Entanglement dynamics of two nitrogen vacancy centers coupled by a nanomechanical resonator

0.89 MB·

by  Z. Toklikishvili

Entanglement dynamics oftwo nitrogenvacancy centers coupled by ananomechanical resonator
Z.Toklikishvili,1L.Chotorlishvili,2S.K.Mishra,3 S.Stagraczynski,2M.Schüler,2A.R.P.Rau,4 andJ.Berakdar2
1Department of Physics, TbilisiStateUniversity, Chavchavadze av. 3, 0128, Tbilisi, Georgia
2Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06120 Halle, Germany
3DepartmentofPhysics,IndianInstituteofTechnology, BanarasHinduUniversity,Varanasi-221005, India
4DepartmentofPhysicsandAstronomy,LouisianaStateUniversity,BatonRouge, Louisiana70803-4001, USA
In this paper we study the time evolution of the entanglement between two remote NV Centers (nitrogen
vacancy in diamond) connected by adual-mode nanomechanical resonator withmagnetic tipson bothsides.
Calculatingthenegativityasameasurefortheentanglement,wefindthattheentanglementbetweentwospins
oscillateswithtimeand canbemanipulated by varying theparametersof thesystem. Weobserved thephe-
nomenonofasuddendeathandtheperiodicrevivalsofentanglementintime. Forthestudyofquantumdeco-
7
herence,weimplementaLindbladmasterequation.Inspiteofitscomplexity,themodelisanalyticallysolvable
1
0 underfairlyreasonableassumptions,andshowsthatthedecoherenceinfluencestheentanglement, thesudden
2 death,andtherevivalsintime.

n
a I. INTRODUCTION time-varyingmagneticfieldisasuperpositionoftwodifferent
J
frequencies. Such a magnetic field connects the two remote
8
NV centers and causes an indirect interaction between them
1 Coherent steering of the dynamics of quantum systems
throughananomechanicalresonator. Thisinteractioninvolve
has always been a subject of intense research due to its
all the three levels of the NV centers. In previous studies,
] wide spread applications in devices and setups where quan-
h the"dark"superpositionstatewasdecoupled.Usingthedual-
p tum coherence is relevant. Notable examples are nano- moderesonatorallowconsideringthecasewhenboth"bright"
- electromechanical devices1,2, quantum resonators3–5, two-
nt and three-level quantum systems6–10, and confined cold and"dark"superpositionstatesareinvolvedinthedynamics.
a atoms11–15. In recent years, much attention has been paid
We will demonstrate that in spite of the complexity, with
u to nitrogen-vacancy (NV) impurities in diamonds and to
q realisticassumptions,thismodelisanalyticallysolvableeven
the possibilities of their usage in quantum computing and
[ in the presenceof quantumdecoherence. The paper is orga-
entanglement16–27.Thesedefectscanbeconsideredasathree-
nizedasfollows: InsectionII,westudythedynamicsofthe
1 levelsystem or spin triplets(S = 1). The advantageofNV-
dual-modecantilever, in section IIIwe presentan exactana-
v basedsystemsisthattheyhavealongcoherencetimeevenat
1 room temperaturesand they are easily manipulatedby light, lytical solution for the case of one and two NV centers. We
7 derivethe Hamiltonianofthe non-directinteractionbetween
electricormagneticfields.
0 NVcentersmediatedbythedual-modecantilever. Insection
5 Recent developments in fabrication technologies of IV, we study the time dependence of the system’s entangle-
0 nanomechanical resonators made it possible to implement ment,andinthelastsectionwestudytheproblemofquantum
1. the coherent connection between the motion of the magne- decoherenceusingaLindbladmasterequation.
0 tized resonator tip and the individual point defect centers in
7 diamond19. The resonators also allow to link two NV cen-
1 ters of diamonds.20,26 Previousworksstudied mainlythe en-
v: tanglementbetweentwoNVcenters,whichareconnectedby
i a resonator in vibration. The resonator is assumed to have
X
only one fundamental frequency of vibration and only two
r statesoftheNVcenterspintripletsareentangledwiththeres-
a
onator. Moderntechnologiesallow producingnanomechani-
calresonatorswithtwoorseveralfundamentalfrequenciesof II. DYNAMICSOFTHEDUAL-MODERESONATOR
vibrations28–30. Thedual-moderesonatorlinkallowsthepar-
ticipation of all the three spin states of NV centers. In this
paper,westudythequantumcorrelationbetweentworemote
A prototype model of a nanomechanical resonator con-
NV Centersthrougha dual-modenanomechanicalresonator.
nected to two NV centers of diamond is shown in Fig. 1.
Twomagnetictipsareattachedtobothsidesoftheresonator,
A smallhigh-frequencycantileveris attachedatthe endofa
whichisplacedsymmetricallybetweentwoNVcenters.
largerlow-frequencycantilever,whichinturnisattachedtoa
The physical set-up consists of two cantilevers in series, base. Two magnetic tips are attached to the small cantilever
both excited at their effective resonant frequencies to pro- and two spin-one systems (NV centers) denoted by ‘1’ and
duce a tip response as a superposition of a low- and a high- ‘2’areplacedatdistancesh andh fromtherespectivemag-
1 2
frequencyoscillationstate. Sincetheresonatorisattachedto netic tips. The model representsa union of two consecutive
thetopoftwomagnetictips,thevibrationcausesatimevary- resonators. A mechanical analogue in terms of springs and
ing magnetic field. The vibrating resonator tip can be con- massesisshowninFig.2. Theequationofmotionoftworods
sidered as the superposition of two oscillations, so that the inthemechanicalmodelcanberepresentedbythefollowing
2
spin1
Physical system
h1
k1 , Q1 k2 , Q2
h2
base
spin2
z
x
y
FIG.1.(coloronline)Ananomechanicalresonator,connectedtotwo FIG. 2. (color online) A mechanical model of the system, which
NVvacancycentersofdiamond. Twomagnetictipsareattachedat representsacombinationofspringsandmasses.
theendoftheresonator,whichoscillatesinthedirectionofthezaxis.
