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Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2013, Article ID 517395, 7 pageshttp://dx.doi.org/10.1155/2013/517395

Research ArticleA Time-Splitting and Sine Spectral Method for Dynamics ofDipolar Bose-Einstein Condensate

Si-Qi Li, Xiang-Gui Li, and Dong-Ying Hua

School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China

Correspondence should be addressed to Xiang-Gui Li; [email protected]

Received 28 May 2013; Accepted 9 July 2013

Academic Editor: Fedele Lizzi

Copyright © 2013 Si-Qi Li et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A two-component Bose-Einstein condensate (BEC) described by two coupled a three-dimension Gross-Pitaevskii (GP) equationsis considered, where one equation has dipole-dipole interaction while the other one has only the usual s-wave contact interaction,in a cigar trap. The time-splitting and sine spectral method in space is proposed to discretize the time-dependent equations forcomputing the dynamics of dipolar BEC. The singularity in the dipole-dipole interaction brings significant difficulties both inmathematical analysis and in numerical simulations. Numerical results are given to show the efficiency of this method.

1. Introduction

The achievement of the Bose-Einstein condensation of dilutegases in 1995 marked the beginning of a new era in atomic,molecular, and optical physics.That has attractedmuch atten-tion both theoretically and experimentally. Most of theirproperties of these trapped quantum gases are governed bythe interactions between particles in the condensate [1]. In thelast several years, there has been an investigation for realizinga newkind of quantumgaseswith the dipolar interaction, act-ing between particles having a permanent magnetic or elec-tric dipole moment. The experimental realization of a BECof 52Cr atoms [2, 3] at the University of Stuttgart in 2005 gavenew impetus to the theoretical and the numerical investiga-tions on these novel dipolar quantum gases at low temper-ature. Recently more detailed and controlled experimentalresults have been obtained, illustrating the effects of phaseseparation in a multicomponent BEC [4–6]. In these papers,the studies of the binary condensates were limited to the caseof s-wave interaction, while a great deal of attention has beendrawn recently to the dipolar BEC.

In this work, a numerical method for computing thedynamics of the two-component dipolar BEC is considered,where one equation has dipole-dipole interaction and theother has only the usual s-wave contact interaction. However,since the dipole-dipole interaction is of long range, ani-sotropic, and partially attractive and the computational cost

in three dimensions high, the nontrivial task of achieving andcontrolling the dipolar BEC is thus particularly challenging.

This paper is organized as follows. In Section 2, a numer-ical method for computing ground states is presented. InSection 3, numerical results are reported to verify the effi-ciency of this numerical method. Finally, some concludingremarks are drawn in Section 4.

2. Numerical Method for Computingthe Dynamics

2.1.TheNonlocal Gross-Pitaevskii Equation. The two-compo-nent dipolar BEC, confined in a cigar trap, is described bytwo coupled Gross-Pitaevskii equations. As far as the dipolarinteraction is concerned, a convolution term is introduced[7–9] to modify the classical Gross-Pitaevskii equation,which results in the following differential-integral equations(1). Since the transition metal has a magnetic dipole interac-tionwhile the alkalimetal does not have, we take into accountthis factor in this system. We take Cr as component 1 and Rbas component 2 [10]. Then the GP equations for this systemcan be written as

𝑖ℎ𝜕𝜑

1( ⃗𝑟, 𝑡)

𝜕𝑡= [−

ℎ2

2𝑚 1∇2+ 𝑉

1+ 𝑈

1

𝜑1

2

+ 𝑈12

𝜑2

2

+𝑉dip ∗𝜑1

2

]𝜑1,

2 Advances in Mathematical Physics

𝑖ℎ𝜕𝜑

2( ⃗𝑟, 𝑡)

𝜕𝑡= [−

ℎ2

2𝑚 2∇2+ 𝑉

2+ 𝑈

2

𝜑2

2

+ 𝑈12

𝜑1

2

]𝜑2,

(1)

where 𝜑1, 𝜑

2are the wave functions of components one and

two, respectively. The interatomic and the intercomponent s-wave scattering interactions are described by 𝑈

𝑗(𝑗 = 1, 2)

and 𝑈12, respectively, with the following expressions [11]:

𝑈𝑗=

4𝜋ℎ2𝑎𝑗

𝑚𝑗

, 𝑈12

=

2𝜋ℎ2𝑎𝑗

𝑚1𝑚

2/ (𝑚

1+ 𝑚

2), 𝑗 = 1, 2, (2)

where 𝑎𝑗is the scattering length of component 𝑗 and 𝑎

12

is that between components 1 and 2. Here ℎ is the Planckconstant,𝑚

𝑗is the mass of the atom of component 𝑗, 𝑗 = 1, 2,

and 𝑉𝑗is the external trapping potential confining the gas.

