Table Of ContentKibble–Zurek mechanism in a trapped ferromagnetic Bose–Einstein condensate
Hiroki Saito1, Yuki Kawaguchi2, and Masahito Ueda3
1Department of Engineering Science, University of Electro-Communications, Tokyo 182-8585, Japan
2Department of Applied Physics and Quantum-Phase Electronics Center, University of Tokyo, Tokyo 113-0032, Japan
3Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
Spontaneous spin vortex formation in the magnetic phase transition of a trapped spin-1 Bose–
Einstein condensate is investigated using mean-field theory. In a harmonic trapping potential, an
3 inhomogeneous atomic density leads to spatial variations of the critical point, magnetization time
1 scale, andspincorrelation length. TheKibble–Zurekphenomenaareshown toemergeeveninsuch
0 inhomogeneous systems, when thequenchof thequadraticZeeman energy is fast enough. Forslow
2
quench, the magnetized region gradually expands from the center of the trap pushing out spin
n vortices, which hinderstheKibble–Zurek mechanism from occurring. A harmonic trap with aplug
a potential is also taken into account.
J
1
2 I. INTRODUCTION dynamics of a spin-1 BEC confined in a harmonic trap-
ping potential to show that the KZ mechanism can be
] observedin realistic experiments. We will show that the
s Symmetry breaking phase transitions are considered
a inhomogeneity of the trapped system has two effects on
to play crucial roles in the early universe. As the hot
g the KZ properties. The first one is caused by the spatial
universe cooled down, the phase transitions broke the
- dependence of the spin correlation length. The number
t symmetries of the vacuum fields. Since causally discon-
n of spin vortices created by the KZ mechanism depends
nected regions acquire independent values of the order
a onthespincorrelationlength,andtherefore,dependson
u parameter in the course of the phase transition, topo-
the position. The secondone originatesfromthe compe-
q logical defects can be left behind1, such as monopoles,
tition between two velocities. Since the density is high
at. strings,anddomainwalls. Itwasproposedthatthis cos- around the center of the atomic cloud, the magnetiza-
mologicalscenariooftopologicaldefectformationcanbe
m tionstartsfromthecenterandthemagnetizedregionex-
tested by the normalfluid–superfluid phase transitionof
pands outward. If this expansion velocity is slower than
- liquid helium2. Such a mechanism of topological defect
d the velocity of the spin wave, the magnetized region can
formationiscalledKibble–Zurek(KZ)mechanism,which
n be causally connected with the region that is going to
o has been studied in a wide variety of systems3–11.
magnetize, and the KZ mechanism breaks down. A plug
c Bose–Einsteincondensates(BECs)ofatomicgasesare potential applied to the center of the trap is shown to
[
highly controllable quantum systems and suitable for resolve this problem.
1 studyingtheKZmechanisminacontrolledmanner. The This paper is organized as follows. Section II formu-
v BEC transition breaks the U(1) symmetry for a single- lates the problem and provides mean-field and Bogoli-
0 componentsystem,andquantizedvorticescanbeformed ubov analyses. Sections IIIA and IIIB show the numer-
7
7 by the KZ mechanism. This has been demonstrated in icalresults for suddenquench andgradualquenchof the
4 the experiments reported in Refs.12,13. Spinor BECs magnetic field, respectively. Section IIIC examines the
(BECs of atoms with spin degrees of freedom) have a
. case of a harmonic potential with a plug potential. Sec-
1
rich variety of magnetic phases with different symmetry tion IV concludes this paper.
0
groups, and thus have various kinds of topological de-
3
fects14,15. In the experiments reported in Refs.16,17, the
1
: transitionfromthepolarstatetotheferromagneticstate II. MEAN-FIELD ANALYSIS OF SPIN
v in a spin-1 87Rb was observed, which was controlled by CORRELATIONS
i
X anexternalmagneticfield. Formationofspinvorticesby
the KZ mechanism in this magnetic transition has been
r WeconsiderbosonicatomswithmassM andhyperfine
a investigated in Refs.18–21. It is predicted that the KZ
spin F = 1 confined in an external potential V (r).
