Quantum Fluctuations of Spacetime Generate Quantum Entanglement

Eur. Phys. J. C (2018) 78:954 https://doi.org/10.1140/epjc/s10052-018-6433-5

Regular Article - Theoretical Physics

Quantum ﬂuctuations of spacetime generate quantum entanglement between gravitationally polarizable subsystems

Shijing Cheng1, Hongwei Yu1,a,JiaweiHu1,b 1 Department of Physics, Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, Hunan, China

Received: 25 February 2018 / Accepted: 11 November 2018 / Published online: 19 November 2018 © The Author(s) 2018

Abstract There should be quantum vacuum fluctuations of ations is the flight time fluctuations of a probe light signal spacetime itself, if we accept that the basic quantum princi- from its source to a detector [6Ð8]. Another expected effect ples we are already familiar with apply as well to a quan- is the Casimir-like force which arises from the quadrupole tum theory of gravity. In this paper, we study, in linearized moments induced by quantum gravitational vacuum fluctua- quantum gravity, the quantum entanglement generation at the tions [9Ð13], in close analogy to the Casimir and the Casimir- neighborhood of the initial time between two independent Polder forces [14,15]. gravitationally polarizable two-level subsystems caused by In the present paper, we are concerned with yet another fluctuating quantum vacuum gravitational fields in the frame- effect associated with quantum fluctuations of spacetime, work of open quantum systems. A bath of fluctuating quan- that is, quantum entanglement generation by quantum fluc- tum vacuum gravitational fields serves as an environment tuations of spacetime. Quantum entanglement is crucial to that provides indirect interactions between the two gravita- our understanding of quantum theory, and it has many inter- tionally polarizable subsystems, which may lead to entangle- esting applications in various novel technologies. However, ment generation. We find that the entanglement generation a significant challenge to the realization of these quantum is crucially dependent on the polarizations, i.e, they cannot technologies with quantum entanglement as a key resource get entangled in certain circumstances when the polariza- is the environmental noises that lead to the quantum to classi- tions of the subsystems are different while they always can cal transition. In the quantum sense, one environment that no when the polarizations are the same. We also show that the physical system can be isolated from is the vacuum that fluc- presence of a boundary may render parallel aligned subsys- tuates all the time. On one hand, there are obviously vacuum tems entangled which are otherwise unentangled in a free fluctuations of matter fields, and on the other hand, there space. However, the presence of the boundary does not help should also be quantum vacuum fluctuations of spacetime in terms of entanglement generation if the two subsystems itself if we accept that the basic quantum principles we are are vertically aligned. already familiar with apply as well to a quantum theory of gravity. Currently, there are discussions with vacuum fluctuations of quantized matter fields as inevitable environmental 1 Introduction noises that cause quantum decoherence [16]. Like matter fields, the fluctuations of gravitational fields, Recently, gravitational wave signals from black hole merging i.e. the fluctuations of spacetime itself, may also cause quan- systems have been directly detected [1Ð4], and this confirms tum decoherence. Since gravitation can not be screened the prediction based on Einstein’s general relativity over a unlike electromagnetism, gravitational decoherence is uni- hundred years ago [5]. Naturally, one may wonder what hap- versal. Different models for gravitational decoherence have pens if gravitational waves are quantized. One of the con- been proposed [17Ð26], see Ref. [27] for a recent review. sequences, if gravity is quantized, is the quantum fluctua- However, decoherence is not the only role the environment tions of spacetime itself. A direct result of