1 Introduction

Quantum vacuum fluctuation, which is an inevitable consequence of quantum theory, may induce some observable effects. The electromagnetic Casimir–Polder (CP) interaction is one of the well-known examples, which arises from the dipole–dipole interaction between two neutral atoms or molecules induced by electromagnetic vacuum fluctuations [1]. Such interatomic or intermolecular CP effects have been widely studied in different circumstances and found to be influenced significantly by the atomic or molecular states [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16], the external electromagnetic fields [17,18,19,20,21,22,23,24,25,26,27] and boundaries [28, 29]. For instance, the CP interaction potential behaves as \(r^{-6}\) and \(r^{-7}\) in the near and far regimes respectively when the atoms are in their ground states [1], while it decreases as \(r^{-3}\) and \(r^{-1}\) in the near and far regimes respectively when the atoms are prepared in an entangled state [15] or placed in external electromagnetic fields [17, 19].

Likewise, in the gravitational case, there may also exist a CP-like quantum gravitational quadrupole–quadrupole interactions between non point-like objects if one accepts that basic quantum principles are applicable to gravity. Unfortunately, a full theory of quantum gravity is elusive at present. Nonetheless, one can still study low energy quantum gravitational effects in the framework of effective field theory or linearized quantum gravity. Similar to the electromagnetic case, the CP-like quantum gravitational quadrupole–quadrupole interaction potentials also have been found to be significantly different when the non point-like objects are prepared in different states or placed in certain environments. For example, the gravitational CP-like quadrupole–quadrupole potential between two ground-state objects behaves as \(r^{-10}\) and \(r^{-11}\) in the near and far regimes respectively [30,31,32,33], while it decreases as \(r^{-5}\) and \(r^{-1}\) in the near and far regimes respectively when the objects are in an entangled state or in external gravitational radiations [34,35,36]. Moreover, in the presence of gravitational boundaries, the quantum gravitational quadrupole–quadrupole interaction between two ground-state objects is found to be always strengthened as compared with the case without boundaries [37].

The CP-like quantum gravitational interactions discussed above are obtained respectively under a particular condition. Since the entanglement, the external gravitational radiations and gravitational boundaries can all enhance the quantum gravitational quardrupole–quardrupole interaction in vacuum, a natural question arises as to whether such quantum gravitational effects can be more significant under certain composite circumstances. Physically, the gravitational CP-like interaction for two entangled objects in vacuum is a second-order effect while it is a fourth-order effect for objects in their ground states or in an external gravitational field, due to the fact that the former is related to the single-graviton processes while the latter is relevant to the two-graviton processes. Hence, the second-order quantum effect correlated to the single-graviton processes is expected preferentially and the composite case of two entangled objects together with gravitational boundaries is thus taken into account first. Note here that, although the reflection of gravitational waves by ordinary materials is hardly possible [38], it might be realized by quantum matter such as superconducting films [39, 40].

In this paper, we explore the quantum gravitational quardrupole–quardrupole interaction between two entangled non point-like objects in the presence of both Dirichlet and Neumann boundaries based on the method proposed by Dalibard, Dupont-Roc, and Cohen-Tannoudji (DDC) [41, 42]. First, we give a brief description of the formulae for the interaction between the two objects. Then, we obtain the general expression of the interaction energy between the two objects in the presence of Dirichlet and Neumann boundaries respectively, and discuss these results in specific cases. Throughout this paper, the Latin indices run from 1 to 3 and the Einstein summation convention for repeated indices is assumed.

2 Basic equations

We consider two entangled non point-like objects (labeled as A and B) coupled with the fluctuating gravitational fields in vacuum in the presence of gravitational plane boundaries. For simplicity, the objects A and B are modeled as two-level systems with the ground and excited states being \(|g\rangle \) and \(|e\rangle \) respectively, and the corresponding energy spacing is labeled as \(\omega _0\). The total Hamiltonian of the system is

$$\begin{aligned} H=H_S+H_F+H_I, \end{aligned}$$
(1)

where \(H_{S}\) denotes the Hamiltonian of the two-level systems (A and B), \(H_{F}\) denotes the Hamiltonian of the gravitational fields in vacuum, and \(H_I\) denotes the interaction Hamiltonian between the non point-like objects and the gravitational fields. Here \(H_I\) takes the form

$$\begin{aligned} H_I=-\frac{1}{2}Q^{A}_{ij}E_{ij}(\vec x_A)-\frac{1}{2}Q^{B}_{ij}E_{ij}(\vec x_{B}), \end{aligned}$$
(2)

