Towards Communication in a Curved Spacetime Geometry ✉ ✉ Qasem Exirifard 1 , Eric Culf1 & Ebrahim Karimi 1

Home , Light cone, Shapiro time delay, Speed of light

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https://doi.org/10.1038/s42005-021-00671-8 OPEN Towards communication in a curved spacetime geometry ✉ ✉ Qasem Exirifard 1 , Eric Culf1 & Ebrahim Karimi 1

The current race in quantum communication – endeavouring to establish a global quantum network – must account for special and general relativistic effects. The well-studied general relativistic effects include Shapiro time-delay, gravitational lensing, and frame dragging which all are due to how a mass distribution alters geodesics. Here, we report how the curvature of spacetime geometry affects the propagation of information carriers along an arbitrary geodesic. An explicit expression for the distortion onto the carrier wavefunction in terms of the

1234567890():,; Riemann curvature is obtained. Furthermore, we investigate this distortion for anti de Sitter and Schwarzschild geometries. For instance, the spacetime curvature causes a 0.10 radian phase-shift for communication between Earth and the International Space Station on a monochromatic laser beam and quadrupole astigmatism; can cause a 12.2% cross-talk between structured modes traversing through the solar system. Our ﬁnding shows that this gravitational distortion is signiﬁcant, and it needs to be either pre- or post-corrected at the sender or receiver to retrieve the information.

1 Department of Physics, University of Ottawa, Ottawa, ON, Canada. ✉email: [email protected]; [email protected]

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hotons, electromagnetic waves, are widely used in classical the effects of the curvature at the vicinity of the Earth. Here, we and quantum communication since they do not possess present a computationally simple method to calculate the dis- P ’ electric charge or rest mass. However, a photon s traits, e.g. tortion by the curvature of any spacetime geometry on any group and phase velocity, wavelength, linear and optical angular localised wavepacket. momentum, are modified inside or during propagation through a In flat spacetime geometry, the propagation of each polarisa- linear or a nonlinear medium. These traits are governed by tion of photon is isomorphic to the propagation of a massless Maxwell’s equations, which are the relativistic quantum field scalar field. Based on our understanding of the Einstein equiva- theory of the U(1) gauge connection. Understanding how these lence principle, we expect that studying a massless scalar field optical properties are altered upon propagation is a key element theory also captures some features of a photon’s propagation in for any optical communication network. In optical communica- curved spacetime geometry. Therefore, in this study, we consider tion, the sender and the receiver, namely Alice and Bob, use one the propagation of a massless scalar in the bulk of the paper. In or several internal photonic degrees of freedom, such as wave- Supplementary Note 2, we prove that in the Lorentz gauge, each length, polarisation, transverse mode or time-bins, to share linear polarisation of the photon in a curved spacetime geometry information, including a ciphertext and the secret key to decrypt gets corrected as if it were a massless scalar field. It is reported the ciphertext. The propagation, e.g. through fibre, air or that the Riemann tensor quantum mechanically alters the wave- underwater channels, causes these photonic degrees of freedom to packet propagating along a geodesic. The alteration operators be altered, and thus causes undesired errors on the shared depend on the geodesic and components of the Riemann tensor information. Therefore, the alteration to those degrees of freedom on the geodesic. The alteration is calculated for examples for any communication channel needs to be considered and well including the spacetime geometry around the Earth and the Sun. examined. Sharing information with a longer range or with – moving objects, e.g. satellites, airplanes, submersibles1 5, requires Results and discussion the optical beam to not only traverse through a medium, but in a We start by considering a relativistic massless scalar field ψ ≔ ψ few cases, also in the fabric of the spacetime geometry, where (xμ) that propagates in a curved spacetime geometry xμ = (t, xi) 6–9 general relativistic effects manifest . Effects associated to the = (t, x, y, z) with an arbitrary Riemann curvature tensor Rabcd. change of the geodesic due to a mass distribution, such as Shapiro The units are chosen such that the speed of light in vacuum and time-delay10, gravitational lensing11 and frame dragging12, are = ℏ = – the reduced Planck constant are equal to one, i.e. c 1 and 1. well studied and observed13 16. Let us consider a localised wavefunction (information carrier) We explore the propagation of relativistic wavepackets along whose size is small compared to the curvature of the spacetime an arbitrary null geodesic in a general curved spacetime geometry, geometry. At the leading order, therefore, the carrier can be and show how the curvature of the spacetime geometry distorts treated as a massless point-like particle that travels along a null the wavepacket as it travels along the null geodesic. Different geodesic γ, see Fig. 1a. We choose the local Fermi coordinates20 methods are used to tackle this study. For instance, the four- along the geodesics in order to compute the quantum relativistic dimensional Klein-Gordon equation is approximated to a simple corrections. The metric’s components in the Fermi coordinates two-dimensional partial differential equation by ignoring all the can be expanded in terms of the components of the Riemann 17,18 multi-polar modes . In particular, in derivation of Eq. (7) tensor Rabcd and its covariant derivatives evaluated on the geo- from Eq. (6) in ref. 17, the second term on the left-hand side of desic, see Fig. 1b,c. The expansion of the metric up to quadratic Eq. (41) of ref. 19 has not been taken into account, therefore,17,18 order in the transverse coordinates of the geodesic of a massless cannot claim to reproduce all the effects of a curved spacetime particle γ is given by21, 19 ℓ h geometry. In ref. , all the multi-polar modes are presented þ À þ 2 2 ¼ þ δ a b À a bð Þ only at the level of the equations; however, the upper value of ds 2dx dx abdx dx Rþaþbx x dx ℓ = 100 on the multi-polar modes is considered to compute the 4 þ 1 þ b cð aÞþ b cð a bÞ þ ¼ ; solution. On the surface of the earth, a narrow beam with an Rþbacx x dx dx Racbdx x dx dx initial width of 10 cm and a large value of Rayleigh range requires 3 3 taking into account the contribution of multi-polar modes up to ð1Þ pffiffiffi at least ℓ = 109. So the solutions presented in ref. 19 do not take where x ± ¼ðx3 ± tÞ= 2 represent the Dirac light-cone into consideration all the multi-polar modes required to calculate + coordinates22 in the Fermi coordinates (x is always tangent to

