D 103, 044017 (2021)

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Noise and decoherence induced by

Sugumi Kanno ,1,2 Jiro Soda,3 and Junsei Tokuda 3 1Department of , Osaka University, Toyonaka 560-0043, Japan 2Department of Physics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan 3Department of Physics, Kobe University, Kobe 657-8501, Japan

(Received 12 October 2020; accepted 24 December 2020; published 9 February 2021)

We study noise and decoherence induced by gravitons. We derive a Langevin equation of geodesic deviation in the presence of gravitons. The amplitude of noise correlations tells us that large squeezing is necessary to detect the noise. We also consider the decoherence of spatial superpositions of two massive particles caused by gravitons in the vacuum state and find that gravitons could give a relevant contribution to the decoherence. The decoherence induced by gravitons would offer new vistas to test quantum in tabletop experiments.

DOI: 10.1103/PhysRevD.103.044017

I. INTRODUCTION equation in order to obtain the noise in the detectors. An understanding of the of gravity has been a The noise is usually associated with the decoherence central issue in physics since the discovery of general induced by between a system and relativity and quantum . Nevertheless, no one has gravitons [11]. Thus, as an approach to testing quantum succeeded in constructing quantum of gravity. In gravity, it would be important to understand the noise induced particular, the existence of gravitons is still obscure [1].In by gravitons and then the decoherence caused by the noise. these situations, it is legitimate to doubt the necessity of The decoherence due to gravity in the context of quantum canonical of gravity [2]. Hence, it is worth superposition of massive objects has been investigated [12].1 seeking an experimental evidence of . The effect of a on the Usually, theorists explore the field of quantum gravity of nonrelativistic particles was investigated by using the at energy scales near the Planck scale. However, it is far influence functional method and it is shown that the beyond the capacity of the current or future particle decoherence due to a gravitational field is effective in accelerators. Instead, cosmological observations have been the energy eigenstate basis [14]. Moreover, based on the exploited for probing high-energy physics. In fact, cosmo- approach, the decoherence rate was logical observations suggest that the large scale structure of derived under the Markovian approximation (the assumption the stems from the quantum fluctuations during that the correlation time is very short) [15]. The quantum the inflationary stage. It is natural to consider that primor- Markov master equation for gravitating was derived in dial gravitational waves are also generated directly from [16]. Following the paper of decoherence in the context of the quantum fluctuations. Hence, one possible approach to electromagnetic dynamics [17], the effect of the gravitational testing quantum gravity is to study the nonclassicality of bremsstrahlung on the destruction process of quantum super- primordial gravitational waves [3,4]. Recently, as an alter- position has been considered [18]. More explicit formulation native approach, tabletop experiments are drawing attention along this direction has been given in [19]. The formalism is [5–7]. Remarkably, based on the development of quantum further applied to the system of atoms [20]. Recently, this information, several ideas to test the quantum nature of possibility was discussed again under the Markovian gravity through laboratory experiments are proposed [8,9]. assumption [21]. The decoherence due to quantum fluctua- More recently, as a new probe of gravitons, noise in the tions of geometry caused by gravitons is also discussed in lengths of the arms of gravitational wave detectors is [22]. Since the Markovian approximation cannot be applied discussed by using a path integral approach [10]. One of to the decoherence caused by the gravitational bremsstrah- our goals in this paper is to derive the quantum Langevin lung, the non-Markovian decoherence process is analyzed in [23]. In the above papers, the minimal coupling of a metric to a particle has been considered. However, from the point of Published by the American Physical Society under the terms of view of the , the does the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, 1The cosmological decoherence due to thermal gravitons can 3 and DOI. Funded by SCOAP . be found in [13].

2470-0010=2021=103(4)=044017(14) 044017-1 Published by the American Physical Society SUGUMI KANNO, JIRO SODA, and JUNSEI TOKUDA PHYS. REV. D 103, 044017 (2021)

2 2 i j not feel gravity. Hence, the deviation of the geodesics ds ¼ −dt þðδij þ hijÞdx dx ; ð2:1Þ is studied in [24]. Thus, when considering the decoherence i due to gravitons, it would be necessary to take into where t is the time, x are spatial coordinates, δij and hij are account both the non-Markov process and the equivalence the Kronecker delta and the metric perturbations which principle. satisfy the transverse traceless conditions hij;j ¼ hii ¼ 0. In this paper, we study the and decoherence The indices ði; jÞ run from 1 to 3. Substituting the metric in order to probe gravitons and ultimately quantum gravity. Eq. (2.1) into the Einstein-Hilbert action, we obtain the First, we target quantum noise in gravitational wave detectors. quadratic action When gravitational waves arrive at the interferometer, Z Z the suspended mirrors interact with the gravitational waves. 1 ffiffiffiffiffiffi 1 4 p− ≃ 4 _ ij _ − ij;k The mirror interacts with an environment of gravitons 2κ2 d x gR 8κ2 d x½h hij h hij;k; ð2:2Þ quantum mechanically. By using Fermi normal coordinates, we evaluate the effect of quantum noise induced by gravitons 2 where κ ¼ 8πG and a dot denotes the derivative with on the suspended mirrors. We show that the noise in the respect to the time. We can expand the metric field h ðxi;tÞ squeezed state can be sizable. The results we obtain by using ij in terms of the Fourier modes the quantum Langevin equation are consistent with those derived by using the path integral method in [10]. 2κ X i ffiffiffiffi A ik·x A k Second, as our main goal in this paper, we consider a hijðx ;tÞ¼p hkðtÞe eijð Þ; ð2:3Þ tabletop experiment by using two massive particles, one of V k;A which is superposed spatially, so called, the ’ A k of Schrödinger s cat. Without using the Markovian where we introduced the polarization tensor eijð Þ nor- assumption, we give a formula for the decoherence rate A k B k δAB malized as eij ð Þeijð Þ¼ . Here, the index A denotes of the superposition induced by gravitons. We then evaluate the linear polarization modes A ¼þ; ×. Note that we the decoherence rate for some simple configurations of consider finite volume V ¼ LxLyLz and discretize the k- superposition states and show that the decoherence due to mode with a width k ¼ð2πn =L ; 2πn =L ; 2πn =L Þ gravitons could be a relevant contribution. To explore the x x y y z z where n ¼ðn ;n ;n Þ are integers. Substituting the for- decoherence process due to gravitons would be a first step x y z mula (2.3) into the quadratic action (2.2), we get toward discovery of gravitons in a laboratory. Z   The organization of the paper is as follows: In Sec. II,we X 1 1 describe geodesics in the background and derive a ≃ _ A k _ A k − 2 A k A k Sg dt 2 h ð ;tÞh ð ;tÞ 2 k h ð ;tÞh ð ;tÞ ; Langevin type equation of the system by eliminating the k;A environment of gravitons. In Sec. III, we evaluate the noise ð2:4Þ correlation functions and show that the noise can be if the gravitons are in the squeezed state. In where we used k ¼jkj. Sec. IV, we discuss the decoherence induced by gravitons and detectability of gravitons. We give a formula for the decoherence rate and evaluate it for simple cases. The final B. Action for two test particles section is devoted to the conclusion. A detailed calculation When gravitational waves arrive at the laser interferom- of a momentum integral is presented in the Appendix A and eters, the suspended mirrors interact with the gravitational the derivation of decoherence functionals is given in waves. Let us regard the mirror as a point particle for Appendix B. We work in the natural unit: c ¼ ℏ ¼ 1. simplicity. A single particle, however, does not feel the gravitational waves because of Einstein’s equivalence II. IN THE principle at least classically. To see the effect of the GRAVITON BACKGROUND gravitational waves, we need to consider two massive particles and measure the geodesic deviation between them. In this section, we present a model to study quantum In this subsection, we evaluate the effect of gravitational mechanics in the graviton background. It gives rise to the waves on the two particles by introducing an appropriate basis for studying the noise and the decoherence due to low coordinate system called the Fermi normal coordinates energy gravitons. In particular, we derive the quantum along one of their geodesics γτ (see Fig. 1). The Fermi Langevin equation. normal coordinate system represents a local inertial frame. The dynamics of the other geodesic of particle γτ0 is A. Gravitational waves described by the position xiðtÞ¼ξiðtÞ in the vicinity of We consider gravitational waves in . the point Pð0;tÞ and ξi represents the deviation. The metric describing gravitational waves in the transverse The action for the two test particles along the geodesics traceless gauge is expressed as γτ; γτ0 is given by

