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Dissipative Stern-Gerla h re ombination experiment Thiago R. Oliveira and A. O. Caldeira Departamento de Físi a da Matéria Condensada, Instituto de Físi a Gleb Wataghin,Universidade Estadual de Campi∗nas and Caixa Postal 6165, Campinas, SP, CEP 13083-970, Brazil The possibility of obtaining the initial pure state in a usual Stern-Gerla h experiment through there ombinationofthetwo emerging beamsisinvestigated. Wehaveextendedthepreviouswork of Englert, S hwinger and S ully [2℄ in luding the (cid:29)u tuations of the magneti (cid:28)eld generated by a properly hosen magnet. As a result we obtained an attenuation fa tor to the possible revival of oheren e when the beams are perfe tly re ombined . When the sour e of the magneti (cid:28)eld is a SQUID(super ondu ting quantum interferen e devi e) the attenuation fa tor an be ontrolled by 6 external ir uitsandthespinde oheren edire tlymeasured. FortheproposedSQUIDwithdimen- 0 sions in the s ale of mi rons the attenuation fa tor has been shown unimportant when ompared 0 with theintera tion timeof the spinwith themagnet. 2 n I. INTRODUCTION of the wave fun tion have been disregarded. a J In another manus ript [6℄ a model for a dissipative Stern-Gerla hexperiment was proposed without expli it 1 The Stern-Gerla h experiment gives an experimental 1 eviden e of the quantum nature of the spin of a parti le. referen e to the origin of the (cid:29)u tuations. It basi ally + dealtwithaStern-Gerla happaratusinsideadissipative 1 WhSexn a beam of spin 1/2parti les, in the eigenstate| zˆi mediumand,besides,didnot onsiderthere ombination of , goes through a variable magneti (cid:28)eld in the v pro ess. dire tion and the out oming parti les are dete ted on 0 In this manus riptwe intend to makeafully quantum 7 a s reen, we observe the presen e of two distin t peakzˆs des ription of the magneti (cid:28)eld and study the e(cid:27)e ts 0 orrespondingto the spinsin thepositiveand negative of its unavoidable (cid:29)u tuations in the spatial part of the 1 dire tion. 0 wavefun tion. Wealsopayspe ialattentiontotheorigin Moreover,besidesgivinganeviden eofthespinquan- 6 of these (cid:29)u tuations by presenting a model where the tization, this example is onsidered the paradigm of the 0 magneti (cid:28)eldisgeneratedbyapairofSQUIDswherethe / measurement pro ess. The beam goes into t+he magSnet magneti (cid:29)uxisknowntoobeyaLangevinequation. We h x x in a pure state, for example, the eigenstate | i of , p Sz will be only interested in the spin ohSeren e that an be andasfarasthemeasurementof is on erneditisde- x - Sz measured through the mean value of and arises from t s ribedasasuperpositionoftheeigenstatesof . Later, n the o(cid:27)-diagonalterms ofthe redu eddensity operatorof as a result of the measurement pro ess, the quantum a the system in the spin spa e. u state ollapses into one of the eigenstates