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

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Towards Communication in a Curved Spacetime Geometry ✉ ✉ Qasem Exirifard 1 , Eric Culf1 & Ebrahim Karimi 1 ARTICLE 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 geo- desic. 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] COMMUNICATIONS PHYSICS | (2021) 4:171 | https://doi.org/10.1038/s42005-021-00671-8 | www.nature.com/commsphys 1 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00671-8 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+. 2 COMMUNICATIONS PHYSICS | (2021) 4:171 | https://doi.org/10.1038/s42005-021-00671-8 | www.nature.com/commsphys COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00671-8 ARTICLE 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.
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