Fig. 1 Schematics of communication in a general curved spacetime geometry and proper chosen coordinates. a Two parties, namely Alice and Bob, communicate in a general curved spacetime geometry. Alice encodes her message in a sequence of information carriers and sends them to Bob. The information traverses through the spacetime over a geodesic γ. The physical traits of information are distorted by the curvature of the spacetime, causing errors to the ﬁeld wavepacket. b The shown null geodesic γ. Close to γ, at the local coordinates, the metric is approximately pseudo-Euclidean. c The associated Fermi coordinates, the null-geodesic-path is mapped to x+.

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δ ^γ_ ω fi thenullgeodesicFig.1b), ab is the Kronecker delta, is the tangent where cp,ℓ,n( )aredened based on the initial and boundary con- ∈ ð aÞ¼ð À; aÞ Oω Qω Qω of the null geodesic and a, b {1, 2}, and x x x and the ditions. The operators, , U and N encodes how the curvature curvature components are evaluated on γ. The tree-level action of a of the spacetime geometry distorts the wavepacket. They are given by, fi ψ massless scalar eldR pinffiffiffiffiffiffiffi a general curved spacetime geometry is ω iω ~ i ~ ½ψ¼1 4 À μν∂ ψ∂ ψ O ¼À G xaxb þ G xa∂b þ G ∂a∂b; given by S 2 dx gg μ ν ,whereg is the determinant ab ab ω ab μ ν ∈ 2 of the metric gμν,and , {±, a}. Since the tree-level action is ω2 ψ ω i À 2 À a À ~ a quadratic in terms of , it is quantum mechanically exact, which Q ¼ 1 À ðÞx GÀÀ À ix ωx GÀ þ x GÀ ∂ ; ð5Þ fi U ω 2 a a can be veriR ed byR lookingpffiffiffiffiffiffiffiffiffiffi atR its generating function, À ð ½ψþ 4 À ψÞ À ½ψ i.e. Z½J¼ Dψe i S d x det gJ = Dψe iS À Dψ repre- Qω ¼ ÀG þ G a þ 2i G~ ∂a; x ÀÀ Àax Àa sents the integration overall field configurations and J is the source N ω field. Its exact effective action, as defined by the Legendre trans- G G~ G~ Γ ψ = where ab, ab and ab are integrals of the components of the Rie- formation of ln Z, coincides with the tree-level action, i.e. [ c] S mann tensor R evaluated on the geodesic—see Supplementary ψ φ ” fi þaþb [ c]. For a general action, c resembles a "classical eld whose Note 1: action is given by Γ[ψ ], while Γ[ψ ] encapsulates all the quantum Z Z Z c c τ τ ~ τ loop corrections. The exact effective action includes both the clas- G ¼ τ ; G~ ¼ τ G ; G~ ¼ τ G~ ; ab d Rþaþb ab d ab ab d ab sical and quantum effects. The classical effects are those that can be 0 0 0 reproduced by motion of a point-like particle along the geodesic; ð6Þ therestarequantum.Theeffectiveactionofafreephotonpro- τ fi pagating in curved spacetime geometry coincides to the tree-level where is the af ne parameter on the geodesic. The distortions action, therefore, we omit the subscript c. provided by Eq. (5) is the solution to the exact quantum effective The massless scalar (quantum) field ψ obeys the (covariant) action and cannot be reproduced by motion of a point-like particle À1 1 μν along a geodesic. They do not exist in flat spactime geometry, so they &ψ ¼ðÀ Þ 2∂ ðÀ Þ2 ∂ ψ ¼ wave equation, g μ g g ν 0. In the manifest a set of quantum effects in curved spacetime geometry. A Fermi coordinates, gμν can be viewed as a perturbation to the similar approach can be used to find the wavefunction of a massive Minkowski metric, inducing expansion series for the inverse and fi μν = ημν + εδ μν + ε2 scalar particle. The physical degrees of U(1) gauge elds get corrected determinantpffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the metric: g g O( ) and as if they were scalar fields, see Supplementary Note 2 and Supple- ð À Þ¼εδ þ ðε2Þ ε ln det g g O . is the systematic perturbation mentary Note 3. Supplementary Note 4 presents the operators in the parameter introduced to keep track of the perturbation series, Hilbert space that corresponds to these corrections. The following which means that all the components of the Riemann tensor in subsections show how these operators distort the physical informa- Eq. (1) are multiplied