Vacuum Rabi oscillation in nonzero-temperature open cavity

0.18 MB·English

by  Patrycja Stefańska

Vacuum Rabi oscillation in nonzero-temperature open cavity
Patrycja Stefan´ska,1 Marcin Wilczewski,2 and Marek Czachor2
1Zespo´ l Fizyki Atomowej — Katedra Fizyki Atomowej i Luminescencji, Politechnika Gdan´ska, 80-233 Gdan´sk, Poland
2Katedra Fizyki Teoretycznej i Informatyki Kwantowej, Politechnika Gdan´ska, 80-233 Gdan´sk, Poland
Published as: Open Syst. Inf. Dyn. 18 (2011) 363-373
DOI: 10.1142/S123016121100025X
Comparison of theory of Rabi oscillations with experiment [M. Wilczewski and M. Czachor, Phys.
Rev. A79, 033836 (2009)] suggeststhatcavitylifetimeparametersobtained inmeasurementswith
manyphotonsmay bemuchsmaller thanthoseapplicable toalmost vacuumstatesoflight. Inthis
contextweshowthattheconclusionremainsunchangedevenifonetakesamorerealisticdescription
2 of theinitial state of light in cavity.
1
0 PACSnumbers: 42.50.Lc,42.50.Dv,32.80.Ee,32.80.Qk
2

n PHnL
a I. INTRODUCTION 1
J
0 The1996Bruneet al. experimentonvacuumRabios- 0.8
1 cillation [1, 2] was analyzed in [3] by means of several 0.6
alternative models of atom-reservoir interaction. The
0.4
] study was motivated by difficulties with fitting the data
h
0.2
p bytheoreticalcurves,aproblemaddressedearlierbyvar-
- ious authors [4–6]. Agreement with experimental Rabi n
t 0 1 2 3 4
n oscillation data was then obtained but for the price of
a a cavity quality factor that was 500 times bigger than
FIG. 1: Probabilities of vacuum (n=0) and 1-photon initial
u the one reported in [1]. A part of open questions thus
states (n = 1) effectively dominate the initial thermal prob-
q
remained. ability distribution at 0.8K. Contributions from n ≥ 1 were
[
Thesolutionsofmasterequationsdiscussedin[3]were not taken into account in [3].

2 easier to find than in standard approaches because the
v formalismwasbasedonjump operatorsgeneratingtran-
4 sitions between the dressed states of the atom-field sys- the behavior of the system. In order to understand the
1
tem. Such a construction is more consistent with the issueonecoulditerativelysolveequationsinvolvingmore
0
generaltheoryofopensystems[7,8]thanthepopularap- and more doublets and more transitions between them,
1
. proachbased on jumps between the atomic (hence bare) and compare predictions with the data. If inclusion of a
2
states [9], and is mathematically simpler. In the context next doublet would not produce visible modifications of
1
of quantum optics it appearedonly relatively recently in Rabi oscillations, it would be justified to conclude that
0
1 [10]. Another reason why it was possible to find exact truncation of the Hilbert space to a subspace spanned
: solutionswasthat the atom-fieldsystemwasassumedto by a given number of dressed state has sufficiently well
v
startfromthe initialphotonic vacuumstate. Intermsof approximated the thermal state.
i
X dressed states the initial state belonged to the subspace Thegoalofthepresentpaperisperformthistestonthe
r of the first dressed-state doublet. datafrom[1]. We willseethatinclusionofthenextdou-
a The latter assumption was not very realistic. Light in bletofdressedstatesessentiallycomplicatescalculations,
the cavity was initially in thermal state at T = 0.8K. but does not really change agreement with experiment.
The analysis from [3] not only took into account emis- Weconcludethatsolutionoftheproblemofthe“wrong”
sions from the dressed states downward on the energy cavity Q factor will not be achieved by taking more re-
ladder, but also thermal excitations from vacuum to the alistic initial states. The true physical mechanism must
two dressed states, as well as thermal long-wave fluctu- be therefore different.
ations between the dressed states from the first doublet.