Generally, that is, 𝑉𝑗( ⃗𝑟) = (𝑚

𝑗/2)(𝜔

2

𝑗𝑥𝑥2+ 𝜔

2

𝑗𝑦𝑦2+ 𝜔

2

𝑗𝑧𝑧2)

with 𝜔𝑗𝑝

(𝑝 = 𝑥, 𝑦, 𝑧) representing the trap frequency in𝑥, 𝑦, and 𝑧 directions, respectively. The local mean-field𝑈𝑗|𝜑

𝑗|2 represents the s-wave interaction. 𝑉dip is the long-

range isotropic dipolar interaction potential between twodipoles and it is defined by

𝑉dip =𝜇0𝜇2

dip

4𝜋⋅1 − 3( ⃗𝑟 ⋅ ⃗𝑛)

2/| ⃗𝑟|

2

| ⃗𝑟|3

=

𝜇0𝜇2

dip

4𝜋⋅1 − 3cos2𝜃

| ⃗𝑟|3

,

(3)

where 𝜃 is the angle between the polarization axis ⃗𝑛 andthe relative of two atoms (i.e., cos 𝜃 = ⃗𝑛 ⋅ ⃗𝑟/𝑟, 𝑟 = | ⃗𝑟| =√𝑥2 + 𝑦2 + 𝑧2). The wave function 𝜑

𝑖(𝑥, 𝑡) is normalized ac-

cording to ‖𝜑𝑖‖2= ∫

𝑅3|𝜑

𝑖( ⃗𝑟, 𝑡)|

2

𝑑 ⃗𝑟 = 𝑁𝑖, 𝑖 = 1, 2, where 𝑁

𝑖is

the number of the atoms in the dipolar BEC.

2.2. Time-Splitting and Sine Spectral Numerical Method forDynamics. The system (1) can be made dimensionless andsimplified by adopting a unit system where the units forlength, time, and energy are given by 𝑎

0, 1/𝜔

0, and ℎ𝜔

0, re-

spectively, with 𝜔0= min{𝜔

𝑗𝑥, 𝜔

𝑗𝑦, 𝜔

𝑗𝑧}, 𝑎

0= √ℎ/𝑚

1𝜔0[12].

By introducing the dimensionless variables 𝑡 = 𝑡/𝜔0,

⃗𝑟

= ⃗𝑟/𝑎0, 𝜑

𝑗= 𝑎

3/2

0𝜑𝑗, we obtain the dimensionless GP

equations in 3D from (1) as follows:

𝑖𝜕𝜑

1

𝜕𝑡= [−

1

2∇2+ 𝑉

1( ⃗𝑟) + 𝛽

11

𝜑

1

2

+𝛽12

𝜑

2

2

+ 𝜆 (𝑉dip ∗𝜑

1

2

) ] 𝜑

1,

𝑖𝜕𝜑

2

𝜕𝑡=[−

1

2∇2+ 𝑎

𝑚𝑉

2( ⃗𝑟) + 𝛽

21

𝜑

1

2

+ 𝛽22

𝜑

2

2

] 𝜑

2,

(4)

where 𝛽11

= 4𝜋𝑎1𝑁

1, 𝛽

12= ((1 + 𝑎

𝑚)/𝑎

𝑚)2𝜋𝑎

12𝑁

2, 𝛽

21=

((1 + 𝑎𝑚)/𝑎

𝑚)2𝜋𝑎

12𝑁

1, 𝛽

22= (4𝜋𝑎

2/𝑎

𝑚)𝑁

2, 𝑎

𝑚= 𝑚

2/𝑚

1,

𝜆 = 𝑚1𝑁

1𝜇0𝜇2

dip/3ℎ2𝑎0, and𝑉dip = (3/4𝜋) ⋅ ((1−3 cos

2𝜃)/𝑟

3).

