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 [6–8]. 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 [9–13], 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 fluctu- ations 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 [1–4], 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 [17–26], 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 pro- vided 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 [28–44]. Here the bath can be b e-mail: [email protected] either in the vacuum state [28–34], or in a thermal state [35– 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 entangle- ment 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 grav- itational 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 Heisenberg–Coulomb 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 polar- izable 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 Bose–Einstein 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 [48–55]. 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 Born–Markov 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 Kossakowskl–Lindblad 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 .