Signatures of correlation of spacetime fluctuations in laser interferometers

Domain: physics
Published: 2025-12-23
Editorial outcome: minor_revision

Abstract
Spacetime fluctuations (SFs), a common feature of proposed gravity models, could be detected using laser interferometers. To advance this effort, we provide the correspondence between expected interferometer output signals and gravity models. We consider three classes of SFs, characterised by the decay behaviours and symmetries of their two-point correlation functions. For each, we identify the low- and high-frequency behaviour of the outputs and their dependence on the interferometer’s length. Capturing these requires sensitivity over a broad frequency range that spans the light-round-trip frequency, as provided by laboratory-scale setups, whereas detecting the presence or absence of SFs can be done with high narrow band sensitivity at the light-round-trip frequency. Our approach applies to interferometers with arm cavities, such as the km-long LIGO detectors, and those without, like laboratory-scale setups QUEST and GQuEST. Finally, we constrain the strength and correlation scale of SFs by comparing our modelled signals with experimental data.

Introduction
A wide spectrum of ideas has been considered1,2 to understand the fundamental nature of gravity. Some, such as the idea of spacetime fluctuations (SFs), form a leitmotif in this effort. Since its first proposal by Wheeler3, SFs have been extensively examined4 in the context of a quantum description of gravity, as well as in different semiclassical models of gravity5–11 and the study of stochastic gravitational waves12,13. Spacetime has thus been hypothesised to be, for instance, classical but stochastic14, or classical but emerging from underlying quantum entanglement15, or holographic and having quantum perturbations16. These hypotheses suggest different mathematical forms for the correlation functions of SFs.

The scale of the correlations in SFs differs widely across models, ranging from the Planck length scale in effective field theories17 to long-range correlations in holographic models18,19. The latter hypothesise observable effects such as the violation of Lorentz invariance20, gravitational decoherence2,21, blurring of astronomical objects22, and interferometric noise23–25. Of these, the last has garnered attention through the development of gravitational-wave detectors26, and the Holometer27,28 experiment. Recently, sophisticated laboratory-scale Michelson laser interferometers (MLIs), such as QUEST29 and GQuEST25, which incorporate new quantum technologies, aim to search for SFs.

Unambiguous detection of SFs would constitute a breakthrough in understanding the fundamental nature of gravity. To enable such a detection, experimental designs require estimates of the strength and bandwidth of the expected interferometric output signals so they can be suitably optimised. Further, computing the interferometric output signal can help in understanding how SF signals, which are typically broadband, can be distinguished from instrumental noise. Moreover, quantitative theoretical predictions of the output signal are essential either to rule out the presence of SFs in interferometric data or, if the underlying gravity theory includes free parameters, to constrain those parameters. While many theoretical works have modelled the interferometric output signal to identify possible SF signatures10,16,17,30–32, they only offer model-specific predictions.

We take a more expansive approach in studying SFs. Irrespective of the classical or quantum description of a phenomenon, correlations in physical processes tend to decay either exponentially or polynomially with increasing separation r between two spacetime points33,34. Exponential decay of correlations typically emanates from underlying physics that is short-ranged. In the context of gravity, this captures both quantum15 and semiclassical11 models. On the other hand, polynomial decay of correlations of the form r−η (η∈R+), emanate from long-range interactions. In the context of gravity, such correlations of SFs are expected to decay as a reciprocal of distance, i.e., 1/r16,17,22,35 or possibly with a higher inverse power-law such as r−4/336. Furthermore, internal symmetries of the spacetime metric are reflected in induced symmetries of the correlation function, such as factorisation into spatial and temporal parts. This factorisation occurs in SF models such as the Oppenheim model10, a generalised Károlyházy model36, the continuous spontaneous localisation model37,38, and the Diósi-Penrose model39–41.

Thus motivated, we consider three possible classes of two-point correlation functions of the SFs: (a) factorised into spatial and temporal correlations, both decreasing with increasing spatial and temporal separation (for instance, a semiclassical model10); (b) an inverse of the separation between the two spacetime points (for instance, an SF model obeying the wave-equation in 3+1 dimensions16,22,35), with subclasses refining the definition of the separation as either a spatial or a spacetime separation; and (c) an exponential decay with the separation between the two spacetime points (for instance, due to quantum entanglement15, and in semiclassical11 models), with subclasses as in (b). Finding experimental evidence for or against one of the above classes of correlation functions describing SFs would constitute a breakthrough in elucidating the fundamental nature of gravity.

In this work, corresponding to each class of SFs, we identify three characteristic signatures: the low-frequency behaviour, the high-frequency behaviour and the L dependence of the interferometric output signal, where L is the arm length of the MLI. In order to observe all three signatures, it is sufficient if the MLIs are sensitive over two decades, as in the case of laboratory-scale MLIs in the light-round-trip frequency fLRT = c/(2L). However, sensitivity over a broader range could allow observation of the three signatures to a limited extent. The projected sensitivities of laboratory-scale MLIs such as QUEST and GQuEST span the fLRT, unlike LIGO. Therefore, we find that these laboratory-scale MLIs allow observation of more signatures than LIGO in principle, thus providing more information on the class of the underlying correlation function and aiding in distinguishing between the classes (a)–(c).