In addition the wave functions in (4) satisfy ∫𝑅3|𝜑

1|2

= 1,

∫𝑅3|𝜑

2|2

= 1. By using the following formula [13]

1

𝑟3(1 −

3( ⃗𝑛 ⋅ ⃗𝑟)2

𝑟2) = −

3

4𝜋𝛿 ( ⃗𝑟) − 𝜕

𝑛𝑛(1

𝑟) , (5)

where 𝛿( ⃗𝑟) is the Dirac delta function and 𝜕𝑛𝑛

= ⃗𝑛 ⋅ ∇( ⃗𝑛 ⋅ ∇),we can get

𝑉dip ∗𝜑

1

2

= −𝜑

1

2

− 3𝜕𝑛𝑛

(𝜉) , (6)

where

𝜉 ( ⃗𝑟, 𝑡) =1

4𝜋∫𝑅3

1

⃗𝑟 − ⃗𝑟

⋅𝜑

1( ⃗𝑟

, 𝑡)

2

𝑑 ⃗𝑟. (7)

And it is easy to see that

∇2𝜉 = −

𝜑

1( ⃗𝑟, 𝑡)

2

. (8)

Plugging (6) into (4) and noticing (7) and (8), we can refor-mulate GPE (4) into the Schrödinger-Poisson type system

𝑖𝜕𝜑

1

𝜕𝑡= [−

1

2∇2+ 𝑉

1+ (𝛽

11− 𝜆)

𝜑1

2

+ 𝛽12

𝜑2

2

− 3𝜆𝜕𝑛𝑛

(𝜉) ] 𝜑1,

𝑖𝜕𝜑

2

𝜕𝑡= [−

1

2∇2+ 𝑉

2+ 𝛽

21

𝜑1

2

+ 𝛽22

𝜑2

2

] 𝜑2,

∇2𝜉 = −

𝜑1 (⃗𝑟, 𝑡)

2

.

(9)

In practice, the whole space problem is usually truncated intoa bounded computational domain Ω = [𝑎, 𝑏] × [𝑐, 𝑑] × [𝑒, 𝑓]with the homogeneous Dirichlet boundary condition. Let

Ω𝑀𝐾𝐿

= {(𝑗, 𝑘, 𝑙) | 𝑗 = 1, 2, . . . ,𝑀 − 1, 𝑘 = 1, 2,

. . . , 𝐾 − 1, 𝑙 = 1, 2, . . . , 𝐿 − 1} ,

Ω0

𝑀𝐾𝐿{(𝑗, 𝑘, 𝑙) | 𝑗 = 0, 1, . . . ,𝑀, 𝑘 = 0, 1, . . . , 𝐾,

𝑙 = 0, 1, . . . , 𝐿} .

(10)

Choose the spatialmesh size asℎ𝑥= (𝑏−𝑎)/𝑀, ℎ

𝑦= (𝑑−𝑐)/𝐾

and ℎ𝑧= (𝑓− 𝑒)/𝐿 and define 𝑥

𝑗= 𝑎+ 𝑗ℎ

𝑥, 𝑦

𝑘= 𝑐+ 𝑘ℎ

𝑦, and

𝑧𝑙= 𝑒 + 𝑙ℎ

𝑧, 𝑗, 𝑘, 𝑙 ∈ Ω0

𝑀𝐾𝐿. Then denote the space 𝑌

𝑀𝐾𝐿=

span{𝜙𝑝𝑞𝑠

( ⃗𝑟), (𝑝, 𝑞, 𝑠) ∈ Ω𝑀𝐾𝐿

} with

𝜙𝑝𝑞𝑠

(𝑟) = sin (𝑢𝑝(𝑥 − 𝑎)) sin (𝑢

𝑞(𝑦 − 𝑐)) sin (𝑢

𝑠(𝑧 − 𝑒)) ,

(11)

where 𝑢𝑝= 𝑝𝜋/(𝑏 − 𝑎), 𝑢

𝑞= 𝑞𝜋/(𝑑 − 𝑐), and 𝑢

𝑠= 𝑠𝜋/(𝑓 − 𝑒).

We propose a time-splitting sine pseudo-spectral methodfor computing the dynamics of the BEC [12].

