The angular momentum of a magnetically trapped atomic condensate

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by  P. Zhang

The angular momentum of a magnetically trapped atomic condensate
P. Zhang,1 H. H. Jen,1 C. P. Sun,2 and L. You1,3
1School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
2Institute of Theoretical Physics, Chinese Academy of Science, Beijing 100080, China
3Center for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China
(Dated: February 4, 2008)
7
0 Foranatomiccondensateinanaxiallysymmetricmagnetictrap,thesumoftheaxialcomponents
0 oftheorbitalangularmomentumandthehyperfinespinisconserved. InsideanIoffe-Pritchardtrap
2 (IPT) whose magnetic field (B-field) is not axially symmetric, the difference of the two becomes
surprisingly conserved. In thispaper we investigate therelationship between thevalues of the sum
n
ordifferenceangularmomentumsforanatomiccondensateinsideamagnetictrapandtheassociated
a
gaugepotentialinducedbytheadiabaticapproximation. Ourresultprovidessignificantnewinsight
J
intothe vorticity of magnetically trapped atomic quantum gases.
9
1
PACSnumbers: 03.75.Mn,03.75.Lm,03.65.Vf,67.57.Fg

]
r
e A magnetic trap constitutes one of the key enabling This paper is organized as follows. We first consider
h
technologies for the recent successes in atomic quan- a spin-1 condensate in a QT. Making use of an effective
t
o tum gases [1]. The most commonly employed mag- energy functional appropriate for the adiabatic approxi-
.
t netic traps includes the quadrupole trap (QT) as in the mation [5], we prove Jz [ 1,1] with the actual value
a ∈ −
magneto-optical-trap (MOT) configuration [2] and the determined by the angle between the z-axis and the di-
m
Ioffe-Pritchard trap (IPT) [3]. The direction of the B- rection of the B-field. We then generalize to the spin-F
-
d field formingthe magnetic trapis generallya functionof case. Finally, our result is extended to an IPT.
n the spatial position. For a trapped atom, its hyperfine The Hamiltonian of a spin-1 atomic condensate with
o spin adiabatically follows the changing B-field direction, N atoms in a magnetic trap is H = H +H with the
S I
c
and the atom remains aligned (or anti-aligned) with re- single atom part
[
spect to the local B-field. As a result of the adiabatic
2 approximation, the center of mass motion of a magneti- H = ψˆ†(~r) ∇2 +µ g B(~r)F~ nˆ(~r) ψˆ(~r)d~r,
v cally trapped atom experiences an induced gauge poten- S Z (cid:20)−2M B F · (cid:21)
6
tial from the changing B-field [4, 5].
7 and the atom-atom interaction Hamiltonian
3 Inavarietyofmagnetictraps,e.g.,inaQT,theB-field
1 isinvariantwithrespecttorotationsalongafixedz-axis. H = ψˆ(z)†(~r)ψˆ(z)†(~r′)Vmn(~r,~r′)ψˆ(z)(~r)ψˆ(z)(~r′)d~rd~r′.
1 I m n pq p q
As the cse of single particle dynamics [6], such a SO(2) Z
6 mX,n,p,q
symmetry leads to the conservation of the z-component
0
/ Jz (= Lz +Fz) of the total angular momentum or the ψˆ(~r) = [ψˆ(z)(~r),ψˆ(z)(~r),ψˆ(z)(~r)]T denotes the annihila-
t −1 0 1
a sum of the z-components of the atomic spatial angular tion field operator for the z-quantized F -component of
m momentumL~ andthehyperfinespinF~. Adifferentsym- z
m,n,p,q = 0, 1. M is the atomic mass, and µ is the
B
- metry exists for an IPT giving rise to a corresponding Bohr magneto±n. B(~r) and nˆ(~r) denote the strength and
d
n conserved quantity Dz (= Lz − Fz), the difference of direction of the local B-field. ~ = 1 is assumed. The
o Lz and Fz. To our knowledge, this surprising property Lande g factor is gF=1 = 1/2.
c has never before been identified explicitly. We thus feel Within the mean field a−pproximation, the field opera-
v: obliged to present this letter because it significantly af- tor ψˆ(~r) is replaced by its average ψˆ(~r) . To introduce
i fectsthevorticalpropertiesofaglobalcondensateground the adiabatic approximation, we dehfine ia group of nor-
X
state in a magnetic trap.
