arxiv:2110.13990v1 [gr-qc] 26 oct 2021

Gravitational distortion on photon state at the vicinity of the Earth

Qasem Exirifardlowast and Ebrahim Karimidagger

Department of Physics University of Ottawa 25 Templeton St Ottawa Ontario K1N 6N5 Canada

As a photon propagates along a null geodesic the space-time curvature around the geodesicdistorts its wave function We utilise the Fermi coordinates adapted to a general null geodesic andderive the equation for interaction between the Riemann tensor and the photon wave function Theequation is solved by being mapped to a time-dependent Schrodinger equation in (2+1) dimensionsThe results show that as a Gaussian time-bin wavepacket with a narrow bandwidth travels over a nullgeodesic it gains an extra phase that is a function of the Riemann tensor evaluated and integratedover the propagation trajectory This extra phase is calculated for communication between satellitesaround the Earth and is shown to be measurable by current technology

I INTRODUCTION

Einstein gravity is a geometric theory for gravitywherein energymass distribution curves its surroundingspace-time geometry and particles propagate along thegeodesics of the curved geometry The light bending bya gravitational source manifesting that photons propa-gate along null geodesics was first observed by Edding-ton et al [1] with 30 precision and the effect has nowbeen measured with the precision of 001 [2ndash5] In lightbending as well as all other observed effects of Einsteingravity the photon is treated as a point-like particle[6]The photon however is governed by the rules of quan-tum mechanics where particle-wave duality is manifestAs a photon moves along the geodesic its quantum wavefunction interacts with the curvature of the space-timegeometry around the geodesic and gets distorted When-ever the photon wave function is used as an informationcarrier [7ndash11] the distortion affects the communicationchannel and may introduce errors The current race toestablish a quantum network in space [12ndash14] may needto take into account how the curvature of the space-timegeometry affects the wave function of the photon as itmoves along a geodesic Approximations where all themulti-polar modes are neglected are used to calculatethe Green function for the propagation which resultedin a measurable distortion [15ndash18] Jonsson et al [19]has kept the first few multi-polar modes to calculatethe distortion for a photon scattered from a black holeThough none of these methods return the exact distor-tion they highlight that the distortion is a substantial ef-fect In Ref [20] the Fermi coordinates along a geodesicis adapted and a distortion of measurable magnitude isreported for communication between the Earth and theInternational Space Station However turbulence dueto the Earthrsquos atmosphere adds noise and may not al-low measurement of the effect Here we aim to calcu-late the gravitational distortion for communication per-

lowastElectronic address qexirifauottawacadaggerElectronic address ekarimiuottawaca

formed between two satellites around the Earth whereatmospheric effects are absent The paper is organisedas follows Section II considers a photon that propagatesalong a null geodesic in a general curved space-time ge-ometry The photon is approximated to a time-bin wavepacket with a very narrow frequency line The Fermi-coordinates adapted to a null geodesic are utilised to cal-culate the interaction between the photonrsquos wave functionand the curvature of the space-time geometry around thegeodesic We show that as the photon propagates alongthe null geodesic it gains an extra phase that is a func-tion of the Riemann tensor evaluated and integrated overthe null geodesic This extra phase which depends on thespace-time geometry is calculated for communication be-tween satellites near the Earth The results are shown inSection III and Section IV provides the discussion onhow to measure this extra phase

II PREDICTING A GENERAL GEOMETRICPHASE

The photon moves on the null geodesic γ We adaptthe Fermi frame wherein the metric on the geodesic coin-cides to the Minkowski metric and Levi-Cevita symbolvanishes too

gmicroν |γ = ηmicroν (1)

Γmicroνη|γ = 0 (2)

We represent the coordinates in the Fermi frame by(xplusmn x1 x2) where xplusmn = 1radic

2(x3 plusmn ct) are the Dirac light-

cone coordinates while x+ is tangent to γ [21] Themetric around the geodesic in the Fermi coordinates upto quadratic order in the transverse coordinates is givenby [22]

ds2 = 2dx+dxminus + δab dxadxb minusR+a+bx

axb(dx+)2

minus 4

3R+bacx

bxc(dx+dxa)minus 1

3Racbdx

bxc(dxadxb)

+ O(xaxbxc) (3)

where xa = (x1 x2) and xa = (xminus xa) and all thecurvature components (R+a+b R+acd and Rabcd) are

arX

iv2

110

1399

0v2

[gr

-qc]

23

Mar

202

2

2

evaluated on γ and Einsteinrsquos notation is used whereintwice appearance of an index variable in a single termmeans summation over that index and δab is the Kro-necker delta We approximate the space-time geometryaround the Earth to the Schwarzschild space-time geom-etry Therefore the effective Lagrangian of a masslesspoint particle propagating on a null geodesic can be givenby

L = minus(

1minus m

r

)t2 +

r2

1minus mr

+ r2(θ2 + sin2θ ϕ2

) (4)

where dot presents variation with respect to the affine pa-rameter on the geodesic m = (2GNMoplus)c2 and Moplus isthe mass of the Earth We choose the units such that thespeed of light in vacuum is set to 1 ie c = 1 and m = 1Due to the spherical symmetry without loss of generalitywe can choose the equatorial plane θ = π2 and θ = 0to describe any given geodesic at all time The cyclicvariables of ϕ and t lead to invariant quantities r2ϕ = l(1 minus 1r)t = E where l and E are constant values Weconsider null geodesics reaching the asymptotic infinityand set E = 1 Due to the form of the Lagrangianits Legendre transformation which is the Lagrangian it-self is invariant We consider a null geodesic and set

L = 0 that gives |r| =radic

1minus 1r2

(1minus 1

r

)l2 Let et er eϕ

and eθ represent the normalized unit vectors in t r θ ϕcoordinates The normalized unit vectors in the Fermicoordinates can be chosen as

e+ =fradic2

(+et + rer + rfϕeϕ) (5a)

eminus =1radic2f

(minuset + rer + rfϕeϕ) (5b)

e1 = minusrfϕer + reϕ e2 = eθ (5c)

where f =radic

1minus 1r The components of the Riemanntensor in the Fermi coordinates should be computed bythe tensor transformation law [23]

Rαβγδ = Rmicroprimeνprimeσprimeτ prime(eα)microprime(eβ)ν

prime(eγ)σ

prime(eδ)

τ prime (6)

Utilizing the abstract method employed in [20] identifiesthe non-vanishing components of the Riemann tensor inthe Fermi frame

R+minus+minus =3l2(r minus 1)

2r6minus 1

r3 (7a)

R+2+2 = minusR+1+1 =3l2

4r5 (7b)

R+minus+1 =3radic

2l

2r4r (7c)

Next we consider the electromagnetic potential Amicrowhose field strength is given by Fmicroν = partmicroAνminuspartνAmicro Thedynamics of the electromagnetic potential in a curvedspace-time geometry endowed with metric gmicroν around

the geodesic and is given by

Γ[Amicro] = minus1

4

intd4x gmicromicro

primeννprimeFmicroνFmicroprimeνprime (8a)

gmicromicroprimeννprime =

1

2

radicminusdet g (gmicromicro

primegνν

primeminus gmicroν

primegνmicro

prime) (8b)