with ε, and ε is treated as an infinitesimal ε = tion encoded in wavepackets traveling along couple of examples of parameter. At the end of the computation, we set 1. This null geodesics in the Solar system and around the Earth. technique helps us to systematically perform perturbations for We now study the distortion operators in a couple of spacetime small curvatures. Utilising the perturbation gives, geometries, including de Sitter and Schwarzschild spacetime ð0Þ μν μν 2 fi &ψ ¼ & ψ þ εð∂μðδg ∂νψÞþη ∂νδg∂μψÞþOðε Þ; ð2Þ geometries. We rst study the de Sitter and anti de Sitter spacetime geometries because their symmetry allows one to immedi- ð0Þ μν 2 where & ¼ η ∂μ∂ν ¼ 2∂À∂þ þ ∇ ? is the d’Alembert operator ately write down the components of the Riemann tensor in Fermi fl ∇2 ¼ ∂2 þ ∂2 coordinates evaluated on the geodesic. in the at spacetime geometry, and ? 1 2. The perturbative nature of Eq. (2) seeks for a series expansion, ψ = ψ(0) + εψ(1) + O(ε2). Here, ψ(0) satisfies the scalar wave equation in the flat space- de Sitter and anti de Sitter spacetime geometries. The de Sitter time geometry □(0)ψ(0) = 0, and the perturbed term to the wave- and anti de Sitter space-times are maximally symmetric and the function, ψ(1),yields, Riemann tensor at any given event in the spacetime in any ð0Þ ð1Þ μν ð0Þ μν ð0Þ coordinates, including the Fermi coordinates, is given by & ψ ¼À∂μðδ g ∂νψ Þþη ∂νδ g∂μψ : ð3Þ Rμνμ0ν0 ¼ Λðgμμ0 gνν0 À gμν0 gνμ0 Þ. The value of Λ determines dif- − Λ We assumeR the Fourier expansion in terms of x variable, ferent geometries: > 0 represents the de Sitter spacetime geo- ψð0Þ ¼ ω ð0Þð þ; aÞ iωxÀ ð0Þ fi Λ ÀÁd f ω x x e ,wheref ω satis es the paraxial equa- metry; < 0 represents the anti de Sitter spacetime geometry; and 2 ð0Þ Λ = 0 is the Minkowki spacetime geometry. R+−+− = Λ is the tion 2iω∂þ þ ∇ ? f ω ¼ 0. This implies that the solutions, given by the paraxial approximation in optics23,24, are exact. The paraxial only non-zero component for Rþaþb evaluated on the geodesic, and thus the correction operators, Eq. (5) are, equation is isomorphic to the Schrödinger equation, and its solutions 2 (the transverse and longitudinal parts) can be expressed in the form ω ω i ω À þ ω þ À O ¼ 0; Q ¼ 1 À ðÞx 2 Λx ; Q ¼ Λx x : ð7Þ of Laguerre-Gauss (LG) modes (with an azimuthally symmetric U ω 2 N intensity profile) or Hermite-Gaussian (HG) wavepackets25.We consider a wavepacket wherein the field is slowly varying, and assume Let us consider a Gaussian wavepacket with normal distribution fi ω ω σ ψ ¼ ð0Þð þ; 1; 2Þ that the metric does not signi cantly change inside the wavepacket. for around 0 with the width of , i.e. Alice f x x x (0) (0) ðσ ÀÞ2 Therefore, all derivatives of ∂μψ , except ∂−ψ , are negligible, and iω xÀ À x −− e 0 e 2 . The validity of the perturbative solution demands the leading term in the right hand side of Eq. (3)is∂−(δ g ∂−ψ), jΛjω2 ∣Λ∣≪σ2 σ ≪ ω ÀÀ ð1Þ a b that 0, and 0. The wavepacket after the where δ g ¼Àgþþ ¼ Rþþx x .Therefore,Eq.(3) reduces to, ð ÀÞ2σ2 a b ω ω ω À ð0Þ À x ψ ’ð þ εQ 0 þ εQ 0 Þ i 0x 2 ð Þ ð Þ ð Þ propagationω isω Bob 1 U N e f e . & 0 ψ 1 ¼À a b∂2 ψ 0 : ð Þ 0 0 Rþaþb x x À 4 Q and Q change the wavepacket amplitude and phase, R N U ω − ð Þ À ð Þ jQ 0 ψ j = σ ψð1Þ ¼ ω 1 ð þ; aÞ iωx 1 respectively. The maximum of N Alice occurs at x ±1/ . Here, d f ω x x e with f ω being the correction to τ ω Requiringpffiffi it to be smaller than 1 yields T < A where the structure function for the frequency of .Wehavefoundthe τ ¼ðσ Þ=jΛj τ solutions to Eq. (4)—see the Supplementary Note 1 for more detail A e . A is the maximum time that the wavepacket feels the curvature of the spacetime geometry and keeps its onthederivation.Thesolutionis, ω Z Q 0 amplitude intact. U alters the phase ofp theffiffiffi wavepacket. The À ÀÁ ω ψð μÞ¼ ω ∑ ðωÞ iωx þ εðOω þ Qω þ Qω Þ ð0Þ; maximum of jQ 0 ψ j occurs at xÀ ¼ ± 2=σ. Requiring it to x d cp‘n e 1 U N f ω U Alice pffiffi p;‘;n τ τ ¼ðστ Þ=ð ω Þ be smaller than 1 results in T < φ where φ A e 2 0 .