where \(Q^{\xi }_{ij}\) is the quadrupole moment operator of object \(\xi \) (\(\xi =A,B\)), and \(E_{ij}\) is the gravitoelectric tensor of the fluctuating gravitational fields in vacuum defined as \(E_{ij}=-c^2 C_{0i0j}\) by an analogy between the linearized Einstein field equations and the Maxwell equations  [43,44,45,46,47,48,49], where \(C_{\mu \nu \alpha \beta }\) is the Weyl tensor and c the speed of light. Under the weak-field approximation, the metric tensor for the fluctuating gravitational fields can be expanded as a sum of the flat spacetime metric and a linearized perturbation \(h_{\mu \nu }\). Then, the gravitoelectric tensor \(E_{ij}\) can be expressed as

$$\begin{aligned} E_{ij}=\frac{1}{2}\ddot{h}_{ij}. \end{aligned}$$
(3)

where a dot denotes the derivative with respect to time t. For the pure vacuum case, the quantized metric perturbations can be expressed as (in the transverse traceless gauge) [50]

$$\begin{aligned} h_{ij}= & {} \sum _{\vec p, \lambda }h^{\lambda }_{ij,\vec p} =\sum _{\vec p, \lambda }\sqrt{\frac{\hbar G}{\omega c^2\pi ^2}}[ a_{\lambda }(\omega )e^{(\lambda )}_{ij}(\vec p) e^{i(\vec {p}\cdot \vec {x}-\omega t)}\nonumber \\{} & {} +\text {H.c.} ], \end{aligned}$$
(4)

where \(h^{\lambda }_{ij,\vec p}\) is the gravitational field mode with wave vector \(\vec p\), \(a_{\lambda }(\omega )\) represents the annihilation operator of the fluctuating gravitational fields, \(\lambda \) denotes the polarization states, \(\omega =c|\vec p|= c(p_{x}^2+p_{y}^2+p_{z}^2)^{1/2}\), \(e^{(\lambda )}_{ij}(\vec p)\) are polarization tensors, and H.c. denotes the Hermitian conjugate.

We assume that the two objects are in their maximally entangled states, i.e., the symmetric or antisymmetric state,

$$\begin{aligned} \psi _{\pm }=\frac{1}{\sqrt{2}}(|g_{A}\rangle |e_{B}\rangle \pm |e_{A}\rangle |g_{B}\rangle ), \end{aligned}$$
(5)

and denote the vacuum state of the gravitational field as \(|0\rangle \). To investigate the quantum gravitational quadrupole–quadrupole interaction potential between two entangled objects in vacuum, the second-order DDC formalism has ever been exploited and shown to be really convenient, see in Ref. [34]. Hence, in the presence of gravitational boundaries, we employ this formalism again to calculate interaction energy shift between objects A and B, which is [34]

$$\begin{aligned} \Delta E_{AB}= & {} -\frac{i}{4}\int ^{t}_{t_{0}}d t'\chi ^{F}_{ijkl}(\vec x_{A}(t),\vec x_{B}(t'))C^{AB}_{ijkl}(t,t')\nonumber \\{} & {} +(A\rightleftharpoons B \ \text {terms}). \end{aligned}$$
(6)

Here \(C^{AB}_{ijkl}(t,t')\) and \(\chi ^{F}_{ijkl}(\vec x_{A}(t),\vec x_{B}(t'))\) are respectively the statistical functions introduced for the gravitational field and the objects, which take the form

$$\begin{aligned} C^{AB}_{ijkl}(t,t')= & {} \frac{1}{2}\langle \psi _{\pm }|\{Q^{AF}_{ij}(t),Q^{BF}_{kl}(t')\}|\psi _{\pm }\rangle , \end{aligned}$$
(7)
$$\begin{aligned} \chi ^{F}_{ijkl}(\vec x(t),\vec x(t'))= & {} \frac{1}{2}\langle 0|[E^{F}_{ij}(\vec x(t)), E^{F}_{kl}(\vec x(t'))] |0\rangle , \end{aligned}$$
(8)

where the label “F” denotes the free part of the operator, i.e., the part presents even in the absence of interaction.

