Quan um mechanical desc ip ion o S e n-Ge lach expe imen s
G. Po el,1F. Ba anco,2S. C uz-Ba ios,3,1 and J. Gómez-Camacho1
1Depa amen o de Física A ómica, Molecula y Nuclea , Apa ado Pos al 1065, 41080 Se illa, Spain
2Depa amen o de Física Aplicada 3, E.S.I. Isla de la Ca uja, Se illa, Spain
3Depa amen o de Física Aplicada 1, E.U.P. Vi gen de Á ica, Se illa, Spain
共Recei ed 21 Sep embe 2004; published 27 May 2005兲
The mo ion o neu al pa icles wi h magne ic momen s in an inhomogeneous magne ic ield is desc ibed in
a quan um mechanical amewo k. The alidi y o he semiclassical app oxima ions which a e gene ally used
o desc ibe hese phenomena is discussed. App oxima e exp essions o he e olu ion ope a o a e de i ed and
compa ed o he exac calcula ions. Focusing and spin- lip phenomena a e p edic ed. The eliabili y o S e n-
Ge lach expe imen s o measu e spin p ojec ions is assessed in his amewo k.
DOI: 10.1103/PhysRe A.71.052106 PACS numbe 共s兲: 03.65.Sq, 03.65.Ta, 03.65.Wj, 03.65.Nk
I. INTRODUCTION
The S e n-Ge lach expe imen consis s in aking a beam
o pa icles ha ha e a neu al elec ic cha ge, bu a ini e
magne ic momen , and passing hem h ough an inhomoge-
neous magne ic ield. The obse ed esul is ha he pa icles
de lec di e en ly depending on he spin p ojec ion along he
magne ic ield. So, by measu ing he de lec ion, one can in e
he alue o he spin p ojec ion o he pa icles along he
di ec ion o he magne ic ield. The desc ip ion o his phe-
nomenon is done wi h he ollowing assump ions.
共i兲The spin p ojec ion along he zaxis, aken along he
magne ic ield a he cen e o he beam, is conse ed.
共ii兲Pa icles wi h di e en spin p ojec ions along he z
axis, as hey go h ough he inhomogeneous magne ic ield,
su e a o ce in he zdi ec ion ha is gi en by he p oduc o
he magne ic momen imes he g adien o he ield imes he
spin p ojec ion.
This is wha we will call he ex book desc ip ion o he
S e n-Ge lach expe imen 关1–4兴. Thus, conside ing he pa -
icle posi ion as a poin e and he spin p ojec ion as he quan-
um p ope y o be measu ed, he S e n-Ge lach se up is as-
socia ed wi h a measu emen ope a o on he spin s a e which
has as eigen alues he spin p ojec ions along he zaxis. Un-
de he ex book desc ip ion, he S e n-Ge lach expe imen
co esponds o an “ideal” measu emen , in he sense o on
Neumann 关5兴, because he quan um s a e is no modi ied by
he measu emen p ocess when i is an eigens a e o he mea-
su ing appa a us. Besides, i is “comple ely eliable,” in he
sense discussed in 关6兴, because he posi ion is comple ely
co ela ed wi h he spin p ojec ion.
Howe e , when he expe imen is in es iga ed in mo e
de ail, he si ua ion becomes mo e complica ed. As he mag-
ne ic ield has ze o di e gence, hen i is no possible o ha e
a g adien o he ield only in one di ec ion. This p oduces
e ms in he Hamil onian ha can change he spin o he
inciden pa icle. A de ailed in es iga ion o hese e ec s was
made in a ecen publica ion 关7兴, making use o he concep
o cohe en in e nal s a es 关8兴in a semiclassical app oach. In
his app oach, i is shown ha he quan um mechanical wa e
unc ion which desc ibes he mo ion o a sys em wi h in e -
nal deg ees o eedom can be app oxima ed by a single a-
jec o y only o ce ain in e nal s a es which a e called co-
he en in e nal s a es. These in e nal s a es e ol e in ime
acco ding o an e olu ion ope a o which is de e mined by
he in e ac ion e alua ed along he ajec o y. The cohe en
in e nal s a es, in he case o he S e n-Ge lach expe imen s,
a e s a es wi h de ini e p ojec ion along he di ec ion o he
magne ic ield. This di ec ion may a y depending on he
posi ion o he pa icle, because he magne ic ield is no
homogeneous.
The main esul o 关7兴is ha , indeed, when a beam o
pa icles goes h ough a S e n-Ge lach magne , he di e en
spin p ojec ions de ia e depending on he spin p ojec ion.
Howe e , when he size o he beam is no e y small com-
pa ed o he ange o inhomogenei y o he magne ic ield,
addi ional e ec s occu .
共i兲The e is a ocusing e ec , so ha he pa icles de ia -
ing in he di ec ion in which he ield dec eases end o ocus,
while hose going in he di ec ion o inc easing ield end o
de ocus.
共ii兲The e a e some pa icles wi h a gi en spin p ojec ion
which de ia e as hose wi h a di e en spin p ojec ion. So
he S e n-Ge lach se up is no , e en in heo y, a “comple ely
eliable” measu ing appa a us.
共iii兲The e a e some pa icles, wi h a de ini e spin p ojec-
ion along he quan iza ion axis, which change he spin p o-
jec ion as hey go h ough he magne . So he S e n-Ge lach
se up is no an “ideal” measu emen appa a us, as successi e
measu emen s will no gi e exac ly he same esul s.
This is wha we will call he semiclassical desc ip ion o
he S e n-Ge lach expe imen . No e ha i we associa e he
pa icle posi ion a e he magne as a “poin e ,” which gi es
he esul o he measu emen o he spin p ojec ion along he
zaxis, hen we conclude ha , in he semiclassical desc ip-
ion, he S e n-Ge lach expe imen is no an ideal measu e-
men , because i can al e he spin p ojec ion, o a comple ely
eliable one, because he posi ion is no always co ela ed
wi h he spin p ojec ion.
These conclusions we e ob ained in a semiclassical
amewo k, in which he mo ion o he pa icles was de-
sc ibed by classical ajec o ies which depended in he spin
p ojec ion along he magne ic ield ha hey encoun e ed.
