Pozas-Kerstjens, Alejandro and Louko, Jorma and Martín-Martínez, Eduardo

12 Pages·2017·0.28 MB·English

by  Christine Middleton

PHYSICAL REVIEW D 95, 105009 (2017)
Degenerate detectors are unable to harvest spacelike entanglement
Alejandro Pozas-Kerstjens,1 Jorma Louko,2 and Eduardo Martín-Martínez3,4,5,*
1ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology,
08860 Castelldefels (Barcelona), Spain
2School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
3Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
4Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
5Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo,
Ontario N2L 2Y5, Canada
(Received 8 March 2017; published 30 May 2017)
Weshow,underaverygeneralsetofassumptions,thatpairsofidenticalparticledetectorsinspacelike
separation,suchasatomicprobes,canonlyharvestentanglementfromthevacuumstateofaquantumfield
when they have a nonzero energy gap. Furthermore, we show that degenerate probes are strongly
challengedtobecomeentangledthroughtheirinteractionthroughscalarandelectromagneticfieldsevenin
full light contact. We relate these results to previous literature on remote entanglement generation and
entanglement harvesting, giving insight into the energy gap’s protective role against local noise, which
prevents the detectors from getting entangled through the interaction with the field.
DOI: 10.1103/PhysRevD.95.105009
I. INTRODUCTION detectors have also been shown to display the same
qualitative behavior when they harvest entanglement from
Itiswellknownthatthegroundstateofaquantumfield
the quantum field [13]. Along these lines, entanglement
contains entanglement between different regions of space-
harvesting is not a fragile phenomenon: it has been proven
time. This is so even if the regions are spacelike separated
robust against uncertainties in the synchronization and
[1,2]. Moreover, this entanglement can be extracted (or
spatial configuration of the particle detectors [14]. The
harvested) into pairs of particle detectors through local
variety of situations in which the phenomenon of entangle-
interactions of each detector with the field (again, even in
ment harvesting has been found relevant has motivated
spacelike separation), leading to the entanglement of
works analyzing the experimental feasibility of implement-
initially uncorrelated detectors [3–5] even for arbitrary
ing timelike and spacelike entanglement harvesting proto-
spatial separation and smooth switching profiles [6].
cols in both atomic and superconducting systems [15–17].
Thisphenomenon,knownasentanglementharvesting,is
Entanglement harvesting is affected by local noise.
very sensitive to the properties of the spacetime back-
For example, a sudden switching of the detector-field
ground (e.g., its geometry [7] or its topology [8]).
interaction (which locally excites the detectors) is ineffi-
Entanglement harvesting has been proposed as a means
cient for harvesting spacelike entanglement since the local
tobuildsustainablesourcesofentanglement(viaentangle-
noise overshadows the correlations harvested from the
mentfarmingprotocols[9]),andhasbeenproventobevery
field. In contrast, if the interaction is switched on adia-
sensitive to the state of motion of the detectors and the
batically, it has been shown that it is possible to harvest
boundaryconditionsonthefieldonwhichitisperformed.
entanglement with arbitrarily distant spacelike separated
Thishasledtoproposalsofapplicationsinmetrologysuch
detectors [6]. To harvest spacelike entanglement from
asrangefinding[10]orasaverysensitivemeanstodetect
arbitrarily long distances, the detectors’ energy gaps (the
vibrational motion [11].
energy difference between ground and first excited state)
Entanglement harvesting has been proven to be substan-
havetobeincreasedproportionallytotheseparationofthe
tially independent of the particular particle detector model
detectors to shield them from local excitations that would
employed: there are no notable qualitative differences
overwhelmtheharvestingofcorrelations(seeRef.[18]for
between simplified Unruh-DeWitt (UDW) models in its
a thorough study).
differentvariants.Namely,itwasshowninRef.[12]thatan
It has been observed that temperature also prevents
Unruh-DeWitt detector coupled to the amplitude or to the
entanglement from being harvested [19], particularly for
momentum of a scalar field yields qualitatively similar
spacelike separation between the detectors. This can be
results to those of a fully featured hydrogenlike atom
understoodascausedbythedecayofquantumcorrelations
coupled to the electromagnetic field. Harmonic oscillator
in a quantum field with temperature.
Remarkably, and in contrast to this, it was shown by
*[email protected] Braun [20,21] that, even with zero energy gap, spin-1=2
2470-0010=2017=95(10)=105009(11) 105009-1 © 2017 American Physical Society
POZAS-KERSTJENS, LOUKO, and MARTÍN-MARTÍNEZ PHYSICAL REVIEW D 95, 105009 (2017)
systems in timelike separation could entangle through II. UNRUH-DEWITT DYNAMICS AND
their interaction with thermal baths and quantum fields in ENTANGLEMENT HARVESTING
thermal states. This mechanism was initially proposed as a
In a typical scenario of entanglement harvesting [3,4],
meansofcreatingentanglementbetweendistantparties[20],
two localized quantum systems interact with the vacuum
but a closer examination of the problem revealed that the
state of a field. We model the interaction between an
more interesting phenomenon of spacelike entanglement
individual inertial smeared detector and a massless scalar
harvesting—in which not even indirect communication
field in an (nþ1)-dimensional flat spacetime with the
through the field is possible and none of the detectors can
UDWparticledetectormodel[22].Thismodelcapturesthe
knowoftheexistenceoftheother—wasnotpossibleinthe
fundamental features of the light-matter interaction in
casesstudiedinRef.[21].Theseresultsraisethequestionof
scenarios where angular momentum exchange does not
whatisspecialintheregimesanalyzedinRefs.[20,21]that
play a fundamental role [12,23,24]. More relevant to our
preventsspacelikeentanglementharvesting.Inprinciple,and
case, the UDW model has been explicitly proven to yield
with no additional data, one could have thought of three
qualitativelyidenticalresultsinentanglementharvestingto
possible suspects for the lack of spacelike entanglement
those with fully featured hydrogenoid atoms interacting
harvesting in the setups in Refs. [20,21]: (1) the use of
with the electromagnetic field (in particular, see Ref. [12]
thermal backgrounds as opposed tothevacuumstate of the
for this last claim). For technical reasons, we assume
field, (2) the particular switching functions utilized (recall
throughoutn≥2.Thecasen¼1wouldrequireadditional
that switching can strongly influence the ability to harvest
inputforhandlingthewell-knowninfrareddivergencesofa
entanglement[5,6])or(3)thefactthat[20,21]onlyanalyze
masslessfieldintwospacetimedimensions.Wemakesome
degeneratetwo-levelsystems(withzerogapbetweenground
explicitcommentsaboutthe 1þ1-dimensional casewhen
and excited states).
we discuss some of our results.