of the latter. Anotherpointthatisworthmentioninghereisthefa t q A question that arisesis: ould we get the initial pure that we will be aiming at a problem quite di(cid:27)erent from : v state of the system if we re ombine the two beams from thestandardoneintheareaofdissipativesystems. Usu- i the Stern-Gerla h experiment? allypeoplewanttogetridofde oheren eforma ros opi X This idea of the re ombination of the beams, through quantum me hani al variables whereas we will try to in- r a theso- alledStern-Gerla hInterferometer(SGI),isanold reasethee(cid:27)e tofde oheren eonami ros opi variable. one,buthasalwaysbeentreatedinaqualitativeway[1℄. The reason for this is to fully test the existing theory of One of the (cid:28)rst quantitative studies was presented in a de oheren eas unequivo ally due to the sour esof noise series of three arti les [2, 3, 4℄, in whi h the authors ex- and/or dissipation known to be oupled to the system. II amined the possibility of re ombination and on luded Thepaperisorganizedasfollows: inSe . wepresent that the pre ision of ontrol of the magneti (cid:28)eld is of amodelfortheSGIwherethemagneti (cid:28)eldisgenerated III fundamentalimportan etothe re onstru tionoftheini- byapairofSQUIDs. InSe . weevaluatethedynam- tialpurestate. However,inthosearti lestheauthorsdid i softheredu eddensityoperatorinthespinspa eusing IV nottakeintoa ounttheoriginofthemagneti (cid:28)eldand the Feynman-Vernon path integral approa h. In Se . its possible sour es of (cid:29)u tuations. we analyze the oheren e revival in the SGI. Finally, we V Inamorere entstudy[5℄thisproblemhasbeen revis- summarize our results in Se . ited and the quantum nature of the magneti (cid:28)eld taken into a ount. However, only the oupling of the uniform II. THE STERN-GERLACH part of (cid:28)eld to the spin of the parti le has been onsid- INTERFEROMETER eredandthee(cid:27)e tsofits(cid:29)u tuationsonthespatialpart The SGI an be divided in two parts: the (cid:28)rst one is ∗ responsiblefor separatingthe beams and the se ond one Ele troni address: troi(cid:28).uni amp.br forre ombiningthem. Wewillfollowthebasi modeland 2 f(t) = µ∂B/∂z µ where and is the magneti moment of the parti le. Here we are not onsidering the uniform partofthe magneti (cid:28)eldthat isonlyresponsibleforthe zˆ pre essionthespinaboutthe dire tionandhasalready been treated in [5℄. For a perfe tly noiseless anti-symmetri (cid:28)eld with a pie ewise onstant gradient we observe loss and revival of the spin oheren e through the behavior of the o(cid:27)- diagonalelements of the redu ed density operator in the T spin spa e, as shown in Figure 1. There, is the experiment time and the revival in the middle of the experiment is due to the re ombination of the wave pa kets in Figure 1: Loss and revival of oheren e in a SGI without (cid:29)u tuations. the momentum spa e. The verti al lines show four dif- ferent regions of a SGI. In