# Band structures of a dipolar Bose-Einstein condensate in one-dimensional lattices

**ABSTRACT** We derive the effective Gross-Pitaevskii equation for a cigar-shaped dipolar Bose-Einstein condensate in one-dimensional lattices and investigate the band structures numerically. Due to the anisotropic and the long-ranged dipole-dipole interaction in addition to the known contact interaction, we elucidate the possibility of modifying the band structures by changing the alignment of the dipoles with the axial direction. With the considerations of the transverse parts and the practical physical parameters of a cigar-shaped trap, we show the possibility to stabilize an attractive condensate simply by adjusting the orientation angle of dipoles. Some interesting Bloch waves at several particle current densities are identified for possible experimental observations.

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**ABSTRACT:**The anisotropy of dipole-dipole interaction is revealed by energy band, tunneling dynamics and stabilities of a dipolar condensate in one-dimensional optical lattices. It is demonstrated that the Bloch band structure, the tunneling rate between Bloch bands and the stabilities of Bloch states can be controlled by adjusting the effective aspect ratio of the condensate and the dipolar orientation.Chinese Physics Letters 01/2012; 29(2). · 0.92 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We show that magnetic dipolar interactions can stabilize superfluidity in atomic gases, but the dipole alignment direction required to achieve this varies depending on whether the flow is oscillatory or continuous. If a condensate is made to oscillate through a lattice, damping of the oscillations can be reduced by aligning the dipoles perpendicular to the direction of motion. However, if a lattice is driven continuously through the condensate, superfluid behavior is best preserved when the dipoles are aligned parallel to the direction of motion. We explain these results in terms of the formation of topological excitations and tunnel barrier heights between lattice sites.Physical Review A 02/2013; 87(2). · 3.04 Impact Factor - SourceAvailable from: arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**We study properties of a trapped dipolar Bose-Einstein condensate (BEC) in a circular ring or a spherical shell using the mean-?eld Gross-Pitaevskii equation. In the case of the ring-shaped trap we consider different orientations of the ring with respect to the polarization direction of the dipoles. In the presence of long-range anisotropic dipolar and short-range contact interactions, the anisotropic density distribution of the dipolar BEC in both traps is discussed in detail. The stability condition of the dipolar BEC in both traps is illustrated in phase plot of dipolar and contact interactions. We also study and discuss the properties of a vortex dipolar BEC in these traps.Physical Review A 05/2012; 85(5). · 3.04 Impact Factor

Page 1

Band structures of a dipolar Bose-Einstein condensate in one-dimensional lattices

YuanYao Lin,1Ray-Kuang Lee,1Yee-Mou Kao,2and Tsin-Fu Jiang3,*

1Institute of Photonics Technologies, National Tsing-Hua University, Hsinchu 300, Taiwan

2Department of Physics, National Changhua University of Education, Changhua 500, Taiwan

3Institute of Physics, National Chiao Tung University, Hsinchu 300, Taiwan

?Received 14 February 2008; published 25 August 2008?

We derive the effective Gross-Pitaevskii equation for a cigar-shaped dipolar Bose-Einstein condensate in

one-dimensional lattices and investigate the band structures numerically. Due to the anisotropic and the long-

ranged dipole-dipole interaction in addition to the known contact interaction, we elucidate the possibility of

modifying the band structures by changing the alignment of the dipoles with the axial direction. With the

considerations of the transverse parts and the practical physical parameters of a cigar-shaped trap, we show the

possibility to stabilize an attractive condensate simply by adjusting the orientation angle of dipoles. Some

interesting Bloch waves at several particle current densities are identified for possible experimental

observations.

DOI: 10.1103/PhysRevA.78.023629PACS number?s?: 03.75.Lm, 05.30.Jp

I. INTRODUCTION

The Bose-Einstein condensate ?BEC? in an optical lattice

has provided a versatile and controllable platform to study

the condensed-matter-like properties by the atomic quantum

gases ?1?. As electrons in a crystal lattice, matter waves in

laser-induced optical lattices have many similar but ubiqui-

tous interesting features due to the nonlinear atom-atom in-

teractions in BECs. Bloch waves with discrete eigenenergy

are the stationary solutions for BEC in periodic potentials.