mechanism can also be tested by the Mott transition of trap
ThemagneticfieldBisappliedinthezdirection,andthe
coldatomsinanopticallattice22,solitonformationinthe
linear and quadratic Zeeman effects change the energies
BEC transition in a one-dimensional gas23, a miscible–
of spin sublevels m= 1 by
immiscibletransitioninabinaryBEC24,andamagnetic ±
transition in an antiferromagnetic spinor BEC25. µ2B2
In Ref.21, we studied the KZ mechanism in the mag- p= gFµBB, q = B , (1)
∓ 4E
hf
netic transition of a spin-1 87Rb BEC and numerically
demonstrated the KZ scaling properties. However, the respectively,whereg isthe hyperfineg factor,µ isthe
F B
numericalsimulationsinRef.21wererestrictedtothesys- Bohrmagneton,andE isthehyperfinesplittingenergy.
hf
tems with uniform atomic density. In the present paper, For 87Rb atoms, g = 1/2 for F = 1 and E /h 6.8
F hf
≃
we perform numerical simulations of the magnetization GHz. The interactionbetweenatoms is characterizedby
2
spin-independent and spin-dependent interaction coeffi- of the broken axisymmetry state (8) is (F2 +F2)1/2 =
x y
cients given by (1 q2/q2)1/2.
− c
We study the stability of the polar state (7) using the
4π¯h2a +2a 4π¯h2a a
c = 0 2, c = 2− 0, (2) Bogoliubov analysis. Substituting
0 1
M 3 M 3
respectively,where a is the s-wavescatteringlengthfor ψ0(r,t) = e−ic0n0t/h¯√n0, (10)
S
otwfoa0co=llid1in01g.8aatoBmasnwditah2t=ota1l0s0p.i4naBS2.6WfoeruFse =the1va87luRebs ψ±1(r,t) = e−ic0n0t/h¯ k √1V eik·ra±1,k(t), (11)
atoms, where a is the Bohr radius. X
B
We employthe mean-fieldtheoryatzerotemperature. into the GP equation (6), where V is the volume of the
The state of the system is described by the macroscopic system, and keeping the first order terms in a 1,k, we
wave functions ψ (r,t). The mean-field energy is given obtain ±
m
by
da 1,k(t)
¯h2 i¯h ±dt =(εk+q+c1n0)a±1,k(t)+c1n0a∗∓1,−k(t),
E = dr ψ 2+V +mp+m2q ψ (12)
Z (cid:20) m m∗ (cid:18)−2M∇ trap (cid:19) m where εk =¯h2k2/(2M). The solution is given by
X
c c
+ 0ρ2+ 1F F , (3) Ekt εk+q+c1n0 Ekt
2 2 · a 1,k(t) = cos i sin a 1,k(0)
(cid:21) ± (cid:18) ¯h − Ek ¯h (cid:19) ±
where c n E t
1 0 k
ρ(r,t) = ψ1 2+ ψ0 2+ ψ 1 2, (4) −(cid:18)i Ek sin ¯h (cid:19)a∗∓1,−k(0), (13)
| | | | | − |
F(r,t) = ψm∗ fmm′ψm′, (5) where
mX,m′ Ek = (εk+q)(εk+q qc). (14)
−
with f = (f ,f ,f ) being the spin-1 matrices. The dy-
x y z When q q , E pis real for all k, and Eq. (13) is an
c k
namics is given by the Gross-Pitaevskii(GP) equation, ≥
oscillating function. In this case, the polar state (7) is
∂ψm δE stable against small deviations. When q < qc, Ek is
i¯h = , (6) imaginaryfor0<ε <q q. Themodeswithimaginary
∂t δψm∗ E exponentially gkrow, cw−hich make the polar state (7)
k
where the right-hand side indicates functional deriva- dynamically unstable.