the (cid:28)rst part the beams are zˆ a elerated in opposite dire tions along . The se ond approximationsof[2℄, onsideringthatthebeamssplitin and third parts are due to the empty interval between zˆ the dire tion due to the magneti (cid:28)eld gradient whi h the magnets where the dire tion of the a eleration for also points along this dire tion. As referred to a given ea hbeamsuddenly hanges. Inthelastpartthea eler- origin, half way from two spe i(cid:28) magnets (see arrange- ation of ea hbeam returns to its original value and they ment below), the magneti (cid:28)eld should depend on y in are (cid:28)nally re ombined as they emerge from the set up. ananti-symmetri waytoenableaperfe tre ombination of the two beams. In order to solve the problem we will yˆ onsider that the velo ity in the dire tion is onstant Wenowintrodu ethe(cid:29)u tuationsofthemagneti (cid:28)eld whi h allows us to repla e its position dependen e by a to the problem. In order to do this we will onsider a time dependen e. SGI where ea h of the two magnets is formed by a pair Our model Hamiltonian is SQUIDsarrangedasshownin(cid:28)gure2. Withthismodel p2 we obtain a (cid:28)eld on(cid:28)guration very similar to that pro- H = σ f(t)z, z 2m − (1) posed in [2℄ and des ribed above. I Z I I Y X Z I X Figure 2: Lateral view of the proposed apparatus for theISGand themagneti (cid:28)eld lines in a frontal view. The total (cid:29)ux inside a SQUID has its dynami s de- at zero external (cid:28)eld. We are onsidering that there s ribed by the model Hamiltonian given by [7℄ is no tunnelling or thermal a tivation of the (cid:29)ux variable to its neighboring minima. Thus, the (cid:29)ux is Hsq+osc = 2PCφ2 + 2ΦL′2 + 2Pmk2 Los0 il=lati[n1g/Lar+ou2nπdi0/anΦlo0 ]a−l1misiniamnume(cid:27)e ltoisveetiondnuΦ t0ana ned, 0 k (cid:20) k X that arises from a se ond order expansion around the +mkωk2 x + Ck Φ′ 2 , minimum of the ele tromagneti potential energy. i0 is 2 (cid:18) k mkωk2 (cid:19) # (2) the riti al urrent for the SQUID. For this approxima- tions to be valid we should be sure that we have many 2πi L/Φ 1 (ω) = minima, in other words: 0 0 ≫ . As a matter wπith Ckt2heδ(ω spωe )tra=l ηωfun tion Φ′ J of fa t we shall assume that ea h SQUID is arrying 2 k mkωw − k . Here is the (cid:29)u tu- a persistent urrent orresponding to the metastable atiPon about one of the many metastable (cid:29)ux values 3 n=1 minimum at . For an original Ohmi spe tral fun tion, whi h is Here, it shouldbe emphasizedthat weareonly taking known to hold for SQUIDs, we obtain the following ef- intoa ounttheparametersofonesingleSQUIDofea h fe tive spe tral fun tion: pair, say, the smaller one. Operationally it means that ηω (ω)= the parti le traje tories are always lose to that same Jeff 1+ ω 2+ ω 4 (7) Ω Ω′ ir uit over half of the experiment time. In the se ond half of the experiment the same applies to the se ond η = ǫ2L2/R Ω = 1/ (cid:0)L2(cid:1)/R2(cid:0) 2(cid:1)CL Ω′ = pair of SQUIDs. The presen e of the larger SQUID in w1/it√hCL 0R , C 0 − 0 and 0 ea hpairisonlytomimi the(cid:28)eldlinesofausualStern- . Here and arep, respe tively, the resistan e Gerla happaratus. Webelievethatthisrequirement an and apa itan e of the jun tion of the SQUID. The ef- be dropped at the expense of using e(cid:27)e tive parameters fe tive spe tral fun tion is Ohmi Ωwith aΩn′ew uto(cid:27) fre- for the oupled SQUIDs in a Hamiltonian of the same quen y given by the minimum of and . form as (2). CFor10t−y1p2iF a,lLSQ1U0I−D1s0Hw,ei hav1e0−th5Ae,followRing 1vΩalues: 0 Therefore, we are dealing with a parti le oupled to a ∼ a(z) ∼ ∼ and ∼ . To magneti (cid:28)eld that is produ ed by a (cid:29)ux whi h, on its obta1i0n−5 10w−e3wmill2suppose a SQUID with dimensions turn, is oupled to a bath of harmoni os illators. Now of × and will estimate the total (cid:29)ux us- if wewrite the magneti (cid:28)eldin terms ofthe (cid:29)uxwewill ing(cid:0)the (cid:28)eld in t(cid:1)he middle10o−f8tmwo2in(cid:28)nite wires whi h is have the parti le indire tly oupled to thΦe =batBh(tzh)rAou(gzh) multiplied by the areazof . Comparing this with the (cid:29)uxB. (Tz)his an be done if we writez Ath(ez(cid:29))ux at a distan e wae(z a)n obtazin=an10e−x3pmression for where is theAm(za)gneti (cid:28)eld at as seen by the a(z)=an1d01t3hmen−3estimate . For we have parti les whereas is an e(cid:27)e tive area rossed by L 10−10.HThese sηpe= i(cid:28)1 0a−t4i0oKnsgfso−r1the apparatus 0 the(cid:28)eldlinesthroughwhi hΦthemagneti (cid:29)uxisexa tly give m∼= 1,8 10an−d25kg . For γ o−o1per given by the total value ofz inside the ring. Sin e the atoms, 10×15 , the relaxation time, , is latter does not depend on we have of the order of se onds. We end this se tion with this proposal for a model for the SGI that in orporates unavoidable sour es of (cid:29)u tu- ∂A ∂B B(z) +A(z) =0, ations of the magneti (cid:28)eld. In our model these (cid:29)u tua- ∂z ∂z (3) tions are modelled by a oupling to a bath of os illators following [7℄. Now we have to evaluate the redu ed den- that an be manipulated to give sity operator of the system in the spin spa e to see the behavior of the spin oheren e through its o(cid:27)-diagonal ∂B 1 ∂A elements. =a(z)Φ, a(z)= . ∂z with A(z)2 ∂z (4) III. THE REDUCED DENSITY OPERATOR Using of the last expression, the total Hamiltonian reads Our model of a pair of SQUIDs generating the mag- P2 P2 Φ′2 neti (cid:28)eld for the SGI obeys the following Hamiltonian ′ φ H = +ǫσ nΦ z+ǫσ Φz+ + part+sq+osc 2m z 0 z 2C 2L0 H =H0+HI +HR, (8) P 2 m ω2 C 2 + k + k k x + k Φ′ , 2m 2 k m ω2 with Xk " k (cid:18) k k (cid:19) # p2 H = +σ f (t)z, 0 z 0 (5) 2m (9) H = σ z C x ǫ=µa I z k x and (10) with . k X Nowwehavetheparti le oupledindire tlytothebath p2 m ω2 C2 H = k + k kx2 + k (σ z)2 . oelfimosin ialltaetothrse.