But it did not take into account the presence of the re-
maining bands of dressed states in the initial state. The II. OPEN-CAVITY MODEL
first doublet appears with probability around 0.95, but
the second doublet has probability higher than 0.045. It We employ the standard Jaynes-Cummings Hamilto-
looks like downwardtransitions from the second doublet nian H =~Ω in exact resonance,
are processes of the same order as the upward thermal
ω
transitions at 0.8K from exact vacuum to the first dou- Ω = 0 (e e g g )
blet. Sowhatremainedunclearin[3]wastowhatextent 2 | ih |−| ih |
thefactthattheinitialstatewasthermalwasinfluencing +ω0a†a+g ae g +a† g e . (1)
| ih | | ih |
(cid:0) (cid:1)
2

E+ The decay coefficients from Fig. 2 satisfied
γ3 γc
E–
γ2 γb γ1 γa γa = e−~(ωk0T+g)γ1 ≈0.0466327γ1, (6)
γb = e−~(ωk0T−g)γ2 0.0466328γ2, (7)
≈
E0 γc = e−2k~Tgγ3 0.999997γ3, (8)
≈
FIG.2: Energylevelsanddecaycoefficientsusedinthegener-
alization of theScala model [10] discussed in [3]; E± =~Ω±,
E0=~Ω0. Thermalexcitationsfromthegroundstatetothefirsttwo
dressed states involve proportionality factors e−~(ωk0T−g)
that are of the same order as p1. Since p1 measures
The initial state at T =0.8 K is
the probability of occurrence of e,1 e,1 in the initial
| ih |
thermal mixture, it simultaneously determines probabil-
ρ(0) = p0 e,0 e,0 +p1 e,1 e,1 +..., (2)
| ih | | ih | ities of finding the next two dressed states. Accordingly,
transitions determined by γ and γ should be regarded
where p0 = 0.952381, p1 = 0.0453515, p2 = 0.00215959. a b
Let us note that ∞j=1pj = 1−p0 = 0.047619 is of the afrsomprothceessseescoonfdthderessasemde-sotardteerdoausbdloetw:nward transitions
orderofp1. TheaPnalysisperformedin[3]assumedjumps
betweendressedstateswithtransitioncoefficientstypical
of T = 0.8 K, but the initial state was approximated by
1
tρr(a0n)si=tio|nes,0biehetw,0e|e.nTthheedsroelsusteiodnstgaitveesnsihnow[3n] ienmFpliogy.e2d, |Ω2+i = √2 |g,2i+|e,1i , (9)
(cid:0) (cid:1)
i.e. those involving e,0 and the ground state: 1
| i Ω2− = g,2 e,1 . (10)
| i √2 | i−| i
1 (cid:0) (cid:1)
Ω+ = g,1 + e,0 , (3)
| i √2 | i | i
(cid:0) (cid:1)
1
Ω = g,1 e,0 , (4) It is therefore justified to regard Fig. 3 as more realistic
−
| i √2 | i−| i than Fig. 2. The open-cavity generalization of the Scala
