Noise and decoherence induced by gravitons

We study quantum noise and decoherence induced by gravitons. We derive a Langevin type equation of geodesic deviation in the presence of gravitons. We calculate the noise correlation in squeezed coherent states and find that the squeezed state enhance it compared with the vacuum state. We also consider the decoherence of spatial superpositions of massive objects caused by gravitons in the vacuum state and find that gravitons might give the leading contribution to the decoherence. The decoherence induced by gravitons would offer new vistas to test quantum gravity in tabletop experiments.


Introduction
An understanding of the nature of gravity has been a central issue in physics since the discovery of general relativity and quantum mechanics. Nevertheless, no one has succeeded in constructing quantum theory of gravity. In particular, the existence of gravitons is still obscure [1]. In these situations, it is legitimate to doubt the necessity of canonical quantization of gravity [2]. Hence, it is worth seeking an experimental evidence of quantum gravity.
Usually, theorists explore the field of quantum gravity at energy scales near the Planck scale.
However, it is far beyond the capacity of the current or future particle accelerators. Recently, tabletop experiments of quantum gravity are drawing attention [3,4,5]. Remarkably, based on the development of quantum information, several ideas to test the quantum nature of gravity through a laboratory experiment are proposed [6,7]. It is also argued that noise in the lengths of the arms of gravitational wave detectors can be a probe of gravitons [8] and may be a hint of quantum gravity.
The noise is usually associated with the decoherence which is induced by quantum entanglement between a system and gravitons. Thus, as an approach to testing quantum gravity, it is important to understand the noise induced by gravitons and then the decoherence caused by the noise profoundly.
The decoherence due to gravity in the context of quantum superposition of massive objects has been investigated. For instance, the decoherence rate was derived based on effective field theory approach [9]. The decoherence due to quantum fluctuations of geometry caused by gravitons is discussed in [10]. Following the paper in the context of electromagnetic dynamics [11], the effect of gravitons on the destruction process of quantum superposition has been studied [12,13,14,15,16,17,18,19]. In these literature, an open quantum system in the graviton reservoir has been studied. The interaction between a quantum mechanical system and an environment of gravitons produces the noise and leads to the decoherence of the system [20].
In this paper, we focus on the quantum noise and decoherence to probe gravitons and ultimately quantum gravity. Firstly, we target on gravitational wave detectors. When gravitational waves arrive at the laser interferometer, the suspended mirrors interact with the gravitational waves. The mirror interacts with an environment of gravitons quantum mechanically. We will evaluate the effect of quantum noise induced by gravitons on the suspended mirrors. We show that the noise in the squeezed state can be sizable. Secondly, we consider a tabletop experiment by using two massive particles, one of which is superposed spatially, so called, the quantum state of Schrödinger's cat. It would be interesting to study how the entanglement is created and how a spatial superposition of a massive object is destroyed by gravitons. We give a simple estimate of the decoherence rate induced by gravitons. We expect that these studies could give a hint to perform a tabletop experiment for finding the evidence of quantum nature of gravity or gravitons in the future.
The organization of the paper is as follows: In section 2, we describe geodesics in the graviton background and derive a Langevin type equation of the system by eliminating the environment of gravitons. In section 3, we evaluate the noise correlation functions and show that the noise can be observable if the gravitons are in the squeezed state. In section 4, we discuss the decoherence induced by gravitons and detectability of gravitons. The final section is devoted to the conclusion. A detailed calculation of a momentum integral is presented in the Appendix.

Quantum mechanics in the graviton background
In this section, we present a model to study quantum mechanics in the graviton background. It gives rise to the basis for studying the noise and the decoherence due to low energy gravitons. In particular, we derive the quantum Langevin equation.

Gravitational waves
We consider gravitational waves in the Minkowski space. The metric describing gravitational waves in the transverse traceless gauge is expressed as where t is the time, x i are spatial coordinates, δ ij and h ij are the Kronecker delta and the metric perturbations which satisfy the transvers traceless conditions h ij, j = h ii = 0. The indices (i, j) run from 1 to 3. Substituting the metric Eq. (2.1) into the Einstein-Hilbert action, we obtain the quadratic action where κ 2 = 8πG and a dot denotes the derivative with respect to the time. We can expand the metric field h ij (x i , t) in terms of the Fourier modes where we introduced the polarization tensor e A ij (k) normalized as e * A ij (k)e B ij (k) = δ AB . Here, the index A denotes the linear polarization modes A = +, ×. Note that we consider finite volume V = L x L y L z and discretize the k-mode with a width k = (2πn x /L x , 2πn y /L y , 2πn z /L z ) where n = (n x , n y , n z ) are integers. Substituting the formula (2.3) into the quadratic action (2.2), we get Note that we used k = |k|. We see that a gravitational wave consists of an infinite number of harmonic oscilators.