spacetime fluctu- plays. In certain circumstances, the indirect interactions provided by the common bath may also create rather than destroy entanglement between the subsystems via spontaneous emis- a e-mail: [email protected] sion and photon exchange [28Ð44]. Here the bath can be b e-mail: [email protected] either in the vacuum state [28Ð34], or in a thermal state [35Ð 123 954 Page 2 of 8 Eur. Phys. J. C (2018) 78 :954 ω ω 38]. In Ref. [39], a general discussion showed that two inde- = σ (1) + σ (2), HS 3 3 (2) pendent atoms in a common bath can be entangled during a 2 2 Markovian, completely positive reduced dynamics. In Ref. ( ) ( ) where σ 1 = σ ⊗ σ , σ 2 = σ ⊗ σ , with σ (i = 1, 2, 3) [40], this approach has been applied in the study of the entan- i i 0 i 0 i i being the Pauli matrices, σ the 2 × 2 unit matrix, and ω the glement generation for two atoms immersed in a common 0 energy level spacing. H is the Hamiltonian of the external bath of massless scalar fields, and found that entanglement F gravitational fields, the details of which are not needed here. generation can be manipulated by varying the bath tempera- H describes the quadrupolar interaction between the grav- ture and the distance between the two atoms. This work was I itationally polarizable subsystems and the fluctuating gravi- further generalized to the case in the presence of a reflect- tational fields, which can be expressed as ing boundary [41,42], which showed that the presence of a boundary may offer more freedom in controlling entanglement generation. =−1 (1)( ) ( (1)( )) + (2)( ) ( (2)( )) , HI Qij t Eij x t Qij t Eij x t In this paper, we study whether a common bath of fluctuat- 2 (3) ing gravitational fields may generate entanglement between subsystems just as matter fields do. We consider a system (α) where Q (α = 1, 2) is the induced quadrupole moment composed of two gravitationally polarizable two-level sub- ij of the αth subsystem, and E =−∇∇ φ with φ being the systems in interaction with a bath of quantum gravitational ij i j gravitational potential. The quadrupolar interaction Hamilto- fields in vacuum. We will show that in certain conditions, nian (3) can be obtained as follows. The energy of a localized the two subsystems may get entangled due to the fluctuat- mass distribution ρ (x) in the presence of an external grav- ing gravitational fields. Like matter fields, the fluctuations m itational potential (x) is of gravitational fields are also expected to be modified if a boundary is present. Although it is generally believed 3 that gravitational waves can hardly be absorbed or reflected, V = ρm(x)(x)d x. (4) there have been proposals that the interaction between gravitational waves and quantum fluids might be significantly When (x) varies slowly over the region where the mass is enhanced compared with normal matter (see Ref. [45]fora located, it can be expanded as review), and recently there have been interesting conjectures that superconducting films may act as mirrors for gravita- ∂( ) ∂2( ) ( ) = ( ) + x0 + 1 x0 +··· , tional waves due to the so-called HeisenbergÐCoulomb effect x x0 xi xi x j (5) ∂xi 2 ∂xi ∂x j [46,47]. Therefore, we are interested in how the conditions for entanglement generation due to quantum fluctuations of so the quadrupolar interaction term reads spacetime is modified if such gravitational boundaries exist. ∂2 1 3 HI = d xρm(x)xi x j . (6) 2 ∂xi ∂x j 2 The basic formalism Since ∇2 = 0 in an empty space, the equation above can be rewritten as The system we consider consists of two gravitationally polarizable two-level subsystems, which are weakly coupled with 1 HI =− QijEij, (7) a bath of fluctuating quantum gravitational fields in vacuum. 2 In principle, the gravitationally polarizable system can be any quantum system that carries nonzero quadrupole polar- where izability. For example, a BoseÐEinstein condensate (BEC) is 3 1 2 such a system of which the quadrupole polarizability can be Qij = d xρm(x) xi x j − δijr , (8) estimated [12]. The Hamiltonian of the whole system takes 3 the following form and