Ou mo i a ion he e is o see whe he he same conclusions
hold when he ull quan um mechanical p oblem is consid-
e ed. In Sec. II we o mula e he ime-dependen quan um
PHYSICAL REVIEW A 71, 052106 共2005兲
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mechanical p oblem o a wa e packe going h ough a S e n-
Ge lach magne and discuss he alidi y o he ex book and
semiclassical app oaches. In Sec. III we p esen he nume i-
cal solu ion o he quan um mechanical p oblem. In Sec. IV
we in es iga e se e al analy ic app oxima ions o he p ob-
lem, conside ing he alidi y o he concep o cohe en in-
e nal s a es. In Sec. V we discuss he in e p e a ion o S e n-
Ge lach expe imen s as measu emen s de ices. Sec. VI is o
a summa y and conclusions.
II. QUANTUM MECHANICAL FORMULATION
We wan o in es iga e he e ec o an inhomogeneous
magne ic ield on he e olu ion o a quan um wa e packe .
The si ua ion ha we will conside is a magne ic ield ha
has componen s in he Xand Zdi ec ions, bu no in he Y
di ec ion. This magne ic ield has a leng h L, and i can be
w i en as
B
ជ
=共B0+B1Z兲u
ជ
z−B1Xu
ជ
x,0艋Y艋L.共1兲
We use he capi al le e s X,Y,Z,T o ep esen magni udes
wi h dimensions. Lowe case x,y,z, co espond o dimen-
sionless quan i ies. We neglec bo de e ec s a ound Y=0 o
Y=L. No e ha his ield ul ills ⵜB
ជ
=0 and also ⵜ⫻B
ជ
=0,as
should be expec ed o a magne ic ield in he egion whe e
he e a e no cu en s. These condi ions we e no ul illed in
he case discussed in ex books such as 关1–3兴.
The Hamil onian which desc ibes a non ela i is ic neu al
pa icle which en e s in his ield is gi en by
H=PX
2+PY
2+PZ
2
2M−
␮
B
ជ
·I
ជ
,共2兲
whe e
␮
is he magne ic momen and I
ជ
is he spin ope a o .
We conside now a wa e packe 兩⌿共T兲;m0典which en e s
in o his ield. Ini ially, he wa e packe can be cha ac e ized
in coo dina e space as a Gaussian which is mo ing in he y
di ec ion, while he ini ial spin p ojec ion along he Zaxis is
m0:
具XYZ,m兩⌿共T=0兲;m0典=Nexp
冉
−X2+Y2+Z2
2
␴
2
冊
⫻exp共ikyY兲
␦
共m,m0兲.共3兲
No e ha , neglec ing he e ec s o he bo de , he Ycompo-
nen o he wa e unc ion is no a ec ed by he in e ac ion.
Bo de e ec s will be ele an when he ansi ime, which
is he ime ha he pa icle akes o go om he si ua ion in
which he ield anishes 共B
ជ
=0兲 o he si ua ion whe e he
ield is s a iona y 共B
ជ
⯝B0u
ជ
z兲, is sho compa ed o he p e-
cession ime o he spin in he magne ic ield. An es ima e o
he ansi ime is =B0/共B1 兲. The p ecession ime is p
=ប/共B0
␮
兲. I can be seen ha , o ealis ic cases, he ansi
ime, al hough sho compa ed o he ime ha he beam
spend wi hin he magne , is always la ge han he p ecession
ime. So he spins o he pa icles o he beam ha e he ime
o adap o he magne ic ield in which hey en e .
The wa e unc ion is gi en by a wa e packe ha can be
ac o ized in o a Ycomponen and an 共X,Z兲componen . The
Ycomponen will e ol e eely inside he magne , because
he Hamil onian does no ha e any in e ac ion e m which
depends on Y, once ha he bo de e ec s 共occu ing a Y
=0 and Y=L兲a e ound o be negligible.
No e ha he wa e packe will s ay wi hin he magne ic
ield du ing a ime
␶
=L/ y, whe e y=បky/M. Assuming ha
he size o he wa e packe
␴
is e y small compa ed o L
and aking in o accoun ha he ansi ime is much
smalle han T, we can conside ha he magne ic ield s a s
a T=0 and inishes a T=
␶
. So we ocus on sol ing he
wo-dimensional ime-dependen p oblem, which co e-
sponds o calcula ing he ime e olu ion be ween he ime
T=0 and T=
␶
in a Hamil onian
H=PX
2+PZ
2
2M−
␮
B
ជ
·I
ជ
,共4兲
conside ing ha he ini ial wa e unc ion is
具XZ;m兩⌽共T=0兲;m0典=Nexp
冉
−X2+Z2
2
␴
2
冊
␦
共m,m0兲.共5兲
I is con enien o make use o dimensionless a iables. So
we de ine x=X/
␴
,z=Z/
␴
, =T/
␶
, and h=H
␶
/ប. Then, he
equa ion o mo ion becomes
h兩⌽共 兲;m0典=id
d 兩⌽共 兲;m0典.共6兲
The dimensionless Hamil onian can be w i en as h=h0+ ,
wi h
h0=A
2共px
2+pz
2兲, =−S关Iz共z+z0兲−Ixx兴,共7兲
whe e px=−id/dx,pz=−id/dz, and he dimensionless pa am-
e e s A,S,z0a e
A=ប
␶
M
␴
2,S=
␮
B1
␶
␴
ប,z0=B0
␴
B1.共8兲
The adiaba ici y pa ame e Ais he a io o he in e ac ion
ime
␶
o he na u al ime o expansion o he Gaussian
packe . The sepa a ion pa ame e Sis he a io o he mo-
men um change induced by he magne ic ield g adien di-
ided by he momen um wid h o he Gaussian packe . The
inhomogenei y pa ame e z0de e mines he ela i e change
o he magne ic ield in he ange o he Gaussian. No e ha
in he posi ion 共x=0,z=−z0兲, he magne ic ield anishes.
No e ha he p oduc AS=
␮
B1
␶
2/M
␴
is independen o ប.
This magni ude is ela ed o he de ia ion o he beam in he
magne . Fo a gi en ajec o y, which is de e mined by a
ixed alue o he p oduc AS, he classical limi is eached
as S→⬁and A→0. No e ha his co esponds o making
ប→0 in Eqs. 共8兲.