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τφ is the maximum time that a wavepacket can feel the curvature and the phase: τ of the spacetime geometry and keep its phase intact. For T > φ, À ε ðÞσ À 2 ÀÁ x 1 1 À x 2 information about phase is lost at perturbation. This may point to δA ¼À À e 2 þ O ε “ ” 2 2 a gravitational decoherence , and its possible consequence on 2 ra rb fi 26 ð9Þ anti de Sitter/conformal eld theory correspondence demands ω ðÞσ À 2 ÀÁ i 0 1 1 À 2 À x 2 attention. τ represents the amount of time of interaction with δχ ¼ À ðÞσx e 2 þ O ε ; A 4σ2 r2 r2 the curvature that the wavepacket can keep its amplitude intact. a b τ ≪ τ We observe that φ A. So the phase changes sooner than the where rb is the location of Bob. The maximum alteration to the change in the amplitude. − amplitude andp phaseffiffiffi of the Gaussian wavepacket occur at x = ±1/σ and xÀ ¼ ± 2=σ, respectively. For a Gaussian wavepacket Schwarzschild spacetime geometry. We choose the standard that propagates radially close to the Earth, the maximum alteration to spherical coordinates r, θ, φ where geodesics are extrema of, jδA j¼ 1 1 À 1 the amplitude and phase are, respectively, max : σ 2 2 2 ra rb ω δχ ¼ 0 mÈ 1 À 1 À1 and : σ2 2 2 ,wherem⊕ is the Schwarzschild radius m 2 m _ 2 max 2 ra r L ¼À 1 À t_ þ 1 À r_2 þ r2ðθ þ sin2θφ_ 2Þ ; ð8Þ b r r of Earth. ν = Δν = δχ ¼ : For 0 456 THz and 1 kHz, max : 0 10 rad. Supplementary Note 8 provides further details on choosing these and m = 2GM• is the Schwarzschild radius, M• is the mass of the values. blackhole and G is the gravitational constant—the units are such that As a final example, we examine the weak regime of gravity m = 1. The components of the Riemann tensor in the Fermi coor- when the beam possesses well-defined transverse modes—see dinates adapted to a general null geodesic of Schwarzschild spacetime Fig. 2a. The wavepacket carrying a well-defined transverse mode geometry are derived in the Supplementary Note 5. Figure 2ashows traverses through space and reaches to the minimum distance of l several null geodesics that go very close to a blackhole. We first to the central mass—here, we assume l is large. We now consider consider that the wavepacket propagates along the radial direction, a specific wavepacket, a Hermite-Gauss transverse mode ðσ ÀÞ2 Fig. 2b, where the only non-zero components of the Riemann tensor ð0Þ À þ 1 2 iω xÀ À x þ 1 2 f ω; ;‘; ðx ; x ; x ; x Þ¼e 0 e 2 HG ; ðx ; x ; x Þ—Her- is R+−+− = −1/r3. This is the same component that appeared in the p q m n de Sitter spacetime geometry. The radial geodesic has l =0, and its mite-Gauss modes are used to extend the communication À 27 Oω ¼ Qω ¼À i À ðÞωx 2 1 À 1 alphabet beyond bits, i.e. 0 and 1 . The longitudinal and correction operators are 0, U ω 1 2 2 , 2 2 ra r frequency distributions are assumed to be Gaussian. When σ is Qω ¼À1 1 À 1 À large, the dominant correction operator is calculated to be: and 2 2 x , where Alice is located at ra.Thesecor- N 2 ra r rection terms do not contain derivatives of the spatial transverse : 2 ÀÁ O ¼À9 5ia 2 À 2 ; ð Þ coordinates. Thus, the Riemann tensordoesnotaffectthespatial x y 10 fi zR transverse pro le of the wavepacket. This is due to the symmetry, as pffiffiffi pffiffiffi the radial geodesic inherits the static and spherical symmetry of the ¼ = ð þÞ ¼ = ð þÞ where x 2x1 w x andrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiy 2x2 w x are dimensionless background. Figure 3a shows the amplitude and phase of a Gaussian þ 2 time-bin signal. Figure 3b depicts the alternations in the amplitude coordinates, wðxþÞ¼w 1 þ x is the beam radius, z ¼ 0 zR R