malized scalar wave functions ϕ (~r):
u
r
a Our work is focused on a detailed investigation of the
relationship between the gauge potential and the associ- ψˆ(~r) = √NξB(b,~r)ϕ (~r), (1)
b
h i
ated values of Jz (Dz) in a magnetic trap. For a spin-F b=X0,±1
condensate,duetotheappearanceoftheadiabaticgauge
potential,thepossiblevaluesofJ orD arerestrictedto where ξB(b,~r) is the eigenstate of the B-quantized spin
z z
adefinite region[ F,F]. Thegaugepotentialofourfor- componentF~ nˆ(~r)witheigenvalueb,satisfyingtherela-
mulation is direct−ly related to the effective trap rotation tionsF~ nˆ(~r)ξ·B(b,~r)=bξB(b,~r)andξB†(b,~r)ξB(b′,~r)=
·
studied earlier in Ref. [5]. While Ho and Shenoy mainly δb,b′. Inthez-quantizedrepresentation,ittakesthe form
studied the orbital angular momentum component in an ξB(b,~r) = [ξB (b,~r),ξB(b,~r),ξB(b,~r)]T. In this study,
−1 0 1
IPT [5], instead we concentrate on the conserved quan- it is important to distinguish ξB(b,~r) from the eigen-
tity, the sum (J ) or difference (D ) for a QT or a IPT. states ξ (0, 1) of F with eigenvalues 0, 1. In explicit
z z z z
± ±
2
form, we have ξ ( 1) = [1,0,0]T, ξ (0) = [0,1,0]T, and onlyonenormalizedgroundstate(uptoglobalphasefac-
z z
−
ξ (1)=[0,0,1]T. tors)asinthesituationconsideredhereforacondensate,
z
A magnetic dipole precesses around the direction of a it has to be a common eigenstate for all rotation opera-
B-field. Majorona transitions between different ξB(b,~r) tors exp[θ∂/(∂φ)], i.e., an eigenstate of i∂/(∂φ). Thus,
statescanbe neglectedwhenB(~r)is largeenough. Thus we take ϕ (~r) = ϕ˜ (ρ,z)exp(isφ). On substituting into
g g
theatomichyperfinespinadiabaticallyfreezesinthelow- Eq. (2),weobtainthez-quantizedmeanfield ψˆ(~r) for
g
field seeking state ξB( 1,~r) during the trapped center the ground state h i
−
of mass motion, and ϕ (~r) = ϕ(~r) and ϕ (~r) = 0.
−1 0,+1
Similarly the z-quantized mean field becomes ψˆ(~r) g =√NξB( 1,~r)ϕ˜g(ρ,z)exp(isφ), (5)
h i −
ψˆ(~r) ψˆ(~r) =√NξB( 1,~r)ϕ(~r), (2) which is an eigenstate of Jz with an eigenvalue s, i.e.,
≈h i −
with ϕ(~r) a B-quantized scalar function. Jz ψˆ(~r) g =s ψˆ(~r) g, (6)
h i h i
Substituting Eq. (2) into the expression of H, we can
obtain the expression of the condensate energy E as because of Eq. (4).
ad
a functional of the scalar wave function ϕ(~r): Ead[ϕ] = Equation (5) and the expansion hψˆ(~r)ig =
N ϕ∗(~r) ϕ(~r)d~r. Here defined as [4, 5]: ψˆ(z)(~r) ξ (m) gives
Ead Ead m=0,±1h m ig z
R P
ad = [−i∇+A~(~r)]2 +µBB(~r)+W +Vo+ g2 ϕ2. (3) hψˆm(z)ig =√NξmB(−1,~r)ϕg(~r)=bm(ρ,z)ϕ˜g(ρ,z)ei(s−m)φ,(7)
E 2M 2 | |
The gauge potential induced by the adiabatic approx- with bm = ξz†(m)e−iFyβ(ρ,z)ξz(−1). The individual spin
components of state Eq. (7) naturally carry topological
imation is A~(~r) = iξB†( 1,~r) ξB( 1,~r) and W =
( ξB†( 1,~r) ξB( −1,~r) −A~ A~)/∇(2M)−. Thetrappoten- windings as a direct result of the conservation of Jz.