3 Dirichlet boundary condition

Now let us consider the quantum gravitational quadrupole–quadrupole potential between two entangled non point-like objects when a Dirichlet boundary is present. For simplicity, this plane boundary is assumed to be placed at \(z=0\), so that the metric perturbations satisfies \(h^{\lambda }_{ij,\vec p}|_{z=0}=0\) and the gravitational field mode \(h^{\lambda }_{ij,\vec p}\) can be expressed as

$$\begin{aligned} h^{\lambda }_{ij,\vec p}= & {} \sqrt{\frac{\hbar G}{2c^2\omega \pi ^2}}\left\{ a_{\lambda }(\omega )\left[ e^{(\lambda )}_{ij}(\vec p) e^{i(\vec {p}\cdot \vec {x}-\omega t)}-e^{(\lambda )}_{ij}(\vec {p}_{-})\right. \right. \nonumber \\{} & {} \times \left. \left. e^{i(\vec {p}_{-}\cdot \vec {x}-\omega t)}\right] +\text {H.c.} \right\} , \end{aligned}$$
(9)

where \(\vec {p}_{-}=\{p_x,p_y,-p_z\}\). Then, according to Eqs. (3) and (8), the statistical function of the gravitational field \(\chi ^{F}_{ijkl}(\vec x_{A}(t),\vec x_{B}(t'))\) in the presence of a Dirichlet boundary can be expressed as

$$\begin{aligned}{} & {} \chi ^{F}_{ijkl}(\vec x(t),\vec x(t'))=\frac{1}{8}\langle 0|[\ddot{h}_{ij}(\vec x(t)),\ddot{h}_{kl}(\vec x(t'))]|0\rangle \nonumber \\{} & {} \quad =\frac{\hbar c G}{16\pi ^2}\int d^{3}\vec p \ p^{3} \sum _{\lambda } \Big [e_{ij}^{(\lambda )}(\vec p)e_{kl}^{(\lambda )}(\vec p)e^{i\vec p\cdot \vec r} \nonumber \\{} & {} \qquad + \, e_{ij}^{(\lambda )}(\vec p_{-})e_{kl}^{(\lambda )}(\vec p_{-}) e^{i\vec p_{-}\cdot \vec r_{-}}\nonumber \\{} & {} \qquad -\, e_{ij}^{(\lambda )}(\vec p)e_{kl}^{(\lambda )}(\vec p_{-})e^{i\vec p\cdot \vec r_{-}} \nonumber \\{} & {} \qquad -\, e_{ij}^{(\lambda )}(\vec p_{-})e_{kl}^{(\lambda )}(\vec p)e^{i\vec p_{-}\cdot \vec r} \Big ] \left( e^{-i\omega \Delta t'}-e^{i\omega \Delta t'} \right) , \end{aligned}$$
(10)

where \(\Delta t'=t-t'\), \(\vec r=\{x-x',y-y',z-z'\}\) and \(\vec r_{-}=\{x-x',y-y',z+z'\}\). Note here that \(r=|\vec r|\) is the distance between the two objects and \(r_{-}=|\vec r_{-}|\) is the distance between an object and the image of another one. Here the summation of polarization tensors in the TT gauge gives [50]

$$\begin{aligned}{} & {} \sum _{\lambda }e^{(\lambda )}_{ij}(\vec p) e^{(\lambda )}_{kl}(\vec p')=\delta _{ik}\delta _{j l}+\delta _{il}\delta _{j k}- \delta _{ij}\delta _{k l}\nonumber \\{} & {} \quad +{\hat{p}}_{i}{\hat{p}}_{j}{\hat{p}}^{'}_{k}{\hat{p}}'_{l}+{\hat{p}}_{i}{\hat{p}}_{j}\delta _{k l}+ {\hat{p}}'_{k}{\hat{p}}'_{l}\delta _{ij}\nonumber \\{} & {} \quad -{\hat{p}}_{i}{\hat{p}}'_{k}\delta _{j l}-\, {\hat{p}}_{i}{\hat{p}}'_{l}\delta _{j k}-{\hat{p}}_{j}{\hat{p}}'_{k}\delta _{i l}-{\hat{p}}_{j}{\hat{p}}'_{l}\delta _{i k} , \end{aligned}$$
(11)

where \({\hat{p}}_i\) is the i-th component of the unit vector \(\vec p_i /|\vec p|\). From this summation of polarization tensors, we can obtain