∂2 H = H + H + H . 1 2 S F I (1) Eij =− + δij∇ . (9) ∂xi ∂x j 3

Here HS represents the Hamiltonian of the two gravita- In general relativity, Eij is deﬁned as the Weyl tensor Ci0 j0, tionally polarizable two-level subsystems, which can be which can be shown to coincide with the expression given described as in Eq. (9) in the Newtonian limit. Here Eij = Ci0 j0 and 123 Eur. Phys. J. C (2018) 78 :954 Page 3 of 8 954

=−1 kl ˆ = / its dual tensor Bij 2 iklC j0 are the gravito-electric with ki ki k. and gravito-magnetic tensors which satisfy the linearized We assume that initially the two subsystems are decou- Einstein field equations organized in a form similar to the pled with the quantum gravitational fields, i.e. ρtot(0) = Maxwell equations [48Ð55]. ρ(0) ⊗|00|, where ρ(0) denotes the initial state of the In the present paper, we are considering two quantum sub- two subsystems, and |0 the vacuum state of the gravita- systems in interaction with quantum gravitational vacuum tional fields. The density matrix of the total system satisfies (α) fluctuations, so the quadrupole moment Qij ought to be a the Liouville equation quantum operator. In the interaction picture, the quadrupole ∂ρ (t) operator can be written as tot =−i[H,ρ (t)]. (15) ∂t tot (α)( ) = (α)σ −iωt + (α)∗σ iωt , Qij t qij −e qij +e (10) Under the BornÐMarkov approximation, the reduced den- where σ+ = |+−|, σ− = |−+| are the raising and lower- sity matrix of the two gravitationally polarizable subsystems (α) = | (α)| ρ(t) = Tr [ρ (t)] satisfies the KossakowsklÐLindblad ing operators respectively, and qij 0 Qij 1 is the tran- F tot sition matrix element of the quadrupole operator of the αth master equation [56,57], (α) subsystem, which is symmetric and traceless, i.e., q = 0 ii ∂ρ( ) (α) = (α) t =−[ ,ρ( )]+D[ρ( )], and qij q ji , thus leaving only five independent compo- i Heff t t (16) (α) (α) (α) (α) (α) ∂t nents q11 , q22 , q12 , q13 and q23 . The quadrupole moments are induced by the fluctuating gravitational fields. In this where paper, we assume that the gravitational quadrupole polar- 2 3 izabilities are real, and so are the quadrupole moments. The = − i (αβ)σ (α)σ (β), Heff HS Hij i j (17) gravito-electric field Eij is also supposed to be quantized. 2 α,β=1 i, j=1 The metric for a flat spacetime metric with a perturbation = η + η can be expressed as gμν μν hμν, where μν denotes and the flat spacetime metric, and hμν a linearized perturbation. 2 3 In the transverse traceless (TT) gauge, the spacetime pertur- D[ρ( )]=1 (αβ) t Cij bation can be quantized as [7] 2 α,β= , = 1 i j 1 (β) (α) (α) (β) (α) (β) h = [a ,λe (k,λ)f + H.c.], (11) × σ ρσ − σ σ ρ − ρσ σ . ij k ij k 2 j i i j i j (18) k,λ Introducing the Fourier and Hilbert transforms of the gravita- (αβ) − 1 i(k·x−ωt) ( − ) = where f = (2ω(2π)3) 2 e is the field mode and tional field correlation function Gijkl t t k ( , ) ( , ) eμν(k,λ)is the polarization tensor with ω =|k|=(k2+k2+ Eij t xα Ekl t xβ , x y 1 ∞ 2) 2 λ (αβ) ω (αβ) kz . Here H.c. denotes the Hermitian conjugate, labels the G (ω) = dtei t G (t), (19) ¯ = = π = ijkl ijkl polarization state, and units in which h c 32 G 1 −∞ have been adopted. From the definition of E ,wehave ∞ ij K(αβ)(ω) = ( ) iωt (αβ)( ) ijkl dt sgn t e Gijkl t 1 ¨ −∞ E = h , (12) (αβ) ij ij ∞ G (λ) 2 P ijkl = dλ , (20) ∂ πi −∞ λ − ω where a dot means ∂t . The correlation function can then be obtained as [7] where P denotes the principal value. The coefficient matrix (αβ) (αβ) 1 3 3 Hij and Cij can then be expressed as Eij(x)Ekl(x )= d k eij(k,λ)ekl(k,λ)ω 8(2π)3 λ (αβ) = (αβ)δ − (αβ) δ − (αβ)δ δ , Cij A ij iB ijk 3k A 3i 3 j (21) ik·(x−x) −iω(t−t) ×e e , (13) (αβ) = A(αβ)δ − B(αβ) δ − A(αβ)δ δ , Hij ij i ijk 3k 3i 3 j (22) where where e (k,λ)e (k,λ) = δ δ + δ δ − δ δ + kˆ kˆ kˆ kˆ ij kl ik jl il jk