Fig. 2 Propagation of wavepacket and associated null geodesics in the Schwarzschild spacetime geometry. a The null geodesics (dashed curves) for beams that propagate very close to the event horizon—we set the event horizon at 1. b Schematic of wavepacket propagation radially in the Schwarzschild spacetime geometry. Alice and Bob are located at ra and rb, respectively, while θ and φ are polar and azimuthal angles of the standard spherical coordinates.

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Fig. 3 Propagation of Gaussian wavepacket along radial direction in the Schwarzschild spacetime geometry. a Amplitude (purple) and phase (orange) of the initial Gaussian beam at the sender (Alice), respectively, presented by Aða:u:Þ and χ(a. u. ). Their units, a. u. , are chosen such that both are normalised to their maximum values. b The changes in the amplitude (δA) and the phase (δχ) of the Gaussian wavepacket around frequency ω0 with width of σ transmitted over the radial null geodesic in the Schwarzschild geometry between ra and r = rb, the geometry is shown in Fig. 2b. The units are chosen where the Schwarzschild radius and light speed are one.

+ 1 2 Fig. 4 Propagation of Hermite-Gaussian modes in a weak gravitational field. Hermite-Gaussian mode HGm,n(x , x , x ) is not shape invariant under propagation in the Schwarzschild spacetime geometry, and both the intensity and phase profiles modify. a Intensity and phase distributions of the first 9 HG modes m, n ∈ {0, 1, 2} prior to the free-space propagation. b The intensity ∣εψ(1)∣2 and phase arg εψð1Þ distributions of the corrected term, for the first 9 HG modes m, n ∈ {0, 1, 2}, after the propagation in a weak gravitational field. The rows and columns are associated to m ∈ {0, 1, 2} and n ∈ {0, 1, 2}, respectively. The dimensionless coordinates are used for these plots.