$$\begin{aligned}{} & {} \sum _{\lambda }e^{(\lambda )}_{ij}(\vec p) e^{(\lambda )}_{kl}(\vec p) e^{i \vec p \cdot \vec r} \nonumber \\{} & {} \quad =\frac{1}{p^4}[(\delta _{ik}\delta _{j l}+\delta _{il}\delta _{j k}-\delta _{ij}\delta _{k l})\nabla ^4 +(\partial _i\partial _j\delta _{kl}\nonumber \\{} & {} \qquad +\,\partial _k\partial _l\delta _{ij}-\partial _i\partial _k\delta _{jl} \nonumber \\{} & {} \qquad - \, \partial _i\partial _l\delta _{jk}-\partial _j\partial _k\delta _{il} -\partial _j\partial _l\delta _{ik})\nabla ^2+\partial _i\partial _j\partial _k\partial _l]e^{i \vec p \cdot \vec r} \nonumber \\{} & {} \quad = \frac{1}{p^4} H^{r}_{ijkl} e^{i \vec p \cdot \vec r}, \end{aligned}$$
(12)

and

$$\begin{aligned}{} & {} \sum _{\lambda }e^{(\lambda )}_{ij}(\vec p) e^{(\lambda )}_{kl}(\vec p_{-}) e^{i \vec p \cdot \vec r_{-}}\nonumber \\{} & {} \quad =\frac{1}{p^4}\sigma _{km}\sigma _{ln}[(\delta _{im}\delta _{j n}+\delta _{i n}\delta _{j m} -\delta _{ij}\delta _{m n})\nabla ^4\nonumber \\{} & {} \qquad +\,(\partial _i\partial _j\delta _{m n} +\partial _m\partial _n\delta _{ij}\nonumber \\{} & {} \qquad -\,\partial _i\partial _m\delta _{j n}-\partial _i\partial _n\delta _{j m}-\partial _j\partial _m\delta _{i n} -\partial _j\partial _n\delta _{i m}) \nabla ^2\nonumber \\{} & {} \qquad +\,\partial _i\partial _j\partial _m\partial _n]e^{i \vec p \cdot \vec r_{-}} \nonumber \\{} & {} \quad = \frac{1}{p^4}\sigma _{km}\sigma _{ln} H^{r_{-}}_{ijmn} e^{i \vec p \cdot \vec r_{-}}, \end{aligned}$$
(13)

where \(H^{\rho }_{ijkl}\) denotes the differential operator to \(\rho \), \(\nabla ^2=\partial _i\partial ^i\), and \(\sigma _{ij}=\delta _{ij}-2\delta _{i3}\delta _{j3}\). Substituting Eqs. (12) and (13) into Eq. (10) and performing the integration in the spherical coordinate, we obtain

$$\begin{aligned}{} & {} \chi ^{F}_{ijkl}(\vec x(t),\vec x(t'))= \frac{\hbar c G}{2\pi }\int _{0}^{\infty }d p \nonumber \\{} & {} \qquad \times \left[ H^{r}_{ijkl}\frac{\sin {p r}}{r} -\sigma _{km}\sigma _{ln}H^{r_{-}}_{ijmn} \frac{\sin {p r_{-}}}{r_{-}} \right] \nonumber \\{} & {} \qquad \times (e^{-i\omega \Delta t'}-e^{i\omega \Delta t'} ) \nonumber \\{} & {} \quad =-\frac{i\hbar c G}{2}\Bigg [ H^{r}_{ijkl}\frac{1}{r}[\delta (r-c\Delta t')-\delta (r+c\Delta t')] \nonumber \\{} & {} \qquad - \sigma _{km}\sigma _{ln} H^{r_{-}}_{ijmn} \frac{1}{r_{-}}[\delta (r_{-}-c\Delta t')-\delta (r_{-}+c\Delta t')] \Bigg ] .\nonumber \\ \end{aligned}$$
(14)

As for the statistical function \(C^{AB}_{ijkl}(t,t')\) of the objects, its form can be obtained as

$$\begin{aligned} C^{AB}_{ijkl}(t,t')=\pm \frac{1}{2}{\hat{Q}}^{A}_{ij}{\hat{Q}}^{B}_{kl}(e^{-i\omega _{0}\Delta t'} +e^{i\omega _{0}\Delta t'}), \end{aligned}$$
(15)

where the sign ± correspond to the symmetric and antisymmetric states respectively, \({\hat{Q}}^{\xi }_{ij}=e^{i\omega _{0}t}\langle g_{\xi }|Q^{\xi F}_{ij}|e_{\xi }\rangle \), \({\hat{Q}}^{\xi *}_{ij}=e^{-i\omega _{0}t}\langle e_{\xi }|Q^{\xi F}_{ij}|g_{\xi } \rangle \), and \({\hat{Q}}^{\xi }_{ij}={\hat{Q}}^{\xi *}_{ij}\) is assumed.