Here gmicroν is the inverse of the metric and det g is itsdeterminant Note that gmicromicro

primeννprime has all the symmetries ofFmicroνFmicroprimeνprime under exchange of its indices The functionalvariation of the action with respect to the gauge fieldgives its equation of motion ie

partmicro

(gmicromicro

primeννprimeFmicroprimeνprime)

= 0 (9)

which we would like to solve for a photon that travelsalong a null geodesic We choose the Fermi coordinatesadapted to the null geodesic Eq (3) to describe thespace-time at the vicinity of the geodesic where the com-ponents of the Riemann tensor are given in Eq (7) Atthe vicinity of the Earth the components of the Riemanntensor are minimal We therefore treat them as a per-turbation We introduce ε as the systematic parameterof the perturbation In other words we add a factor of εto all terms in Eq (3) where the components of the Rie-mann tensor are present At the end of the computationwe set ε = 1 The ε-perturbation to the electromagneticpotential and gmicromicro

primeννprime follow

Amicro = A(0)micro + εA(1)

micro +O(ε2) (10a)

gmicromicroprimeννprime = g(0)micromicroprimeννprime + εg(1)micromicroprimeννprime +O(ε2) (10b)

Equation (9) at the leading order in ε can be simplifiedto

(0)A(0)micro + partmicropartνA(0)ν = 0 (11)

Henceforth ηmicroν is utilized to move up or down the indices

ie A(0)ν = ηνλA(0)λ partmicro = ηmicroνpartν where ηmicroν represents

the Minkowski metric in Dirac coordinates We choosethe Lorentz gauge partνA

(0)ν = 0 which simplifies the

equation for A(0) to (0)A(0)micro =

(2part+partminus +nabla2

perp)A

(0)micro = 0

where nabla2perp is the Laplace operator in x1 and x2 direc-

tions nabla2perp = part2

1 + part22 Utilizing the Fourier expan-

sion of the gauge field in terms of the variable xminus ie

A(0)micro =

intdωf

(0)micro (ω x+ xa)eiωx

minusleads to

(2iωpart+ +nabla2perp)f (0)

micro = 0 (12)

which is referred to as the paraxial Helmholtz equationNote that due to the definition of eminus in Eq (5b) the grav-itational red-shift is already encoded in the Fourier ex-

pansion of A(0)micro We refer to f

(0)micro (w x+ xa) as the struc-

ture function of the photon with frequency ω in modemicro The structure mode can be expanded in terms of theHermite-Gaussian or Laguerre-Gaussian modes For thepurpose of communication we are interested in a field

3

configuration that can be understood as a perturbationmodulated over a frequency that holds

|part+f(0)ν | ω|f (0)

ν | |partminusf (0)ν | ω|f (0)

ν | (13)

which is the same as the paraxial approximation in opticsEmploying Eq (13) in the Lorenz gauge condition yields

ωf(0)+ + part+f

(0)minus + part1f

(0)1 + part2f

(0)2 = 0 (14)

The paraxial approximation conditions ie Eq (13)imply the following perturbative solutions

f(0)+ = 0 (15a)

part+f(0)minus + partaf (0)

a = 0 (15b)

where partaf(0)a = part1f

(0)1 + part2f

(0)2 is used We solve

Eq (15b) for f(0)minus This leaves f

(0)a as the physical modes

which can be perceived as the distribution of the photonrsquospolarization This means that each polarization of pho-ton that we choose to represent by Ψ satisfies Eq (12)The paraxial wave equation Eq (12) can be rewrittenas minus 1

2ωnabla2perpΨ = ipart+Ψ which is the Schrodinger equation

for a particle with a rest mass of ω inldquo2+1rdquo dimensionswhere x+ plays the role of time

Utilizing Eqs (10) and (11) in Eq (9) yields the equa-tion of motion for A(1)

(0)A(1)micro = minuspartν(g(1)micromicroprimeννprimeF

(0)microprimeνprime

) (16)

where Lorenz gauge condition is assumed on A(1) tooThe propagation of a photon in a smooth space-timegeometry holds |partλg(1)microν | ω|g(1)microν | Therefore thederivative of the components of the metric on the righthand side of Eq (16) can be neglected and thus we

have (0)A(1)micro = minusg(1)micromicroprimeννprimepartνF(0)microprimeνprime We express F


in terms of A(0)microprime

(0)A(1)micro = minusg(1)micromicroprimeννprime(partνpartmicroprimeAνprime minus partνpartνprimeAmicroprime) (17)

The paraxial approximation expressed in Eq (13) im-plies that the dominant term on the right hand side ofEq (17) is the one that part2

minus acts on A(0) Keeping onlythe dominant term results

(0)A(1)micro =(g(1)minusminusmicroα minus g(1)minusαmicrominus

)part2minusA

(0)α (18)

Due to gauge symmetry and the chosen Lorentz gaugewe choose to solve Eq (18) for micro = minus 1 2 SubstitutingEq (15) into Eq (18) results in

(0)A(1)+ = 0 (19a)

(0)A(1)a = minusg(1)minusminuspart2

minusA(0)a (19b)

g(1)minusminus can be expressed in terms of the components ofthe Riemann tensor and thus we have

(0)A(1)i = minusR+a+b x

axbpart2minusA

(0)i (20)

We would like to consider the Fourier transformationof A

(1)i with respect to the variable xminus ie A

(1)i =int

dωf(1)i (ω x+ xa)eiωx

minus where f

(1)i is the correction to

the structure function of mode i Utilizing the Fouriertransformations in Eq (20) yields

(2iωpart+ +nabla2perp)f

(1)i =(

minusR+minus+minuspart

2ω + 2iR

+minus+minusaxapartω + ω2R

+a+bxaxb

)ω2f

(0)i

(21)

We observe that different physical modes ie differ-ent i are not coupled at the sub-leading order There-fore without loosing generality we consider one physicalmode and we set i = 1 However all the results thatwe will calculate will be equally valid for i = 2 We

consider f(0)1 in the form of f

(0)1 = A(ω)fmn(ω x+ xa)

where fmn is the Hermite-Gaussian mode and A(ω) isthe amplitude that we choose as a normal distributionaround ω = ω0 with the width of σ for the first po-

larization of photon A(ω) = 1radicσπ

14

exp(minus (ωminusω0)2

2σ2

) We

consider a time-bin wavepacket with a narrow bandwidthsuch that the first term on the right hand side of Eq (21)is the dominant term Since σ is very small we can utilise

f(0)1 asymp A(ω)fmn(ω0 x

+ xa) This approximation allows

us to neglect partωfmn in derivatives of f(0)1 with respect

to ω Size of the wave packet is identified by two pa-rameters Its size perpendicular to its trajectory is givenby the width of the beam while its size in direction ofpropagation is proportional to c

