DYNAMICAL SYSTEMS SERIES B
Volume 19, Number 4, June 2014 pp. 1119–1128
A KINETIC ENERGY REDUCTION TECHNIQUE AND CHARACTERIZATIONS OF THE GROUND STATES OF SPIN-1
BOSE-EINSTEIN CONDENSATES
Liren Lin
Institute of Mathematics, Academia Sinica Taipei, 10617, Taiwan
I-Liang Chern
Department of Applied Mathematics and Center of Mathematical Modeling and Scientific Computing, National Chiao Tung University
Hsinchu, 30010, Taiwan and
Department of Mathematics, National Taiwan University Taipei, 10617, Taiwan
(Communicated by Yuan Lou)
Abstract. We justify some characterizations of the ground states of spin-1 Bose-Einstein condensates exhibited from numerical simulations. For ferromag-netic systems, we show the validity of the single-mode approximation (SMA). For an antiferromagnetic system with nonzero magnetization, we prove the vanishing of the mF = 0 component. In the end of the paper some remaining
degenerate situations are also discussed. The proofs of the main results are all based on a simple observation, that a redistribution of masses among different components will reduce the kinetic energy.
1. Introduction. At ultra low temperature, massive bosons could occupy the same lowest-energy state and form the so-called Bose-Einstein condensates (BECs). This phenomenon was predicted by Bose and Einstein in 1925, and was first realized on several alkali atomic gases in 1995 by laser cooling technique [1, 7, 11]. In early experiments, the atoms were confined in magnetic traps. In this situation the spin degrees of freedom are frozen. Through the mean-field approximation the system is then described by a scalar wave function, which satisfies the Gross-Pitaevskii (GP) equation [10,15,24]. In contrast, in an optically trapped atomic BEC all hyperfine spin states can be active simultaneously, and a spin-F BEC is then described by a vector wave function Ψ = (ψF, ψF −1, · · · , ψ−F)T, where the j-th component
corresponds to the mF = j hyperfine state [27, 28, 22, 4,14]. The theory of such
spinor BEC was first developed independently by several groups [23,16,18]. After these early studies, spinor BEC has become an area of great research interest.
2010 Mathematics Subject Classification. Primary: 35A15, 35Q55; Secondary: 35Q40. Key words and phrases. Spin-1 BEC, ground state, single-mode approximation, nonlinear sys-tem, mass redistribution.
This work was partially supported by the National Science Council of the Republic of China under the grants NSC99-2115-M-002-003.
1.1. Mathematical model for spin-1 BEC. For a spin-1 BEC, the vector wave function Ψ = (ψ1, ψ0, ψ−1)T satisfies a generalized GP equation:
i~∂tΨ =
δE
δΨ∗, (1)
where the Hamiltonian is given by E[Ψ] := Z D ~2 2ma X j |∇ψj|2+ V (x)|Ψ|2+ cn 2 |Ψ| 4+cs 2 |Ψ ∗SΨ|2 dx. Here D is a domain in Rd
, ~ is the reduced Planck constant, ma is the atomic mass,
V is a locally bounded real-valued function representing the trap potential, Ψ∗ is the Hermitian of Ψ, and S = (Sx, Sy, Sz) is the triple of spin-1 Pauli matrices:
Sx= 1 √ 2 0 1 0 1 0 1 0 1 0 , Sy= i √ 2 0 −1 0 1 0 −1 0 1 0 , Sz= 1 0 0 0 0 0 0 0 −1 .
