Influence of relativistic effects on satellite-based clock synchronization

0.56 MB·English

by  Jieci Wang

Influence ofrelativisticeffects onsatellite-based clocksynchronization
Jieci Wang1,2 , Zehua Tian1, Jiliang Jing1 and Heng Fan2
∗ †
1 Department of Physics, Collaborative Innovation Center for Quantum Effects,
and Key Laboratory of Low Dimensional Quantum Structures and Quantum
Control of Ministry of Education, Hunan Normal University,
Changsha, Hunan 410081, People’s Republic of China
2 Beijing National Laboratory for Condensed Matter Physics, Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
6
Clocksynchronizationbetweenthegroundandsatellitesisafundamentalissueinfuturequantumtelecom-
1
munication,navigation,andglobalpositioningsystems.Here,weproposeaschemeofnear-Earthorbitsatellite-
0
basedquantumclocksynchronizationwithatmosphericdispersioncancellationbytakingintoaccountthespace-
2
timebackground oftheEarth. Twofrequencyentangledpulsesareemployed tosynchronizetwoclocks, one

r atagroundstationandtheotheratasatellite. Thetimediscrepancy ofthetwoclocksisintroduced intothe
a
pulsesbymovingmirrorsandisextractedbymeasuringthecoincidencerateofthepulsesintheinterferometer.
M
WefindthatthepulsesaredistortedduetoeffectsofgravitywhentheypropagatebetweentheEarthandthe
satellite,resultinginremarkablyaffectedcoincidencerates.Wealsofindthattheprecisionoftheclocksynchro-
3
nizationissensitivetothesourceparametersandthealtitudeofthesatellite. Theschemeprovidesasolution

forsatellite-basedquantumclocksynchronizationwithhighprecision,whichcanberealized,inprinciple,with
]
h currenttechnology.
p
- PACSnumbers:03.67.Hk,04.20.-q,06.30.Ft,42.50.-p
t
n
a
u I. INTRODUCTION by quantum mechanics, and there is also a security advan-
q tage[34]. On the otherhand, time dilation[32] is a concern
[ because of relativistic effects of the Earth on the QCS. The
High precision synchronization of clocks plays an impor-
influenceofrelativisticeffectsonquantumsystems[35–43]is
4 tantroleinmodernsocietyandscientific research[1, 2]; ex-
v amplesincludenavigation,globalpositioning,testsofgeneral afocusofstudyinrecentyearsbecausesuchstudiesprovide
8 insights into some key questions in quantum mechanics and
relativitytheory,longbaselineinterferometryinradioastron-
7 relativity, such as nonlocality, causality, and the information
omy,aswellasgravitationalwaveobservation.Twostandard
4 paradox of black holes. Relativistic effects are particularly
1 classical protocols for clock synchronization are Einstein’s
significantforthequantumversionsoftheEddingtonscheme
0 synchronizationscheme[3]andEddington’sslowclocktrans-
[6–8]becauseonemustassumethatthetransferisperformed
. fer [4]. The former requires operational exchange of light
1 “adiabaticallyslowly”[44]andthespacetimeisflatsuchthat
pulses between the distant clocks and the latter is based on
0
relativistic effects are negligible. However, time dilation in-
5 sending a locally synchronizedclock from one part to other
duced by Earth’s spacetime curvature is experimentally ob-
1 parts. Recently, quantum strategies have been exploited to
servedfor a changein heightof 0.33m[32] and thus cannot
: improvetheaccuracyofclocksynchronization. Afewquan-
v beneglected.
tumclocksynchronization(QCS)proposals[5–16]andexper-
i
X iments[17] are reported. Itisshownthatthe schemesbased In this paper we propose a practical scheme for satellite-
r onquantummechanicscan gainsignificantimprovementsin based QCS. We let two observers, Alice and Bob, exchange
a
precisionovertheirclassicalcounterparts. twofrequencyentangledpulsesbetweenagroundstationand
Ononehandseveralsatellite-basedquantumopticsexperi- asatellite. Theinfluenceofgravitationalred-shiftonthefre-
quency of a pulse can be eliminated by an opposite gravi-
ences[18]arefeasiblewithcurrenttechnology,suchassatel-
lite quantum communication [19–27], and quantum tagging tational blue-shift. By assuming the clocks have the same
precision [5, 7, 8], clock synchronizationcan be realized by
[28], as well as gravity probes using beam interferometers
[29, 30] and atomic clocks [31, 32] to test the principle of identifying the time discrepancies. In our scheme, the time
equivalence. Among these experiments, the synchronization discrepancies are introduced through adding or subtracting
of clocks between a satellite and a ground station [33] is an optical path differences into the pulses and the coincidence
essentialstep. Besides, asatellite-basedquantumnetworkof rate of the pulses in the interferometer as a function of the
clocks is promising to act as a single world clock with un- time discrepancy between the clocks is measured. In actual
precedented stability and accuracy approaching the limit set satellite-basedquantuminformationprocessingtasks[19,21–
23, 26, 27, 45], and similarly in protocolsof QCS, the main
errors are induced by photon-loss and the dispersion effects
oftheatmospherethroughwhichthepulsestravel.Therefore,
∗[email protected] weemployfrequencyentangledlight,insteadoftheentangled
†[email protected] N00Nstate whichisvulnerableforphoton-loss[46], aswell
2
asthedispersioncancellationtechnology[8,47,48]toelimi- on the surfaceof the Earth, we can agree thatthe collapsing
natetheinfluenceoftheatmosphericscattering. Wefindthat timeisidenticalt = ta = tb,evenifAliceandBob’reading
thecoincidencerateofinterferometryisremarkablyaffected times are different τa = τb at this moment. From Eq. (1)
6
bythespacetimecurvatureoftheEarth. Wealsofindthatthe we can see thatthe delaysare proportionalto the time inter-
precisionoftheclocksynchronizationissensitivetothelight valbetweenthestartingtimeτi forthemovingofthemirrors
0
sourceparameters. andthe time t when the photonsare detected. If the propor-
Theoutlineofthepaperisasfollows. InSec. IIwebriefly tionality constant v in Eq. (1) and the starting time reading
introducethesketchoftheexperimentalsetup. InSec. IIIwe τi on Alice’s and Bob’s clocks are identical, the quantity of
0
discuss howthe Earth’sspacetime effectsthe propagationof OPD at Alice’s point will be zero after an exchange period.