Substituting Eqs. (10) and (15) into Eq. (6), the interobject quantum gravitational interaction potential in the presence of a Dirichlet boundary can then be obtained as

$$\begin{aligned} \Delta E_{AB}= & {} {\mp }\frac{G}{4}{\hat{Q}}^{A}_{ij} {\hat{Q}}^{B}_{kl}\Big [\theta (c\Delta t-r)H^{r}_{ijkl} \frac{\cos {p_0r}}{r}\nonumber \\{} & {} -\, \theta (c\Delta t-r_{-})H^{r_{-}}_{ijkl} \frac{\cos {p_0r_{-}}}{r_{-}} \Big ], \end{aligned}$$
(16)

where we have introduced \(p_0=\omega _0/c\) and \(\Delta t= t-t_0\). \(\theta (x)\) is a step function, which equals to 1 when \(x>0\) but 0 when \(x<0\). Let us note here that

$$\begin{aligned}{} & {} H^{r}_{ijkl} \frac{\cos {p_0r}}{r} =\frac{1}{r^5}[(\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk}-\delta _{ij}\delta _{kl}\nonumber \\{} & {} \quad + \, {\hat{r}}_i {\hat{r}}_j\delta _{kl}+{\hat{r}}_k {\hat{r}}_l\delta _{ij}-{\hat{r}}_i {\hat{r}}_k\delta _{jl}\nonumber \\{} & {} \quad -\, {\hat{r}}_i {\hat{r}}_l\delta _{jk}-{\hat{r}}_j {\hat{r}}_k\delta _{il}-{\hat{r}}_j {\hat{r}}_l\delta _{ik}\nonumber \\{} & {} \quad +\, {\hat{r}}_i {\hat{r}}_j{\hat{r}}_k {\hat{r}}_l)r^4 p_0^4 \cos {p_0 r}\nonumber \\{} & {} \quad +\, 2(-\delta _{ik}\delta _{jl}-\delta _{il}\delta _{jk}+\delta _{ij}\delta _{kl}\nonumber \\{} & {} \quad -\, {\hat{r}}_i {\hat{r}}_j\delta _{kl}-{\hat{r}}_k {\hat{r}}_l\delta _{ij}+2{\hat{r}}_j {\hat{r}}_k\delta _{il}\nonumber \\{} & {} \quad +\, 2{\hat{r}}_j {\hat{r}}_l\delta _{ik}+2{\hat{r}}_i {\hat{r}}_k\delta _{jl}+2{\hat{r}}_i {\hat{r}}_l\delta _{jk}\nonumber \\{} & {} \quad -\, 5{\hat{r}}_i {\hat{r}}_j{\hat{r}}_k {\hat{r}}_l)r^3 p^3_0 \sin {p_0 r}\nonumber \\{} & {} \quad +\, (-3\delta _{ik}\delta _{jl}-3\delta _{il}\delta _{jk}+\delta _{ij}\delta _{kl}+3{\hat{r}}_i {\hat{r}}_j\delta _{kl}\nonumber \\{} & {} \quad +\, 3{\hat{r}}_k {\hat{r}}_l\delta _{ij}+9{\hat{r}}_j {\hat{r}}_k\delta _{il}\nonumber \\{} & {} \quad +\, 9{\hat{r}}_j {\hat{r}}_l\delta _{ik}+9{\hat{r}}_i {\hat{r}}_k\delta _{jl}+9{\hat{r}}_i {\hat{r}}_l\delta _{jk}\nonumber \\{} & {} \quad -\, 45{\hat{r}}_i {\hat{r}}_j{\hat{r}}_k {\hat{r}}_l) r^2 p^2_0\cos {p_0 r}\nonumber \\{} & {} \quad +\, 3(\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk}+\delta _{ij}\delta _{kl}\nonumber \\{} & {} \quad -\, 5{\hat{r}}_i {\hat{r}}_j\delta _{kl}-5{\hat{r}}_k {\hat{r}}_l\delta _{ij}-5{\hat{r}}_j {\hat{r}}_k\delta _{il}\nonumber \\{} & {} \quad -\, 5{\hat{r}}_j {\hat{r}}_l\delta _{ik}-5{\hat{r}}_i {\hat{r}}_k\delta _{jl}\nonumber \\{} & {} \quad -\, 5{\hat{r}}_i {\hat{r}}_l\delta _{jk}+35{\hat{r}}_i {\hat{r}}_j{\hat{r}}_k {\hat{r}}_l)r p_0 \sin {p_0 r}\nonumber \\{} & {} \quad +\, 3(\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk}+\delta _{ij}\delta _{kl}-5{\hat{r}}_i {\hat{r}}_j\delta _{kl}\nonumber \\{} & {} \quad -\, 5{\hat{r}}_k {\hat{r}}_l\delta _{ij}-5{\hat{r}}_j {\hat{r}}_k\delta _{il}\nonumber \\{} & {} \quad -\, 5{\hat{r}}_j {\hat{r}}_l\delta _{ik}-5{\hat{r}}_i {\hat{r}}_k\delta _{jl}-5{\hat{r}}_i {\hat{r}}_l\delta _{jk}\nonumber \\{} & {} \quad +\,35{\hat{r}}_i {\hat{r}}_j{\hat{r}}_k {\hat{r}}_l)\cos {p_0 r}], \end{aligned}$$
(17)