σ We assume cσ is much

larger than the width of package and the wavepacket isextended in the direction of propagation where the firstterm on the right hand side of Eq (21) becomes thedominant term Keeping only the dominant term yields


(1)1 =

minusR+minus+minus

(ω2(ω minus ω0)2

σ4+ω(4ω0 minus 5ω)

σ2

)f

(0)1 (22)

where 2f(0)1 on the right hand side of Eq (22) is also

neglected Equation (22) can be perceived as the pertur-bation of (

2iωpart+ +nabla2perp)

Ψ = 2εωV (x+)Ψ (23)

where

Ψ = f(0)1 + εf

(1)1 +O(ε2) (24a)

V (x+) = minus(ω(ω minus ω0)2

2σ4+

4ω0 minus 5ω

2σ2

)R+minus+minus(x+)

(24b)

Equation (23) can be rewritten as

ipart+Ψ =

(minus 1

2ωnabla2 + εV (x+)

)Ψ (25)

4

which is the Schrodinger equation for a particle with amass ldquoωrdquo in (2+1) dimensions with a time-dependent po-tential where the potential is only a function of time Forany Ψ(0) that satisfies ipart+Ψ(0) = minus 1

2ωnabla2Ψ(0) the per-

turbative solution to Eq (25) is Ψ = Ψ(0) (1 + εχ(x+))

where χ = minusiint x+

0dτ V (τ) This implies that the correc-

tion to the structure function f(1)1 is expressed in term

of the structure function f(0)1 ie

f(1)1 = if (0)

(ω(ω minus ω0)2

2σ4+

4ω0 minus 5ω

2σ2

)G (26)

where G the geometrical factor is given by

G =

int x+

0

dτR+minus+minus(τ) (27)

Here τ is the affine parameter on the geodesic andx+ = 0 is the wave-packet initial plane Recalling that

f(1)1 and f

(0)0 are the Fourier transformation of A

(1)1 and

A(1)0 allows us to integrate over ω and obtain the elec-

tromagnetic field

A1 =radic

2σπ14 eminus

(σxminus)2

2 +iω0xminusfmn(ω0 x

+ xa)

times(

1minus iω0G2

(xminus)2

) (28)

where only the dominant term is kept and we set ε =1 (note that ε is a dummy parameter to systematicallytrack the perturbation) We could have chosen i = 2 toobtain the same expression for the second polarizationie A2 We therefore observe that as a Gaussian time-bin wave-packet with sharp width of σ around frequencyof ω0 travels over the geodesic and it gains an extrageometric phase that is given by

χg = minusω0G2

(xminus)2 (29)

where G is the integration of ldquo+minus+minusrdquo component of theRiemann tensor evaluated on the geodesic as defined inEq (27) Equation (29) is in accord with [20] whereinthe equations are solved by a different method Let itbe emphasized that χg is the change in the phase of aGaussian beam with the width of σ There exist somedifficulties associated with measuring χg at the far tail(|σxminus| ge 5) of the Gaussian beam because the amplitudedecreases exponentially and it would not be easy to gen-erate a Gaussian beam whose far tail remains Gaussiantoo To avoid these problems we suggest to measure χgaround the peak of the Gaussian beam or equivalentlyfor |σxminus| 1 In doing so it is convenient to re-expressχg to

χg = minusω0G2σ2

(σxminus)2 (30)

and note that χg is measured for |σxminus| 1 Equa-tion (30) explicitly shows that the maximum measurable

Intial amplitude Intial phase

- σ2

ω0χg

-15 -10 -05 05 10 15σ x-

02

04

06

08

10

FIG 1 The initial amplitude (shown in blue) the initialphase which chosen to be zero (shown in red) and the geo-

metric phase multiplied by minus σ2

ω0G(shown in purple) in term

of σxminus

value of χg depends on σ Figure 1 depicts the ampli-tude the initial phase and the change in the phase interm of σxminus for minus15 le σxminus le 15

III GEOMETRIC PHASE FORCOMMUNICATION BETWEEN SATELLITES

Let us consider a communication link where Alice(sender) and Bob (receiver) are on different satellites lo-cated at radii of r = a and b from the centre of the Earthrespectively where a le b In this section we assume thatAlice and Bob are stationary with respect to the standardspherical coordinates of the Schwarzschild geometry Innext section we utilise relativistic Doppler shift to gen-eralise the result to the case that Alice and Bob are notstationary We use α to represent the angular separationof the two satellites considering the lines from satellitesto the centre of Earth α is the angle between these twolinesFirst case as shown in Fig 2a for r ge 0

The ldquo+minus+minusrdquo component of the Riemann tensor eval-uated on the geodesic is given in Eq (7a) which for large

r can be approximated to R+minus+minus = 3l2

2r5 minus1r3 Applying

the same approximation on r holds r =radic

1minus l2

r2 The

geometrical factor defined in Eq (27) then is given by

G =

int b

a

dr

rR+minus+minus =

radicb2 minus l22b3

minusradica2 minus l22a3

(31)

Here l is the minimum of the distance between the centreof the Earth and the line or the extrapolation of the lineconnecting Alice and Bob ie l = ab sinαradic

a2+b2minus2ab cosα The

5

(a) During the entire journey of the signal r gt 0 (b) At the beginning r lt 0 then r gt 0

FIG 2 Alice at radius a at one instant sends a signal toward Bob at radius b with a lt b The signal propagatesalong a null geodesic that can be approximated to a straight line Notice that l is the minimum distance between the centreof the Earth and the line or the extrapolation of the line connecting Alice and Bob α is the angle between lines connectingAlice and Bob to the centre of the Earth Alice and Bobrsquos trajectories are not fixed There exist two scenarios depicted aboveIn the first scenario r ge 0 while in the second case r can be negative

geometrical phase Eq (30) is then given by

χg = +ω0cmoplus(b2 minus a2)

4(σab)2

radic(aminus b cosα)2

a2 + b2 minus 2ab cosα

(σxminus

c

)2

(32)where the Schwarzschild radius of the Earthmoplus = 2GMoplus

c2 = 887 millimeters and c is recov-ered Equation (32) for α = 0 coincides to the resultreported in [20] for radial communication between theEarth and the International Space Station

Second case as shown in Fig 2b For α ge arccos abas depicted in fig 2b r can be negative in some parts ofthe geodesic For the geometrical factor defined in (27)therefore we can write

G =

intdτR+minus+minus = minus

int l

a

dr

|r|R+minus+minus +

int b

l

dr

|r|R+minus+minus

= 2

int a

l

dr

|r|R+minus+minus +

int b

a

dr

|r|R+minus+minus (33)

that leads to

χg = minusω0cmoplus(b2 + a2)

4(σab)2


a2 + b2 minus 2ab cosα(σxminus

c)2

(34)