So Ψ∗SΨ denotes the vector (Ψ∗SxΨ, Ψ∗SyΨ, Ψ∗SzΨ). Also note that |Ψ| denotes
the Euclidean length (P
j|ψj|2)1/2, and similiarly for |∇ψj| and |Ψ∗SΨ|. The
pa-rameters cn and csare real constants given by
cn =4π~ 2 3ma (a0+ 2a2), cs=4π~ 2 3ma (−a0+ a2),
where a0and a2are respectively the s-wave scattering lengths for scattering channels
of total hyperfine spin zero and spin two. The parameter cn characterizes the
spin-independent interaction, and the parameter cscharacterizes the spin-exchange
interaction. For cn < 0 (resp. cn> 0), the spin-independent interaction is attractive
(resp. repulsive). For cs < 0 (resp. cs > 0), the spin-exchange interaction is
ferromagnetic (resp. antiferromagnetic). Typical examples of ferromagnetic and antiferromagnetic systems are87Rb and23Na condensates.
The generalized GP equation (1) implies two conserved quantities: (C1) Z D |Ψ|2= N, (C2) Z D |ψ1|2− |ψ−1|2 = M,
where N is the total number of atoms and M is the total magnetization. For the system to be nontrivial, we assume N > 0. We also assume |M | < N (note that obviously |M | ≤ N ), for if |M | = N the system reduces to a single component BEC, which is a trivial case for all considerations in this work. Now we say Ψ is a ground state if it is a minimizer of E under the above two constraints.
1.2. Innovation and organization. In researches concerning ground states of spin-1 BEC, the following ansatz was often adopted:
ψj= γjψ for each j, (2)
where γjare constants and ψ is a function independent of j. This is called the
single-mode approximation (SMA) in the physics literature [18,13,26,17,25,12]. It has been found [29] from numerical simulations that ground states obey the SMA exactly for ferromagnetic systems (and does not in general for antiferromagnetic ones), and hence can effectively be characterized as one-component systems. The first goal of this paper is to analytically confirm this observation. On the other hand, for
antiferromagnetic systems, we will show that ψ0 ≡ 01 when M 6= 0, another
well-known phenomenon from numerical simulations [3, 9] not being rigorously proved before. For the degenerate case M = 0, however, the SMA is again valid while ground states are not unique, and ψ0 does not necessarily vanish. It’s interesting
that although the two phenomena (SMA and vanishing of ψ0) look quite irrelevant
to each other, they can be proved by the same simple principle, that a redistribution of masses between different components will decrease the kinetic energy.
The paper is organized as follows. Section 2 is the preliminary, where we reformu-late the mathematical model more precisely, and then provide a result of maximum principle which is crucial in justifying the expected characterizations. In Section 2.2 the idea of mass redistribution is introduced. Sections 3 and 4 treat respectively the ferromagnetic and antiferromagnetic systems.
2. Preliminary. For notational simplicity, let’s redefine E[Ψ] = Z D X j |∇ψj|2+ V |Ψ|2+ cn|Ψ|4+ cs|Ψ∗SΨ|2 .
This causes no loss of generality for the phenomena we are going to investigate. The admissible class is
C =nΨ ∈ H1(D) ∩ L4(D) ∩ L2(D, V dx)3 Ψ satisfies (C1) and (C2) o , where L2(D, V dx) consists of all functions f such thatR
DV |f |
2< ∞. Let u denotes
(u1, u0, u−1). We also define
A = {u ∈ C | uj≥ 0 for each j} ;
A1= {u ∈ A | u = (γ1f, γ0f, γ−1f ) for some constants γj and some function f } ;
A2= {u ∈ A | u0≡ 0} .
Let’s also use γ to denote (γ1, γ0, γ−1), so that (γ1f, γ0f, γ−1f ) can be abbreviated
as γf .
In Section 2.1, we introduce a common reduction which asserts that to study ground states we can simply consider A instead of C. Indeed, A consists just the amplitudes of elements in C. And A1 (resp. A2) corresponds to the set of all
elements obeying the SMA (resp. with vanishing zeroth components). For the moment, we do not consider any boundary condition for simplicity. See the remark after Theorem3.1.