(cid:0) (cid:1)
Ω0 = g,0 . (5) et al. model [10], corresponding to Fig. 3, reads
| i | i
ρ˙ = ρ= i[Ω,ρ]
L −
1 1 1 1
+γ1 Ω0 Ω+ ρΩ+ Ω0 Ω+ Ω+ ,ρ +γa Ω+ Ω0 ρΩ0 Ω+ Ω0 Ω0 ,ρ
(cid:26)2| ih | | ih |− 4h| ih | i+(cid:27) (cid:26)2| ih | | ih |− 4h| ih | i+(cid:27)
1 1 1 1
+γ2 Ω0 Ω− ρΩ− Ω0 Ω− Ω− ,ρ +γb Ω− Ω0 ρΩ0 Ω− Ω0 Ω0 ,ρ
(cid:26)2| ih | | ih |− 4h| ih | i+(cid:27) (cid:26)2| ih | | ih |− 4h| ih | i+(cid:27)
1 1 1 1
+γ3 Ω− Ω+ ρΩ+ Ω− Ω+ Ω+ ,ρ +γc Ω+ Ω− ρΩ− Ω+ Ω− Ω− ,ρ
(cid:26)2| ih | | ih |− 4h| ih | i+(cid:27) (cid:26)2| ih | | ih |− 4h| ih | i+(cid:27)
1 1 1 1
+γ4 Ω+ Ω2+ ρΩ2+ Ω+ Ω2+ Ω2+ ,ρ +γ5 Ω2− Ω2+ ρΩ2+ Ω2− Ω2+ Ω2+ ,ρ
(cid:26)2| ih | | ih |− 4h| ih | i+(cid:27) (cid:26)2| ih | | ih |− 4h| ih | i+(cid:27)
1 1 1 1
+γe Ω2+ Ω2− ρΩ2− Ω2+ Ω2− Ω2− ,ρ +γ6 Ω− Ω2+ ρΩ2+ Ω− Ω2+ Ω2+ ,ρ
(cid:26)2| ih | | ih |− 4h| ih | i+(cid:27) (cid:26)2| ih | | ih |− 4h| ih | i+(cid:27)
1 1 1 1
+γ7 Ω+ Ω2− ρΩ2− Ω+ Ω2− Ω2− ,ρ +γ8 Ω− Ω2− ρΩ2− Ω− Ω2− Ω2− ,ρ .
(cid:26)2| ih | | ih |− 4h| ih | i+(cid:27) (cid:26)2| ih | | ih |− 4h| ih | i+(cid:27)
(11)
Expressing the initial condition ρ(0)= c ρ as a lin- vectors. Twenty of them are easy to find and determine
j j j
ear combination of eigenvectors of , Pρj = Λjρj, we
find the solution ρ(t) = jcjeΛjtρj.LWLe need 25 eigen-
P
3

E2+ ρ16 = Ω+ Ω− , (23)
γ5 γe γ4 E2– Λ16 = |−i(Ωih+−|Ω−)− γ1+γ2+4 γ3+γc,
γ7 γ6 ρ17 = |Ω+ihΩ0|, γ1+γ3+γa+γb (24)
γ8 E+ Λ17 = −i(Ω+−Ω0)− 4 ,
γ3 γc ρ18 = |Ω−ihΩ0|, (25)
E– Λ18 = i(Ω− Ω0) γ2+γa+γb+γc,
− − − 4
γ2 γb γ1 γa ρ19 = Ω− Ω+ , (26)
| ih |
E0 γ1+γ2+γ3+γc
Λ19 = i(Ω+ Ω−) ,
− − 4
FIG. 3: E2+ =~Ω2+,E2− =~Ω2−, E± =~Ω±, E0=~Ω0. ρ20 = Ω− Ω2− , (27)
| ih |
γ2+γ7+γ8+γc+γe
Λ20 = i(Ω2− Ω−) ,
− − 4
off-diagonalmatrix elements(in the dressed-statebasis),
ρ21 = Ω− Ω2+ , (28)
| ih |
ρ6 = |Ω2+ihΩ2−|, γ4+γ5+γ6+γ7+γ8+γ(e12) Λ21 = i(Ω2+−Ω−)− γc+γ2+γ44+γ5+γ6,
Λ6 = −i(Ω2+−Ω2−)− 4 , ρ22 = Ω0 Ω2+ , (29)
| ih |
ρ7 = |Ω2+ihΩ+|, (13) Λ22 = i(Ω2+ Ω0) γa+γb+γ4+γ5+γ6,
γ1+γ3+γ4+γ5+γ6 − − 4
Λ7 = −i(Ω2+−Ω+)− 4 (14) ρ23 = Ω0 Ω2− , (30)
| ih |
γa+γb+γ7+γ8+γe
Λ23 = i(Ω2− Ω0) ,
− − 4
ρ8 = Ω2+ Ω− , (15)
| ih | γc+γ2+γ4+γ5+γ6 ρ24 = |Ω0ihΩ+|, (31)
Λ8 = −i(Ω2+−Ω−)− 4 , Λ24 = i(Ω+ Ω0) γ1+γ3+γa+γb,
− − 4
ρ9 = Ω2+ Ω0 , (16)
| ih | γa+γb+γ4+γ5+γ6 ρ25 = |Ω0ihΩ−|, (32)
Λ9 = i(Ω2+ Ω0) , γ2+γa+γb+γc
− − − 4 Λ25 = i(Ω− Ω0) .
