Fluctuations of Spacetime and Holographic Noise in Atomic Interferometry

General Relativity and Gravitation (2011) DOI 10.1007/s10714-010-1137-7

RESEARCHARTICLE

Ertan Gokl¨ u¨ · Claus Lammerzahl¨ Fluctuations of spacetime and holographic noise in atomic interferometry

Received: 10 December 2009 / Accepted: 12 December 2010 c Springer Science+Business Media, LLC 2010

Abstract On small scales spacetime can be understood as some kind of spacetime foam of fluctuating bubbles or loops which are expected to be an outcome of a theory of quantum gravity. One recently discussed model for this kind of spacetime fluctuations is the holographic principle which allows to deduce the structure of these fluctuations. We review and discuss two scenarios which rely on the holographic principle leading to holographic noise. One scenario leads to fluctuations of the spacetime metric affecting the dynamics of quantum systems: (i) an apparent violation of the equivalence principle, (ii) a modification of the spreading of wave packets, and (iii) a loss of quantum coherence. All these effects can be tested with cold atoms. Keywords Quantum gravity phenomenology, Holographic principle, Spacetime fluctuations, UFF, Decoherence, Wavepacket dynamics

1 Introduction

The unification of quantum mechanics and General Relativity is one of the most outstanding problems of contemporary physics. Though yet there is no theory of quantum gravity. There are a lot of reasons why gravity must be quantized [33]. For example, (i) because of the fact that all matter fields are quantized the interactions, generated by theses fields, must also be quantized. (ii) In quantum mechanics and General Relativity the notion of time is completely different. (iii) Generally, in General Relativity singularities necessarily appear where physical laws are not valid anymore. Such singularities probably can be avoided by a theory of Quantum Gravity. Approaches like string theory or loop quantum gravity make some general predictions which may serve as guidance for experimental search. The difficulty

E. Gokl¨ u¨ C. Lammerzahl¨ ZARM University Bremen Am Fallturm 28359 Bremen, Germany [email protected] 2 E. Gokl¨ u,¨ C. Lammerzahl¨ is that in many cases the precise strength of the various expected effects are not known. One class of expected effects of a quantized theory of gravity is related to a nontrivial quantum gravity vacuum (depicted by Wheeler as “spacetime foam”) which can be regarded as a fluctuating spacetime. Spacetime fluctuations could lead to a minimal observable distance setting an absolute bound on the accuracy of the measurement of distances and defining a fundamental length scale [29; 34]. For instance, corresponding additional noise sources in gravitational wave interferometers have been considered [3], which has also been analyzed in the context of an experiment with optical cavities [47]. A spacetime foam may also violate the principle of equivalence as was suggested by Ellis et al. [18]. Another prediction of quantum gravity models are so called deformed dispersion relations [8]–[16]. In the work of Hu and Verdauger [28] classical stochastic fluctuations of the spacetime geometry were analyzed stemming from quantum fluctuations of matter fields in the context of a semiclassical theory of gravity. This leads to a stochastic behavior of the metric tensor. Furthermore, the effects of fluctuations of the spacetime geometry which lead to, e.g., lightcone fluctuations, redshift, and angular blurring were discussed in [19]. Spacetime fluctuations can also lead to the decoherence of matter waves which was discussed in [10; 44; 56] and [15]. The analysis in [17] takes into account nonconformal fluctuations of the metric and yields a modified inertial mass which is subject to the stochastic properties while—generalizing this work—in [20] the implications on a modified mass and an apparent violation of the weak equivalence principle were analyzed. In this paper we will lay special emphasis on the implications of a holographic noise. The reason for doing so is a special model of holographic noise advocated by C. Hogan who claimed that it could explain the so-called “GEO600 mystery noise”. Later, after having used different readout methods at GEO600, a fundamental holographic noise could be ruled out in the region from 150 to 300 Hz. However, it is claimed by Hogan that the GEO600 interferometer is a well suited and highly sensitive device, in principle, for detecting effects of a holographic spacetime. In the next sections we will discuss how a holographic spacetime leads to a (special kind) of fluctuations of spacetime, how an uncertainty of transverse positions emerges and what effects can be detected with cold atoms, atomic- and gravitational wave interferometers. In addition, we will also shortly point out the main differences of the two holographic models of spacetime. The paper is organized as follows: We start with an overview of the holographic principle and present two scenarios leading to holographic noise. The first scenario leading to metrical fluctuations has been extensively discussed by Ng et al. [39; 40], and the second is devoted to the work of Hogan [26], where the holographic noise manifests as transversal fluctuations of the position of the beam splitter in GEO600 (transversal with respect to the direction of the laser beam). Hereafter, we will discuss how fluctuations of the metric modify the dynamics of quantum systems. We will show that this leads to several modifications: (i) the inertial mass of quantum particles is modified accompanied by an apparent violation of the weak equivalence principle. (ii) The spreading of wave packets is affected by spacetime fluctuations and (iii) quantum particles are subject to decoherence. The discussion concerning the type of metrical fluctuations (noise Fluctuations of spacetime 3 scenario) is as general as possible and we will also show how holographic noise affects all aforementioned modifications of quantum dynamics. In all cases we discuss related interferometry experiments with cold atoms.

2 Models of holographic noise

In the following we shortly review how the notion of “holographic” noise appears in the physics of spacetime in an operational sense and introduce two different models. The ﬁrst describes metric ﬂuctuations and the second can be motivated by some kind of emergent spacetime.