The resulting band structures are identified by the Brillouin

zones. With the nonlinear mean-field Gross-Pitaevskii equa-

tion, the swallow-tailed loop structure at the boundary of the

Brillouin zone was first predicted by a simple two-state

model ?2?. Later on, an exact solution of such loop behaviors

in the band structure was found for a particular kind of one-

dimensional lattice ?3,4?, and was further studied numeri-

cally by a detailed many-mode expansion method ?5,6?. The

atomic band structures are related to the dynamics and sta-

bility of the condensates. The new property has attracted

intensive investigations, including the nonlinear Landau-

Zener tunneling ?2,7?, the Bloch oscillation ?8,9?, and the

stability of Bloch waves ?6,10?.

In 2005, a new species of dipolar BEC was realized in

addition to the alkali-metal atoms BEC systems since the

first realization in 1995. This dipolar system uses chromium

atoms,

moment of 6 Bohr magneton which is larger than that of the

alkaline atom ?11–14?. More recently,

duced with an all-optical method ?15?. For the dipolar BEC,

there is then an extra dipole-dipole interaction between at-

oms in addition to the known contact interaction in the BECs

of alkali-metal atoms. The dipole-dipole interaction is aniso-

tropic and long-ranged. So, there are new tunable parameters

from this interaction. There are renewed interests in the di-

polar BEC due to the dipole force. Namely, an unusual prop-

erty of double-peak order parameter for the dipolar BEC un-

52Cr. Each chromium atom has a magnetic dipole

52Cr BEC was pro-

der certain environments was demonstrated ?16?; and the

effects of the dipolar interaction to the quantum phase tran-

sition temperature was also explored ?17?. The stability,

ground state, and excitations of the dipolar BEC in a trap

potential were already investigated in the literature ?18–20?.

It was found that the Luttinger-liquid phase persists for a

wide range of density in one-dimensioanl dipolar gas ?21?.

The solidlike to liquidlike phase change of the one-

dimensional dipolar system ground state with respect to the

linear density by the quantum Monte Carlo method was

shown ?22?. The signature of one-dimensional dipolar gas in

the Super-Tonks-Girardeau regime was studied ?23?. The

ground state phase diagram of the two-dimensional dipolar

gas was also investigated ?24?. Applying the Bose-Hubbard

model to the system of dipolar BEC in a two-dimensional

optical lattice, the possibility of several quantum phases for

the ground state with different aspect ratios was elucidated

?25?. An extension to dipolar spinor BEC was also explored

recently ?26?. More interestingly, a novel structure of dipolar

bosons in a planar array of one-dimensional tubes was pro-

posed ?27?.

The goal of this paper is to investigate the band structures

of the dipolar BEC in a quasi-one-dimensional optical lattice.

As shown in Ref. ?25?, in the case of extreme quantum re-

gime, there are several possible phases for the ground state

for dipolar Bosons in optical lattice. Instead of using the

Bose-Hubbard model with long-range force, mean-field

theory is applied here as the cases of nondipolar bosons in

lattice and our system parameters are within the range of

superfluid phase. The number density of atoms we consid-

ered will be quite large and without significant fluctuations.

In such a scenario, even though the effects of the transverse

confinement could wash out some of the ground states, but

effectively with the regime of a mean-field approach one

only obtains a modified one-dimensional Gross-Pitaevskii

equation with coefficients depending on the geometry of the

ground state. Instead of using the ideal pure one-dimensional

lattice model ?2–6?, in this work we use practical physical

parameters of a cigar-shaped trap and take into account the

effects of transverse parts. As mentioned above, with the

dipolar potential, the effects of the spatial distribution of the

*tfjiang@faculty.nctu.edu.tw

PHYSICAL REVIEW A 78, 023629 ?2008?

1050-2947/2008/78?2?/023629?8?