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waves will be suppressed because of the equivalence principle. Hence, we consider the action of the form Z  X X  m 2 m κ ≃ ξ_i ffiffiffiffi ̈A k A k ξiξj Sp dt 2 þ 2 p h ð ;tÞeijð Þ ; γτ0 V A k≤Ωm ð2:9Þ

0 where the metricPhijð ;tÞ is replaced by the Fourier modes in Eq. (2.3) and represents the mode sum with the k≤Ωm Ω ∼ ξ−1 UV cutoff m . We see a qubic derivative interaction FIG. 1. Two neighboring timelike geodesics (γτ; γτ0 ) separated appear in the above action. by ξi are depicted in the blue lines and the green lines show two neighboring spacelike geodesics (γ ; γ 0 ) orthogonal to the geo- s s C. Particles in an environment of gravitons desic γτ in the spacelike hypersurface Σs. We introduce the Fermi normal coordinate system using the orthogonal geodesics at the From Eqs. (2.4) and (2.9), the total action S ¼ S þ S 0 g p point Pð ;tÞ. we consider is given by Z   X 1 1 Z Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ≃ _ A k _ A k − 2 A k A k S dt 2 h ð ;tÞh ð ;tÞ 2 k h ð ;tÞh ð ;tÞ − τ − − ξi ξ_ μξ_ ν k;A Sp ¼ m d ¼ m dt gμνð ;tÞ ; ð2:5Þ Z   γ γ X X τ0 τ0 m 2 m κ ξ_i ffiffiffiffi ̈A k A k ξiξj þ dt 2 þ 2 p h ð ;tÞeijð Þ : μ i V k≤Ω where ξ ¼ðt; ξ ðtÞÞ. Note that we omit the action for A m γ the particle along τ because it is in the inertial frame and ð2:10Þ then the action has no dynamical variables. The metric gμν up to the second order of xi in the Fermi coordinates is Note that the geodesic deviation of particles in the graviton computed as background is studied in [10,22,24]. Now we canonically quantize this system. We can expand the 4 2 ≃ −1 − i j 2 − j k i field hˆ Aðk;tÞ, whose time evolution is governed by the ds ð R0i0jx x Þdt 3 R0jikx x dtdx I   quadratic action, in terms of the creation and annihilation 1 δ − k l i j operators as þ ij 3 Rikjlx x dx dx : ð2:6Þ ˆ A k ˆ k ˆ † −k h I ð ;tÞ¼aAð ÞukðtÞþaAð ÞukðtÞ; ð2:11Þ Here the Riemann tensor is evaluated at the origin xi ¼ 0 in the Fermi normal coordinate system. Substituting the where the creation and annihilation operators satisfy the metric (2.6) into the action (2.5), the action for the two standard commutation relations2 particles up to the second order of ξi is expressed as ˆ k ˆ † k0 δ δ ˆ k ˆ k0 Z   ½aAð Þ; aA0 ð Þ ¼ k;k0 AA0 ; ½aAð Þ; aA0 ð Þ 2 m _i m i j † † S ≃ dt ξ − R0 0 ð0;tÞξ ξ : ð2:7Þ ˆ k ˆ k0 0 p 2 2 i j ¼½aAð Þ; aA0 ð Þ ¼ ð2:12Þ γτ0

and ukðtÞ denotes a mode function properly normalized as Because the Riemann tensor R0i0j is gauge invariant at the leading order in the metric fluctuation hij, we can evaluate _ − _ − ukðtÞukðtÞ ukðtÞukðtÞ¼ i: ð2:13Þ it in the transverse traceless gauge to get R0i0jð0;tÞ¼ −̈ 0 2 hijð ;tÞ= . We then finally obtain the action for the The Minkowski vacuum j0i is defined by aˆ AðkÞj0i¼0, geodesic deviation with choosing the mode function as Z   m 2 m 1 S ≃ dt ξ_i þ ḧ ð0;tÞξiξj : ð2:8Þ u ðtÞ¼pffiffiffiffiffi e−ikt ≡ uMðtÞ: ð2:14Þ p 2 4 ij k 2 k γτ0 k

Notice that, when considering waves with wavelengths smaller than the characteristic separation ξ 2 ð3Þ 0 length , an approximation (2.6) cannot be used. δk;k0 is replaced by δ ðk − k Þ when taking the infinite However, we expect that the effect from such gravitational volume limit Lx;Ly;Lz → ∞.