2.1 Bounds on length measurements

We start with the question how precisely we can measure a distance l when quantum mechanics is taken into account. This problem was introduced by Salecker and Wigner [46] who carried through a gedankenexperiment and showed that length measurements are bounded by the mass of the measuring device. The measuring device is composed of a clock, a photon detector, and a source (e.g. laser), which altogether possesses a mass m and a typical size a. A mirror is placed at a distance l where the light coming from the source is reﬂected. Then it turns out that by virtue of the quantum mechanical uncertainty relation the uncertainty of the detector velocity ∆v is related to the position uncertainty ∆x1 of the detector according to

h¯ ∆v = . (1) 2m∆x1

After the light round trip time T = 2l/c the detector moved by T∆v which yields for the total position uncertainty of the detector, by adding up linearly,

hT¯ ∆xtot = ∆x1 + T∆v = ∆x1 + . (2) 2m∆x1

Minimizing this expression w.r.t. ∆x1 yields the minimum position uncertainty of the detector given by r hl¯ δl = (∆x ) = 2 , (3) qm tot min mc which is the result of the famous Salecker and Wigner Gedankenexperiment [46]. This has a connection with the so-called standard quantum limit [12] which is important for the precision of measurements in gravitational wave detectors. There- fore, in principle, the error in the position measurement can be reduced arbitrarily by increasing the mass m. However, this is limited by the requirement that the detector should not turn into a black hole. This condition calls for the inclusion of gravity in the analysis of the measurement uncertainty. 4 E. Gokl¨ u,¨ C. Lammerzahl¨

From the requirement that the detector must not be a black hole we set l > a > rS, where rS denotes the Schwarzschild radius, and one can suspect a ﬁrst estimate for the total measurement uncertainty. By inserting l & rS we get

δlqm & lP. (4)

p 3 Interestingly, the uncertainty is limited by the Planck length lP = Gh¯/c (≈ 10−35m), giving a hint, that this length can be regarded as a fundamental, minimal length scale for the geometry of spacetime. However, in the following we will see that the uncertainty is larger than in the original Salecker Wigner case, when, right from the start, we allow for gravitational effects in the measurement procedure. Since the measurement apparatus is a massive object with mass m it induces a curvature of spacetime changing the optical path of the photons. This ‘uncertainty’ is calculable but, as we are inter- ested in minimizing this effect and as—what will be shown in the following—it cannot be minimized arbitrarily and therefore is always existent, we will interpret the gravitational contribution to the total uncertainty as an additional uncertainty. Following the approaches in [2] and [38] we consider a spherically symmetric detector and the spacetime is characterized by the Schwarzschild metric, then the optical path from the light traveling from the detector (starting at the position a > rS) to a point l > a is the well known result

l Z dρ l − rS c∆t = = (l − a) + rS log , (5) 1 − rSρ a − rS a which can be approximated for l > a rS leading to the gravitational contribution δlg to the total uncertainty of the length measurement

l − rS l δlg = rS log ≈ rS log . (6) a − rS a

l ≥ a l ≥ Gm total uncertainty If we measure a distance 2 then δ g c2 and the reads r hl¯ Gm δl = δl + δl = 2 + . (7) tot qm g mc c2

By minimizing the total error w.r.t. the mass m of the detector this yields the minimal uncertainty

2 1/3 (δltot)min = 3(lpl) . (8)

In contrast to the ﬁrst naive estimate (4) where the measurement uncertainty was bound linearly by the Planck length, we see that the total error is now larger due to the scaling with the cubic root. Because the Planck length is so short the uncertainty in the distance is still very small. For example, considering the whole uni- 10 −15 verse (l ≈ 10 light years) leads to a tiny effect of (δltot)min ≈ 10 m. Fluctuations of spacetime 5

2.2 Connection to the holographic principle

The holographic principle was ﬁrst proposed by Gerard ’t Hooft [50] and was later given a string-theory interpretation by Leonard Susskind [49]. A good overview is given in [11]. The holographic principle states that the information content of a volume in spacetime is limited and related by the area enclosing this volume in Planck units. Alternatively we can state that although the world appears to have three spatial dimensions, its contents in terms of its degree of freedom can be encoded on a two–dimensional surface, like a hologram. In the following we will argue that the maximum entropy of a volume of space bounded by an area should be proportional to the area, instead of the volume. This result stems from black hole physics and thermodynamics, and motivates the holographic principle.

Entropy bounded by volume Suppose that a region of space with volume V is discretized by means of a lattice with spacing lp, equipped on each site with a spin 3 which can be in two states (that is, one information bit per lattice site V = lp). Then 3 the number of distinct quantum states distributed to n = V/lp sites reads N(V) = 2n. Its entropy is given by the logarithm and therefore the maximum entropy in this volume V yields

V S = lnN(V) = 3 ln2. (9) lp

Thus, if the energy density is bounded, the maximum entropy scales with the volume of the region. This result is valid for any theory in which the laws of nature are local [49] and what one usually presumes. Instead, there is reason to believe that the correct result which accounts for a quantum theoretical description of gravity (hence, a quantum gravitational description of spacetime) is a function of the area. The arguments were brought by Beckenstein [7] and are connected with the entropy of black holes given by their surface.

Entropy bounded by surface Consider a system with entropy S0 inside a spherical region Γ bounded by a surface with area A, where its mass must be less than that for a black hole. Imagine a spherically symmetric shell of matter collapsing onto the system with the right amount of energy so that it forms together with the system a black hole ﬁlling the region Γ . The entropy is now given by the 2 surface, that is by the Beckenstein-Hawking expression S = AkB/(4lp), where kB is the Boltzmann constant. If the second law of thermodynamics applies, then 2 dS > 0, therefore S0 6 S and hence S0 6 AkB/(4lp). This motivates the holographic principle, which states here that the maximum entropy of a region of space, hence the maximum amount of information, which can be stored in a volume is bounded by its surface area in Planck units,

A kB S = 2 . (10) lp 4 6 E. Gokl¨ u,¨ C. Lammerzahl¨

Fundamental measurement uncertainty After these preparatory remarks we are now in the position to derive the measurement uncertainty δl in an alternative way by means of the holographic principle, as was done by Ng et al. [39; 40]. Suppose we have a spatial cubic region with volume V = l3 which is parti- tioned into small cubes. The smallest possible cube can be as small as (δl)3 given by the measurement accuracy δl. We assume that one degree of freedom (which we will identify with information) is encoded in each of these cubes. The question is how many degrees of freedom N = V/(δl)3 can be put into the cubic region with volume V? According to the holographic principle the maximum amount of information (maximum N) is bounded by the surface area l2 of the cube with volume V in units of Planck-area. That leads to,

l3 l2 3 ≤ 2 (11) δl lp and we get for the uncertainty δl

2 1/3 δl ≥ (lpl) , (12) which is the same result like that obtained by the Gedankenexperiment combining Salecker-Wigner experiment and Gravity (8). The consistency of both results (8) and (12) for the uncertainties in length measurements gives us a hint that a microscopic description of spacetime may lead to modifications of geometry. In such a quantum description of geometry it is expected that spacetime has a foamy structure leading to spacetime fluctuations. Since we included gravity in the extended Salecker–Wigner Gedankenexperiment we can suppose that the holographic noise is a possible low–energy remnant of a quantum description of spacetime. In this limit the fluctuations can be modeled by fluctuations of the metric so that the fluctuations of a length measurement δl translate to metrical fluctuations according to

2/3 δgµν ≥ (lp/l) aµν , (13) where aµν is a tensor of order O(1).