Advances in Mathematical Physics 3

From 𝑡 = 𝑡𝑘to = 𝑡

𝑘+1, the GP equation (9) is solved by

three steps. First, we solve

𝑖𝜕𝜑

1( ⃗𝑟, 𝑡)

𝜕𝑡= −

1

2∇2𝜑1( ⃗𝑟, 𝑡) ,

𝑖𝜕𝜑

2( ⃗𝑟, 𝑡)

𝜕𝑡= −

1

2∇2𝜑2( ⃗𝑟, 𝑡) ,

𝜑1( ⃗𝑟, 𝑡) |

𝑟∈𝜕Ω= 0, 𝜑

2( ⃗𝑟, 𝑡) |

𝑟∈𝜕Ω= 0

(12)

from 𝑡𝑘to 𝑡

𝑘+1/2, followed by solving the nonlinear ODE

𝑖𝜕𝜑

1

𝜕𝑡= [𝑉

1+ (𝛽

11− 𝜆)

𝜑1

2

+ 𝛽12

𝜑2

2


(𝜉)] 𝜑1,

𝑖𝜕𝜑

2

𝜕𝑡= [𝑉

2+ 𝛽

21

𝜑1

2

+ 𝛽22

𝜑2

2

] 𝜑2,

∇2𝜉 = −

𝜑1 (⃗𝑟, 𝑡)

2

,

𝜑1( ⃗𝑟, 𝑡) |

𝑟∈𝜕Ω= 0, 𝜑

2( ⃗𝑟, 𝑡) |

𝑟∈𝜕Ω= 0,

⃗𝑟 ∈ Ω 𝑡𝑛≤ 𝑡 ≤ 𝑡

𝑛+1

(13)

for one time step. Again, we solve (12) from 𝑡𝑘+1/2

to 𝑡𝑘+1

.Suppose the exact solutions are

𝜑1(𝑥

𝑗, 𝑦

𝑘, 𝑧

𝑙, 𝑡

𝑛)

= 𝜑𝑛

1𝑗𝑘𝑙=

𝑀−1

∑

𝑝=1

𝐾−1

∑

𝑞=1

𝐿−1

∑

𝑠=1

𝜑1𝑝𝑞𝑠

(𝑡𝑛) sin

𝑗𝑝𝜋

𝑀sin

𝑘𝑞𝜋

𝐾sin 𝑙𝑠𝜋

𝐿,

(14)

𝜑2(𝑥

𝑗, 𝑦

𝑘, 𝑧

𝑙, 𝑡

𝑛)

= 𝜑𝑛

2𝑗𝑘𝑙=

𝑀−1

∑

𝑝=1

𝐾−1

∑

𝑞=1

𝐿−1

∑

𝑠=1

𝜑2𝑝𝑞𝑠

(𝑡𝑛) sin

𝑗𝑝𝜋

𝑀sin

𝑘𝑞𝜋


𝐿.

(15)

Substitute (14), (15) into (12); we can find that

𝑖𝑑

𝑑𝑡𝜑1𝑝𝑞𝑠

(𝑡𝑛) =

𝑢2

𝑝+ 𝑢

2

𝑞+ 𝑢

2

𝑠

2𝜑1𝑝𝑞𝑠

(𝑡𝑛) ,

𝑖𝑑

𝑑𝑡𝜑2𝑝𝑞𝑠

(𝑡𝑛) =

𝑢2

𝑝+ 𝑢

2

𝑞+ 𝑢

2

𝑠

2𝜑2𝑝𝑞𝑠

(𝑡𝑛) ,

(𝑝, 𝑞, 𝑠) ∈ Ω0

𝑀𝐾𝐿,

(16)

which can be solved exactly, and we obtain

𝜑(1)

1𝑗𝑘𝑙=

𝑀−1

∑

𝑝=1

𝐾−1

∑

𝑞=1

𝐿−1

∑

𝑠=1

𝑒−𝑖𝜏((𝑢

2

𝑝+𝑢2

𝑞+𝑢2

𝑠)/4)

× 𝛼𝑝𝑞𝑠

sin𝑗𝑝𝜋

𝑀sin

𝑘𝑞𝜋


𝐿,

𝜑(1)

2𝑗𝑘𝑙=

𝑀−1

∑

𝑝=1

𝐾−1

∑

𝑞=1

𝐿−1

∑

𝑠=1


2

𝑝+𝑢2

𝑞+𝑢2

𝑠)/4)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 52030405060708090100

t

𝜑1

𝜑2

Figure 1: The energy evolution according to 𝑡.