IV ON MEASURING THE GEOMETRICPHASE NEAR EARTH

In the following we would like to evaluate the geomet-rical phase for a set of parameters to see if the geometri-cal phase can be detected in communication between twosatellites around the Earth In so doing we first wouldlike to generalise the result of the previous section to thecase that Alice and Bob are not stationary

Alice at position of ra prepares a time-bin Gaussianpulse with the mean frequency of ωA and line-width ofσA The pulse that Alice produces in Alicersquos rest frameis given by

AAlice = A0Alicee

minus (σAxminus)2

2 +iωAxminus (35)

Alice moves with velocity of ~vA with respect to thelocal Riemann coordinates at ra which is stationarywith respect to the standard spherical coordinates in theSchwarzschild geometry In the local Riemann coordi-nates at ra since the source of the pulse moves withvelocity of ~v the pulse at the event of its generation isgiven by

Aa = A0aeminus (σax

minus)2

2 +iωaxminus (36)

where ωa = ∆~vaωA σa = ∆~vaσA and ∆~va stands forthe relativistic Doppler shift We should still transformthis pulse to the Fermi coordinates Noticing the factorof f and 1f in the right-hand side of Eq (5a) and (5b)

6

the pulse in the Fermi coordinates at the event of itsgeneration can be derived from the pulse in the localRiemann coordinates by scaling the frequencies

A1 = A0eminus(σxminus)2

2 +iω0xminus (37)

where

ω0 = ∆~vaf(ra)ωA σ = ∆~vaf(ra)σA (38)

Here f =radic

1minus 1r see the text after Eq (5c) As thepulse moves toward Bobrsquos geodesic it gains an extra ge-ometric phase At the time of its detection the pulse inthe Fermi coordinates is given by


2 +iω0xminus

exp(minusiχg) (39)

where χg is given in Eq (30) Bob is moving with velocity~vB with respect to the local Riemann coordinates at ~rbThe pulse that Bob observes can be obtained by trans-forming the beam from the Fermi coordinates to the localRiemann coordinates and then to the Bobrsquos rest frameBob in his rest frame observes

ABob = A0eminus(σBx

minus)2

2 +iωBxminus

exp(minusiχg) (40)

where

ωB =f(ra)

f(rb)∆~va∆~vbωA σB =

f(ra)

f(rb)∆~va∆~vbσA (41)

Here ∆(~va)∆(~vb) accounts for relativistic Doppler shiftwhile f(ra)f(rb) describes the gravitational red-shiftEquations (38) and (41) can be utilised to re-express thegeometric phase by

χg = minusωBf(rb)G2∆~vbσ

2B

(σBxminus)2 (42)

where (38) and (41) are used to express ω0 in Eq (30)in term of ωB We notice that Bob can interpret χgas a time-dependent phase modulated over the Gaus-sian time-bin wave packet with the mean frequency ωBand line-width of σB [24] So Bob can measure it Forterrestrial satellites with velocities less than 104 mph|∆~v minus 1| le 10minus5 For satellites around the Earth|f(rb)minus 1| le 10minus9 So in measuring the geometric phasewith a precision larger than 0001 percent the Dopplerand gravitational effects can be neglected and the ge-ometric phase that Bob observes can be approximatedto

χg = minus ωG2σ2

(σxminus)2 (43)

where ω and σ respectively respectively represent themean frequency and the line-width

It is worth noting that in our notation a plane-wave in

x+-direction is expressed as eiωxminus

= ei ωradic

2(x3minusct)

In op-tics however a plane-wave in x3-direction is represented

by eiω(x3minusct) Thus what we define as a frequency isradic2 times the notation in optics The geometrical phase

presented in Eq (32) in the standard optic notation isχsg1 = f1(α) y2 where

y =σ(x3 minus ct)

c (44a)

f1(α) =

radic2ν0cmoplus(b2 minus a2)

16π(σab)2


a2 + b2minus 2ab cosα(44b)

where ω0 is replaced withradic

2νs02π instead of ν02π and

σ is changed toradic

2σ The geometrical factor presentedin Eq (34) in the standard optical notation reads χsg2 =

f2(α) y2 where

f2(α) = minusradic

2ν0cmoplus(b2 + a2)

16π(σab)2



(45)and y is defined in Eq (44a) In order to understandhow the geometric factor depends on the Newton Gravi-tational constant Heisenberg constant and speed of lightin vacuum we treat the photon as a particle entity withan energy of E0 = ~ν0 and variance of ∆E0 = ~σ Thegeometric phase then can be expressed by

χg = l2pf(a b α)times (Moplusc2)E0

(∆E0)2 (46)

where l2p = GN~c3 is the Planckrsquos length while f(a b α)

encodes the geodesicrsquos details and has the unit dimen-

sion of the inverse length squared and (Moplusc2)E0

(∆E0)2 encodes

properties of the pulse Near the Earth l2pf can be esti-

mated to be at the order of 10minus84 The factor of (Moplusc2)E0

(∆E0)2

can become arbitrary large in the limit of ∆E0 rarr 0 butthis divergence points to the break of the perturbativemethods in calculating the geometric phase and demandsnon-perturbative derivation of the geometric phase Theultra-stable lasers with bandwidth of 5 mHz at 194 THzreported in [25ndash27] gives rise to (Moplusc

2)E0

(∆E0)2 at the order

of 1094 We however notice that σ should satisfy someother conditions Here δχ is calculated by perturbativemethods so a value should be chosen for σ that resultsin |χg| le 1 The length of the wave packet in the direc-tion of the propagation is given by c

σ and the employedmethod has assumed that the Riemann tensor is con-stant within the wave packet We also have implicitlyassumed that the whole of the wave packet propagates inthe space before its detection The choice of bandwidthof 5 mHz at 194 THz violates these conditions Howeverbandwidth at a few kHz satisfies these conditions and (asshown below) leads to a measurable value of χg

In order to consistently neglect the effect of atmosphereon the geometrical factor let us consider communicationbetween satellites in space and choose a = 7 000 km andb = 7 500 km respectively We notice that commercialportable continuous lasers with a line-width of 1 Hz at

7

-05 05α

-025

-020

-015

-010

-005

005

χ(α)

y2

FIG 3 The geometrical phase is a time-dependent phase

quadratic in y = σ(x3minusct)c

modulated on a time-bin Gaussianwave-packet The coefficient of the quadratic term dependson the distance between two satellites and their apparent an-gle as seen from the centre of the Earth This figure de-picts the dependency of the coefficient of the quadratic termin the geometrical factor for satellites at a = 7 000 km andb = 7 500 km and for Gaussian time-bin communication per-formed with ν0 = 287times1015 Hz σ = 316 kHz The verticalaxis is the coefficient of the quadratic term in the geometricfactor

χgy2

the horizontal axis is the apparent angle

the wavelength of 657 nm exists [28] These can be usedto construct a sharp Gaussian time-bin with line-widthof σ = 316 kHz for ν0 = 287 times 1015 Hz Using thesenumerical values simplify f1 and f2 to

f1(α) =0055times |75 cosαminus 7|radic

10525minus 105 cosα (47a)

f2(α) = minus0802times |75 cosαminus 7|radic10525minus 105 cosα

(47b)