2.3 Holographic model by Hogan

Wave picture of spacetimeIn Hogan‘s model of holographic uncertainty [26] the conjecture is that a so called “holographic quantum geometry of spacetime“ forms null sheets with two spatial dimensions perpendicular to the propagation direction of spacetime wavefunctions |ψST i. Locally, in a frame of reference, the pro- jection of the two dimensional sheet on this direction generates the third spatial dimension—it is therefore an apparent entity. A measurement of positions of two bodies is now subject to the spacetime wavefunctions revealing the properties of this spacetime. In order to describe the position measurement of the relative position of two bodies b and b (e.g. test masses in an interferometer) which are 1 2 described by means of the wavefunctions ψb1 and ψb2 , a position operatorx ˆ is applied which measures positions along the x-axis. Fluctuations of spacetime 7

Then the quantum mechanical amplitude for obtaining a pair of position results is given by [27],

hx1|x2i = ψb2 |xˆ2|ψST ψST |xˆ1|ψb1 , (14) showing that the properties of the so deﬁned spacetime impinge upon position measurements. This is translated by Hogan to a uncertainty relation for position measurements for two times t1 and t2 according to

[xˆ(t1),xˆ(t2)] = −ilpL, (15) where L = c ·(t2 −t1), for example, is the arm length of an interferometer as mea- sured by a light beam and lp is the Planck length. Note that the x−axis is perpendicular to the null trajectory of the spacetime wavefunction and therefore the measurement procedure shows a transverse position indeterminacy

δx(t1)δx(t2) > lpL. (16)

2 2 2 The overall error yields δx = δx(t1) + δx(t2) and the conjecture is, that a holographic uncertainty principle for transverse positions at events of null spacetime separations is obtained by minimizing the total error, hence setting δx(t1) = δx(t2), yielding p δx > lpL. (17) This resembles Eq. (12) but the difference between both holographic models of spacetime is that in Ng‘s approach, the uncertainty in length measurements is ori- ented along the longitudinal direction and scales with the cubic root of the length L, whereas in Hogans model we obtain a transverse length uncertainty, exhibit- ing a random-walk behavior. This transverse behavior and the different power-law dependence shows that Hogan‘s approach is not connected to ﬂuctuations of the metric but resembles a random-walk like ﬂuctuating metric.

Motivation from Matrix theoryNext, one needs a description for the dynamics of the spacetime wavefunctions in order to quantify possible deviations from the non- holographic case, obtained by measuring distances. For this, Hogan refers to a speciﬁc model of matrix theory [5]. The conjecture that spacetime wavefunctions generate the z-direction is suggested by this model from which Hogan deduces a wave equation for the spacetime wavefunction u(x,y,z), where in a lab frame z = ct, ∂ 2u ∂ 2u 4πi ∂u 2 + 2 + = 0. (18) ∂x ∂y cλ0 ∂z

Here λ0 is identified with a minimum wavelength given by the radius R of a com- pactified M dimension. This equation is analogous to a special form of the paraxial form of the Helmholtz equation (paraxial equation), (∇2 + k2)E(x) = 0, when applying the ansatz E(x) = u(x,y,z)exp(−ikz) and after having carried out an appropriate approximation. The circumference 2πR determines now the minimum wavelength by setting k = 1/R = 2π/λ0 in emergent spacetime being the inverse 8 E. Gokl¨ u,¨ C. Lammerzahl¨ wavelength of the “carrier wave” exp(−ikz) of the wavefunction u(x,y,z). By requiring that his description of spacetime agrees with black hole thermodynamics one identifies the minimum wavelength λ0 with the Planck-length by means of λ0 = 2lP. The solution to the paraxial wave equation is given by a Gaussian function, which is, for example, utilized to describe Gaussian beams in electrodynamics. Let z = 0 be the center of this beam, then one observes that along the z-axis the p beam forms a “waist” with half-width w0 = λ0zR/π at z = 0,√ with zR being the Rayleigh range where the beam width reaches w(±zR) = w0 2, and shows a broadening perpendicular to the z-direction for z > 0. It is important to note that in the context of position measurements of the relative distance of two bodies the minimum diffraction uncertainty w0 is now identified with the transverse uncertainty in Eq. (17) thus giving the heuristic uncertainty relation (15) a foundation inspired by Matrix theory. Furthermore, it has to be emphasized that this transverse uncertainty is common to all test particles and generated by the holographic spacetime.