2.1. Reduction from C to A. Given Ψ ∈ C. Let ujeiθj be the polar form of ψj
for each j. Then, by formally2 differentiating the θ
j’s, it’s easy to check that
E[Ψ] = Z D X j
(|∇uj|2+ u2j|∇θj|2) + V |u|2+ cn|u|4
+ cs h 2u20 u21+ u2−1+ 2u1u−1cos (θ1− 2θ0+ θ−1) 2 + (u21− u2 −1)2 i . (3)
1We use “f ≡ g” to stress f is “identically” equal to g. We shall also usually only use “=” for
equalities of functions later on, possibly in the sense of almost everywhere. There is no true point to distinguish them in this paper.
2 The differentiation is formal since the fact that ψ
j∈ H1alone doesn’t imply we can choose
For Ψ to be a ground state, we thus require the following:
The θj’s are constants and cos (θ1− 2θ0+ θ−1) = ±1 for cs≶ 0. (4)
And then E[Ψ] = Z D X j
|∇uj|2+ V |u|2+ cn|u|4+ cs
h 2u20(u1± u−1)2+ (u21− u 2 −1) 2i , (5) where the plus-minus sign ± corresponds to cs≶ 0.
Let’s now define E : A → R, E[u] is given by the right-hand side of (5). We use G to denote the set of all minimizers of E over A. We will not rigorously establish a correspondence between ground states and elements in G. For example, we will not justify the validity of the differentiation of the θj’s in (3). Also note that if some
component of a ground state Ψ vanishes, then (4) needs not be satisfied. Despite these problems, we claim that the assertion “every ground state obeys the SMA (2)” does be equivalent to “every element in G lies in A1.” Similarly, the assertion
that “every ground state Ψ has ψ0≡ 0” is equivalent to “every u in G lies in A2.”
We shall omit the proofs of these facts. (See [21].) Without loss of generality, we henceforth consider E and G instead of the original model.
For convenience let’s use H to denote the integrand of E, i.e. E[u] =RDH(u).
We also write H = H1+ H2, where
H1(u) = X j |∇uj|2+ cs h 2u20(u1± u−1)2+ (u21− u 2 −1)2 i , H2(u) = V |u|2+ cn|u|4.
This splitting of H is only for convenience of later discussion.
The Euler-Lagrange system for u ∈ G is given by the following coupled Gross-Pitaevskii equations: (µ + λ)u1= Lu1+ 2csu20(u1± u−1) + u1(u21− u 2 −1) µu0= Lu0+ 2csu0(u1± u−1)2 (µ − λ)u−1= Lu−1+ 2csu20(u−1± u1) + u−1(u2−1− u21) , (6)
where L = −∆ + V + 2cn|u|2, and λ and µ are the Lagrange multipliers induced
by the constraints (C1) and (C2). We remark that in this paper we do not involve ourselves in the problem of existence. To best illustrate the simplicity of our method, we just assume there is a ground state. (See [20,8,2] for related concerns of existence problem). Also note that u ∈ G is continuously differentiable by standard regularity theorem.
The following lemma will be crucial in our characterizations of ground states. Lemma 2.1. If u ∈ G, then for each j, either uj ≡ 0 or uj> 0 on all of D.
Proof. Let K be a compact subset of D. By subtracting respectively Qjuj, j =
1, 0, −1, from the three equations in (6) with large enough constants Qj, and using
the assumption uj ≥ 0, it’s easy to verify that each uj satisfies
∆uj+ hjuj≤ 0
for some function hj which is non-positive on K. Thus either uj > 0 or uj≡ 0 on
K by the strong maximum principle. Since K ⊂ D is arbitrary, the assertion of the lemma holds.
2.2. A kinetic-energy-reducing redistribution. Consider an n-tuple of non-negative functions f = (f1, f2, ..., fn) ∈ (H1(D))n. Let g = |f |. It’s well-known (see
e.g. [19], Theorem 7.8) that |∇g|2≤P
k|∇fk|2. In fact, X k |∇fk|2− |∇g|2= 1 g2 X j<k |fj∇fk− fk∇fj|2 on where g > 0 0 on where g = 0. (7)
This convexity inequality for gradients has a simple while interesting generalization, when f2
1, f22, . . . , fn2 do not sum to a single g2, but instead are redistributed into
multiple parts. To be precise, we give the following definition.