©2008 The American Physical Society 023629-1

Page 2

atoms must be treated with special attentions. Our calcula-

tion follows the method of many-mode expansion in Refs.

?5,6? and the convergent results are shown. We found that the

transverse part modifies the coupling constant, and the

dipole-dipole interaction can be reduced to an effective con-

tact term with adjustable parameter ?, which is the angle

between the aligned dipoles and the axial direction. Interest-

ingly, the band structures can be tuned by adjusting the angle

?. This is a special property of the band structures for the

dipolar BEC in optical lattices; and it is also possible to

adjust the angle ? such that a BEC with attractive interaction

can be stable without collapse.

Besides the ultracold atomic systems, artificial periodic

structures with the modulation in the refractive index in a

Kerr-type nonlinear material, known as nonlinear photonic

crystals, also share the same knowledge of such loop struc-

tures. For example, experimental observation of photonic

Bloch oscillations and Zener tunneling were reported in a

photorefractive material with configurable two-dimensional

square lattices ?28?. In terms of nonlinear wave packets, pe-

riodic potentials of optical lattices in a condensate also sup-

ported the unique solution, called gap solitons ?29?. Gap soli-

tons have attracted a great deal of attention due to their

controllable interaction and robust evolution uninhibited by

collapse. The swallow-tail structures are related to the split

even- and odd-numbered periodic soliton arrays ?6,30?. Since

the band structures determine the dispersion relation for op-

tical waves, the results in this work also provide useful in-

formation for the studies of wave packets inside nonlocal

nonlinear photonic crystals, such as their mobilities and in-

teractions ?31,32?.

The rest of the paper is organized as follows: In Sec. II,

the formulation of a dipolar BEC in quasi-one-dimensional

optical lattices was derived from the three-dimensional

Gross-Pitaevskii equation with a cigar-shaped trap. In Sec.

III, we describe the method of solution based on many-mode

expansions; and the results with several sets of physical pa-

rameters are presented in Sec. IV, especially some interesting

cases of Bloch waves with zero particle current density. Fi-

nally, the discussion and conclusions are presented.

II. FORMULATION

We consider the mean-field equation for a system of N

aligned dipolar atoms in a quasi-one-dimensional lattice de-

scribed by the time-independent Gross-Pitaevskii equation,

???r ?? =?−?2

+ Ngd?d3r?Vd?r ?,r??????r????2???r ??.

Here Vtrap=1

=?0?m

and ?mis the atomic magnetic dipole moment. Note that in a

cigar-shaped trap, the trap frequency for the axial direction is

much smaller than the transverse frequency ??and is thus

neglected. More specifically, we take the recently realized

52Cr dipolar BEC system as a model case. With millions of

2m?2+ Vtrap+ Ngs???r ???2

?1?

2m??

2?x2+y2?+V0sin2??z

d?, gs=4??2a/m, gd

2/4?, m is the atomic mass, a is the scattering length,

BEC atoms in a cigar-shaped trap modulated by hundreds of

lattice sites and the dipoles aligned in the direction p ˆ, besides

the constant gd, the dipole-dipole interaction potential in the

above equation is written as

1 − 3?p ˆ · e ˆrr??2

Vd?r ?,r??? =

?r ? − r???3

,

where e ˆrr?=

tact term in the dipole-dipole potential Vd. Because under the

mean-field theory, an atom has been modeled as a hard

sphere with radius of the scattering length and the overlap of

two atoms is not considered.

In the cigar-shaped trap, we may assume the transverse

part of the solution stays in the ground state of the harmonic

potential, that is,

r ?−r??

?r ?−r???is the unit vector. Note that there is no con-

??r ?? = ?g?x,y???z?,

2/??Lw

?2?

where ?g?x,y?=e−?x2+y2?/2Lw

monic potential in the x-y directions, and Lw=??/m??is the

length scale of the transverse wave function. We approximate

the z-direction wave function in the Bloch form:

??z? = ?B?z?eikz,

where ?B?z+d?=?B?z? is a periodic function for the lattice

constant d. The independent wave number k lies in the first

Brillouin zone, ?−?/d,?/d?. We use the scaled units ?=m

=1 unless otherwise stated. For hundreds of lattice sites, the

above assumptions are valid and used often ?33,34?. We label

the site index as j=−M,−M+1,...,−2,−1,0,1,2,...,M

−1,M. Then the Fourier expansion is used due to the peri-

odicity in the Bloch function:

2is the ground state of har-

?3?