Holographic noise signal in interferometers This uncertainty in relative position has impacts on gravity wave interferometers. It is quantiﬁed by the spectral density and must be calculated by the solution u(x,z) to Eq. (18). For interferometers, this function is interpreted as a wavefunction of position. One calculates now the time correlation of this signal and ﬁnally performs a Fourier transformation to frequency space ω yielding

lp S( f ) = (1 − cos( f / fc)), (19) π(2π f )2 where f = ω/(2π), fc = c/(4πL) and L is the length of one interferometer arm. In the low frequency limit, that is, for f c/(2L) this gives a ﬂat spectrum

t L2 S( f ) ≈ 2 p , (20) π

−44 where tp ≈ 10 s is the Planck time. Note that this spectrum is independent of any parameter and only incorporates the size of the interferometer. The effect of this specific model of holographic spacetime is such, that the holographic signal induces a transverse uncertainty of the position of the beamsplitter relative to the mirrors, where the light is reflected in the interferometer. Comparing the spectrum of holographic noise with an equivalent strain spectral density of gravitational waves finally yields √ −1q −1 −22 h( f ) = N 2tp/π = N 1.84 × 10 / Hz, (21) where N is the average number of photon round trips in the interferometer√ arms. For GEO600, N = 2 and the noise strain is h = 0.92 × 10−22/ Hz which lies in the region around 500 Hz and is the most sensitive frequency region of this gravitational wave detector. Fluctuations of spacetime 9

3 Modiﬁed quantum dynamics

In the model [20] spacetime is regarded as a fluctuating entity which consists of a classical fixed background on which Planck scale fluctuations are imposed ηµν + δgµν . In the low–energy limit they appear as classical fluctuations of spacetime and we model this as perturbations of the metric up to second order

gµν (x,t) = ηµν + hµν (x,t), (22) gµν (x,t) = η µν − hµν (x,t) + h˜ µν (x,t), (23) where |hµν | 1, greek indices run from 0 to 3 and latin indices from 1 to 3. ˜ µν µκ λν The second order perturbations are given by h = ηκλ h h where indices are µν raised and lowered with ηµν and η . We consider a scalar ﬁeld φ(x) non–minimally coupled to gravity as given by the action Z 2 2 1 4 √ µν ∗ m c ∗ S = d x g g ∂µ φ(x) ∂ν φ(x) − + ξR(x) φ(x) φ(x) , (24) 2 h¯ 2 where g = −detgµν is the determinant of the metric and x = (x,t). Here R(x) is the Ricci scalar and ξ a numerical factor. There are three values of ξ which are of particular interest [9]: (i) ξ = 0 is 1 the minimally coupled case, (ii) ξ = 6 is required by conformal coupling and can also be derived from the requirement that the equivalence principle is valid for the 1 propagation of scalar waves in a curved spacetime [48]. (iii) ξ = 4 originates from the squaring the Dirac equation. Variation of the action with respect to φ(x) yields the Klein–Gordon equation non–minimally coupled to the gravitational ﬁeld

2 2 µν m c g Dµ ∂ν φ(x) − + ξR(x) φ(x) = 0, (25) h¯ 2 where Dµ is the covariant derivative based on the metric gµν .

3.1 Non–relativistic limit

Because of the fact that in atomic interferometers typical velocities are far below the speed of light c we only need the Schrodinger¨ equation without relativistic corrections. For this, we have to calculate the non–relativistic approximation of the modiﬁed Klein–Gordon equation by expanding the wavefunction φ(x,t) = eiS(x,t)/h¯ in powers of c2

2 −2 S(x,t) = S0(x,t)c + S1(x,t) + S2(x,t)c + ··· , (26) according to the scheme worked out by Kiefer and Singh [32]. We give now a sketch of the calculations which were done in [20]. One performs a step-by-step calculation to orders c4 and c2 and inserts the expressions obtained for the func- 0 tions Si into the equation of motion to order c , where non-hermitian terms appear in the Hamiltonian. In order to get rid of them one can choose a modified scalar 10 E. Gokl¨ u,¨ C. Lammerzahl¨ product in curved space or alternatively keep the usual ‘flat’ euclidean scalar product and transform the operators and the wavefunction accordingly [35]. We proceed with the latter possibility and finally arrive at 1/4 h¯ 2 −1/4 h¯ 2 m ih¯∂ ψ = − (3)g ∆ (3)g ψ − ξR(x)ψ + g00ψ t 2m cov 2m 2 1 + ih¯∂ ,gi0 ψ. (27) 2 i

Here ∆cov represents the Laplace–Beltrami operator, {·,·} is the anticommutator, (3) 00 g is the determinant of the 3–metric gi j and g includes the Newtonian potential U(x). The Hamiltonian is manifest hermitian with respect to the chosen ‘ﬂat’ scalar product.

3.2 The effective Schrodinger¨ equation

It is assumed that the particle described by the modified Schrodinger¨ equation has its own finite spatial resolution scale. Consequently, only an averaged influence of spacetime fluctuations can be detected. This is quantified by means of the spatial average h···iV of the modified Hamiltonian leading to an effective Hamiltonian h¯ 2 H = − δ i j + αi j(t)∂ ∂ + ξ hRi − mU(x). (28) 2m i j V where the tensorial function αi j(t) consists of the spatial average of squares of the metrical fluctuations. The tensorial function αi j(t) and the Ricci scalar can be split into a positive definite time–average part and a fluctuating part according to i j i j i j α (t) = α˜ + γ (t) and hRiV (t) = r˜+ r(t) (29) with i j hγ (t)iT = 0 and hr(t)iT = 0. (30) Finally, this leads to an effective Schrodinger¨ equation h¯ 2 h¯ 2 ih¯∂ ψ(x,t)=− δ i j+α˜ i j∂ ∂ ψ(x,t)− r˜ψ(x,t)−mU(x)ψ(x,t), (31) t 2m i j 2m where the fluctuating phase γ(t)+r(t) was absorbed into the wave function which is still denoted by the same symbol. We will show in the following that this Schrodinger¨ equation leads to an apparent violation of the weak equivalence principle [20], to a modified dynamics of wavepackets [15] and to a decay of coherences in energy representation [21]. Sincer ˜ is constant over the averaging scale it just represents a constant shift of energy what can be omitted. Therefore we will operate in the following with the effective Hamiltonian h¯ 2 H = − δ i j + α˜ i j∂ ∂ − mU(x). (32) 2m i j Fluctuations of spacetime 11

4 An apparent violation of WEP

It will be shown now that the ﬂuctuation tensor α˜ i j leads to a renormalized inertial mass implying an apparent breakdown of the Weak Equivalence Principle (WEP). We identify the mass term in the Hamiltonian (32) as a renormalized tensorial inertial mass by virtue of 1 (m˜ i j)−1 ≡ δ i j + α˜ i j, (33) m what can alternatively be written as

i −1 m˜ i ≡ m · (1 + αp) , (34) D E i j 1 2 3 i i j ˜ii where α˜ = diag αp,αp,αp and αp = δ h(0) . The index p stands for ”par- Vp 1/3 ticle” characterized by its spatial resolution scale λp = Vp . This kind of anoma- lous inertial mass tensor was extensively studied by Haugan in connection with the principle of equivalence (EP) [23]. There, in contrast to the above result, this quantity is postulated in order to model violations of EP and appears in the energy function of a composite body subject to an external gravitational potential. All spacetime dependent ﬂuctuation properties of the inertial mass are now encoded in the quantities αi which are given by the averaged quadratic perturba- 2 tions σii . When we identifym ˜ j = mi and m = mg as inertial and gravitational mass giving for the ratio of both masses i mg i = 1 + αp , (35) mi p we can conclude that the metric ﬂuctuations utilized in our model lead to an apparent violation of the weak equivalence principle.