h1andh2aredistances(approximately25nm)fromthemagnetictip
tothefirstandsecondNVspin. tions: 2πν1−0 ω , 2πν2−0 ω . Giventheseestimateswe
Q1 ≪ 1 Q2 ≪ 2
canrewriteEq.(1)as
equations29:
m z¨ = (k +k )z +k z ,
1 1 1 2 1 2 2
−
d2z (t) m z¨ = k z +k z , (2)
m1 dt12 =−k1[z1(t)−z0(t)]+k2[z2(t)−z1(t)] 2 2 − 2 2 2 1
whichcanfurtherbesimplifiedto
2πν dz (t)
1 0 1
m1 − ,
− Q1 dt z¨ = ω2z +ω2 z ,
d2z2(t) 2πν2 0dz2(t) 1z¨ =− 1ω21(z +21z2), (3)
m2 dt2 =−k2[z2(t)−z1(t)]−m2 Q− dt 2 − 2 2 1
2
+Fts[z2(t)], (1) where ω1 = (k1+k2)/m1, ω21 = k2/m1 and ω2 =
k /m .
where m and m are the effective masses of the first and 2 2 p p
1 2 Thegeneralsolutionofthe system ofequationsEq. 3has
thesecondrod,k1 andk2arespringconstants(vibratingrods pthefollowingform:
obey Hooke’s law). z (t) and z (t) are the coordinates of
1 2
the first and second rod tips, Q and Q are quality factors,
1 2 z =Acos(Ωt+φ),
1
ν and ν are free-resonantfrequencies of the first and
th1e−0second2ro−d0. F [z (t)] is the externalforceacting onthe z2 =Bcos(Ωt+φ). (4)
ts 2
second rod tip. The system can be excited by flapping the
HereA,B andΩareconstantsandcanbecalculatedbysub-
stembase,z isthelocationofthefirststembaseattimet.
0 stitutingEq.4intoEq.3as
We will study the dynamical equation Eq. 1 by employ-
ing severalassumptions: First, let usassume that the base is (ω2 Ω2)A ω2 B =0,
stopped at z0 = 0 and there is no magnetic interaction be- ω12A−+(ω2−Ω221)B =0. (5)
tween magnetic tips and spins: F [z (t)] = 0. Before pro- − 2 2 −
ts 2
ceeding further, it is useful to estimate the range of realistic Thenon-trivialsolutionforΩ,obtainedbysolvingtheabove
parameters for the cantilever26. A typical cantilever fabri- equations,is
cated out of Si(100) has a Young’s modulus Y = 130 GPa
andadensityρ=2.33 103kg/m3. Inthedual-modehybrid ω2+ω2 (ω2 ω2)2+4ω2ω2
× Ω2 = 1 2 ± 1− 2 2 21, (6)
cantilever (see Fig. 1), the length of the two different parts 1,2 2
is L 15 103nm, L 9 103 nm, width w =300nm, p
1 2 1
w2 =2≈00nm· andthicknes≈sd1·=30nm,d2 =20nm,masses where Ω1 and Ω2 are the system’s own frequencies. They
m1 = 3.5 10−16kgandm2 = 10−16kg. Usingequations differfromtheresonancefrequenciesω1andω2ofthesystem
× becauseofthecouplingtermω ω inEq.6.
2πν1−0 = 1.82 Y3dρ212L121 and 2πν2−0 = 1.82 Yρd222L122, TheratioofamplitudesAan2dB21atΩ =Ω1canbewritten
we therefore obtqain the frequencies πν1 0 = 0q.54 MHz, as
πν =1.7MHz. Theestimatedspringc−onstantsare: k =
2 0 1
Y×wL−131×d31 =3·10−4kg/s2,k2 = Y×wL232×d32 =2·10−4kg/s2. BA = ω12−ω22+ (ω212ω−2 ω22)2+4ω22ω221 ≡κ1,
The quality factor is large19, Q1,2 105, and the frequen- (cid:18) (cid:19)Ω1 p 21
≈ (7)
ciesareω = k1+k2 = 1.2MHz,ω = k2 = 1.4MHz.
1 m1 2 m2
Thus, by desigqning geometricalcharacterisqtics of the hybrid where κ1 is completelydeterminedby the parametersof the
dual cantilever, one can easily achieve the following condi- system and does not depend on the initial conditions. It is
3
calledthedistributioncoefficientamplitudeforΩ1. Similarly |+1> δ+ 3 ωe |e>
wefindthedistributioncoefficientforΩ as
2 ~BS
0 z
BA = ω12−ω22− (ω212ω−2 ω22)2+4ω22ω221 ≡κ2, |-1> Ω+ δ- 2 ωd |d1>
(cid:18) (cid:19)Ω2 p 21 ω
(8) Ω 0
-
a) |0> b) 1 ω |g>
andthegeneralsolutionofthecoupledequationsEq.4canbe g
writtenas:
FIG.3.(coloronline)(a)Theconfigurationoftheenergylevels.Ω±
stand for theRabi frequencies between the ground and theexcited
z =A cos(Ω t+φ )+A cos(Ω t+φ ),
1 1 1 1 2 2 2
levels(0 , 1 ) and(0 , 1 ). δ± denotethedetuning between
z =κ A cos(Ω t+φ )+κ A cos(Ω t+φ ). (9) | i |− i | i | i
2 1 1 1 1 2 2 2 2 themicrowavefrequencyω0andthetransitionfrequencies. (b)The
transitionfromtheground statetothebright andthe darkstatesis
This means that the oscillations of each stem tip are the su-
shown.
perpositionof two harmonicoscillationswith frequenciesof
vibrationgivenbyEq. 6.
Letusdefinetwonormalcoordinatesxandysuchthatx= expressed as zˆ = a10(aˆ1 +aˆ†1)+a20(aˆ2 +aˆ†2) and the in-
A1cos(Ω1t+φ1)andy =A2cos(Ω2t+φ2).Intermsofthese teraction between the magnetic tip and the spin is expressed
normalcoordinates,thesolutiontakesonasimplerform, as
z1 =x+y, Hsr =h¯(λ1(aˆ1+aˆ†1)+λ2(aˆ2+aˆ†2))Sz, (16)
z =κ x+κ y, (10)
2 1 2
where λ = g µ G a , i = 1,2, the gyromagneticratio
i s B m i0
andΩ1,2canbewrittenas gs 2,andai0 = ¯h/2MiΩi aretheamplitudezero-point
≈
fluctuations. The system is placed in an external magnetic
Ω21,2 = k1m+k+2(κm1,2κ−2 1)2. (11) fieldB0alongthezpdirection.Iftheconditions 2πQν11−0 ≪ω1,
1 2 1,2 2πν2−0 ω donotholdtrue, insteadofEq. (2)oneneeds
Q2 ≪ 2
TheHamiltonianofthesystemcanbewrittenas to solve directly Eq. (1). The difference between Eq. (1)
and Eq. (2) is the weak damping term and this damping of
1
= (m +m κ2)(x˙2+Ω2x2) the cantilever’s oscillation leads to quantum decoherence.
