August 10, 2008 / Vol. 6, No. 8 / CHINESE OPTICS LETTERS 611
Effects of external magnetic trap on two dark solitons of
a two-component Bose-Einstein condensate
Hong Li (
ooo
÷÷÷
)
1,2
and D. N. Wang
3
1
Department of Physics, Hubei Normal University, Huangshi 435002
2
School of Mechanical and Electronic Engineering, Huangshi Institute of Technology, Huangshi 435003
3
Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong
Received December 28, 2007
Two dark solitons are considered in a two-component Bose-Einstein condensate with an external magnetic
trap, and effects of the trap potential on their dynamics are investigated by the numerical simulation.
The results show that the dark solitons attract, collide and repel periodically in two components as time
changes, the time period depends strictly on the initial condition and the potential, and there are obvious
self-trapping effects on the two dark solitons.
OCIS codes: 300.6250, 140.7010.
doi: 10.3788/COL20080608.0611.
It is well known that the Gross-Pitaevskii equation
(mean-field model) supports solitonic solutions. Exact
analytical solutions to one and two-dimensional (2D)
mean-field models describing dynamics of Bose-Einstein
condensates (BECs) ar e investigated, and the interaction
in differ e nt components is studied for a misc ible case
[1−5]
.
Bright solitons have been detected only very rec e ntly us-
ing an optical red-detuned laser beam along the axial di-
rection of the sample to impose a transverse confinement,
while dark solitons of Bose condensed ato ms were exper-
imentally observed few years ago. A dark soliton in a
BEC is a macroscopic e xcitation of the condensate with
a correspo nding po sitive scattering length, and is charac-
terized by a loc al density minimum and a phase gradient
of the wave function at the position of the minimum
[6]
.
The stability of dark solitons, which a re defined as sta-
tionary o bjects, in a nonlinear system is a critical prop-
erty. Behavior of a BEC strongly depends on parameters
of trap potentials which can be controlled by either the
intensity or geometry of laser beams. For example, the
matter wave dynamics can b e effectively manipulated by
external fields. The harmonic trap induces a dynamical
instability of the solito n, culminating in sound emission.
The interactio n of dark solito ns may be strongly repul-
sive, and the cross-phase modulation has an important
influence on bo th the formation and interaction of the
dark solitons
[7−9]
.
A multi-co mponent BEC presents novel and fundamen-
tally different scenarios. In particular, it is observed that
the BEC can reach a n equilibrium state characterized by
the separation of the species in different domains, and
the creation of dark solitons in a two-component BEC is
investigated. The magnetic trap provides the principle
increase of phase space density to the BEC transition,
and the atoms are cooled in the magnetic trap using en-
ergetically selective spin transitions
[10]
. A relevant inter-
esting issue is to learn how to control the motion of the
condensates and different types of dark solitons in the
multi-component BEC, then the question arises how one
could affect or even g uide the motion of the dark soli-
tons. In this letter, the effects of external trap o n two
dark solito ns of a two-component BEC are investigated
by the numerical simulation, and some novel results are
obtained.
The two vector components are evolving under the
Gross-P itaevskii equation, which are the macroscopic
wave functions of Bose-condensed atoms in two different
internal sta tes . By rescaling process, the general equa-
tions may easily be put into the dimensionless form,
and the procedure results in the following coupled
Schr¨odinger equations
[11,12]
j
∂u
1
∂t
+
1
2
∂
2
u
1
∂z
2
− [|u
1
|
2
+ g
1
|u
2
|
2
+ V (z) + µ
1
]u
1
= 0,
(1a)
j
∂u
2
∂t
+
1
2
∂
2
u
2
∂z
2
− [|u
2
|
2
+ g
2
|u
1
|
2
+ V (z) + µ
2
]u
2
= 0,
(1b)
where u
i
(i = 1, 2) are the condensate wave-functions,
µ
i
(i = 1, 2) are the chemical potentials which are
connected to the number of atoms of the condensate,
g
i
(i = 1, 2) are the interactio n s trengths describing
the inter-atomic interaction between two components o f
the B EC, V (z) is the normalized confining potential of
the components in the longitudinal direction (z direc-
tion). In the present work, we a ssume a cylindrical
highly anisotropic trappe d potential, and under this cir-
cumstance, one can approximate the field as Eqs. (1a)
and (1b)
[13]
. Namely, the transverse potential is much
stronger than the longitudinal one, it is natural to reduce
the full three-dimensional (3D) Schr¨odinger equation to
an effective one-dimensional (1D) Schr¨odinger equation.
Different approaches were proposed to achieve this pur -
pose under different co nditions
[14]
.
Additionally, we assume tha t the intera ction strengths
satisfy g
1
= g
2
= g, and the assumption matches well
with the experimental conditions. The µ
i
(i = 1, 2)
term can be eliminated by using the transformation
u
i
→ u
i
exp(−jµ
i
t) (i = 1, 2), and the coupled 1D equa-
1671-7694/2008/080611-04
c
2008 Chinese Optics Letters
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