tive. In the rotating frame of the spin space (ψ If the initial state is prepared in the stable polar state
1
e ipt/h¯ψ ), the linear Zeeman terms in Eq. (6) c±an b→e (7) with q q and q is decreased to q <q , the system
∓ 1 c c
eliminate±d, and we neglect them in the following calcu- becomes dy≥namically unstable and the transverse mag-
lations. netizationF emerges16,17. UsingEq.(11)withEq.(13),
To study the behaviors of the system analytically, we the correlati±on function of the transverse magnetization
consider a uniform system with density ρ = n . When F =F iF is calculated to be
0 x y
c < 0 and q > 0, which is the case of spin-1 87Rb, the ± ±
1
F (r,t)F (r ,t)
ground state of Eq. (3) satisfying F =0 is given by + ′
z h − i
ψ1 0 = 2Vn0 cosE¯hkt +iεkE+q sinE¯hkt 2e−ik·(r−r′)
ψ0 =√n0eiα 1 (7) Xk (cid:12)(cid:12) k (cid:12)(cid:12)
ψ−1 0 × h|a1,k(0)(cid:12)(cid:12)|2i+h|a−1,−k(0)|2i , (cid:12)(cid:12) (15)
for q >qc and where (cid:2) indicates the average(cid:3)with respect to dif-
h···i
eiβ1 1 q faenrdenttheinrmitiaallflvuaclutueastiao±n1s,,±rke(s0i)d.uaTlhateoymisncilnudtheeqmua=ntum1
ψ1 2 − qc ±
ψ =√n eiα 1 q1+ q (8) states, and other experimental noises, and therefore we
ψ 0 0 √2 qc assumethat a 1, k(0)areindependent complex random
−1 e−iβq12 1− qqc ncounmtabienrss.thFeorex±tphoe±ndeynntiaamllyicaglrloywuinngstfaabclteormeoxdpe(s2, EEq.t(/1¯h5)),
q | k|
for q q , where which has a sharp peak at the most unstable wave num-
c
≤
ber k . We can thus approximate Eq. (15) as21
mu
q =2c n , (9)
c 1 0
| |
F (r,t)F (r ,t)
+ ′
andα andβ are arbitraryphases. The states in Eqs.(7) h − i
t 1
amnedtry(8)staatree27c,alrleedspethcteivpeolyl.arTshtaetetraanndsvebrrsoekemnaganxeistyizma-- ∝ dkexp τ 1− 4ξc2orr∆k2 +ik·(r−r′) ,
Z (cid:20) (cid:18) (cid:19) (cid:21)
tion of the polar state (7) is (F2+F2)1/2 = 0 and that (16)
x y
3
where ∆k =k k , and τ and ξ are defined by the imaginary time propagation method. We add small
mu corr
−
random noises to the initial state of ψ to trigger the
1
2|Ek|t = t 1 1ξ2 ∆k2 +O(∆k4). (17) magnetization. We takea sufficiently la±rgespaceso that
¯h τ − 4 corr the boundary condition does not affect the results.
(cid:18) (cid:19)
We definethe transverseandlongitudinalautocorrela-
For qc/2 < q < qc, the most unstable wave number is tion functions as
k =0 with
mu
F 2dr F2dr
τ = ¯h , (18) GT = R | ρ+2|dr , GL = R ρz2dr. (26)
2 q(q q)
c
− We also define thRe transverse autocorRrelation function
p¯h2 2q qc along a circle with radius r as
ξ = − . (19)
corr
sM q(qc−q) 2π F 2(r,θ)dθ
G (r)= 0 | +| . (27)
Inthiscase,theintegralinEq.(16)foratwo-dimensional T R 2πρ2(r,θ)dθ
system can be performed to yield 0
The transverse spin windiRng number along a circle with
t τ r r 2
hF+(r,t)F−(r′,t)i∝exp(cid:18)τ − |tξ−c2orr′| (cid:19). (20) radius r is defined as1 2π ∂
w(r)= argF (r,θ)dθ. (28)
For q <qc/2, the most unstable wave number is 2π Z0 ∂θ +
2M q
c
k = q (21)
mu ¯h2 2 − A. Sudden quench
r
(cid:16) (cid:17)
with
We first investigate the magnetization dynamics for
¯h suddenquenchofthe quadraticZeemanenergyto q =0.
τ = , (22)
Thiscorrespondstothe situationinwhichthestable po-
q
c
lar state is preparedat sufficiently large q, and the mag-
8h¯2qc 2q netic field is suddenly switched off at t =0. Figure 1(a)
ξ = − . (23)
corr s M q2 showsthetimeevolutionoftheautocorrelationfunctions
c
G (t) and G (t). The transverse magnetization starts
T L
The integral in Eq. (16) for a two-dimensional system to grow at t 100 ms and the longitudinal magnetiza-
≃
becomes tion follows. Figures 1(b)–1(e) show the profiles of the
transverse magnetization. The transverse magnetization
F+(r,t)F (r′,t) kJ0(k r r′ )e−4tτξc2orr∆k2dk, emerges around the center and grows outward. This is
h − i∝ | − | because the growth time in Eq. (22) is inversely propor-
Z
(24)
tionaltotheatomicdensityandthemagnetizationgrows
whereJ istheBesselfunction. Fork r r 1,this
0 mu| − ′|≫ fast at which the density is large. The total density dis-
expression can be evaluated to be
tribution ρ(r) is almost unchanged during the time evo-
lution, since c is much larger than c .