We do not want the signal to enter the atmosphere belowthe altitude of 400 km where the air density is about10minus12 kgm3 Thus the gravitational effects are a coupleof orders larger than the atmospherersquos diffraction Thisconstrains α to |α| le 0675 where the saturation occurswhen the line connecting the satellites is tangent to theorbit with a radius of 6800 km Figure 3 depicts the valueof the geometrical phase divided by y2 for the allowedrange of α The dependency on α is very non-trivial Forthe chosen parameters the value of χ[α]y2 ranges from005 to minus0187 This means that the geometrical phaseat y = 1 depending on the value of α varies in therange of 005 to minus0187 Radians which can be measuredOne may choose smaller values of σ to obtain a largergeometrical phase It however should be noted that thegeometrical phase is calculated by perturbative methodsSo the choices leading to a large geometric factor can notbe supported by perturbative methods developed in thiswork The choice of σ = 316 kHz for ν0 = 287times1015 Hzand a = 7 000 km and b = 7 500 km as depicted inFigure 3 leads to a measurable geometric phase consistent

with a perturbative calculation

V CONCLUSIONS

As the photonrsquos wave-function travels along a nullgeodesic it interacts with the Riemann tensor aroundthe geodesic The interaction distorts the photonrsquos wave-function Here the Fermi coordinates along the nullgeodesic have been utilized The equations for the U(1)gauge field theory in the Fermi coordinates have beencalculated The equation for the interaction between theRiemann tensor and the photonrsquos wave-function has beenderived and mapped to a time-dependent Schrodingerequation in (2 + 1) dimensions It has been shown thatas a Gaussian time-bin wave-packet with a sharp widthof σ around the frequency of ω0 travels over the nullgeodesic it gains an extra geometric phase given byχg = minusω0G

2σ2 (σxminus)2 where G is ldquo+minus+minusrdquo componentof the Riemann tensor in the Fermi coordinates eval-uated on and integrated over the null geodesic G =intR+minus+minus(x+)dx+ where x+ represents the coordinate

of Fermi frame tangent to the central null geodesic andthe integration is performed from the event of generationof the pulse to the event of its detection

The space-time geometry outside the Earth has beenapproximated by the Schwarzschild space-time geome-try The geometrical phase has been calculated for asignal sent between two satellites located at radii ofa and b respectively The current commercial ultra-stable continuous-wave lasers (wavelength of 657 nm andσ = 316 kHz) have been utilized to calculate the geo-metrical phase between satellites at radii 7 000 km and7 500 km It has been shown that for the chosen range ofthe parameters the geometrical phase within the peak ofthe Gaussian pulse varies from 005 tominus0187 Radians asdepicted in Fig 3 This illustrates that the predicted ge-ometrical phase can be measured by the currently avail-able commercial devices The geometrical phase calcu-lated in the current work is consequent of applying quan-tum field theory in curved space-time geometry Thethree paradigms of special relativity general relativityand quantum mechanics are equally important in thisderivation It therefore is a prediction of how Einsteinrsquosgravity ldquotalksrdquo to the quantum realm Hence measuringthis phase will provide the first experimental datum onif and how gravity affects the quantum realm

Acknowledgments

This work was supported by the High Throughputand Secure Networks Challenge Program at the Na-tional Research Council of Canada the Canada ResearchChairs (CRC) and Canada First Research ExcellenceFund (CFREF) Program and Joint Centre for ExtremePhotonics (JCEP) We thank Felix Hufnagel for proof-reading the paper

8

[1] Daniel Kennefick Testing relativity from the 1919eclipsemdasha question of bias Physics Today 6237 2009

[2] D E Lebach B E Corey I I Shapiro M I RatnerJ C Webber A E E Rogers J L Davis and T A Her-ring Measurement of the Solar Gravitational Deflectionof Radio Waves Using Very-Long-Baseline Interferome-try Phys Rev Lett 751439ndash1442 1995

[3] S S Shapiro J L Davis D E Lebach and J S Gre-gory Measurement of the Solar Gravitational Deflectionof Radio Waves using Geodetic Very-Long-Baseline Inter-ferometry Data 1979-1999 Phys Rev Lett 921211012004

[4] E Fomalont S Kopeikin G Lanyi and J BensonProgress in measurements of the gravitational bendingof radio waves using the vlba The Astrophysical Jour-nal 699(2)1395ndash1402 Jun 2009

[5] S B Lambert and C Le Poncin-Lafitte Determining therelativistic parameter γ using very long baseline interfer-ometry Astronomy amp Astrophysics 499(1)331ndash335 Apr2009

[6] CM Will The Confrontation between General Relativ-ity and Experiment Living Reviews in Relativity 1742011

[7] R Ursin F Tiefenbacher T Schmitt-ManderbachH Weier T Scheidl M Lindenthal B BlauensteinerT Jennewein J Perdigues P Trojek B OmerM Furst M Meyenburg J Rarity Z Sodnik C Barbi-eri H Weinfurter and A Zeilinger Entanglement-basedquantum communication over 144 km Nature Physics3(7)481ndash486 Jul 2007

[8] Juan Yin Yuan Cao Yu-Huai Li Sheng-Kai Liao LiangZhang Ji-Gang Ren Wen-Qi Cai Wei-Yue Liu Bo LiHui Dai Guang-Bing Li Qi-Ming Lu Yun-Hong GongYu Xu Shuang-Lin Li Feng-Zhi Li Ya-Yun Yin Zi-QingJiang Ming Li Jian-Jun Jia Ge Ren Dong He Yi-LinZhou Xiao-Xiang Zhang Na Wang Xiang Chang Zhen-Cai Zhu Nai-Le Liu Yu-Ao Chen Chao-Yang Lu RongShu Cheng-Zhi Peng Jian-Yu Wang and Jian-Wei PanSatellite-based entanglement distribution over 1200 kilo-meters Science 356(6343)1140ndash1144 2017

[9] Juan Yin Yuan Cao Yu-Huai Li Ji-Gang Ren Sheng-Kai Liao Liang Zhang Wen-Qi Cai Wei-Yue Liu Bo LiHui Dai Ming Li Yong-Mei Huang Lei Deng Li LiQiang Zhang Nai-Le Liu Yu-Ao Chen Chao-Yang LuRong Shu Cheng-Zhi Peng Jian-Yu Wang and Jian-WeiPan Satellite-to-ground entanglement-based quantumkey distribution Physical Review Letters 119200501Nov 2017

[10] Soren Wengerowsky Siddarth Koduru Joshi FabianSteinlechner Julien R Zichi Bo Liu Thomas ScheidlSergiy M Dobrovolskiy Rene van der Molen JohannesW N Los Val Zwiller Marijn A M Versteegh Al-berto Mura Davide Calonico Massimo Inguscio AntonZeilinger Andre Xuereb and Rupert Ursin Passivelystable distribution of polarisation entanglement over 192km of deployed optical fibre npj Quantum Information65 January 2020

