Generalized Gross-Pitaevskii equation adapted to the $U(5)\supset SO(5)\supset SO(3)$ symmetry for spin-2 condensates

0.27 MB·English

by  Y. Z. He

Generalized Gross-Pitaevskii equation adapted to the U(5) SO(5) SO(3) symmetry
⊃ ⊃
for spin-2 condensates
Y. Z. He1, Y. M. Liu2 and C. G. Bao1,
∗
1The State Key Laboratory of Optoelectronic Materials and Technologies,
School of Physics and Engineering, Sun Yat-Sen University, Guangzhou, P. R. China and
2Shaoguan University, Shaoguan, 512005, P.R. China
5 A generalized Gross-Pitaevskii equation adapted to the U(5) ⊃ SO(5) ⊃ SO(3) symmetry has
1 been derived and solved for the spin-2 condensates. The spin-textile and the degeneracy of the
0 ground state (g.s.) together with the factors affecting the stability of the g.s., such as the gap and
2 theleveldensityintheneighborhoodoftheg.s.,havebeenstudied. Basedonarigoroustreatmentof
thespin-degreesoffreedom, thespin-textilescanbeunderstoodinaN-bodylanguage. Inaddition
n
a to the ferro-, polar, and cyclic phases, the g.s. might in a mixture of them when 0<M <2N (M
J is the total magnetization). The great difference in the stability and degeneracy of the g.s. caused
by varying ϕ (which marks the features of the interaction) and M is notable. Since the root mean
2
square radius R is an observable, efforts have been made to derive a set of formulae to relate
1 rms
Rrms and N, ω(frequency of the trap), and ϕ. These formulae provide a way to check the theories
] with experimentaldata.
s
a
PACSnumbers: 03.75.Hh,03.75.Mn,03.75.Nt
g
Keywords: spin-2condensates, Gross-Pitaevskii equation, spin-textile of the ground state, stability of the
-
t groundstate
n
a
u I. INTRODUCTION is considered as being conserved, which depends on how
q
the system is experimentally prepared. The influence of
.
t M on the spin-textiles and the degeneracy of the g.s. is
a Since the success of trapping cold atoms via optical
m found to be great.
trap,[1] the study of condensates of atoms with nonzero
After we have obtained the spatial wave functions,
- spin is a hot topic. There are numerous literatures
d physical quantities related to the spatial degrees of free-
dedicated to the study of the ground state (g.s.). For
n dom can be known. In particular, we have calculated
spin-2 condensates, the g.s. was found to have three
o the root mean square radius R . Since R is an ob-
c phases, namely, the ferro-phase, polar phase and cyclic rms rms
servable,theoreticalresultsandexperimentaldatacanbe
[ phase, depending on the features of the interaction.[2–4]
comparedwitheachother. Thisisanotablewaytocheck
On the other hand, it has been revealed that the total
1 thetheory. TheGPequationwillbefirstlysolvedanalyt-
Hamiltonian of the spin-2 systems is associatedwith the
v ically under the Thomas-Fermiapproximation(TFA). A
6 U(5) SO(5) SO(3) symmetry.[5, 6] Since there are
⊃ ⊃ number of formulae relating the physicalquantities have
5 elegant theory dealing with this symmetry, it is worthy
been thereby obtained, and the underlying physics can
3 to derive a generalized Gross-Pitaevskii (GP) equation
3 adapted to the U(5) SO(5) SO(3) symmetry. Since beunderstoodinananalyticalway. Then,strictnumeri-
0 ⊃ ⊃ calcalculationsareperformedtocheckthevalidityofthe
thesolutionsofthis equationcouldprovidethe detailsof
. TFA.
1 the low-lying states, we believe that the understanding
0 to the spin-2 condensates could be thereby enriched.
5
We consider the case that the magnetic field B = 0.