Validi y o he semiclassical desc ip ions
We will now discuss he alidi y o he semiclassical and
ex book desc ip ions o he S e n-Ge lach expe imen . I
should be no iced ha , in gene al, a beam o pa icles is no
gi en by a pu e quan um mechanical s a e, bu a he by a
POTEL e al. PHYSICAL REVIEW A 71, 052106 共2005兲
052106-2
mix u e o small quan um wa e packe s. Fo de ini eness, we
conside ha ini ially one has a dis ibu ion o pa icles de-
sc ibed as a Gaussian mix u e, o ange
␴
m, o small Gauss-
ian wa e packe s o ange
␴
. The beam p o ile will hen be
cha ac e ized by a Gaussian o ange
␴
=冑
␴
m
2+
␴
2. The con-
di ions equi ed, in o de o jus i y he semiclassical desc ip-
ion done in 关7兴a e he ollowing.
共a兲The inhomogenei y o he magne ic ield o e he
quan um size o he wa e packe should be small:
␴
B1ⰆB0.
This implies ha z0Ⰷ1.
共b兲The momen um change should be la ge compa ed o
he quan um sp ead o he beam momen um:
␮
B1
␶
Ⰷប/
␴
.
This implies ha SⰇ1.
No e ha hese condi ions a e e y well sa is ied in eal-
is ic si ua ions o S e n-Ge lach expe imen s. Howe e , he
alidi y o he ex book desc ip ion equi es also he a mo e
s ingen condi ion
␴
B1ⰆB0, which equi es a e y s ong
ield B0o , al e na i ely, a e y hin beam.
The pu pose o his wo k is o in es iga e he ull quan-
um solu ion o his p oblem o alues o he pa ame e s z0
and Swhich a e no necessa ily e y la ge, so ha he semi-
classical and ex book desc ip ions become dubious. Ne e -
heless, in o de o ha e a e e ence o compa e he quan um
calcula ion, we ecall he expec ed esul s in he ex book
desc ip ion. The ajec o y o he cen e o he wa e packe
inside he magne is gi en by he exp ession
zm共 兲= 1/2共SA兲m 2,共9兲
which depends on he spin p ojec ion m. No e ha , a e he
in e ac ion 共 =1兲, he posi ions o he cen e o he wa e
packe s o each spin p ojec ion a e gi en by zm共1兲
=SAm/2 and hei eloci ies a e z
˙m共1兲=SAm/2. I , a e he
in e ac ion, he beam e ol es eely du ing a ime d, hen he
posi ions o he cen e o he wa e packe s a e expec ed o be
gi en by
zm共 d兲=共1/2 + d兲共SA兲m.共10兲
As a ypical alue o he d i ime dwe will conside he
ime necessa y o each he posi ion zm=−z0, o he spin
p ojec ion m=−1/2,
d=2z0/共SA兲− 1/2. 共11兲
Thus we would expec ha , a e a d i ime d, pa icles
wi h spin p ojec ion m=1/2 should appea a ound z=z0,x
=0, and pa icles wi h spin p ojec ion m=−1/2 should ap-
pea a ound z=−z0,x=0.
III. NUMERICAL CALCULATIONS
We conside he sca e ing o a spin-1/2 pa icle. We ex-
pand he wa e unc ion in o wo componen s, which ha e
de ini e spin p ojec ions along he zaxis,
具xz;m= 1/2兩⌽共 兲;m0典=
␣
共x,z, 兲ei Sz0/2,
具xz;m= − 1/2兩⌽共 兲;m0典=
␤
共x,z, 兲e−i Sz0/2,共12兲
and he Sch ödinge equa ion o he 共x,z兲plane can be w i -
en as
冤
A
2共px
2+pz
2兲−S
2zS
2x
S
2xA
2共px
2+pz
2兲+S
2z
冥
冋
␣
共x,z, 兲
␤
共x,z, 兲
册
=id
d
冋
␣
共x,z, 兲
␤
共x,z, 兲
册
,共13兲
whe e
␣
共x,z, 兲and
␤
共x,z, 兲a e he componen s o he
spino in he basis o he eigens a es o Iz. The nume ical
solu ion o his equa ion has al eady been pe o med by Ga -
away and S enholm 关9兴. Howe e , hey conside ed he case
in which z0was la ge, so hei nume ical esul co esponded
o he ex book in e p e a ion. A simila p oblem has been
add essed by F anca e al. 关10兴, bu hey made use o he
adiaba ic app oxima ion, neglec ing he kine ic ene gy du -
ing he in e ac ion ime.
To ollow ou app oach we mus i s w i e bo h compo-
nen s o he spino as linea combina ions o ha monic oscil-
la o unc ions, so ha
␣
共x,z, 兲=兺
nm anm共 兲
␾
n共x兲
␾
m共z兲,
FIG. 1. P obabili y dis ibu ion o an unpola ized wa e packe
a e going h ough an inhomogeneous magne ic ield. No e he
ocusing e ec o he lowe componen , which co esponds p e-
dominan ly o m=−1/2. The uppe igu e co esponds o A=0.5,
S=4. The lowe igu e is o A=0.1, S=20, which is close o he
classical limi .
QUANTUM MECHANICAL DESCRIPTION OF STERN-…PHYSICAL REVIEW A 71, 052106 共2005兲
052106-3
␤
共x,z, 兲=兺
nm bnm共 兲
␾
n共x兲
␾
m共z兲,共14兲
whe e
␾
n共x兲and
␾
m共z兲a e he ha monic oscilla o eigen-
s a es o o de nand min he xand zdi ec ions, espec i ely.
To calcula e he ime-dependen coe icien s anm共 兲and
bnm共 兲o he expansion, i is na u al o ew i e Eq. 共13兲in
e ms o he well-known c ea ion and des uc ion ope a o s
ax=1
冑2共x+ipx兲,ax
†=1
冑2共x−ipx兲,
az=1
冑2共z+ipz兲,az
†=1
冑2共z−ipz兲.共15兲
Thus, subs i u ing he ope a o s 共15兲in o Eq. 共13兲, we ob ain
he desi ed sys em o o dina y coupled di e en ial equa ions
o he coe icien s o he expansion o
␣
共x,z, 兲and
␤
共x,z, 兲:
a
˙nm =iA
4关an+2,m冑共n+1兲共n+2兲+an−2,m冑n共n−1兲+an,m+2冑共m+1兲共m+2兲+an,m−2冑m共m−1兲−2anm共n+m+1兲兴
+iS
2冑2关an,m+1冑m+1+an,m−1冑m−共bn+1,m冑n+1+bn−1,m冑n兲e−iSz0 兴,
b
˙nm =iA
4关bn+2,m冑共n+1兲共n+2兲+bn−2,m冑n共n−1兲+bn,m+2冑共m+1兲共m+2兲+bn,m−2冑m共m−1兲−2bnm共n+m+1兲兴
+iS
2冑2关−bn,m+1冑m+1−bn,m−1冑m−共an+1,m冑n+1+an−1,m冑n兲eiSz0 兴,共16兲
whe e he o e do s ands o di e en ia ion wi h espec o
he dimensionless pa ame e . This sys em is sol ed using a
ou h-o de Runge-Ku a me hod. The numbe o ha monic
oscilla o basis unc ions needed in he calcula ion was ypi-
cally o he o de o 40 in each coo dina e.