In this paper we address this question and show that the
The UDW interaction Hamiltonian is given by
lack of spacelike entanglement harvesting is not due to
thethermalbackgroundortothenatureoftheswitching.The Z
X
culpritisthegaplessnatureofthedetectors.Weprovethatit HˆðtÞ¼ λνXνðtÞ dnxSνðx−xνÞμˆνðtÞϕˆðt;xÞ: ð1Þ
isimpossibleforapairofidenticalinertialgaplessdetectors
ν
to harvest any amount of entanglement from spacelike
separatedregionseveninthevacuumstateofa scalarfield In this expression, the label ν∈fA;Bg identifies the
inflatspacetime,andarguethattheproofshouldcarryoverto detectorandλν isthecouplingstrengthofdetectorνtothe
thecaseofentanglementharvestingwithhydrogenlikeatoms scalar field ϕˆðt;xÞ. The field can be written as a sum of
from the electromagnetic field [12].
plane-wave modes as
After an introduction to the formalism of entanglement
Z
harvestingandthenotationtobeusedthroughoutthepaper
in Sec. II, we divide the proof in two parts: in Sec. III we ϕˆðt;xÞ¼ pffiðffi2ffiffidffiπffinffiÞffikffinffiffi2ffiffiffijffikffiffijffiffi½aˆkeik·xþaˆ†ke−ik·x(cid:2); ð2Þ
prove that when the time intervals of interaction of each
individual detector with the field do not overlap, gapless
detectorscannotharvest anyentanglementatall,regardless where aˆk and aˆ†k are bosonic annihilation and creation
of their specific spatial shape, their relative separation (not operators of a field mode with momentum k, and
only spacelike, but also timelike or lightlike) or the total k·x¼−jkjtþk·x. μˆνðtÞ is the monopole moment of
amount of time of interaction with the field, and then in detector ν, given by
Sec. IV we give the proof that spacelike entanglement
harvesting is not possible in the case when the periods of μˆνðtÞ¼eiΩνtσˆþν þe−iΩνtσˆ−ν ð3Þ
interaction have nonzero overlap, which requires the extra
assumption of the shapes being spherically symmetric. In [σˆþandσˆ−aretheusualSU(2)ladderoperators].HereΩνis
Sec.VweextendtheresultsinSecs.IIIandIVtodetectors the energy gap between the two levels of detector ν:
interacting with an electromagnetic field through a realistic XνðtÞ is the switching function that controls the duration
dipole-typelight-matterinteraction.InSec.VIwealsoshow andstrengthoftheinteraction.SνðxÞisthesmearingfunction
that very short and strong “deltalike” switching functions ofthedetectorsthatcanbeassociatedtotheirspatialextension
cannotharvestentanglementatallregardlessofenergygaps, and shape (e.g., for a hydrogenoid atom it is connected
regimeofseparationorsmearingofthedetectors.Finally,in to theground and excited statewave functions[12]).
Sec. VII we conclude by providing the physical interpreta- As usual in entanglement harvesting scenarios, the
tion of the results: as was already noted in Ref. [6], the detectors, initially completely uncorrelated and in their
energygapshieldsfromlocalexcitationsofthedetectorsand ground state, couple to the field, and after the coupling
its absence allows for any local noise to overcome the (controlled by the switching function), they end up in a
nonlocal excitations produced by the vacuum fluctuations. final state given by
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DEGENERATE DETECTORS ARE UNABLE TO HARVEST … PHYSICAL REVIEW D 95, 105009 (2017)
ρˆAB ¼TrϕˆðUˆjψ0ihψ0jUˆ†Þ; ð4Þ SidAen¼ticSaBl,≡soS.thWateλal¼soλta≡keλt,haendcotuhpelisnwgitcsthriennggtfhusncttoiobnes
A B
whereTr denotesthepartialtracewithrespecttothefield tobeidenticaluptoatimeshift,sothatXνðtÞ≡Xðt−tνÞ,
degrees oϕ^f freedom. Here where tν is the time at which the interaction of detector ν
and the field begins.
(cid:3) Z (cid:4) The detectors’ state ρˆ after the interaction is a two-
Uˆ ¼T exp −i ∞dtHˆðtÞ ð5Þ qubit X-state [4,6]. We qABuantify the entanglement in this
−∞ state with the negativity (a faithful entanglement measure
for a system of two qubits [25]). To the first nontrivial
is the time evolution operator and the initial state of the
perturbative order in the coupling strength, the negativity
detectors-field system is taken to be
takes the simple form [5,6]
jψ0i¼jgAi⊗jgBi⊗j0ϕˆi: ð6Þ Nð2Þ ¼maxð0;jMj−LÞþOðλ4Þ. ð7Þ
We consider detectors that have identical energy gaps Notethatthroughoutthispaperweareusingthenotationin
and identical spatial shapes, so that Ω ¼Ω ≡Ω and Ref. [6]. The functions L and M are
A B
Z Z Z Z
∞ ∞
L¼λ2 dt1 dt2Xðt1ÞXðt2ÞeiΩðt1−t2Þ dnx1 dnx2Sðx1ÞSðx2ÞWnðt2;x2;t1;x1Þ
−∞ −∞
Z Z Z
jS~ðkÞj2 ∞ ∞
¼λ2 dnk 2jkj −∞dt1Xðt1ÞeiðjkjþΩÞt1 −∞dt2Xðt2Þe−iðjkjþΩÞt2
Z (cid:5)Z (cid:5)
¼λ2 dnkjS~ðkÞj2(cid:5)(cid:5)(cid:5) ∞dtXðtÞeiðjkjþΩÞt(cid:5)(cid:5)(cid:5)2; ð8Þ
2jkj −∞
Z Z Z Z
M¼−λ2 ∞dt1 t1 dt2 dnx1 dnx2Sðx1−xAÞSðx2−xBÞeiΩðt1þt2ÞWnðt1;x1;t2;x2Þ
−∞ −∞
×½Xðt1−t ÞXðt2−t ÞþXðt1−t ÞXðt2−t Þ(cid:2)
A B B A
Z Z Z
¼−λ2 dnkjS~2ðjkkÞjj2eik·ðxA−xBÞ −∞∞dt1 −∞t1 dt2e−iðjkj−ΩÞt1eiðjkjþΩÞt2
×½Xðt1−t ÞXðt2−t ÞþXðt1−t ÞXðt2−t Þ(cid:2); ð9Þ
A B B A
the Wightman function of the free scalar field in n spatial entanglement from the field (i.e., for the negativity of the
dimensions is given by jointstateρˆ tobenonzeroafterinteractingwiththefield)
AB
the correlation term M must overcome the local noise L.