Definition 2.2. Let f be as above, and let g = (g1, g2, ..., gm) be an m-tuple of
nonnegative functions. We say g is a square redistribution of f if g2` =
n
X
k=1
a`kfk2 for ` = 1, 2, . . . , m, (8)
where a`k are constants, a`k≥ 0, and P m
`=1a`k= 1 for each k = 1, 2, . . . , n.
Note that g = |f | is the only square redistribution of f for m = 1. In general we have the following result.
Theorem 2.3. For any square redistribution g of f as in Definition2.2, we have (a) |g| = |f |, (b) Pm `=1|∇g`| 2≤Pn k=1|∇fk| 2.
Proof. (a) follows by summing (8) over ` = 1, 2, . . . , m. For fixed `, apply the convexity inequality for gradients to the vector (√a`1f1,
√ a`2f2, . . . , √ a`kfk), we obtain |∇g`|2≤ n X k=1 a`k|∇fk|2.
And (b) follows by summing this inequality over ` = 1, 2, . . . , m.
Remark. We can naturally generalize the idea to p-th power redistribution, which may be useful in studying systems with p-Laplacian terms.
To save notation, in the following we shall omit the adjective “square” and simply say “redistribution”. Since the square of the amplitude of a wave function represents the distribution of its mass density, a redistribution of u ∈ A means a redistribution of the masses between its three components. If u ∈ A and v = (v1, v0, v−1) is a
redistribution of u, Theorem2.3(a) says the shapes of their total mass distribution are the same. In particular, v satisfies the first constraint (C1), and H2(v) = H2(u).
These facts together with (b), which causes a reduction of the kinetic energy, allow us to give a simple and unified approach to our problems.
3. Ferromagnetic systems. In this section we assume cs < 0, and the goal is
to prove the validity of SMA, that is G ⊂ A1. The idea is to find, for u ∈ A,
a redistribution of u in A1 which has no larger energy than u, and then try to
Now given any u ∈ A. It’s easy to see that a redistribution of u lies in A1 if and
only if it can be expressed as γ|u|, where γ = (γ1, γ0, γ−1) is a triple of nonnegative
constants satisfying ( γ2 1+ γ 2 0+ γ 2 −1 = 1 γ12− γ2 −1= M/N. (9) Let Γ denote the set containing all such γ:
Γ :=γ ∈ R3 γj≥ 0 for each j, γ satisfies (9) .
Then
H1(γ|u|) = |∇|u||2+ csP (γ)|u|4,
where
P (γ) = 2γ20(γ1+ γ−1)2+
M2 N2.
For the redistributed γ|u| to have no larger energy than u, the best candidate is obviously obtained by maximizing P (γ). It’s easy to check that
max
γ∈ΓP (γ) = P (γ ?) = 1,
where the maximizer γ?= (γ?
1, γ0?, γ−1? ) is uniquely given by γ1?= 1 2 1 +M N , γ0?= s 1 2 1 − M 2 N2 , and γ?−1=1 2 1 − M N . We can now state our main theorem of this section.
Theorem 3.1. Assume cs< 0. If u ∈ G, then u = γ?|u|.
Proof. Since γ?|u| is a redistribution of u, H
2(u) = H2(γ?|u|). Hence
H(u) − H(γ?|u|) = H1(u) − H1(γ?|u|) =: Dk+ Ds,
where Dk= X j |∇uj|2− |∇|u||2≥ 0 by (7), and Ds= cs2u20(u1+ u−1)2+ (u21− u 2 −1)2 − cs|u|4= −cs(u20− 2u1u−1)2≥ 0.