× 𝛼

𝑝𝑞𝑠sin

𝑗𝑝𝜋

𝑀sin

𝑘𝑞𝜋


𝐿,

(𝑗, 𝑘, 𝑙) ∈ Ω0

𝑀𝐾𝐿,

(17)

where

𝛼𝑝𝑞𝑠

=8

𝑀𝐾𝐿∑

𝑝𝑞𝑠

𝜑𝑛

1𝑗𝑘𝑙sin

𝑗𝑝𝜋

𝑀sin

𝑘𝑞𝜋


𝐿,

𝛼

𝑝𝑞𝑠=

8

𝑀𝐾𝐿∑

𝑝𝑞𝑠

𝜑𝑛

2𝑗𝑘𝑙sin

𝑗𝑝𝜋

𝑀sin

𝑘𝑞𝜋


𝐿.

(18)

Equations (12) will be discretized in space by sine pseudo-spectral method and integrated in time [14]. Next, we willshow that (13) can be solved exactly.

In fact, for 𝑡 ∈ [𝑡𝑛,𝑡𝑛+1

], multiplying (13) by theconjugation of 𝜑(𝑟, 𝑡), that is, 𝜑(𝑟, 𝑡), we get

𝑖𝜕𝜑

1

𝜕𝑡𝜑1= [𝑉

1+ (𝛽

11− 𝜆)

𝜑1

2

+ 𝛽12

𝜑2

2

−3𝜆𝜕𝑛𝑛

(𝜉) ] 𝜑1𝜑1,

𝑖𝜕𝜑

2

𝜕𝑡𝜑2= [𝑉

2+ 𝛽

21

𝜑1

2

+ 𝛽22

𝜑2

2

] 𝜑2𝜑2,

(19)

and we also have

−𝑖𝜕𝜑

1

𝜕𝑡𝜑1= [𝑉

1+ (𝛽

11− 𝜆)

𝜑1

2

+ 𝛽12

𝜑2

2

−3𝜆𝜕𝑛𝑛

(𝜉) ] 𝜑1𝜑1,

−𝑖𝜕𝜑

2

𝜕𝑡𝜑2= [𝑉

2+ 𝛽

21

𝜑1

2

+ 𝛽22

𝜑2

2

] 𝜑2𝜑2.

(20)

Therefore, subtracting (13) from (14), one obtains

𝑖𝑑𝜑𝑖 (

⃗𝑟, 𝑡)

2

𝑑𝑡= 0, 𝑖 = 1, 2, (21)

which implies that𝜑𝑖 (

⃗𝑟, 𝑡)

2

=𝜑𝑖 (

⃗𝑟, 𝑡𝑘)

2

, 𝑖 = 1, 2 ∀𝑘, 𝑡𝑘≤ 𝑡 ≤ 𝑡

𝑘+1. (22)


−50

5

−5 05

0

0.1

0.2t = 0

−50

5

−5 05

0

0.1

0.2t = 1

−50

5

−5 05

0

0.1

0.2 t = 3

−50

5

−5 05

0

0.1

0.2 t = 5

Figure 2: The wave function evolution according to time. Surface plots for |𝜑1(𝑥, 0, 𝑧, 𝑡)|

2 at different times.

Substituting (22) into (13), we get a linear ODE

𝑖𝜕𝜑

1( ⃗𝑟, 𝑡)

𝜕𝑡= [𝑉

1+ (𝛽

11− 𝜆)

𝜑1 (⃗𝑟, 𝑡𝑘)

2

+𝛽12

𝜑2 (⃗𝑟, 𝑡𝑘)

2


(𝜉)] 𝜑1( ⃗𝑟, 𝑡) ,

𝑖𝜕𝜑

2( ⃗𝑟, 𝑡)

𝜕𝑡= [𝑉

1+ 𝛽

21

𝜑1 (⃗𝑟, 𝑡𝑘)

2

+𝛽22

𝜑2 (⃗𝑟, 𝑡𝑘)

2

] 𝜑2( ⃗𝑟, 𝑡)

(23)

which can be solved exactly. Integrating (19) from 𝑡𝑘to 𝑡, one

gets

𝜑(2)

1𝑗𝑘𝑙= 𝜑

(1)

1𝑗𝑘𝑙

× 𝑒−𝑖𝜏[𝑉1+(𝛽11−𝜆)|𝜑1( ⃗𝑟,𝑡𝑛)|

2+𝛽12|𝜑2( ⃗𝑟,𝑡𝑛)|

2−3𝜆𝜕𝑛𝑛(𝜉

𝑛)],

𝜑(2)