Wnðt;x;t0;x0Þ¼h0ϕˆjϕˆðt;xÞϕˆðt0;x0Þj0ϕˆi; ð10Þ Ourobjectiveistoprovethatidenticalzero-gapdetectors
cannot harvest entanglement from spacelike separated
and the Fourier transform of the smearing function is regions of the field.
Z From now on we consider gapless detectors, Ω¼0, so
1
S~ðkÞ¼pffiffiffiffiffiffiffiffiffiffiffi dnxSðxÞeik·x: ð11Þ thatthemonopolemoment(3) becomestimeindependent.
ð2πÞn We also take the switching function X to have compact
support, writing
We have used the time translation invariance of (cid:6)
Wnðt;x;t0;x0Þ to write (8) in a way that makes explicit XðtÞ¼ χðtÞ for 0≤t≤T; ð12Þ
that L is independent of the beginning of the interaction 0 otherwise
with the field tν.
It is already discussed in Refs. [4,5], and with our where T >0 is the duration of each detector’s interaction
notation in Refs. [6,12], that the term L corresponds to withthefield.Weemphasizethatthetimestν,atwhichthe
local excitations of each detector, while M accounts for interaction of each detector with the field begins, remain
correlationsbetweenbothdetectors.Therefore,Eq.(7)has arbitrary.TheseinitialtimeshavedroppedoutofL(8)but
an intuitive physical meaning: for two detectors to harvest they appear in M (9). Similarly, we emphasize that the
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POZAS-KERSTJENS, LOUKO, and MARTÍN-MARTÍNEZ PHYSICAL REVIEW D 95, 105009 (2017)
spatial points xν, at which the detectors are centered, have where the changes of variables t¼t1−t and t0 ¼t2−t
B A
dropped out of L (8) but they appear in M (9). have been performed.
This yields the following conclusion: when there is no
overlap between the time intervals of individual inter-
III. NONOVERLAPPING SWITCHINGS
actions with the field, the nonlocal, entangling term is
When the switching functions’domains do not overlap, always upper-bounded by the local one and therefore
thetimeintegralsinthenonlocalterm(9)greatlysimplify. Nð2Þ ¼maxð0;jMj−LÞ¼0 for any—compactly sup-
There are two summands in this term, which require ported or not—smearing function of the (recall gapless)
separate study. detectors and any compactly supported, nonoverlapping
In the first summand the integrand is nonzero for t1 ∈ switchings.
½t ;t þT(cid:2)andt2 ∈½t ;t1 ≤minðt ;t ÞþT(cid:2).Withoutloss This means that gapless inertial comoving detectors
A A B A B
of generality, let us assume that detector B is switched on with the same switching functions are unable to harvest
afterdetectorAhasbeenswitchedoff(i.e.,t >t þT).In any entanglement regardless of their relative positioning
B A
this case, because of the nested nature of the integrals, the (spacelike,timelikeorlightlike)fromevenarbitrarilyclose
region of integration over t2 is limited by the support of regions if they are switched on at different times with no
Xðt1−t Þ.SincedetectorBisswitchedonafterdetectorA overlapbetweenthetimeintervalsinwhicheachindividual
A
isswitchedoff,theregionofintegrationovert2 liesoutof detector interacts with the field.
the support of Xðt1−t Þ, and therefore the integral We stress that this is the case even for gapless detectors
A
evaluates to 0 regardless of the specific shape of χðt2Þ. whichareinregionsthatcanbeconnectedbylight.Thisis
In the second summand, in contrast, the integrand true even if the smearing functions overlap (which means
is supported in t1 ∈½t ;t þT(cid:2) and t2 ∈½t ;t1 ≤ having effectively zero distance between the detectors).
B B A
minðt ;t ÞþT(cid:2). Now, in the case that detector B is Although this proof assumed that the switchings were
A B
switched on after detector A has been switched off, the the same for both detectors, numerical evidence for a
effective region of integration over t2 after taking into generality of compactly supported switching functions
account the supports of Xðt1−t Þ and Xðt2−t Þ is suggests that the detectors are unable to harvest entangle-
B A
½t ;t þT(cid:2).Thismeansthatwecandenestthetwointegrals, ment also in the case of switchings of different duration
A A
Z Z T ≠T . We highlight that this is true for detectors in
A B
∞dt1 t1 dt2e−ijkjðt1−t2ÞXðt1−tBÞXðt2−tAÞ timFeilnikalel,y,spnaoctiecleiktehaotrtehvisenprloigohftcliakreriessepoavreartioton.the case of
−∞Z −∞Z
∞ ∞ 1þ1dimensionsifweaddaninfraredcutoff.Evenwithan
¼ dt1 dt2e−ijkjðt1−t2ÞXðt1−t ÞXðt2−t Þ; ð13Þ infraredcutoff,theidentity(14)stillholdsinthesameway
B A
−∞ −∞
as in (15), so the inability of gapless detectors to harvest
entanglement applies also to this case.
wheretheequalityfollowsbecauseallthevaluesoft2inthe
support of Xðt2−t Þ are strictly smaller than the smallest
A
value of t1 in thesupport of Xðt1−t Þ. IV. OVERLAPPING SWITCHINGS
B
Now, using the fact that the modulus of an integral is
We now explore the case when the time intervals of
upper bounded by the integral of the modulus of the
interaction overlap, either partially or totally. For this
integrand, i.e.,
scenario, numerical evidence shows that entanglement har-
(cid:5)Z (cid:5) Z vesting is possible in general for timelike and lightlike
(cid:5) (cid:5)
(cid:5)(cid:5) dxfðxÞ(cid:5)(cid:5)≤ dxjfðxÞj; ð14Þ separations, sowe focus onthe harvesting of entanglement
from spacelike separated regions and ask the following
question: can two gapless detectors harvest entanglement
we see that fromthefieldvacuumwhiletheyremainspacelikeseparated?