[11] Felix Hufnagel Alicia Sit Frederic Bouchard Ying-wen Zhang Duncan England Khabat Heshami Ben-jamin J Sussman and Ebrahim Karimi Investigationof underwater quantum channels in a 30 meter flume

tank using structured photons New Journal of Physics22(9)093074 September 2020

[12] Juan Yin Yu-Huai Li Sheng-Kai Liao Meng Yang YuanCao Liang Zhang Ji-Gang Ren Wen-Qi Cai Wei-YueLiu Shuang-Lin Li Rong Shu Yong-Mei Huang LeiDeng Li Li Qiang Zhang Nai-Le Liu Yu-Ao ChenChao-Yang Lu Xiang-Bin Wang Feihu Xu Jian-YuWang Cheng-Zhi Peng Artur K Ekert and Jian-WeiPan Entanglement-based secure quantum cryptographyover 1120 kilometres Nature 582(7813)501ndash505 2020

[13] Daniele Dequal Luis Trigo Vidarte Victor Roman Ro-driguez Giuseppe Vallone Paolo Villoresi Anthony Lev-errier and Eleni Diamanti Feasibility of satellite-to-ground continuous-variable quantum key distributionnpj Quantum Information 7(1) Jan 2021

[14] Juan Yin Yuan Cao Yu-Huai Li Sheng-Kai Liao LiangZhang Ji-Gang Ren Wen-Qi Cai Wei-Yue Liu Bo LiHui Dai Guang-Bing Li Qi-Ming Lu Yun-Hong GongYu Xu Shuang-Lin Li Feng-Zhi Li Ya-Yun Yin Zi-QingJiang Ming Li Jian-Jun Jia Ge Ren Dong He Yi-LinZhou Xiao-Xiang Zhang Na Wang Xiang Chang Zhen-Cai Zhu Nai-Le Liu Yu-Ao Chen Chao-Yang Lu RongShu Cheng-Zhi Peng Jian-Yu Wang and Jian-Wei PanSatellite-based entanglement distribution over 1200 kilo-meters 2017

[15] David Edward Bruschi Tim Ralph Ivette FuentesThomas Jennewein and Mohsen Razavi Spacetime ef-fects on satellite-based quantum communications PhysRev D 90(4)045041 2014

[16] David Edward Bruschi Animesh Datta Rupert UrsinTimothy C Ralph and Ivette Fuentes Quantum esti-mation of the Schwarzschild spacetime parameters of theEarth Phys Rev D 90(12)124001 2014

[17] David Edward Bruschi Symeon Chatzinotas Frank KWilhelm and Andreas Wolfgang Schell Spacetime ef-fects on wavepackets of coherent light Phys Rev D104(8)085015 2021

[18] David Edward Bruschi and Andreas W Schell Gravita-tional redshift induces quantum interference 9 2021

[19] Robert H Jonsson David Q Aruquipa Marc CasalsAchim Kempf and Eduardo Martın-Martınez Commu-nication through quantum fields near a black hole PhysRev D 101(12)125005 2020

[20] Qasem Exirifard Eric Culf and Ebrahim Karimi To-wards Communication in a Curved Spacetime GeometryCommun Phys 4171 2021

[21] P A M Dirac Forms of relativistic dynamics Reviewof Modern Physics 21392ndash399 Jul 1949

[22] Matthias Blau Denis Frank and Sebastian Weiss Fermicoordinates and penrose limits Classical and QuantumGravity 23(11)3993ndash4010 may 2006

[23] FK Manasse and CW Misner Fermi Normal Coordi-nates and Some Basic Concepts in Differential GeometryJournal of Mathematical Physics 4735ndash745 1963

[24] H Hansen T Aichele C Hettich P Lodahl A ILvovsky J Mlynek and S Schiller Ultrasensitivepulsed balanced homodyne detectorapplication to time-domain quantum measurements Opt Lett 26(21)1714ndash1716 Nov 2001

[25] W Zhang et al Ultrastable Silicon Cavity in a Contin-uously Operating Closed-Cycle Cryostat at 4 K Phys

9

Rev Lett 119(24)243601 2017[26] D G Matei et al 15 microm Lasers with Sub-10 mHz

Linewidth Phys Rev Lett 118(26)263202 2017[27] T Kessler C Hagemann C Grebing T Legero

U Sterr F Riehle M J Martin L Chen and J Ye

A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity Nature Photonics 6(10)687ndash692October 2012

[28] httpsmenlosystemscomproductsultrastable-lasers

I Introduction

II Predicting a general geometric phase

III Geometric phase for communication between satellites

IV On measuring the geometric phase near Earth

V Conclusions

Acknowledgments

References

2

evaluated on γ and Einsteinrsquos notation is used whereintwice appearance of an index variable in a single termmeans summation over that index and δab is the Kro-necker delta We approximate the space-time geometryaround the Earth to the Schwarzschild space-time geom-etry Therefore the effective Lagrangian of a masslesspoint particle propagating on a null geodesic can be givenby

L = minus(

1minus m

r

)t2 +

r2

1minus mr

+ r2(θ2 + sin2θ ϕ2

) (4)

where dot presents variation with respect to the affine pa-rameter on the geodesic m = (2GNMoplus)c2 and Moplus isthe mass of the Earth We choose the units such that thespeed of light in vacuum is set to 1 ie c = 1 and m = 1Due to the spherical symmetry without loss of generalitywe can choose the equatorial plane θ = π2 and θ = 0to describe any given geodesic at all time The cyclicvariables of ϕ and t lead to invariant quantities r2ϕ = l(1 minus 1r)t = E where l and E are constant values Weconsider null geodesics reaching the asymptotic infinityand set E = 1 Due to the form of the Lagrangianits Legendre transformation which is the Lagrangian it-self is invariant We consider a null geodesic and set

L = 0 that gives |r| =radic

1minus 1r2

(1minus 1

r

)l2 Let et er eϕ

and eθ represent the normalized unit vectors in t r θ ϕcoordinates The normalized unit vectors in the Fermicoordinates can be chosen as

e+ =fradic2

(+et + rer + rfϕeϕ) (5a)

eminus =1radic2f

(minuset + rer + rfϕeϕ) (5b)

e1 = minusrfϕer + reϕ e2 = eθ (5c)

where f =radic

1minus 1r The components of the Riemanntensor in the Fermi coordinates should be computed bythe tensor transformation law [23]

Rαβγδ = Rmicroprimeνprimeσprimeτ prime(eα)microprime(eβ)ν

prime(eγ)σ

prime(eδ)

τ prime (6)

Utilizing the abstract method employed in [20] identifiesthe non-vanishing components of the Riemann tensor inthe Fermi frame

R+minus+minus =3l2(r minus 1)

2r6minus 1

r3 (7a)