?B?z? = ?

?=−?M

?M

c?ei??2?z/d?,

?4?

where all the mode coefficients c?are real numbers ?5?; and

with the normalization condition,

?

−?M+?1/2??d

= ?2M + 1?d??

?M+?1/2?d

???z??2dz = ?2M + 1??

one site

??B?z??2dz

2?= 1,

?

c?

?5?

one can obtain a condition for all the coefficients. Usually,

only few modes in the expansion are enough to give conver-

gent results. For the illustrations demonstrated later, we coin

the expansions with M=1,2,3,... as the 3−,5−,7−,...

mode approximation, respectively.

We multiply Eq. ?1? with ?g?x,y? and integrate over x and

y directions to obtain

?? − ??????z? =?−?2

2m

+ Ngd?dxdy?g?x,y?2?Vd?r ?,r???

????r????2d3r????z?.

d2

dz2+ V0sin2??z

d?+

Ngs

2?Lw

2???z??2

?6?

LIN et al.

PHYSICAL REVIEW A 78, 023629 ?2008?

023629-2

Page 3

We can see that the transverse part modifies the coupling

constant of the contact interaction term; but even with the

approximation in the transverse part, the system is still hard

to solve due to the nonlocal dipolar interaction. The treat-

ment in the conjugate space ?16? is applied to solve Eq. ?6?.

We first define the nonlocal potential as

Vnonlocal?r ?? =?Vd?r ?,r??????r????2d3r?.

?7?

Here the dipole-dipole interaction potential Vddepends on

the relative position r ?−r??. Thus the above integral is in con-

volutional type. It can be calculated by inverse Fourier trans-

form of the product of the Fourier transform of ???r ???2and

Vd?r ??. The Fourier transform of Vd?r ?? is

Ud?q ?? =?d3rVd?r ??eiq ?·r ?

= − 4??1 − 3 cos2???sin?qa?

?qa?3−cos?qa?

?qa?2?

? −4?

3?1 − 3 cos2??,

?8?

where ? is the angle between the vector q ? and the aligned

dipoles. The last form comes from the mean-field theory be-

cause the value of qa is small practically ?16?. The density

function in momentum space is defined as

n?q ?? =?d3r???r ???2eiq ?·r ?

??dxdy?g?x,y?2ei?q1x+q2y??dz???z??2eiq3z

? n??q1,q2?nz?q3?,

?9?

where with the transverse approximation, we have

n??q1,q2? = e−?Lw

2/4??q1

2+q2

2?.

?10?

Using the periodic property,

?

?j+1/2?d

?j+3/2?d

???z??2eiq3zdz = eiq3d?

?j−1/2?d

?j+1/2?d

???z??2eiq3zdz,

?11?

we find

nz?q3? = 2 sin?2M + 1?q3d

2 ?

?− 1??+?c?c?

1

?2M + 1?q3d

+ ?

?,?;???

2??? − ??/d + q3?.

?12?

Then, from the convolutional theorem, we have

8?3?d3qUd?q ??n?q ??e−iq ?·r ?.

Vnonlocal?r ?? =

1

?13?

In this way, we can define in Eq. ?6? the effective axial po-

tential as

Veff?z? ??dxdy?g?x,y?2Vnonlocal?r ??.

?14?

After some reduction steps, we obtain

6?2?e−??q1

Veff?z? =− 1

2+q2

2?Lw

2/2??1 − 3 cos2??nz?q3?e−iq3z

? V1?z? + V2?z?,

?15?

where we define

and

V2?z? =

1

2?2?e−??q1

2+q2

2?Lw

2/2?nz?q3?cos2?e−iq3zd3q. ?17?