photons. InSec. IVwestudythefeasibilityofsatellite-based Therefore,thefinalOPDistotallyproducedfromthestarting
QCSandhowtheeffectoftheEarth’sgravitywilldisturbit. time discrepancy∆τ = τb τa between Alice’s clock and
0 − 0
In the last section we discuss the experimentalfeasibility of Bob’sclock. Thenthesignalandidlerpulsesareinterferedat
ourschemeandgiveabriefsummary. thebeamsplitter (BS)andclickthedetectors. We willshow
in section IV that the final differenceof opticalpath lengths
isaffectedbyafactordependingonthestartingtimediscrep-
II. THESCHEME ancy∆τ. BymeasuringthephotoncoincidencerateP atthe
c
outputports1and2ofthebeamsplitter,onemayacquirevery
The sketch of our proposal for satellite-based QCS is de- precise informationon the OPD in the two arms. Thus, it is
scribed in Fig. (1). The quantum optical technology of our sufficient to measure the photon coincidence rate to recover
proposal is based on the Hong, Ou, and Mandel (HOM) in- the exact time discrepancybetween Alice’s clock and Bob’s
terferometer [52]. We assume that Alice works on the sur- clock. Then Alice tells Bob by classical communication to
face of the Earth (r = r ) with her own clock, while Bob adjusthisclockaccordingto thetime discrepancy. By using
A
works on a satellite at constant radius r = r > r . The this scheme, Alice and Bob may check how much the accu-
B A
clocks have the same accuracy thus the clock synchroniza- racy is disturbedby the gravity inducedspacetime curvature
tionproblemisreducedtotheproblemofidentifyingthetime oftheEarth,andinverselytheycanpreciselymeasurethecur-
discrepancy between the clocks. Alice sends two frequency vaturevia an atomic clock, whichis the mostaccuratesetup
entangled beams produced by a parametric down converter currentlyavailableintheworld.
crystal (PDC) source to Bob and Bob bouncesthem back to
Alice again. Those two pulses are named signal (S) beam
and idler (I) beam, respectively. By exchanging entangled III. EARTH’SSPACETIMEONTHEPROPAGATIONOF
PHOTONS
beamsbetweenAliceandBob,a“conveyorbelt”[8]fortime
informationisestablished. After propagatingthroughdiffer-
entopticalpaths, the signalandidlerbeamsare interferedat Now we describe the propagation of photons from the
the50/50beamsplitterandbemeasuredbythedetectors. To Earth to a satellite by taking gravity of the Earth into con-
introducetimeinformationintothebeams,AliceandBobuse sideration. The Earth’s spacetime curvature will influence
movingmirrorswith constantspeed v to add or subtractop- the lightpulsesduringtheir propagationbetweenthe ground
ticalpathdifferences(OPD)tothebeams[8]. AliceandBob station and the satellite. We know that the Earth rotates
cometoanagreementonthestartingtimeτ0 oftheirmirrors slowly with an angular velocity at the equator of ωE =
in advance. Since they do not have a synchronizedclock to 7.2921151247 10−5rad/s or at a linear speed of vE =
×
startwith,theycanonlystartthemirrorsattimeτ0i(i = a,b) 465 ms−1, which is much slower than the speed of light.
relativetothetimereadingsoftheirownclock,whicharedif- Therefore,theSchwarzschildmetric[49–51]isasufficientap-
ferentduetodifferentlocations. AsdescribedinFig. (1),Al- proximationfortheEarth’sspacetime,ashasbeendiscussed
iceatthegroundstationstartamovingmirrortoaddanOPD in[27,45,49]. TheSchwarzschildmetricisgivenby[50,51]
δlI(t)totheidlerbeam[8]andto subtract thesameamount r r
ofaOPDδlaS(t)fromthesignalbeamwhichwasbouncedback ds2 = −(1− rs)dt2+(1− rs)−1dr2
fromBob. AtthesatelliteBobsubtractsanOPDδlbI(t)from +r2(dθ2+sin2θdϕ2), (2)
theidlerbeamandaddsanidenticalOPDδlS(t)tothesignal
b
beam.ThelineartimedependentOPDsaregivenby whererS := 2Gc2M istheEarth’sSchwarzschildradius,M is
themassoftheEarth,cisthespeedoflightinvacuo,andG
δlaI(t)=v(t−τ0a), δlbI(t)=−v(t−τ0b), isthegravitationalconstant.