R+2+2 = minusR+1+1 =3l2

4r5 (7b)

R+minus+1 =3radic

2l

2r4r (7c)

Next we consider the electromagnetic potential Amicrowhose field strength is given by Fmicroν = partmicroAνminuspartνAmicro Thedynamics of the electromagnetic potential in a curvedspace-time geometry endowed with metric gmicroν around

the geodesic and is given by

Γ[Amicro] = minus1

4

intd4x gmicromicro

primeννprimeFmicroνFmicroprimeνprime (8a)

gmicromicroprimeννprime =

1

2

radicminusdet g (gmicromicro

primegνν

primeminus gmicroν

primegνmicro

prime) (8b)

Here gmicroν is the inverse of the metric and det g is itsdeterminant Note that gmicromicro

primeννprime has all the symmetries ofFmicroνFmicroprimeνprime under exchange of its indices The functionalvariation of the action with respect to the gauge fieldgives its equation of motion ie

partmicro

(gmicromicro

primeννprimeFmicroprimeνprime)

= 0 (9)

which we would like to solve for a photon that travelsalong a null geodesic We choose the Fermi coordinatesadapted to the null geodesic Eq (3) to describe thespace-time at the vicinity of the geodesic where the com-ponents of the Riemann tensor are given in Eq (7) Atthe vicinity of the Earth the components of the Riemanntensor are minimal We therefore treat them as a per-turbation We introduce ε as the systematic parameterof the perturbation In other words we add a factor of εto all terms in Eq (3) where the components of the Rie-mann tensor are present At the end of the computationwe set ε = 1 The ε-perturbation to the electromagneticpotential and gmicromicro

primeννprime follow

Amicro = A(0)micro + εA(1)

micro +O(ε2) (10a)

gmicromicroprimeννprime = g(0)micromicroprimeννprime + εg(1)micromicroprimeννprime +O(ε2) (10b)

Equation (9) at the leading order in ε can be simplifiedto

(0)A(0)micro + partmicropartνA(0)ν = 0 (11)

Henceforth ηmicroν is utilized to move up or down the indices

ie A(0)ν = ηνλA(0)λ partmicro = ηmicroνpartν where ηmicroν represents

the Minkowski metric in Dirac coordinates We choosethe Lorentz gauge partνA

(0)ν = 0 which simplifies the

equation for A(0) to (0)A(0)micro =

(2part+partminus +nabla2

perp)A

(0)micro = 0

where nabla2perp is the Laplace operator in x1 and x2 direc-

tions nabla2perp = part2

1 + part22 Utilizing the Fourier expan-

sion of the gauge field in terms of the variable xminus ie

A(0)micro =

intdωf

(0)micro (ω x+ xa)eiωx

minusleads to

(2iωpart+ +nabla2perp)f (0)

micro = 0 (12)

which is referred to as the paraxial Helmholtz equationNote that due to the definition of eminus in Eq (5b) the grav-itational red-shift is already encoded in the Fourier ex-

pansion of A(0)micro We refer to f

(0)micro (w x+ xa) as the struc-

ture function of the photon with frequency ω in modemicro The structure mode can be expanded in terms of theHermite-Gaussian or Laguerre-Gaussian modes For thepurpose of communication we are interested in a field

3


|part+f(0)ν | ω|f (0)


ν | (13)


ωf(0)+ + part+f

(0)minus + part1f

(0)1 + part2f

(0)2 = 0 (14)


f(0)+ = 0 (15a)


a = 0 (15b)


(0)1 + part2f










) (16)









)part2minusA

(0)α (18)


(0)A(1)+ = 0 (19a)


minusA(0)a (19b)



axbpart2minusA

(0)i (20)



(1)i =int


minus where f




(1)i =(


2ω + 2iR


+a+bxaxb

)ω2f

(0)i

(21)



(0)1 = A(ω)fmn(ω x+ xa)



14


2σ2

) We









(1)1 =

minusR+minus+minus

(ω2(ω minus ω0)2


σ2

)f

(0)1 (22)




Ψ = 2εωV (x+)Ψ (23)

where

Ψ = f(0)1 + εf

(1)1 +O(ε2) (24a)


2σ4+

4ω0 minus 5ω

2σ2

)R+minus+minus(x+)

(24b)


ipart+Ψ =

(minus 1


)Ψ (25)

4








f(1)1 = if (0)

(ω(ω minus ω0)2

2σ4+

4ω0 minus 5ω

2σ2

)G (26)


G =

int x+

0



f(1)1 and f


(1)1 and


tromagnetic field

A1 =radic

2σπ14 eminus

(σxminus)2


+ xa)

times(

1minus iω0G2

(xminus)2

) (28)


χg = minusω0G2

(xminus)2 (29)


χg = minusω0G2σ2

(σxminus)2 (30)



- σ2

ω0χg

-15 -10 -05 05 10 15σ x-

02

04

06

08

10




of σxminus








1minus l2

r2 The


G =

int b

a

dr

rR+minus+minus =

radicb2 minus l22b3


(31)



5





4(σab)2



(σxminus

c

)2




G =


int l

a

dr

|r|R+minus+minus +

int b

l

dr

|r|R+minus+minus

= 2

int a

l

dr

|r|R+minus+minus +

int b

a

dr


that leads to


4(σab)2



c)2

(34)




AAlice = A0Alicee

minus (σAxminus)2

2 +iωAxminus (35)



minus)2

2 +iωaxminus (36)


6



2 +iω0xminus (37)

where


Here f =radic



2 +iω0xminus

exp(minusiχg) (39)



minus)2

2 +iωBxminus

exp(minusiχg) (40)

where

ωB =f(ra)


f(ra)




2B

(σBxminus)2 (42)


χg = minus ωG2σ2

(σxminus)2 (43)




= ei ωradic

2(x3minusct)




y =σ(x3 minus ct)

c (44a)

f1(α) =


16π(σab)2







f2(α) y2 where

f2(α) = minusradic


16π(σab)2





(∆E0)2 (46)




(∆E0)2 encodes



(∆E0)2


2)E0





7

-05 05α

-025

-020

-015

-010

-005

005

χ(α)

y2




χgy2






(47b)



V CONCLUSIONS





Acknowledgments


8



























9






I Introduction




V Conclusions

Acknowledgments

References

3


|part+f(0)ν | ω|f (0)


ν | (13)


ωf(0)+ + part+f

(0)minus + part1f

(0)1 + part2f

(0)2 = 0 (14)


f(0)+ = 0 (15a)


a = 0 (15b)


(0)1 + part2f










) (16)









)part2minusA

(0)α (18)


(0)A(1)+ = 0 (19a)


minusA(0)a (19b)



axbpart2minusA

(0)i (20)



(1)i =int


minus where f




(1)i =(


2ω + 2iR


+a+bxaxb

)ω2f

(0)i

(21)



(0)1 = A(ω)fmn(ω x+ xa)