To determine the value of s or J for the ground
|∇ − ∇ − |− · z
tialµBB(~r)isaugmentedbyVo(ρ,z)fromothersources, state, we first compute the gauge potential A~(~r) for a
e.g.,anopticalpotentialwithrotationalsymmetryalong QT. With the expression of ξB( 1,~r), we find A~(~r) =
the z-axis. Under the approximation of contact pseudo- −
cosβ(ρ,z)eˆ /ρ. Using the expressions for A~(~r), , and
potentials,trappedatomscollideinthesamespinaligned φ Ead
E , it is easy to show that for a scalar wave function
state. Thus the g term is proportionalto the scattering ad
2 ϕ˜ (ρ,z)exp(imφ) with any integer m, the energy func-
length a in the total spin channel F =2. g
2 tot tional E satisfies
We show below that the gauge potential A~(~r) con- ad
strains the values of Jz for a condensate ground state. E [ϕ˜ (ρ,z)eimφ]=E [ϕ˜ ]+∆E [ϕ˜ ], (8)
ad g ad g m g
In cylindrical coordinate (ρ,φ,z), the B-field of a QT is
expressed as B~(~r) = B′(ρeˆ 2zeˆ ). An optical poten- with E [ϕ˜ ]= d~rϕ˜∗(ρ,z)2[(m2+4mcosβ)/(2Mρ2)].
ρ− z m g | g |
tial Vo(ρ,z) is introduced to push atoms away from the InadditiontothRe centrifugaltermproportionaltom2, a
region of small B(~r), eliminating the deadly Majorona term linear in m appears due to the A~ term in .
ad
·∇ E
transitions [7, 8]. Because this B-field is cylindrically In the following we show that the above linear term
symmetric, F~ nˆ(~r) commutes with the z-component of is important for the value s, which we determine with
·
the total atomic angularmomentum Jz =Lz+Fz. This a variational approach. Because Ead takes its minimal
allows us to choose ξB( 1,~r) as the common eigenstate value in the state ϕ˜gexp[isφ], we have Ead[ϕ˜geisφ]
of F~ nˆ(~r) and Jz with−any possible eigenvalues. In this Ead[ϕ˜gei(s±1)φ]. Together with Eq. (8), we find the ne≤c-
study· we choose ξB( 1,~r) for convenience to satisfy essaryconditionsatisfiedby s: ∆Es[ϕ˜g] ∆Es±1[ϕ˜g] or
− s+C 1/2, where the coefficient C is≤defined as
J ξB( 1,~r)=0. (4) | |≤
z
− d~r ϕ˜∗(ρ,z)2[cosβ(ρ,z)/ρ2]
C = | g | .
This constraint on Jz limits the respective values for Lz R d~r ϕ˜∗(ρ,z)2[1/ρ2]
or F , in a sense equivalent to a gauge choice. The ac- | g |
z R
tual value of Jz for the ground state of a condensate is When the correlation between cosβ(ρ,z) and ρ−2
determined by appropriate system parameters. Explic- is neglected, the factor is approximated by C
itly,asimplerotationgivesξB( 1,~r)=exp[iφ]exp[ iF~ d~r ϕ˜∗(ρ,z)2cosβ(ρ,z), i.e., the expectation value o≈f
− − · | g |
eˆφβ(ρ,z)]ξz(−1). The angle between the z-axisandnˆ(~r) Rcosβ(ρ,z) for state ϕ˜g. Since |C|<1, we have |s+C|≥
is β(ρ,z)=arccos[ 2z/ ρ2+4z2]. s 1, which is the same as s 1 or s [ 1,1].
− | | − | | ≤ ∈ −
The B-quantized groupnd state scalar mean field wave WithoutthegaugepotentialA~(~r),wewouldalwayshave
function is denoted as ϕg(~r), determined from a mini- ∆Em[ϕ˜g] = m2 d~r|ϕ˜∗g(ρ,z)|2(2Mρ2)−1 ≥ 0. Thus the
mization of Ead[ϕ]. The cylindrical symmetry of both value of s definiRtely would be zero. Therefore, the ap-
B~(~r) and V assures E [ϕ(ρ,φ,z)] = E [ϕ(ρ,φ+θ,z)] pearance of a non-zero valued s [ 1,1] arises due to
o ad ad
∈ −
for any θ. Due to this SO(2) symmetry, if there exists the induced gauge potential.