We ha e pe o med calcula ions using ypical alues o
A=0.5, S=4, and z0=4. This co esponds o a case in which
he magne ic ield anishes a a dis ance o 4
␴
. The ime o
he in e ac ion is such ha he wid h o he beam would
inc ease by a ac o o 冑1+A2. The magne ic ield g adien is
such ha each componen o he magne ic ield will acqui e
a momen um o Sប/2
␴
, in opposi e di ec ions. As a compa i-
son, we ha e also conside ed calcula ions wi h A=0.1, S
=20, and z0=4, which p oduce he same de ia ion o he
beam, bu a e close o he classical limi .
A e he in e ac ion, we conside a d i ime d, gi en by
Eq. 共11兲, du ing which he sys em e ol es in he ee Hamil-
onian, so ha he cen e o he m=±1/2wa e packe would
each he poin z=±z0, acco ding o he ex book desc ip ion.
In Fig. 1 we ep esen he p obabili y dis ibu ion o a
wa e packe , co esponding ini ially o an unpola ized beam.
This is gi en by
P0共x,z兲=1
2兺
mm0
兩具x,z;m兩⌽共 兲;m0典兩2.共17兲
The ocusing e ec can be clea ly seen by compa ing he
shape o he dis ibu ions o he uppe and lowe compo-
nen s, which co espond p edominan ly o m=1/2 and m
=−1/2, espec i ely. The e ec o he ocusing is inc eased
as Adec eases and Sinc eases. So we ha e con i med ha
he ocusing e ec ha was p edic ed in he semiclassical
calcula ion in 关8兴is a genuine esul ha appea s in he quan-
um mechanical calcula ion, al hough i is di used i he
adiaba ici y pa ame e Ahas a sizable alue. I should be
no iced ha his ocusing e ec was also ound in he calcu-
la ions p esen ed in 关9兴.
In con as o he ex book desc ip ion, e en i he ini ial
beam has a de ini e spin p ojec ion along he zaxis, a e he
sca e ing p ocess his spin p ojec ion can change. We ha e
e alua ed he p obabili y ha he pa icles change hei spin
p ojec ion along he zaxis. I should be no iced ha he
p obabili y o going om spin up o spin down is no exac ly
he same as ha o going om spin down o spin up. Fo he
e e ence case 共A=0.5, S=4, z0=4兲, we ob ain ha p共1/2,
−1/2兲=0.0166 and p共−1/2,1/2兲=0.0198.
The spin- lip phenomenon also appea s in he semiclassi-
cal desc ip ion, because no all he pa icles ha compose he
beam see he magne ic ield along he zaxis. The semiclas-
sical spin- lip p obabili y is p共1/2,−1/2兲=p共−1/2,1/2兲
=0.0156, which depends only on he alue o z0. This is in
good quali a i e ag eemen wi h he quan um calcula ions. In
Fig. 2 we ep esen he spa ial dis ibu ion o he spin- lip
p obabili y. No e ha he spin- lip p obabili y anishes o
pa icles coming ou along he zaxis. The spa ial dis ibu ion
o he spin- lip p obabili y is in quali a i e ag eemen wi h
he semiclassical calcula ion, which becomes mo e accu a e
as one makes he limi A→0, S→⬁, wi h AS cons an .
The esul s o ou calcula ions can be summa ized as ol-
lows: When a beam o pa icles, desc ibed by a Gaussian
POTEL e al. PHYSICAL REVIEW A 71, 052106 共2005兲
052106-4
wa e unc ion and wi h a gi en spin p ojec ion along he z
axis, goes h ough an inhomogeneous magne ic ield, mos o
he pa icles sca e as expec ed in he ex book desc ip ion.
Howe e , a sizable ac ion o hem, which depends on z0
共abou 2% o z0=4兲, su e a change o he spin p ojec ion
共spin lip兲. F om hese pa icles ha su e spin lip, abou
hal sca e in he same di ec ion as he majo i y o he pa -
icles and he o he hal sca e in he opposi e di ec ion. We
can conclude ha he spin- lip e ec desc ibed in he semi-
classical desc ip ion, which was no p esen in he ex book
desc ip ion o S e n-Ge lach expe imen s, is suppo ed by
he ull quan um mechanical calcula ions. Also, we con i m
ha he S e n-Ge lach expe imen , when conside ed as a
measu emen appa a us o he spin p ojec ion, is no an ideal
measu emen 共because he e is spin lip兲and i is no ully
eliable 共because he e is no an exac co ela ion be ween
he ini ial spin p ojec ion and he inal posi ion o he pa -
icle兲.
Howe e , he e a e quali a i e ea u es o he ull quan-
um mechanical esul , such as he di e ence be ween up-
down and down-up spin- lip p obabili ies, ha a e no
p esen in he semiclassical desc ip ion and equi e u he
in es iga ion.
IV. APPROXIMATE TREATMENTS
Ha ing sol ed nume ically he p oblem, we will conside
se e al app oxima e ea men s o imp o e ou unde s and-
ing o he phenomena unde conside a ion. The s a ing poin
is he exac e olu ion ope a o and he ee e olu ion ope a-
o
U共 兲= exp关−i共h0+ 兲 兴,U0共 兲= exp共−ih0 兲.共18兲
I should be no iced ha h0and do no commu e. Thus, a
p io i he e is no a single basis o spin s a es whe e he
e olu ion ope a o is diagonal. Ne e heless, i can be a gued
ha he in e ac ion domina es o e he ee Hamil onian h0.
Tha would indica e ha he eigens a es o , which a e
s a es wi h de ini e spin p ojec ion along he magne ic ield
共and hence cohe en in e nal s a es兲, should play an impo -
an ole in he app oxima e solu ion o his p oblem. In his
sec ion we de i e se e al app oxima e exp essions which
make use o expansions o he exac e olu ion ope a o in
e ms o and h0and i s commu a o s.