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ˆ A k D. Langevin type equation of geodesic deviation An expectation value of h I ð ;tÞ can be nonzero for generic quantum states, such as coherent states as we will The variation of the action Eq. (2.10) with respect to hA see in Sec. III A 1. In this case, it may be convenient and ξi gives the following equations of for the ˆ A to divide the gravitational perturbation h I around the operators in the : Minkowski background into a “classical” piece h ðk;tÞ ≡ cl κ 2 ˆ A k “ ” ̈ˆ 2 ˆ m d ˆ ˆ hh I ð ;tÞi and a quantum piece as h Aðk;tÞþk h Aðk;tÞ¼ pffiffiffiffi eAðkÞ fξ iðtÞξ jðtÞg; 2 V ij dt2 δˆ A k ˆ A k − k h I ð ;tÞ¼h I ð ;tÞ hclð ;tÞ: ð2:15Þ ð2:16Þ ˆ ˆ ̈ κ X X ̈ Here, hXi denotes an expectation value of an X for ξˆ iðtÞ¼pffiffiffiffi eA ðkÞhˆ Aðk;tÞξˆ jðtÞ: ð2:17Þ a given quantum state. A precise value of the expectation V ij A k≤Ωm value depends on the given quantum state. We refer to δhˆ as the gravitational quantum fluctuations, i.e., gravitons, in the Equation (2.16) is solved by standard Green’s function presence of the classical gravitational perturbations hcl. techniques. Specifically we consider the setup where the Similarly, we promote the position ξiðtÞ to the operator interaction between hˆ and ξi is turned on at t ¼ 0. Under ξˆ iðtÞ below. this setup, the formal solution of Eq. (2.16) is given by

Z κm t sin k t − t0 d2 ˆ A k A k δˆ A k ffiffiffiffi A k 0 ð Þ ξˆ i 0 ξˆ j 0 h ð ;tÞ¼hclð ;tÞþ h I ð ;tÞþ p eij ð Þ dt 2 f ðt Þ ðt Þg: ð2:18Þ 2 V 0 k dt0

The last nonhomogeneous solution describes the gravitational waves emitted from the particles. Substituting the formal solution Eq. (2.18) into Eq. (2.17), we have 1 κ X X ξ̈i ̈cl 0 ξˆ j − ffiffiffiffi ξj 2 A δˆ A k ðtÞ¼2 hijð ;tÞ ðtÞ p ðtÞ k eij h I ð ;tÞ V A k≤Ω m Z κ2m X t d2 − ξj − 0 − 0 ξˆ k 0 ξˆ l 0 ðtÞ ½PikPjl þ PilPjk PijPkl dt k sin kðt t Þ 2 f ðt Þ ðt Þg 4V 0 dt0 k≤Ωm κ2 X 2 m ˆ j d ˆ k ˆ l þ ½P P l þ P lP − P P lξ ðtÞ fξ ðtÞξ ðtÞg; ð2:19Þ 4V ik j i jk ij k dt2 k≤Ωm where we have defined 2κ X cl i ≡ ffiffiffiffi A k A k ik·x hijðx ;tÞ p eijð Þhclð ;tÞe ; ð2:20Þ V k;A ¯ ¯ ¯ and introduced the projection tensor Pij ¼ δij − kikj orthogonal to the unit wave number ki ¼ ki=k and used X 1 A A k k P l l − l eij ð Þeklð Þ¼2 ½ ikPj þ Pi Pjk PijPk : ð2:21Þ A P Here, the UV-regulated mode sum k≤Ω in the second and the third lines of (2.19)Pcan be performed byR taking the m 3 Ω 3 continuum limit of the k-mode by removing the width introduced in Eq. (2.3): 1=V → 1= 2π m d k. The k≤Ωm ð Þ momentum integral is computed in Appendix A, and the result is   1 κ2 2 5 ̈ˆ i ̈cl ˆ j m ˆ j d ˆ k ˆ l ξ ðtÞ − h ð0;tÞξ ðtÞþ δ δ l þ δ lδ − δ δ l ξ ðtÞ fξ ðtÞξ ðtÞg 2 ij 40π ik j i jk 3 ij k dt5   κ2 2 4 ˆ ˆ j m ˆ j d ˆ k ˆ l ¼ −δN ðtÞξ ðtÞþ Ω δ δ l þ δ lδ − δ δ l ξ ðtÞ fξ ðtÞξ ðtÞg ð2:22Þ ij 20π2 m ik j i jk 3 ij k dt4 where we have defined

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κ X X ˆ ffiffiffiffi 2 A ˆ A Below we compute the noise correlation functions when the δNijðtÞ ≡ p k e ðkÞδh ðk;tÞ: ð2:23Þ V ij I graviton is in a squeezed- and discuss the A k≤Ωm Minkowski vacuum as a special case. The coherent state or The third term on the left-hand side represents the of squeezed state can be realized when the squeezing param- radiation reaction. On the right-hand side, the first term is eter or the coherent parameter goes to zero, respectively. the random force induced by gravitons, which is nothing but quantum noise. The last term is not relevant in the 1. Squeezed coherent states discussions of the noise and the decoherence. The sche- matic expression of Eq. (2.22) without the last term is The definition of the jζ;Bi is ˆ obtained in [10]. The noise of gravitons δNij always exists if the gravitational waves are quantized. If the noise is jζ;Bi ≡ SˆðζÞDˆ ðBÞj0i; ð3:3Þ detected, it would be evidence that gravity is quantized.

III. QUANTUM NOISE INDUCED BY GRAVITONS where SˆðζÞ and Dˆ ðBÞ are the squeezing and the displace- A. Quantum noise correlations ment operators, respectively. They are expressed by

In this section, we compute the amplitude of the noise for  X  various quantum states. 1 † † SˆðζÞ ≡ exp ðζaˆ ðkÞaˆ ð−kÞþζ aˆ ðkÞaˆ ð−kÞÞ ; The anticommutator correlation function of δN t can k A A k A A ijð Þ V k;A be computed by using Eqs. (2.21) and (2.12) in the infinite ð3:4Þ volume limit Lx;Ly;Lz → ∞ as   κ2 2   δ ˆ δ ˆ 0 δ δ δ δ − δ δ hf NijðtÞ; Nklðt Þgi ¼ 2 ik jl þ il jk ij kl 1 X 10π 3 ˆ ≡ ˆ † k − ˆ k Z DðBÞ exp ðBkaAð Þ BkaAð ÞÞ ; ð3:5Þ Ω V m k;A 6 0 × dkk Pδhðk; t; t Þ; ð3:1Þ 0 where ζ ≡ r exp½iφ and r is the squeezing parameter. where we defined the anticommutator symbol f·; ·g as k k k k ˆ ˆ ˆ ˆ ˆ ˆ Bk is the coherent parameter. We assume that the parameter fX; Yg ≡ ðX Y þY XÞ=2, and Pδ is given by h ζk and Bk only depend on k and are independent of the direction of k. These operators are unitary, and satisfy the δ A k δ A0 k0 0 δ δ 0 hf hI ð ;tÞ; hI ð ;tÞgi ¼ AA0 kþk0;0Pδhðk; t; t Þ: ð3:2Þ following relations:

† † † φ ˆ ˆ ζ ˆ k ˆ ζ ˆ ˆ k − ˆ −k i k D ðBÞS ð ÞaAð ÞSð ÞDðBÞ¼ðaAð ÞþBkÞ cosh rk ðaAð ÞþBkÞe sinh rk; ð3:6Þ

† † † † − φ ˆ ˆ ζ ˆ −k ˆ ζ ˆ ˆ −k − ˆ k i k D ðBÞS ð ÞaAð ÞSð ÞDðBÞ¼ðaAð ÞþBkÞ cosh rk ðaAð ÞþBkÞe sinh rk: ð3:7Þ

The of the above operators become

† † φ 0 ˆ ˆ ζ ˆ k ˆ ζ ˆ 0 − i k h jD ðBÞS ð ÞaAð ÞSð ÞDðBÞj i¼Bk cosh rk Bke sinh rk; ð3:8Þ

† † † − φ 0 ˆ ˆ ζ ˆ −k ˆ ζ ˆ 0 − i k h jD ðBÞS ð ÞaAð ÞSð ÞDðBÞj i¼Bk cosh rk Bke sinh rk: ð3:9Þ

δˆ A k On the other hand, the transformation of the operator h I ð ;tÞ is given by definition as

ˆ † ˆ † ζ δˆ A k ˆ ζ ˆ ˆ † ˆ † ζ ˆ A k ˆ ζ ˆ − 0 ˆ † ˆ † ζ ˆ A k ˆ ζ ˆ 0 D ðBÞS ð Þ h I ð ;tÞSð ÞDðBÞ¼D ðBÞS ð Þh I ð ;tÞSð ÞDðBÞ h jD ðBÞS ð Þh I ð ;tÞSð ÞDðBÞj i: ð3:10Þ

ˆ A k ˆ k ˆ † k Because h I ð ;tÞ consists of the aAð Þ and aAð Þ, we see that the right-hand side of the above relation is independent of the coherent parameter Bk and we have

† † † φ † − φ ˆ ˆ ζ δˆ A k ˆ ζ ˆ ˆ k − ˆ −k i k ˆ −k − ˆ k i k D ðBÞS ð Þ h I ð ;tÞSð ÞDðBÞ¼ðaAð Þcoshrk aAð Þe sinhrkÞukðtÞþðaAð Þcoshrk aAð Þe sinhrkÞukðtÞ sq ˆ k sq ˆ † −k ¼ uk ðtÞaAð Þþuk ðtÞaAð Þ ˆ † ζ ˆ A k ˆ ζ ¼ S ð Þh I ð ;tÞSð Þ; ð3:11Þ

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0 sq sq 0 where the mode function in the squeezed state is given Pδhðk; t; t Þ¼Re½uk ðtÞuk ðt Þ in terms of that in the Minkowski space in Eq. (2.14) 1 such as ¼ ½cosfkðt − t0Þg cosh 2r 2k k sq − φ 0 ≡ M − i k M − cosfkðt − t Þ − φ g sinh 2r : ð3:14Þ uk ðtÞ uk ðtÞ cosh rk e uk ðtÞ sinh rk: ð3:12Þ k k ˆ φ Hence, the anticommutator correlation function of δNijðtÞ In general, the squeezing parameter rk and the phase k in the squeezed coherent state in Eq. (3.1) becomes depend on k. However, for simplicity, we regard these independent of the coherent state such as variables as constants. Then, plugging this into Eq. (3.13), we obtain ˆ ˆ 0 hζ;BjfδNijðtÞ; δNklðt Þgjζ;Bi ˆ ˆ 0   hζ;BjfδN ðtÞ; δN lðt Þgjζ;Bi κ2 2 ij k  δ δ δ δ − δ δ 2 6 ¼ 2 ik jl þ il jk ij kl κ Ω 2 10π 3 m δ δ δ δ − δ δ Ω − 0 φ Z ¼ 2 ik jl þ il jk ij kl Fð mðt t Þ;r; Þ; Ω 20π 3 m 6 sq sq 0 × dkk Re½uk ðtÞuk ðt Þ; ð3:13Þ 0 ð3:15Þ where where

1 Fðx; r; φÞ¼ ½fð5x4 − 60x2 þ 120Þ cos x þ xðx4 − 20x2 þ 120Þ sin x − 120g cosh 2r x6 − fð5x4 − 60x2 þ 120Þ cos ðφ − xÞ − xðx4 − 20x2 þ 120Þ sin ðφ − xÞ − 120 cos φg sinh 2r: ð3:16Þ

Note that Fðx; r; φÞ converges to zero for large x and the where function Fðx; r; φÞ for small x can be expanded as 1 ≡ 5 4 − 60 2 120 1 10 FðxÞ 6 ½ð x x þ Þ cos x Fðx; r; φÞ¼ ðcosh 2r − cos φ sinh 2rÞ − sin φ sinh 2rx x 6 7 þ xðx4 − 20x2 þ 120Þ sin x − 120: ð3:20Þ 25 − ðcosh 2r − cos φ sinh 2rÞx2 þ Oðx3Þ: 4 Note that, for small x, the function FðxÞ can be expanded as ð3:17Þ 1 25 FðxÞ¼ − x2 þ Oðx4Þ: ð3:21Þ Ω 6 4 We find that quantum noise correlations increase as m increases. Comparing this with the result of the squeezed coherent state, we see the quantum noise correlations are enhanced 2. The Minkowski vacuum state exponentially by the squeezing parameter. For comparison, let us see the correlation functions of the quantum noise in the Minkowski vacuum state which is B. Detectability of the quantum noise → 0 obtained by taking rk and then we have In this subsection, we roughly estimate the effective h δNˆ 1 strain eff corresponding to the quantum noise ij and 0 M M 0 0 Pδ k t − t discuss the detectability of the quantum noise. For a given hðk; t; t Þ¼Re½uk ðtÞuk ðt Þ ¼ 2 cosf ð Þg: k quantum state, the amplitude of the quantum noise in the ð3:18Þ frequency domain can be characterized as