The V1is easily worked out as

V1?z? = −

2

3Lw

2???z??2.

?18?

For V2?z?, we need more manipulations. Due to the cylindri-

cal symmetry, we can choose p ?=?sin ?,0,cos ?? without loss

of generality, where ? is the angle between the direction of

dipoles and the z axis, i.e.,

cos2? = ?q1sin ? + q3cos ??2/?q1

We first integrate over ?q1,q2? in polar coordinates, with

?q1,q2?=k?cos ?,sin ??, and obtain

?e?−?q1

= ??

0

2+ q2

2+ q3

2?.

2+q2

2?Lw

2/2?cos2?dq1dq2

?

e−?k2Lw

2/2??k2sin2? + 2q3

2cos2??

2

k2+ q3

kdk.

?19?

The final approximated format for V2?z? is derived in the

following:

2??e−iq3znz?q3??I?q3?sin2? + 2J?q3?cos2??dq3

V2?z? =

1

? sin2????z??2/Lw

where we define I?q3? and J?q3? as

I?q3? ??

0

2,

?20?

?

e−?k2Lw

2/2?

k2

k2+ q3

2kdk =

1

Lw

2+q3

2

2eq3

2Lw

2/2Ei?−q3

2Lw

2

2?,

?21?

J?q3? ??

0

?

e−?k2Lw

2/2?

q3

2

k2+ q3

2kdk =

1

Lw

2− I?q3?,

?22?

with the exponential integral,

Ei?− x? = −?

x

?e−t

tdt

with x ? 0.

?23?

We can see that for small value of q3, I?q3??1/Lw

J?q3??0; while for a large value of q3, I?q3?→2/?q3

2and

2Lw

4?

BAND STRUCTURES OF A DIPOLAR BOSE-EINSTEIN…

PHYSICAL REVIEW A 78, 023629 ?2008?

023629-3

Page 4

?0 and J?q3??1/Lw

dominant value at smaller values of q3. Finally we obtain the

effective one-dimensional equation for the dipolar BEC at-

oms in a quasi-one-dimensional optical lattice:

2. The expression of nz?q3? shows the

?? − ??????z? =?−?2

2m

2???z??2?

d2

dz2+ V0sin2??z

d??24?

3gdP2?cos ??????z?,

+N

Lw

gs

2?−2

?24?

where P2is the Legendre polynomial of the first kind. We

arrived at the conclusion that with the approximated trans-

verse part, the dipolar effects can be manipulated into an

effective contact term with adjust parameter ?.

III. METHOD OF SOLUTION

In the previous section, we derived the effective Gross-

Pitaevskii equation in Eq. ?24? for condensate dipolar atoms

in a quasi-one-dimensional lattice. In the following, we em-

ploy the many-mode expansion method for the Bloch func-

tions to solve the corresponding energy band structures ?5?.

The energy functional of the effective one-dimensional sys-

tem is

E??? =?

2?

all sites?

?2

2m?d?

dz?

3gdP2?cos ??????z??4?dz,

2

+ V0sin2??z

d????z??2

+1

2

1

Lw

gs

2?−2

?25?

here we have defined

−1−0.50

k

0.51

0

2

4

6

8

10

12

?

FIG. 1. ?Color online? The band diagram with V0=1, ?

=0.7377 ?rad?, and ngeff=1. The blue ?lowest solid line?, red

?dashed lines?, and black ?upper solid lines? lines are calculated by

1-mode, 3-mode, and 5-mode, respectively. It shows that all the

calculated results are close to each other when only the lowest three

bands are considered.

−0.50 0.5

0.5

1

1.5

|ΨB(z)|

A

−0.50 0.5

0.5

1

1.5

|ΨB(z)|

B

−0.50 0.5

0.5

1

1.5

z/π

|ΨB(z)|

B′

−0.50 0.5

0

0.5

1

1.5

2

C

−0.50 0.5

0

0.5

1

1.5

2

C′

−0.50 0.5

0

0.5

1

1.5

2

z/π

D

0 0.20.40.60.81

0.8

1

1.2

1.4

1.6

1.8

2

2.2

k

?