ij kl i j k l (αβ) = 1 G(αβ)(ω) + G(αβ)(−ω) , λ A 16 + kˆ kˆ δ + kˆ kˆ δ − kˆ kˆ δ (αβ) 1 (αβ) (αβ) i j kl k l ij i l jk B = G (ω) − G (−ω) , (23) ˆ ˆ ˆ ˆ ˆ ˆ 16 − ki kkδ jl − k j kl δik − k j kkδil, A(αβ) = 1 K(αβ)(ω) + K(αβ)(−ω) , (14) 16 123 954 Page 4 of 8 Eur. Phys. J. C (2018) 78 :954 (αβ) 1 (αβ) (αβ) B = K (ω) − K (−ω) , (24) 3.1 Entanglement generation in the free Minkowski 16 spacetime with We assume the two gravitationally polarizable subsystems are placed at the points x = (0, 0, 0) and x = (0, 0, L) 3 1 2 (αβ) (α)∗ (β) (αβ) respectively, where L denotes the separation between the two G (ω) = q q G (ω), (25) ij kl ijkl subsystems, as shown in Fig. 1. The explicit expressions of i, j,k,l=1 the coefficients Ai and Bi in Eq. (28) are given in appendix 3 = = K(αβ)(ω) = (α)∗ (β)K(αβ)(ω). A, see Eqs. (A4)Ð(A6). Since A1 B1 and A2 B2,it qij qkl ijkl (26) , , , = is clear that the criterion for entanglement generation (28) i j k l 1 2 > becomes A3 0. That is, entanglement can be generated as (α) long as A , which is dependent on q and ωL, is nonzero. Here the effective Hamiltonian Heff corresponds to the 3 ij Lamb-like shift and an environment induced direct coupling When the polarizations of the two subsystems are the (α) =−(α) =|| (α) = between the two subsystems. In the following, we neglect same, for example, when q11 q22 q or q11 − (α) =| | this term and concentrate on the effects produced by the dis- q33 q and other components being zero, we plot, in sipative part D[ρ(t)]. Fig. 2, the coefficient A3 as a function of ωL in the unit of 5 0, with 0 =|q|ω /π.AsωL grows, A3 oscillates around zero with a decreasing amplitude, and it approaches zero as ωL →∞. That is, when the separation of the two subsys- 3 The condition for entanglement generation tems is infinitely large, they are always separable. Even for finite separations, there are some special values of ωL that In this section, we investigate whether the two subsystems give A3 = 0, so entanglement can not be generated at those become entangled or not at a neighborhood of the initial separations. time with the help of the partial transposition criterion [58, When the polarizations of the two subsystems are the 59], i.e., a two-atom state ρ is entangled if and only if the (1) = (2) = same, i.e., qij qij qij, there are no solutions to operation of the partial transposition of ρ does not preserve its F = K = 0 (see Eqs. (A5)Ð(A6)) for any given L, i.e., positivity. We assume that the initial state is ρ(0) = |++|⊗ irrespective of the polarization of the two subsystems, the |−−|, which is pure and separable, where |− and |+ are entanglement between the two subsystems with the same the ground and excited state respectively. In this case, the polarization can always be generated for a given separation. partial transposition criterion can be found to be equivalent to Therefore, when the polarizations of the two subsystems are the following condition, i.e., entanglement can be generated the same, they can always get entangled if the separation is at the neighbourhood of t = 0 if and only if [39,40] finite, except for a series of special values of ωL which are the zero points of A3. Similar conclusions have been drawn ( ) ( ) ( ) in the case of two independent atoms coupled with mass- μ|C 11 |μν|(C 22 )T |ν < |μ|Re(C 12 )|ν|2, (27) less scalar (matter) fields [40]. This suggests that there is no difference between spacetime fluctuations and matter fields where the subscript T denotes the matrix transposition, and the three-dimensional vectors μi , νi are μi = νi = {1, −i, 0}. For brevity, we make the following substitution in (11) (22) (12) (21) Eq. (21), A → A1, A → A2, and A , A → A3, and do the same to the coefficients B(αβ). Plugging the new (αβ) expressions of Cij into Eq. (27), leads to