F (r,t)F (r ,t) 0 1
h + − ′ i Many spin vortices can be seen in Figs. 1(c)–1(e) (the
t τ r r′ 2 holes in the F profiles, around which argF rotate
∼cos(kmur−π/4)exp τ − |tξ−2 | . (25) by 2π). In t|er+m|s of the spin components in E+q. (8), β
(cid:18) corr (cid:19) ±
changesby 2πaroundthevortexcore,whichisoccupied
±
by the m=0 component. Such a spin vortex is called a
III. NUMERICAL RESULTS polar-corevortex. Thespinwindingnumberw(r)defined
inEq.(28)representsthedifferencebetweenthenumbers
We restrictourselvesto two-dimensional(2D) systems of polar-core vortices with opposite circulations within
confinedinaharmonicpotentialV =Mω2(x2+y2)/2 the radius r.
trap
with ω/(2π) = 2 Hz. When the system is tightly con- We note that the spin vortices are produced by two
fined in the z direction and the thickness of the cloud distinctmechanismsinFig.1withq =0: theKZmecha-
is smaller than the spin healing length (typically a few nism and the spin conservation dynamics21,29. Since the
micrometers), the spin dynamics are effectively 2D. We spin correlationfunction in Eq. (25) has a finite correla-
assume that the thickness in the z direction is 1 µm, tionlengthξ ,thedirectionsofmagnetizationatrand
corr
and use 2D interaction coefficients as c2D =c /(≃1µm). r are independent for r r ξ , giving rise to the
j j ′ | − ′|≫ corr
We numerically solve the 2D GP equation using the KZ mechanism. On the other hand, when q =0, the to-
pseudospectralmethod28. The initialstate isthe ground talmagnetization Fdrmustbeconservedatzero,since
state of Eq. (3) with ψ = 0, which is obtained by Eq. (3) is invariant with respect to spin rotation in the
1
± R
4
(a) 0.8 15
0.6 GT
L
G
, T 0.4 10
G
0.2 GL 2〉
w
0 〈
0 200 400 600 800 1000 5
t [ms]
(b) t = 0 ms
0
0 50 100 150 200
r [µm-1]
FIG. 2: Variance of the winding number hw2(r)i along the
circumferenceofacircleofradiusrforsuddenquench,where
(c) t = 140 ms theparameters are thesame as thosein Fig. 1. Thedata for
each r is taken when GT(r) exceeds 0.1. The average h···i
is taken over 400 runs of simulations for the different initial
statesproducedbyrandomnumbers. Thedashedcurveisthe
least square fit of Eq.(29).
zationthusoccursinsuchawaythatthelocalmagnetiza-
tionisconservedatzero,i.e.,spintexturesareformed30.
(d) t = 200 ms
Among various spin textures, the polar-core vortices are
favorable, since the excess energy at the defect can be
minimized31. This is the second mechanism of the spin
vortex formation in Fig. 1. Thus, to see the effect of
the KZ mechanism, we must take spatial region much
larger than ξ . The correlation length is ξ 10
corr corr
≃
µm around the center of the trap and ξ 20 µm at
corr
≃
r =200 µm for Fig. 1.
(e) t = 400 ms
We consider the r-dependence of the spin winding
numberw(r). AccordingtotheKZtheory,thenumberof
domainsalongthecircleofradiusris r/ξ andhence
corr
w2(r) r/ξ . Substitutingq =0an∼dq =2c n (r)
corr c 1 TF
intoξ ∼ inEq.(23),wheren (r) R2 r2|wi|thR
corr TF ∝ TF− TF
being the Thomas–Fermi radius, we obtain
−π π w2(r) r R2 r2. (29)
0 1 ∝ TF−
q
To compare Eq. (29) with the numerical simulation, we
FIG. 1: (a) Time evolutions of the autocorrelation func-
performmanyrunsoftimeevolutionwithdifferentinitial
tions GT(t) and GL(t) for sudden quench to q = 0. (b)–(e) randomnoises,andtaketheaverageofw2(r)withrespect
Selns)apasnhdotistsofdtihreecttrioannsavregrsFe+m(ra,gtn)e(trizigahtitonpa|Fne+ls()r.,tT)|h(eleuftnpitano-f to the runs, which is shown in Fig. 2. Since the time at
|F+(r,t)| is 1.2×1014 m−2. The field of view of each panel whichthemagnetizationemergesdependsonr,eachw(r)
is 400×400 µm. The number of atoms is N = 107. See the is calculated when GT(r) in Eq. (27) exceeds a certain
supplementarydata fileforthemovieshowing thedynamics. value (0.1 in Fig. 2). The numerical result and Eq. (29)
(circles and dashed curve in Fig. 2, respectively) are in
good agreement, where the fitting parameter is only the
rotating frame ψ e ipt/h¯ψ . In the present case, proportionality coefficient in Eq. (29).