Action for two test particles
When gravitational waves arrive at the laser interferometers, the suspended mirrors interact with the gravitational waves. Let us regard the mirror as a point particle for simplicity. A single particle, however, does not feel the gravitational waves because of the Einstein's equivalence principle at least classically. To see the effect of the gravitational waves, we need to consider two massive particles and measure the geodesic deviation between them.
In this subsection, we evaluate the effect of gravitational waves on the two particles by introducing an appropriate coordinate system called the Fermi normal coordinates along one Figure 1: Two neighboring timelike geodesics (γ τ , γ τ ) separated by ξ i are depicted in the blue lines and the green lines show two neighboring spacelike geodesics (γ s , γ s ) orthogonal to the geodesic γ τ in the spacelike hypersurface Σ s . We introduce the Fermi normal coordinate system using the orthogonal geodesics at the point P (0, t).
of their geodesics γ τ (See Figure 1). The Fermi normal coordinate system represents a local inertial frame. The dynamics of the other geodesic of particle γ τ is described by the position x i (t) = ξ i (t) in the vicinity of the point P (0, t) [23] and ξ i represents the deviation.
The action for the two test particles along the geodesics γ τ , γ τ is given by where ξ µ = (t, ξ i (t)). Note that we omit the action for the particle along γ τ because it's in the inertial frame and then the action has no dynamical variables. The metric g µν up to the second order of x i in the Fermi coordinates is computed as Here the Riemann tensor is evaluated at the origin x i = 0 in the Fermi normal coordinate system. Substituting the metric (2.6) into the action (2.5), the action for the two particles up to the second order of ξ i is expressed as Because the Riemann tensor R 0i0j is gauge invariant at the leading order in the metric fluctuation h ij , we can evaluate it in the transverse traceless gauge to get R 0i0j (0, t) = −ḧ ij (0, t)/2. We then finally obtain the action for the geodesic deviation Notice that, when considering gravitation waves with wavelength smaller than the characteristic separation length ξ, an approximation (2.6) cannot be used. However, we do not need to take into account the effect from such gravitational waves in this study because of the equivalence principle. Thus, we consider the action of the form where the metric h ij (0, t) is replaced by the Fourier modes in Eq. (2.3) and k≤Ωm represents the mode sum with the UV cutoff Ω m ξ −1 . We see a qubic derivative interaction appeared in the above action.

Particles in an environment of gravitons
From Eqs. (2.4) and (2.9), the total action S = S g + S p we consider is given by Now we canonically quantize this system. We can expand the interaction picture field h A I (k, t), whose time evolution is governed by the quadratic action, in terms of the creation and annihilation operators aŝ where the creation and annihilation operators satisfy the standard commutation relations 1 and u k (t) denotes a mode function properly normalized aṡ The Minkowski vacuum |0 is defined byâ A (k) |0 = 0, with choosing the mode function as (2.14) For later convenience, we define a "classical" piece h cl (k, t) ≡ ĥ A I (k, t) and a "quantum" piece as Here, X denotes an expectation value of an operatorX for a given quantum state. In this way, one can describe the gravitational quantum fluctuation δĥ on top of a given classical gravitational wave background h cl . We may identify δĥ as gravitons. Similarly, we promote the position ξ i (t) to the operatorξ i (t) below.

Langevin type equation of geodesic deviation
The variation of the action Eq. (2.10) with respect to h * A and ξ i gives the following equations of motion for the operators in the Heisenberg picture: The last nonhomogeneous solution describes the gravitational waves emitted from the particles. Substituting the formal solution Eq. (2.18) into Eq. (2.17), we havë where we have defined 20) and introduced the projection tensor P ij = δ ij −k ikj orthogonal to the unit wave number Here, the UV-regulated mode sum k≤Ωm in the second and the third line of (2.19) can be performed by taking the continuum limit of the k-mode by removing the width introduced in Eq.
where we have defined The third term on the left hand side represents the force of radiation reaction. On the right hand side, the first term is the random force induced by gravitons, which is nothing but quantum noise. The second term is not relevant in the discussions of the noise and the decoherence.  and (2.12) in the infinite volume limit L x , L y , L z → ∞ as where we defined the anticommutator symbol {· , ·} as {X,Ŷ } ≡ (XŶ +ŶX)/2, and P δh is given by Below we compute the noise correlation functions when the graviton is in a squeezed-coherent state and discuss the Minkowski vacuum as a special case. The coherent state or squeezed state can be realized when the squeezing parameter or the coherent parameter goes to zero, respectively.