(1) A photon can be properly modeled by a wave packet of
δlaS(t)=−v(t−τ0a), δlbS(t)=v(t−τ0b), electromagneticfields[53]withadistributionFω(KK,)0 ofmodes
peaked around the frequencies ω [54, 56], where K =
whereτa andτb arethestartingtimes(propertime)asmea- K,0
0 0 A,B labels either Alice or Bob. The annihilation operator
suredbyAlice’sandBob’sclocks,respectively.InEq.(1)tis
foraphotonfromthepointofviewofAliceorBobtakesthe
thetime point(coordinatetime)ofthecoincidencedetection
form
of the signal and idler photons, i.e., the time when the pho-
tsotantseqisuainnsttuamntasntaetoeuisslymceoalslaupresedd. bByythaessmumeaisnugretmheenqtusamntaudme aωK,0(tK)= +∞dωΩKe−iωKtKFω(KK,)0(ωK)aωK, (3)
Z0
3
FIG.1:Sketchoftheexperimentalsetupforsatellite-basedQCS.Alicesendstwofrequencyentangledbeamsproducedbyaparametricdown
convertercrystal(PDC)sourcetoBobwhobouncesthembacktoAliceagain. AliceaddsanOPDδlI(t)totheidlerbeam[8]andsubtracts
a
thesameamountofOPDδlS(t)fromthesignalbeamwhichwasbouncedbackfromBob.AtthesatelliteBobsubtractsanOPDδlI(t)from
a b
theidlerbeamandaddsanidenticalOPDδlS(t)tothesignalbeam.Finally,weletthepulsesinterfereatthe50/50beamsplitter(BS)andbe
b
detectedatthemeasurementsetups.ThecoincidenceratePcismeasuredasafunctionofthetimediscrepancyτ0b−τ0abetweentheclocks.
where ω is the physical frequencies as measured in their IV. SPACETIMECURVATUREONSATELLITE-BASED
K
labs. At time τ Alice prepares a wave packet F(A) and QCS
A ωA,0
sendsit to Bob whowill receiveit at a differentpropertime
τ = ∆τ + f(r )/f(r )τ , where f(r ) = 1 rs, TheemittedsignalandidlerbeamsfromthePDCinitially
∆Bτ is propagation tBime ofAtheAwave packet,Aand the−facrtAor shareanentangledstate[8]
p
f(r )/f(r )τ indicatestherelativistictimedilation.The
B A A
wave packet received is modified due to the spacetime cur- ψ = dω dω F(ω ,ω )a (ω )a (ω )0 , (6)
pvatureofthe Earthandtakesthe formFω(BB,)0. By employing | i Z 1 2 1 2 † 1 † 2 | i
the definitionof propertime, it is easy to show thatthe time
where a (ω ) and a (ω ) are creation operators of the first
† 1 † 2
evolution for the modes has the form i∂ φ = ω φ ,
τK ωK K ωK and the second photons, respectively. For a monochromatic
whereφ(ωuK) are the quantumstatesof the modescorrespond- pumpthisstatecanberewrittenas
ingtotheoperatorsa [27]. Thisequationindicatesthatthe
ωK
physicalfrequencyω measuredby anobserverat theposi-
K ψ = dω F (ω) ω +ω ω ω , (7)
tion r is ω = f(r ) 1/2ω. Then we find that Bob will | i ω0 | 0 iI| 0− iS
K K K − Z
receive a mode with frequencyω = f(rA)ω if a sharp
frequencymodewithωA wassentBbyAqlicfe(.rTBh)isAisthewell- pwuhleseres,|aωnidSFaωn0d(ω|ω)iiIstahreestpheectsrtaaltedsisftroirbuthtieonsifgunnacltiaonndoifdtlheer
known gravitationalred-shifteffect, which was predictedby down-converted photons [47, 48]. Now let us briefly dis-
Einsteinin1911andexperimentallyverifiedin1960[55]. In cuss how quantum entanglement is useful to quantum clock
our scheme such a classical gravitational effect is designed synchronizationandtheadvantageofthequantumclocksyn-
to be eliminated by an opposite gravitational blueshift fac- chronizationscheme.Themainadvantageofaquantumstrat-
tor f(rB) because the signal will be sent downward from egy is that we can employ quantum uncertainty and coher-
f(rA) encetimeofthefrequencyentangledphotons. FromEq. (7)
asaqtellitetotheEarth.
