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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 remaining 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 experimental 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 cavity 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 − − [1] M.Brune,F.Schmidt-Kaler,A.Maali,J.Dreyer,E.Ha- [6] R. 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