− − 4
ρ10 = Ω2− Ω0 , (17)
| ih |
γa+γb+γ7+γ8+γe
Λ10 = i(Ω2− Ω0) ,
− − − 4
ρ11 = Ω2− Ω− , (18)
| ih |
Λ11 = i(Ω2− Ω−) γ2+γ7+γ8+γc+γe, The remaining five eigenvectors ρj, j = 1,2,3,4,5 are
− − − 4 related to the diagonal matrix elements of ρ,
ρ12 = Ω2− Ω+ , (19)
| ih |
γ1+γ3+γ7+γ8+γe
Λ12 = i(Ω2− Ω+) ,
− − − 4
ρ13 = Ω2− Ω2+ , (20)
| ih |
Λ13 = i(Ω2+−Ω2−)− γ4+γ5+γ6+4 γ7+γ8+γe, ρj = xj|Ω2+ihΩ2+|+yj|Ω2−ihΩ2−|+zj|Ω+ihΩ+|
ρ14 = Ω+ Ω2+ , (21) +vj|Ω−ihΩ−|+wj|Ω0ihΩ0| (33)
| ih |
γ1+γ3+γ4+γ5+γ6
Λ14 = i(Ω2+ Ω+) ,
− − 4
ρ15 = Ω+ Ω2− , (22)
| ih |
γ1+γ3+γ7+γ8+γe
Λ15 = i(Ω2− Ω+) , The corresponding eigenvalue problem is equivalent to
− − 4
4
γ4+γ5+γ6 γe 0 0 0 xj xj
1 γ5 γ7+γ8+γe 0 0 0  yj   yj 
γ4 γ7 γ1+γ3 γc γa zj = Λj zj . (34)
2 γ6 γ8 γ3 γ2+γc γb  vj   vj 
 0 0 γ1 γ2 γa+γb  wj   wj 
The eigenvalues are the atom in its ground state reads finally
Λ1 = 0, (35a) pg(t) = B1(x1+y1+z1+v1+2w1)eΛ1t
Λ2 = 0.25 ω+δ+√θ , (35b) +B2(x2+y2+z2+v2+2w2)eΛ2t
− (cid:16) (cid:17) +B3(x3+y3+z3+v3+2w3)eΛ3t
Λ3 = −0.25(cid:16)ω+δ−√θ(cid:17), (35c) +B4(x4+y4+z4+v4+2w4)eΛ4t
Λ4 = 0.25 ζ+ξ+√κ , (35d) +B5(x5+y5+z5+v5+2w5)eΛ5t
−
Λ5 = 0.25(cid:0)ζ+ξ √κ(cid:1), (35e) 0.025e−γ4+γ5+γ6+4γ7+γ8+γetcos2√2gt
− − −
(cid:0) (cid:1) 0.475e−γ1+γ2+4γ3+γctcos2√2gt. (40)
where −
Let us stress that the above solution is found under the
κ = γ6((δ+ω)2 4(γ2(γ3+γa) (36a) assumption that the atom-field coupling g is constant in
−
+ (γa+γb)(γ3+γc)+γ1(γ2+γb+γc))), time. We know, however, that the atom interacts with
themodewhosespatialprofileisGaussianwithwidthw.