we can see that although the sum frequency 2ω is certain,
However, such a nonlinear gravitationaleffect is found to 0
the down-shifted frequencies ω ω and ω + ω are
influence the fidelity of the quantum channel between Alice | 0 − iS | 0 iI
highlyuncertain. The frequenciesare largelydeterminedby
andBob[27,45]andwillinevitablyaffecttheaccuracyofthe
the pass bands of the interference filters that inserted in the
satellite-based QCS as well. The mode a¯ received by Bob
′ down-shiftedbeams[52]. Thesepassbandshavebeenfound
canbedecomposedintermsofthemodeapreparedbyAlice
ontheorderof5 1012Hz,whichcorrespondstoacoherence
andanorthogonalmodea [27,45,57] ×
⊥ time foreach photononthe orderof 100fs[52]. Therefore,
a¯ =Θa+ 1 Θ2a , (4) it is able to measure a time interval of or better than the co-
′
− ⊥ herencetimeofthephotons(50fs), withanaccuracyof1fs
whereΘisthewavepacketopverlapbetweenthedistributions (10 15s).Also,themostimportanttaskforthequantumclock
Fω(BB,)0(ωB)andFω(AA,)0(ωB), syn−chronizationistodeterminethetimeintervalbymeasuring
therate atwhichphotonsare detectedin coincidence,which
+
Θ= ∞dω F(A)⋆(ω )F(B) (ω ), (5) relatestothecoherencelengthofthephotonwavepacketand
A ωA,0 B ωB,0 B
Z0 entanglementofthephotons.
whichdescribesthefidelityofthechannelbetweenAliceand As stated before, Alice adds an OPD δlI(t) to the idler
a
Bob. ForaperfectchannelonehasΘ=1. beam[8]andsubtracts thesameamountofOPDδlS(t)from
a
4
the signal beam which was bouncedback from Bob. At the BobandtheBS,andΘ isthemodeoverlapbetweenthedis-
2
satelliteBobsubtractsanOPDδlI(t)fromtheidlerbeamand tributionswhenthepulseisreceivedfromthesatellite. InEq.
b
addsanidenticalOPDδlS(t)tothesignalbeam.Thenwelet (11) the terms κS and κS describe the effect of the disper-
b t f
the pulses interfere at the 50/50 BS and be measured in the siveatmosphereonthesignalbeamontheirwaytoandfrom
HOMinterferometer. ThedetectedcoincidencerateP atthe thesatellite,respectively. NoticethattheDopplershiftintro-
c
detectorsisgivenbytheMandelformula[58] duced by the first mirror is ω/χ and the frequency itself is
revaluatedatω againbecausethesecondmirrormovesinthe
Pc ∝Ztdt1dt2 hψ|E1(−)E2(−)E2(+)E1(+)|ψi, (8) opplipedostiotethdeiriedclteiornbe[a8m]. pTrhoepaagnaatlioognopursocperossc,eydiuerledsincganthbeefianpa-l
operatortransformations
wheret isthe interactiontimeof thedetectors, andtheelec-
tromagneticfieldsattimet attheoutputofthebeamsplitter
canbedefinedby j a′I′(ω)=Θ1Θ2aI(ω)e−iωΥ3+iκIt(ω/χ)+iκIf(ω)+Λa⊥I (ω),
Ej(+) = i dω s4~πωζc aj(ω)e−iω(tj−xj/c) dwihstearnecΥe3be=tw1e2+eββn(tτh0ea−BSτ0ba−ndxct0h)e−PD(LC/χso+urLce′)i/scm. Bucehcasumsealtlheer
Z than the distance between the satellite and the Earth, we as-
= (−) †,forj =1,2, (9) sumethatL = L′. Attheoutputofthe50/50BS,themodes
(cid:16)Ej (cid:17) are found to be a1(ω1) = √12[ia′I′(ω1) e−iω1δl/c +a′S′(ω1)]
where ζ is the beam cross section, and xj is the position of anda2(ω2) = √12[ia′S′(ω2)+a′I′(ω2) e−iω2δl/c], whereδl is
the moving mirrors detector. The relationship between the thedelayintroducedtorelatethecoincidencerateP withthe
c
outputandinputfieldsofthebeamofthemovingmirrorscan path length. Then the coincidence rate P defined in (8) is
c
beobtainedbyperformingLorentztransformationsonthein- obtainedas
put fields [8]. For example, the factor aj(ω)e−iω(τ0j−xj0/c)
a√ftχearjt(hχeωb)eea−miωs1o2−βuβt(flτo0jw−xfj0r/ocm), twhehemreovβin=g mvcirriosrstheevoLlovreesnttoz Pc ∝Z dω1dω2hψ|a†1(ω1)a†2(ω2)a1(ω1)a2(ω2)|ψi
transformationfactor,χ = 1+β denotestheDopplershiftin- = dω dω 0a (ω )a (ω )ψ 2 . (12)
1 β 1 2|h | 1 1 2 2 | i|
troducedbythemovingmirr−ors,andxj isthedistanceofthe Z
0
beam’sdelay. ForthetimedelaysdefinedinEq. (1),wehave Thematrixelement 0a (ω )a (ω )ψ in(12)isgivenby
1 1 2 2
h | | i