Z 1 Substituting this into Eq. (3.1), we get ∞ 2 ij 2πift δNðfÞ ≡ dthfδN ðtÞ; δNijð0Þgie : ð3:22Þ −∞ ˆ ˆ 0 h0jfδN ðtÞ; δN lðt Þgj0i ij  k  κ2Ω6 2 From Eq. (2.22), it is found that the response of ξi to the m 0 δ δ l δ lδ − δ δ l F Ω t − t ; ¼ 20π2 ik j þ i jk 3 ij k ð mð ÞÞ presence of the classical gravitational wave and the ̈ 0 δ quautum noise is proportional to hclðt; Þ and NðtÞ, ð3:19Þ respectively. Here we omitted spatial indices. This suggests

044017-6 NOISE AND DECOHERENCE INDUCED BY GRAVITONS PHYS. REV. D 103, 044017 (2021) that we can discuss the detectablity of the noise by using ≡ 2π −2δ the effective strain heffðfÞ ð fÞ NðfÞ in the fre- quency domain. Let us start with the Minkowski vacuum state. In this case, the amplitude of the quantum noise in the frequency domain can be computed as

5 ð2fÞ2 δNðfÞ¼π2 ; ð3:23Þ Mpl

∼ 1018 where the reduced Planck mass is Mpl GeV. The corresponding effective strain is then

1  1 ð2fÞ2 −42 f 2 −1 ≈ 2 10 2 heffðfÞ¼ × 1 Hz : ð3:24Þ Mpl Hz FIG. 2. Timelike curves of two massive particles γτ and γτ0 , and the γτ0 is in a superposition state of two spatially separated For instance, the characteristic frequency of LIGO is locations ξ1ðtÞ and ξ2ðtÞ. The superposition state survives ξ ≠ ξ 0 around 100 Hz. Then the amplitude of quantum noise 1ðtÞ 2ðtÞ for the duration

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In the Schrödinger picture, the unitary time evolution of the superposition state is expressed as

jΨðt0Þi → jΨðtÞi¼jξ1ðtÞiþjξ2ðtÞi; ð4:1Þ where jξ1ðtÞi and jξ2ðtÞi are approximately eigenstates ˆ i ˆ i i of the operator ξ satisfying ξ jξ1ðtÞi ¼ ξ1ðtÞjξ1ðtÞi and ˆ i i ξ jξ2ðtÞi ¼ ξ2ðtÞjξ2ðtÞi. The superposition state lasts for 0 ξi ξi ∉ 0

hξ2ðt Þjξ1ðt Þi 2 −Γ ≡ f f t ¼ tf = in this configuration. The schematic diagram of exp½ ðtf Þ : ð4:3Þ hξ2ðt0Þjξ1ðt0Þi this configuration is depicted in Fig. 3. The maximum 2 Δξ 2 ≡ Δ separation at t ¼ tf= is ðtf= Þ¼vtf L. The loss of quantum coherence between jξ1ðtÞi and jξ2ðtÞi The decoherence rate Eq. (4.4) in the Minkowski −Γ ≪ 1 occurs when exp½ ðtfÞ . We can compute the vacuum state is expressed as decoherence functional by integrating out the gravitons as6 2 6 2 Z Z m Ω ξ tf tf 2 Z Γ ≈ m 0Δξ Δξ 0 Ω − 0 m tf ðtf Þ 2 2 dt dt ðtÞ ðt ÞFð mðt t ÞÞ: Γ ≈ Δ ξiξj 30π M 0 0 ðtfÞ 8 dt ð ÞðtÞ pl Z 0 t ð4:6Þ f 0 k l 0 ˆ ˆ 0 × dt Δðξ ξ Þðt ÞhfδNijðtÞ; δNklðt Þgi: ð4:4Þ 0 Here we used Eq. (3.19). Substituting Eq. (4.5) into 8 i j i j i j Eq. (4.6), we obtain Here Δðξ ξ ÞðtÞ¼ξ1ðtÞξ1ðtÞ − ξ2ðtÞξ2ðtÞ denotes a differ- ξi ξj ence of ðtÞ ðtÞ in the superposition. The value of 2m2v2 ΔðξiξjÞðtÞ is determined by an experimental setup. We Γðt Þ ≈ ðΩ ξÞ2GðΩ t Þ; ð4:8Þ f 5π2M2 m m f consider some simple configurations of the superposition pl state in order to evaluate the rate of decoherence explicitly in the next subsection. where GðxÞ is given by    2 x B. Decoherence rate for simple configurations GðxÞ ≡ 1 þ sinðxÞ − 8 sin 3x 2 of the superposition state     1 2 32 x For simplicity, we assume that ξi ðtÞ¼ξ ðtÞδi1 for − 10 a a þ 2 3 cosðxÞ 3 cos 2 þ : ð4:9Þ a ¼ 1, 2 and that ξ ≡ ðξ1ðtÞþξ2ðtÞÞ=2 is independent x of time. We then consider the configuration of the super- 7 position state with the separation ΔξðtÞ ≡ ξ2ðtÞ − ξ1ðtÞ as 8In terms of the maximum separation length ΔL, Eq. (4.8) can  be written as 2 0 ≤ 2 vt for