(cid:29)uFxollvoawriianbglethinetphreesH raimptiilotonnoiafn[8a℄ndwew riatne R Xk (cid:20)2mk 2 k 2mkωk2 z (cid:21) (11) a new one where the parti le is oupled dire tly to new os illators with an e(cid:27)e tive spe tral fun tion. The new It des ribes the dynami s of a parti le submitted to a Hamiltonian is linear potential and also oupled to a bath of os illators both in aspin dependent way. The problemof aparti le 2 in a linear potential and oupled to a bath of os illators P2 P˜ 2 m˜ ω˜2 C˜ H˜ = +σ f z+ k + k k x˜ + k σ z has already been treated in the literature and the only 2m z 0 k 2m˜k 2 k m˜kω˜k2 z !  di(cid:27)eren ehere isthespin dependen ethatwasabsentin X [6℄. However,sin ewehaveonly onespinoperatorin (8)   (6) it a ts as a parameter in our problem and we will use a 4 slightly modi(cid:28)ed Feynman-Vernon formalism to solve it. spa e are written as Wearenotgoingtoshowallthedetails,ofthe al ulation for whi h the reader should onsult [7, 9, 10℄. For separable initial onditions, where the inter- ′ ′ ′ ′ ′ ′ ′ a tion btetw=een0the system anxd′R′tSheρ(b0a)thy′Qis′St′urne=d ρss′(x,y,t)= dxdy Jss′(x,y,t;x,y ,0)hxS|ρ1(0)|y S i, oxn′Saρt (0) y′S′ ,R′onρe(0h)asQ′h | | i Z Z 1 2 h | | ih | | iandthefourelementsof (12) the redu ed density operator of the system in the spin with Jss′(x,y,t;x′,y′,0)= dRdR′dQ′Ks(x,R,t;x′,R′,0)hR′|ρ2(0)|Q′iKs∗′(y,R,S′,t;y′,Q′,0) (13) Z Z Z whi h propagates tRhe,Rre′d,Qu ′ed density operator in time. (N-dimensionalve tors)ofthebathofos illators. Within Inthisexpressions arearbitrary on(cid:28)gurations the path integral formalism it is written as Jss′(x,y,t;x′,y′,0)= xDx(t′) yDy(t′)e~i S0s[x(t′)]−S0s′[y(t′)] Fss′[x(t′),y(t′)] Zx′ Zy′ (cid:16) (cid:17) (14) with R Q Fss′[x(t′),y(t′)] = dRdR′dQ′ρ2(R′,Q′,0) DR(t′) DQ(t′) Z Z Z ZR′ ZQ′ e~1 SIs[x(t′),R(t′)]−SIs′[y(t′),Q(t′)] e~1 SRs[R(t′)]−SRs′[Q(t′)] × (cid:16) (cid:17) (cid:16) (cid:17) (15) being the so- alled in(cid:29)uen e fun tional, whi h ontains in whi h all the in(cid:29)uen e of the bath on the system. It has been evaluated before [9℄ and the resulting expression is Fss′[x(t′),y(t′)]=e−~1Φss′[x(t′),y(t′)], (16) im t ′ ′ ′ ′ ′ ′ ′ ′ Φss′[x(t),y(t)] = [Sx(0) S y(0)] dt γ(t)[Sx(t) S y(t)]+ 2 − − Z0 im t t′ ′ ′′ ′ ′ ′ ′ ′′ ′′ ′ ′′ + dt dt [Sx(t) S y(t)]γ(t t )[Sx˙(t )+S y˙(t )]+ 2 − − Z0 Z0 t t′ ′ ′′ ′ ′ ′ ′ ′′ ′′ ′ ′′ + dt dt [Sx(t) S y(t)]α (t t )[Sx(t ) S y(t )] , R − − − (17) Z0 Z0 with ∞ 2 (ω) γ(t) = θ(t) J cosωt dω mπ ω (18) Z0 5 and we have an Ohmi ase, eq [7℄, given by ′ ′′ 1 ∞ ~ω ′ ′′ (ω)= ηω forω <Ω αR(t t )= (ω)coth cosω(t t )dω. J 0 forω >Ω (20) − π Z0 J (cid:18)2KT(cid:19) − (cid:26) (19) Ω All we have to know is the spe tral fun tion of the where we have introdu ed a uto(cid:27) frequen y . With bath, this we obtain (ω)= π Ck2 δ(ω ωk), Jss′(x,y,t;x′,y′,0)= xDx(t′) yDy(t′) e~1βss′[x(t′),y(t′)] J 2 k mkωk − Zx′ Zy′ X (21) that ompletely hara terizes it. In our proposed model with t m η βss′[x(t′),y(t′)] = i dt′ x˙2 y˙2 f0(t′)(Sx