xj = x for all j. Then the beams are sent from the Earth
0 0 1
tothesatelliteandareinfluencedbythedissipationoftheat- 0a (ω )a (ω )ψ = (Θ Θ )2δ(ω +ω 2ω )
1 1 2 2 1 2 1 2 0
mosphereandthespacetimecurvature. Theformerproduces h | | i 2 −
a phase discrepancy iκjt(ω) and the latter can be described eiϕφ(ω1−ω0) 1−e2i(ω1−ω0)Υ4−i∆κ(ω1) , (13)
by Eq. (5). Taking the signal beam as an example, the full
h i
procedure can be expressed by the transformations between whereΥ = 4β (τb τa) δl/c,andϕisanoverallphase
theannihilationoperatorswhenthephotonsaresentfromthe 4 1+β 0 − 0 −
term that will disappear by taking the modulus [8], and the
Earthtothesatellite
contributionofthedispersiontermsis
a′S(ω) = Θ1√χaS(cid:18)χsff((rrBA))ω(cid:19)e−irff((rrAB))ωΥ1+iκSt(ω) +∆κκIt(ω(ω)′=)−κκStSf(ω()ωχ−1)κ+If(κωSf)(+ωχ)κ−If(ωκIt′)(−ωχ)κ,St(ω′) (14)
f(r )
+ 1 Θ2a χ A ω , (10)
q − 1 ⊥S(cid:18) sf(rB) (cid:19) iwnghethreroωu′g=hd2iωff0er−enωt.oIpfttihcaelppraotphesrtaireessoufcthhethbaetaκmSs=proκpIagaantd-
whereΥ = 2β (τb x /c) L/candΘ isthewavepacket t f
overlapb1etwe1e−nβthe0d−istr0ibuti−onswhenthe1 pulseissentfrom κSf = κIt, the dispersion effect of the atmosphere is erased.
Suchconditionscanbesatisfiedbyallowingthe“from”idler
theEarthtothesatellite.Wecanseethattheannihilationoper-
beamtopropagateatadistancelessthanthespatialinhomo-
atora (ω)atAlice’slaboratoryevolvesintoa (ω)whenob-
S ′S geneities of the atmosphere from the “to” signal beam and,
servingatBob’slaboratory,whereListhedistancebetween
equivalently,byallowingthe“to”idlerbeamtopropagateless
theEarthandthesatellite. Again,theannihilationoperatorfor
thanthe“from”signalbeam[8]. Here, the“from”beamde-
thesignalbeamevolvesintoa (ω)beforeenteringthe50/50
′S′ notes the beam from Alice to Bob and vice versa. Then the
BS
dispersionsufferedbyoneofthephotonscancancelthatsuf-
a′S′(ω) = Θ1Θ2aI(ω)eiωΥ2+iκSt(ω)+iκSf(ω/χ) ftoertaeldlybcyoitnhceidoethnetrafptherotporno.paTghateisnegttwhrooupghhotdoinffserceanntroepmtiacianl
+Λa (ω), (11)
⊥S paths. SubstitutingEq.(13)intoEq. (12),wecanobtain
where Υ2 = 12+ββ(τ0a − τ0b + xc0) + (L/χ + L′)/c, Λ = P =(Θ Θ )2 dω 2(1 cos[2ω(4v∆τ δl)]), (15)
Θ2 1−Θ21 + 1−Θ22, L′ denotes the distance between c 1 2 Z |F| − c 1+β −
p p
5
0
W
0
8.´10-6 5´1014
6.´10-6 1´1015
p 0.0015
D 4.´10-6 Dp
Coincidencerate 0.0010
2.´10-6 disturbance 0.0005
0.0000
0 0
1´107 2´107 3´107 4´107 5´107
5´1012
r Σ
B
1´1013
FIG.2: (Coloronline) Therelativedisturbanceof coincidencerate FIG.3: (Coloronline)Therelativedisturbanceofcoincidence rate
∆pasafunctionofthealtitudeofthesatelliterB.Theparametersof ∆p as a function of the source parameters ω0 and σ for the fixed
thePDClightsourcearefixedasω0 =700THzandσ=100MHz. distancerB =6.771×106m(LEO).
where F = Fω0(ω). From Eq. (15) we can see thatthe co- In Fig. (2), we plotthe relative disturbanceof the Earth’s
incidence rate Pc is directly related to the time discrepancy spacetimecurvatureonthecoincidencerate∆p asafunction
∆τ = τ0b −τ0a of Alice’s and Bob’s clocks. That is to say, of the distance rB between the satellite and the Earth’s core
Alice’s and Bob’s clocks can be synchronized by using the forthefixedlightsourceparameters. Itisshownthat∆ in-
p
measuredcoincidencerateofinterferometry. creases as the distance increases, i.e., the accuracy of clock
To be explicit we next only consider the case in synchronizationdependsonthealtitudeofthesatellite,which
which Fω0(ω) is a Gaussian wave packet Fω0(ω) = also verifies that the spacetime curvature remarkably influ-
√421πσ2e−(ω−4σω20)2, whereσ is the Gaussian width. The wave evnercyesditfhfeerreunntnfrinogmotfhattheofaRtoemf.ic[8c],loinckwsh[i3c2h].theThcoisinrceisduelntcies
packetoverlapsΘ andΘ arefoundtobe
1 2 rateisindependentofthedistanceLbetweenAlice andBob
whenthespacetimecurvatureoftheEarthisnotconsidered.