2 + > . A3 B1 B2 A1 A2 (28)

Now we investigate the condition for entanglement generation between a pair of gravitationally polarizable subsystems both in a free space and in a space with a Dirichlet boundary. In the presence of a boundary, we will consider cases of subsystems aligned parallel to and perpendicular to the Fig. 1 Two gravitationally polarizable subsystems separated from boundary respectively. each other by a distance of L in the z-axis 123 Eur. Phys. J. C (2018) 78 :954 Page 5 of 8 954 +(ω2 2 − ω4 4 + ) (1) (2) + (1) (2) , L L 3 q11 q22 q22 q11 (31) and ( ) ( ) K = 4(−3 + 2ω2 L2)q 1 q 2 1 12 12 ( ) ( ) ( ) ( ) +(2ω2 L2 − 9) q 1 q 2 + q 1 q 2 11 11 22 22 −( + ω2 2) (1) (2) + (1) (2) . 3 2 L q11 q22 q22 q11 (32) It is obvious that when (1) (2) = (1) (2) =− (1) (2), q11 q11 q22 q22 q12 q12 (33)

A3 is always zero, i.e., the entanglement generation cannot Fig. 2 The coefficient A as a function of ωL. The red dashed line 3 happen. Otherwise, the value of A3 oscillates around zero and green solid line correspond to two different polarizations of the ω (α) =− (α) =| | (α) =− (α) = as L varies with a changing amplitude. This reveals a clear subsystems respectively, i.e. q11 q22 q and q11 q33 |q|, with other components being zero difference between entanglement generation by vacuum fluctuations of massless scalar (matter) fields and that of the spacetime, that is, entanglement generation may never hap- fluctuations when the entanglement generation is concerned pen for certain polarizations if only spacetime fluctuations with two subsystems of the same gravitational polarization. are considered. When the polarizations of the two subsystems are differ- = = ent, it can be shown by solving the equations F K 0 3.2 Entanglement generation for two gravitationally that the two subsystems may not get entangled for any given polarizable subsystems aligned parallel to the boundary separations when the components of quadrupole moments of the two subsystems satisfy the following conditions simulta- Now we concentrate on the effect of a Dirichlet boundary neously, (recall our comments in the Introduction for gravitational ( ) ( ) ( ) ( ) ( ) q 1 = 0, q 1 q 2 + q 1 q 2 = 0, boundaries ) on entanglement generation between two grav- 33 13 13 23 23 itationally polarizable subsystems in a bath of fluctuating (1) (2) + (1) (2) − (2) = . 2 q12 q12 q11 q11 q22 0 (29) gravitational fields. We assume that the Dirichlet boundary is located at y = 0inthexoz plane, and the two subsys- Note that we do not distinguish the two subsystems here, tems separated from each other by a distance L is placed i.e. the superscripts (1) and (2) in the equations above can (α) along the z direction at a distance of y, see Fig. 3 .The be exchanged. Recall that q is traceless, so q = 0 indi- ij 33 coefficients A , A , A and B , B , B can be calculated (α) =− (α) 1 2 3 1 2 3 cates that q11 q22 . The components of the quadrupole from Eqs. (23) and (25) with the Fourier transform of Eq. moment can be interpreted as follows. The diagonal com- (A7), and the results are lengthy, so we do not give them ponents q11, q22 and q33 respectively represent the contri- = explicitly here. As before, we will examine when A3 0 butions to the total mass quadrupole moment from the mass to determine whether entanglement can be generated or not, distributed along the x axis, y axis and z axis, and the off- = ( = , , ) since Ai Bi i 1 2 3 still holds in the present circum- diagonal components q12, q13 and q23 respectively represent stances. In this case, the conditions that entanglement cannot the contributions to the total mass quadrupole moment from be generated for any L are solved as follows the mass distributed in the xoy, xozand yoz plane. Therefore, (1) (2) = , (1) (2) = , (1) (2) = , when the mass distributions of the two subsystems induced by q12 q12 0 q13 q13 0 q12 q13 0 quantum gravitational vacuum fluctuations satisfy the con- (α) = (α) = (α) = (α = ), q22 q33 q23 0 1or2 (34) dition (29), the two subsystems remain disentangled. As an example, we consider the case when the two sub- Now we consider the same example as before, i.e. when ( ) xoy q 1 = the two subsystems are only polarizable in the xoy plane, systems are only polarizable in the plane, i.e. 3i ( ) ( ) (2) 1 = 2 = q = 0. In this case, we have q3i q3i 0. In this case, we have 3i 1 F1 sin (ωL) + K1 cos (ωL)ωL = 1 [ (ω ) + (ω )ω ], A = A3 F1 sin L K1 cos L L (30) 3 π 5 128π L5 128 L M sin (ωR) + N cos (ωR)ωR where + 1 1 , (35) R9 = ( − ω2 2 + ω4 4) (1) (2) F1 4 3 3 L L q12 q12 where the explicit expressions of M1 and N1 are given in + − ω2 2 + ω4 4 (1) (2) + (1) (2) 9 5 L L q11 q11 q22 q22 appendix (see Eqs. (A9)Ð(A10)). Obviously, if entanglement 123 954 Page 6 of 8 Eur. Phys. J. C (2018) 78 :954