H 2 1 2 1 1 Usually, NV centersare characterizedby a low decoherence
+ 1(m +m κ2)(y˙2+Ω2y2). (12) rate. However, in order to take into account environmental
2 1 2 2 2 effects,insectionVweutilizeaLindbladmasterequationfor
NVcentresandaddresstheproblemofquantumdecoherence
IntroducingthemassesM = m +m κ2 andmomenta
1,2 1 2 1,2 morecompletely.
P =M x˙,P =M y˙,theHamiltoniancanbeexpressedas
x 1 y 2
1 1 1 1
= P2+ M Ω2x2+ P2+ M Ω2y2.(13)
H 2M x 2 1 1 2M y 2 2 2
1 2
III. NVCENTERSCOUPLEDTOADUAL-MODE
In this way, the Hamiltonian of the nanomechanical res- CANTILEVER
onator can be considered as the sum of two noninteract-
ingharmonicoscillators. Thequantum-mechanicalHamilto-
Inthissection,wepresentananalyticalsolutionforthesin-
nian operator can be written by replacing x, y, P and P
x y gleandtwoNVcenterscoupledtothedual-modecantilever.
by quantum-mechanical operators xˆ, yˆ, Pˆx and Pˆy, respec- We derivetheHamiltonianofnon-directinteractionbetween
tively. Afurthertransformationintocreationandannihilation
NVcentersmediatedviathedual-modecantilever.
operators30leadsto
1 1
H=¯hΩ1(aˆ†1aˆ1+ 2)+¯hΩ2(aˆ†2aˆ2+ 2), (14) A. One-spincase
where aˆ aˆ have the usualmeaning. If we neglectthe zero
1 2 The total spin of a nitrogen vacancy center in diamond is
point vibrations, the nanomechanical resonator may be de-
S = 1, with the three spin sub-states, m = 1, 0 and +1
s
scribedbytheHamiltonian: −
being separated from each other by the frequencyω /2π
0
2.88GHz19. AnaddedexternalmagneticfieldB shifts 1≃
Hnr =¯hΩ1aˆ†1aˆ1+¯hΩ2aˆ†2aˆ2. (15) and +1 states(Zeemanshift)proportionallyt0oB0Sz|.−Thei
| i
The magnetic tip movement generates a magnetic field spinpartoftheHamiltonian NV reads
H
B~ G zˆ, whereG isthemagneticfield gradientand
|zˆitsipt|he≃tipmlocation opermator. Since the tip peak vibrationis H = ¯hδ i i + ¯hΩi 0 i +(i 0 , (17)
NV i
asuperpositionoftwoharmonicwaves,thezˆoperatorcanbe − | ih | 2 | ih | | ih |
i= (cid:18) (cid:19)
X± (cid:0) (cid:1)
4
where δ and Ω denote the detunings and the Rabi fre- where
quences±of the tw±o transitions. We consider here the case
2
w∆hΩe(nBth)e),RwabhierfereΩqueinsctiheesRΩa±biartreannsoittieoqnufarle,q(uΩe±ncy=inΩz0er±o H0 =Hnr+ HNi V, (21)
0 0 i=1
X
magnetic field and ∆Ω(B0) = µBB0. In addition, we con- i =¯hω Ri +¯hω Ri +¯hω Ri ,
siderthecasewhenthedetuningsareequal19 δ = δ = δ. HNV g 11 d 22 e 33
+
Theschematicsof thetransitionis shownin Fig. 3(a)−. With and
this assumption, we can calculate the eigenfunctions of the 2
Hamiltonian as d = 1 (Ω 1 Ω +1 ), e = V = i , (22)
| 1i Ω√2 +|− i− −| i | i 0 Hsr
cosθ b + sinθ 0 and g = cosθ 0 sinθ b , where i=1
1 1 X
| i | i | i | i − | i
|b1i= Ω√12(Ω−|−1i+Ω+|+1i)andtan(2θ)=−√2Ω/δ, Hsir =h¯(λg(aˆ1R2i1+aˆ†1R1i2)+λe(aˆ2R3i2+aˆ†2R2i3)).
2arΩe2s=upeΩr2+po+sitΩio2−ns. Iotfsh"boruilgdhbt"enbote=dt(hatst1ate+s|b1+ia1nd)/|d√12i sTphien,sruepseprescctriivpetlsy.iT=he1i,n2tesratacntidonfoorftthheefisprsitnsanwditthhethesemcoangd-
| i | − i | i
and "dark" d = ( 1 + 1 )/√2 states19: b = netic tips of the resonator results in an indirectcoupling be-
1
| i | − i − | i | i
Ω0|bi−∆ΩΩ(B)|di, |d1i = Ω0|di+∆ΩΩ(B)|bi. The correspond- tweTehnetHheamspilitnosn.ianoftheindirectinteractionbetweentheNV
ing eigenfrequenciesare ω = δ, ω = δ+√δ2+2Ω2 and
d − e − 2 spins can be evaluated using the Fröhlich method35(set ¯h =
ωg = −δ−√δ22+2Ω2. Notethatωg <ωd <ωe. 1):
Inthediagonalbasis,theHamiltonianassumesthefollow-
i 0
ingform: = dt[V (t),V (0)] (23)
eff ′ 0 ′ 0
H 2
Z−∞
NV =¯hωg g g +¯hωe e e +¯hωd d1 d1 . (18) and
H | ih | | ih | | ih |
In this diagonal basis, the z-component of the spin is Sz = V0(t)=e−iH0tV0(0)eiH0t. (24)
−Ω+ΩΩ2− sinθ(|d1ihg|+|gihd1|)−cosθ(|d1ihe|+|eihd1|) . Intherotating-waveapproximation,wewriteEqs. (22)-(24)
Sinceωg <ωd <ωewecanrelabeltheenergylevels g , d1 as
(cid:0) | i | (cid:1)i
and e as 1 , 2 , and 3 , respectively. We also introduce
| i | i | i | i ˆ = ˆ +Vˆ, (25)
theoperatorsR = i j , obeyingR R = R δ where eff 0
ij ij kl il kj H H
| ih |
i,j =1,2,3.AfterimplementingtheoperatorsRij andusing where
Eqs.(15),(16),(17),and(18),werewritethetotalHamiltonian
ˆ =α(nˆ (R1 +R2 ) (nˆ +1)(R1 +R2 ))+
ofthesysteminamoreconvenientform H0 1 11 11 − 1 22 22
+β(nˆ (R1 +R2 ) (nˆ +1)(R1 +R2 )), (26)
3 22 22 − 3 33 33
= + + ,
Hs Hnr HNV Hsr and
=¯hω R +¯hω R +¯hω R , (19)
HNV g 11 d 22 e 33 Vˆ = α(R1 R2 +R2 R1 ) β(R1 R2 +R2 R1 ).