We can use he coo dina es
␳
=冑共z+z0兲2+x2,
␤
= a c an x
共z+z0兲,共19兲
and e e he spin componen s o he di ec ion o he mag-
ne ic ield a each posi ion:
IB=Izcos共
␤
兲−Ixsin共
␤
兲,IT=Izsin共
␤
兲+Ixcos共
␤
兲.
共20兲
In e ms o hese a iables, he ini ial s a e can be exp essed
as
具
␳
␤
;m兩⌽共 =0兲;m0典
=Nexp
冉
−
␳
2−2
␳
z0cos
␤
+z0
2
2
冊
␦
共m,m0兲
共21兲
and h0and ake he exp essions
h0=A
2共p
␳
2+
␳
−2p
␤
2兲, =−S
␳
IB,共22兲
whe e p
␳
and p
␤
a e he momen a associa ed wi h
␳
and
␤
.
The ele an commu a o s a e he ollowing:
关h0, 兴=iAS共p
␳
IB−兵p
␤
,IT其/2
␳
兲,共23兲
关关h0, 兴, 兴=−AS2共IB
2+IT
2−兵p
␤
,Iy其/2兲.共24兲
No e ha 关关h0, 兴,h0兴=0 and 关关关h0, 兴, 兴,h0兴=0. Fo spin-
1/2 pa icles, I=1/2, IB
2=IT
2=1/4.
A. Adiaba ic app oxima ion
The simples app oxima ion o he e olu ion ope a o
consis s in neglec ing comple ely he e ec o h0. This leads
o he adiaba ic app oxima ion, gi en by
U共 兲⯝exp共−i 兲= exp共i S
␳
IB兲.共25兲
No e ha his exp ession conse es he p ojec ion o he spin
along he di ec ion o he magne ic ield. Thus, i is con e-
nien o expand he ini ial spin s a e in o s a es 兩n典which
ul ill IB兩n典=n兩n典. This can be done conside ing he o a ion
o an angle
␤
a ound he yaxis which akes he z axis o he
di ec ion o he magne ic ield. Thus, he adiaba ic exp es-
sion o he wa e unc ion a e he in e ac ion becomes
具
␳
␤
;m兩⌽共 兲;m0典=Nexp
冉
−
␳
2−2
␳
z0cos
␤
+z0
2
2
冊
⫻兺
ndnm
1/2共
␤
兲exp共in
␳
S 兲dnm0
1/2 共
␤
兲.
共26兲
No e ha his exp ession is equi alen o Eq. 共3.3兲in 关10兴,
FIG. 2. Con ou plo o he p obabili y dis ibu ion o he spin-
lip componen 共spin up o spin down兲o he wa e unc ion. The
maximum is 3.3⫻10−4.
QUANTUM MECHANICAL DESCRIPTION OF STERN-…PHYSICAL REVIEW A 71, 052106 共2005兲
052106-5

whe e hey expanded he wa e unc ion in componen s ha
had de ini e spin p ojec ions along he local magne ic ield.
This exp ession con ains he quali a i e ea u es desc ibed in
he nume ical calcula ion. The e is a spin- lip p obabili y, as
m⫽m0. The ocusing e ec appea s when his adiaba ic
wa e unc ion unde goes a ee e olu ion du ing a ime d
a e he in e ac ion. Howe e , du ing he in e ac ion ime,
he p obabili y dis ibu ion is ozen.
B. Pseudoadiaba ic app oxima ion
The nex app oxima ion consis s in neglec ing he com-
mu a o 关h0, 兴. This leads o he pseudoadiaba ic app oxi-
ma ion, gi en by
U共 兲⯝exp共−i 兲exp共−i h0兲= exp共i S
␳
IB兲U0共 兲.共27兲
This exp ession also conse es he p ojec ion o he spin
along he di ec ion o he magne ic ield, bu s a ing om a
wa e unc ion ha has e ol ed eely du ing he in e ac ion
ime . The wa e unc ion has an analy ic exp ession gi en
by
具
␳
␤
;m兩⌽共 兲;m0典=Nexp
冉
−
␳
2−2
␳
z0cos
␤
+z0
2
2共1+iA 兲
冊
⫻兺
ndnm
1/2共
␤
兲exp共in
␳
S 兲dnm0
1/2 共
␤
兲.
共28兲
The di e ence o his exp ession wi h he adiaba ic one
lies in he ac ha he Gaussian wa e packe ge s wide
du ing he in e ac ion ime, by a ac o 冑1+A2, which is he
widening o he ee wa e packe du ing he in e ac ion ime.
C. Cohe en -s a e app oxima ion
We conside he expansion o he e olu ion ope a o up o
he hi d o de commu a o . The ollowing ela ions can be
de i ed:
U共 兲⯝exp
冉
共−i 兲3
6关关关h0, 兴, 兴兴
冊
exp共−i 兲
⫻exp
冉
共−i 兲2
2关h0, 兴
冊
U0共 兲.共29兲
This exp ession is he basis o an analy ic ea men o he
wa e unc ion. Fo ha pu pose, we no e ha he dominan
e ms in he e olu ion ope a o a e hose which conse e he
spin p ojec ion along he di ec ion o he magne ic ield. The
s ongly oscilla ing ac o exp共−i 兲 ends o cancel he e ms
ha do no conse e IB. We e ain in he expansion only
hose e ms which commu e wi h IB. This leads o he exp es-
sion
U共 兲⯝exp共−i 3AS2/12兲exp共i S
␳
IB兲exp共−i 2ASp
␳
IB兲U0共 兲.
共30兲
The ope a o exp共−i 2ASp
␳
IB兲, when ac ing on eigens a es o
IB, gene a es a displacemen in
␳
, which is gi en by
␳
=
␳
i
+ 2ASIB. This leads o an analy ic exp ession o he wa e
unc ion, gi en by
具
␳
␤
;m兩⌽共 兲,m0典
= exp共iAS2 3/12兲N兺
ndnm
1/2共
␤
兲exp共in
␳
S 兲冑
␳
n
␳
⫻exp
冉
−
␳
n
2−2
␳
nz0cos
␤
−z0
2
2共1+iA 兲
冊
dnm0
1/2 共
␤
兲,共31兲
whe e
␳
n=
␳
−nAS 2/2. This wa e unc ion conse es he
spin p ojec ion along he di ec ion o he magne ic ield.