S′y) (Sx S′y)(Sx˙ +S′y˙) + 2 − − − − 2 − Zt0 ht′ (cid:0) (cid:1) i ′ ′′ ′ ′ ′ ′ ′′ ′′ ′ ′′ dt dt [Sx(t) S y(t)]α (t t )[Sx(t ) S y(t )]. R − − − − (22) Z0 Z0 (S =S′ = 1) (S =S′ = 1) Computing the diagonal ± and the o(cid:27)-diagonals 6 ± terms we obtain t m η β [x(t′),y(t′)] = i dt′ x˙2 y˙2 f (t′)(x y) (x y)(x˙ +y˙) + d 0 2 − ∓ − − 2 − Zt0 ht′ (cid:0) (cid:1) i ′ ′′ ′ ′ ′ ′′ ′′ ′′ dt dt [x(t) y(t)]α (t t )[x(t ) y(t )] R − − − − (23) Z0 Z0 and t m η β [x(t′),y(t′)] = i dt′ x˙2 y˙2 f (t′)(x+y) (x+y)(x˙ y˙) + od 0 2 − ∓ − 2 − Zt0 ht′ (cid:0) (cid:1) i ′ ′′ ′ ′ ′ ′′ ′′ ′′ dt dt [x(t)+y(t)]α (t t )[x(t )+y(t )]. R − − (24) Z0 Z0 t Sowehaveadi(cid:27)erentdynami sforthediaqgo=na(lxa+ndy)o/(cid:27)2- βo′d = dt′ mq˙ξ˙−ηξq∓2f0(t′)q (28) diagoξn=alxtermys. De(cid:28)ning new xo(otr)dinateys(t) Z0 h i and − wede ouple the and traje tories and and get t t′ q ξ ′′ ′ ′′ ′ ′ ′′ ′′ Jd(od)(q,ξ,t;q′,ξ′,0) = Zq′ Dq(t′)Zξ′ Dξ(t′) (25) βod =4Z0 dt Z0 dt q(t)αR(t −t )q(t ). (29) e~1[iβd′(od)−βd′′(od)] Evaluating the integral in the usual way we expand it about the lassi al traje tory and solve the remaining fun tional integral to obtain with Jd(od)(q,ξ,t;q′,ξ′,0)=e~1[iβd′(colads)s−βd′′(colda)ss]G(q,ξ,t;q′,ξ′,0) t (30) β′ = dt′ mq˙ξ˙ ηξq˙ f (t′)ξ , d − ∓ 0 (26) with ′′ Z0t ′ h t′ ′′ ′ ′ i′′ ′′ mγeγt β = dt dt ξ(t)α (t t )ξ(t ), G(t)= , d R − (27) 2π~sinhγt (31) Z0 Z0 6 βd′class = ξqL′−(t)+ξ′q′L+(t)−ξq′N′(t) L±(t)=mγ[coth(γt)±1], (39) ξ qM(t) X(t)ξ Z(t)ξ , − ∓ ∓ (32) βo′dclass = ξqξL′q−N(t()t)+ξ′2qX′L(+t)(tq)−2ξZq′(Mt)(qt′), e−γt − ∓ ∓ (33) N(t)=mγ , sinhγt (40) 1 β′′class = ξ2A(t)+2ξξ′B(t)+ξ′2C(t) , d 2 (34) eγt (cid:2) (cid:3) M(t)=mγ , and sinhγt (41) β′′class =2 q2A(t)+2qq′B(t)+q′2C(t) . od (35) ξ′,Inξ(tt)h=e ξabove(cid:2) expressions we have usedq(cid:3) ξ(0) = X(t)= sineh−(γγtt) tdt′f0(t′)eγt′sinhγt′, (42) and the same notation holds for . We have Z0 also de(cid:28)ned the following fun tions and A(t) = e−2γt tdt′ tdt′′eγ(t′+t′′)sinhγt′ sinh2γtZ0 Z0 Z(t)= 1 tdt′f (t′)eγt′sinhγ(t t′). ×αR(t′−t′′)sinhγt′′, (36) sinh(γt)Z0 0 − (43) For an initial Gaussian pa ket entered at the origin, 1 − x2 C(t) = 1 tdt′ tdt′′eγ(t′+t′′)sinhγ(t t′) ψ(x)= √2πσ e 4σ2 , (44) sinh2γt − Z0 Z0 ′ ′′ ′′ p α (t t )sinhγ(t t ), R × − − (37) we have ′ ′ 1 −q′2 −ξ′2 ρ(q ,ξ ,0)= e 2σ2 e 8σ2 . B(t) = e−γt tdt′ tdt′′eγ(t′+t′′)sinhγt′ √2πσ2 (45) sinh2γt Z0 Z0 ′ ′′ ′′ The diagonal element of the redu ed density operator α (t t )sinhγ(t t ), R × − − (38) for the system be omes ρ (q,ξ,t) = πG(t)exp M2 q Z 2 ξ2 1A + N2σ2 1 σ2NL ~B 2 + d ra (−4a~2 (cid:18) ∓ M(cid:19) − (cid:20)2 ~ 2~2 − 4a~4 +− (cid:21) (cid:0) (cid:1) i M i Z +~ L−− 2a~2 σ2NL+−~B qξ± ~ X+ 2a~2 σ2NL+−~B ξ (46) (cid:20) (cid:21) (cid:20) (cid:21) (cid:27) (cid:0) (cid:1) (cid:0) (cid:1) with 1 L2σ2 ~C ~2 a= + + + , ~2 2 2 8σ2 (47) (cid:18) (cid:19) whereas the o(cid:27)-diagonal element is 7 π 2σ2N2 σ2L2 2 B2 ρ (q,ξ,t) = G(t) exp 1 + + A q2+ od a − ~2 − 2a~2 ~ 