We also show that MLIs with arm cavities have a significant advantage in detecting the bare presence or absence of SFs. This is due to the peak in the interferometric output signal for MLIs with arm cavities at their fLRT. We note that for LIGO, fLRT ≈ 37.5 kHz, which is outside the frequency span of the publicly available data. GWTC-3 run42 in LIGO reports data only up to 5 kHz. We await new results from experimental runs that report at a higher frequency range.

Our approach is agnostic to the microscopic origins of the SFs. It is important to emphasis that the microscopic origins of the SFs differ widely between the different gravity models, ranging from quantum entanglement of degrees of freedom resulting in suitable moments of the metric15 to geontropic16 or stochastic11 fluctuations of the metric itself. Our approach requires as input only the two-point correlation of the spacetime metric from any model of gravity and the geometry of the MLI. This enables us to compute the output signal power spectral density (PSD) of the MLI on which all our conclusions are based. Our approach also accommodates computing the signal PSD for MLIs with arm cavities, such as LIGO. Whether the presence of arm cavities provides any advantage towards detection has been much debated lately16,18,43. This debate even led to the exclusion of arm cavities from experimental designs of recent MLIs25,28,29. We now settle the debate for any SF from classes (a)–(c).

In addition, we compare the computed output signal PSD with the measured output signal from the Holometer and QUEST MLI experiments to constrain the free parameters in classes (a)–(c). We also compare our constraints with prior ones16.

Results
In this paper, we present three main results, namely, distinguishing between correlation classes (a)−(c) using three characteristic signatures identified in the output signal PSD of the MLI without arm cavities, the advantage enjoyed by the MLI with arm cavities in detecting the bare presence or absence of spacetime fluctuations, and constraining the free parameters in the correlation functions from different gravity models using the measured output signal from the Holometer and QUEST MLI experiments. To this end, we use the approach described in the “Methods” section to obtain the PSD of the optical path difference between the two arms of the MLI in the presence of random fluctuations w(r) at r≡(t, x, y, z) in the spacetime metric (Eq. (4)). Thus obtained, the interferometric output signal PSD for MLI without (Eq. (15)) and with (Eq. (22)) arm cavities involves the correlation function w(t1, r1)w(t2, r2) = ΓS ρ(t12, r12) (Eq. (8)) where X12 = X1 − X2 (X=t, r) with ri ∈ {x, y, z} (i=1, 2) and ΓS is the strength of the SFs. Subsection “Correlation classes” in the “Methods” section lists the correlation functions defining the three correlation classes (a)−(c). Further, for the ease of comparison, we consider dimensionless frequency and PSDs defined in the “MLI output signal PSDs” subsection: ν (Eq. (24)), SNC (Eq. (25)), and SC (Eq. (26)). Note that we have two types of dimensionless PSDs: SNC independent of the correlation length ℓr, catering to correlation class (a) which is also independent of ℓr, and SC which depends on ℓr for correlation classes (b) and (c). From Eqs. (15, 17 and 22), it is clear that both SNC(ν) and SC(ν) are independent of ΓS.

Distinguishing correlation functions
To list the three characteristic signatures of the interferometric output signal PSD corresponding to each correlation class, we first consider the case of a simple MLI without arm cavities, as in Fig. 3(a). We present the analytical expressions for the PSDs corresponding to each class (For more details, see Sec. V of the Supplementary Information for class (a) and Sec. VI of the Supplementary Information for classes (b) and (c)).

Low-frequency limit ν≪1: For class (b) (Fig. 1 (b)), the PSDs are proportional to ν2 with SC(ν=0)=0 in this limit. We analytically find the constants of proportionality as 2/(3π) and 1/2 for ρIS and ρIST respectively (see, Sec. VI of the Supplementary Information for details). This can also be understood intuitively, especially in the case of ρIS. In this particular case, the fluctuations satisfy the wave equation and therefore, the 4-d Fourier transform (Eq. (18)) of ρ involves a Dirac delta function that forces 2πf=c|k| (see Eq. (81) of the Supplementary Information for the exact form). This leaves the interferometer response function (Eq. (19)) as the only frequency-dependent contribution to the PSD, which, up to a constant factor, is L2f2|T+(x)T+(z)|2=4 in this limit. The f2 in this term produces the quadratic trend of PSD in class (b).

For classes (a) and (c), the PSDs are almost flat in a log-log plot in the limit ν≪1 in Fig. 1(a), (c). We find the numerical fit in this limit for both cases to be of the form e−αν2 where α∈R is found numerically. In class (a), the exact numerical fit is SNC(ν≪1)≈0.275e−5.2νπ2. This fit is independent of the value of L (see Sec. V of the Supplementary Information for details). In class (c), the value α in e−αν2 depends on the specific value of the ratio κ=ℓr/L. For instance, α=4/π for κ=0.01 (see Sec. VI of the Supplementary Information for details).