14


2σ2

) We









(1)1 =

minusR+minus+minus

(ω2(ω minus ω0)2


σ2

)f

(0)1 (22)




Ψ = 2εωV (x+)Ψ (23)

where

Ψ = f(0)1 + εf

(1)1 +O(ε2) (24a)


2σ4+

4ω0 minus 5ω

2σ2

)R+minus+minus(x+)

(24b)


ipart+Ψ =

(minus 1


)Ψ (25)

4








f(1)1 = if (0)

(ω(ω minus ω0)2

2σ4+

4ω0 minus 5ω

2σ2

)G (26)


G =

int x+

0



f(1)1 and f


(1)1 and


tromagnetic field

A1 =radic

2σπ14 eminus

(σxminus)2


+ xa)

times(

1minus iω0G2

(xminus)2

) (28)


χg = minusω0G2

(xminus)2 (29)


χg = minusω0G2σ2

(σxminus)2 (30)



- σ2

ω0χg

-15 -10 -05 05 10 15σ x-

02

04

06

08

10




of σxminus








1minus l2

r2 The


G =

int b

a

dr

rR+minus+minus =

radicb2 minus l22b3


(31)



5





4(σab)2



(σxminus

c

)2




G =


int l

a

dr

|r|R+minus+minus +

int b

l

dr

|r|R+minus+minus

= 2

int a

l

dr

|r|R+minus+minus +

int b

a

dr


that leads to


4(σab)2



c)2

(34)




AAlice = A0Alicee

minus (σAxminus)2

2 +iωAxminus (35)



minus)2

2 +iωaxminus (36)


6



2 +iω0xminus (37)

where


Here f =radic



2 +iω0xminus

exp(minusiχg) (39)



minus)2

2 +iωBxminus

exp(minusiχg) (40)

where

ωB =f(ra)


f(ra)




2B

(σBxminus)2 (42)


χg = minus ωG2σ2

(σxminus)2 (43)




= ei ωradic

2(x3minusct)




y =σ(x3 minus ct)

c (44a)

f1(α) =


16π(σab)2







f2(α) y2 where

f2(α) = minusradic


16π(σab)2





(∆E0)2 (46)




(∆E0)2 encodes



(∆E0)2


2)E0





7

-05 05α

-025

-020

-015

-010

-005

005

χ(α)

y2




χgy2






(47b)



V CONCLUSIONS





Acknowledgments


8



























9






I Introduction




V Conclusions

Acknowledgments

References

4








f(1)1 = if (0)

(ω(ω minus ω0)2

2σ4+

4ω0 minus 5ω

2σ2

)G (26)


G =

int x+

0



f(1)1 and f


(1)1 and


tromagnetic field

A1 =radic

2σπ14 eminus

(σxminus)2


+ xa)

times(

1minus iω0G2

(xminus)2

) (28)


χg = minusω0G2

(xminus)2 (29)


χg = minusω0G2σ2

(σxminus)2 (30)



- σ2

ω0χg

-15 -10 -05 05 10 15σ x-

02

04

06

08

10




of σxminus








1minus l2

r2 The


G =

int b

a

dr

rR+minus+minus =

radicb2 minus l22b3


(31)



5





4(σab)2



(σxminus

c

)2




G =


int l

a

dr

|r|R+minus+minus +

int b

l

dr

|r|R+minus+minus

= 2

int a

l

dr

|r|R+minus+minus +

int b

a

dr


that leads to


4(σab)2



c)2

(34)




AAlice = A0Alicee

minus (σAxminus)2

2 +iωAxminus (35)



minus)2

2 +iωaxminus (36)


6



2 +iω0xminus (37)

where


Here f =radic



2 +iω0xminus

exp(minusiχg) (39)



minus)2

2 +iωBxminus

exp(minusiχg) (40)

where

ωB =f(ra)


f(ra)




2B

(σBxminus)2 (42)


χg = minus ωG2σ2

(σxminus)2 (43)




= ei ωradic

2(x3minusct)




y =σ(x3 minus ct)

c (44a)

f1(α) =


16π(σab)2







f2(α) y2 where

f2(α) = minusradic


16π(σab)2





(∆E0)2 (46)




(∆E0)2 encodes



(∆E0)2


2)E0





7

-05 05α

-025

-020

-015

-010

-005

005

χ(α)

y2




χgy2






(47b)



V CONCLUSIONS





Acknowledgments


8



























9






I Introduction




V Conclusions

Acknowledgments

References

5





4(σab)2



(σxminus

c

)2




G =


int l

a

dr

|r|R+minus+minus +

int b

l

dr

|r|R+minus+minus

= 2

int a

l

dr

|r|R+minus+minus +

int b

a

dr


that leads to


4(σab)2



c)2

(34)




AAlice = A0Alicee

minus (σAxminus)2

2 +iωAxminus (35)



minus)2

2 +iωaxminus (36)


6



2 +iω0xminus (37)

where


Here f =radic



2 +iω0xminus

exp(minusiχg) (39)



minus)2

2 +iωBxminus

exp(minusiχg) (40)

where

ωB =f(ra)


f(ra)




2B

(σBxminus)2 (42)


χg = minus ωG2σ2

(σxminus)2 (43)




= ei ωradic

2(x3minusct)




y =σ(x3 minus ct)

c (44a)

f1(α) =


16π(σab)2







f2(α) y2 where

f2(α) = minusradic


16π(σab)2





(∆E0)2 (46)




(∆E0)2 encodes



(∆E0)2


2)E0





7

-05 05α

-025

-020

-015

-010

-005

005

χ(α)

y2




χgy2






(47b)



V CONCLUSIONS





Acknowledgments


8



























9






I Introduction




V Conclusions

Acknowledgments

References

6



2 +iω0xminus (37)

where


Here f =radic



2 +iω0xminus

exp(minusiχg) (39)



minus)2

2 +iωBxminus

exp(minusiχg) (40)

where

ωB =f(ra)


f(ra)




2B

(σBxminus)2 (42)


χg = minus ωG2σ2

(σxminus)2 (43)




= ei ωradic

2(x3minusct)




y =σ(x3 minus ct)

c (44a)

f1(α) =


16π(σab)2







f2(α) y2 where

f2(α) = minusradic


16π(σab)2





(∆E0)2 (46)




(∆E0)2 encodes



(∆E0)2


2)E0





7

-05 05α

-025

-020

-015

-010

-005

005

χ(α)

y2




χgy2






(47b)



V CONCLUSIONS





Acknowledgments


8



























9






I Introduction




V Conclusions

Acknowledgments

References

7

-05 05α

-025

-020

-015

-010

-005

005

χ(α)

y2




χgy2






(47b)



V CONCLUSIONS





Acknowledgments


8



























9






I Introduction




V Conclusions

Acknowledgments

References

8



























9






I Introduction




V Conclusions

Acknowledgments

References

9






I Introduction




V Conclusions

Acknowledgments

References

arxiv:2110.13990v1 [gr-qc] 26 oct 2021

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