Hsr =¯h(λg(aˆ1R21+aˆ†1R12)+λe(aˆ2R32+aˆ†2R23)). − 12 21 12 21 − 23 32 32 23(27)
HereHnristheHamiltonianofthenanomechanicalresonator Intheabovenˆi = aˆ†iaˆi isthemeanphotonnumberoperator,
(see Eq.(15), is the Hamiltonian of the NV-center and α = λ2/∆ , β = λ2/∆ , ∆ = Ω ω , ∆ = Ω +ω .
HNV g 1 e 2 1 1− e 2 2 g
istheHamiltonianofinteractionbetweenthem,obtained In the next section we demonstrate the phenomenon of the
sr
Hintherotatingwaveapproximation:λ = λ1Ω−Ω+sinθ and early-stage disentanglementin the system (the entanglement
g − Ω2
λ = λ2Ω−Ω+cosθ. TheaboveHamiltonianrepresentsagen- suddendeath). Importantlyforaparticularratiobetweenthe
erealizedJayΩn2es–Cummingsmodel7,34. Thelevelsandtransi- parametersα=β,meaningthat Ω1+ √2Ω2+δ2 δ /2=
−
tionsareshowninFig.3(b). tan2 θ Ω2 − √2Ω2+δ2 +δ(cid:0)/2th(cid:0)e entanglement(cid:1)in the
system acquires a persistent value. Here Ω are the fre-
1,2
(cid:0) (cid:1)(cid:0) (cid:0) (cid:1)
quencies of the dual mode cantilever and Ω is the Rabi fre-
quency.Togetherwiththedetuningδandtheangletan(2θ)=
B. Two-spincase
√2Ω/δalltheseparameterscanbecontrolledbythesizeof
−
thecantileverandtheappliedmagneticfield.
Consider the case of two identical spins (NV centers)
placedonthedifferentsidesofaresonatorandcoupledbyit.
LetusassumethatthetwospinshavethesameRabitransition IV. MEASUREOFTHEENTANGLEMENT:NEGATIVITY
frequency,Ωi = Ω ,detuningδi = δ,andinteractioncon-
stantswithth±ereson±atorλig,e = λ±g,e, wherethesuperscripts Theamountofentanglementsharedbetweenthetwospins
i=1,2representthespins. TheHamiltonianforthetwospin canbemeasuredbythe"negativity"whichisdefinedas
casecanbewrittenas
χ χ
N(ρ)= | i|− i, (28)
= +V , (20) 2
H H0 0 Xi
5
whereχ ’s are the eigenvaluesof the partialtransposedden-
i
sitymatrixwithrespecttothespin .Ifχ >0then χ χ = 0.5
1 i i i
| |−
0,however,ifχ < 0, then χ χ = 2χ andthenega-
i i i i
| |− −
tivity 0.4
N(ρ)= χ . (29)
i
− 0.3
χXi<0 ρ)
(
Thetimedependenceofthenegativitycanbecalculatedusing N
0.2
theSchrödingerequation(with¯h=1):
β
d 1
i ψ =Hˆeff ψ , (30) 0.1 2
dt| i | i
3
where the initial state can be considered in its most general 0 4
form
0 1 2 3 4 5 6 7
ψ =a 1 1 +a 1 2 +a 1 3 +a 2 1 +a 2 2
| i 1| i| i 2| i| i 3| i| i 4| i| i 5| i| i t
+a 2 3 +a 3 1 +a 3 2 +a 3 3 . (31)
6 7 8 9
| i| i | i| i | i| i | i| i
Here i j i j , the kets corresponding to the first FIG.4. (coloronline)Negativityasafunctionoftime,whenα=1
and th|eis|eico≡nd|spiin⊗, a|nid a , a a are coefficientsto be kHz and β = 1,2,3,4 kHz. The initial conditions are a2(0) =
1 2 9
determined. ··· a6(0) = 1/√2ora4(0) = a8(0) = 1/√2andalltheothercoeffi-
cientsareequaltozero.Thenegativityvanisheswhenα=1,β =2
UsingtheSchrödingerequation,wecanwrite:
orwhenα = 1,β = 4att = π +πk,k = 0,1,2.... Thetimeis
2
da measuredinmilliseconds.
1
i =2αn a ,
1 1
dt
da
i 2 = αa +βn a αa , Using these coefficients, we can write the density matrix ρ
2 3 2 4
dt − −
withelementsρ = a a . Forthisdensitymatrix,wecan
da nm n ∗m
i 3 =αn a β(n +1)a , easilycalculatethepartialtransposematrixwithrespecttothe
1 3 3 3
dt − spin . The elements of the partial transpose matrix are con-
1
da
4 nectedtotheelementsρ ofthedensitymatrixasfollows:
i = αa +βn a αa , nm
4 3 4 2
dt − −
ida5 = 2α(n +1)a +2βn a , hi1,j2|ρTspin1|k1,l2i≡hk1,j2|ρ|i1,l2i. (34)
1 5 3 5
dt − TheeigenvaluesofthematrixρTspin1 canbecalculatedandthe
da6 negativeeigenvaluesusedinEq.(29)tocalculatethetimede-
i = α(n +1)a βa βa ,
1 6 6 8
dt − − − pendenceofnegativity. ThevalueoftheNegativityishighly
da7 dependenton thechoiceof initialconditions,i.e., thechoice
i =αn a β(n +1)a ,
dt 1 7− 3 7 ofCi,i=1 9.