3
We now generalize our result to atoms with an arbi- π
traryF andinside anyaxiallysymmetric B-fields. Anal- (a) φ
ogouslywecanprovethatthevaluesofJ intheground
z
state satisfies the necessary condition 0 θ(φ)
|s−ηFFC|≤1/2, (9) −π0 F z=1 π 0 π 0F z=−1 π 2π
and s [ F,F] with η = sign(g ). The result of s
∈ − F F ∈ (b)
[ F,F] and the conservation of J is independent of the
z
− 15
form of the atomic interaction potential. Although its ρ
strengthg2 doesaffectthewavefunctionshape,thuscan 10
influence the value of s through the factor C.
z
The condition Eq. (9) also allows for a rough esti-
−5 0 5 −5 0 5 −5 0 5
mate of L . A straightforward calculation gives L =
z z
h i
s η F d~r ϕ˜∗(ρ,z)2cosβ(ρ,z) for the spinor mean 15 (c)
− F | g |
field ψˆ(~rR) g. Neglecting the correlation between ρ−2 0 y
h i
and cosβ(ρ,z) as before, the value of L becomes
z
h i
approximately s η FC, which lies always in the re- −15
F F =1 F =−1
gion [ 1/2,1/2] −according to Eq. (9). Therefore, the x x x
− −10 0 10 −10 0 10 −10 0 10
value L , or the weighted averageof the winding num-
z
h i
bers, is generally a small number, despite the wind-
ing number s m itself, for the component ψˆ(z)(~r) , FIG. 1: (Color online). (a) The phases of the z-quantized
− h m ig components hψˆ(z)i (left), hψˆ(z)i (middle), and hψˆ(z)i (right
may take any integer in the region [ 2F,2F]. We find 1 0 −1
− panel), as functions of the azimuth angle φ; (b) the density
F = η F d~r ϕ˜∗(ρ,z)2cosβ(ρ,z) from the expres-
h zi F | g | distributions |hψˆ(z)i|2 (left), |hψˆ(z)i|2 (middle), and |hψˆ(z)i|2
sion of hLzi, Ra qualitative reflection that atomic hyper- (right panel) of 1the z-quantized0components as functio−ns1 of
fine spin is aligned (gF > 0) or anti-aligned (gF < 0) ρandz;(c) theintegrated densitydistributionsR |hψˆ(x)i|2dz
with respect to the local B-field. 1
Ourresultaboveallowsforthedirectcreationofvortex (left), R |hψˆ0(x)i|2dz (middle), and R |hψˆ−(x1)i|2dz (right panel)
of the x-quantized components as functions of x and y. The
states in a quadrupole trapped atomic condensate. For
unitsfor x,y, ρ,and z in (b) and (c) are all arbitrary.
example, assume a spin-1 condensate in a QT plus an
“optical plug” [8] satisfies V (ρ,z) = V (ρ, z), then we
o o
−
findC =0ands=0duetothespatialreflectionsymme- mediately note that ψˆ(x)(~r) is a superposition of vor-
m g
tryaboutthex y plane. The groundstate components tex states with definhite windiing numbers 0 or 1, e.g.,
twhψiˆoit±(nzh1,)(wt~rhi)enigdloinwthgfienneul−dmausbeteeorkmsina∓gt1icaaatloclcmyosrcdaairrnergyttrpoaepErpsqeis.dt(en7ne)ta.rcIutnhraeredxnd-tiys- dhψeˆn0(xs)i(t~ry)idgis=trib(√utNio/n√o2f)h[bψˆ−0(x1,±)(ρ1(,~rz))iegiφa−s sbh1o(wρ,nz)ine−F±iφig].. T1(hce)
clearly illustrates the interference pattern along the eˆ
φ
plane at z = 0 because B(~r) is an increasing function
| | direction. As is demonstrated in Fig. 1(c), the mid-
of z. The populations for the three z-quantized states,
dlepanelfor ψˆ(x)(~r) 2 clearlydisplaysthedoublepeak
determinedby ξ0B(−1,~r) andξ±B1(−1,~r), areofthe same structure alon|hg t0he azii|muthaldirection, arisingfrom the
order of magnitudes. Therefore, when a ground state
interferenceofthe termsproportionalto ei±φ. Thus,if a
condensate in the “plugged” QT is created, its 1 com-
± Stern-GerlachB-fieldisusedtoseparatethex-quantized
ponents ψˆ(z)(~r) aresinglequantizedvortexstatesand
h ±1 ig components ψˆ(x)(~r) , a superposition of vortices with
can be directly resolved with a Stern-Gerlach B-field as h m ig
different winding numbers would be obtained.
used in Ref. [9].