However, u ∈ G, thus we must have Dk = Ds= 0. From (7), this occurs if and only
if
uj∇uk− uk∇uj= 0 for j 6= k ; (10)
u20− 2u1u−1= 0. (11)
Since we assume the total number of atoms N > 0, from Lemma 2.1, at least one uj is strictly positive in D. Assume u1> 0 on D, then (10) implies
∇ (u0/u1) = ∇ (u−1/u1) = 0. (12)
Since D is connected, (12) implies u0 and u−1 are both constant multiples of u1.
This shows u ∈ A1. The same conclusion holds obviously if u0 > 0 or u−1 > 0.
That u must be γ?|u| then follows either by (11) or by the fact that γ?is the unique
Remark. We can add more assumptions in the definition of A for Theorem3.1to hold. The only thing we need to take care is that we need γ?|u| ∈ A whenever u ∈ G, so that E[u] ≤ E[γ?|u|] is not violated. For example, we can consider a homogeneous boundary condition (e.g. homogeneous Dirichlet or Neumann boundary condition) for u ∈ A, since then γ?|u| also satisfies the same condition.
Theorem 3.1 implies that searching for ground states of ferromagnetic spin-1 BEC can be reduced to a single-component minimization problem. Precisely, define the single-component admissible class
As= {|u| | u ∈ A } =u ∈ H1(D) ∩ L4(D) ∩ L2(D, V dx) u ≥ 0, R Du 2= N , (13) and define Es[u] =RD|∇u| 2+ V u2+ (c n+ cs)u4 for u ∈ As, Gs= {u ∈ As | Es[u] = min v∈AsEs[v] } .
We have the following characterization. Corollary. G = {γ?u | u ∈ Gs}.
Proof. If u ∈ G, then u = γ?|u| by Theorem3.1. To see |u| ∈ Gs, note that
Es[|u|] = E[γ?|u|] ≤ E[γ?v] = Es[v]
for every v ∈ As. Conversely if u ∈ Gs, we want to show γ?u ∈ G. This is true since
E[γ?u] = Es[u] ≤ Es[|v|] = E[γ?|v|] ≤ E[v] for every v ∈ A.
4. Antiferromagnetic systems and some degenerate cases. The main focus of this section is the phenomenon u0 ≡ 0. After justifying it in Section 4.1, some
degenerate situations are also discussed in Section 4.2.
4.1. Justification of the vanishing phenomenon. Assume cs> 0 in this
sub-section. We want to show that any ground state must have a vanishing zeroth component. Similar to the approach in the previous section, we want to find an appropriate redistribution eu of u ∈ A so thateu ∈ A2 and E[eu] ≤ E[u]. Now, not
as before, the assumptionu ∈ Ae 2 alone doesn’t give rise to a definite hint for what
e
u should be. In view that suchu satisfies |e u| = |u| and hence (C1), as a guess, wee try just imposing the additional assumption thatu also satisfiese
e u21−ue
2
−1= u21− u2−1,
so that (C2) is also satisfied byu automatically. This results in only one possibility,e that is e uj = r u2 j+ u2 0 2 for j = 1, −1. (14)
It’s fortunate that it works.
Proof. For u ∈ A, letu ∈ Ae 2 be its redistribution defined by (14). Then H(u) − H(u) = De k+ Ds, where Dk = X j |∇uj|2− X j |∇euj|2≥ 0 by Theorem2.3(b), and Ds= 2csu20(u1− u−1)2≥ 0.
by direct computation. Assume u ∈ G, then we must have u20(u1− u−1)2= 0.
(That Dk = 0 is also true but is not needed here.) By Lemma2.1, either u0 ≡ 0
or u1≡ u−1. However since we assume M 6= 0, we cannot have u1≡ u−1, and the
assertion follows.
4.2. Some degenerate situations. The requirement M 6= 0 in Theorem 4.1 is necessary. In fact, for M = 0, SMA is again valid while ground states are not unique, and u0≡ 0 is not necessarily the case. Precisely, consider the minimization
problem (recall that As is defined by (13))
min v∈As Z D |∇v|2+ V v2+ c nv4 . (15)
We have the following characterization.