1
II. INHERENT SYMMETRY AND THE
: The symmetry-adapted GP equation is derived firstly.
v GROSS-PITAEVSKII EQUATION
Then,by solvingthe GP equation,both the spin-textiles
i
X and the spatial wave functions of the low-lying states
can be known. The emphasis is placedon the g.s.. Since AcondensatewithN spin-2 neutralatomstrappedby
r
a the stability of the g.s. is affected by its environment, an isotropic harmonic potential 1mω2r2 is considered,
2
in addition to the g.s. itself, special effort is made to in which the spin-orbit coupling is assumed to be weak.
studytheenvironment,namely,thewidthofthegap(the It is further assumed that the trap is not so weak (say,
energy difference between the first excited state and the ω 100s−1) and the spin-dependent interaction is weak
≥
g.s.) and the level-density in the neighborhood of the so that the spin-modes are much lower than and sepa-
g.s.. This is a topic less studied before. Besides, the ratedfromthe spatiallyexcited modes. This is the basic
degeneracy of the g.s. is an important feature which is assumption of this paper. In other words, the following
also studied in this paper. The total magnetization M resultsholdonlyforthesystemsadaptedtothisassump-
tion. In this case a group of excited states distinct in
their spin-modes would emerge in the neighborhood of
the g.s.. In each of these states (including the g.s.) all
∗Correspondingauthor: [email protected] theparticleshaveacommonspatialwavefunctionwhich
2
is most advantageous to binding and adapted to a spe- are bound by the following conditions: (i) N 0, (ii)
free
≥
cific spin-textile. Thus one of these states with a specific N S 2N , (iii) S = 2N 1 is not allowed,
free free free
≤ ≤ −
spin-mode γ can be in general written as and of course (iv) M S.[5, 8]
| |≤
Letus define g =g g , g =g g , andg =
04 0 4 24 2 4 (024)
N 1(g +g +g ). Then, t−he eigenenerg−y associated with
Ψ = φ (r )Θ , (1) 3 0 2 4
γ Y γ j γ Θ is [5]
γ
j=1
E˜ =a N+a N(N+4)+g v(v+3)+g S(S+1), (4)
where φ is the commonspatialwavefunction andΘ is vS 1 2 v S
γ γ
a totalspin-state whichis now unknown. If the particles
where g = ( 7g +10g )/70, g = ( g )/14, a =
did not fall into the same state but have some of them 11g v1 g −5g04 ,and24a = S1 g −12g4 +1g 1 .
gettingexcited,thetotalenergywouldbetherebyhigher. 15 04−42 24−2 (024) 2 −15 04−42 24 2 (024)
The group Ψγ totally form a band which is called the NotethatE˜vS doesnotdependonntri implyingthatthe
ground ban{d. }The states not in the ground band will levels might be degenerate as shown below.[8]
contain various spatially excited modes and will not be InsertingEq.(1)intotheSchr¨odingerequation,wehave
studied in this paper.
When ~ω and λ = ~/(mω) are used as units for N N
p φ (r )Θ H E φ (r )Θ =0, (5)
energy and length (where m is the mass and is given at hY γ j γ| − γ|Y γ j γi
the one of 87Rb), the total Hamiltonian is j=2 j=1
where the integration covers all the degrees of freedom
H =Xh(i)+XVij, (2) exceptr1. Fromthis equationandmaking use of Eq.(3),
i i<j it is straight forward to obtain the symmetry-adapted
GP equation for the normalized φ (r ) as
where h(i) = 1 2 + 1r2. V = δ(r r ) g Pij. γ 1
−2∇i 2 i ij i − j Ps s s
Psij = Pms|(χ(i)χ(j))smsih(χ(i)χ(j))sms|, χ(i) is the [h+ 2 E˜ φ φ ǫ ]φ =0, (6)
spin-state of the i-th particle, and the two spins of i and N vS ∗γ γ − γ γ
j arecoupledtothecombinedspinsanditsz-component
ms, s = 0, 2, or 4. Obviously, Psij is the projector onto where ǫγ is the chemical potential. After solving Eq.(6)
thes-channel. gs isthestrengthrelatedtothescattering we canobtainthe totalenergyEγ =Nǫγ−Vtot,γ, where
length of the s-channel. V = φ 4dr E˜ is the total interaction energy.