Theatomispropagatingthroughthecavitywhichmakes,
θ =(ζ ξ)2+4γ5γe (36b) effectively, the coupling time-dependent. A method of
−
taking this into account was discussed in detail in [3].
Assuming that the length of the cavity is d we obtain
ω =γ1+γ2+γ3, ζ =γ4+γ5+γ6, (36c) probability appropriate for comparison with experimen-
tal data
ξ =γ7+γ8+γe, δ =γa+γb+γc. (36d) pg(t) = B1(x1+y1+z1+v1+2w1)eΛ1t
+B2(x2+y2+z2+v2+2w2)eΛ2t
Explicit forms of the eigenvectors can be found in the +B3(x3+y3+z3+v3+2w3)eΛ3t
Appendix.
Areasonableapproximationoftheinitialthermalstate +B4(x4+y4+z4+v4+2w4)eΛ4t
is given by +B5(x5+y5+z5+v5+2w5)eΛ5t
0.025e−γ4+γ5+γ6+4γ7+γ8+γetcos 2√2g√πwt
ρ(0) = 0.95e,0 e,0 +0.05e,1 e,1 (37) − d
= 0.95|(cid:16)|Ω+ihihΩ+||+|Ω−|ihΩi−h| | −0.475e−γ1+γ2+4γ3+γctcos(cid:0)2√2g(cid:0)√πwdt(cid:1). (41(cid:1))
Ω+ Ω− Ω− Ω+ Let us recall that the data shown in [1] were plotted as
− | ih |−| ih |(cid:17) a function of an effective time t = √πwt. Since it is
eff d
+ 0.05(cid:16)|Ω2+ihΩ2+|+|Ω2−ihΩ2−| [m3]o,rweecornesvceanlieentthefodrautas ttootw.oFrkig.in4tsehromwssovfactutuhmanRtaebffi
Ω2+ Ω2− Ω2− Ω2+ . (38) oscillationpredicted by our morerealistic scenario(solid
− | ih |−| ih |(cid:17)
line) as compared to experimental data and predictions
Aftersomewhatlengthybutsimplecalculationsonefinds of the simplified model from [3] (dotted).
that
ρ(t) = 2 B1eΛ1tρ1+B2eΛ2tρ2+B3eΛ3tρ3+B4eΛ4tρ4
+(cid:0)B5eΛ5tρ5 +B6eΛ6tρ6+B13eΛ13tρ13 III. CONCLUSIONS
+B16eΛ16tρ(cid:1)16+B19eΛ19tρ19. (39)
The effort devoted to solving the more realistic case
The coefficients are explicitly given in the Appendix. apparentlydidnotpay: Thenewcurvedoesnotdescribe
Probability pg(t) = pg,0(t)+pg,1(t)+pg,2(t) of finding thedataanybetter. Theapproximationmadein[3]turns
5
p HtL 5
g
A2j = εiklmxizkvlwm (j =1,..,5)
1.0
X
{i,k,l,m}=1
{i,k,l,m}6=j
0.8
0.6 5
A3k = εijlmxiyj vlwm (k =1,..,5)
X
0.4 {i,j,l,m}=1
{i,j,l,m}6=k
0.2
5
t A4l = εijkmxiyj zkwm (l =1,..,5)
0.0000 0.0001 0.0002 0.0003 0.0004 0.0005
X
{i,j,k,m}=1
{i,j,k,m}6=l
FIG. 4: The solid line represents (41). The dotted curve is
the prediction from [3]. The parameters are γ1 = γ2 =γ4 =
γ6 =γ7 =γ8 =17.73 Hz,γa =γb =ǫγ1, γ3 =γc =γ5 =γe = 5
0.07g, g=47π103 Hz. t is the truetime.