Proposition 4.2. If cs> 0, M = 0 or cs= 0, then
G =n t,p1 − 2t2, tu
0 ≤ t ≤ 1/ √
2, u is a minimizer of (15)o. Proof. Note that since M = 0, from (9), γ ∈ Γ implies
γ = t,p1 − 2t2, t
for some 0 ≤ t ≤ 1/√2. Now it’s easy to see that for any u ∈ A and γ ∈ Γ we have
H(γ|u|) = |∇|u||2+ V |u|2+ cn|u|4.
Thus, H(γ|u|) is independent of γ ∈ Γ, and H(γ|u|) ≤ H(u) obviously. The proof that u ∈ G implies u must be one of the γ|u| is much the same as in the proof of Theorem3.1, and we omit it.
In contrast to the above theorem, SMA is almost never the case when M 6= 0. Proposition 4.3. Assume cs> 0 and M 6= 0, then u ∈ G ∩ A1implies u1 and u−1
are constants. And this is possible only if V is constant.
Proof. By Theorem4.1, the Euler-Lagrange system (6) is reduced to the following two-component system: ( (µ + λ)u1= Lu1+ 2csu1(u21− u 2 −1) (µ − λ)u−1= Lu−1+ 2csu−1(u2−1− u21), (16) where L = −∆ + V + 2cn(u21+ u2−1).
Recall that we assume −N < M < N , thus, for j = 1, −1, uj > 0 on D. So
u ∈ A1 implies u−1 = κu1 for some constant κ > 0. Also note that κ 6= 1 since
M 6= 0. The system (16) then gives the following two equations for u1:
(µ + λ)u1= −∆u1+ V u1+ 2cn(1 + κ2)u31+ 2cs(1 − κ2)u31; (17)
(µ − λ)u1= −∆u1+ V u1+ 2cn(1 + κ2)u31+ 2cs(κ2− 1)u31. (18)
Now (17) minus (18) gives λu1= 2cs(1 − κ2)u31. Since u1> 0 on D, we get
u1=
s λ 2cs(1 − κ2)
.
In particular u1and u−1= κu1are constants. Hence ∆u1 = 0, and then (17) plus
(18) gives
µu1= V u1+ 2cn(1 + κ2)u31,
from which we get
V = µ − 2cn(1 + κ2)u21= µ −
cn(1 + κ2)
cs(1 − κ2)
λ, which is also a constant.
Acknowledgments. The authors acknowledge the National Center for Theoretical Sciences of the Republic of China for the supports of this research.
REFERENCES
[1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell,Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269 (1995), 198–201. [2] W. Bao and Y. Cai, Ground states of two-component Bose-Einstein condensates with an
internal atomic Josephson junction, East Asian J. Appl. Math, 1 (2011), 49–81.
[3] W. Bao and F. Y. Lim,Computing ground states of spin-1 Bose-Einstein condensates by the normalized gradient flow, SIAM J. Sci. Comput., 30 (2008), 1925–1948.
[4] M. D. Barrett, J. A. Sauer and M. S. Chapman, All-optical formation of an atomic Bose-Einstein condensate, Phys. Rev. Lett., 87 (2001), 010404.
[5] F. Bethuel and X. Zheng, Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal., 80 (1988), 60–75.
[6] J. Bourgain, H. Brezis and P. Mironescu, Lifting in Sobolev spaces, Journal d’Analyse Math´ematique, 80 (2000), 37–86.
[7] C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet,Evidence of Bose-Einstein conden-sation in an atomic gas with attractive interactions, Phys. Rev. Lett., 75 (1995), 1687–1690.
[8] D. Cao, I.-L. Chern and J. Wei,On ground state of spinor Bose-Einstein condensates, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 427–445.