···
da Letusconsidertheinitialstateofthesystemtobe
8
i = α(n +1)a βa βa ,
1 8 8 6
dt − − −
ψ =a (0)1 2 +a (0)2 3 , (35)
ida9 = 2β(n +1)a . (32) 2 | i| i 6 | i| i
dt − 3 9 with a (0) = a (0) = 1/√2 and a (0) = a (0) =
2 6 1 3
Theequationsfora2anda4aswellasfora6anda8showthe a4(0) = a5(0) = a7(0) = a8(0) = a9(0) = 0.
sameoscillations,whichisobviousasthepairofcoefficients For this choice of initial state, we can calculate the non-
a anda relatestates 1 2 and 2 1 suchthata 1 2 zero elements of the density and the eigenvalues of its par-
2 4 2
a4 2 1 and the pair o|fic|oiefficie|ntis|ai6 and a8 rela|tei|stiat⇔es tial transpose with respect to spin1. The partial transpose
2|3i|aind 3 2 suchthata 2 3 a 3 2 .Thesolutions matrix has only one negative eigenvalue given as χ =
6 8
o|fie|qiuation|si(|32i)are | i| i⇔ | i| i √ρ24ρ42+ρ26ρ62+ρ48ρ84+ρ68ρ86 and the negativity
−
fromEq. (29)comesouttobe
a1 =C1e−2iαn1t,
a2 =C2e−iβn3t+C4ei(2α−βn3)t, N = (36)
a3 =C3e−i(αn1−β(n3+1))t, 6−cos4αt−cos4βt+2cos2(α−β)t+2cos2(α+β)t.
a4 = C2e−iβn3t+C4ei(2α−βn3)t, p 4√2
−
a5 =C5e−2i(α(n1+1)−βn3)t, It should be noted that if α = β then N(ρ) = 0.5. We
a =C eiα(n1+1)t+C ei(2β+α(n1+1))t, obtainthesameresultifwetaketheinitialconditiona4(0)=
6 6 8
a (0) = 1/√2 and a (0) = a (0) = a (0) = a (0) =
a7 =C3e−i(αn1−β(n3+1))t, a8(0) = a (0) = a (01) = 0. InF2ig. 4,the3timedepe5ndence
a8 = C6eiα(n1+1)t+C8ei(2β+α(n1+1))t, of6thenega7tivityiss9hownforachoiceoftheinitialconditions
−
a =C e2iβ(n3+1)t. (33) a (0) = a (0) = 1/√2 or a (0) = a (0) = 1/√2 and the
9 9 2 6 4 8
6
0.5 0.5
γ γ
0.45 1.00 d 0.45 1.00 e
0.4 0.75 0.4 0.75
0.50 0.50
0.35 0.35
0.3 0.3
) )
(ρ 0.25 (ρ 0.25
N N
0.2 0.2
0.15 0.15
0.1 0.1
0.05 0.05
0 0
0 1 2 3 4 5 6 0 1 2 3 4 5 6
t t
0.5 0.5
α β
0.45 1 0.45 1
0.4 2 0.4 2
3 3
0.35 0.35
0.3 0.3
) )
(ρ 0.25 (ρ 0.25
N N
0.2 0.2
0.15 0.15
0.1 0.1
0.05 0.05
0 0
0 1 2 3 4 5 6 0 1 2 3 4 5 6
t t
FIG.5. Timeevolutionofthenegativityofthesysteminthepresenceofdissipationisshown. (a)α = β = γe = 1andγd isvaried. (b)
α=β=γd =1andγeisvaried.(c)β=1,γe =γd =0.5,andαisvaried.(d)α=1,γe =γd =0.5,andβisvaried.
restofthecoefficientsbeingzero. Wesee thatthenegativity where n and m are integers. n + m + 1 and n m have
−
oscillates in time and the oscillations are more rapid as the different parity. If the first is even, then the second is odd
differencebetweenαandβ increases. Inthiscase,thevalue andviceversa. Thus,negativitybecomeszerowhentheratio
ofthenegativityisboundedbetween0and0.5. α/β isequaltoafractionwhosedenominatorisevenandthe
It is interesting to find the instances where the negativity numeratorisoddorviceversa.InFig.4,weseethatN(ρ)=0
vanishes resulting in zero entanglement between the spins. when α isequalto 1 or 1 andt= π +πk,k =0,1,2...
β 2 4 2
FromEq.(36)weinfer Itshouldbenotedherethatifwetaketheinitialconditions
as a (0) = a (0) = a (0) = a (0) = 1/2 and the rest of
2 4 6 8
cos2t(α+β)cos2t(α β) cos2t(α+β) thea tobezero,thenegativitywillbeN(ρ) = 1/2. Foran
− − − i
cos2t(α β)=3. initial state with all the a being the same and equal to 1/3,
i
− −
(37) the eigenvaluesof the partialtransposeof the densitymatrix
cannotbe calculated analyticallyexceptfor the case α = β.
Itiseasytoseethattheaboveequationholdstruewhen Forthiscase,thenegativityis
4
cos2t(α+β)= 1, N(ρ)= ( sinαt sinαt), (40)
− 9 | |−
cos2t(α β)= 1. (38)
− − whichcanbeexpressedintheform
Thesolutionoftheseequationsdeliverstheconditionforthe
0, 2πk t (2k+1)π
choiceofαandβ forazeronegativityas N(ρ)= α ≤ ≤ α
(8sinαt, (2k+1)π <t< 2(k+1)π
α n+m+1 9 α α
= , (39)
β n m where k is an integer. The maximum of the negativity is
−
7
N (ρ) = 8 at t = π + 2πk We find that the entan- and R1 = R I , R2 = I R , with I the three-
max 9 −2α α ij ij ⊗ 3 ij 3 ⊗ ij 3
glementiszeroduringthetimeinterval 2πk t (2k+1)π, dimensional unit matrix. The most general solution of the
α ≤ ≤ α master equation (41) for an arbitrary initial state is given in
however,there is a partialentanglementsharingbetweenthe
spinsduringtheinterval (2k+1)π <t< 2(k+1)π. theappendix. Infig. 5wehaveshowntheeffectofdamping
α α ontheentanglementbetweenthetwospinsbyvaryingallthe
parametersofinterest.Fig.5(a)showsthedynamicsofentan-
glement for differentvalues of γ with the other parameters
V. LINDBLADEQUATION d
fixedasα = β = γ = 1. Weseethattheentanglementde-
e
caysmoresharplyduetotheincreaseindecoherenceparame-
We consider two three-levelatoms (NV centers), with the
terγ .Thesamecanbeseenbyvaryingtheotherdecoherence
energylevelsofeachNV centerasin Fig.(3). We havedis- d
parameterγ infig.5(b). Aswesee,duetothedecoherence,
cussedabovethesetwoNVcentersinteractingwitheachother e
entanglement’ssuddendeathissmearedout.
indirectlyviaananomechanicalresonator,aschemedescribed
in ref. 27. We now introducedampingby assumingthatthe
excited levels 2 and 3 decay to the ground state 1 , and
| i | i | i
a direct transition between the excited levels is not allowed.