1 ∓ 1
however, the con±serv→ation law i±s more strict because of
the finite spin correlation length. Since the spin direc-
tions at r and r are independent for r r ξ , B. Gradual quench
′ ′ corr
| − | ≫
not only the total magnetization but also the local mag-
netization Fdr integrated over the size of ξ We next consider the case of gradual quench of the
local ∼ corr
must be conserved in each spatial region. The magneti- magneticfield. ThequadraticZeemanenergyqislinearly
R
5
decreased in the time scale τ as
Q (a) 100
q(t)=q (1 t/τ ), (30)
0 Q
−
for 0 < t < τQ and q(t) = 0 for t τQ. As seen in the 10
≥
previous subsection, the critical value q (r) for magne-
tization depends on the position r, andcq0 is chosen to 2〉w
be the maximum of q (r). We define the time T(r) at 〈
c
1
which q reaches a local critical value as q(T(r)) =q (r).
c
Themagnetizationatpositionr isexpectedtoemergeat
t=T(r)+∆t(r) satisfying2
0.1
∆t(r) τ(r,t). (31) 100 1000
∼ τ [ms]
Q
UsingEqs.(18)and(31)withq ∆t(r)/[q (r)τ ] 1,we (b) τ = 1s
0 c Q Q
≪
obtain t=500ms t=600ms t=700ms
¯h2 1/3
∆t(r) τ1/3. (32)
∼(cid:20)qc(r)q0(cid:21) Q 1
Substituting this time into Eq. (19), we obtain the τ -
Q (c) τ = 3s
dependence of the correlationlength as Q
t=500ms t=1000ms t=1500ms
¯h2 q (r)q 1/6 0
ξ (r) c 0 τ1/3. (33)
corr ∼sMq0 (cid:20) ¯h2 (cid:21) Q
The winding number thus obeys
w2(r) τ−1/3. (34) FIG. 3: (a) Variance of the winding number hw2(r)i along
∝ Q the circumference of a circle of radius r = 200 µm (circles),
Figure3(a)shows w2(r) obtainedbynumericalsim- r = 250 µm (squares), and r = 300 µm (triangles) for grad-
h i ual quench given by Eq. (30). The data for each r is taken
ulations of the GP equation (6). The variance of the
windingnumberhw2(r)iisroughlyproportionaltoτQ−1/3 wruhnesnoGfsTi(mruwl)aetixocnesedfosr0t.1h.eTdihffeeraevnetraigneithia·l··sitaitsetsapkerondouvceerd1b0y0
froerti1ca0l0amrgsu<∼mτeQnt<∼(314)s.,wFohricτhQa<gre1e0s0wmitsh, tthheeaabssouvmepthtieoon- rTahnedonmumnbuemrboefrsa.toTmhse disasNhed=lin10e8i.s p(rbo)p,o(rct)ionSanlaptoshτoQ−ts1/o3f.
foofrt/suτQdd≪en 1quiesnvciholiantetdh;ehlwim2(i∼tr)oifaτpQproac0h.esFothreτQva>lue1s tThheetararrnoswvesrtsreamceatghneetviozartteioxnm|Fot+io(rn,.tT)|hfeorunτQit=of1|Fs+a(rn,dt)3|sis.
s, w2(r) significantly deviates from τ→−1/3 and stee∼ply 3.4×1014 m−2. The field of view of each panel is 300×300
h i Q µm. Seethesupplementarydatafiles forthemoviesshowing
dropsinFig.3(a). Tounderstandthisbehavior,wecom-
thedynamics.
pare the dynamics of F (r) for τ = 1 s and τ = 3 s
+ Q Q
| |
shown in Figs. 3(b) and 3(c). When τ = 1 s, new spin
Q
vortices are produced one after another as the magneti-
The transverse spin wave for the broken axisymmetry
zation grows outward. When τ = 3 s, by contrast, the
Q state (8) is a phonon-like mode in the limit of k 0,
spinvortexcreatedaroundthecenterispushedoutward, whose velocity is given by14,27 →
and no new spin vortices are created at the front of the
magnetization growth. q
In the dynamics in Figs. 3(b) and 3(c), there are two vs = , (37)
2M
characteristic velocities: the velocity v at which the r
m
magnetization front spreads out and the sound velocity where q(t) in Eq. (30) should be used on the right-hand
vs ofthespinwave. Theformerisroughlyobtainedfrom side. If vm is alwaysfaster thanvs, the regionthatis go-
ing to magnetize is causally disconnected with the mag-
q(t)=q (r (t)), (35)
c M netized region, and therefore the KZ mechanism works.