To talkproperlyaboutspacelike separation,weconsider
(cid:5)Z
jMj¼λ2(cid:5)(cid:5)(cid:5) dnkjS~2ðjkkÞjj2eik·ðxA−xBÞ dCeotnecctroerteslyw,idtheteacrtboirtrsarAy caonmdpBacthlyavesupfipnoitretedchasmraecaterrinisgtisc.
Z Z (cid:5) lengths of R and R respectively. In analogy with
∞ ∞ (cid:5) A B
× dt1 dt2eijkjðt1−t2ÞXðt1−t ÞXðt2−t Þ(cid:5)(cid:5) Eq.(12),thesmearingfunctionsofthedetectorsaregivenby
B A
−∞ −∞ (cid:6)
≤λ2Z dnkjS~2ðjkkÞjj2 SνðxÞ¼ s0νðxÞ foothrejrxwji≤se12Rν: ð16Þ
(cid:5)Z Z (cid:5)
(cid:5) ∞ ∞ (cid:5)
×(cid:5)(cid:5) dt dt0eijkjðt−t0ÞXðtÞXðt0Þ(cid:5)(cid:5)¼L; ð15Þ Forthefollowingproof,wefurthermoreassumethatthe
−∞ −∞ shapes of the detectors are spherically symmetric, which
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DEGENERATE DETECTORS ARE UNABLE TO HARVEST … PHYSICAL REVIEW D 95, 105009 (2017)
Z Z Z Z
aEmq.o(u1n1t)sotnolysadyeipnegntdhsatonthtehiernFoorumrieorfttrhaenFsfoourrmierasvagriivaebnlebky. M¼−λ2 ∞dt1 t1 dt2 dnx1 dnx2
−∞ −∞
Explicitly, writing (8) and (9) in spherical coordinates ×Sðx1−xAÞSðx2−xBÞWnðt1;x1;t2;x2Þ
L¼λ2Z ∞djkjZ dΩn−1jkjn−2jS~ðj2kjÞj2(cid:5)(cid:5)(cid:5)(cid:5)Z ∞dtXðtÞeijkjt(cid:5)(cid:5)(cid:5)(cid:5)2; ×½Xðt1−tAÞXðt2−tBÞþXðt1−tBÞXðt2−tAÞ(cid:2)ð:22Þ
0 −∞
ð17Þ
Giventhatthesmearingandswitchingfunctionsarereal,
the only element that can make M complex is the
Z Z Wightman function W . Remarkably, the imaginary part
M¼−λZ2 0∞djZkj dΩn−1jkjn−2jS~ðj2kjÞj2eik·ðxA−xBÞ (otfhetheexWpeicgthattimonanvafulunectoinof)nthWencðotm;xm;tu0;taxt0oÞriosfptrhoepfoiretlidonaatlthtoe
× ∞dt1 t1 dt2e−ijkjðt1−t2Þ Rpoeifn.t[s18ðt];],xÞ and ðt0;x0Þ. Namely [see, e.g., Eq. (23) in
−∞ −∞
×½Xðt1−tAÞXðt2−tBÞþXðt1−tBÞXðt2−tAÞ(cid:2): h0ϕˆj½ϕˆðt;xÞ;ϕˆðt0;x0Þ(cid:2)j0ϕˆi¼2iIm½Wnðt;x;t0;x0Þ(cid:2): ð23Þ
ð18Þ
The commutator between field observables (and in
particular, the field commutator) is only supported inside
The spherical symmetry of the smearing allows us to
their respective light cones (this property is known as
performtheintegrationovertheangularvariablesthatappear
microcausality).Therefore,forspacelikeseparatedregions
inEqs.(17)and(18).Onthe one hand,theintegralsinthe
the imaginary part of the Wightman function as given by
local term (8) straightforwardly evaluate to the surface of
the (n−1)-sphere, while on the other handtheintegrals in Eq. (23) vanishes and the nonlocal term described by
Eq. (22) is real. This means, from (22), that M is real.
thenonlocalterm(9)areslightlylessstraightforwardandare
Armed with this information about M, we look at it in
computed explicitly in AppendixesA and B. The resulting
expressions for L andM are the form (20). Since the hypergeometric functions in
Eq. (20) are real and M itself is real, we conclude that
L¼λ2Z0∞djkjjkjn−2jS~ðjkjÞj2Γðnπ=n22ÞRe½T0ðjkj;TÞ(cid:2); ð19Þ oaswlnloliytwchsthiunesgrteoaanlredppalraratcdeoiafTllTΔytΔðjtskðyjjmk;Tjm;ÞTebÞtryiccRonest½mrTibeΔuatrtðeijnksgj;toTuÞMn(cid:2)dfeo.rrTathnhiyes
condition that the detectors are spacelike separated.