tot,γ γ vS
R | |
We introduce an operator defined in the total spin- Thewholespectrumofthegroundbandcanbeobtained
space as V˜ g Pij. It has been proved that fromthe set E . Obviously,this equationis a general-
≡ Pi<jPs s s { γ}
V˜ is a linear combination of a set of Casimir opera- ization of the one for spin-1 condensates given in ref.[9]
tors belonging to a chain of nested algebra as U(5) It is also a more accurate version for the one given as
SO(5) SO(3).[5, 6] Consequently, the eigenstates an⊃d Eq.(74) in ref.[4], in which the spin-dependent interac-
the eige⊃nenergies of V˜ can be exactly known as tion has been neglected.
V˜Θ =E˜ Θ . (3)
γ vS γ
III. THOMAS-FERMI APPROXIMATION
Now, γ represents a group of quantum numbers v, n ,
tri
S, and M, where S and M are the total spin and its Z- SinceN isusuallylarge,Eq.(6)canbe solvedby using
component,ν andntri willbeexplainedbelow,andΘγ theTFA.Neglectingthe kineticterminEq.(6),when~ω
≡
ΘvntriSM istherelatedeigenstate. Forconvenience,M ≥ and λ are used as units, the normalized TFA solution is
0 is assumed. φ = 15 1 (r/r )2 and ǫ =r2/2, where
When two spins are coupled to zero, they form a sin- γ q8πr03p − 0 γ 0
glet pair θ (ij) (χ(i)χ(j)) . When three spins are
coupled topzaeirro, th≡ey form a tr0iplet and can be approx- r =( 15 E˜ )1/5, (7)
0 vS
2πN
imately denoted as θ (χχχ) (this notation is ex-
tri 0
≈
act when N ).[7] In the triplet every pair of spins is the TF-radius. From this solution we can obtain the
→ ∞
are coupled to 2. It turns out that the singlet pair and root mean square radius R r2 1/2 = 3/7r , and
tripletarebasicbuilding blocks. Aneigenstatemaycon- the total energy rms ≡ h i p 0
tain n singlet pairs and n triplets. The quantum
pair tri
number v ≡ N − 2npair is named the seniority, which E = 1(3257)1/5N3/5(E˜ )2/5 = 5NR2 . (8)
is the number of particles not in the singlet pairs. The γ 7 27π2 vS 6 rms
number of particles neither in the pairs nor the triplets
is N N 2n 3n , the total spin S is con- Thus,thetotalenergyisdirectlyrelatedtothesize(mea-
free pair tri
≡ − −
tributed by them. It has been proved that, when npair, sured by Rrms).
n , S and M are given, the totally-symmetric eigen- When N is large, we know from Eq.(4) that E˜ is
tri vS
state Θ is unique. The four quantum numbers proportionaltoN2. Accordingly,R N1/5 andE
vntriSM rms ∝ γ ∝
3
does not directly depend on M, the condition S M
61000 ≥ | |
restricts the possible choice of S, and E is thereby af-
III: (N, M, d ) gs
FERRO fected. E versus ϕ with M given at five values are
I: (N, 2N, 1) gs
59000 FERRO shown in Fig.1. There are two critical points ϕ =
a
tan 1( 7(N+M/2+3) ) and ϕ = tan 1( 7 ).[5] When ϕ
− 10(N+M/2 2) b − −10
) 57000 goesth−rougheithe−rϕ orϕ aphasetran−sitionwilloccur.
-E (hgs POLAR Tthheismpehaennfioemldentohneowryaas(ifnouwnhbdicahsϕear=lytaasnin1(20070)baynudstinhge
55000 5 a − 10
sameϕ werefound.[2] The presentmore acc−urateresult
3 4 b
will tend to the old one when N is large).