We now extend our result for an axially symmetric
Thequalitativeexampleaboveis confirmedbythe nu-
magnetictrapto the widely usedIPT whoseB-fieldpos-
merical solution for a condensate of 5 106 23Na atoms
× sesses a different symmetry. In the region near the z-
in a quadrupole plus a plug trap. We take B′ = 22
axis, B~(~r) = B′[cos(2φ)eˆ sin(2φ)eˆ + heˆ ], the an-
Gauss/cm and Vo = Uoexp[ ρ2/σ2] with Uo = (2π)8 ρ − φ z
− × gle β(ρ,z) between the local B-field and the z-axis sat-
104Hz and σ = 7.4µm. The ground state distribution
p = d~r ψˆ(z)(~r) 2 is found to be p = 27.2% and isfies cosβ(ρ,z) = h/ ρ2+h2. In this case Jz is no
i |h i i| ±1 longer conserved due tpo the lack of the SO(2) symme-
p0 =4R5.6%. The phase and density distributions for the
try. However, we find that D is now conserved be-
three components hψˆ0(z,±)1(~r)ig are shownin Fig. 1 (a, b). cause it commutes with F~ B~(~r)z. Therefore, we can se-
We also can expand the groundstate hψˆ(~r)ig in terms lect the low field seeking hy·perfine spin state ξB(ηFF,~r)
of the eigenstates ξx(m) of Fx with eigenvalues m: as the eigenstate of Dz with an eigenvalue ηFF, the
hψˆ(~r)ig = m=0,±1√Nhψˆm(x)(~r)igξx(m). We then im- same spin state as used in [5], again define−d through
P
4
a rotation ξB(η F,~r) = exp[ iF~ nˆ β]ξ (η F). For is quite challenging inside an IPT. This was considered
F ⊥ z F
− ·
h > 0, we find the induced gauge potential becomes quantitatively by Ho and Shenoy [5], who obtained an
A~(~r)= η F(1 cosβ(ρ,z))eˆ /ρ. approximategaugepotentialresemblinganeffectivetrap
F ϕ
− −
Adopting the same notation as before we denote rotation when expanded to the first order of ρ. How-
ψˆ(~r) = √Nϕ (~r)ξB(η F,~r). Interestingly we find ever, their expansion easily fails away from the z-axis.
g g F
h i
E [ϕ(ρ,φ,z)]=E [ϕ(ρ,φ+θ,z)]remainssatisfied,and A numerical discussion on this challenge is provided in
ad ad
the ground state takes the form ϕ = ϕ˜ (ρ,z)exp(iuφ). [10]. The more general constraint of D = η F, i. e.,
g g z F
6 −
Therefore ψˆ(~r) is the eigenstate of D with an eigen- the necessary condition Eq. (10) of the angular momen-
g z
h i
value d = u η F, and its components ψˆ(z)(~r) = tum difference Dz, was not obtained in [5, 10]. From
F m g
− h i a direct calculation, it can be proved that if C is ap-
√Nb′ (ρ,z)ei(m+u−ηFF)φ analogously carry a persistent
m proximated by d~r ϕ˜∗(ρ,z)2cosβ(ρ,z) and d = η F,
cb′murr=enξtz†(wmit)he−aiFywβiξnzd(ηinFgFn).umThbiesrremsu+ltiusc−onηsFisFte.ntHweitrhe wwehihchavceahnLnziev=eRrdb+e|gηgrFeFatRerd|~rth|ϕ˜a∗gn(ρ1,/z2)|2a[c1co−rdcionsg6βt(oρ,FEz)q],.
thegroundstatevortexphasediagramforanF =1con-
(10).
densate found numerically in the z =0 plane of an IPT.
TheconservationofD asfoundbyus,however,callsfor Insummary,wehaveinvestigatedtheangularmomen-
z
asimplerlabellingofeachvortexphasebecauseonlyone tum of a magnetically trapped condensate. Inside an
of three integers (m , m , m ) is independent, as with axially symmetric trap such as a QT, the total angular
1 0 −1
Eq. (15) of Ref. [10]. momentum Jz along the symmetric z-axisis found to be
Following the same reasoning as before, we find conserved,while the angularmomentumdifference Dz is
conserved in an IPT. Both conservation laws reflect the
d+ηFF(1 C) 1/2, (10) underlying symmetries of the traps’ magnetic fields, and
− ≤
(cid:12) (cid:12) thevaluesofJ orD inthegroundstatesaredetermined
(cid:12) (cid:12) z z
for d = ηFF, and d [ F,F] or the value of Dz in the by the gauge potential A~(~r). In the global ground state,
6 ∈ −
ground state lies in the region [ F,F].
the corresponding eigenvalues of J and D are limited
− z z
In an IPT, atoms are trapped near the z-axis where
to [ F,F] with the precise values directly related to
the B-field is essentially along the z-axis direction and ∈ −
the angle between the local B-field and the z-axis.