High-frequency limit ν≫1: As is evident from Fig. 1 (a)−(c), the PSDs for all correlation classes decrease with increasing ν. However, the rates at which they decay are starkly different. In Sec. VI of the Supplementary Information, we analytically justify why we find the following decay rates of SC(ν): ∝1/ν for ρIS, ∝1/ν for ρIST, and ∝1/ν2 for ρES. We also present SC(ν) vs ν for ρES in Fig. 4. Though the PSD in this case is evidently dependent on κ, and therefore ℓr, we use SNC(ν) to showcase the high-frequency behaviour, which is always ∝1/ν2 irrespective of the value of κ (as this is not very apparent from Fig. 1(c)). We also note here that the onset of this decay is delayed as the ratio κ decreases.

The other decay rates, such as SC ∝1/ν2 for class (a) and SC(ν)≈0.03ν−0.3e−0.01ν for ρEST, are obtained through numerical fits. Of these, the functional form of the latter is obtained by considering the possible form of the cosine transform of ρEST (see Sec. VI of the Supplementary Information for details).

Dependence on L and κ: For class (a), we analytically show that SNC(ν) is independent of L in Eq. (68) of the Supplementary Information, and that SC(ν) for class (b) is independent of L in Eqs. (69, 70) of the Supplementary Information. For class (c), we analytically show that SC(ν) depends on κ in Eqs. (74-76) of the Supplementary Information. Figure 1(c) illustrates this κ-dependence of the PSDs.

As shown above, the different classes of spacetime fluctuations produce three characteristic signatures in their corresponding output signal PSDs. These can be used to identify the nature of the underlying SFs from interferometric data. Here, it is worth pointing out another feature that is present in the scaled PSDs of the three classes is the oscillatory behaviour at ν>1. However, this feature originates from the interferometer response (See Eqs. (49-50) of the Supplementary Information for details) and is not a characteristic signature of the different classes.

To observe all three signatures characteristic of each correlation class, the interferometer should ideally be sensitive in the range of ν≈0.1 to ν≈10. In QUEST with L=3 m (resp., GQuEST with L=5 m), the sensitive bandwidth is designed to span from 1 MHz to 250 MHz29 (resp., 8 MHz to 40 MHz25), with a corresponding span of 0.03⩽ν⩽78 (resp., 0.42⩽ν⩽2.1). This illustrates that the bandwidths of both QUEST and GQuEST would allow observation of all three signatures, although the narrower bandwidth of GQuEST could limit the observation of the low- and high-frequency signatures to some extent.

On the other hand, experimental data from LIGO cover the frequency range only from about 10 Hz to 10 kHz, corresponding roughly to 0.0004 < ν < 0.4. However, for completeness, we discuss the low- and high-frequency behaviour of the PSD for LIGO in Sec. VIII of the Supplementary Information.

To summarise, QUEST and GQuEST, with their broader bandwidths, allow observation of all the characteristic signatures that could help in distinguishing between correlation functions using their interferometric PSD data.

Detecting SFs
To highlight the advantage LIGO enjoys in detecting the SFs, we list three key features of the interferometric output signal of the MLIs with arm cavities (see Fig. 3(b) for the geometry, and see Fig. 2 for the PSDs) for the different correlation classes. For further details, see Sec. VIII of the Supplementary Information.

(1) For any fluctuation described by a Gaussian random process, the signal PSD (Eq. (22)) of MLIs with arm cavities has peaks at ν=mπ/2, for m=1, 2, 3, ..., of magnitude T4M/(1−RM)6. For instance, in LIGO with T=1−R=0.014 of the input mirror of the arm cavity, the magnitude of the peak is ≈3.2×105.

(2) For class (b), SC for an MLI without arm cavities is independent of L, with a global maximum at ν=1. Therefore, in this class, the Fabry-Pérot cavity response (Eq. (23)) enhances the signal strength (Eq. (22)) at every frequency ν. The strongest signal is at ν=π/2 (equivalently, f≈37.5 kHz for LIGO), as expected. This is illustrated in the top panel of Fig. 2. This is also consistent with prior work16. A similar argument holds for class (a) as its SC for an MLI without arm cavities is also independent of L.

(3) For class (c), SC is directly proportional to κ. We compare SC for an MLI without arm cavities in two different setups: LIGO with L=4000 m, and QUEST with L=3 m. It is evident from the values of L that the ratio κ in QUEST is far greater than that in LIGO for a given ℓr. Thus, SC ∝ κ implies that the LIGO signal is reduced with respect to that of QUEST by the factor ≈10−3 (i.e., the ratio L of QUEST to that of LIGO). This should be considered in conjunction with Feature 1 in this list (the presence of a peak at ν=π/2 in the LIGO signal is always enhanced by the Fabry-Pérot cavity gain ≈105). Therefore, when measuring SF with any ℓr in these setups, SC at ν=π/2 of LIGO has a peak that exceeds the SC of QUEST. This is illustrated in the bottom panel of Fig. 2.

It is thus evident that for the classes (a)−(c) considered, LIGO has a clear advantage over QUEST and GQuEST in detecting the presence of SFs.