6 For the derivation of (4.4), see Appendix B. Note that 0 < ΔL

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∼ O 1 Γ ≪ 1 Since GðxÞ ð Þ, we find ðtfÞ as long as the momentum of the massive particle is much smaller ≪ ≈ 4 10−6 than the Planck mass, mv Mpl × g, and then decoherence does not occur. On the other hand, for ≫ mv Mpl, the decoherence happens. Let us estimate the τ Γ τ 1 decoherence time dec defined by ð decÞ¼ in this case. Γ τ 1 In order to hold ð decÞ¼ , GðxÞ has to be less than unity. Thus the decoherence time can be computed by using GðxÞ ≃ x4=288 (x ≪ 1)as   M 1=2 Ω τ ∼ 10 × pl : ð4:11Þ m dec mv FIG. 4. A plot of GðxÞ. GðxÞ grows polynomially for x ≪ 1 and We see the decoherence time is Ω τ ≪ 1 for the case then oscillatory behavior converges to order unity for x ≳ 10. m dec that the momentum of the massive particle is much larger ≫ than the Planck mass mv Mpl. The behavior of GðxÞ is shown in Fig. 4. The function GðxÞ As another simple configuration of the superposition 4 6 grows polynomially as GðxÞ ≃ x =288 þ Oðx Þ for x ≪ 1 state, we consider a system that the previous configuration and shows a damping oscillation around GðxÞ¼1 for large Eq. (4.5) repeats N times as Ω x. As we explained in Sec. II B, the cutoff m satisfies Ω ξ ∼ O 1 Ω ξ 1 N−1 m ð Þ. If we take m ¼ , then Eq. (4.8) reads X ΔξðtÞ¼ Δξðt; kÞ; ð4:12Þ 2 2 2 k¼0 Γ ≈ m v Ω ðtf Þ 5π2 2 Gð mtf Þ: ð4:10Þ Mpl with

8 2 ≤ 1 2 < vt for kðTs þ TcÞ

The configuration of the superposition state of (4.12) is 2m2v2 Γ t ≈ N G Ω T ; : shown in Fig. 5. The duration of the superposition state is ð fÞ × 5π2 2 ð m sÞ ð4 14Þ Mpl Ts and each configuration repeats with a time interval Tc. ≫ We assume Tc Ts so that each configuration is inde- − 1 pendent. Under this assumption, we have with tf ¼ NTs þðN ÞTc. If we compared with Eq. (4.10), we see an enhancement factor N come in. Thus, if we take a sufficiently large N, decoherence occurs Γ ≫ 1 ( ðtf Þ ) even when the momentum of the massive ≲ particle is smaller than the Planck mass, mv Mpl. This system might be possible to get carried out in a tabletop experiment with upcoming technology and would open up the possibility of the first detection of the graviton-induced decoherence. Thus, we expect that the decoherence caused by the noise of gravitons would offer new vistas to test quantum gravity in tabletop experiments. We note that the decoherence rate is enhanced for squeezed states. As we showed in Sec. III B, gravitons produced during inflation experience large squeezing. FIG. 5. The separation of the configuration of the superposition Hence, it would be interesting to consider the decoherence state as a function of time. The blue solid line indicates the configuration given in Eq. (4.12). The configuration of Eq. (4.5) induced by the primordial gravitons. We should also notice repeats N times as time evolves. The duration of the superposition that our analysis of the superposition state can only be ξ−1 state is Ts and each configuration repeats in particular time applicable for a frequency range lower than . Hence, interval Tc. our estimation of the decoherence rate is a lower bound.

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We should assess the decoherence rate by taking into of superposition states. In the simplest configuration given Ω ξ 1 account the contribution from a frequency range m > . in Eq. (4.5), we found that the decoherence does not occur unless we consider the momentum of the massive particle is V. CONCLUSION much larger than the Planck mass (c ¼ ℏ ¼ 1). As the second simplest configuration, we considered the system In this paper, we studied the physical effects induced by that repeats the first configuration Eq. (4.5) N times as time gravitons on the system of two massive particles. We first evolves as shown in Fig. 5. We found that the decoherence derived the effective equation of motion for the geodesic rate is enhanced by the factor N and decoherence happens deviation between two particles, that is, the Langevin-type even when the momentum of the massive particle is smaller equation in (2.22). We found that gravitons give rise to the than the Planck mass if we take a sufficiently large N. The noise to the dynamics of particles. Note that Eq. (2.22) system we considered might be possible to get carried out without the last term agrees with the one in [10].We in a tabletop experiment with upcoming technology and calculated the noise correlation in squeezed coherent would open up the possibility of the first detection of the states and demonstrated that the squeezed states enhance graviton-induced decoherence. We expect that the graviton- it compared with the vacuum state. We then discussed the induced decoherence would offer new vistas to test quan- detectability of the noise of gravitons by gravitational tum gravity in tabletop experiments. wave detectors. It turned out that the amplitude of the In order to perform a tabletop experiment to find noise of gravitons in the case of the Minkowski vacuum is gravitons, we need a more concrete setup for the experi- too small to be detected by the current detectors. However, ment. Correspondingly, we should clarify to what extent in the squeezed state, the noise of gravitons is enhanced by other sources of decoherence contribute and also refine the the exponential factor of the squeezing parameter. This is method for analyzing the decoherence induced by grav- consistent with the amplitude of the stochastic gravita- itons. We leave these issues for future work. tional waves from inflation that experience the large squeezing. ACKNOWLEDGMENTS We then explored the effect of gravitons on the process of decoherence in a laboratory. We considered a system of S. K. was supported by the Japan Society for the two massive particles, one of which is in a superposition Promotion of Science (JSPS) KAKENHI Grant state of two spatially separated locations as shown in Fig. 3 No. JP18H05862. J. S. was in part supported by JSPS and investigated the loss of coherence of the superposition KAKENHI Grants No. JP17H02894 and No. JP17K18778. state caused by gravitons. In order to estimate the J. T. was supported by JSPS Postdoctoral Fellowship decoherence rate, we presented two simple configurations No. 202000912.

APPENDIX A: MOMENTUM INEGRAL Here, we calculate the integral in Eq. (2.19): 1 κ X X ξ̈i ̈cl 0 ξˆ j − ffiffiffiffi ξˆ j 2 A δˆ A k ðtÞ¼2 h ijð ;tÞ ðtÞ p ðtÞ k eij h I ð ;tÞ V A k≤Ω Z m Z 2 Ω 2 κ m 1 m t d − ξˆ j 3 − 0 − 0 ξˆ k 0 ξˆ l 0 3 ðtÞ d k½PikPjl þ PilPjk PijPkl dt k sin kðt t Þ 2 f ðt Þ ðt Þg 4 ð2πÞ 0 dt0 Z κ2 1 Ω 2 m m 3 ˆ j d ˆ k ˆ l þ d k½P P l þ P lP − P P lξ ðtÞ fξ ðtÞξ ðtÞg: ðA1Þ 4 ð2πÞ3 ik j i jk ij k dt2

By using the following angular integrals, Z Z Z 4 4π Ω 4π Ω i j πδij Ω i j k l δijδkl δikδjlþδilδjk d ¼ ; d k k ¼ 3 ; d k k k k ¼ 15 ð þ Þ; ðA2Þ we find Z   8π 2 Ω − δ δ δ δ − δ δ d ½PikPjl þ PilPjk PijPkl¼ 5 ik jl þ il jk 3 ij kl : ðA3Þ