2∆ ϑ2ω02
Θ1(2) =s1+∆1(212()2)e−4σ2[1+∆21(2)], (16) peaInkfFriegq.ue(n3c),ywωe0panlodtbthaendrweliadttihveσdoisftuthrbeaPnDceC.∆Iptiosvsehrotwhne
that the disturbance of accuracy depends sensitively on the
where ∆ = 1 ϑ and the signs occur for r < r bandwidthσ ofthesource,whichissimilartotheflatspace-
1(2) A B
± ±
timecase[8]. However,herewe findthatthedisturbanceon
or r > r . In Eq. (16), we define ϑ = f(rA) 1
A B f(rB) − accuracy also depends on the peak frequency of the pulses,
and ωA,0 = ωB,0 = ω0 is assumed. The mqodes will be whichisdifferentfromthatof[8]wheretheaccuracyisinde-
perfectly overlapped (Θ = 1) when Alice and Bob are in
pendentof the peak frequency. In this paper we are particu-
a flat spacetime f(r ) = f(r ) = 1. For the typical
A B larlyinterestedintwotypicalcases,inwhichtheQCSareper-
resources used in quantum optics experiments, the relation
formedbetweenthegroundstationandeitheralowearthorbit
ϑ ≪ (ϑσω0)2 ≪ 1 should be satisfied [27, 45, 59], which satellite(LEO),orageostationaryearthorbit(GEO)satellite,
yieldsΘ1 = Θ2 ∼ 1− ϑ82σω202. Thenwe find thatthe coinci- respectively.
dencerateP hastheform The LEO case: The typical distance from the Earth to a
c
LEO satellite is about 400km, which yields r = 6.371
A
Pc =(1− ϑ82σω202)4[1−e−2σ2(δl−41v+∆βτ)2/c2]. (17) 1S0ch6mwaraznsdchrilBd ra=dius6.o7f7t1he×E1ar0t6hmis. rSC=ons9idmemrin,git itshaftoutnh×de
that δ 1(rs rs) = 4.17 10 11. We employ a
We can see that the coincidence rate Pc has the factor (1 typical∼PD−C2soruBrce−wriAth a wavelen×gth o−f 369.5 nm (corre-
−
ϑ82σω202)4 compared to that of the flat spacetime case, where sponding to ω0 = 812THz) and σ = 100MHz (the relation
Pcf = 1−e−2σ2(δl−41v+∆βτ)2/c2 [8]. We define the effect of δpea≪k fr(eδqωσu0e)n2cy≪an1d bisansdatwisifidethd)i.s aAvaliilgahbtles,ofuorrceexwamithplesu,cihn
spacetimecurvatureontheaccuracyofclocksynchronization
trappedionexperiments[59]. Therelativedisturbanceofthe
astherelativedisturbanceofcoincidencerate
spacetime curvature on the coincidence rate is obtained as
∆ = Pcf −Pc =1 (1 ϑ2ω02)4. (18) ∆Lp =5.73993×10−8. Theachievementofanopticallattice
p Pf − − 8σ2 clock with accuracy at the 10−18 level has been reported in
c
Refs. [60,61]. Ifwewouldliketosynchronizetwoclocksup
Itisnowclearthattherelativedisturbanceofcoincidencerate toatimediscrepancyofτ = 1ns(asin[8], oramuchlower
∆ dependsonthespacetimeparameterϑandthecharacter- level τ = 100ns as presented in [7]), the correction of the
p
isticsofthePDCsource. Earth’sspacetimecurvatureeffectwillreach10 17sduringa
−
6
singlesynchronizationprocess.Suchacorrectioniscompara- bustfrequencyentangledpulsesoflightbytakingtheeffects
bletotheaccuracyoftheatomclocksandthusshouldbecon- of the spacetime curvature of the Earth into consideration.
sidered for the QCS between clocks in future satellite-based The spacetime background of the Earth is described by the
applications. Therefore, we can safely arrive at the conclu- Schwarzschildmetric,andthequantumopticspartofourpro-
sion that the spacetime curvature is not negligible when the posal is based on the HOM interferometer. By eliminating
synchronizationisperformedbyLEOsatellites. thegravitationalredshiftandblueshiftofthelaserpulsesand
The GEO case: The typical distance L between a GEO theatmosphericdispersioncancellation,theaccuracyofclock
satellite andthe groundis about3.6 107m. Therefore,the synchronizationin ourquantumschemecanbe veryhighby
×
distancebetweentheEarthandthesatelliter isaboutr = showing that ∆ is close to unity. Our proposal can be im-
B B p
42.371 106m,whichyieldsδ 1(rs rs)=6 10 10. plemented, in principle, with current available technologies.