0.951

1.94

2.02

A

B

B’

C’

C

D

FIG. 2. ?Color online? Part of the lowest band diagram with V0=1, ?=0.7377 ?rad?, and ngeff=1. The red dashed and black solid lines

are derived from 3-mode and 5-mode models. The absolute value of the Bloch wave function of some interested modes is also plotted.

LIN et al.

PHYSICAL REVIEW A 78, 023629 ?2008?

023629-4

Page 5

?

all sites

???z??2dz = N.

?26?

In the following, we will use the recoiled energy Er

=?2?2/?2md2? as the energy unit, and L?=d/?? as the length

unit. The unit cell is then ?−?/2,?/2? and the quasimomen-

tum k lies in ?−1,1?. The energy density functional per site ?

becomes

??

−??/2?

+geff???

2

???? =1

?/2??d?

???z??4?dz,

dz?

2

+ V0sin2z???z??2

?27?

with the definition of number density per site

n =1

??

−?/2

??/2?

???z??2dz,

?28?

where the effective coupling constant is

geff??? =

1

Lw

2LEr?

gs

2?−2

3gdP2?cos ???.

?29?

In the many-mode method, the order parameter is written in

the Bloch form:

??z? = eikz?B?z?,

?B?z + ?? = ?B?z?,

?B?z? =?n?

?

c?ei2?z.

?30?

The energy functional per atom can be recast as

? =?

?

c?

2?k + 2??2+1

2V0−1

?c?c?c??c????+??,?+???,

4V0?

?

?c??c?+1+ c?−1??

+n

2geff?1 +?

?,???

??,??

?31?

where in the double summations, we have ??? and ??

???, respectively. For a fixed number of modes in the Fou-

rier expansion, the energy functional ??k? is minimized with

respect to the coefficients c?’s for a given value of quasimo-

mentum k.

For the numerical calculations, we assume that the optical

lattice is constructed with lasers of wavelength 800 nm, and

the lattice constant becomes d=400 nm. We are considering

the chromium atoms with atomic magnetic dipole moment of

6 Bohr magneton, and trap frequency ??=2??712.5 Hz

from experimental parameters ?11–14?. Thus the transverse

length scale is Lw=0.523 ?m. With these parameters, we

have

geff??? = 9.868? 10−5?a/aB? − 1.4913? 10−3P2?cos ??,

?32?

where aBis the Bohr radius. Note that even for the case of

attractive contact interaction ?negative scattering length?, be-

cause −0.5?P2?cos ???1, it is possible to tune the angle ?

between the aligned dipoles with cylindrical axis, such that

the geffis positive for the appropriate value of scattering

length, the system will still be stable. Physically it means

that the dipole-dipole interaction is repulsive and overcomes

the attractive contact interaction with the chosen value of ?,

then the 3-body collisional loss induced collapse can be pre-

vented. Also, we can tune the scattering length a and align

−1 −0.50 0.51

2

2.5

3

3.5

4

4.5

5

5.5

6

k

?

−0.020 0.02

5

5.05

−0.10 0.1

5

5.1

5.2

5.3

5.4

0.050.1 0.15

5.36

5.4

5.44

5.48

B’

B

A

D

C

C’

(b)

(a)

FIG. 3. ?Color online? ?a? Part of the first excited band diagram

with V0=1, ?=0.7377 ?rad?, and ngeff=1. The red dashed and black

solid lines are derived from 3-mode and 5-mode models. ?b? The

absolute value of the Bloch wave function of some interested modes

is also plotted.

−1 −0.50

k

0.51

2

4

6

8

10

?

−0.100.1

5.28

5.3

5.32

−0.500.5

0

0.5

1

1.5

z/π

|ΨB(z)|

−0.500.5

0.96

0.98

1

1.02

1.04

z/π

|ΨB(z)|

A

B

FIG. 4. ?Color online? Part of the second excited band diagram

with V0=1, ?=0.7377 ?rad?, and ngeff=1. The red dashed and black

solid lines are derived from 3-mode and 5-mode models. The abso-

lute value of the Bloch wave function of some interested modes is

also plotted.