Then we can perform the momentum integral as

044017-10 NOISE AND DECOHERENCE INDUCED BY GRAVITONS PHYS. REV. D 103, 044017 (2021) Z Z Z Ω Ω m 3 m 2 d k½PikPjl þ PilPjk − PijPkl¼ dkk dΩ½PikPjl þ PilPjk − PijPkl 0   8π 2 δ δ δ δ − δ δ Ω3 ¼ 15 ik jl þ il jk 3 ij kl m: ðA4Þ

If we define the limit representation of Dirac’s delta function as

sin Ω ðt − t0Þ fðt − t0Þ¼ m → Ω → ∞πδðt − t0Þ; ðA5Þ t − t0 m we have

− 0 Ω fðt t Þjt0¼t ¼ m; sin Ω t − 0 m ≃ 0 ≠ 0 fðt t Þjt0¼0 ¼ for t ; t d − 0 0 0 fðt t Þ ¼ ; dt 0 t ¼t d sin Ω t − Ω t cos Ω t − 0 m m m ≃ 0 Ω → ∞ 0 fðt t Þ ¼ 2 as m ; ðA6Þ dt t0¼0 t

d2 1 − 0 − Ω3 02 fðt t Þ ¼ m; dt 0 3 t ¼t d2 2 sin Ω t − 2Ω t cos Ω t − Ω2 t2 sin Ω t − 0 m m m m m ≃ 0 Ω → ∞ 02 fðt t Þ ¼ 2 as m : ðA7Þ dt t0¼0 t

For Eqs. (A6) and (A7), after smearing the functions with an appropriate window function, these quantities vanish. The momentum integral in the second line of the right-hand side of Eq. (A1) is written by the function fðt − t0Þ of this form Z Z Z Ω 2 3 2 m t d t d d 2 0 − 0 ξi ξj 0 − 0 ξi 0 ξj 0 dkk dt k sin kðt t Þ 2 f ðtÞ ðtÞg ¼ dt 3 fðt t Þ 2 f ðt Þ ðt Þg: ðA8Þ 0 0 dt0 0 dt0 dt0

Using the above results, we can evaluate as follows: Z     t d3 d2 d2 d2 t d d3 t 0 − 0 ξi 0 ξj 0 − 0 ξi 0 ξj 0 − − 0 ξi 0 ξj 0 dt 3 fðt t Þ 2 f ðt Þ ðt Þg ¼ 2 fðt t Þ 2 f ðt Þ ðt Þg fðt t Þ 3 f ðt Þ ðt Þg 0 dt0 dt0 dt0 dt0 dt0 dt0   0 Z 0 d4 t t d5 − 0 ξi 0 ξj 0 − 0 − 0 ξi 0 ξj 0 þ fðt t Þ 04 f ðt Þ ðt Þg dt fðt t Þ 05 f ðt Þ ðt Þg dt 0 0 dt 1 d2 d4 π d5 ¼ − Ω3 fξiðt0Þξjðt0Þg þ Ω3 fξiðt0Þξjðt0Þg − fξiðt0Þξjðt0Þg: ðA9Þ 3 m dt02 m dt04 2 dt05

Then Eq. (A1) becomes

1 κ X X ξ̈i ̈cl 0 ξˆ j − ffiffiffiffi ξˆ j 2 A δˆ A k ðtÞ¼2 h ijð ;tÞ ðtÞ p ðtÞ k eij h I ð ;tÞ V A k≤Ω  m   κ2 1 2 4 π 5 m j d i 0 j 0 d i 0 j 0 þ ξ ðtÞ δ δ l þ δ lδ − δ δ l Ω fξ ðt Þξ ðt Þg − fξ ðt Þξ ðt Þg : ðA10Þ 4 5π2 ik j i jk 3 ij k m dt04 2 dt05

This is the Eq. (2.22).

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APPENDIX B: INFLUENCE FUNCTIONAL where we assume that jΨ1ðtÞi and jΨ2ðtÞi are approximately METHOD the eigenstates of the current density operator Jˆ μðx;tÞ. μ We refer to the respective eigenvalues as J1ðx;tÞ¼ In this Appendix, we briefly explain the influence μ ðρ1ðx;tÞ; J1ðx;tÞÞ and J2ðx;tÞ¼ðρ2ðx;tÞ; J2ðx;tÞÞ,and functional method, particularly focusing on its applica- μ tion to the current context. In Appendix B1, we consider we treat an electric current J as an external current. Writing the (QED) setup which is the the of the particle as q, we can write these current μ x μ δ x − x same as the setup discussed in [17], and obtained the densities as Jað ;tÞ¼qvaðtÞ ð aðtÞÞ. We then have μ x ≠ μ x 0 same result. We then generalize the discussion to J1ð ;tÞ J2ð ;tÞ only for

1. Influence functional in QED jΨaðt0Þi ¼ cajΨðt0Þi;c1 þ c2 ¼ 1; ðB2Þ We consider the superposition state of an electrically charged particle which is analogous to the setup we discussed in Sec. IV, and briefly explain how we can for a ¼ 1, 2 without loss of generality. Here, ca can be compute the coherence of the superposition state after complex in general. Obviously, this setup is quite similar to integrating out . The setup and the discussion in this the gravitational setup discussed in Sec. IV. Appendix is essentially based on [17]. We now discuss how photons affect the coherence Specifically, we consider the superposition state of an between these states. In the Coulomb gauge which is electrically charged particle such as an electron, and the i specified by the condition ∂iA ðx;tÞ¼0, the time respective world lines of the particle are parametrized by evolution of the system is governed by the action S ¼ t; x1 t and t; x2 t . Corresponding 4-velocities are A J ρ ð ð ÞÞ ð pðffiffiffiffiffiffiffiffiffiffiffiffiffiÞÞ S½ ; þScoul½ which is defined by μ _ _ 2 vaðtÞ ≡ ð1; xaðtÞÞ= 1 − xa for a ¼ 1, 2, which are nor- η μ ν −1 malized as μνvava ¼ . Z   Suppose that the superposition state survives for 4 1 2 S½A; J ≡ d x − ð∂μAðxÞÞ þ AðxÞ · JðxÞ ; 0 ¼ 1. In this setup, the time evolution of the quantum state in the R where x ¼ðx;tÞ. The term d4xðA · JÞ encodes the Schrödinger picture can be written as influence of the external current JðxÞ on the time evolution. S expresses the influence of the Coulomb energy on Ψ → Ψ Ψ Ψ coul j ðt0Þi j ðtÞi¼j 1ðtÞiþj 2ðtÞi; ðB1Þ the system. Using the standard path-integral expression of the unitary time evolution in the Lagrangian form, one has Z ρ A J Ψ iScoul½ a DA A x iS½ ; aΨ A j aðtf Þi ¼ cae j ð ;tfÞie 0½ ;

for a ¼ 1; 2: ðB4Þ

Q Here, Ψ0½A ≡ xhAðx; 0ÞjΨðt0Þi is the initial wave functional. jAðx;tÞi denotes an eigenstate of the gauge ˆ ˆ field AðxÞ in the Schrödinger picture: AðxÞjAðx;tÞi ¼ Aðx;tÞjAðx;tÞi.9 Using Eq. (B4), we obtain the path- integral expression of the decoherence functional as