It is also evident from our arguments that this advantage of LIGO is not guaranteed for all correlation functions. For instance, let us consider a class of correlation functions for which SC ∝ κ2 for an MLI without arm cavities. For such a class of correlation functions, using arguments similar to those used in discussing Feature 3, we can see that the peak in the LIGO signal does not exceed the PSD of QUEST. Therefore, LIGO does not enjoy an advantage in detection for such a class. This line of thinking might help in understanding some gravity models that predict LIGO should not enjoy any advantage in detecting SFs18,43.

Comparison with experimental data
We compare our results with experimental data from the Holometer [44, Fig. 12], a now-retired MLI experiment with L=40 m, and QUEST [45, Fig. 4], a table-top MLI experiment (L=1.8 m) that has just produced its first data. Note that we use QUEST data from the full measurement bandwidth, only part of which is published in Fig. 4 of 45.

While both the Holometer and QUEST experiments comprise twin MLIs designed to be cross-correlated, we restrict the present comparison to single-MLI PSDs for generality, avoiding additional assumptions on the correlation length of the SFs. We also do not consider data from LIGO or other gravitational-wave (GW) detectors, as these instruments operate and record data in a bandwidth that spans a relatively small range of ν and does not include the highest magnitude of the signal PSD, which is at ν=π/2.

The data from the Holometer and QUEST is obtained using the measurement technique in ref. 46, so the output signal sensitivity is improved by removing quantum shot noise, and potentially revealing stationary signals and classical noise produced in the MLI. The data is dominated by classical noise, and no signal was detected (i.e., the three characteristic signatures mentioned earlier cannot be discerned in the data). Thus, we can constrain the two parameters ΓS and κ (equivalently, ℓr). To obtain the constraints, we consider the magnitude of the PSD SC(ν=νi) of the SFs described by each correlation class to be smaller than the experimentally measured spectral density Smeas(ν=νi), at frequencies {νi}. This experimentally measured Smeas(ν=νi) contains negligible measurement uncertainty for the data considered. For instance, in the Holometer (respectively, QUEST) experiment, the constraint is obtained by SC(ν=νi) ⩽ Smeas(ν=νi) at every νi in the range 0.4⩽ν⩽4 (resp., 0.38⩽ν⩽1.8). The constraints are listed in Table 1 and plotted in Fig. 5 for classes (a)−(c). It is to be noted that the plot (Fig. 5) is a log-log plot of the parameter space {ΓS, κ}. Each grid line on the ΓS-axis marks an increase by a factor of 10, while that on the κ-axis corresponds to an increase by a factor of 100.2 ≈ 1.6.

(a) Factorised correlation function: Considering ρF (Eq. (10)) is independent of ℓr, the constraint obtained involves only ΓS. We know that SC is independent of ΓS and L from Eq. (68) of the Supplementary Information. Therefore, the constraint is given by
ΓS ⩽ min [Smeas(νi)/SC(νi)] (1)
The constraints obtained using the above expression are quoted in Table 1 and plotted in Fig. 5 (a). We note that we consider 10−4 ⩽ κ ⩽ 5×10−2. This range is chosen such that the two following conditions are satisfied: (1) κ ≪ 1 to ensure that the path difference remains a stationary random process, and (2) κ ≫ λ/L, where λ is the wavelength of light (~10−6 m for Holometer and QUEST), to ensure that the eikonal approximation remains valid. There are no constraints on the fluctuation strength considering ρF from prior work.

(b) Inverse correlation function: The constraint in each subclass is given by
ΓS κ ⩽ min [Smeas(νi) / (L3 SC(νi))] (2)
where SC(ν) is given by Eqs. (69, 70) of the Supplementary Information with m=S (respectively, m=ST) for the subclass (b1) (resp., (b2)). Equation (2) yields the constraints listed in Table 1 and plotted in Fig. 5 (b1), (b2).

In the case of the subclass (b1), the constraints are compared with the constraint α ⩽ 0.6 in the Pixellon model16. This is because
α ≈ (2/3) (ΓS κ L / 2πℓP)1/2 (3)
where ℓP is the Planck length. In 16, the constraint on α is obtained by comparing the expected signal PSD with the cross-spectral density (CSD) data from the twin interferometers in the Holometer (~10−41 m2 Hz−1). This is in contrast to our comparison with the PSD data from a single interferometer (~10−39 m2 Hz−1 for the Holometer), yielding the constraint α ⩽ 40. The two orders of magnitude difference between the constraints on α obtained using data from the Holometer can be explained by a corresponding difference between the magnitude of the CSD and PSD from the same experiment. In order to use the CSD data, an implicit assumption that the correlation length is larger than the distance between the two interferometers is made. We avoid such an assumption by restricting our comparison to the PSD data of the single interferometer. In subclass (b2), there are no constraints from prior work.

(c) Exponential correlation function: The constraints are obtained by setting SC(ν=νi) ⩽ Smeas(ν=νi) at every νi. Here we numerically compute SC(ν=νi) = ΓS ℓr L2 SC(ν=νi)/c using Eqs. (74-76) of the Supplementary Information. A numerical fit to such constraints yields the expressions listed in Table 1. In this class, there are no constraints from prior work.