53000 FERRO CYCLIC
II: Whenϕ is in(0,ϕa), i.e.,in region-I,the g.s. does not
1
(M /2, M, 1) 2 depend on M and will have (v,S,d)=(N,2N,1), where
51000 d is the degeneracy of the state. Since all the spins align
0.0 0.5 a 1.0 b 1.5 2.0 along a common direction in this g.s., it is in the ferro-
/
phase.[2–4]
vFeIrGsu.s1:ϕ.(cNolo=r 1o2n0l0in0ea)nTdhωe=gro3u00nsd−s1taatreeaensseurmgyedEg(sth(einsa~mωe) peWndhseonnϕMisanind(wϕiall,ϕhab)v,ei(.ve.,,Si,nd)re=gi(oMn-/II2,,Mthe,1g)..s.Sindcee-
in the following figures except specified). The total magneti- v = M/2 and S = M, all the M/2 unpaired spins must
zation is given at five cases as M = 0, 1000, N, 2N −1000,
align along a common direction so that the total spin
and 2N, respectively, for the curves ”1” to ”5”. There are
can be maximized (i.e., S can be equal M). Thus each
three regions of ϕ separated by ϕ = tan−1( 7(N+M/2+3) )
a −10(N+M/2−2) g.s. in region-II is a mixture of a group of aligned spins
andϕb =tan−1(−−170)markedbythetwoverticaldottedlines. together with the (N M/2)/2 singlet pairs, namely, a
In each region the g.s. has its specific (v,S,d) marked in the mixture of ferro-phase−and polar phase. When M = 0,
region, where d is the degeneracy. In region I, E is M-
gs (v,S,d)=(0,0,1),thusitisinpurepolarphase(wehave
independent. In I and II, the g.s. is not degenerate (d= 1).
assumed that N is even, otherwise S = 2). When M is
InIIIthedegeneracydependsonM seriously. Forthecurves
larger,morespinsareintheferro-phase,andaccordingly
”1” to ”5”, d=1, 167, 2001, 167, and 1, respectively.
E gets higher as shown in the figure. When M = 2N,
gs
(v,S,d)=(N,2N,1), andthe g.s. is in pure ferro-phase.
N7/5. ThusthesizeincreaseswithN veryslowly,butE When ϕ is in (ϕb,2π), i.e., in region-III, the g.s. de-
γ
pends also on M and has (v,S,d) = (N,M,d). Note
increases with N faster than linearly.
On the other hand, it is noted that, in the units of ~ω that these g.s. have v = N, therefore they do not con-
tain any singlet pairs, but the spins in triplets are al-
and λ, all the strengths g √ω˜, where ω˜ is the magni-
s
tude of ω, i.e., ω = ω˜s 1. ∝Therefore E˜ √ω˜. When lowed. Itis recalledthat, inregionsI andII,the g.s. has
− vS
∝ S = 2v. It implies that all the unpaired spins must be
ω-independent units are used, the size is described by
R λ (ω˜) 2/5, i.e., a stronger trap leads to a smaller aligned, and therefore there is no room for the triplets.
rms −
∝ Whereas in III, the number of triplets n can have dif-
size. Similarly, one can prove that the chemical poten- tri
tial ǫ ~ω (ω˜)6/5, and the total energy E ~ω (ω˜)6/5. ferent choices in a specific domain under the constraints
γ γ
∝ ∝ of symmetry given in points (i) to (iv) in the paragraph
Thus the total energy will increase with ω a little faster
above Eq.(4) (while those spins not in the triplets are
than linearly.