where \({\hat{r}}_{i}\) is the i-th component of the unit vector \(\vec r_i/r\). The first term in Eq. (16) corresponds to the interaction between two entangled objects in vacuum [34], while the second term is the additional contribution due to the Dirichlet boundary which can be regarded as the interaction between an object and the image of the other one. It shows that, both the two terms exist simultaneously only when the distance r and \(r_{-}\) are smaller than the distance characterized by \(c\Delta t\), which means that such interaction potential appears only when one object (as well as its image) is in the light cone of the other.

3.1 Two special cases

Now Let us discuss the interaction potential in some special circumstances. The first case is that the two objects are placed perpendicular to the Dirichlet boundary. We take \(\vec r=\{0,0,r\}\) and, for convenience, assume \(r<c\Delta t\). When the two-object system is very close to the boundary, i.e., \(r \sim r_{-}\), the interaction potential Eq. (16) becomes

$$\begin{aligned} \Delta E_{AB}= & {} {\mp }\frac{\hbar G \omega _0}{r^5} \alpha (8p_0^3 r^3\sin {p_0 r} +\,24 p_0^2 r^2 \cos {p_0 r}\nonumber \\{} & {} -48p_0 r \sin {p_0 r}-48\cos {p_0 r}), \end{aligned}$$
(18)

where the isotropic gravitational polarizability \(\alpha =\alpha _{ijkl}= {\hat{Q}}_{ij}{\hat{Q}}_{kl}/\hbar \omega _0\) defined in Ref. [34] has been employed again. In the near regime, i.e., \(r\ll c/\omega _{0}\), the leading term of the potential Eq. (18) takes the form

$$\begin{aligned} \Delta E_{AB}\simeq \pm \frac{48\hbar G \omega _0}{r^5} \alpha \cos {p_0 r}, \end{aligned}$$
(19)

while in the far regime, i.e., \(r\gg c/\omega _{0}\), it becomes

$$\begin{aligned} \Delta E_{AB}\simeq \mp \frac{8\hbar G \omega _0^4}{r^2 c^3} \alpha \sin {(p_0 r+\phi _1)}, \end{aligned}$$
(20)

where \(\phi _1=\arcsin \frac{3}{\sqrt{9+p_0^2 r^2}}\). This shows that, when the two entangled objects are placed perpendicular to the Dirichlet boundary, the interobject quantum gravitational quadrupole–quadrupole interaction behaves as \(r^{-5}\) in the near regime and \(r^{-2}\) in the far regime. Compare to the case without gravitational boundary [34], the Dirichlet boundary increases the near-regime potential about 2.3 times in the leading order since the coefficient in the pure vacuum case is 21, and modifies the \(r^{-1}\) far-regime behavior of the interaction in the absence of boundary.