A5m = εijklxiyj zkvl (m=1,..,5)
X
{i,j,k,l}=1
{i,j,k,l}6=m
out to be physically reasonable. This is in a sense good
newssinceinclusionofyethigherdresseddoubletswould
requiresolvingalgebraicequationsoforderhigherthan5,
5
andthustherewouldbepracticallynochanceforfinding
χ= ε x y z v w
exact solutions. ijklm i j k l m
On the other hand, this also means that refined the- {i,j,kX,l,m}=1
oretical analysis of Rabi oscillation has not brought us
For eigenvalues Λ (k =1,2,3,4,5) we define eigenvec-
any closer to understanding why probability of atomic k
tors with coordinates x ,y ,z ,v ,w
exited state does not decay to zero as fast as expected k k k k k
{ }
on the grounds of cavity lifetime reported in [1]. The
cavity quality seems to be better than that assumed by x1 = 0
Brune et al. Perhaps, as suggested in [3], the key el- y1 = 0
ement is in opening of the cavity and inclusion of the
z1 = γ2γa+γcγa+γbγc
long-wave transitions within a given doublet of dressed
states. Theappropriatejumpoperatorsoccurifonemod- v1 = γ3γa+γ1γb+γ3γb
ifies the cavity-reservoir coupling. The required interac- w1 = γ1γ2+γ2γ3+γ1γc
tion with the reservoir has stronger dependence on the
numberofphotonsinside ofthe cavity. Measurementsof For k =2,3 we have
cavity lifetimes are typically performed with more cav-
ity photons than in Rabi oscillation experiments. It is x = 0
k
possible that the same cavity has a much longer lifetime
y = 0
k
if only a few photons are present. The problem requires
further experimental studies. zk = 32γ1 az +( 1)k√κbz
−
vk = 32γ1(cid:0)av+( 1)k√κbv(cid:1)
−
Appendix wk = 32γ1(cid:0)aw+(−1)k√κbw(cid:1)
(cid:0) (cid:1)
and if k =4,5 we have
Coefficients introduced in previous section
are: B6 =B13 = 0.025 and B16 =B19 = 0.025,
whereas for n=1−,2,3,4,5 we have: − xk = 16γ1γ6 cx+( 1)k√θdx
(cid:16) − (cid:17)
1
Bn = χ[0.2375(A3n+A4n)+0.0125(A1n+A2n)] yk = 32γ1γ5γ6(cid:16)cy+(−1)k√θdy(cid:17)
where zk = 32γ1γ6 cz +( 1)k√θdz
(cid:16) − (cid:17)