2a~ − r (cid:26)(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) i B σ2NL + + ~ (ξM ∓2Z) 2a~ − 2a~2 +(ξL−±2X) q+ (cid:20) (cid:20) (cid:21) (cid:21) (ξM 2Z)2 2σ2L NB ∓ + q2 . − 16a~2 − a~3 ) (48) 1 So we have obtained an analyti al expression for the redu ed density operator of the system in the spin spa e, 0.8 when the initial spatial part of the wave fun tion is a Gaussian entered at the origin. Now we are going to Ž analyze this result. r 0.6 HtL od 0.4 IV. SPIN COHERENCE IN THE SGI 0.2 The positions of the enterof the pa ketsaregiven by the probability density 0 1 2 3 4 T(cid:144)4 2 1 1 Z ρ (q,0,t)= exp q , d √2πσ˜(t) − 2σ˜(t)2 ∓ M (49) (cid:20) (cid:21) Figure 3: Behaviour of the exponent of (53), showing the possible re overy of1 0o−h9esren e. T , the experiment time, is with hereof theorder of . 1 ~ 2a(t) σ˜(t)= . 0.9 M(t) (50) p 0.8 σ˜(St)o,asexpe ted,wehaveaGaussianpa ketwithwidth 0.7 whose enter is des ribed by hHtL 0.6 z(t)= Z(t) = 1 tdt′f (t′) 1 e−2γ(t−t′) , 0.5 0 M(t) 2mγ − Z0 (cid:16) (cid:17) 0.4 (51) 0.3 whi histhe lassi altraje toryofaparti lesubje t to a 0 1 2 3 4 f0(t) ηv T(cid:144)4 for e and to the vis ous for e . ToseehowpurethestateisasitevolvesintheSGIwe h(t) will investigate the o(cid:27)-diagonal elements of the density Figure 4: Fa tor showing the irreversible loss of oher- oKpTerator~iγnthespinspa e. AIn(tth)eBhig(th)tempeCra(ttu)relimit, en e. T is thesameas above. ≫ , the fun tions , and an be easily evaluated and yield 2σ2N2 σ2L2 2 B2 a′(t) = 1 + A 1 B σ2NL 2 ~2 − 2a~2 − ~ 2a~ − ρ (t) = h(t)exp Z + X (cid:18) (cid:19) (cid:18) (cid:19) od (−a′~2 (cid:20) (cid:18)2a~ − 2a~2 (cid:19)− (cid:21) +2σ2L+NB. a~3 (54) 1 ∆z˜ 2 −2(cid:18)σ˜(t)(cid:19) ) (52) In the non-dissipative limit, γ → 0, h(t) goes to one and the exponential re overs the non-dissipative expres- with sion for the redu ed density operator. Therefore, we an say that the irreversibleloss of oheren eis in the fa tor h(t) M(t) . In(cid:28)gures3and4weshowthh(et)qualitativebehavior h(t)= 2~ a(t)a′(t) and (53) of the exponential term and the fa tor. In these graphs we an see that the re ombination p 8 0.9 would only make matters even worse. 0.8 Although we have only stressed the possible e(cid:27)e t of 0.7 (cid:28)eld (cid:29)u tuations on the re ombination one should bear 0.6 inmind thattheperfe tre ombinationisbyitselfavery rodHtL0.5 di(cid:30) ult task to perform. 0.4 0.3 V. SUMMARY 0.2 In this work we have investigated the e(cid:27)e t of spin 0 1 2 3 4 T(cid:144)4 oheren e, dis ussing experiments with beams of parti- les subje t to magneti (cid:28)elds. We have extended the analysis of the Stern-Gerla h interferometer adding un- Figure 5: Behaviour of the non-diagonal element in the spin avoidable (cid:29)u tuations of the magneti (cid:28)eld through the spa e. oupling of the system to a bath of harmoni os illators. The introdu tion of these (cid:29)u tuations adds a fa