The time evolution of such a system is given by the master VI. CONCLUSIONS
equation(LindbladEquation)38,39:
dρ OneofthemainchallengesfortheNVspin-basednanome-
= i[Vˆ,ρ]+Lρ, (41)
dt − chanical resonator is to achieve a high degree of entangle-
ment between NV spins. The success of this proposal nat-
whereVˆ isthe(27),andLρisthedampingterm38,39: urally depends on the strength of the coupling between NV
γ spins. On the other hand, single NV spins S = 1 are a
Lρ= e 2R1 ρR1 R1 ρ ρR1 +
2 13 31− 33 − 33 tripletstate with two characteristictransitionfrequenciesbe-
+γd(cid:0)2R1 ρR1 R1 ρ ρR1 (cid:1)+ (42) tweenthethreetripletstates. Therealizationofthecontrolled
2 12 21− 22 − 22 transitionsbetweenallthethreelevelsrequirestwo-frequency
+γe(cid:0)2R2 ρR2 R2 ρ ρR2 (cid:1)+ nanomechanical resonator with a special type of dual fre-
2 13 31− 33 − 33 quency cantilever. The frequency characteristics of the can-
+γ(cid:0)d 2R2 ρR2 R2 ρ ρR(cid:1)2 . tileverdependsontheparticularchoiceofthecantilever’sge-
2 12 21− 22 − 22 ometry.Apropersetupofthedualcantileverhelpstuningthe
Thespontaneousemis(cid:0)sionofatoms1and2fromth(cid:1)eirexcited oscillationfrequenciestotheresonancefrequencies.Thedual
states 3 to the ground states 1 is described by the spon- cantilever is supplemented by the magnetic tips. Thus, the
taneou|sdiecay rate γ , similarly|γi is the spontaneousdecay oscillation of the cantilever leads to the indirect interaction
e d
rateofexcited 2 totheground 1 . Forthederivationofthe between the NV spins, while the direct interaction is small.
masterequation|,iwechoosetheb|asiis: Hence, the entanglementbetween the NV spins is fueled by
the dynamicsof the cantilever. In spite of the complexityof
1 0 0 thesystem(twocoupledNVcentersandadualfrequencycan-
1 = 0 , 2 = 1 , 3 = 0 (43) tilever), the modelis analyticallysolvable. In orderto study
| i 0! | i 0 ! | i 1 ! decoherence, we utilized the Lindblad master equation. An
exact analyticalsolution of the Lindblad equationshows the
andtheR = i j operatorsinthisbasishavethefollowing
ij influence of the decoherence. A prominenteffect of the en-
| ih |
forms:
tanglement’ssuddendeathissmearedoutbydecoherence.
1 0 0 0 0 0
R = 1 1 = 0 0 0 ,R = 2 2 = 0 1 0 ,
11 22
| ih | 0 0 0 ! | ih | 0 0 0 !
0 0 0 0 0 1
ACKNOWLEDGMENTS
R = 3 3 = 0 0 0 ,R = 1 3 = 0 0 0 ,
33 13
| ih | 0 0 1 ! | ih | 0 0 0 !
We acknowledge financial support from by Deutsche
0 1 0 0 0 0
ForschungsgemeinschaftSFB762.
R = 1 2 = 0 0 0 ,R = 2 1 = 1 0 0 ,
12 21
| ih | 0 0 0 ! | ih | 0 0 0 !
0 0 0 0 0 0
R = 2 3 = 0 0 1 ,R = 2 1 = 0 0 0 ,
23 32
| ih | 0 0 0 ! | ih | 0 1 0 ! VII. APPENDIX
0 0 0
R = 3 1 = 0 0 0 ThemostgeneralsolutionoftheLindbladequation(41)for
31
| ih | 1 0 0 ! anarbitraryinitialstateisgivenas(ρij ontheright-handside
8
aretheinitialvalues): ρ76(t)=e−γd+22γet(ρ76cosβt iρ78sinβt);
−
ρ75(t)=ρ75e−2γd2+γet;
ρ74(t)=e−γd+2γet γ74cosαt iγ72sinαt)+
−
+e−γd+2γet cosα(cid:0)t eγd+2γetF74(t)cosαt+

Z
ρ99(t)=ρ99e−2γet; +sinαtZ eγd+2γetF74(t)sinαt!;
ρ98(t)=e−γd+23γet(ρ98cosβt iρ96sinβt); ρ73(t)=ρ73e−γet;
−
ρ97(t)=ρe−23γet; ρ72(t)=e−γd+2γet γ72cosαt−iγ74sinαt)+
ρ96(t)=e−γd+23γet(ρ96cosβt−iρ98sinβt); +ie−γd+2γet cosα(cid:0)t eγd+2γetF74(t)sinαt
ρ95(t)=ρ95e−(γd+γe)t; Z −
ρ94(t)=e−(γ2d+γe)t(ρ94cosαt−iρ92sinαt); sinαt eγd+2γetF74(t)cosαt ;
ρ93(t)=ρ93e−23γet; − Z !