A1i = 5 εjklmyj zkvlwm (i=1,..,5) vk = 32γ1γ6(cid:16)cv+(−1)k√θdv(cid:17)
{{jj,,kkX,,ll,,mm}}=6=1i wk = 128γ1γ6(cid:16)cw+(−1)k√θdw(cid:17)
6
where
az = 2γ5(γ6γ7−γ4γ8) ω2−δ2−2γ2(δ+ω) cz = 4γ2((ζ−ξ)(γ4γb−γ6γa)+2γ5(γ8γa−γ7γb))
+ 4(γ2γa+γcγa+γ(cid:0)bγc) +(γ6(δ+ω 2ξ) + (ξ+ζ−2δ+2γc)((2γ5γ7−γ4(ζ−ξ))
−
−+ 2γγ45δγ28)(cid:0)4ωγ26γγ2c2++22(γδ6γ+c((cid:1)ωω)−δ4)γ−c(γ4aγ+2γaγ(bγ)4+γ6) ×− (θξ(γ+4(ζξ−+23γζ2−−22δγ)c−)+2(2γγ2cγ(4γ+6(ζγ5−γ7ξ)+−γ62γγc5)γ)8))
κγ(cid:0)6(γ4−(δ+3ω 2ξ) 2(−γ2γ4+γ5γ7+(cid:1)γ(cid:1)6γc)) cv = −θ(γ6(−2(δ+ω−γ2−γc)+3ζ+ξ)−2(γ3γ4
− − − + γ5γ8))+( 2δ+ζ+ξ+2γc)(2γ3((ζ ξ)
av = 4γ5(γ4γ8 γ6γ7)(γ3(ω δ)+2(γ3γc γ1γb)) − −
− − − (γ4+γ6)+2γ5(γe ξ)) (ζ+ξ 2γ1)
κγ6(γ6(δ+ω 2ξ) 2(γ3γ4+γ5γ8 × − − −
− − − (γ6(ζ ξ) 2γ5γ8)) 4γ1((ζ ξ)
γ6(γ2+γc)))+((δ+ω 2ξ)γ6 2γ5γ8) × − − − −
− − − (γ4γb γ6γa)+2γ5(γ8γa γ7γb))
(2γ4(γ3(ω δ)+2(γ3γc γ1γb)) × − −
×+ γ6((γ1+γ3− γa γb)2 −(γ2+γc)2+4γ1γa)) cw = γ4+γ2γ6)−(γ1γ2+γ1γc+γ2γ3)((ζ −ξ)
− − − (γ4+γ6) 2γ5γ8)+γ5γ7(2γ2(γ1+γ3)
aw = 2 (γ6(δ+ω 2ξ) 2γ5γ8)(γ1γ4(δ+ω) × −
+ γ(cid:0)2γ6(δ−ω)−−2γ2(−γ1γ4+γ3γ4−γ2γ6) d =− 2−(γ(ξ1(+ζ+ζ ξ−δ 2γωc)))−ζ2γ2γξ52γ8(ζ+ξ)
x
− 2γ1γc(γ4+γ6))+2γ5(γ6γ7−γ4γ8) θ(δ+ω −ξ −2ζ)(cid:0)+η−ζ) (cid:1)
(2(γ1γ2+γ3γ2+γ1γc) γ1(δ+ω)) − − −
× − d = 2(δ+ω ξ ζ)(ξ+ζ) η θ
y
+ κγ6(γ1γ4+γ2γ6) − − − −
bz = 4γ6((γ1+γ3)(γ4((cid:1)ξ ω)+γ2γ4+γ5γ7) dz = 4(γ5γ7(ξ+ζ−δ−γ2)+ζ(γ4(δ−ζ+γ2)+γ6γc)
− γ5(γ8γc+γ4γe) (γ2γa+γc(γa+γb))(γ4+γ6))
γa(γ1γ4+γ2γ6)+(ω ξ+γc)γ6γc − −
− − dv = 4( γ5(γ3γ7 γ2γ8+γ6γe) (γ4+γ6)(γ3γa
γc(γ3γ4+γ5γ8)) − − −
− + γb(γ1+γ3))+γ5γ8(ξ+ζ δ ω+γc)
bv = 4γ6(γ3γ4(ω ξ+γc) γb(γ1γ4+γ2γ6) − −
− − + ζ(γ3γ4+γ6(δ+ω ζ γ2 γc)))
+ (γ5γ8+γ6(ξ γ2 γc))(γ2+γc) − − −
− − dw = γ2γ6(ζ ω) γ2(γ3γ4 γ2γ6+γ5γ8)
γ3(γ5γ7+γ6γc)) − − −
− + γ1γ4(ζ γ2 γc) γ1(γ5γ7+γ6γc)
bw = 4γ6(γ2γ6(δ ξ) γ2(γ1γ4+γ3γ4 γ2γ6 − − −
− − − η = γ2(4γ3 2(ξ+ζ 2γa))
+ γ5γ8) γ1(γ4(ξ δ ω+γc) γ5γ7 γ6γc)) − −
c = η ζ2 −ξ2 +θ(θ−+η− ξ2 − − + (ξ+ζ−2γ3−2γc)(ξ+ζ−2δ+2γc)
x
+ ζ((cid:0)4ξ −4δ(cid:1) 4ω+5ζ))− + 2γ1(2γ2−(ξ+ζ−2δ+2γa))
− −
c = θ(2(δ+ω) 3(ξ+ζ)) η(ξ+ζ)
y
− −
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