It follows from Eqs. (36) and (37) that this condition is
wherer (t)istheradiusofthemagnetizationfront. Us-
M
satisfied for
ing the Thomas–Fermi density distribution, the right-
hand side is q (r (t)) = 2c n (r (t)) q [1
c M 1 TF M 0
rM2 (t)/RT2F]. The velocity vm i|s t|hus given by ≃ − τQ < 2MRTF. (38)
s q0
dr (t) R
M TF
v = = . (36)
m dt 2√τ t For the parameters in Fig. 3, the right-hand side of this
Q
6
thepotentialV (r)hasaminimumatr 250µmand
100 trap
≃
the atomic density becomes maximalaroundthis radius.
1
As a result, the magnetization starts from the annulus
around r 250 µm. Therefore, magnetic domains on
≃
2〉w 10 slope = −1/3 0 tmheischraandiisumsairseeaxlpweacytsedcatuosawlloyrkdisecvoennnfeocrtelda,rgaendτQt.heFKigZ-
〈 ure4showstheresultsofthenumericalsimulations. The
density at r 250 µm is almost the same as the density
≃
atthesameradiusinthesystemofFig.3,and w2(r) is
h i
similar to the corresponding data in Fig. 3(a) (squares)
for τ < 1 s. However, w2(r) in Fig. 4 obeys the KZ
1 100 τ Q1 0[m00s] 10000 scalinQg∼τQ−1/3 eveInV.forCτQOhN>∼C1LsU,iaSsIOexNpSected.
FIG. 4: Variance of the winding number hw2(r)i along the
We have investigated the spin vortex formation due
circumferenceofacircleofradiusr=250µm(dashedcircle)
to the KZ mechanism in a quenched ferromagnetic BEC
for gradual quench given by Eq. (30). The potential has a
confinedinatrappingpotential. Sincetheatomicdensity
form of Eq. (39). The data for each r is taken when GT(rw)
isinhomogeneousinaharmonictrap,thespincorrelation
exceeds 0.1. The average h···i is taken over100 runs of sim-
length depends on the radius r. In fact, the numerical
ulations for the different initial states produced by random
numbers. The dashed line is least square fit of the plots by simulations showed that the spin winding number de-
usingafunctionproportionaltoτ−1/3. Thenumberofatoms pends on r, which was in good agreementwith the theo-
Q
isN =108. Theinsetshowstheinitialdensityprofilewiththe retical prediction (Fig. 2). When the quadratic Zeeman
unitofdensity2.4×1014 m−2 andthefieldofview900×900 energyqisgraduallyquenchedwiththetimescaleτQ,the
µm. magnetized region gradually expands from the center to
theperipheryoftheatomiccloud. Iftheexpansionveloc-
ityismuchfasterthanthespinwavevelocity,thesystem
inequality is 1.7 s, which agrees well with the time at exhibits the KZ scaling law, and if the former is slower
≃
which the plots in Fig. 3(a) deviates from τ−1/3. than the latter, the KZ scenario breaks down (Fig. 3).
Q
When a plug potential is added to the harmonic trap,
thegeometryofthesystemischangedandtheKZpower
C. Gradual quench with a plug potential law can be observed over a wide range of τQ (Fig. 4).
Intheprevioussubsection,weshowedthattheKZsce-
nario breaks down when the magnetized region expands
slowly. This is because the region that is going to mag-
netize is causally connected to the initially magnetized
Acknowledgments
region at the trap center. To eliminate this effect, we
remove atoms around the trap center by adding a plug
potential as
This work was supported by Grants-in-Aid for Sci-
Vtrap(r)= 1Mω2r2+Ae−r2/d2, (39) e2n2t3i4fi0c11R6,esNeaor.ch227(4N0o2.652,21a0n3d00N5o,. N23o5.402426344)01fr1o4m, Nthoe.
2
Ministry of Education, Culture, Sports, Science and
where the values of the parameters are chosen to be TechnologyofJapan. YKacknowledgesthefinancialsup-
A = 1500¯hω and d = 222 µm. For these parameters, port from Inoue Foundation.
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