× ∼−2 rB−rA × −
Inthiscasetherelativedisturbanceofthespacetimecurvature To be specific, optical sources with the required peak fre-
on the coincidence rate is ∆G = 1.18729 10 5. We find quencyandbandwidthhavebeenachievedbythetrappedion
p × −
thatthedisturbanceofthespacetimecurvatureonthecoinci- experiments [59]. The feasibility of photon exchanges be-
denceratefortheGEOsatellitesbecomesevenmoreremark- tween a satellite and a ground station has been experimen-
able than that of the LEO case. We remark that the current tallydemonstrated[21] bytheMateraLaser RangingObser-
GPS satellites have r 2.7 107m. In the QCS scheme, vatory(MLRO)inItaly.Mostrecently,theyhavereportedthe
B
≈ ×
thespacetimecurvatureisalsoremarkable. operationof experimentalsatellite quantumcommunications
Error analysis: It is to mentioning that the velocity vari- [22] by sending selected satellites laser pulses. Our scheme
ationsof the movingmirrorsmay inducesome errorson the can also be generalized to the quantum clock network cases
coincidence rate. However, the order of magnitudes of the [6,34].Theresultsshouldbesignificantbothfordetermining
movementspeedof the mirrorsis muchsmaller thanthe ve- theaccuracyofclocksynchronizationandforourgeneralun-
locityoflight, letalonethevelocityvariationof themirrors. derstandingoftimediscrepancyinfuturesatellite-basedquan-
To be specific, the typical velocity of the moving mirrors is tumsystems.
10 1m/s. Letussupposethatthisvelocityhasonepercentof
−
variation,say10 3m/s,whichismuchsmallerthantheveloc-
−
ityoflight.Notethattherelativedisturbanceofthespacetime
curvatureonthecoincidencerate isontheorderof10 8 for
−
theLEOsatellitesandoftheorderof10 5fortheGEOsatel-
− Acknowledgments
lites,whichareatleast3ordersofmagnitudeslargerthanthat
of the velocity variationsof the mirrors. Therefore, the sys-
tematicerrorsinducedbythevelocityvariationofthemirrors This work is supported by the National Natural Science
canbesafelyignoredinthescheme. Foundation of China under Grants No. 11305058, No.
11175248, No. 11475061, the Doctoral Scientific Fund
Project of the Ministry of Education of China under Grants
V. DISCUSSIONS No. 20134306120003, Postdoctoral Science Foundation of
China under Grants No. 2014M560129, No. 2015T80146
We have proposeda practicalsatellite-based QCS scheme and the Strategic Priority Research Program of the Chinese
with the advantages of dispersion cancellation and the ro- AcademyofSciences(underGrantNo. XDB01010000).
[1] B.Simons, J.L.Welch, and N.Lynch, IBMResearchReport [11] S.Walter,A.Nunnenkamp,andC.Bruder,Phys.Rev.Lett.112,
No.6505(1988). 094102(2014).
[2] W. Lewandowski, J. Azoubib, and W. J. Klepczynski, Proc. [12] M.H.Xu,D.A.Tieri,E.C.Fine,J.K.Thompson, andM.J.
IEEE87,163(1999). Holland,Phys.Rev.Lett.113,154101(2014).
[3] A.Einstein,Ann.Phys.(Berlin)322,891(1905). [13] Y.-L.Zhang,Y.-R.Zhang,L.-Z.Mu,andH.Fan,Phys.Rev.A
[4] A. S. Eddington, The Mathematical Theory of Relativity, 2nd 88,052314(2013).
edition(CambridgeUniversityPress,Cambridge,1924). [14] J.-D. Yue, Y.-R. Zhang, H. Fan, Phys. Rev. A 92, 032321
[5] R.Jozsa,D.S.Abrams,J.P.Dowling,andC.P.Williams,Phys. (2015).
Rev.Lett.85,2010(2000). [15] M. R.Hush, W.-B.Li, S. Genway, I.L. Sanovsky, and A.D.
[6] V. Giovannetti, S. Lloyd, and L. Maccone, Nature 412, 417 Armour,arXiv:1412.1863.
(2001). [16] V. Ameri, M. Eghbali-Arani, A. Mari, A. Farace, F. Kheiran-
[7] IsaacL.Chuang,Phys.Rev.Lett.85,2006(2000). dish, V. Giovannetti, and R. Fazio, Phys. Rev. A 91, 012301
[8] V.Giovannetti,S.Lloyd,L.Maccone,andF.N.C.Wong,Phys. (2015).
Rev.Lett.87,117902(2001). [17] A.Valenciaetal.,Appl.Phys.Lett.85,2655(2004).
[9] A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, [18] D.Rideoutetal.,Class.QuantumGrav.29,224011(2012).
Phys.Rev.Lett.111,103605(2013). [19] C.Bonatoetal.,Opt.Expr.14,10050(2006).
[10] T. E. Lee, and H. R. Sadeghpour, Phys. Rev. Lett. 111, [20] C.Bonato,A.Tomaello,V.DaDeppo, G.Naletto,andP.Vil-
234101(2013). loresi,NewJ.Phys.11,045017(2009).