[9] J.-H. Chen, I.-L. Chern and W. Wang,Exploring ground states and excited states of spin-1 Bose-Einstein condensates by continuation methods, J. Comput. Phys., 230 (2011), 2222– 2236.
[10] F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari,Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463–512.
[11] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle,Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1995), 3969–3973.
[12] L.-M. Duan, J. I. Cirac and P. Zoller,Quantum entanglement in spinor Bose-Einstein con-densates, Phys. Rev. A, 65 (2002), 033619.
[13] E. V. Goldstein and P. Meystre,Quantum theory of atomic four-wave mixing in Bose-Einstein condensates, Phys. Rev. A, 59 (1999), 3896–3901.
[14] A. G¨orlitz, T. L. Gustavson, A. E. Leanhardt, R. L¨ow, A. P. Chikkatur, S. Gupta, S. Inouye, D. E. Pritchard and W. Ketterle, Sodium Bose-Einstein condensates in the f = 2 state in a large-volume optical trap, Phys. Rev. Lett., 90 (2003), 090401.
[15] E. P. Gross,Structure of a quantized vortex in boson systems, Il Nuovo Cimento Series 10 , 20 (1961), 454–477.
[16] T.-L. Ho,Spinor bose condensates in optical traps, Phys. Rev. Lett., 81 (1998), 742–745. [17] T.-L. Ho and S. K. Yip,Fragmented and single condensate ground states of spin-1 bose gas,
Phys. Rev. Lett., 84 (2000), 4031–4034.
[18] C. K. Law, H. Pu and N. P. Bigelow,Quantum spins mixing in spinor Bose-Einstein conden-sates, Phys. Rev. Lett., 81 (1998), 5257–5261.
[19] E. H. Lieb and M. Loss, Analysis, 2nd edition, Graduate Studies in Mathematics, 14, Amer-ican Mathematical Society, Providence, RI, 2001.
[20] E. H. Lieb, R. Seiringer and J. Yngvason, Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional, Phys. Rev. A, 61 (2000), 043602.
[21] L.-R. Lin, Mass Redistribution and Its Applications to the Ground States of Spin-1 Bose-Einstein Condensates, Ph.D thesis, National Taiwan University, 2013.
[22] H.-J. Miesner, D. M. Stamper-Kurn, J. Stenger, S. Inouye, A. P. Chikkatur and W. Ketterle,
Observation of metastable states in spinor Bose-Einstein condensates, Phys. Rev. Lett., 82 (1999), 2228–2231.
[23] T. Ohmi and K. Machida, Bose-Einstein condensation with internal degrees of freedom in alkali atom gases, Journal of the Physical Society of Japan, 67 (1998), 1822–1825.
[24] L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Soviet Phys. JETP, 13 (1961), 451– 454.
[25] H. Pu, C. K. Law and N. P. Bigelow,Complex quantum gases: Spinor Bose-Einstein conden-sates of trapped atomic vapors, Physica B: Condensed Matter , 280 (2000), 27–31.
[26] H. Pu, C. K. Law, S. Raghavan, J. H. Eberly and N. P. Bigelow,Spin-mixing dynamics of a spinor Bose-Einstein condensate, Phys. Rev. A, 60 (1999), 1463–1470.
[27] D. M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur, S. Inouye, H.-J. Miesner, J. Stenger and W. Ketterle, Optical confinement of a Bose-Einstein condensate, Phys. Rev. Lett., 80 (1998), 2027–2030.
[28] J. Stenger, S. Inouye, D. M. Stamper-Kurn, H. Miesner, A. P. Chikkatur and W. Ketterle, Spin domains in ground-state Bose-Einstein condensates, Nature, 396 (1998), 345–348. [29] S. Yi, O. E. M¨ustecaplıo˘glu, C. P. Sun and L. You,Single-mode approximation in a spinor-1
atomic condensate, Phys. Rev. A, 66 (2002), 011601.
Received October 2012; revised January 2014.
E-mail address: [email protected]