BAND STRUCTURES OF A DIPOLAR BOSE-EINSTEIN…

PHYSICAL REVIEW A 78, 023629 ?2008?

023629-5

Page 6

angle ? such that geff→0. The effects of contact interaction

and dipole-dipole interaction will then cancel out each other

such that the nonlinear effect has totally disappeared.

To show the properties of Bloch bands in dipolar lattices,

we assume the scattering length a to be tuned by the Fesh-

bach resonance technique to a=15.1aBsuch that the contact

term and effective dipolar interaction are comparable to each

other in our model. Thus our effective nonlinear coupling

coefficient in the above equation becomes

geff??? = 1.4913? 10−3?1 − P2?cos ???.

?33?

IV. RESULTS

We first calculate the three lowest band structures for

ngeff=1 and potential height V0=1. The condition can be

reached, for instance, with n=1000 and adjusting the align-

ment angle ? between the dipoles and the cylindrical axis to

be 42.3°. Figure 1 depicts the energy bands calculated with

1-mode, 3-mode, and 5-mode expansions, respectively. We

can see that the difference in results from the 1-mode,

3-mode, and 5-mode is very small. The swallow tail structure

appears near the zone boundary of the ground band and near

the zone center of the first excited band. Figure 2 shows

more detailed results of the ground band. We can see that the

3-mode results show no swallow tail at the zone boundary.

The calculation with at least 5-mode expansion is necessary.

We plot also the amplitude of the Bloch wave function

??B?z?? for several interested values of wave number k. The

use of absolute value is because the Bloch wave is in general

complex. So, for a pure real or pure imaginary Bloch wave

function, the particle current density, j=???k?/??k, vanishes.

The Bloch wave amplitude at k=0 is designated as A. Near

the zone boundary and with k=0.99, they are designated as

B, B?. D corresponds to k=1, that is, at the zone boundary.

Near the tip, they are designated as C and C?. Especially, at

A and D, there is no particle current density.

Figure 3?a? shows the detailed structure of the first excited

band near the zone center while in Fig. 3?b? we show the

Bloch amplitudes. The swallow tail structure for the first

excited band with ngeff=V0=1 appears only near the zone

center instead of near the zone boundary as the ground band.

The amplitude ??B?z?? at the zone boundary is designated as

A. At the value of k=0.01, the Bloch amplitudes are desig-

nated as B and B?, respectively. At the top of the band with

k=0, the Bloch amplitude is designated as D; and near the

tip, they are designated as C and C?. Again, the particle

current density j?k? for A and D vanishes.

In Fig. 4 we plot the detailed band structure of the second

excited band and the corresponding Bloch amplitude for

wave number at the zone center and the zone boundary. At

−0.50 0.5

0

0.5

1

1.5

2

|ΨB(z)|

A

z/π

−0.50 0.5

0.95

1

1.05

z/π

|ΨB(z)|

−10

k

1

2

3

4

5

6

7

8

9

10

?

−0.10 0.1

5.28

5.3

5.32

B

B

A

FIG. 5. ?Color online? The band diagram with V0=1 and ngeff

=2 derived from 3-mode models. Some amplitudes of the Bloch

wave function are also plotted.

−1 −0.50

k

0.51

1

2

3

4

5

6

7

?

−0.50 0.5

0.95

1

1.05

1.1

|ΨB(z)|

−0.50 0.5

0.9

1

1.1

1.2

|ΨB(z)|

−0.50 0.5

0.9

1

1.1

1.2

|ΨB(z)|

−0.50 0.5

0.8

1

1.2

1.4

z/π

|ΨB(z)|

−0.500.5

0.6

0.8

1

1.2

−0.50 0.5

0

0.5

1

−0.50 0.5

0

0.5

1

−0.500.5

0.9

1

1.1

z/π

−0.50 0.5

0.9

1

1.1

−0.50 0.5

0.8

0.9

1

1.1

−0.50 0.5

0.8

1

1.2

−0.50 0.5

0

0.5

1

1.5

z/π

A

A

B

B

C

B’

B’

C

C’

C’

D

D

E

F

F

E

F’

G

G

G’

G’

F’

H

H

FIG. 6. ?Color online? Part of the band diagram with V0=1 and ngeff=2 derived from 3-mode models. The absolute value of the Bloch

wave function of some interested modes are also plotted.

