外磁场陷阱对二元玻色-爱因斯坦凝聚态暗孤子动态的影响

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本文探讨了在外部磁势阱中，两成分玻色-爱因斯坦凝聚态(Bose-Einstein Condensate, BEC)中暗孤子(dark solitons)的动态行为。研究者通过对二元BEC进行数值模拟，发现当暗孤子在两个成分中交互时，它们展现出周期性的吸引、碰撞和排斥现象。这个周期性行为的时间尺度严格依赖于初始条件和磁势阱的具体参数。研究结果揭示了一个重要的自捕获效应，即暗孤子之间的相互作用不仅影响它们的运动轨迹，还导致它们在特定条件下能够自我稳定在一个特定的模式中。暗孤子是BEC理论中的非线性波结构，它们的存在源于BEC中的非线性相互作用和量子效应。在外部磁势阱的作用下，暗孤子的稳定性以及它们之间的相互作用变得复杂，因为磁场可以改变粒子的能量分布，从而影响到它们之间的相互作用势。这个系统的动态行为反映了量子场论中非线性动力学的一个关键特性，对于理解多体系统中非平衡态的形成与演化具有重要意义。文章指出，虽然Gross-Pitaevskii方程（通常用于描述BEC的平均场模型）支持暗孤子的存在，并且理论上可以找到精确的解析解，但实际应用中，如在实验条件下，需要通过数值模拟来精确描绘暗孤子的动态行为，这有助于验证理论预测，并可能发现新的物理现象或效应。总结来说，本文的研究为理解多体量子系统中的复杂行为提供了新的视角，尤其是在控制和操纵暗孤子动态方面具有潜在的应用价值，比如在量子信息处理和量子计算中可能利用这种自组织行为。同时，这项工作也为未来研究者在设计新型量子器件和探索更深层次的量子力学现象提供了有价值的基础。

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

Department of Physics, Hubei Normal University, Huangshi 435002

School of Mechanical and Electronic Engineering, Huangshi Institute of Technology, Huangshi 435003

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]

∂u

∂t

∂

∂z

− [|u

+ g

+ V (z) + µ

= 0,

(1a)

∂u

∂t

∂

∂z

− [|u

+ g

+ V (z) + µ

= 0,

(1b)

where u

(i = 1, 2) are the condensate wave-functions,

(i = 1, 2) are the chemical potentials which are

connected to the number of atoms of the condensate,

(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

= g

= g, and the assumption matches well

with the experimental conditions. The µ

(i = 1, 2)

term can be eliminated by using the transformation

→ u

exp(−jµ

t) (i = 1, 2), and the coupled 1D equa-

1671-7694/2008/080611-04

 2008 Chinese Optics Letters

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