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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 517395, 7 pages
http://dx.doi.org/10.1155/2013/517395
Research Article

A Time-Splitting and Sine Spectral Method for Dynamics of Dipolar Bose-Einstein Condensate

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

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, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A two-component Bose-Einstein condensate (BEC) described by two coupled a three-dimension Gross-Pitaevskii (GP) equations is 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 for computing the dynamics of dipolar BEC. The singularity in the dipole-dipole interaction brings significant difficulties both in mathematical 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 dilute gases in 1995 marked the beginning of a new era in atomic, molecular, and optical physics. That has attracted much attention both theoretically and experimentally. Most of their properties of these trapped quantum gases are governed by the interactions between particles in the condensate [1]. In the last several years, there has been an investigation for realizing a new kind of quantum gases with the dipolar interaction, acting between particles having a permanent magnetic or electric dipole moment. The experimental realization of a BEC of 52Cr atoms [2, 3] at the University of Stuttgart in 2005 gave new impetus to the theoretical and the numerical investigations on these novel dipolar quantum gases at low temperature. Recently more detailed and controlled experimental results have been obtained, illustrating the effects of phase separation in a multicomponent BEC [46]. In these papers, the studies of the binary condensates were limited to the case of s-wave interaction, while a great deal of attention has been drawn recently to the dipolar BEC.

In this work, a numerical method for computing the dynamics of the two-component dipolar BEC is considered, where one equation has dipole-dipole interaction and the other has only the usual s-wave contact interaction. However, since the dipole-dipole interaction is of long range, anisotropic, and partially attractive and the computational cost in three dimensions high, the nontrivial task of achieving and controlling the dipolar BEC is thus particularly challenging.

This paper is organized as follows. In Section 2, a numerical method for computing ground states is presented. In Section 3, numerical results are reported to verify the efficiency of this numerical method. Finally, some concluding remarks are drawn in Section 4.

2. Numerical Method for Computing the Dynamics

2.1. The Nonlocal Gross-Pitaevskii Equation

The two-component dipolar BEC, confined in a cigar trap, is described by two coupled Gross-Pitaevskii equations. As far as the dipolar interaction is concerned, a convolution term is introduced [79] to modify the classical Gross-Pitaevskii equation, which results in the following differential-integral equations (1). Since the transition metal has a magnetic dipole interaction while the alkali metal does not have, we take into account this factor in this system. We take Cr as component 1 and Rb as component 2 [10]. Then the GP equations for this system can be written as where , are the wave functions of components one and two, respectively. The interatomic and the intercomponent s-wave scattering interactions are described by and , respectively, with the following expressions [11]: where is the scattering length of component and is that between components 1 and 2. Here is the Planck constant, is the mass of the atom of component , , and is the external trapping potential confining the gas. Generally, that is, with representing the trap frequency in , , and directions, respectively. The local mean-field represents the s-wave interaction. is the long-range isotropic dipolar interaction potential between two dipoles and it is defined by where is the angle between the polarization axis and the relative of two atoms (i.e., , ). The wave function is normalized according to , , where is the number of the atoms in the dipolar BEC.

2.2. Time-Splitting and Sine Spectral Numerical Method for Dynamics

The system (1) can be made dimensionless and simplified by adopting a unit system where the units for length, time, and energy are given by , , and , respectively, with , [12].

By introducing the dimensionless variables ,  , , we obtain the dimensionless GP equations in 3D from (1) as follows: where , , , , , , and . In addition the wave functions in (4) satisfy , . By using the following formula [13] where is the Dirac delta function and , we can get where And it is easy to see that Plugging (6) into (4) and noticing (7) and (8), we can reformulate GPE (4) into the Schrödinger-Poisson type system In practice, the whole space problem is usually truncated into a bounded computational domain with the homogeneous Dirichlet boundary condition. Let Choose the spatial mesh size as ,   and and define , , and , . Then denote the space with where , , and .

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

From to , the GP equation (9) is solved by three steps. First, we solve from to , followed by solving the nonlinear ODE for one time step. Again, we solve (12) from to .

Suppose the exact solutions are Substitute (14), (15) into (12); we can find that which can be solved exactly, and we obtain where Equations (12) will be discretized in space by sine pseudo-spectral method and integrated in time [14]. Next, we will show that (13) can be solved exactly.

In fact, for , multiplying (13) by the conjugation of , that is, , we get and we also have Therefore, subtracting (13) from (14), one obtains which implies that Substituting (22) into (13), we get a linear ODE which can be solved exactly. Integrating (19) from to , one gets where .

The discrete sine transform coefficients of the vector for are Let and be the approximations of and , respectively, which are the solution of (9). For , a second-order time-splitting and sine spectral method for solving (9) via the standard Strang splitting is [1416] where

3. Numerical Results

3.1. Example with the Same Initial Condition

The confining potential is a cigar potential with . Consider the dynamics of the BEC in the cigar trap. The initial condition is given as follows: Here , , and . We solve this system on with and .

Figure 1 shows energy evolutions of dipole BEC. And the energy is conserved. Figures 2 and 3 show the wave function evolutions according to time.

517395.fig.001
Figure 1: The energy evolution according to .
517395.fig.002
Figure 2: The wave function evolution according to time. Surface plots for at different times.
517395.fig.003
Figure 3: The wave function evolution according to time. Surface plots for at different times.
3.2. Example with the Different Initial Condition

The confining cigar trap potential is . Solve the dynamics problem for a dipolar BEC with the cigar trap. The initial condition is Figure 4 shows energy evolutions of dipole BEC. And the energy is conserved. Figures 5 and 6 show the wave function evolutions according to time.

517395.fig.004
Figure 4: The energy evolution according to .
517395.fig.005
Figure 5: The wave function evolution according to time. Surface plots for at different times.
517395.fig.006
Figure 6: The wave function evolution according to time. Surface plots for at different times.

4. Conclusion

An efficient numerical method is presented for computing the dynamics of the dipolar Bose-Einstein condensates based on two coupled three-dimensional Gross-Pitaevskii equations where one equation has a dipole-dipole interaction potential and the other one has only the usual s-wave contact interaction. Using equality (5), we can reformulate the GPE for dipolar BEC into a Grosss-Pitaevskii-Poisson type system. Numerical examples are given to show the efficiency of our method. The figures show the evolution of the wave function with time. And in all cases, total energy is conserved. The results agree with the previous work [16]. Numerical results are given to demonstrate the efficiency of our numerical method.

Authors’ Contribution

All authors conceived the study, participated in its design and coordination, drafted the paper, participated in the sequence alignment, and read and approved the final paper.

Acknowledgments

This work was supported by National Science Foundation of China (no. 11171032) and Beijing Municipal Education Commission (no. KM201110772017).

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