Z Continuingwiththeproof,weshowinAppendixCthat
M¼−λ2 ∞djkjjkjn−2jS~ðjkjÞj2 πn2
0(cid:3) (cid:4)Γðn=2Þ Re½TΔtðjkj;TÞ(cid:2)¼2πjX~ðjkjÞj2cosðjkjΔtÞ
×0F1 n2;−jkj2jxA4−xBj2 TΔtðjkj;TÞ; ð20Þ ¼Re½T0ðjkj;TÞ(cid:2)cosðjkjΔtÞ: ð24Þ
As the confluent hypergeometric limit function satisfies
where 0F1ða;zÞ is the confluent hypergeometric limit (see 10.14.4 and 10.16.9 in Ref. [26])
function [26],
Z Z j0F1ðα;−x2Þj≤1; ð25Þ
∞ ∞
TΔtðjkj;TÞ¼ dt1 dt2θðt1−t2Þe−ijkjðt1−t2Þ we obtain
−∞ −∞
(cid:5)Z
×½Xðt1ÞXðt2−ΔtÞþXðt1−ΔtÞXðt2Þ(cid:2); jMj¼λ2(cid:5)(cid:5)(cid:5) ∞djkjjkjn−2jS~ðjkjÞj2 2πn2þ1
ð21Þ 0 Γðn=2Þ
(cid:3) (cid:4) (cid:5)
and Δt¼t −t . ×0F1 n2;−jkj2jxA4−xBj2 jX~ðjkjÞj2cosðjkjΔtÞ(cid:5)(cid:5)(cid:5)
B A Z
Acrucialobservationtoprovethatgaplessdetectorswith
overlappinginteractiontimeintervalscannotharvestspace- ≤λ2 ∞djkjjkjn−2jS~ðjkjÞj2 2πn2þ1 jX~ðjkjÞj2 ¼L:
like entanglement is that only the real part of the function 0 Γðn=2Þ
TΔt contributestotheevaluationofMwhenthedetectors ð26Þ
are spacelike separated. To see this, we return to the
expression of M in terms of the Wightman function in This implies that Nð2Þ ¼0 for gapless, spacelike sep-
Eq. (9), which for gapless detectors is arated spherically symmetric detectors for any zero or
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nonzerooverlapbetweenthetimeintervalsofinteractionof where e is the electron charge, the matrix Wðt1;x1;t2;x2Þ
each detector with the field. Hence, combining the results istheistheWightmantensoroftheelectricfieldoperatorEˆ
of this section with those of Sec. III, we see that gapless whose components are given by
detectors with finite, spherically symmetric smearings
interactingforafinitetimewiththefieldcanneverharvest ½W(cid:2)ij ¼Wijðt;x;t0;x0Þ¼h0EˆjEˆiðt;xÞEˆjðt0;x0Þj0Eˆi; ð29Þ
entanglement from spacelike separated regions, independ-
entlyofthespecificwayofinteractingwiththefieldortheir and the vectors Sνt and S(cid:3)νt are respectively the transpose
shape.This,ofcourse,includesasaparticularcasetheuse and Hermitian conjugate of the vector Sν (the spatial
of pointlike detectors, which is the case that is used most
smearing vector) which relates to the ground and excited
often in the literature.
wave functions by
V. ASYVMEMRYETRREILCECVAASNET: TNHOENSRPEHAELRIISCTAICLLY SνðxÞ¼ψ(cid:3)eνðxÞxψgνðxÞ ð30Þ
LIGHT-MATTER INTERACTION [note that this smearing vector is called FνðxÞ in [12]].
Inthissectionweconsidertherealisticcaseofthelight- In the case of atomic switching functions that do not
matter interaction. Namely, the interaction of an atomic overlap, the reasoning used in Sec. III applies: the first
electroninahydrogenlikeatomwiththevacuumstateofan summand of Eq. (28) evaluates to 0 and in the second
electromagneticfieldthroughadipolarcoupling.Ourstudy summand the integrals in time denest, making jMEMj
becomes particularly relevant for transitions between orbi- upper-boundedbyLEM,regardlessofthesmearingvectors
tals of the same quantum number n, which have zero being compactly supported or not. This means that non-
energygap.Inthesimplifiedcaseofpointlikeatoms,there simultaneously interacting hydrogenlike atoms cannot
was numerical evidence that gapless detectors do not harvest any entanglement from the vacuum at all through
allow for entanglement harvesting in spacelike separated transitions of zero energy.
regions [21]. When there is some overlap between the intervals of
Beyond that simplification, the general study of atom- interaction of each individual atom with the field, the
light interactions for arbitrary finite energy gaps was arguments used in Sec. IV would also apply for hypo-
reported in Ref. [12], where the fully featured shape of theticalcompactlysupportedatoms:inthiscase,andsince
the atomic wave functions was taken into account. In the electric field also satisfies microcausality (the electric
particular, it was shown in Ref. [12] that entanglement fieldcommutatoris0forspacelikeseparatedevents),MEM
harvesting from both electromagnetic and scalar fields would also be real for spacelike separations between the
exhibitsthesamequalitativefeaturesdespitethedifference compactly supported atoms. Then, without assuming
in the setups. We now focus on the case of two fully spherical symmetry of the smearing functions, the hyper-
featured hydrogenlike atoms when an energy degenerate geometricfunctioninEq.(20)isreplacedbycombinations
transitionisusedtoharvestentanglementfromthevacuum of spherical Bessel functions. For example, for the zero-
state of the electromagnetic field. energy transition 2s→2p Eqs. (27) and (28) read
For a pair of identical atoms, the negativity takes a
Z
similar form as in the scalar case. Namely, the negativity 3a2 ∞ ða2jkj2−1Þ2
LacqaunidrendoanfltoecrailntMeractetiromnsisbgecivoemnebynoEwq.[(s7e)e.wEhqesre. (th3e1)loacnadl LEM ¼e22π02 0 djkjjkj3ða200jkj2þ1Þ8Re½T0ðjkj;TÞ(cid:2);
(32) in Ref. [12]] ð31Þ
Z Z
LEM ¼e2 ∞dt1 ∞dt2Xðt1ÞXðt2Þ MEM ¼−e23a20cosϑ
Z−∞ Z −∞ Z 2π2
× d3x1 d3x2S(cid:3)tðx2ÞWðt2;x2;t1;x1ÞSðx1Þ; × 0∞djkjjkj3ððaa2020jjkkjj22þ−11ÞÞ28TΔtðjkj;TÞ
Z Z Z Z ð27Þ ×½j0ðjkjjx −x jÞþj2ðjkjjx −x jÞ(cid:2); ð32Þ
A B A B
MEM¼−e2 −∞∞dt1 −∞t1 dt2 d3x1 d3x2 whereϑistheangleoftheaxisofsymmetryofatomB’s2p
×½Xðt1−t ÞXðt2−t Þ orbital with respect to atom A’s orbital.