9Strictly speaking, the state jAðx;tÞi is also an eigenstate of Jˆ, while we have omitted its eigenvalue dependence. We can safely omit this dependence here because both jΨ1ðt Þi and ˆ f FIG. 6. The schematic diagram of each trajectory of a charged jΨ2ðtf Þi are eigenstates of J with an identical eigenvalue 0 J J particle. The superposition state survives for

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right-hand side of (B5) can be obtained by computing all the connected diagrams in which external currents are connected by just as we usually do in FIG. 7. The diagram which contributes to Γ. Two blobs are the the external field method. The diagram which contributes to external sources which are given by the difference of external Γ is shown in Fig. 7, and the result is current densities. External sources are connected by the wavy Z Z Z internal line which is the symmetric of photons. 1 t t Γ f f 0 3 ðtf Þ¼ dt dt d x 2 0 0 Z hΨ2ðt ÞjΨ1ðt Þi −Γ f f 3 ˆ ˆ exp½ ðtf Þ ¼ × d yΔJiðx;tÞhfA ðx;tÞ; A ðy;t0ÞgiΔJkðy;t0Þ; hΨ2ðt0ÞjΨ1ðt0Þi i k Z Z Y DA DA δ A x ðB6Þ ¼ þ − ð þð ;tfÞ x

A J − A J −A x iðS½ þ; 1 S½ −; 2ÞΨ A Ψ A −ð ;tf ÞÞe 0½ þ 0½ − : ˆ ˆ † ˆ ˆ with ΔJ ≡ J1 − J2 and Aðx;tÞ ≡ U ðt; t0ÞAðxÞUðt; t0Þ, ˆ ðB5Þ Uðt; t0Þ is a unitary operator expressing the time evolution in the absence of the external current. Here, we used Δ x 0 ∉ 0 A 0 This is nothing but the modulus of the generating functional Jð ;tÞ¼ for all t ð ;tf Þ, h i¼ , and the following for photons in the Schwinger-Keldysh formalism. Then, the equality

Z Z Y A 0 − A 0 ˆ x ˆ y 0 DA DA δ A x − A x iðS½ þ; S½ −; ÞΨ A Ψ A c x c y 0 hfAið ;tÞ; Akð ;tÞgi ¼ þ − ð þð ;tf Þ −ð ;tfÞÞe 0½ þ 0½ −Ai ð ;tÞAkð ;tÞ; x ðB7Þ

A A where Ac ≡ þþ −. Equation (B6) precisely coincides with the same as the QED case we discussed in Appendix B1 2 except the form of the action. In the gravitational setup, the the one obtained in [17]. Note that when hAi ≠ 0, we need time evolution of this system is governed by the action to replace hfAˆ ðx;tÞ; Aˆ ðy;t0Þgi by its connected piece i k (2.10) in the Fermi normal coordinates, with treating ξi which equals the symmetric two-point function of as an external field whose time dependence is given by δAˆ ≡ Aˆ − Aˆ h i: hand. The last term on the right-hand side of (2.10) is the interaction between gravitons and the system consisting of ˆ x ˆ y 0 δ ˆ x δ ˆ y 0 hfAið ;tÞ; Akð ;tÞgiconnected ¼hf Aið ;tÞ; Akð ;tÞgi: two test particles, which could induce the loss of coherence ðB8Þ after integrating out gravitons. The Hamiltonian of the system is still quadratic in the conjugate momentum of gravitons, and hence the derivation of the path-integral 2. Influence functional in gravity expression of the decoherence functional is parallel to the We consider the system discussed in Sec. IV. It is almost discussion in Appendix B1, leading to obvious that our gravitational setup is formally almost

Z Z Y ξ ξ − h 2ðtfÞj 1ðtf Þi − − i S˜ hþ;ξi −S˜ h ;ξi − −Γ ≈ D þ D δ þ x − x ð ½ ij 1 ½ ij 2ÞΨ þ Ψ exp½ ðtf Þ ¼ ξ ξ hij hij ðhijð ;tf Þ hijð ;tf ÞÞe 0½hij 0½hij ; ðB9Þ h 2ðt0Þj 1ðt0Þi x where S˜½h; ξ is

Z Z   d3k X 1 1 mκ ˜ ξ _ A k _ A k − 2 A k A k ̈A k A k ξi ξj Θ Ω − S½h; ¼ dt 2π 3 2 h ð ;tÞh ð ;tÞ 2 k h ð ;tÞh ð ;tÞþ 2 h ð ;tÞeijð Þ ðtÞ ðtÞ ð m kÞ : ð Þ A ðB10Þ

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Here, we took the infinite volume limit V → ∞. We also omitted the terms which only change the overall phase of jξ1ðtÞi ξ Γ Γ and j 2ðtÞi so that they do not contribute to ðtfÞ. Equation (B9) implies that we can compute ðtfÞ by writing down all the connected diagrams as in the QED case discussed in Appendix B1. The result can be written in terms of the noise of ˆ graviton δNij as

2 Z Z m tf tf Γ ≈ Δ ξiξj 0Δ ξkξl 0 δ ˆ δ ˆ 0 ðtfÞ dt ð ÞðtÞ dt ð Þðt Þhf NijðtÞ; Nklðt Þgi: ðB11Þ 8 0 0

i j i j i j ˆ Here, Δðξ ξ ÞðtÞ ≡ ξ1ðtÞξ1ðtÞ − ξ2ðtÞξ2ðtÞ and δNij is defined by (2.23). In this way, we obtain Eq. (4.4).

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