Squeezed coherent states
The definition of the squeezed coherent state |ζ, B is whereŜ(ζ) andD(B) are the squeezing and the displacement operators, respectively. They are expressed byŜ where ζ k ≡ r k exp[iϕ k ] and r k is the squeezing parameter. The coherent parameter B k is written as B k ≡ |B k | exp[iθ k ]. Here, we assume that the parameter ζ k and B k only depend on k and are independent of the direction of k. These operators are unitary, and satisfy the following relations: The vacuum expectation value of the above operators become On the other hand, the transformation of the operator δĥ A I (k, t) is given by definition aŝ Becauseĥ A I (k, t) consists of theâ A (k) andâ † A (k), we see that the right hand side of the above relation is independent of the coherent parameter B k and we havê where the mode function in the squeezed state is given in terms of that in the Minkowski space in Eq. (2.14) such as Hence, the anticommutator correlation function of δN ij (t) in the squeezed coherent state in Eq. (3.1) becomes independent of the coherent state such as In general, the squeezing parameter r k and the phase ϕ k depend on k. However, for simplicity, we regard these variables as constants. Then, plugging this into Eq. (3.13), we obtain −120 cos ϕ sinh 2r . We find that quantum noise correlations increases as Ω m increases.

The Minkowski vacuum state
For comparison, let us see the correlation functions of the quantum noise in the Minkowski vacuum state which is obtained by taking r k → 0 and then we have Substituting this into Eq. (3.1), we get Note that, for small x, the function F (x) can be expanded as Comparing this with the result of squeezed coherent state, we see the quantum noise correlations are enhanced exponentially by the squeezing parameter.

Detectability of the quantum noise
In this subsection 3.2, we roughly estimate the effective strain h eff corresponding to the quantum noise δN ij and discuss the detectability of the quantum noise. For a given quantum state, the amplitude of the quantum noise in frequency domain can be characterized as From Eq. (2.22), it is found that the response of ξ i to presence of the classical gravitational wave and the quautum noise is proportional toḧ cl (t, 0) and δN (t), respectively. Here we omitted spatial indices. This suggests that we can discuss the detectablity of the noise by using the effective strain h eff (f ) ≡ (2πf ) −2 δN (f ) in frequency domain.
Let us start with Minkowski vacuum state. In this case, the amplitude of the quantum noise in frequency domain can be computed as where the reduced Planck mass is M pl ∼ 10 18 GeV. Corresponding effective strain is then For instance, the characteristic frequency of LIGO is around 100 Hz. Then the amplitude of quantum noise becomes h eff (f )| f ∼100 Hz ∼ 10 −41 Hz −1/2 . The strain sensitivity of LIGO is about 10 −23 Hz −1/2 for f ∼ 100 Hz, so the amplitude of quantum noise is too small to be detected.
However, if gravitational waves are in the squeezed state (or in the squeezed coherent state) when arriving at the detectors, the amplitude of the quantum noise is enhanced by the exponential factor of squeezing parameter as is seen in Eq. (3.16). That is, For instance, if the squeezing parameter is large as much as e r k ∼ 10 22 , the amplitude of the quantum noise at the characteristic frequency of LIGO becomes h eff (f ) ∼ 10 −20 Hz − 1 2 , which is detectably large.

Primordial gravitons
One possible and well-known mechanism to produce gravitons with large quantum fluctuations is inflation. Gravitons produced during inflation experience large squeezing which leads to the detectably large noise amplitudes. 2 In the case of primordial gravitational waves, the relation between the squeezing parameter and the current frequency f is given by where f c is the cutoff frequency. In the case of GUT inflation, we have f c ∼ 10 8 Hz.
In this case, the effective strain at f ∼ 0.1 Hz, which is the characteristic frequency of DECIGO [21,22], reads h eff (f ) ∼ 10 −24 Hz − 1 2 . If we could observe the noise δN ij , it would mean the detection of gravitons.

