Dissipative Stern-Gerlach recombination experiment

0.21 MB·English

by  Thiago R. Oliveira

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 exper-
iment time and the revival in the middle of the experi-
ment 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 re-
du
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).