Thus, he s a es wi h a de ini e spin p ojec ion along he
magne ic ield in each posi ion co espond o he cohe en
in e nal s a es in oduced in Re . 关7兴. So we call his app oxi-
ma ion he cohe en -s a e app oxima ion. No e ha in his
app oxima ion he wa e unc ion no only ge s wide du ing
he in e ac ing egion, bu he componen s wi h di e en al-
ues o IBsepa a e.
D. Symme ized app oxima ion
We can app oxima e he e olu ion ope a o by he ollow-
ing exp ession, which is co ec up o commu a o s o ou h
o de :
U共 兲⯝U0共 /2兲exp兵−i −共−i 兲3关关h0, 兴, 兴/12其U0共 /2兲.
共32兲
Neglec ing he e ms ha do no commu e wi h IB,we
ha e
U共 兲⯝exp共i 3AS2/24兲U0共 /2兲exp共i S
␳
IB兲U0共 /2兲.共33兲
The wa e unc ion can be w i en as
兩⌽共 兲;m0典= exp共iAS2 3/24兲U0共 /2兲兩⌽⬘共 兲;m0典,共34兲
whe e
具
␳
␤
;n兩⌽⬘共 兲;m0典=Nexp
冉
−
␳
2−2
␳
z0cos
␤
+z0
2
2共1+iA /2兲
冊
⫻兺
ndnm
1/2共
␤
兲exp共−in
␳
S 兲dnm0
1/2 共
␤
兲,
共35兲
which, al hough i is no comple ely analy ic, i can be ap-
plied o e alua e he expansion o he wa e unc ion in a
ha monic oscilla o basis. This app oxima ion co esponds o
spli he e ec o U0共 兲du ing he in e ac ion symme ically,
aking hal o i be o e and hal o i a e he in e ac ion.
No e ha he e also he e olu ion associa ed wi h he in e ac-
ion conse es he spin p ojec ion along he magne ic ield.
We call his he symme ized app oxima ion.
E. Compa ison wi h he exac calcula ion
We ha e pe o med calcula ions wi h all he app oxima-
ions. We ind ha he quali a i e cha ac e is ics o he exac
calcula ions discussed abo e, which a e he ocusing e ec
in he componen which goes o nega i e z alues and he
POTEL e al. PHYSICAL REVIEW A 71, 052106 共2005兲
052106-6
p esence o spin- lip componen s, appea in all he calcula-
ions. The quan i a i e di e ences be ween he di e en ap-
p oaches a ise in he momen um dis ibu ion o he spin lip
componen . This comes ou symme ic in he adiaba ic and
pseudoadiaba ic app oxima ions 共same p obabili y dis ibu-
ion o posi i e and nega i e momen a兲and no ully sym-
me ic in he cohe en -s a e o symme ized app oxima ions,
in close ag eemen wi h he exac calcula ions.
To e alua e he quali y o hese app oxima ions, we ha e
calcula ed he a e age o he o e lap be ween he exac and
app oxima e calcula ions. This o e lap is de ined as
O=1
2冏兺
m0
具⌽ex共 =1兲;m0兩⌽ap共 =1兲;m0典冏.共36兲
They a e displayed in Fig. 3, as a unc ion o he adiaba ici y
pa ame e A, o a ixed alue o he p oduc AS=2, which
de e mines he de ia ion o he cen e o he wa e packe in
he magne ic ield, as shown in Eq. 共9兲. The quan i y 1−Ois
abou 10% o a wide ange o alues o A. In pa icula , o
A=0.5 and S=4, 1−0=0.088 o he adiaba ic calcula ion
and 1−O=0.064 o he pseudoadiaba ic calcula ion. On he
con a y, he symme ized and cohe en -s a e app oxima ions
a e much be e , so ha 1−Ois abou 0.1%. In pa icula , o
A=0.5 and S=4, 1−0=0.0015 o he cohe en -s a e and
1−0=0.0006 o he symme ized calcula ions. The eason
o his be e ag eemen a ises om he ac ha he
cohe en -s a e and symme ized calcula ions allow o he
dis o ion in he wa e unc ion p oduced by he magne ic
ield g adien , while o he adiaba ic and pseudoadiaba ic
calcula ions he e ec o he ield con ibu es only o a phase.
In all he calcula ions ha we ha e pe o med, he quali y
o he app oxima ed calcula ions imp o es as one goes om
he adiaba ic o he pseudoadiaba ic o he cohe en s a e and
inally o he symme ized app oxima ions. Globally consid-
e ed, he app oxima ions de e io a e as he p oduc SA ge s
la ge , because hen he e is mo e dis o ion in oduced in he
wa e unc ion due o he combined e ec o he in e ac ion
and he ee Hamil onian.
A e y in e es ing case is he limi A→0, S→⬁ o ixed
alues o AS. Nai ely, one would expec ha he adiaba ic
app oxima ion would be adequa e he e, as he ee Hamil-
onian h0is negligible compa ed o . Howe e , his is no
he case. As shown in Fig. 3, he adiaba ic and pseudoadi-
aba ic app oxima ions a e a he poo , gi ing alues o
1−Oo abou a ew pe cen . The cohe en -s a e and symme-
ized app oxima ions a e e y good o A=0.015, bu hen
hey become wo se o smalle alues o A. Nume ical cal-
cula ions a e e y di icul when Sis la ge, because a la ge
oscilla o basis is needed. An analy ic solu ion o his limi -
ing case would be desi able.
The in e es o his limi case 共A→0, AS cons an 兲is no
only o mal. In nuclea physics he e a e cases in which
weakly bound nuclei in e ac s ongly wi h a ge s du ing a
e y sho ime, so ha he quan um s a e is signi ican ly
dis o ed. The alidi y o he adiaba ic app oxima ion in
hese si ua ions is open o deba e 关12兴.
No e ha in he de ini ion o he o e lap we allow o an
o e all phase di e ence be ween he exac and app oxima e
wa e unc ions. This o e all phase di e ence does no a ec
any obse able. We ind ha he bes app oxima e calcula-
ions 共cohe en s a e and symme ized兲only ep oduce accu-
a ely he phase o he exac wa e unc ion when bo h Aand
Sa e small. We hink ha his is ela ed o he e ec o
highe -o de e ms in he commu a o se ies o he e olu ion
ope a o , which seem o a ec only a global phase in he
wa e unc ion.