Decoherence induced by gravitons
In this section, we consider a tabletop experiment by using two massive particles, one of which is in a superposition state of two spatially-separated locations and then consider the decoherence of the superposition state caused by the quantum noiseN ij . We assume that the timelike geodesic γ τ of a mass m is spatially superposed across the distance ξ 2 − ξ 1 as in Figure 2. Such a superposition state is described by where |ξ 1 and |ξ 2 are eigenstates of the operatorξ i satisfyingξ i |ξ 1 = ξ i 1 |ξ 1 andξ i |ξ 2 = ξ i 2 |ξ 2 . To discuss the rate of decoherence, we apply the influence functional method [24]. The influence functional exp[iΦ] represents the time evolution of density operator of the system, which is expressed by The decoherence rate Γ is described by the time evolution of the influence phase functional iΦ such as Then, the decoherence rate Γ is given by We may then define the decoherence time τ dec by Γ(τ dec ) = 1.  5) and if this quantity becomes order one, then the decoherence becomes effective. Here, we considered the Minkowski vacuum and estimated the noise amplitude in the time domain as 6) and assumed that the quadratic component of ∆(ξ i ξ j ) is of the order of ξ 2 . Under this condition, the decoherence time is computed as where we used the reduced Planck mass M pl ∼ 10 −5 g. We notice that where ∆E ∼ m(Ω m ξ) 2 is the energy difference of the quantum superposition. This is consistent with the result in [10].

Decoherence due to gravitons in a tabletop experiment
In this subsection, we suppose that gravitational fields are quantized and then behaves as the noise as in Eq. (4.6). This setup enables us to perform a tabletop experiment of quantum gravity through the decoherence of the superposition caused by the quantized gravitational fields. 3 Let us consider some proposed tabletop experiments. In [6], the separation of quantum superposition of a massive object with the mass m = 10 −14 kg is given by ∆ξ = 250 µm.
Since the cutoff Ω m satisfies the relation Ω m ξ = 1 , we have Ω m ∼ 10 12 Hz in this case. By using Eq. (4.7), we estimate the time of decoherence due to the noise of gravitons would be τ dec 10 −5 s. In this paper, the time of collisional and thermal decoherence is estimated as 3.5 s under the temperature 0.15 K and the pressure 10 −15 Pa. It turns out that the decoherence due to the noise of gravitons takes place faster than the collisional and the thermal decoherence in this experiment. Another paper [7] considered the mass m = 10 −12 kg and the separation ∆ξ ∼ 1 µm. For these parameters, we find Ω m = 10 14 Hz. Hence, the time of decoherence due to the noise of graviton is estimated as τ dec 10 −9 s. Since the time-scale for the decoherence due to entanglement between two quantum systems is expected to be in the range of µs to ms in this paper, the decoherence induced by the noise of gravitons takes place faster than their estimation.
The decoherence caused by the noise of gravitons would offer new vistas to test quantum gravity in the tabletop experiment.

Conclusion
We considered quantum mechanics in the graviton background. We derived the Langevin type equation of geodesic deviation in Eq. (2.22). We found that the gravitons give rise to the noise to the dynamics of particles. We also found that the force of radiation reaction came in this system. We calculated the noise correlation in squeezed coherent states and find that the squeezed state enhance it compared with the vacuum state. We then discussed the detectability of the noise of gravitons. It turned out that the amplitude of the noise of gravitons in the case of the Minkowski vacuum is too small to be detected by the current detectors. However, in the squeezed state, we found that the noise of gravitons is enhanced by the exponential factor of squeezing parameter and then we may be able to detect the noise of gravitons for sufficiently large squeezing. Hence, the primordial gravitational waves that experience the large squeezing during inflation tend to make the amplitude of the noise sizable.
The noise is usually associated with the decoherence. The decoherence is a phenomena of the appearance of classical world from a quantum world. It is often argued that gravity has a universal role in the process of the decoherence. In particular, a superposition state of two spatially-separated locations has attracted interest of researchers as an approach to testing quantum gravity. We estimated the time of decoherence by gravitons and found that gravitons might be in fact the leading source of decoherence for some setup. In other words, we may be able to test quantum gravity by tabletop experiments through the process of decoherence due to the noise of gravitons in the future. The Hanbry-Brown-Twiss interferometers measure the intensity-intensity correlation functions (fourth order correlation functions) [27,28] and supply experimental evidence for quantum nature of gravitons [29,30], hence we expect the higher order correlation functions of the noise might contain the further information of the nonclassicality of gravitational waves. In this paper, we have simply estimated the amplitude of the noise of gravitons and the time scale of the decoherence. We leave the issue of more precise calculations in realistic experimental setups for future work.