The second case is that the two objects are placed parallel to the Dirichlet boundary. In the limit of \(r\sim r_{-}\), the interaction potential Eq. (16) becomes

$$\begin{aligned}{} & {} \Delta E_{AB}={\mp }\frac{\hbar G \omega _0}{r^5} \alpha (2p_0^4 r^4\cos {p_0 r} \nonumber \\{} & {} \quad +\,6 p_0^2 r^2 \cos {p_0 r}-18p_0 r \sin {p_0 r}-18\cos {p_0 r}), \end{aligned}$$
(21)

where \(\vec r=\{r,0,0\}\) has been applied. In the near regime, i.e., \(r\ll c/\omega _{0}\), the leading term of Eq. (21) becomes

$$\begin{aligned} \Delta E_{AB}\simeq \pm \frac{18\hbar G \omega _0}{r^5} \alpha \cos {p_0 r}, \end{aligned}$$
(22)

while in the far regime, i.e., \(r\gg c/\omega _{0}\), it reduces to

$$\begin{aligned} \Delta E_{AB}\simeq \mp \frac{2\hbar G \omega _0^5}{r c^4} \alpha \cos {p_0 r}. \end{aligned}$$
(23)

In this case, the interobject quantum gravitational quadrupole–quadrupole interaction behaves as \(r^{-5}\) and \(r^{-1}\) in the near and far regimes, respectively. Compare to the case without gravitational boundary [34], the Dirichlet boundary weakens the near-regime interaction potential, but increases the far-regime potential 2 times in the leading order since the coefficient in the pure vacuum case is 1.

4 Neumann boundary condition

For the Neumann boundary condition, the field mode satisfies \(\partial _{z} h^{\lambda }_{ij,\vec p}|_{z=0}=0\) and thus can be written as

$$\begin{aligned} h^{\lambda }_{ij,\vec p}= & {} \sqrt{\frac{\hbar G}{2c^2\omega \pi ^2}}\left\{ a_{\lambda }(\omega )\bigg [e^{(\lambda )}_{ij}(\vec p) e^{i(\vec {p}\cdot \vec {x}-\omega t)}\right. \nonumber \\{} & {} \left. + e^{(\lambda )}_{ij}(\vec {p}_{-}) e^{i(\vec {p}_{-}\cdot \vec {x}-\omega t)}\bigg ]+\text {H.c.}\right\} . \end{aligned}$$
(24)

From the above equation, one can show that the statistical function of the gravitational field \(\chi ^{F}_{ijkl}(\vec x_{A}(t),\vec x_{B}(t'))\) becomes

$$\begin{aligned}{} & {} \chi ^{F}_{ijkl}(\vec x(t),\vec x(t'))= \frac{\hbar c G}{2\pi }\int _{0}^{\infty }d p \nonumber \\{} & {} \quad \times \left[ H^{r}_{ijkl}\frac{\sin {p r}}{r} +\sigma _{km}\sigma _{ln}H^{r_{-}}_{ijmn} \frac{\sin {p r_{-}}}{r_{-}} \right] \nonumber \\{} & {} \quad \times (e^{-i\omega \Delta t'}-e^{i\omega \Delta t'} ) \nonumber \\{} & {} \quad =-\frac{i\hbar c G}{2}\Big [ H^{r}_{ijkl}\frac{1}{r}[\delta (r-c\Delta t')-\delta (r+c\Delta t')] \nonumber \\{} & {} \quad +\, \sigma _{km}\sigma _{ln} H^{r_{-}}_{ijmn} \frac{1}{r_{-}}[\delta (r_{-}-c\Delta t')-\delta (r_{-}+c\Delta t')] \Big ] ,\nonumber \\ \end{aligned}$$
(25)

and the interaction potential Eq. (6) then reads

$$\begin{aligned} \Delta E_{AB}= & {} {\mp }\frac{G}{4}{\hat{Q}}^{A}_{ij} {\hat{Q}}^{B}_{kl}\Bigg [\theta (c\Delta t-r)H^{r}_{ijkl} \frac{\cos {p_0r}}{r} \nonumber \\{} & {} \quad + \theta (c\Delta t-r_{-})H^{r_{-}}_{ijkl} \frac{\cos {p_0r_{-}}}{r_{-}} \Bigg ]. \end{aligned}$$
(26)

Obviously, the boundary-dependent interaction energy exists also only when an object as well as its image are in the light cone of the other one, similar to the Dirichlet case.