coupled to S =M). This leads to the degeneracy of the
g.s.[8] When M = 0 the g.s. has S = 0 and therefore
N = 0 (point (ii)). Thus, all the particles are in the
IV. GROUND STATE AND ITS PHASE free
triplets(i.e.,inapurecyclicphase),andhenceforthd=1
(we have assumed that N is a multiple of 3 to simplify
It is obvious from Eq.(4) that which pair of v and S the discussion). When M = 2N, the g.s. has S = 2N
is more advantageous to binding depends on the com- andtherefore N =N. Thus,neither the pairsnor the
free
petition between gv and gS, which depend on g04 and triplets emerge (i.e., in a pure ferro-phase), and hence-
g24, and they are considered as variable. Therefore, we forthd=1also. WhenM isneither0nor2N,theg.s. is
introduce a variable parameter ϕ to manifest the com- ingeneraldegenerate. Inparticular,whenM =N, N
free
petition. In the units of ~ω and λ, the strengths are can be ranged from M/2 to M (each step is 3). In this
assumed as g04 = u(ω˜)1/2cosϕ, g24 = u(ω˜)1/2sinϕ, case the degeneracy d is maximized (the curve ”3”in III
g(024) = 2.5u(ω˜)1/2, and u = 10−3 (For a comparison, has d=2001). The appearance of highly degenerate g.s.
the realistic strengths for 87Rb are g = 0.28u(ω˜)1/2, in the region III is a notable feature.
24
g = 0.41u(ω˜)1/2, and g = 2.39u(ω˜−)1/2). In fact, It is reminded that the polar phase is composed of
04 (024)
−
it has been shown previously that the phase of the g.s. s = 0 pairs. Therefore, when g is less positive than g
0 2
depends on ϕ.[2–4] and g , the g.s. will prefer this phase. The cyclic phase
4
Under the TFA, the ground state energy E is just is composedofs=0 triplets in whicheverypair ofspins
gs
the lowest E given by Eqs.(8) and (4). Although E must be coupled to s = 2. Therefore, when g is less
γ γ 2
4

positive, the g.s. will prefer this phase. While the ferro-
1.0
phase is composed of aligned spins, in which every two
must be coupled to s = 4. Therefore, when g is less
4
positive, the g.s. will prefer this phase. Thus the phase 0.8
transitions manifested above reflects the competition of
theinteractionsofthethrees-channels. Thecompetition ) I II III
0.6
mdeupsetnbdes aolnignMedbisecdaeutseermthineedleabsytMnu.mber of spins that -E (hgap 5
0.4
When M = 2N is given, the ferro-phase is the only
4
choice disregarding how the interaction is as shown by
3
the uppermost curve in Fig.1. When N is sufficiently 0.2
large the terms 1/N can be neglected, then we have
∼ 1 2
E˜ (a +g +4g )N2 0.0
vS ≈ 2 v S 0.0 0.5 a 1.0 b 1.5 2.0
1
= N2[ (cosϕ+sinϕ)+1.25]u(ω˜)1/2. (9) /
−6
FIG. 2: (color on line) The energy gap E (in ~ω) versus
gap
From this formula we know that E˜ has a minimum ϕ. The parameters are the same as in Fig.1, except that the
vS curve”5” has M =2N −4 instead of M =2N.
at ϕ = π/4 and a maximum at ϕ = 5π/4 where
E˜ = (15√2 4)N2u(ω˜)1/2. Since the total energy is
vS 12√∓2
proportional to N3/5(E˜ )2/5 as shown in Eq.(8), with the g.s. is very stable in the ferro-phase. In II and III,
vS
N =12000andourparameters,theminimumatϕ=π/4 theupmostcurveinFig.2hasM =2N 4. Accordingly,
has E = 51612~ω, and the maximum at ϕ = 5π/4 has (v ,S )=(N 2,2N 4) and(N,2N −4), respectively.