So we see om hese app oxima ions ha a c ucial ea-
u e o hem is he ac ha he mos ele an e ms in he
e olu ion ope a o conse e he spin p ojec ion along he lo-
cal di ec ion o he magne ic ield. This is he basis o he
semiclassical calcula ion pe o med in 关7兴, in which he
s a es wi h de ini e spin p ojec ions along he local magne ic
ield we e aken as cohe en in e nal s a es, and hence hei
mo ion could be desc ibed in e ms o ajec o ies.
Despi e he ac ha he app oxima ions discussed he e,
especially he cohe en -s a e and symme ized app oxima-
ions, a e e y accu a e, hey do no desc ibe an impo an
e ec o he exac e olu ion ope a o . In all he app oaches
desc ibed he e, he sca e ing ampli udes o gi en spin p o-
jec ions along he yaxis 共 he beam axis兲a e equal, up o a
phase ac o , o he ampli udes in which he spin p ojec ions
a e e e sed. This is a esul o he ac ha only e ms which
commu e wi h IBa e allowed in he expansion o he e olu-
ion ope a o .
V. REEXAMINING THE STERN-GERLACH
EXPERIMENTS
In he ex book desc ip ion o he S e n-Ge lach expe i-
men , he de lec ion o he beam gi es in o ma ion o he
spin p ojec ion along he zaxis, which is he one ha poin s
along he magne ic ield a he cen e o he beam. The de-
lec ion o he beam is no sensi i e o he spin componen s
along o he di ec ions. I , o a spin-1/2 pa icle, he ini ial
FIG. 3. O e laps o he app oxima e wa e unc ions wi h he
exac one, as a unc ion o he adiaba ici y pa ame e , o SA=2.
The alue 1−O=0 co espond o pe ec ag eemen . The solid line
is he adiaba ic app oxima ion, he dashed line is he pseudoadi-
aba ic app oxima ion, he do ed line is he cohe en -s a e app oxi-
ma ion, and he do -dashed line is he symme ized app oxima ion.
QUANTUM MECHANICAL DESCRIPTION OF STERN-…PHYSICAL REVIEW A 71, 052106 共2005兲
052106-7
spin poin s along he xaxis, mx=+1/2, he ex book desc ip-
ion would indica e ha he pa e n o sca e ed pa icles
would be comple ely equi alen o ha one p oduced by a
mix u e o 50% mz=+1/2 and 50% mz=−1/2 pa icles. The
same would be ue o mx=−1/2. So a S e n-Ge lach expe i-
men is no expec ed o gi e any asymme y be ween di e -
en spin p ojec ions pe pendicula o he zaxis.
To in es iga e his ques ion, we de ine he asymme y o
a gi en axis as he di e ence in he p obabili ies o inding
he sca e ed pa icles in a gi en posi ion in he 共z,x兲plane
o he wo spin p ojec ions. Thus, we ha e
Az共x,z兲=兺
mm0m0
⬘
具x,z;m兩⌽共 兲;m0典具x,z;m兩⌽共 兲;m0
⬘典*具m0兩
␴
z兩m0
⬘典,
共37兲
Ax共x,z兲=兺
mm0m0
⬘
具x,z;m兩⌽共 兲;m0典
⫻具x,z;m兩⌽共 兲;m0
⬘典*具m0兩
␴
x兩m0
⬘典,共38兲
Ay共x,z兲=兺
mm0m0
⬘
具x,z;m兩⌽共 兲;m0典
⫻具x,z;m兩⌽共 兲;m0
⬘典*具m0兩
␴
y兩m0
⬘典.共39兲
No e ha , in he s anda d desc ip ion o he S e n-Ge lach
expe imen , he spin p ojec ion along he zaxis is conse ed,
and hus he asymme ies Axand Ayshould anish a all
poin s. This is no he case. As shown in Fig. 4共b兲, he e is a
di e ence in he pa e n o pa icles sca e ed depending on
he spin p ojec ion along he xaxis. This e ec is ound o
depend on he inhomogenei y o he magne ic ield, which is
de e mined by z0=B0/B1
␴
.I z0is la ge, he inhomogenei y
o he magne ic ield explo ed by he beam is small and so is
Ax. This asymme y can be calcula ed, wi h a ious deg ees
o accu acy, making use o he app oxima e ea men s dis-
cussed he e. I can also be calcula ed wi h he semiclassical
ea men o 关7兴. The o igin o his asymme y can be unde -
s ood by a guing ha he mo ion in an inhomogeneous mag-
ne ic ield conse es he spin p ojec ion along he local mag-
ne ic ield, which has a di e en di ec ion o he di e en
pa s o he wa e unc ion. This links wi h he concep o
cohe en in e nal s a es, which we e in oduced in Re . 关8兴.
The calcula ions in Fig. 4共a兲show also ha he e is an
asymme y Aywhich means ha he e is a dependence o he
spin p ojec ion along he yaxis. This is a dynamical e ec ,
which does no appea in he semiclassical desc ip ion. In
ac , in he analy ic app oxima ions p esen ed he e, he alue
o Ay anishes a e he in e ac ion. Only a e allowing o
some ime o ee e olu ion do non anishing alues o Ay
de elop. The o igin o his asymme y a ises om he e m
AS2p
␤
Iywhich appea s in he double commu a o 关关h0, 兴, 兴.
The e ec o his e m can be unde s ood because p
␤
is he
gene a o o o a ions in he 共x,z兲plane, a ound he poin x
=0, z=−z0, whe e he ield anishes. The e ec o his e m
in he expansion o he e olu ion ope a o would gene a e a
o a ion in he wa e unc ion a ound he poin whe e he ield
anishes, which will be opposi e o he di e en spin p o-
jec ions along he zaxis. Indeed, his e ec compe es wi h
he in e ac ion =S
␳
IB, which ends o p ese e he spin p o-
jec ion along he di ec ion o he ield. The esul o his
compe i ion is ha he magni ude o he asymme y depends
on he a io AS/z0. No e ha he asymme y Ayis associa ed
o he dynamically gene a ed e m AS2p
␤
Iy. This e m de-
pends on he spin p ojec ion Iybu is independen o Yo PY.
So he mo ion in he Ydi ec ion is una ec ed by he dynam-
ics, and hence i is gi en by he ee e olu ion o he Y
componen o he ini ial wa e packe .