1 ω 2 = 28–32 2 0w0 is the Rayleigh range, a ra/l is a scaling parameter, w0 is addition to known well-studied gravitational decoherence ,a the beam radius at Alice’s position—see Supplementary Note 6 decoherence occurs. We observe that, in addition to the known for more details. This operator contains coordinate parameters x decoherence of a bipartite entangled system when each particle and y, and thus, alters both the amplitude and phase of the traverses through a different gravitational field gradient32,33,a transverse modes upon propagation. The correction for the solar coherent beam decoheres when different segments of the spatial system, when Alice and Bob are at the mean Earth-Sun distance spread of the wave experience different tidal gravitational field from the Sun and the wavepacket passes at l = 2R⊙, and for gradients. The phenomenon we are reporting also occurs for = 2 = 9 — zR 2.8 km × a 1.34ÀÁ × 10 m, remains perturbative and is given geodesics passing very close to the event horizon see Supple- ð Þ μ μ εψ 1 ð Þ¼ : 2 À 2 ψ ð Þj þ mentary Note 7. by x i 0 10 x y Alice x x ¼T . (0) μ ψ The amplitude and phase of the mode ψ (x ) and the Finally, it is noteworthy that photon pairs entangled, e.g. correction εψ(1)(xμ) for a few Hermite-Gaussian modes are shown entangled in spatial, frequency or temporal modes, would be in Fig. 4. As seen, these alterations on the modes are considerable. affected by the curved spacetime geometry whenever they are fi For instance, it causes up to 12.2% crosstalk between HG0,3 and shared between two parties, namely Alice and Bob. The nal state M N of the entangled photon, indeed, is given by applying the non- HG0,1 modes. The crosstalk between mode and mode is ÀÁ 2 U ¼ þ εðO þ Q þ Q Þ given by N jM . Therefore, these perturbative alterations local operatorsÀÁ 1 U N onto the entangled U U ψ — U U need to be accounted for when information is encrypted in the states, A B entangled here, A and B are associated spatial modes. The action of curved spacetime geometry on the with the correction operators at Alice and Bob’s places, wavepacket is linear. Therefore, a target beam that does not respectively. possess the information can be used as a reference to monitor the distortion of the information carrier, and an active system can be Conclusion employed for compensating the distortion in real-time—in conjunction, results in retrieving the original information. We have presented how the curvature of the spacetime geometry Moreover, whenever ε ≃ 1, the higher-order terms of correction affects the propagation of an arbitrary wavepacket along a general need to be considered. For instance, for z <28km×a2, the geodesic in a general curved spacetime geometry. The effect is R beyond classical general relativity, residing in the same category as correction becomes larger than 1 and we need to take into 34 account higher ε terms. Taking into account all the corrections is Hawking radiation . A set of linear operators are presented that tantamount to knowing the Riemann tensor in whole of the encode the effect of the curvature. The corrections to the infor- spacetime geometry, a piece of knowledge which is not attainable. mation carrier wavepacket are investigated in cases of de Sitter Thus, we tend to argue that once the perturbation breaks, in (anti de Sitter) and Schwarzschild spacetime geometries. It has been shown that the corrections accumulate overtime and distort

COMMUNICATIONS PHYSICS | (2021) 4:171 | https://doi.org/10.1038/s42005-021-00671-8 | www.nature.com/commsphys 5 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00671-8 the wavepacket. The gravitational distortion, therefore, needs to be 27. Krenn, M. et al. Communication with spatially modulated light through accounted for in quantum communication performed over long turbulent air across Vienna. N. J. Phys. 16, 113028 (2014). distances in a curved spacetime geometry. 28. Pang, B. H., Chen, Y. & Khalili, F. Y. Universal decoherence under gravity: a perspective through the equivalence principle. Phys. Rev. Lett. 117, 090401 (2016). Data availability 29. Stefanov, V., Siutsou, I. & Mogilevtsev, D. Gravitational dephasing in The authors declare that the data supporting the findings of this study are available spontaneous emission of atomic ensembles in timed Dicke states. Phys. Rev. D. within the paper and its supplementary information file. 101, 044042 (2020). 30. 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