E =gs60125~ω. This explains the origin of the dip and Singcegthey are−also la−rge, the gap is −also high. When
gs
the highest peak in Fig.1. M = 2N 4, the g.s. is very close to be in ferro-phase
−
Since the root mean square radius is an observable, andtherefore is stable. However,whenM decreases,the
it is desirable to study R so that the theory can be stability reduces. In particular, when M =0, (v ,S ) =
rms g g
checked via experimental measurement. According to (0,0) and (N,0) in II and III, respectively. In this case
Eq.(7), R is proportional to (E )1/2, thus the vari- E 0.00012 was found in both II and III as shown
rms gs gap ≤
ationofR versusϕ withthe minimum andmaximum by the lowest curve in Fig.2. Thus the g.s. in polar and
rms
ispredicted. Inparticular,whenϕ=π/4(5π/4)andthe cyclic phases with a small M are highly unstable.
unit λ is replaced by µm, the minimum (maximum) in (ii) E in everyregionappears as a peak, namely, in
gap
Fig.1isassociatedwithR =3.53(3.81)µm. Inregion themiddlepartitishigherbutverylowwhenϕiscloseto
rms
II and III, it is predicted that the increase of M would the borders. For Fig.2 the g.s. inI has (v,S)=(N,2N),
leadto the increaseofthe size. This is alsoa pointto be while the first excited state has (v,S) = (N,2N 2)
−
checked. if ϕ 0.4722π. Thus the excitation from (N,2N) to
≤
(N,2N 2) is caused by a change in S. However, when
−
ϕ 0, accordingly g 0. Therefore the gap is zero.
S
→ →
V. STABILITY OF THE GROUND STATE On the other hand, the first excited state has (v,S) =
(N 2,2N 4)when(0.4722π<ϕ ϕ ). FromEq.(4),
a
− − ≤
Itisbelievedthatthestabilityoftheg.s. isassuredby we know the gap
the gap E , namely, the energy difference between the
first excitegdapstate and the g.s.. The former has its (v,S) E˜N,2N E˜N 2,2N 4
− − −
slightlydifferentfromthatoftheg.s.,(v ,S ),andcanbe 4 1
g g = [7(N + )g +10(N 2)g ]. (10)
easilyobtainedfromEq.(4). AnexampleofEgapisshown 70 2 04 − 24
in Fig.2. Note that when M = 2N, (v,S) = (N,2N) is
the only choice, no choice other than (N,2N)is allowed. One can prove that the gap is zero when ϕ = ϕa′
≡
Thus, under M-conservation, there are no excited spin- tan−1 7(1N0(+N1/22)). Since ϕa′ is extremely close to ϕa, this
modes. Hence, in order to show the gap for a very large explain−s the−decline of the gap when ϕ ϕ . Based
a
→
M, the curve ”5” in Fig.2 is not given as M = 2N but on Eq.(4), the decline of the peak in II and III can be
M =2N 4. Fig.2 has the following features: similarly explained. Thus, in the neighborhoods of the
−
(i) Since Eq.(4) manifests that the energy is nearly borders, the g.s. is highly unstable.
v2 and S2, the gap would be in general large if both In addition to the gap, another factor that could af-
∝
v and S are large. In this case, a small deviation in fect the stability of the g.s. is the level density in the
g g
(v,S) will cause a great change in energy. Whereas if neighborhood of the lowest level. This density can be
one of them is zero, the gap would be much lower. In calculated based on Eq.(8). As an example, the num-
region I the g.s. is in the ferro-phase with (v ,S ) = ber of levels with excitation energy 0.1~ω is given in
g g
≤
(N,2N). Therefore,thegapisveryhighandaccordingly Table I, in which seven choices of ϕ and four choices of
5
M are chosen. The other parameters are the same as in notasdenseasinthepolarphase. Notethat,whenM is
Fig.1. If the degeneracy of a level is d, then the number notcloseto0or2N (say,0.05 2N M 0.95 2N),it
× ≤ ≤ ×
contributed by this level is d. has been mentioned that the states in cyclic phase could
be highly degenerate. The great degeneracy contributes
to the level-density substantially (say, in the last row
TABLE I: The number of levels lower than 0.1~ω versus M
of Table I, the number 2001 arises completely from the
and ϕ. ϕ and ϕ denote the borders, δ = 0.01π. The first
a b degeneracy of the g.s.).
column gives theregion of ϕ.