4.1 Holographic noise and other noise scenarios

In analogy to the analysis of fluctuations as function of time carried through in [45] we can associate a spectral density S(k) with the variance, 1 Z 1 Z (σ 2)i j = d3k(S2(k,t))i j = d3xh˜i j(x,t), (36) Vp Vp 3 (1/λp) Vp which allows us—in principle—to calculate the metric fluctuation components α˜ i j. 2 i j β i j i j As a result we then obtain (σ ) ≈ (lP/λp) a , where a is a diagonal tensor of order O(1). The special model [39; 40] which we discussed in Sect. 2 can now be applied. According to this, the outcome of any experiment measuring length intervals is influenced by spacetime fluctuations such that perturbations of this β quantity scales like (lP/l) , where β ≥ 0. As pointed out in [40] common values for β are 1,2/3,1/2. 12 E. Gokl¨ u,¨ C. Lammerzahl¨

We emphasize that by the inclusion of β = 2/3 (corresponding to the fluctua- 2 1/3 tions of the length δl ≈ (lpl) ) a holographic noise scenario is also included in our model of spacetime fluctuations. As a consequence, the modified kinetic term of the averaged Hamilton operator reads

2 β ! h¯ i j lP i j Hkin. = − δ + a ∂i∂ j. (37) 2m λp

This gives for our ﬂuctuation scenario

m m˜ = . (38) i β ii 1 + (lP/λp) a

In terms of the ﬂuctuation scenario (37) the violation factor αp now yields

β i lP ii αp = a . (39) λp

The modiﬁed inertial massm ˜ i is now dependent on the ﬂuctuation scenario β and on the type of particle via the characteristic resolution scale λp.

4.2 Experimental consequences

For typical atom interferometer experiments we consider the comparison of a 87Rb 39 atom to a K atom and take for the resolution scales the typical atom radii λRb ≈ −10 2λK ≈ 10 m. Then we can estimate the Eotv¨ os¨ factor to be on the order of

m g − m g η = 2 Rb K (40) mRbg + mKg l β l β ≈ p − p . (41) λRb λK

In our model this yields

−12 ηβ=1/2 = 10 (random walk noise), (42) −16 ηβ=2/3 = 10 (holographic noise), (43) −24 ηβ=1 = 10 (anticorrelation). (44)

However, it is worth to mention that the random walk scenario β = 1/2 may be tested by high–precision atomic interferometry in the near future as it seems not to be too far off from experimental possibilities. Fluctuations of spacetime 13

5 Spreading of wavepackets

5.1 Approximation of the kinetic term

In the preceding section we analyzed the modified kinetic term in (28) and its implication on the weak equivalence principle. By using spatial averaging and a modified time evolution operator we showed that the effect of non-minimal coupling, given by the Ricci scalar, leads to a fluctuating phase of the wavefunction. However, now we will start again with the full Hamiltonian of (27) and analyze the impact of the modified kinetic term on wavepacket spreading by deriving a modified master equation [21]. All contributions of the kinetic part as well as those terms in the anticommutator of Eq. (27) which do not include derivatives can be summed up together with the Ricci scalar in a stochastic scalar interaction term V(x,t) appearing in the Schrodinger¨ equation

h¯ 2 ih¯∂ ψ = − gi j(x,t)∂ ∂ + ∂ gi j∂ ψ + ihg¯ i0∂ ψ +V(x,t)ψ, (45) t 2m i j i j i where for simplicity we omitted the Newtonian potential g00. Our conclusions will not be affected by taking a Newtonian gravitational field in to account. Since we employ spacetime fluctuations acting on a spacetime scale being much smaller than typical length- and timescales on which the wavefunction ψ(x,t) varies we approximate the Hamiltonian in such a way that terms containing first- and second derivatives of the fluctuating quantities (hi j and the trace h) dominate in the Schrodinger¨ Eq. (45) allowing us to neglect the other contributions. In addition, we note that in the following derivation of the averaged master equation only second moments of V will appear, see Eq. (50). Therefore we can neglect second order terms already on the level of the Schrodinger¨ equation leading to

h¯ 2 ih¯∂ ψ = − ∆ψ +V(x,t)ψ, (46) t 2m where ∆ is the Laplace operator in Cartesian coordinates and where the random scalar interaction V(x,t) to ﬁrst order now reads

h¯ 2 1 V(x,t) = − ξR − ∆h , (47) 2m 4 where h is the first-order spatial part of the trace tr(gµν ). Compared to the second term in the fluctuating potential (47) the non-minimal coupling term ξR has the same structure and the same order of magnitude. Thus this contribution to the Schrodinger¨ Hamiltonian and to the analysis of spacetime fluctuations cannot be neglected in the context of our model and must be taken into consideration in the following calculations. 14 E. Gokl¨ u,¨ C. Lammerzahl¨

5.2 Stochastic properties

We have to specify the statistical properties of the fluctuating potential V(x,t) in order to derive an effective master equation. Owing to the fluctuating metric this term will be interpreted as a Gaussian random function. We will now perform the calculations for one dimension as the effective master equation will be derived for this specific case and from now on the spatial coordinate is denoted by x. This yields for the fluctuating potential

h¯ 2 V(x,t) = ∂ 2h (x,t). (48) 8m x 11 As the quantities of interest are the metric perturbation terms, we choose the following statistical conditions hV(x,t)i = 0 (49) 0 0 2 0 0 V(x,t)V(x ,t ) = V0 δ(t −t )g(x − x ). (50)