A B Note that, despite the fact that the hypergeometric
×S tðx1−x ÞWðt1;x1;t2;x2ÞS ðx2−x Þ function appearing in the scalar nonlocal term [see
A A B B
þXðt1−t ÞXðt2−t Þ Eq. (20)] has been substituted by a combination of
B A
spherical Bessel functions, this combination can still be
×SBtðx1−xBÞWðt1;x1;t2;x2ÞSAðx2−xAÞ(cid:2); ð28Þ upper-bounded by 1 (and the same occurs in the gapped
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DEGENERATE DETECTORS ARE UNABLE TO HARVEST … PHYSICAL REVIEW D 95, 105009 (2017)
casestudiedinRef.[12]).Thismeansthatalsointhiscase FortheswitchingfunctionspecifiedbyEq.(33)thelocal
the magnitude of the nonlocal term jMEMj is upper- and nonlocal terms Eqs. (8) and (9) read
bounded by the local term LEM, which means that no Z (cid:5)Z (cid:5)
evnactaunugmlemweintht cdanegebneerhaatreveasttoemdifcromprotbheeseilfecttrhoemiragrnadeitiacl L¼λ2η2 dnkjS~2ðjkkÞjj2(cid:5)(cid:5)(cid:5) −∞∞dtδðtÞeiðjkjþΩÞt(cid:5)(cid:5)(cid:5)2
functions were compactly supported. This argument con- Z
jS~ðkÞj2
tains, as a special case, that studied numerically in ¼λ2η2 dnk ; ð34Þ
Refs.[20,21]wheretheatomswereassumedtobepointlike. 2jkj
One must however note that the atomic wave functions
Z
ofanelectroninahydrogenlikeatomdonothavecompact jS~ðkÞj2
support.Instead,theradialwavefunctionsdecayexponen- M¼−λ2η2 dnk 2jkj eik·ðxA−xBÞ
Z Z
tiallywiththedistancetotheatomiccenterofmass.Forthis
reason, one may be tempted to argue that the atoms can × ∞dt1 t1 dt2e−ijkjðt1−t2ÞeiΩðt1þt2Þ
never be placed in spacelike-separated regions due to the −∞ −∞
always-existent overlap of their atomic wave functions, ×½δðt1−t Þδðt2−t Þþδðt1−t Þδðt2−t Þ(cid:2): ð35Þ
A B B A
which make the imaginary part of the Wightman function
contribute, albeit suppressed by a factor of the overlap In the case of nonsimultaneous switchings t ≠t , the
A B
between the wave functions. Nevertheless, for the imple- argumentinSec.IIIusedforevaluatingthetimeintegralsof
mentation proposed in Ref. [20] with two quantum dots thenonlocalterm(35)applies:ifdetectorBisswitchedon
separatedbyadistanceofd¼10 nm≈190a0(wherea0is after detector A, the first summand evaluates to 0 while in
theBohrradius),tRheoverlapbetweenthewavefunctionsis the second the integrals denest. Integration over the time
on the order of djxjjxj2ψAðjxjÞψBðjxjÞ≈e−190≈10−83, variables then leads to the expression
which is definitely negligible as compared with the Z
ednisttaanngcleemsceanletst(hfaotragadpeptaeidledatsotumdsyoconuhlodwhtahrevensotncaotmthpoascet M¼−λ2η2 dnkjS~2ðjkkÞjj2eik·ðxA−xBÞe−ijkjðtB−tAÞeiΩðtBþtAÞ:
supportcannotberesponsibleforentanglementharvesting,
ð36Þ
checkSec.IVCofRef.[12]).IntheexamplesofRef.[12]
the atoms were declared effectively spacelike when sepa- The magnitude of this expression satisfies
ratedby104Bohrradiiandtheirinteraction(withGaussian
tshwaintcnhiinneg)timweassthsheotirmt eensocualgehofsodutrhaattio1n0o4fat0h=ecinwtearsacmtioorne. jMj¼(cid:5)(cid:5)(cid:5)(cid:5)λ2η2Z dnkjS~2ðjkkÞjj2eik·ðxA−xBÞe−ijkjðtB−tAÞeiΩðtBþtAÞ(cid:5)(cid:5)(cid:5)(cid:5)
Inthatexample,theoverlapbetweenthewavefunctionsof
thetwoatomswasoftheorderof10−4343,whichiseffectively Z jS~ðkÞj2(cid:5)(cid:5) (cid:5)(cid:5)
0 for all practical purposes. Since the harvesting of entan- ≤λ2η2 dnk 2jkj (cid:5)eik·ðxA−xBÞe−ijkjðtB−tAÞeiΩðtBþtAÞ(cid:5)
glement due to the atomic wave function overlap is negli- Z
jS~ðkÞj2
gible, our results carry over to the light-atom interaction. ¼λ2η2 dnk ¼L ð37Þ
2jkj
VI. INSTANTANEOUS SWITCHINGS
so again in this case Nð2Þ ¼0, regardless of the specific
Finally,letusexplorethecaseinwhichgappeddetectors shape of the detectors, their relative distance, and addi-
interactforaninfinitesimalamountoftimewiththefieldbut tionally now the energy gap.
with an infinite strength. This case is relevant due to its Whentheindividualinteractionsofthedetectorswiththe
similarities with a gapless detector case: In the case of a fieldcoincide,i.e.Δt¼0,Eq.(36)becomesmathematically
delta switching, during the time of interaction the free ambiguous,duetotheargumentofaDiracdeltacoinciding
dynamics of the detectors is effectively halted (roughly with a limit of the integral. For sufficiently symmetric
speaking the free Hamiltonian becomes negligible with regularizations of theDiracdeltas, we however have
respecttothedeltastrengthoftheinteractionHamiltonian).
Z Z
Tfuhnicstioinntseraction is modeled by Dirac delta switching 2 ∞dt1 t1 dt2e−ijkjðt1−t2ÞeiΩðt1þt2Þδðt1−t Þδðt2−t Þ
A A
−∞ −∞
XνðtÞ¼ηδðt−tνÞ; ð33Þ ¼e2iΩtA; ð38Þ
where η is a constant with dimensions of time. This and we give in Appendix D two examples of such regula-
switching allows us to obtain analytical closed-form rizations.Withtheinterpretation(38),thenonlocaltermM
expressions even for Ω≠0. becomes
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POZAS-KERSTJENS, LOUKO, and MARTÍN-MARTÍNEZ PHYSICAL REVIEW D 95, 105009 (2017)
Z
jS~ðkÞj2 Therefore,we attribute the inabilityof gaplessdetectors
M¼−λ2e2iΩtA dnk 2jkj eik·ðxA−xBÞ: ð39Þ to harvest entanglement to the fact that, as shown in
previous studies [6,12], the energy gap has a protective
Again,themagnitudeofthistermisboundedfromabove role that shields from local noise allowing for nonlocal
by the local term L, so Nð2Þ ¼0 and entanglement excitations that entangle the detectors. In the absence of a
harvesting is not possible in the limit when the switching gapbetweentheenergylevels,eventhesmoothestswitch-
becomesvery shortandintense,regardlessof theshapeor ings (those that create the smallest amount of local noise)
sizeoftheprobes,theirrelativedistanceor,inthisspecific break the entanglement between the detectors.