Methods
Modelling light propagation
To investigate the effects of a fluctuating spacetime, we consider an isotropic spacetime metric gβγ (β,γ=0,1,2,3) of the form
g00 = −1+2w(r), gij=δij, g0i=gi0=0, (4)
where i,j=1,2,3 and w(r) is a random process in r≡(t, x, y, z), with w≪1. Here 0 (resp., i∈{1, 2,3}) corresponds to the timelike (resp., spacelike) component. While we consider fluctuations in every gβγ (see Sec. I of the Supplementary Information for details). To model the propagation of light of frequency Ω and wavelength λ=2πc/Ω in the given spacetime manifold, we solve for the electromagnetic tensor in the relativistic wave equation (RWE) (see Sec. I of the Supplementary Information with ref. 51 therein for details). This is subject to the following assumptions.

Assumption (i) Setting length and time scales: The correlation scales in length and time of the metric fluctuations w need to be longer than λ and 2π/Ω respectively. This effectively sets the wavelength as the smallest length scale in the system, i.e., the eikonal approximation. This allows us to neglect diffraction due to the fluctuations.

Applying the eikonal approximation (k=2π/λ→∞), the RWE reduces to the light propagating along the null geodesic for any general spacetime metric52. For light propagating along, say, the z-axis, the electric field solution for the RWE (see Sec. I of the Supplementary Information for details) is
E(r(t)) = Ein(x,y)eikΦ(r(t)), (5)
where
Φ(r(t)) = ct−z+ΦF(r(t)), (6a)
and Ein(x,y) is the input transverse profile of the beam. (6b)

Assumption (ii) Slowly varying envelope approximation (SVEA): The above solution (5) also uses the SVEA, which is consistent with the eikonal approximation. It assumes a very small rate of metric-fluctuation-induced phase fluctuations ∂tΦF ≪ c and ∂iΦF ≪ 1 (i=x,y,z).

We now introduce the assumptions on the random SF process w.

Assumption (iii) Stationarity: w(r) is a stationary Gaussian random process with the expectation values,
w=0, and (7)
w(t1, r1)w(t2, r2) = ΓS ρ(t12, r12), (8)
where X12 = X1 − X2 (X=t, r) with r1, r2 ∈ {x, y, z} and ΓS being dimensionless. Subsequently, we consider the correlation function ρ(t12, r12) to be from the classes (a)−(c) to obtain the corresponding MLI output as illustrated in Fig. 1.

Assumption (iv) Isotropy: The two-point correlation function ρ is
ρ(δ0, {δ1, δ2, δ3}) = ρ(δ0, {δ3, δ1, δ2}) = ρ(δ0, {δ3, δ2, δ1}) = ..., (9)
for any separation δμ (μ=0,1,2,3). We consider the correlation length of ρ to be ℓr for all three spatial dimensions, and the correlation time to be ℓr/c.

Correlation classes
We now list the different correlation classes, which completes the description of the spacetime fluctuations.

(a) Factorised correlation function: This class of correlation functions of the form
ρF(cΔt, Δr) = e−|Δr|/ℓr δ(Δt), (10)
is motivated by the Oppenheim model10, where Δr is any 3-vector in space and Δt is any time interval. We note that multiplicative factors consistent with Assumption (iv) are suitably introduced such that the delta-correlated ρF remains dimensionless. Also, it is evident that ρF is independent of ℓr. Thus, the only parameter in this correlation class is the fluctuation strength ΓS that scales the correlation function (Eq. (8)).

(b) Inverse correlation functions: This class, a subset of the polynomial decay of the correlations, is motivated by models such as those of Karolyhazy35, and Zurek16, as well as effective field theories17. We consider two sub-classes where the correlations decay as a function of spatial separation and spacetime separation, respectively, with the specific form of the correlation functions defining the subclass listed below.

(b1) Spatial separation:
ρIS(cΔt, Δr) = ℓr/|Δr| Θ(|Δr|−c|Δt|), (11)
where the Heaviside theta function Θ(|Δr|−c|Δt|) vanishes for all pairs of time-like separated spacetime points. Such correlation is found in models that assume the fluctuations satisfy the wave equation16,17,35.

(b2) Spacetime separation:
ρIST(cΔt, Δr) = ℓr / (|Δr|2 − c2|Δt|2)1/2 Θ(|Δr|−c|Δt|). (12)
This is a generalisation of (b1). Carrying forward the step function from (b1), the sign convention in the spacetime separation is chosen such that the correlation remains a real-valued function.

While we use two parameters to characterise strength, ΓS and the scale ℓr, in this class, the function depends only on the product ΓS ℓr, rendering the parameters degenerate.

(c) Exponential correlation functions: This class of correlation functions covers models motivated by entanglement between holographic degrees of freedom15 or a mesoscopic interpretation of gravity11. We again list the correlation functions for two sub-classes based on
(c1) Spatial separation:
ρES(cΔt, Δr) = e−|Δr|/ℓr Θ(|Δr|−c|Δt|). (13)
(c2) Spacetime separation:
ρEST(cΔt, Δr) = e−(|Δr|2 − c2|Δt|2)1/2/ℓr Θ(|Δr|−c|Δt|). (14)
Class (c) is a true two-parameter model, unlike classes (a) and (b). Classes (b) and (c) cannot be factorised into spatial and temporal functions.