0 2 2 0 0 We introduced g(x − x ) = ∂x ∂x0C(x − x ), where C(x − x ) is a spatial correla- h¯ 2 tion function and defined V0 = 8m h0, while h0 is the strength of the fluctuations. It has been shown by Heinrichs [24; 25] that the dynamics of wavepackets are unaffected by temporal correlations of the random function h(x,t) if they are sufficiently small. As we are concerning Planck-scale effects (Planck time tp and Planck length lp) we can adopt these arguments. Therefore we have chosen a δ– correlation with respect to time. However, at the moment the spatial correlation function C(x − x0) will be left unspecified. As this is a model for spacetime fluctuations and because of the lack of a complete understanding of the microscopic structure of spacetime one has the freedom to characterize the statistical properties of the terms h with a variety of stochastic models given here by the correlator C(x − x0).

5.3 Effective master equation

Most physical quantities of interest—in our case the mean square displacement— can be conveniently calculated in terms of the reduced density matrix hρ(x0,x,t)i and its evolution equation. For the sake of simplicity we restrict all the calculations to the one–dimensional case. We adopt the calculation techniques of [30; 31] which are used in conjunction with quantum diffusion of particles in dynamical disordered media. We deﬁne the density operator ρ(x0,x,t) = ψ∗(x0,t)ψ(x,t) (51) and set up the master equation according to

2 2 0 ih¯ ∂ ∂ 0 i 0 0 ∂t ρ(x ,x,t) = − ρ(x ,x,t) + V(x ,t) −V(x,t) ρ(x ,x,t). 2m ∂x2 ∂x02 h¯ (52) Fluctuations of spacetime 15

The effective equation of motion can now be derived by applying the average over the fluctuations. Terms like hV(x,t)ρ(x0,x,t)i do not factorize in general as the density operator ρ(x0,x,t) is a functional of the fluctuation potential, hence ρ[V]. Because we have Gaussian fluctuations, we can apply the Novikov theorem [41; 42], which leads in our case to

Z Z δρ(x0,x,t) V(x,t)ρ(x0,x,t) = dt00 dx00δ(t −t00)g(x − x00) . (53) δV(x00,t00)

The functional derivatives are calculated using the formal solution of the master equation. Inserting this into the averaged Eq. (52) we obtain the effective master equation for the reduced density matrix hρ(x0,x,t)i

∂ ih¯ ∂ 2 ∂ 2 ρ(x0,x,t) = − − ρ(x0,x,t) ∂t 2m ∂x02 ∂x2 V 2 − 0 g(0) − g(x − x0) ρ(x0,x,t) . (54) h¯ 2 This is the modiﬁed effective master equation subject to the following discussions.

5.4 Modiﬁed wavepacket spreading

It is convenient to calculate the solution by means of Laplace- and Fouriertrans- forms. Hence, we convert the partial differential Eq. (54) to an ordinary, inhomogeneous ﬁrst-order differential equation. For practical reasons we introduce new coordinates according to X = x + x0 and Y = x − x0 and after having applied both transformations we get

2 m V0 m ∂Y R(K,Y,s)− s + (g(0) − g(Y)) R(K,Y,s)=− R(K,Y,t = 0), 2hK¯ h¯ 2 hK¯ (55) where is the s is the Laplace-variable, while R(K,Y,t = 0) is the Fourier transform of hρ(X,Y,t = 0)i and represents now the inhomogeneous part of the ordinary differential equation. This equation can easily be integrated and yields

msY msY R(K,Y,s) = exp + G(Y) R(K,Y ,s) − exp + G(Y) (56) 2hK¯ 0 2hK¯ Y m Z msY 0 mG(Y 0) × dY 0 exp − R(K,Y 0,t = 0)exp − , hK¯ 2hK¯ 2hK¯ 0 where the term R(K,Y0,s) = R(K,Y = 0,s) represents the initial value of the solu- V 2 tion R(K,Y,s) and we deﬁned G(Y) = 0 R Y dY 0 (g(0) − g(Y 0)). h¯ 2 0 16 E. Gokl¨ u,¨ C. Lammerzahl¨

The mean squared displacement of the particle can be expressed in Fourier space as

2 2 1 ∂ σx (t) = − 2 R(K,Y0,t) . (57) 8 ∂K K=0 Therefore the quantity of interest which has to be calculated is the initial value of R at Y = 0, corresponding to the diagonal element of the density matrix ρ(x0,x,t). The solution R(K,Y,s) is a quadratically integrable function and must vanish for Y → ∞. Therefore the r.h.s. of Eq. (56) must be zero in this limit leading to the Laplace-transformed quantity

∞ Z 2hK¯ R(K,Y ,s) = 2 dτ exp(−sτ)R K, τ,t = 0 exp(−G(τ)), (58) 0 m 0

m where we applied the substitution τ = 2hK¯ Y. Inverting this equation gives

 τ  2 Z 2h¯|K| V0 0 2h¯|K| 0 R(K,Y0,τ)=2R K, τ,t = 0 exp− dτ g(0)−g τ  m h¯ 2 m 0 (59) and the mean squared displacement (57) reads

1 σ 2(t) = e−G(Y)R (Y) ∂ 2G(Y) − (∂ G(Y))2 + 2∂ G(Y)∂ lnR (Y) x 4 0 k k k k 0 2 2 −∂k lnR0(Y) − (∂k lnR0(Y)) K=0 , (60)

h¯|K| 2h¯|K| where Y = 2 m τ and R0(Y) = R(K, m τ,t = 0). Note that the unperturbed solution is given by the last two logarithmic terms, where for K = 0 we get exp(−G(Y))R0(Y) = 1, if the initial state R0 is given by a Gaussian wavepacket. The information about the ﬂuctuation scenario is given by the correlation function appearing in G(Y). In our case we regard even correlation functions G(Y) = G(−Y) excluding a preferred direction in space. We can now calculate the mean squared displacement of a wavepacket and choose a Gaussian correlator