Fig. 4 Two independent gravitationally polarizable subsystems are placed on the z-axis with a separation L. A Dirichlet boundary is located in the xoy plane, and the distance to the closer one is z Fig. 3 Two independent gravitationally polarizable subsystems are placed along the z direction with a separation L, which are at a distance y from a Dirichlet boundary in the xoz plane

pendent gravitationally polarizable two-level subsystems in generation cannot happen for any given L and y, the corre- interaction with a bath of fluctuating quantum gravitational sponding coefficients satisfy F1 = K1 = M1 = N1 = 0, fields in vacuum both with and without a boundary. The and then we have partial transposition criterion has been applied to determine ( ) ( ) ( ) ( ) whether entanglement can be generated or not at the begin- q 1 q 2 = 0, q 1 q 2 = 0. (36) 12 12 11 11 ning of evolution. In the free space case, when the polariza- Comparing the result above with that obtained in the corre- tions of the two subsystems are the same, the two subsys- sponding case without a boundary, we find the condition Eq. tems can always get entangled as long as the separation is (36) is a special case of Eq. (33). That is, if the components finite. This is similar to what happens when the fluctuations (1) (2) = (1) (2) = of scalar (matter) fields are considered. When the polariza- of the quadrupole moments satisfy q11 q11 q22 q22 − (1) (2) = tions of the two subsystems are different, they cannot get q12 q12 0, the two subsystems can be entangled in the presence of a boundary, while they remain separable in the entangled in certain circumstances, whatever the separation free space. is. This is in sharp contrast with the case of massless scalar (matter) fields. In the presence of a boundary, we find that in 3.3 Entanglement creation for two gravitationally some of the cases where entanglement cannot be generated polarizable subsystems aligned vertical to the boundary in a free space, the presence of a boundary placed parallel to the alignment of the subsystems may render the subsystems Now we investigate the case when the alignment of the entangled, but this does not happen for a boundary placed two gravitationally polarizable subsystems is vertical to the vertically. boundary. We assume that the Dirichlet boundary is placed Acknowledgements This work was supported in part by the NSFC at z=0 in the xoy plane and the two subsystems are placed on under Grants No. 11435006, No. 11690034, and No. 11805063. the z-axis with a separation L, the distance from the boundary to the nearer subsystem being z, see Fig. 4. Open Access This article is distributed under the terms of the Creative Following the same procedures, we find that the conditions Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, that entanglement generation cannot happen for any given and reproduction in any medium, provided you give appropriate credit L and z are the same with that obtained in the case with- to the original author(s) and the source, provide a link to the Creative out a boundary, see Eq. (29). That is, when the quadrupole Commons license, and indicate if changes were made. 3 moments of the two subsystems satisfy certain conditions Funded by SCOAP . such that entanglement cannot be generated in the free space, it cannot be generated in the presence of a boundary placed vertically to the alignment of the subsystems either. Appendix A 4 Summary The appendix mainly shows some calculations of the coeffi- In this paper, we have investigated the entanglement genera- cients Ai and Bi in Eq. (28) for the condition of entanglement tion at the neighborhood of the initial time between two inde- generation. 123 Eur. Phys. J. C (2018) 78 :954 Page 7 of 8 954

