two atoms were a closed system.