7
[21] P.Villoresietal.,NewJ.Phys. 10,033038(2008). (2014).
[22] G.Valloneetal.,Phys.Rev.Lett.115,040502(2015). [43] M.Ringbauer, M.A.Broome, C.R.Myers,A.G.White,and
[23] C.Ervenetal.,NewJ.Phys.14,123018(2012). T.C.Ralph,NatureCommunications5,4145(2014).
[24] J.G.Rarity,P.R.Tapster,P.M.Gorman,andP.Knight,NewJ. [44] V.Giovannetti,S.Lloyd,L.Maccone,andM.S.Shahriar,Phys.
Phys.,4,82(2002). Rev.A65,062319(2002).
[25] C. Kurtsiefer, P.Zarda, M. Halder, H. Weinfurter, P.M. Gor- [45] D.E.Bruschi,A.Datta,R.Ursin,T.C.Ralph,andI.Fuentes,
man,P.R.Tapster,andJ.G.Rarity,Nature419,450(2002). Phys.Rev.D90,124001(2014).
[26] J.Y.Wangetal.,NaturePhotonics7,387(2013). [46] V.Giovannetti1, S.Lloyd, andL.Maccone, NaturePhotonics
[27] D. E. Bruschi, T. C. Ralph, I. Fuentes, T. Jennewein, and M. 5,222(2011).
Razavi,Phys.Rev.D90,045041(2014). [47] A.M.Steinberg,P.G.Kwiat,andR.Y.Chiao,Phys.Rev.Lett.
[28] A.Kent,WilliamJ.Munro,andTimothyP.Spiller,Phys.Rev. 68,2421(1992).
A84,012326(2011). [48] A.M.Steinberg,P.G.Kwiat,andR.Y.Chiao,Phys.Rev.A45,
[29] A.Brodutch,A.Gilchrist,T.Guff,AlexanderR.H.Smith,and 6659(1992).
DanielR.Terno,Phys.Rev.D91,064041(2015). [49] W.-T.LiaoandS.Ahrens,NaturePhotonics9,169(2015).
[30] R. Colella, A. W. Overhauser, and S. A. Werner, Phys. Rev. [50] C.W.Misner,K.S.Thorne,andJ.A.Wheeler,Gravitation(W.
Lett.,34,1472(1975). H.FreemanandCompany,SanFrancisco,1973).
[31] D.Kleppner, RobertF.C.Vessot,N.F.Ramsey, Astrophysics [51] R. M. Wald, General Relativity (The University of Chicago
andSpaceScience6,13(1970). Press,Chicago,1984).
[32] C.W.Chou, D.B.Hume, T.Rosenband, andD.J.Wineland, [52] C.K.Hong,Z.Y.Ou,andL.Mandel,Phys.Rev.Lett. 59,2044
Science329,1630(2010). (1987).
[33] V.Giovannetti,S.Lloyd,andL.Maccone,Nature412,417-419 [53] D.E.Bruschi,A.R.Lee,andI.Fuentes,J.Phys.A46,165303
(2001). (2013).
[34] P.Ko´maˆretal.,NaturePhysics10,582(2014). [54] T. G. Downes, T. C. Ralph, and N. Walk, Phys. Rev. A 87,
[35] A.PeresandD.R.Terno,Rev.Mod.Phys.76,93(2004). 012327(2013).
[36] I. Fuentes-Schuller, and R. B. Mann, Phys. Rev. Lett. 95, [55] R.V.Pound,andJr.G.A.Rebka,Phys.Rev.Lett.4,337(1960).
120404(2005). [56] U. Leonhardt, Measuring the Quantum State of Light, Cam-
[37] J.C.Wang,J.F.Deng,andJ.L.Jing,Phys.Rev.A 81,052120 bridgeStudiesinModernOptics(CambridgeUniversityPress,
(2010). Cambridge,2005).
[38] J.C.Wang,andJ.L.Jing,Phys.Rev.A 82,032324(2010);J. [57] P.P.Rohde,W.Mauerer,andC.Silberhorn,NewJ.Phys.9,91
C.Wang,Z.H.Tian,J.L.Jing,andH.Fan,Nucl.Phys.B892, (2007).
390(2015). [58] L.Mandel,andE.Wolf,Opticalcoherenceandquantumoptics
[39] T.G.Downes,I.Fuentes,andT.C.Ralph,Phys.Rev.Lett. 106, (CambridgeUniv.press,Cambridge,1995).
210502(2011). [59] D.N.Matsukevich,P.Maunz,D.L.Moehring,S.Olmschenk,
[40] E. Mart´ın-Mart´ınez, I. Fuentes, and R. B. Mann, Phys. Rev. andC.Monroe,Phys.Rev.Lett. 100,150404(2008).
Lett. 107,131301(2011). [60] N.Hinkleyetal.,Science341,1215(2013).
[41] N.Friisetal.,Phys.Rev.Lett. 110,113602(2013). [61] B.J.Bloometal.,Nature506,71(2014).
[42] J. C. Wang, J. L. Jing, and H. Fan, Phys. Rev. D 90, 025032