Region ϕ Numberof levels
M =0 M =N M =2N −4
I π/4 1 1 1
I ϕ −δ 4 4 2 VI. NUMERICAL SOLUTIONS OF THE
a
II ϕ +δ 23059 7 2 SYMMETRY ADAPTED GP-EQUATION
a
II π 20596 1 1
II ϕ −δ 167171 19 2
b BymakinguseoftheTFAwehaveobtainedanalytical
III ϕ +δ 3372 7999 2
b solutions, thereby the related physics can be understood
III 3π/2 885 2001 1 inananalyticalway. Inordertoevaluatetheaccuracyof
the TFA, the equation with the kinetic energy included
is solved numerically and the results are given below for
From the table we know that a comparison. Firstly, we found that the curves for E
gs
(i)WhenϕisinregionIorM iscloseto2N (i.e.,inor from the numerical solutions as a whole is an upward
close to the ferro-phase), the density of low-lying states shiftofthoseplottedinFig.1. Itimpliesthattheamount
is ratherdiffuse. Togetherwiththe biggap,both factors of total kinetic energies contained in various states with
assurethe stability of the g.s. (When the number =1 as very different spin-textiles are similar, and the E given
gs
shownin the secondrow,there is no excited states lower under the TFA are correct when N =12000 (except the
than 0.1~ω). omission of the kinetic energy). The shift is about 3.5
(ii) When ϕ is in region II and M is very small (i.e., 103~ω (thus the kinetic energy is about 6% of the tota×l
in or close to the polar phase), the density is extremely energy in our cases). Secondly, we found that the curves
dense. Together with the very small gap, both factors for E from the numerical solutions overlapthe curves
gap
lead to a highly unstable g.s.. This situation can be from TFA plotted in Fig.2. It implies that the amount
greatly improved when M gets larger. of kinetic energies contained in the g.s. and in the first
(iii) When ϕ is in region III and M is not very large excited state is similar. Thus the spin-excitation does
(say, M N), the low-lying spectrum is also dense but not remarkably affect the spatial motion.
≤
TABLE II: Root Means Square RadiusR in µm from nu-
rms
merical calculation versus N, ϕ, and M with ω = 300s−1.
The values from TFA are given inside theparentheses.
N ϕ M R N ϕ M R N ϕ M R
rms rms rms
1200 π/4 0,N,2N 2.48(2.23) 6000 π/4 0,N,2N 3.19(3.07) 12000 π/4 0,N,2N 3.62(3.53)
π 0 2.57(2.35) π 0 3.34(3.24) π 0 3.80(3.72)
N 2.58(2.36) N 3.36(3.25) N 3.81(3.73)
2N 2.60(2.38) 2N 3.39(3.29) 2N 3.85(3.77)
5π/4 0 2.56(2.33) 5π/4 0 3.33(3.22) 5π/4 0 3.78(3.70)
N 2.58(2.35) N 3.35(3.25) N 3.80(3.73)
2N 2.62(2.40) 2N 3.42(3.32) 2N 3.88(3.81)
Furthermore, the size of the system is expected to in- the numerical calculation are shown in Table II, where
crease by including the kinetic energy. The results from
6

size is minimized when ϕ=π/4 and is maximized when
3.0 3.5 4.0 4.5 3.0 3.5 4.0 4.5 3.0 3.5 4.0 4.5 5.0
ϕ=5π/4.
(a) M = 0 (b) M = N (c) M = 2N It has been predicted based on the TFA that R
0.7 0.7 rms ∝
N1/5. Tocheckthis relationlog R fromthe numeri-
10 rms
Rog10rms 0.6 2 3 2 3 2 3 0.6 c3acltshoeluctuiorvnessaarerepnloottteedxainctlFyigs.t3raviegrhstuslinloegs.10TNh.eiIrnsl3oapteos
l 1 1 1 depend on N and are plotted in 3d to 3f, respectively.