case, the size of the gap between the energy levels of the As a last comment, these results may also shed some
detectors. light on studies in the context of creation of entanglement
via interaction with a common heat bath through dipolar
VII. SUMMARY AND DISCUSSION couplings[20,21].Inthesestudies,theauthorsawnumeri-
callythatonlywhenoneprobeisdeepinsidethelightcone
In the context of entanglement harvesting [3,5,6] and
of the other (they are in timelike separation) entanglement
creation of entanglement by interaction with a common
can be extracted from the bath to the (gapless) detectors.
heat bath [20,21], we have studied whether degenerate
identicaltwo-levelquantumsystemscouplinglinearlywith
ACKNOWLEDGMENTS
the vacuum state of a scalar field in flat spacetime are
capableofharvestingtheentanglementpresentinspacelike The authors thank Daniel Braun for the interesting
separated regions of the field. We haveestablished several conversations that motivated this work. The work of
results within leading order in perturbation theory. A.P.-K. is supported by Fundación Obra Social “la
First, we have proved that if the time intervals of Caixa,” Spanish Ministerio de Economía y Competitividad
interaction between each individual detector and the field (QIBEQI Grant No. FIS2016-80773-P and Severo Ochoa
haveno overlap the detectors can neverbecome entangled Grant No. SEV-2015-0522), Fundació Privada Cellex and
through their interaction with the field. This result is the Generalitat de Catalunya (SGR875 and CERCA pro-
independent of the shape or size of the detectors (which gram).TheworkofJ.L.issupportedinpartbytheScience
canbeevennotcompactlysupportedinafiniteregion),the and Technology Facilities Council (Theory Consolidated
duration of the interaction or the separation between the Grant No. ST/J000388/1). J.L. thanks the Institute for
probes (timelike, lightlike or spacelike). Quantum Computing at the University of Waterloo for
Second, under the additional assumption of spherical hospitality. The work of E.M.-M. is supported by the
symmetry of the detectors’ smearing functions we have National Sciences and Engineering Research Council of
shown that, although the detectors can harvest timelike Canada through the Discovery program. E.M.-M. also
entanglement, for arbitrary spacelike separations entangle- thankfully acknowledges the funding of his Ontario Early
ment harvesting is impossible in any situation where the Research Award.
time of interaction with the field is finite.
Third, we have shown that considering realistic light-
APPENDIX A: INTEGRATION OVER ANGULAR
matterinteractions,andinparticulartheinteractionoffully
VARIABLES OF THE NONLOCAL TERM
featured hydrogenlike atoms interacting with the electro-
magneticfield,thesamephenomenologyoccurs:asthegap In this appendix we perform the integrations in the
between the atomic levels is scaled down to 0 the gapless generalizedsolidanglevariablesofthevectorkthatappear
detectorsbecomeunabletoharvestspacelikeentanglement in the nonlocal term M in Eq. (18), namely
from the field, and only when the time intervals of the
Z
individualatomicinteractionswiththefieldoverlapcanthe
atoms have a chance of harvesting timelike and lightlike dΩn−1eik·ðxA−xBÞ; ðA1Þ
entanglement.
Finally,wehavealsoshownthatdetectorscoupledtothe
to compare the result to the corresponding integrals in the
field through a deltalike coupling (short and intense local term L, which evaluate to the area of the (n−1)-
coupling strength) are also completely unable to become
sphere, i.e.,
entangled through their interaction with the field in time-
like, spacelike or lightlike regimes at leading order in Z
2πn
panerdt,uirnbathtiiosncatshee,oervye,nreifgtahredylehssavoefatfhineiitresepnaetrigayl gsmape.aTrihnigs dΩn−1 ¼An−1 ¼Γðn=22Þ: ðA2Þ
shouldnotbesurprisingsincethedeltacouplingresembles
a case where the detectors’ internal dynamics are frozen Inndimensionstherearen−1angularvariables,oneof
during the time of interaction, as is the case of zero-gap which (the polar angle ϕn−1) has the range ½0;2πÞ and the
detectors. rest (the azimuthal angles ϕ1;…ϕn−2) have range ½0;π(cid:2).
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DEGENERATE DETECTORS ARE UNABLE TO HARVEST … PHYSICAL REVIEW D 95, 105009 (2017)
Z Z
The solid angle element is therefore ∞ ∞
TLðjkj;TÞ¼ dt1 dt2e−ijkjðt1−t2ÞXðt1ÞXðt2Þ
−∞ −∞
dΩn−1 ¼sinn−2ðϕ1Þsinn−3ðϕ2Þ…sinðϕn−2Þ ¼2πjX~ðjkjÞj2; ðB1Þ
×dϕ1dϕ2…dϕn−1: ðA3Þ
wherethetildedenotestheFouriertransforminthenotation
Let us then begin with the particularly simple case of of Eq. (11). We show that
n¼2forillustration.Choosingthexaxisoftheintegration
frametoalignwithx −x ,theintegraleasilyevaluatesto TLðjkj;TÞ¼Re½T0ðjkj;TÞ(cid:2); ðB2Þ
A B
(see 10.9.4 and 10.16.9 in Ref. [26])
Z 2π where TΔtðjkj;TÞ is given by Eq. (21).
dϕ1eijkjjxA−xBjcosϕ1 ¼2πJ0ðjkjjx −x jÞ Tobeginwith,weseethatforΔt¼0thetwosummands
A B
0 (cid:3) (cid:4) of Eq. (21) coincide, leading to
ðjkjjx −x jÞ2 Z Z
¼2π0F1 1;− A4 B ; T0ðjkj;TÞ¼2 ∞dt1 ∞dt2e−ijkjðt1−t2Þ
−∞ −∞
ðA4Þ
×Xðt1ÞXðt2Þθðt1−t2Þ; ðB3Þ
where F is the confluent hypergeometric limit function.