MLI output signal PSDs
In an MLI (Fig. 3(a)), light propagates from a laser source at the input port A to a detector at the output port B via the two perpendicular arms, denoted by C and D. The 50/50 lossless beamsplitter is denoted by BS and is taken as the origin of the reference frame in our computation. The beamsplitter and the end mirrors are assumed to be perfect due to the very small optical losses that are typical in these high-precision interferometers. We can effectively assume the detector to be at the origin, as the effect of any phase fluctuations suffered by the light after interference at the BS is negligible compared to that of phase fluctuations incurred in the arms in practice. The arm length of the MLI without fluctuations is L and light-round-trip time τ0 = 1/fLRT (we recall that (fLRT)−1 = 2L/c). The PSD of the optical path difference between the two arms (see Sec. II of the Supplementary Information for details) at the detector is then written as a cosine transform from time separation Δτ to frequency f.

S(f) = (c2ΓS / 2π) ∫0τ0 dΔτ σ(Δτ) − ξ(Δτ) cos(2πfΔτ), (15)
where
σ(Δτ) = ∫0τ0 dt1 ∫0τ0 dt2 ρ(c(t1+Δτ−t2), 0, 0, s(t1)−s(t2)) (16a)
ξ(Δτ) = ∫0τ0 dt1 ∫0τ0 dt2 ρ(c(t1+Δτ−t2), s(t1), 0, −s(t2)). (16b)
Also, s(t)=ct if t⩽τ0/2 and s(t)=2L−ct if t>τ0/2. Assumptions (i)−(iv) are used to obtain Eq. (15), with Eq. (16a) using Assumption (iv). We rewrite the PSD in terms of the corresponding interferometer response function χI(f, k1) as an integral over a 3-dimensional reciprocal space (see Sec. III of the Supplementary Information for details).

S(f) = ∫ d3k1 ΓS ρe(2πf, k1) χI(f, k1), (17)
where ρe(2πf, k1) is given by
ρe(2πf, k1) = 1/(2π)4 ∫ d3r12 dt12 ρ(ct12, r12) e−i(2πft12 + k1·r12), (18)
and the response function χI(f, k1) of the MLI (Fig. 3(a)) is
χI(f, k1) = L2 |C1(f, k1) − C2(f, k1)|2, (19)
with
Cj(f, k1) = eifT+(j) Sinc(fT+(j)) + e2πifL Sinc(fT−(j)), (20)
where T±(j)(f, k1) = (πL/c) (1 ± c k1·ej / 2πf), (j=x,z). (21)

We note that the response function corresponding to an MLI with Fabry-Pérot arm cavities (see Fig. 3(b) for the interferometer geometry; also see Sec. IV of the Supplementary Information for details). The signal PSD in terms of this response function is given by
S(f) = χFP(f) ∫ d3k1 ΓS ρe(2πf, k1) χI(f, k1), (22)
where the Fabry-Pérot cavity response is given by
χFP(f) = T4M / (1+RM−2RM cos(2πf/fLRT)), (23)
and TM = 1−RM is the power transmissivity, and RM is the power reflectivity of the mirror M rendering the arm cavity in either arm.

To assess and compare the behaviour of the computed signal PSDs for the different correlation function classes, we consider the dimensionless frequency and PSDs.