ρ92(t)=e−(γ2d+γe)t(ρ92cosαt iρ94sinαt); F74(t)=γdρ85(t)+γeρ96(t);
−
ρ91(t)=ρ91e−γet; 2 eγd+2γetF74(t)sinαt=
ρ89(t)=e−γd+23γet(ρ89cosβt+iρ69sinβt); Z (α β)cos(α β)t+γ sin(α β)t
ρ (t)= 1 ρ (1 cos2βt)+ρ (1+cos2βt)+ =−γdρ85e−γdt − (α−β)2+γd2 − +
88 2 66 − 88 (cid:16) − d
(α+β)cos(α+β)t+γ sin(α+β)t
+iρ68sin2β(cid:0)t−iρ86sin2βt e−(γd+γe)t; + (α+β)2+γd2 +
ρρ8867((tt))==e21−(ργ62d8+(1γe−)t(cρo8s72cβots)(cid:1)β+t+ρ8i6ρ(617+sincoβst2)β; t)− +iγdρ65e−γdt(cid:16)(α−β)sin(d(αα−−ββ))t2−+γγd(cid:17)d2cos(α−β)t −
(α+β)sin(α+β)t γ cos(α+β)t
−iρ88sin2β(cid:0)t+iρ66sin2βt e−(γd+γe)t; − (α+β)2−+γd2 −
d
ρρ8854((tt))==ee−−32γγdd22++γγeett(ρs8in5cαots(−β(cid:1)tiρ+82iρc6o5ssβint+βtρ);62sinβt)+ −γeρ96e−γet(cid:16)(α−β)cos((αα−−ββ))t2++γγee2(cid:17)sin(α−β)t +
(α+β)cos(α+β)t+γ sin(α+β)t
+cosαt(ρ84cosβt(cid:0)+iρ64sinβt) ; + e
(α+β)2+γ2 −
ρρ8823((tt))==ee−−2γγdd+22+2γγeett(ρc8o3scαots(βρ8t2+coi(cid:1)ρs6β3ts+iniβρt6)2;sinβt)+ −iγeρ98e−γet(cid:16)(α−β)sin(e(αα−−ββ))t2−+γγe(cid:17)e2cos(α−β)t −
(α+β)sin(α+β)t γ cos(α+β)t
+sinαt( iρ84cosβ(cid:0)t+ρ64sinβt) ; − e ;
− − (α+β)2+γ2
ρ81(t)=e−γd+2γet(ρ81cosβt+iρ6(cid:1)1sinβt); e (cid:17)
ρ79(t)=ρ79e−32γet; 2 eγd+2γetF74(t)cosαt=
Z
ρ78(t)=e−γd+22γet(ρ78cosβt−iρ76sinβt); =γdρ85e−γdt (α−β)sin((αα−ββ))t2−+γγd2cos(α−β)t +
ρ (t)= 1 2ρ +ρ +ρ +2ρ (cid:16) − d
77 2 77 66 88 99− (α+β)sin(α+β)t γdcos(α+β)t
(cid:18) + − +
−γd2(cid:0)4ρβ626+−γρd288(cid:1) − 2iβγ4dβ(cid:0)2ρ8+6−γd2ρ68(cid:1)(cid:19)e−γet− +iγdρ65e−γdt(α(α+−β)β2)+coγsd(2(αα−ββ))t2++γγd(cid:17)2sin(α−β)t −
−γ2d ρ66−ρ88 2βsin24ββt2−+γγd2cos2βte−(γd+γe)t+ (α+β)cos((cid:16)α+β)t+γdsin−(α+β)td
d +
+−2i1γ2dρ(cid:0)(cid:0)6ρ68+6−ρ8ρ868e(cid:1)(cid:1)−2(γβe+coγsd)24tββ−t2+ρ+9γ9γedd2−si2nγe2tβ;te−(γd+γe)t− −+γ(αeρ+96eβ−)γseitn(cid:16)((α(αα++−βββ))t)2s+inγγ((eαd2αco−−s(ββα))t+2−+βγ)γete2(cid:17)cos(α−β)t +
+ −
(cid:0) (cid:1) (α+β)2+γ2 −
e
(cid:17)
(α β)cos(α β)t+γ sin(α β)t
−iγeρ98e−γet − (α−β)2+γe2 − −
(cid:16) − e
(α+β)cos(α+β)t+γ sin(α+β)t
e
;
− (α+β)2+γ2
e
(cid:17)
9
γ =ρ 1γ ρ γd + γd + ρ69(t)=e−γd+23γet(ρ69cosβt+iρ89sinβt);
72 72− 2 d 65 (α+β)2+γd2 (α−β)2+γd2! ρ (t)= 1 ρ (1 cos2βt)+ρ (1+cos2βt)+
68 86 68
2 −
1 γ γ
+2γeρ98 (α+β)e2+γd2 + (α−β)e2+γd2!+ ρ+6i7ρ(8t8)s=ine2−β(cid:0)(tγ2d−+iγρe6)6t(sρi6n72cβots(cid:1)βet−+(γdi+ργ8e7)sti;nβt);
i α β α+β
1
+ γ ρ − +
2 d 85 (α−β)2+γd2 − (α+β)2+γd2! ρ66(t)= 2 ρ88(1−cos2βt)+ρ66(1+cos2βt)−
i α β α+β iρ68sin2β(cid:0)t+iρ86sin2βt e−(γd+γe)t;
−
+ γ ρ − ;
2 e 96 (α−β)2+γe2 − (α+β)2+γe2! ρ65(t)=e−3γd2+γet(ρ65cosβ(cid:1)t+iρ85sinβt);
1 γd γd ρ64(t)=e−2γd2+γet cosαt(ρ64cosβt+iρ84sinβt)+
γ =ρ + γ ρ + +
74 74 2 d 85 (α+β)2+γd2 (α−β)2+γd2! +sinαt(−iρ62cosβ(cid:0)t+ρ82sinβt) ;
+1γ ρ γe + γe ρ63(t)=e−γd+22γet(ρ63cosβt+iρ(cid:1)83sinβt);
2 e 96 (α+β)2+γd2 (α−β)2+γd2!− ρ62(t)=e−2γd2+γet sinαt( iρ64cosβt+ρ84sinβt)+
−
iγ ρ α−β + α+β + +cosαt(ρ62cosβt(cid:0)+iρ82sinβt) ;
−2 d 65 (α−β)2+γd2 (α+β)2+γd2! ρ61(t)=e−γd+2γet(ρ61cosβt+iρ(cid:1)81sinβt);
+iγ ρ α−β + α+β ; ρ59(t)=ρ59e−(γd+γe)t;
2 e 98 (α−β)2+γe2 (α+β)2+γe2! ρ58(t)=e−3γd2+γet(ρ58cosβt iρ56sinβt);
−
ρ71(t)=γ71e−γ2et−ρ93e−3γ2et + γ2de−2γd2+γet ρρ5576((tt))==ρe−573eγ−d2+2γγed2+t(γρe5t6;cosβt iρ58sinβt);
−
ρ (α−β)sin(α−β)t−γdcos(α−β)t + ρ55(t)=ρ55e−2γdt;
82(cid:16) (α−β)2+γd2 ρ54(t)=e−3γ2dt(ρ54cosαt iρ52sinαt);
−
+(α+β)sin((αα++ββ))t2−+γγd2cos(α+β)t + ρ53(t)=ρ53e−2γd2+γet;
(α β)cos(α β)dt+γdsin(α (cid:17)β)t ρ52(t)=e−3γ2dt(ρ52cosαt−iρ54sinαt);
+iρ62 − (α−β)2+γ2 − − ρ51(t)=ρ51e−γdt;
(α+(cid:16)β)cos(α+β)t−+γdcos(αd +β)t ρ49(t)=e−γd+22γet(ρ49cosαt iρ29sinαt);
+ −
− (α+β)2+γd2 (cid:17) ρ48(t)=e−2γd2+γet sinαt(iρ28cosβt+ρ26ρ28sinβt)+
(α β)cos(α β)t+γ sin(α β)t
+iρ84 − (α−β)2+γd2 − + +cosαt(ρ48cosβt(cid:0)−iρ46sinβt) ;
(α+(cid:16)β)cos(α+β)t−+γdcos(αd +β)t ρ47(t)=e−γd+2γet γ47cosαt+iγ(cid:1)27sinαt +
+ +
(α+β)2+γd2 (cid:17) +e−γd+2γet cosαt(cid:0) eγd+2γetF47(t)cosαt(cid:1)+
(α β)sin(α β)t γ cos(α β)t
d Z
ρ64(cid:16) − (α−−β)2−+γd2 − − +sinαt (cid:0)eγd+2γetF47(t)sinαt ;
(α+β)sin(α+β)t γ cos(α+β)t Z
− d ; F47(t)=γdρ58(t)+γeρ69(t); (cid:1)
−γ71 =ρ71+ρ(9α3++β)2+γd2 (cid:17)! ρ46(t)=e−2γd2+γet cosαt(ρ46cosβt−iρ48sinβt)+
+sinαt(iρ26cosβt(cid:0)+ρ28sinβt) ;
γ γ γ
d d d
+ 2 ρ82(cid:18)(α−β)2+γd2 + (α+β)2+γd2(cid:19)− ρ45(t)=e−23γdt(ρ45cosαt+iρ2(cid:1)5sinαt);
α β α+β
iρ −
− 62 (α β)2+γ2 − (α+β)2+γ2 −
(cid:18) − d d(cid:19)
α β α+β
iρ − + +
− 84 (α β)2+γ2 (α+β)2+γ2
(cid:18) − d d(cid:19)
γ γ
d d
+ρ ;
64(cid:18)(α−β)2+γd2 − (α+β)2+γd2(cid:19)!