4.1 Two special cases

For the special case of two entangled objects placed perpendicular to the Neumann boundary, we take \(\vec r=\{0,0,r\}\) and consider the case when the two-object system is very close to the boundary, i.e., \(r \sim r_{-}\). The interaction potential Eq. (26) becomes

$$\begin{aligned} \Delta E_{AB}= & {} {\mp }\frac{\hbar G \omega _0}{r^5} \alpha (2p_0^4 r^4\cos {p_0 r}-\,4p_0^3 r^3\sin {p_0 r}\nonumber \\{} & {} -6 p_0^2 r^2 \cos {p_0 r}+ 6p_0 r \sin {p_0 r}+ 6\cos {p_0 r}), \end{aligned}$$
(27)

where \(r<c\Delta t\) is assumed. In the near regime, i.e., \(r\ll c/\omega _{0}\), the leading term of the potential Eq. (27) is

$$\begin{aligned} \Delta E_{AB}\simeq {\mp }\frac{6\hbar G \omega _0}{r^5} \alpha \cos {p_0 r}, \end{aligned}$$
(28)

while in the far regime, i.e., \(r\gg c/\omega _{0}\), it takes the form

$$\begin{aligned} \Delta E_{AB}\simeq {\mp }\frac{2\hbar G \omega _0^5}{r c^4} \alpha \cos {(p_0 r+\phi _2)}, \end{aligned}$$
(29)

where \(\phi _2=\arcsin \frac{2}{\sqrt{4+p_0^2 r^2}}\). It shows, when the two objects are placed perpendicular to the Neumann boundary, the interobject quantum gravitational quadrupole–quadrupole interaction behaves as \(r^{-5}\) and \(r^{-1}\) in the near and far regimes, respectively. Compare to the case without boundary [34], the Neumann boundary weakens the near-regime interaction potential, and changes its sign which means the attractive and repulsive properties of the interaction are modified. Also, the far-regime interaction is enhanced about 2 times compared to the pure vacuum case due to the boundary.

For the case when two entangled objects are placed parallel and close to the Neumann boundary, the interaction potential Eq. (26) becomes

$$\begin{aligned} \Delta E_{AB}= & {} {\mp }\frac{\hbar G \omega _0}{r^5} \alpha (4p_0^3 r^3\sin {p_0 r} +\,12 p_0^2 r^2 \cos {p_0 r}\nonumber \\{} & {} - 24p_0 r \sin {p_0 r}- 24\cos {p_0 r}), \end{aligned}$$
(30)

where \(\vec r=\{r,0,0\}\) has been taken. In the near regime, i.e., \(r\ll c/\omega _{0}\), the leading term of Eq. (30) takes the form

$$\begin{aligned} \Delta E_{AB}\simeq \pm \frac{24\hbar G \omega _0}{r^5} \alpha \cos {p_0 r}, \end{aligned}$$
(31)

while in the far regime, i.e., \(r\gg c/\omega _{0}\), it becomes

$$\begin{aligned} \Delta E_{AB}\simeq \mp \frac{4\hbar G \omega _0^4}{r^2 c^3} \alpha \sin {(p_0 r+\phi _1)}. \end{aligned}$$
(32)

It shows, the interobject quantum gravitational quadrupole–quadrupole interaction behaves as \(r^{-5}\) and \(r^{-2}\) in the near and far regimes, respectively. Compare to the case without boundary [34], the Neumann boundary enhances the near-regime interaction potential about 1.1 times in the leading order and modifies the behavior of the interaction in the far regime. Also, one can find that the interaction potentials in these two special cases under Neumann boundary condition are quite different from that of the Dirichlet circumstance. Moreover, it is worth mentioning here that, thought we do not know how to realize the gravitational Neumann boundary condition through specific physical setup, a theoretical exploration may deserve.

5 Discussion

In this paper, we explore the quantum gravitational quadrupole–quadrupole interaction between two entangled non point-like objects in the presence of plane gravitational boundaries, based on the second-order DDC formalism. Two kinds of boundary conditions, i.e., Neumann and Dirichlet, are considered and our result shows that the interaction can be either enhanced or weakened depending on the geometrical arrangement of the two-object system with respect to the boundaries. In the limit when the two-object system is placed very close to the Dirichlet boundary, the near-regime interaction potential is larger than that of the pure vacuum case when the two objects are placed perpendicular to the boundary but smaller when parallel to it, while, in the far regime, such strong and weak relations between potentials are just opposite to that in the near regime. And, there exists a new \(r^{-2}\) far-regime behavior of the interaction potential under the perpendicular configuration. For the case of Neumann boundary, the strong and weak relations between the interaction potentials under perpendicular or parallel configurations and the case without boundary are opposite to the Dirichlet circumstance both in the near and far regimes. Besides, the novel \(r^{-2}\) far-regime behavior occurs for the parallel rather than perpendicular configuration in the presence of Neumann boundary.