ξB(η F,~r) is approximately the eigenstate ξz(η F). L
F F z
Our results provide significant insights into the study
thenisessentiallyalwayszerocorrespondingtoaground
of magnetically trapped condensates. The conservation
state without a vortex. The angular momentum differ-
laws we discuss reveal an important observational con-
ence D then becomes d= η F.
Severzal previous proposa−ls F[11] and experiments [12] sequence: the components ψˆm(z)(~r) g of a condensate
h i
oncreatingvortexstatesunknowinglyhaveusedthefact ground state ψˆ(~r) g automatically carry persistent cur-
h i
that ψˆ(~r) inanIPTisaneigenstateofD . Forexam- rents with different winding numbers. Furthermore, ac-
g z
h i
ple, in the experiment of Ref. [12], spin-1 atoms initially cording to the conditions Eqs. (9) and (10), the values
were prepared in the internal state ξz( 1) (Fz = 1) of Jz or Dz, or the topological winding numbers of the
− −
with no vorticity in its spatial mode. This corresponds spatialwavefunctions,canbecontrolledthroughthean-
to D =1. To create a vortex state, the bias field along gle β(ρ,z) between the local B-field and the z-axis. We
z
the z-axis was adiabatically inverted from the +z to the have shown explicitly for a condensate in a QT with an
z direction. In this process the atomic internal state opticalplug,wheretheatomicpopulationsforthe2F+1
−
was changed from ξz( 1) to ξz(1). If the whole opera- components have approximately the same order of mag-
−
tionisadiabatic,thecommutator[F~ B~(~r,t),D ]=0will nitude. Therefore, vortex states can be present already
z
bemaintained. ConsequentlyD isc·onserved. Intheend inagroundstatecondensatewithoutrequiringadiabatic
z
when the internal state was changed to ξz(1) (F = 1), operations as in [9, 12].
z
itsorbitalangularmomentumbecameLz =Dz+Fz =2 We thank Dr. D. L. Zhou, Mr. B. Sun, Profs. C.
or a double vortex spatial mode as was observed [12]. Raman and W. Ketterle for helpful discussions. This
InRef. [9]whenthebiasfieldisadiabaticallyswitched work is supported by NSF, NASA, and NSFC.
off, the direction of the B-field adiabatically changes to
lie in the x-y plane. D is again conserved during this
z
process (= 1). When complete, the atomic hyperfine
spin state is changed from ξz( 1) to a superposition of
all three components ξz(0, 1)−. As a result, different F [1] T. Bergeman, G. Erez, and H. J. Metcalf, Phys. Rev. A
± z 35, 1535 (1987).
components are then associated with vortex states with
[2] E. L. Raab et al., Phys.Rev.Lett. 59, 2631 (1987).
corresponding L = D +F . When the three internal
z z z [3] D. E. Pritchard, Phys. Rev.Lett. 51, 1336 (1983).
statesareseparated,twoofthemareobservedtocontain [4] C.A.Mead andD.G.Truhlar,J.Chem.Phys.70,2284
vortices with non-zero winding numbers. (1979); C. A.Mead, Phys.Rev.Lett. 59, 161 (1987); C.
Creating a ground state condensate with Dz = ηFF P. Sun and M. L. Ge, Phys.Rev.D 41, 1349 (1990).
6 −
5
[5] T. Ho and V. B. Shenoy, Phys. Rev. Lett. 77, 2595 (2003).
(1996). [10] E. N. Bulgakov and A. F. Sadreev, Phys. Rev.Lett. 90,
[6] T.H.Bergemanetal.,J.Opt.Soc.Am.B6,2249(1989). 200401 (2003).
[7] K.B. Daviset al.,Phys.Rev.Lett. 75, 3969 (1995). [11] T. Isoshima et al.,Phys. Rev.A 61, 063610 (2000).
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(2005). (2002).
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