ν = πf / (2fLRT) (24)
SNC(ν) = (c / ΓS L3) S(f), and (25)
SC(ν) = (c / ΓS ℓr L2) S(f). (26)
Note that we have two types of dimensionless PSDs: SNC independent of ℓr, catering to correlation class (a), which is also independent of ℓr, and SC, which depends on ℓr for correlation classes (b) and (c). From Eqs. (15, 17 and 22), it is clear that both SNC(ν) and SC(ν) are independent of ΓS.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In their manuscript “Signatures of Correlation of Space Time Fluctuations in Laser Interferometers” the authors discuss the\nsignatures of spacetime fluctuations in interferometer experiments and in in particular investigate the characteristics of the\npower spectral density associated with different type of correlations in the fluctuations in the geometry of space and time.\nThey present a careful analysis of three different type of correlations: factorized, inverse, and exponential correlations, where\nin the last two cases they also distinguish between spatial and spacetime separations. The results are written in terms of an\noutput signal that also takes into account the response function associated to the interferometer. The analytical and\nnumerical methods are sound, reasonably standard and are explained in a clear fashion. The obtained output signals for the\npower spectral density are presented concisely in three plots that clearly show the qualitative and quantitative differences\nbetween the different models.\nThe significance of this work lies in understanding how to analyse the data that will be obtained in present day and future\nlaser interferometer experiments. One of these experiments is the QQUEST experiment, which is specifically designed for\nthe purpose of detecting spacetime fluctuations. The possibility of detecting such signatures is of great scientific interest, and\neven when no positive signatures are found, the findings will provide important information about the theoretical models that\nare being developed. I find the paper well written and the results of high enough relevance and interest to recommend\npublication in Nature Communications.", "reviewer_2": "The topic investigated in this manuscript is very significant:\nquantum-gravity-motivated studies of spacetime fluctuations have a very long tradition,\nwith a more intense effort over the last 25 eyars, ignited mainly by Nature 398 (1999) 216\nThis is a research program facing challenges of two types:\non the experimental side the challenge originates from the fact that\nthe relevant effects are expected to be very small\nwhile on the theory side one faces the challenge that the structure of\nthe quantum-gravity problem, while pointing rather clearly to the presence of\nspacetime fluctuations, provides little guidance on how spacetime fluctuations\nshould be formalized and modeled (therefore many alternative formalizations are being studied).\nI formed a positive opinion of this manuscript since its results and observations\nwill prove valuable for the phenomenology of some of the attempted formalizations\nof spacetime fluctuations.\nI recommend the manuscript for publication.", "reviewer_3": "The manuscript \"Signatures of Correlation of Spacetime Fluctuations in Laser Interferometers\" by Sharmila et al. analyzes\nthe effect of spacetime fluctuations on laser interferometers LIGO, QUEST and GQuEST. In particular, they consider three\npossible classes for the correlations of the spacetime fluctuations, that encompass different models (collapse, semiclassical\ngravity, decoherence, effective field theory, holographic dof, etc). They find the corresponding PSDs (power spectral\ndensities), which are expressed in terms of the strength (Gamma) and characteristic length (l_r) of the correlations, the\ninterferometer arm length (L) and the scaled frequency (nu). Depending on the specific class (and subclass) of correlations,\nthe PSDs provide different qualitative behaviors, which would allow to identify the origin of spacetime fluctuations.\nUnfortunately, it is not clear how much the proposed analysis can be experimentally verified. Although citing and\nconstructing ad-hoc PSD for the three experiments mentioned above, the authors completely neglect the direct comparison\nof their results with experimental data. Since the parameters Gamma and l_r are free, I suppose such a comparison would\nhave led to experimental bounds on their possible values. I believe this is a strong shortcoming of the manuscript.\nMinor comments:\n- Introduction: there are several semiclassical models of gravity that are precedent than the cited [5,6] that are worth\nmentioning here. For example, Kafri-Taylor-Milburn, Tilloy-Diosi, Schrodinger-Newton etc\n- Introduction, in place of [2] the authors might want to cite the original papers by Diosi and by Penrose.\n- Introduction, when introducing the classes of correlations (pg 2, top of the left column), the general description does not\ncover the subclasses b1 and c1.\n- Eq.(6) and below, the authors might want to use the standard notation for the indeces: 0 for temporal component, 1-3 for the\nspatial ones.\n- Eq.(8), I suppose Theta is a Heaviside theta function. Please define\n- Above Eq.(12), what is physically tau_0?\n- Eq.(20), T_M and R_M are not defined.\n- Fig.2, are the wrinkles around nu=1 of any interest?\n- Why is the class (a) not discussed in the section \"Detecting SFs\"?\nI will be happy to review the manuscript again, after the authors revise it."}, "author_response": "Point-wise response to Reviewers\nWe thank Reviewers 1 and 2 for their recognition of our work and their recommendation to publish. Our\npointwise response to the comments of reviewer 3 is as follows.\nReviewer 3:\nMajor comment 1: “Unfortunately, it is not clear how much the proposed analysis can be experimentally\nverified. Although citing and constructing ad-hoc PSD for the three experiments mentioned above, the authors\ncompletely neglect the direct comparison of their results with experimental data. Since the parameters Γs and\nℓ are free, I suppose such a comparison would have led to experimental bounds on their possible values. I\nr\nbelieve this is a strong shortcoming of the manuscript.”