10
2 eγd+2γetF47(t)sinαt=
Z
=−γdρ58e−γdt(cid:16)(α−β)cos((αα−−ββ))t2++γγdd2sin(α−β)t + γ47 =ρ47+ 21γdρ58 (α+βγ)d2+γd2 + (α−βγ)d2+γd2!+
(α+β)cos(α+β)t+γ sin(α+β)t
+ d 1 γe γe
(α+β)2+γd2 (cid:17)− +2γeρ69 (α+β)2+γd2 + (α−β)2+γd2!+
(α β)sin(α β)t γ cos(α β)t
−i(γαdρ+56βe)−sγidnt((cid:16)α+−β)t γd(cαo−s−(αβ+)2−β+)tγdd2 − − +2iγdρ56 (α−αβ−)2β+γd2 − (α+αβ+)2β+γd2!+
−
− (α+β)2+γ2 − i α β α+β
d
−γeρ69e−γet (α−β)cos((αα−ββ))t2++γγe2(cid:17)sin(α−β)t + −2γeρ89 (α−β−)2+γe2 − (α+β)2+γe2!;
(α+β)cos(cid:16)(α+β)t+γ si−n(α+β)te ρ44(t)= 1 ρ22+ρ44+2ρ55+ρ88+ρ66 e−γdt
e 2 −
+ +
(α+β)2+γe2 (cid:17) ρ55e−2γdt(cid:0) 1(ρ88+ρ66)e−(γd+γe)t (cid:1)
(α β)sin(α β)t γ cos(α β)t − − 2 −
+iγeρ89e−γet − (α− β)2−+γe2 − − e−γdt
(α+β)sin((cid:16)α+β)t γ co−s(α+β)te − 2 γxcos2αt−iγysin2αt −
e
− (α+β)2−+γe2 (cid:17); −γde2−γ(cid:0)dt cos2αt eγdt(ρ88(t)(cid:1)−ρ66(t))cos2αtdt+
2 eγd+2γetF47(t)cosαt= +sin2αt (cid:0) eγdt(ρ Z(t) ρ (t))sin2αtdt ;
Z 88 − 66
=γdρ58e−γdt(cid:16)(α−β)sin((αα−−ββ))t2−+γγdd2cos(α−β)t + eγdt(ρ8Z8(t)−ρ66(t))sin2αtdt= (ρ88−2(cid:1)ρ66)e−γet
(α+β)sin(α+β)t γdcos(α+β)t Z
+ − 2(α β)cos2t(α β)+γ sin2t(α β)
(α+β)2+γ2 − − − e −
d (cid:17) − 4(α β)2+γ2 −
(α β)cos(α β)t+γ sin(α β)t (cid:18) − e
−iγdρ56e−γdt − (α−β)2+γd2 − − 2(α+β)cos2t(α+β)+γesin2t(α+β) +
(α+β)cos((cid:16)α+β)t+γ sin−(α+β)td − 4(α+β)2+γe2 (cid:19)
d
+ (ρ ρ )
− (α+β)2+γd2 (cid:17) +i 68−2 86 e−γet
(α β)sin(α β)t γ cos(α β)t
+γeρ69e−γet − (α− β)2−+γe2 − + 2(α−β)sin2t(α−β)−γecos2t(α−β)
(α+β)sin(cid:16)(α+β)t γ co−s(α+β)te (cid:18) 4(α−β)2+γe2 −
+ − e + 2(α+β)sin2t(α+β) γecos2t(α+β)
(α+β)2+γ2 − ;
e − 4(α+β)2+γ2
(cid:17) e (cid:19)
(α β)cos(α β)t+γ sin(α β)t
+i(γαeρ+89βe)−cγoest((cid:16)α+−β)t+γ(αsin−−(αβ)+2β+)tγee2 − − Z eγdt(ρ88(t)−ρ66(t))cos2αtdt= (ρ88−2 ρ66)e−γet
e ; 2(α β)sin2t(α β) γecos2t(α β)
− (α+β)2+γ2 − − − − +
e 4(α β)2+γ2
(cid:17) (cid:18) − e
γ27 =ρ27+ 12γdρ56 (α+βγ)d2+γd2 − (α−βγ)d2+γd2!− +2(α+β)sin24t((αα++ββ))2−+γγee2cos2t(α+β)(cid:19)+
(ρ ρ )
−21γeρ89 (α+βγ)e2+γd2 − (α−βγ)e2+γd2!− +i2(α68−2β)8c6ose2−t(γαet β)+γ sin2t(α β)
e
− − −
i α β α+β 4(α β)2+γ2 −
−2γdρ58 (α−β−)2+γd2 + (α+β)2+γd2!− (cid:18)2(α+β)cos2t(α−+β)+γeesin2t(α+β) ;
− 4(α+β)2+γ2
i α β α+β e (cid:19)
γ ρ − + ; ρ ρ
−2 e 69 (α−β)2+γe2 (α+β)2+γe2! γx =ρ22−ρ44+γd 88−2 66
γ γ
e e
+
4(α+β)2+γe2 4(α−β)2+γe2!−
ρ ρ
68 86
iγ −
d
− 2
2(α β) 2(α+β)
− ;
4(α−β)2+γe2 − 4(α+β)2+γe2!