2𝑗𝑘𝑙= 𝜑

(1)

2𝑗𝑘𝑙

× 𝑒−𝑖𝜏[𝑉1+(𝛽11−𝜆)|𝜑1( ⃗𝑟,𝑡𝑛)|

2+𝛽12|𝜑2( ⃗𝑟,𝑡𝑛)|


𝑛)],

(𝑗, 𝑘, 𝑙) ∈ Ω0

𝑀𝐾𝐿,

(24)

where 𝜕𝑛𝑛𝜉𝑛(𝑟)|

𝑗𝑘𝑙= −∑

𝑝𝑞𝑠(𝑢

2

𝑝+ 𝑢

2

𝑞+ 𝑢

2

𝑠)𝛾

𝑝𝑞𝑠sin(𝑗𝑝𝜋/𝑀)

sin(𝑘𝑞𝜋/𝐾) sin(𝑙𝑠𝜋/𝐿).The discrete sine transform coefficients of the vector

𝜙𝑝𝑞𝑠

(𝑟)|(𝑥𝑗 ,𝑦𝑘,𝑧𝑙)

for (𝑝, 𝑞, 𝑠) ∈ Ω𝑀𝐾𝐿

are

𝛾𝑝𝑞𝑠

=1

𝑢2𝑝+ 𝑢2

𝑞+ 𝑢2

𝑠

8

𝑀𝐾𝐿

×

𝑀−1

∑

𝑗=1

𝐾−1

∑

𝑘=1

𝐿−1

∑

𝑙=1

𝜑(1)

1𝑗𝑘𝑙

2

sin(𝑗𝑝𝜋

𝑀) sin(

𝑘𝑞𝜋

𝐾) sin(𝑙𝑠𝜋

𝐿) .

(25)

Let 𝜑𝑛1𝑗𝑘𝑙

and 𝜑𝑛2𝑗𝑘𝑙

be the approximations of 𝜑1(𝑥

𝑗, 𝑦

𝑘, 𝑧

𝑙, 𝑡

𝑛)

and 𝜑2(𝑥

𝑗, 𝑦

𝑘, 𝑧

𝑙, 𝑡

𝑛), respectively, which are the solution of

(9). For 𝑛 = 1, 2, . . ., a second-order time-splitting andsine spectral method for solving (9) via the standard Strangsplitting is [14–16]

𝜑(1)

1𝑗𝑘𝑙=

𝑀−1

∑

𝑝=1

𝐾−1

∑

𝑞=1

𝐿−1

∑

𝑠=1


2

𝑝+𝑢2

𝑞+𝑢2

𝑠)/4)

𝛼𝑝𝑞𝑠

sin𝑗𝑝𝜋

𝑀

× sin𝑘𝑞𝜋


𝐿,

𝜑(1)

2𝑗𝑘𝑙=

𝑀−1

∑

𝑝=1

𝐾−1

∑

𝑞=1

𝐿−1

∑

𝑠=1


2

𝑝+𝑢2

𝑞+𝑢2

𝑠)/4)

𝛼

𝑝𝑞𝑠

× sin𝑗𝑝𝜋

𝑀sin

𝑘𝑞𝜋


𝐿,

𝜑(2)

1𝑗𝑘𝑙= 𝜑

(1)

1𝑗𝑘𝑙𝑒−𝑖𝜏[𝑉1+(𝛽11−𝜆)|𝜑1(𝑟,𝑡𝑛)|

2+𝛽12|𝜑2(𝑟,𝑡𝑛)|


𝑛)],

𝜑(2)

2𝑗𝑘𝑙= 𝜑

(1)

2𝑗𝑘𝑙𝑒−𝑖𝜏[𝑉1+(𝛽11−𝜆)|𝜑1(𝑟,𝑡𝑛)|

2+𝛽12|𝜑2(𝑟,𝑡𝑛)|


𝑛)],

𝜑𝑛+1

1𝑗𝑘𝑙=

𝑀−1

∑

𝑝=1

𝐾−1

∑

𝑞=1

𝐿−1

∑

𝑠=1


2

𝑝+𝑢2

𝑞+𝑢2

𝑠)/4)