tor to h(t) theo(cid:27)-diagonalelementsoftheredu eddensityoperator ould leadto arevivalof oheren e, but ontinually τ inthespinspa ethatde aysinatime ,thede oheren e de reaseswhi hmakesthisrevivalnegligibleasshownin 20 time. Howeverwehaveshownthatevenforavery(cid:16)noisy(cid:17) (cid:28)gure5,where,intheend,wehaveonly %of oheren e environmentprovidedby parti ularly hosen parameters left. foraSQUID thede oheren etime isstill extremelylong Despite the qualitative nature of the graphs, we an ompared to the experiment time. say that in order to have a signi(cid:28) ant revival of oher- When the magneti (cid:28)eld is generatedby ourproposed en e the time of the experiment should be mu h shorter mi rometri SQUID, weobservedthat dissipativee(cid:27)e ts than the de oheren e time, whi h an be extra ted from h(t) arenotimportantwhentheparti lesvelo itiesareofthe theexpressionfor . Sin ethisexpressionisvery um- 1000m/s orderof . We believethat hangingthe SQUID bersome, we have to do it gra(cid:28) ally. parameters it will be possible to observe the loss of spin For the proposed model of a magneti (cid:28)eld generated oheren e by redu ing the SQUID size. In this ase we by a paγir−1of=SQ10U15IDss with dimensions of mi rons,we ob- willhaveamodelofareservoirforagenuinemi ros opi tained . These valuesand theotherτpara1m05es- quantum me hani al variable whi h, in our ase, is the tersalreadyusedgiveusthede oheren etime, ,of 0.1K magneti moment of an atom! for temperatures lose to . In the usual Stern-Gerla h experiment the parti les 1000m/s havevelo itiesofabout 10−6s thatgiveusanintera - A knowledgments tiontimeoftheorderof ,whi hissmallerthanthe de oheren e time. So, for parti les with velo ities of the 1000m/s order of de oheren e an not be observed and T.R.O.,isgratefulforFAPESP(FundaçãodeAmparo will not be important in the SGI. Therefore, in order to à Pesquisa do Estado de São Paulo) whereas A.O.C. a - ontrol the spin oheren e time in an observable way we knowledges the support from CNPq (Conselho Na ional shoulduseSQUIDs ofsmallerdimensionswhi hwill fur- de Desenvolvimento Cientí(cid:28) o e Te nológi o) and the nish us with shorter relaxations times. Larger SQUIDs Millennium Institute for Quantum Information. [1℄ D. Bohm, Quantum Theory (Prenti e-Hall, Englewood [7℄ A.O. Caldeira and A.J. Leggett, Physi a A 121, 587 Cli(cid:27)s, 1951). (1983); Phys.Rev. A 31, 1059 (1985). [2℄ Berthold-Georg Englert et. al., Found. of Phys.18, 1045 [8℄ A. Garg, J.N. Onu hi and V. Ambegaokar, J. Chem. (1988). Phys. 83(9), 4491 (1985). [3℄ J. S hwinger et. al., Z. Phys. D 10, 135 (1988). [9℄ U.Weiss,QuantumDissipativeSystems,WorldS ienti(cid:28) [4℄ M.O. S ully et. al., Phys. Rev. A 40, 1775 (1989). (1993). [5℄ R.doV.Navarro,Master'sthesis(inportuguese)IFGW- [10℄ T.R. Oliveira, Master's thesis (in portuguese) IFGW- UNICAMP (1998). UNICAMP (2002). [6℄ S.BanerjeeeR.Ghosh,Phys.Rev.A62,42105-1(2000).