1 Y 2 C(Y) = √ exp − , (61) 2πa a2 where a represents a ﬁnite correlation length. We choose now for the initial conditions R(K,Y,t = 0) a Gaussian wavepacket. Consequently, the mean squared displacement reads

h¯ 2 5V 2 σ 2(t) = σ 2(0) + t2 + √ 0 t3. (62) x x 2 2 2 7 4m σx (0) 2πm a Fluctuations of spacetime 17

We note that the ﬁrst two terms correspond to the free unperturbed evolution whereas the last term accounts for superdiffusive behavior.1 Asymptotically, the wavepacket dynamics is dominated by the cubic term, namely

2 2 2 5V0 3 σx (t) = σx (0) + √ t , for t → ∞. (63) 2πm2a7 We infer from this fact that spacetime fluctuations lead to superdiffusive behavior of wavepackets and resembles wavepacket spreading in dynamically disordered media [31]. Finally, a short discussion about the length scale a and the fluctuation magnitude (-strength) V0 is necessary. In our model of spacetime we assumed fluctuations with a short wavelength, that is, it can be thought of having Planck wavelength lp. Since we assumed a correlated process and since Planck length appears here as a minimum length-scale, the first estimate would be to identify the correlation length also with lp. As we performed already an averaging over the fluctuations, in order to account for a classical metric and the limited resolution scale of quantum particles, also the correlation length a should be an effective length residing between Planck-length and the minimum resolution scale of the quantum particle. The exact value, of course, can not be deduced by our model, but in principle, could be estimated by measuring the modified expansion properties of wavepackets (see Eq. (63) ). Concerning the fluctuation strength V0 we supposed, for deriving the modified Schrodinger¨ Eq. (45), that it is sufficiently small compared to the minkowskian background metric and possibly exhibits Planck scale. Both quantities, V0 and the correlation length a, are statistically independent of each other and it is therefore possible that the correlation scale is larger than the magnitude of fluctuations and hence the magnitude of the effect of superdiffusion in Eq. (63) is still very small.

5.5 Connection to holographic noise

In the holographic scenario of [39; 40] the ﬂuctuations of the metric δgµν ≥ 2/3 (lp/l) aµν can be translated to a power spectral density (PSD) according to

−5/6 2 1/3 S( f ) = f (clp) , (64) where f is the frequency and c is the speed of light. In terms of a PSD in momentum space this yields

−5/6 2/3 S(k) = k lp , (65) where k is the wavenumber. It can be readily checked that this expression leads 2 1/3 2 R kmax 2 to δl = (l lp) by calculating the variance (δl) = 1/l dk(S(k)) , where kmax is subject to a cut-off given by the Planck length lp and the length l is given by

1 The nomenclature is at follows: For the mean-squared displacement x2 ∝ tν the exponent ν = 1 renders diffusive motion, ν = 2 is a ballistic motion and ν = 3 denotes superdiffusion. 18 E. Gokl¨ u,¨ C. Lammerzahl¨ the experiment. This shows that—by means of a correlation function—a finite, 2/3 nonvanishing correlation length is involved where the correlator scales with lp 2/3 modifying the wavepacket evolution accordingly. Therefore the lp -dependence could be utilized to distinguish the holographic noise scenario for wavepacket spreading from other fluctuations scenarios. However, in order to test these scenarios with high precision we suggest to use cold atoms and long time of flights. In this context it is feasible that test should be carried out in microgravity environments to facilitate the free evolution of wavepackets. This is realized for example by the Bose–Einstein–Condensate of the QUANTUS project [54; 55] where free evolution times up to one second are reached. Furthermore the upcoming PRIMUS project will provide a matter–wave interferometer in free fall which also could be used to perform high–precision measurements of wavepacket spreading.

6 Decoherence due to spacetime ﬂuctuations

6.1 Effective master equation

Having redefined the inertial mass of the particle as described in the preceding Sect. 4 we are left with an effective Schrodinger¨ equation of the form i ∂t |ψ˜ (t)i = − (H + Hp(t))|ψ˜ (t)i, (66) h¯ 0 i j where, however, only the fluctuating part γ (t) enters the Hamiltonian Hp(t) and we write 1 p2 H (t) = γi j(t)p p , H = − mU, (67) p 2m i j 0 2m where pi = −ih¯∂i and H0 is defined with an appropriately renormalized inertial mass in the kinetic term. We start by transforming to the interaction picture and obtain the Schrodinger¨ equation d i ψ˜ (t) = − H˜ p(t)|ψ˜ (t)i, (68) dt h¯ where the tilde labels quantities in the interaction picture. Equation (68) represents a stochastic Schrodinger¨ equation (SSE) involving a random Hamiltonian H˜ p(t) with zero average, hH˜ p(t)i = 0. For a given realization of the random process γi j(t) the corresponding solution of the SSE represents a pure state with the density matrix R˜(t) satisfying the von Neumann equation d i R˜(t) = − H˜ p(t),R˜(t) ≡ L (t)R˜(t), (69) dt h¯ where L (t) denotes the Liouville superoperator. However, if we consider the average over the fluctuations of the γi j(t) the resulting density matrix of the Schrodinger¨ particle, ρ˜(t) = R˜(t) (70) Fluctuations of spacetime 19 generally represents a mixed quantum state. Thus, when considering averages, the dynamics given by the SSE transforms pure states into mixtures and leads to dissipation and decoherence processes. Consequently, the time-evolution of ρ˜(t) must be described through a dissipative quantum dynamical map that preserves the Hermiticity, the trace and the positivity of the density matrix [14]. We derive from the linear stochastic differential Eq. (69) an equation of motion for the average (70) by the cumulant expansion method [52; 53]. To second order in the strength of the fluctuations it yields the equation of motion

t d 1 Z ρ˜(t) = − dt1 H˜ p(t), H˜ p(t1),ρ˜(t) . (71) dt h¯ 2 0 This is the desired quantum master equation for the density matrix of the Schrodinger¨ particle, representing a local ﬁrst-order differential equation with time-dependent coefﬁcients.