𝛽𝑝𝑞𝑠

× sin𝑗𝑝𝜋

𝑀sin

𝑘𝑞𝜋


𝐿,

𝜑𝑛+1

2𝑗𝑘𝑙=

𝑀−1

∑

𝑝=1

𝐾−1

∑

𝑞=1

𝐿−1

∑

𝑠=1


2

𝑝+𝑢2

𝑞+𝑢2

𝑠)/4)

𝛽

𝑝𝑞𝑠

× sin𝑗𝑝𝜋

𝑀sin

𝑘𝑞𝜋


𝐿,

(𝑗, 𝑘, 𝑙) ∈ Ω0

𝑀𝐾𝐿,

(26)


−50

5

−5 05

0

0.1

0.2t = 0

−5 05

−50

50

0.2

0.4t = 1

−50

5

−50 5

0

0.2

0.4t = 3

−5 05

−50

50

0.2

0.4t = 5



where

𝛽

𝑝𝑞𝑠=

8

𝑀𝐾𝐿∑

𝑝𝑞𝑠

𝜑(2)

2𝑗𝑘𝑙sin

𝑗𝑝𝜋

𝑀sin

𝑘𝑞𝜋


𝐿,

𝛽𝑝𝑞𝑠

=8

𝑀𝐾𝐿∑

𝑝𝑞𝑠

𝜑(2)

1𝑗𝑘𝑙sin

𝑗𝑝𝜋

𝑀sin

𝑘𝑞𝜋


𝐿.

(27)

3. Numerical Results

3.1. Example with the Same Initial Condition. The confiningpotential is a cigar potential with 𝑉(𝑟) = (1/2)(𝑥2 + 𝑦2 +0.04𝑧

2). Consider the dynamics of the BEC in the cigar trap.

The initial condition is given as follows:

𝜑1,0

= 𝜑1(𝑥, 𝑦, 𝑧, 0) = 𝜋

−3/4𝛾1/2

𝑥𝛾1/4

𝑧𝑒−(1/2)(𝛾𝑥(𝑥

2+𝑦2)+𝛾𝑧𝑧

2),

𝜑2,0

= 𝜑2(𝑥, 𝑦, 𝑧, 0) = 𝜋

−3/4𝛾1/2

𝑥𝛾1/4

𝑧𝑒−(1/2)(𝛾𝑥(𝑥

2+𝑦2)+𝛾𝑧𝑧

2),

(𝑥, 𝑦, 𝑧) ∈ 𝑅3.

(28)

Here 𝛾𝑥= 𝜔

𝑥/2𝜔

0, 𝛾

𝑧= 𝜔

𝑧/2𝜔

0, and 𝜔

0= min{𝜔

𝑗𝑥, 𝜔

𝑗𝑦, 𝜔

𝑗𝑧}.

We solve this system on [−8, 8]3 with ℎ𝑥= ℎ

𝑦= ℎ

𝑧= 1/2 and

Δ𝑡 = 0.01.Figure 1 shows energy evolutions of dipole BEC. And the

energy is conserved. Figures 2 and 3 show the wave functionevolutions according to time.

3.2. Example with the Different Initial Condition. The confin-ing cigar trap potential is𝑉(𝑟) = (1/2)(𝑥2+𝑦2+0.04𝑧2). Solvethe dynamics problem for a dipolar BEC with the cigar trap.The initial condition is

𝜑1,0

= 𝜑1(𝑥, 𝑦, 𝑧, 0)

= 𝜋−3/4

𝛾1/4

𝑥𝛾1/4

𝑦𝛾1/4

𝑧𝑒−(1/2)(𝛾1𝑥𝑥

2+𝛾1𝑦𝑦

2+𝛾1𝑧𝑧

2),

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50102030405060708090100

t

𝜑1

𝜑2

Figure 4: The energy evolution according to 𝑡.

𝜑2,0

= 𝜑2(𝑥, 𝑦, 𝑧, 0) =

1

2𝜋−3/8

𝑒−(1/2)(𝛾2𝑥𝑥

2+𝛾2𝑦𝑦

2+𝛾2𝑧𝑧

2),

(𝑥, 𝑦, 𝑧) ∈ 𝑅3.

(29)

Figure 4 shows energy evolutions of dipole BEC. And theenergy is conserved. Figures 5 and 6 show the wave functionevolutions according to time.