\nResponse:\nWeagreewiththerefereethatthesuggestedcomparisonaddssignificantscientificvaluetothecurrentmanuscript.\nWe have therefore carried out a comparison of our expected interferometric output signal PSD with the\ndata from the Holometer [1, Fig. 12] and QUEST [2, Fig. 4]. We do not yet consider data from LIGO or\nother gravitational-wave (GW) detectors, as these instruments operate and record data in a bandwidth that\nspans a relatively small range of ν and does not include the highest magnitude of the signal PSD, which is at\nν =π/2. ThecomparisontodatafromQUESTandtheHolometerhasbeenreportedinanewsubsectiontitled\n“Comparison with experimental data”. We have also added both Fig. 5 plotting the exclusion plots in the\nparameter space and Table 1 listing the obtained constraints on Γs and κ for the different correlation classes.\nWe have also compared our constraints with pre-existing ones.\nRelated change in the manuscript:\nAnewsubsectiontitled“Comparisonwithexperimentaldata”,Fig. 5andTable1havebeenaddedtothemain\nmanuscript.\nMinor comment 1: “Introduction: there are several semiclassical models of gravity that are precedent than\nthe cited [5,6] that are worth mentioning here. For example, Kafri-Taylor-Milburn, Tilloy-Diosi, Schrodinger-\nNewton etc”\nResponse:\nReferences [5-9] in the revised manuscript have been added. They are as listed below.\n[5] L. Rosenfeld, On quantization of fields, Nucl. Phys. 40, 353 (1963).\n[6] R. Ruffini and S. Bonazzola, Systems of self-gravitating particles in general relativity and the concept of an\nequation of state, Phys. Rev. 187, 1767 (1969).\n[7] J. R. van Meter, Schr¨odinger–newton ‘collapse’ of the wavefunction, Classical Quant. Grav. 28, 215013\n(2011).\n[8] D. Kafri, J. M. Taylor, and G. J. Milburn, A classical channel model for gravitational decoherence, New J.\nPhys. 16, 065020 (2014).\n[9] A. Tilloy and L. Di´osi, Sourcing semiclassical gravity from spontaneously localized quantum matter, Phys.\nRev. D 93, 024026 (2016).\nMinor comment 2: “Introduction, in place of [2] the authors might want to cite the original papers by Diosi\nand by Penrose.”\nResponse:\nReferences [39-41] in the revised manuscript have been added. They are as listed below.\n[39] L. Di´osi, Gravitation and quantum-mechanical localization of macro-objects, Physics Letters A 105, 199\n(1984).\n[40] L. Di´osi, Models for universal reduction of macroscopic quantum fluctuations, Phys. Rev. A 40, 1165\n(1989).\n1\n[41] R. Penrose, On gravity’s role in quantum state reduction, Gen. Relat. Gravit. 28, 581 (1996).\nMinor comment 3: “Introduction, whenintroducingtheclassesofcorrelations(pg2, topoftheleftcolumn),\nthe general description does not cover the subclasses b1 and c1.”\nResponse:\nWe have added to and corrected the statement in the Introduction. The clause “, with subclasses refining the\ndefinition of the separation as either a spatial or a spacetime separation.” has been added to the definition of\nclass (b). The clause “, with subclasses similar to those in (b).” is added to the definition of class (c).\nMinor comment 4: “Eq.(6) and below, the authors might want to use the standard notation for the indeces:\n0 for temporal component, 1-3 for the spatial ones”\nResponse:\nEquation (6) has been modified accordingly.\nMinor comment 5: “Eq.(8), I suppose Theta is a Heaviside theta function. Please define”\nResponse:\nYes. Thiscommenthasbeenaddressedbyadding“wheretheHeavisidethetafunctionΘ(∥∆⃗ ∥−c|∆ |)vanishes\nr t\nfor all pairs of time-like separated spacetime points.” below Eq. (8).\nMinor comment 6: “Above Eq.(12), what is physically τ ?”\nτ denotes the light-round-trip time. This has been added above Eq. (12).\nM M\nResponse:\nWe have now defined the coefficients by adding “and Tm = 1−Rm is the power transmissivity and Rm is the\npowerreflectivityofthemirrorMrenderingthearmcavityineitherarm.”belowEq. (20). Wehavealsoadded\na definition of the power transmissivity to the figure caption of Fig. 1 to improve clarity.\nMinor comment 8: “Fig.2, are the wrinkles around ν =1 of any interest?”\nResponse:\nThese wrinkles are due to the interferometer response and is unrelated to the specific nature of the spacetime\nfluctuations. Wehaveaddedafootnote“AnotherfeaturethatispresentinthescaledPSDsofthethreeclasses,\nistheoscillatorybehaviouratν >1. Thisfeatureoriginatesfromtheinterferometerresponse[3,Eqs. (49)-(50)]\nandisnotacharacteristicsignatureofthedifferentclasses.”intheparagraphbeginning“Asshownabove,...”\nin Pg. 6.\nMinor comment 9: “Why is the class (a) not discussed in the section “Detecting SFs”?”\nResponse:\nIn both classes (a) and (b), the signal strength in LIGO is enhanced by Fabry-P´erot cavity response. As the\nreasons behind are identical, the following line has been added to item 2 in the “Detecting SFs” subsection. “A\n2\nsimilar argument holds for class (a) as its Snc for an MLI without arm cavities is also independent of L.”\nReferences\n[1] A.Chou, H.Glass, H.R.Gustafson, C.Hogan, B.L.Kamai, O.Kwon, R.Lanza, L.McCuller, S.S.Meyer,\nJ. Richardson, C. Stoughton, R. Tomlin, and R. Weiss, “The holometer: an instrument to probe planckian\nquantum geometry,” Classical Quant. Grav., vol. 34, p. 065005, feb 2017.\n[2] A. Patra, L. Aiello, A. Ejlli, W. L. Griffiths, A. L. James, N. Kuntimaddi, O. Kwon, E. Schwartz,\nH.Vahlbruch,S.M.Vermeulen,K.Kokeyama,K.L.Dooley,andH.Grote,“Broadbandlimitsonstochastic\nlengthfluctuationsfromapairoftable-topinterferometers,”Phys. Rev. Lett.,vol.135,p.101402,Sep2025.\n[3] See Supplemental Material for details regarding the field equations in a fluctuating spacetime, deriving\nthe spectral densities and the corresponding response functions for the two types of setups (aLIGO and\nHolometer), and the behaviour of the PSD for the different correlation function classes."}, {"round": 2, "reviewer_comments": {"reviewer_3": "I am happy with the changes made and I can recommend the manuscript for publication."}, "author_response": ""}]

Decision letter