LIN et al.

PHYSICAL REVIEW A 78, 023629 ?2008?

023629-6

Page 7

this range of nonlinearity, the second excited band structure

is not affected much. The property is similar to a Bloch elec-

tron in periodic potential. The Bloch wave at A, k=0, is a

pure real function while at B, k=1, is a pure imaginary one.

In Fig. 5, the nonlinearity is increased to ngeff=2 and the

potential height is kept as V0=1. We can see that the swallow

tails of the ground and the first excited bands increased in

size; but the second excited band still shows no special struc-

ture. We plot in Fig. 5 also the Bloch amplitude at the zone

center and the zone boundary, designated as A and B, respec-

tively. The two Bloch waves are purely real indicating that

the particle current density vanishes. Figure 6 depicts the

ground and the first excited band structure together with

some interested Bloch amplitudes. For the ground band, A

designates the zone center, B, B? are for the values of k

slightly smaller than k=1 at the zone boundary. C and C?

designate the places near the tip. D designates the top of the

ground band at the zone boundary. For the first excited band,

the swallow structure appears near the zone center. F and F?

have values of k=0.01. G and G? are near the tip, and H

designates the top of the first band at k=0. We further in-

crease the nonlinearity to ngeff=4 and keep the potential

height in V0=1. We can see that the swallow tail structures

increase much in size as shown in Fig. 7.

V. DISCUSSION AND CONCLUSIONS

The realization of the dipolar BEC enables the study of an

ultracold system with long-ranged interatomic interactions.

We present in this work the study of the band structures of a

dipolar BEC under quasi-one-dimensional lattices. In addi-

tion to the long-ranged interaction, we take into consider-

ation the effects of transverse distributions mostly for BECs

in quasi-one-dimensional lattice. We found that the dipole-

dipole interaction and the transverse effects modify the ef-

fective coupling constant. The band diagrams of several val-

ues of nonlinear coupling constant and some interested mode

are shown. On the other hand, the many-mode solution for a

pure one-dimensional model has been studied in detail and

not pursued repeatedly in this paper ?6?.

We found that the alignment angle between the dipoles

and the axial direction provides a tunable parameter for

physical properties through Eq. ?33?. The nonlinear effect,

and hence the swallow tail in band structures, can be

changed easily by adding a small magnetic field to adjust the

alignment angle. From Eq. ?32? of the effective coupling

constant, even for the system with attractive mutual atomic

interaction, it is still possible to adjust the alignment angle ?

to obtain a positive value of geffsuch that the effective mu-

tual interaction becomes repulsive to prevent collapse of the

BEC system. It is also possible to adjust the value of ? such

that the nonlinear effect vanished totally and reduced to an

ordinary Bloch lattice.

In summary, due to the dipole-dipole interaction in the

dipolar BEC in optical lattice, the system becomes versatile

tunable. Comparing to the crystalline lattice in condensed

matter physics, the dipolar BEC in an optical lattice will be

an interesting system in band structure study. With such a

tunable swallow-tailed loop in the band structures, interest-

ing dynamics of the condensates between intraband and in-

terbands are believed to be observed within current experi-

mental technologies.

ACKNOWLEDGMENTS

The work of Y.Y.L. and R.K.L. was supported by the Na-

tional Science Council of Taiwan under Contracts No. 95-

2112-M-007-058-MY3 and No. NSC-95-2120-M-001-006.

The work of T.F.J. was supported by the National Science

Council of Taiwan under Contract No. NSC95-2112-M009-

028-MY2. T.F.J. and Y.M.K. acknowledge the partial support

from the Elite Foundation.

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