4. Conclusion

Anefficient numericalmethod is presented for computing thedynamics of the dipolar Bose-Einstein condensates based ontwo coupled three-dimensional Gross-Pitaevskii equationswhere one equation has a dipole-dipole interaction potentialand the other one has only the usual s-wave contact inter-action. Using equality (5), we can reformulate the GPE fordipolar BEC into a Grosss-Pitaevskii-Poisson type system.Numerical examples are given to show the efficiency of our


−50

5

−5 05

0

0.1

0.2 t = 0

−5 05

−50

50

0.1

0.2t = 1

−50

5

−50 5

0

0.1

0.2t = 3

−5 05

−50

50

0.1

0.2t = 5



−50

5

−5 05

0

0.1

0.2t = 0

−5 05

−50

50

0.1

0.2t = 1

−5 05

−50

50

0.1

0.2t = 3

−5 05

−50

50

0.1

0.2t = 5



method. The figures show the evolution of the wave functionwith time. And in all cases, total energy is conserved. Theresults agree with the previous work [16]. Numerical resultsare given to demonstrate the efficiency of our numericalmethod.

Authors’ Contribution

All authors conceived the study, participated in its design andcoordination, drafted the paper, participated in the sequencealignment, and read and approved the final paper.

Acknowledgments

This work was supported by National Science Foundation ofChina (no. 11171032) and Beijing Municipal Education Com-mission (no. KM201110772017).

References

[1] S. S. ItaevskiiL, Bose-Einstein Condensation, Oxford University,New York, NY, USA, 2003.

[2] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau,“Bose-Einstein condensation of chromium,” Physical ReviewLetters, vol. 94, no. 16, Article ID 160401, 2005.

[3] J. Stuhler, A. Griesmaier, T. Koch et al., “Observation of dipole-dipole interaction in a degenerate quantum gas,” Physical Re-view Letters, vol. 95, no. 15, Article ID 150406, 2005.

[4] V. Schweikhard, I. Coddington, P. Engels, S. Tung, and E. A.Cornell, “Publisher’s note: vortex-lattice dynamics in rotatingspinor bose-einstein condensates,” Physical Review Letters, vol.93, no. 22, Article ID 210403, 2004.

[5] K. M.Mertes, J. W. Merrill, R. Carretero-González, D. J. Frantz-eskakis, P. G. Kevrekidis, and D. S. Hall, ibid. 99, 190402, 2007.

[6] S. B. Papp, J.M. Pino, andC. E.Wieman, ibid. 101, 040402, 2008.


[7] S. Yi and L. You, “Trapped atomic condensates with anisotropicinteractions,” Physical Review A, vol. 61, Article ID 041604,2000.

[8] S. Yi and L. You, “Trapped condensates of atoms with dipoleinteractions,”Physical ReviewA, vol. 63, Article ID 053607, 2001.

[9] S. Yi and L. You, “Calibrating dipolar interaction in an atomiccondensate,” Physical Review Letters, vol. 92, no. 19, Article ID193201, 2004.

[10] K.-T. Xi, J. Li, and D.-N. Shi, “Phase separation of a two-com-ponent dipolar Bose-Einstein condensate in the quasi-one-di-mensional and quasi-two-dimensional regime,” Physical ReviewA, vol. 84, no. 1, Article ID 013619, 2011.

[11] C. J. Pethick andH. Smith, Bose-Einstein Condensation in DiluteGases, Cambridge University Press, Cambridge, UK, 2008.

[12] D.-Y. Hua, X.-G. Li, and J. Zhu, “A mass conserved splittingmethod for the nonlinear Schrödinger equation,”Adance in Dif-ference Equtions, vol. 2012, article 85, 2012.

[13] W. Bao and Y. Cai, “Mathematical theory and numerical meth-ods for Bose-Einstein condensation,” Kinetic and Related Mod-els, vol. 6, no. 1, pp. 1–135, 2013.

[14] W. Bao and Y. Zhang, “Dynamics of the group state and centralvortex states in Bose-Einstein condensation,” MathematicalModels andMethods in Applied Sciences, vol. 15, no. 12, pp. 1863–1896, 2005.

[15] G. Strang, “On the construction and comparison of differenceschemes,” SIAM Journal on Numerical Analysis, vol. 5, pp. 506–517, 1968.

[16] W. Bao, D. Jaksch, and P. A. Markowich, “Numerical solution ofthe Gross-Pitaevskii equation for Bose-Einstein condensation,”Journal of Computational Physics, vol. 187, no. 1, pp. 318–342,2003.

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