FIG. 4. Asymme ies o pa icles pola ized along he y共a兲,x
共b兲, and z共c兲di ec ions. No e ha he maximum asymme y occu s
o pa icles pola ized along he zaxis, bu ha he e a e impo an
asymme ies o pa icles pola ized along he xand yaxes.
POTEL e al. PHYSICAL REVIEW A 71, 052106 共2005兲
052106-8
The ac ha all he asymme ies a e non anishing and
also ha hey ha e di e en beha io as a unc ion o 共x,z兲
leads o an exci ing possibili y. Conside ha we ha e a
beam o pa icles, so ha we do no know hei pola iza ion
s a e. We can make he beam go h ough an inhomogeneous
ield, as desc ibed he e, and de ec he pa e n o sca e ed
pa icles. Le he pola iza ion s a e be desc ibed ini ially as a
densi y ma ix
␳
=1/2共I+px
␴
x+py
␴
y+pz
␴
z兲, whe e p
ជ
is a
ec o which measu es he deg ee and di ec ion o he beam
pola iza ion. Then, he densi y o pa icles de ec ed in he
共x,z兲plane will be p opo ional o
P共x,z兲=P0共x,z兲+1
2关pxAx共x,z兲+pyAy共x,z兲+pzAz共x,z兲兴.
共40兲
This allows us o ob ain all he componen s o he pola iza-
ion ec o om he pa e n o sca e ed pa icles, when a
su icien numbe o pa icles a e de ec ed. No e ha , in con-
as o exp ession 共40兲, he ex book desc ip ion o he
S e n-Ge lach expe imen would be consis en wi h a p ob-
abili y densi y gi en by
P共x,z兲=P0共x,z兲+1
2pzAz共x,z兲,
Az共x,z兲=2P0共x,z兲,z⬎0,
Az共x,z兲=−2P0共x,z兲,z⬍0. 共41兲
This exp ession, when applicable, would allow one o ob ain
in o ma ion only on he alue o pz.
VI. SUMMARY AND CONCLUSIONS
We ha e in es iga ed he mo ion o a pa icle wi h spin in
an inhomogeneous magne ic ield using a quan um mechani-
cal amewo k. Ou aim is o in es iga e in de ail he limi a-
ions o he usual ex book app oach o S e n-Ge lach expe i-
men s, which assumes ha he spin p ojec ion along he
di ec ion o he magne ic ield is conse ed, while di e en
spin componen s acqui e a momen um which depends on he
g adien o he ield.
We ind ha , consis en ly wi h a p e ious semiclassical
analysis, he e is a sizable p obabili y o spin lip, which
depends on he inhomogenei y o he ield. Besides, he e is
a ocusing e ec in he componen ha de ia es owa ds he
di ec ion in which he modulus o he ield dec eases. These
cha ac e is ics a e e y obus and occu in dynamical si ua-
ions which a e a om he semiclassical limi .
Thus, we can conclude ha he S e n-Ge lach expe imen
is no , e en in p inciple, an ideal expe imen , which would
“p ojec ” he in e nal s a e in o he eigen alues o he mea-
su emen ope a o . Mo eo e , he expe imen is no ully e-
liable, as he posi ions o momen a o he pa icles do no
gi e unequi ocal in o ma ion on he spin p ojec ion. The
magni ude ha de e mines how close a S e n-Ge lach expe i-
men is o an ideal eliable measu emen is z0=
␴
B0/B1. Only
when he magne ic ield B0is e y la ge compa ed o i s
g adien o when he size o he beam
␴
is e y small would
he S e n-Ge lach expe imen app oxima e o an ideal eli-
able measu emen .
We ha e in es iga ed di e en app oxima e ea men s o
he exac quan um mechanical p oblem. We ind ha , o a
good app oxima ion, he in e ac ion occu s as i he spin p o-
jec ion along he magne ic ield a each posi ion was con-
se ed. This indica es ha , o each posi ion in he inhomo-
geneous ield, he s a es wi h a gi en spin p ojec ion along
he magne ic ield a e cohe en in e nal s a es. Then, p o-
ided ha he quan um size o he wa e unc ion is small
compa ed o he inhomogenei y o he magne ic ield, i is
meaning ul o app oxima e he mo ion o hese s a es in
e ms o classical ajec o ies. This jus i ies he ea men pe -
o med in 关7兴.
I is in e es ing o no e ha he adiaba ic app oxima ion is
no accu a e, e en in he limi o small A共la ge mass o sho
in e ac ion ime兲, i , a he same ime, he in e ac ion is la ge
so ha i gene a es a ixed de lec ion angle. This obse a ion
can be ele an o cases, such as in nuclea physics 关12兴,in
which, al hough he collision imes a e sho o gua an ee he
alidi y o he adiaba ic app oxima ion, he o ces a e so
s ong o p oduce a ini e de lec ion.
Ou calcula ions indica e ha he S e n-Ge lach expe i-
men is no an ideal measu ing appa a us, in he sense o Re .
关5兴. Howe e , his does no mean ha one canno acqui e
accu a e knowledge om he spin s a e o he p ojec ile by
obse ing he s a is ical esul s o he expe imen . On he
con a y, while an idealized S e n-Ge lach expe imen will
no gi e any in o ma ion o he spin p ojec ion along he xo
yaxis, he analysis o a ealis ic S e n-Ge lach expe imen ,
such as modeled in ou calcula ions, can gi e he alue o all
he componen s o he densi y ma ix ha desc ibes he po-
la iza ion o he beam.
Ou analysis suppo s he idea ha he in e p e a ion o
ealis ic expe imen s does no equi e he use o he educ-
ion p inciple, as discussed by se e al au ho s in 关11兴. Thus,
he in e ac ion be ween he spin and magne ic ield, which is
desc ibed in a pu ely quan um mechanical amewo k, gen-
e a es a co ela ion be ween he spin pola iza ion o he beam
and he inal posi ion o he pa icles o he beam. A mea-
su emen o a su icien ly la ge numbe o hese posi ions
allows one o de e mine he componen s o he densi y ma-
ix o he beam wi h su icien s a is ical accu acy. The e-
duc ion p inciple is no equi ed in his a gumen .
ACKNOWLEDGMENT
This wo k has been pa ially suppo ed by he Spanish
MCyT, P ojec s No. FPA2002-04181-C04-04 and BFM2002-
03315.
QUANTUM MECHANICAL DESCRIPTION OF STERN-…PHYSICAL REVIEW A 71, 052106 共2005兲
052106-9