6.2 Markovian master equation

To proceed further we have to specify the stochastic properties of the random quantities γi j(t). We consider a white noise scenario assuming that the ﬂuctuations are isotropic,

i j γ (t) = σδi jξ(t). (72) This assumption already singles out a certain reference frame, which can be iden- tiﬁed with the frame in which the space averaging has been carried out [20]. In (72) the function ξ(t) is taken to be a Gaussian white noise process with zero mean and a δ-shaped auto-correlation function,

hξ(t)i = 0, hξ(t)ξ(t0)i = δ(t −t0). (73)

2 2 Therefore, the quantity σ has the dimension of time and we set σ = τc. Within the white noise limit the ﬂuctuations thus appear as un-correlated on the time scale of the particle motion and the contributions from the higher-order cumulants vanish. Then the second-order master Eq. (71) becomes an exact equation [51]. Using (72) we ﬁnd

H˜ p(t) = h¯V˜ (t)ξ(t), (74) where √ 2 τc p V˜ (t) = eiH0t/h¯Ve−iH0t/h¯ , V = . (75) h¯ 2m Substitution into the master Eq. (71), employing (73) and transforming back to the Schrodinger¨ picture we ﬁnally arrive at the master equation d i ρ(t) = − [H ,ρ(t)] + D(ρ(t)), (76) dt h¯ 0 20 E. Gokl¨ u,¨ C. Lammerzahl¨ where

1 1 2 D(ρ(t)) = − [V,[V,ρ(t)]] = Vρ(t)V − V ,ρ(t) . (77) 2 2 Equation (76) represents a Markovian quantum master equation for the Schrodinger¨ particle. The commutator term involving the free Hamiltonian H0 describes the contribution from the coherent motion, while the superoperator D(ρ), known as dissipator, models all dissipative effects. We observe that the master equation is in Lindblad form and, thus, generates a completely positive quantum dynamical semigroup [22; 36].

6.3 Estimation of decoherence times

The quantum master Eq. (76) can easily be solved in the momentum representation. To this end, we deﬁne the density matrix in the momentum representation |pi and with the help of the master equation we then ﬁnd d i ρ(p,p0,t) = − E(p) − E(p0)ρ(p,p0,t) dt h¯ τ 2 − c E(p) − E(p0) ρ(p,p0,t), (78) 2h¯ 2 where E(p) = p2/2m. This equation is immediately solved to yield

i (∆E)2τ ρ(p,p0,t) = exp − ∆Et − c t ρ(p,p0,0), (79) h¯ 2h¯ 2 where ∆E = E(p) − E(p0). We see that the diagonals of the density matrix in the momentum representation are constant in time. On the other hand, the coherences 2 2 corresponding to different energies decay exponentially with the rate (∆E) τc/2h¯ . Therefore an associated decoherence time τD is given by

2h¯ 2 h¯ 2 τD = 2 = 2 τc. (80) (∆E) τc ∆E · τc

We see that the dissipator D(ρ) of the master equation leads to a decay of the coherences of superpositions of energy eigenstates with different energies, resulting in an effective dynamical localization in energy space. This feature of the master equation is due to the fact that the fluctuating quantities γi j(t) couple to the components of the momentum operator. In our model the different possible realizations of the wavefunction due to the stochastic fluctuations of spacetime are accounted for by the ensemble average introduced in Eq. (70). In a matter wave experiment this leads to a reduced contrast of the interference fringes when both atom beams are recombined. When the traveling time of both beams is of the order of the decoherence time τD the interference fringes get smeared out significantly indicating the partial destruction of quantum coherence. Fluctuations of spacetime 21

Let us identify the time scale τc that characterizes the strength of the ﬂuctua- tions with the Planck time tp and we set τc = tp = lp/c with the Planck length lp. The expression (80) then yields the estimate

1013s τD ≈ . (81) (∆E/eV)2

We observe that due to the quadratic behavior the decoherence time depends strongly on the scale of the energy difference ∆E. For example, ∆E = 1eV leads to a decoherence time of the order of 300,000 years (1013 s), while ∆E = 1MeV leads to a decoherence time of the order of 10 s. Therefore for typical matter–wave experiments the detection of decoherence effects seems not to be feasible and is far beyond any experimental relevance. However, this result does not rule out in general the experimental decoherence caused by metric ﬂuctuations if one considers, for example, composite quantum objects whose states can be extremely sensitive to environmental noise. In principle the white–noise scenario (73) could be generalized to colored noise including holographic noise. This would lead to additional non–Markovian terms in the master Eq. (76) and we expect, however, that such terms would manifest only as small corrections to the equation for the decoherence time τD.

7 Concluding remarks

We have shown that metric fluctuations affect quantum dynamics where we have left open which noise scenario is realized due to the lack of a final theory of quantum gravity. One candidate is the holographic noise scenario and we have shown that this leads to specific signatures in the quantities of interest, which could be detected by experiments. These are the Eotv¨ os¨ factor, the mean–squared displacement of wave–packets and, when considering Markovian dynamics, the decoherence time of quantum systems. Since the topic of this workshop was to investigate the possibility of the detection of gravitational waves by means of atom interferometers we like to add some remarks about how our work is related to this. The detection of gravitational waves needs exceptional sensitivity and since spacetime fluctuations manifest as tiny effects these improved devices would be feasible to detect metric fluctuations or to rule out our scenario. Furthermore, the possibility that GEO600 may be sensible to holographic noise should give motivation to establish a complement measurement technique to check for holographic noise with an independent method. As a side effect and since gravity couples to all physical systems, this noise may show up in high–precision atom interferometers which are dedicated to the detection of gravitational waves.

Acknowledgments We like to thank K. Danzmann, D. Giulini, C. Hogan, and S. Hild for help- ful discussions. EG gratefully acknowledges the support by the German Research Foundation (DFG) and the Centre for Quantum Engineering and Space–Time Research (QUEST), and CL the support by the German Aerospace Center (DLR) grant no. 50WM0534. 22 E. Gokl¨ u,¨ C. Lammerzahl¨

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