0.5 0.5
Disregarding M and ϕ the slopes tend to 1/5 when N
increases as predicted. It has been predicted based on
0.4 0.4 the TFA that R λ (ω˜) 2/5. To check this relation
rms −
0.20 (d) M = 0 (e) M = N (f) M = 2N 0.20 log R from the n∝umerical solutions are plotted in
slope 00..1168 2 31 2 31 2 31 00..1168 Fi2g/1.045 wvrehmressnusω˜loingc10reω˜a.sesTahsepsrleodpiecsteodf. the curves tend to
−
0.14 0.14
3.0 3.5 4.0 4.5 3.0 3.5 4.0 4.5 3.0 3.5 4.0 4.5 5.0

log10N
VII. SUMMARY
FIG. 3: (color on line) log R versus log N with ω =
300s−1 where R is in µ1m0. rmMs is given a1t0three values.
rms
The generalized GP equation adapted to the U(5)
ϕ=π/4, π,and 5π/4 for thecurves ”1” to ”3”, respectively. ⊃
SO(5) SO(3) symmetry has been derived for spin-2
The lower panels are for the slopes of thecurves.
⊃
condensates. This equation has been solved analytically
under the TFA and by strict numerical calculation. It
1.5 2.0 2.5 3.0 1.5 2.0 2.5 3.0 1.5 2.0 2.5 3.0 was found that the TFA is applicable if N is large (say,
1.0 1.0
N 104). The emphasis is placed on the g.s.. Based
(a) M = 0 (b) M = N (c) M = 2N
≥
on a rigorous treatment of the spin-degrees of freedom,
0.8 0.8
Rrms 3 3 3 thedetailedspin-textiles,i.e.,theferro-,polar,andcyclic
g10 0.6 2 2 2 0.6 phases, and their mixing, are explained in a many-body
lo way and thus the underlying physics can be understood
1 1 1
0.4 0.4 beyond the mean-field-theory. Besides, the variation of
the spin-textiles versus M in regions II and III is no-
0.2 0.2 table. Note that the factors affecting the stability of the
-0.400 -0.400
(d) M = 0 (e) M = N (f) M = 2N g.s., such as the gap and the neighboring level density,
e
slop--00..441005 2 3 2 3 2 3 --00..441005 ttooguecthheedrinwietxhistthinegdleitgeernaetruarceys.oTfhtehseegfa.sc.toirtssealrf,eastruedlieesds
-0.415 1 1 1 -0.415 in detail in this paper. The great difference in the sta-
1.5 2.0 2 .5 3.0 1.5 2.0 2.5 3.0 1.5 2.0 2 .5 3.0 bility and degeneracycausedby varying ϕ (which marks
log10~ the features of the interaction) and M is notable (this
FIG. 4: (color on line) log R versus log ω˜ with N = is true even when ϕ varies within a region, i.e., the g.s.
10 rms 10
12000. RefertoFig.3. Notethattheverticalscalein4dto4f remains in the same phase). We believe that the effect
is very small. It implies that the curves in 4a to 4c are very of these factors would be serious when the temperature
close to straight lines. isverylow. SinceR isanobservable,effortshavealso
rms
been made to clarify the relation between R and N,
rms
ω, and ϕ. This provides a way for checking the theories
the Rrms from TFA are given inside the parentheses and with experimental data.
aresmaller. Fromthe table weknow that the TFA leads
to a 8% reduction of the R (if N = 1200), or a
rms
∼
2% reduction of the R (if N = 12000). Nonethe-
rms
∼ Acknowledgments
less,thetwofeaturesfoundpreviouslyundertheTFAre-
main unchanged, namely, (i) When the g.s. is not in the
ferro-phase, a larger M leads to a larger size. (ii) When The project is supported by the National Basic Re-
the g.s. is in the ferro-phase (ϕ ϕ or M = 2N), the search Programof China under the grant 10874249.
a
≤
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7
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tri 0
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