Quantum interaction between two entangled non point-like objects in the presence of boundaries

We explore, in the framework of linearized quantum gravity, the quantum gravitational quadrupole-quadrupole interaction between two entangled non point-like objects in the presence of both Dirichlet and Neumann boundaries. The results show that, compared to the case without boundaries, the interaction can be either enhanced or weakened depending on the geometrical arrangement of the objects 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^{-2}$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^{-2}$$\end{document} far-regime behavior occurs for the parallel rather than perpendicular configuration in the presence of Neumann boundary.

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 CPlike 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.

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 and |e respectively, and the corresponding energy spacing is labeled as ω 0 . The total Hamiltonian of the system is 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 where Q ξ i j is the quadrupole moment operator of object ξ (ξ = A, B), and E i j is the gravitoelectric tensor of the fluctuating gravitational fields in vacuum defined as E i j = −c 2 C 0i0 j by an analogy between the linearized Einstein field equations and the Maxwell equations [43][44][45][46][47][48][49], where C μναβ 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 μν . Then, the gravitoelectric tensor E i j can be expressed as 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] where h λ i j, p is the gravitational field mode with wave vector p, a λ (ω) represents the annihilation operator of the fluctuating gravitational fields, λ denotes the polarization states, i j ( 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, and denote the vacuum state of the gravitational field as |0 . To investigate the quantum gravitational quadrupolequadrupole 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] Here are respectively the statistical functions introduced for the gravitational field and the objects, which take the form where the label "F" denotes the free part of the operator, i.e., the part presents even in the absence of interaction.

Dirichlet boundary condition
Now let us consider the quantum gravitational quadrupolequadrupole 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 λ i j, p | z=0 = 0 and the gravitational field mode h λ i j, p can be expressed as where p − = {p x , p y , − p z }. Then, according to Eqs. (3) and (8), the statistical function of the gravitational field in the presence of a Dirichlet boundary can be expressed as where Note here that r = | r | is the distance between the two objects and r − = | 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] λ e (λ) wherep i is the i-th component of the unit vector p i /| p|. From this summation of polarization tensors, we can obtain and where H ρ i jkl denotes the differential operator to ρ, ∇ 2 = ∂ i ∂ i , and σ i j = δ i j − 2δ i3 δ j3 . Substituting Eqs. (12) and (13) into Eq. (10) and performing the integration in the spherical coordinate, we obtain As for the statistical function C AB i jkl (t, t ) of the objects, its form can be obtained as where the sign ± correspond to the symmetric and antisymmetric states respectively,Q i j 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 where we have introduced p 0 = ω 0 /c and t = t − t 0 . θ(x) is a step function, which equals to 1 when x > 0 but 0 when x < 0. Let us note here that H r i jkl cos p 0 r r = 1 r 5 [(δ ik δ jl + δ il δ jk − δ i j δ kl +r ir j δ kl +r krl δ i j −r irk δ jl −r irl δ jk −r jrk δ il −r jrl δ ik +r ir jrkrl )r 4 p 4 0 cos p 0 r wherer i is the i-th component of the unit vector 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 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.

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 r = {0, 0, r } and, for convenience, assume r < c t. When the two-object system is very close to the boundary, i.e., r ∼ r − , the interaction potential Eq. (16) becomes where the isotropic gravitational polarizability α = α i jkl = Q i jQkl /hω 0 defined in Ref. [34] has been employed again. In the near regime, i.e., r c/ω 0 , the leading term of the potential Eq. (18) takes the form while in the far regime, i.e., r c/ω 0 , it becomes where φ 1 = arcsin 3 9+ p 2 0 r 2 . This shows that, when the two entangled objects are placed perpendicular to the Dirichlet boundary, the interobject quantum gravitational quadrupolequadrupole 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 ∼ r − , the interaction potential Eq. (16) becomes Gω 0 r 5 α(2 p 4 0 r 4 cos p 0 r + 6 p 2 0 r 2 cos p 0 r − 18 p 0 r sin p 0 r − 18 cos p 0 r ), (21) where r = {r, 0, 0} has been applied. In the near regime, i.e., r c/ω 0 , the leading term of Eq. (21) becomes while in the far regime, i.e., r c/ω 0 , it reduces to In this case, the interobject quantum gravitational quadrupolequadrupole 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.

Neumann boundary condition
For the Neumann boundary condition, the field mode satisfies ∂ z h λ i j, p | z=0 = 0 and thus can be written as From the above equation, one can show that the statistical function of the gravitational field χ F i jkl ( x A (t), x B (t )) becomes and the interaction potential Eq. (6) then reads 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.

Two special cases
For the special case of two entangled objects placed perpendicular to the Neumann boundary, we take r = {0, 0, r } and consider the case when the two-object system is very close to the boundary, i.e., r ∼ r − . The interaction potential Eq. (26) becomes E AB = ∓h Gω 0 r 5 α(2 p 4 0 r 4 cos p 0 r − 4 p 3 0 r 3 sin p 0 r −6 p 2 0 r 2 cos p 0 r + 6 p 0 r sin p 0 r + 6 cos p 0 r ), (27) where r < c t is assumed. In the near regime, i.e., r c/ω 0 , the leading term of the potential Eq. (27) is while in the far regime, i.e., r c/ω 0 , it takes the form where φ 2 = arcsin 2 4+ p 2 0 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 Gω 0 r 5 α(4 p 3 0 r 3 sin p 0 r + 12 p 2 0 r 2 cos p 0 r −24 p 0 r sin p 0 r − 24 cos p 0 r ), where r = {r, 0, 0} has been taken. In the near regime, i.e., r c/ω 0 , the leading term of Eq. (30) takes the form while in the far regime, i.e., r c/ω 0 , it becomes It shows, the interobject quantum gravitational quadrupolequadrupole 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 nearregime 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.

Discussion
In this paper, we explore the quantum gravitational quadrupolequadrupole interaction between two entangled non pointlike 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.