0 1
In fact, the general case is not too difficult to compute where θðxÞ is the Heaviside step function. Using the
either.Innspatialdimensions,onecanchoosetoplaceone identity 1¼θðxÞþθð−xÞ and performing the change of
of the axes of the integration frame aligned with xA−xB, variablest1 ↔t2inthesecondsummandtheresultfollows,
which simplifies the scalar product in the exponential to,
Z Z
ifnotreginrasltsanecvea,lujaktjejxtAo−xBjcosðϕ1Þ. With this choice, the TLðjkj;TÞ¼ ∞dt1 ∞dt2e−ijkjðt1−t2Þ
−∞ −∞
Z dΩn−1eik·ðxA−xBÞ ¼2πZmYn−¼22pΓðffiπffiffimΓþ2ð1m2ÞÞ ¼×ZX∞ðdt1t1ÞXZð∞t2Þd½tθ2ðXt1ð−t1ÞtX2Þðþt2Þθðt2−t1Þ(cid:2)
π −∞ −∞
× dϕ1sinn−2ðϕ1ÞeijkjjxA−xBjcosðϕ1Þ ×θðt1−t2Þðe−ijkjðt1−t2Þþeijkjðt1−t2ÞÞ
0 Z Z
¼2π(cid:3)Yn−2pffiπffiffiΓðm2Þ(cid:4)pffiπffiffiΓðn−21Þ ¼ ∞dt1 ∞dt2Xðt1ÞXðt2Þ
Γðmþ1Þ ΓðnÞ −∞ −∞
m(cid:3)¼2n ð2jkjjx −x jÞ22(cid:4) ×θðt1−t2Þ2Reðe−ijkjðt1−t2ÞÞ
×0F1 2;− A4 B ¼Re½T0ðjkj;TÞ(cid:2): ðB4Þ
(cid:3) (cid:4)
2πn2 n ðjkjjx −x jÞ2
¼Γðn=2Þ0F1 2;− A4 B ;
APPENDIX C: EVALUATION OF ReT
ðA5Þ Δt
InthisappendixweshowthatEq.(21)leadstoEq.(24).
usingagain10.9.4and10.16.9inRef.[26],andnotingthat
Starting from Eq. (21) and changing variables by
Yl t1 ¼t2þs gives
f ≔1 for l<k; ðA6Þ Z Z
i ∞ ∞
i¼k TΔtðjkj;TÞ¼ dt2 dse−ijkjs
−∞ 0
and ×½Xðt2þsÞXðt2−ΔtÞ
Yn−2pffiπffiffiΓðmÞ (1 n≤3 þXðt2þs−ΔtÞXðt2Þ(cid:2): ðC1Þ
2
m¼2 Γðmþ21Þ ¼ Γπðnn−2−231Þ n≥3: ðA7Þ Changing variables in the first summand by μ¼t2þs
and renaming μ¼t2 in the second summand, we obtain
APPENDIX B: TIME INTEGRALS Z Z
∞ ∞
IN THE OVERLAPPING CASE TΔtðjkj;TÞ¼ dμXðμÞ dse−ijkjs
−∞ 0
In this appendix we examine the time integrals in the
local term Eq. (8), given by ×½Xðμ−s−ΔtÞþXðμþs−ΔtÞ(cid:2): ðC2Þ
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POZAS-KERSTJENS, LOUKO, and MARTÍN-MARTÍNEZ PHYSICAL REVIEW D 95, 105009 (2017)
Taking the real part gives
Z Z
∞ ∞
Re½TΔtðjkj;TÞ(cid:2)¼ dμXðμÞ dscosðjkjsÞ½Xðμ−s−ΔtÞþXðμþs−ΔtÞ(cid:2)
−Z∞ 0Z
1 ∞ ∞
¼ dμXðμÞ dseijkjs½Xðμ−s−ΔtÞþXðμþs−ΔtÞ(cid:2)
2
pffiffi−ffiffiffiffi∞Z −∞
2π ∞
¼ dμXðμÞ½eijkjðμ−ΔtÞ½X~ðjkjÞ(cid:2)(cid:3)þeijkjð−μþΔtÞX~ðjkjÞ(cid:2)
2
−∞
¼π½jX~ðjkjÞj2e−ijkjΔtþjX~ðjkjÞj2eijkjΔt(cid:2)
¼2πjX~ðjkjÞj2cosðjkjΔtÞ; ðC3Þ
where the second equality uses the evenness of Xðμ−s−ΔtÞþXðμþs−ΔtÞ in s.
APPENDIX D: REGULARIZATIONS OF INSTANTANEOUS SWITCHING
InthisappendixwepresenttworegularizationsoftheDiracdeltathatare“kink”limitsofswitchingslargelyemployedin
past literature [6] that lead to (38). For notational simplicity, we set t ¼0 and consider the formal expression
A
Z Z
T0ðjkjÞ¼2 ∞dt1 t1 dt2e−ijkjðt1−t2ÞeiΩðt1þt2Þδðt1Þδðt2Þ; ðD1Þ
−∞ −∞
showing that each of the regularizations gives for T0ðjkjÞ the value unity.
1. Top-hat regularization
First, we regard the Dirac delta as a limit of the top-hat function,
(
1 1 if t∈½−ϵ;ϵ(cid:2)
δðtÞ¼ lim 2 2 : ðD2Þ
ϵ→0þϵ 0 otherwise
Then
Z Z
T0ðjkjÞ¼2 ∞dt1 t1 dt2e−ijkjðt1−t2ÞeiΩðt1þt2Þδðt1Þδðt2Þ
−∞ −∞Z Z
¼2ϵl→im0þϵl0→im0þϵ1ϵ0 −ϵϵ==22dt1 −tϵ10=2dt2e−ijkjðt1−t2ÞeiΩðt1þt2Þ
Z
¼2ϵl→im0þϵl0→im0þϵiϵe0−ð12jikϵ0jðjþkjþΩΩÞÞ −ϵϵ==22dt1e−iðjkj−ΩÞt1ð1−e12iðjkjþΩÞð2t1þϵ0ÞÞ
(cid:7) (cid:8)
¼ lim lim 2i 2e−12iϵ0ðjkjþΩÞsin½12ϵðjkj−ΩÞ(cid:2)− sinðΩϵÞ
ϵ→0þϵ0→0þϵϵ0 jkj2−Ω2 jkjΩþΩ2
2i ð−iϵϵ0Þ
¼ lim lim ¼1: ðD3Þ
ϵ→0þϵ0→0þϵϵ0 2
2. Gaussian regularization
Second, we regard the Dirac delta as a limit of the Gaussian function,
δðtÞ¼ lim 1pffiffiffie−4tϵ22: ðD4Þ
2ϵ π
ϵ→0þ
Then
105009-10