A.1: The case in the free Minkowski spacetime The coefﬁcients can be obtained from Eq. (23) and Eq. (25) as In the free Minkowski spacetime, we assume the two gravita- ω5 (1) (1) 2 (1) tionally polarizable subsystems are placed at the points x1 = A = B = (q − q ) + (q 1 1 480π 11 22 11 (0, 0, 0) and x2 = (0, 0, L) respectively, where L denotes − (1))2 + ( (1) − (1))2 the separation between the two subsystems. Straightforward q33 q22 q33 (αβ) calculations of Eqs. (13) and (19) show that G (−ω) = 0, + [( (1))2 + ( (1))2 + ( (1))2] , ijkl 6 q12 q13 q23 (αβ) 5 and the nonzero components of G (ω) can be written as ω ( ) ( ) ( ) ijkl A = B = (q 2 − q 2 )2 + (q 2 2 2 480π 11 22 11 ( ) ( ) ( ) (αβ) ω5 (αβ) − q 2 )2 + (q 2 − q 2 )2 G (ω) = f (ω, L). 33 22 33 ijkl π ijkl (A1) + [( (2))2 + ( (2))2 + ( (2))2] , 6 q12 q13 q23 α = β (11) = (22) 1 When ,wehave f f , where A3 = B3 = [F sin (ωL) ijkl ijkl 128π L5 + K cos (ωL)ωL], (A4) ( ) ( ) ( ) 1 f 11 = f 11 = f 11 = , 1111 2222 3333 15 with ( ) ( ) ( ) 1 11 = 11 = 11 =− , ( ) ( ) ( ) ( ) f1122 f1133 f2233 = ( − ω2 2 + ω4 4) 1 2 + 1 2 30 F 9 5 L L q11 q11 q22 q22 ( ) ( ) ( ) 1 f 11 = f 11 = f 11 = , + ( + ω2 2 − ω4 4) (1) (2) + (1) (2) 1212 1313 2323 20 3 L L q11 q22 q22 q11 ( ) ( ) ( ) ( ) f 11 = f 11 = f 11 = f 11 , + ( − ω2 2 + ω4 4) (1) (2) 1122 2211 3311 3322 4 3 3 L L q12 q12 ( ) ( ) ( ) ( ) ( ) ( ) f 11 = f 11 = f 11 = f 11 = f 11 = f 11 + (− + ω2 2)( (1) (2) + (1) (2)) 1212 1221 1331 2332 2112 3113 24 2 L q13 q13 q23 q23 ( ) ( ) ( ) ( ) = f 11 = f 11 = f 11 = f 11 , (A2) + (ω2 2 − )( (1) (2) + (1) (2) 3223 2121 3131 3232 4 L 3 q33 q11 q11 q33 ( ) ( ) ( ) ( ) ( ) ( ) + q 1 q 2 + q 1 q 2 − 2q 1 q 2 ), (A5) α = β (12) = (21) 33 22 22 33 33 33 and when ,wehave fijkl fijkl , where and (12) f (ω, L) ( ) ( ) ( ) ( ) 1111 K = (2ω2 L2 − 9) q 1 q 2 + q 1 q 2 ωL(−9 + 2ω2 L2) cos (ωL) + (9 − 5ω2 L2 + ω4 L4) sin (ωL) 11 11 22 22 = , ( ) ( ) ( ) ( ) 8ω5 L5 − (3 + 2ω2 L2) q 1 q 2 + q 1 q 2 ( ) 11 22 22 11 f 12 (ω, L) 3333 + (− + ω2 2) (1) (2) −3ωL cos (ωL) + (3 − ω2 L2) sin (ωL) 4 3 2 L q12 q12 = , ( ) ( ) ( ) ( ) ω5 L5 − 8(−6 + ω2 L2)(q 1 q 2 + q 1 q 2 ) ( ) 13 13 23 23 f 12 (ω, L) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1122 + ( 1 2 + 1 2 + 1 2 + 1 2 12 q33 q11 q11 q33 q33 q22 q22 q33 −ωL(3 + 2ω2 L2) cos (ωL) + (3 + ω2 L2 − ω4 L4) sin (ωL) = , − (1) (2)). 8ω5 L5 2q33 q33 (A6) (12)(ω, ) f1133 L ω (ω ) + (− + ω2 2) (ω ) A.2: The case for two subsystems aligned parallel to the = 3 L cos L 3 L sin L , 2ω5 L5 boundary (12)(ω, ) f1212 L ω (− + ω2 2) (ω ) + ( − ω2 2 + ω4 4) (ω ) As for two gravitationally polarizable subsystems aligned = L 3 2 L cos L 3 3 L L sin L , 8ω5 L5 parallel to the boundary, we assume that the boundary is (12)(ω, ) located at y = 0inthexoz plane, and the two subsystems f1313 L ω ( − ω2 2) (ω ) + (− + ω2 2) (ω ) separated from each other by a distance L is placed along the = L 6 L cos L 3 2 L sin L , 4ω5 L5 z direction at a distance of y. With the help of the method (12) = (12), (12) = (12), of images, the correlation functions can be written in the f1111 f2222 f1122 f2211 following form (12) = (12) = (12) = (12), f1133 f2233 f3311 f3322 (12) = (12) = (12) = (12), ( , ) ( , ) = ( , ) ( , ) f1212 f1221 f2112 f2121 Eij t xα Ekl t xβ tot Eij t xα Ekl t xβ free (12) = (12) = (12) = (12) − ( , ) ( , ) , f1313 f2323 f1331 f2332 Eij t xα Ekl t xβ bnd = (12) = (12) = (12) = (12). f3113 f3223 f3131 f3232 (A3) (A7) 123 954 Page 8 of 8 Eur. Phys. J. C (2018) 78 :954

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