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

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.


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 [4][5][6].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.

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 [7][8][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 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  1 ,  2 are the wave functions of components one and two, respectively.The interatomic and the intercomponent swave scattering interactions are described by   ( = 1, 2) and  12 , respectively, with the following expressions [11]: where where  is the angle between the polarization axis ⃗  and the relative of two atoms (i.e., cos  = ⃗  ⋅ ⃗ /,  = | ⃗ | = √ 2 +  2 +  2 ).The wave function   (, ) is normalized ac- , where   is the number of the atoms in the dipolar BEC.
We propose a time-splitting sine pseudo-spectral method for computing the dynamics of the BEC [12].
From  =   to =  +1 , the GP equation ( 9) is solved by three steps.First, we solve from   to  +1/2 , followed by solving the nonlinear ODE for one time step.Again, we solve (12) from  +1/2 to  +1 .Suppose the exact solutions are 2 (  ,   ,   ,   ) 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 pseudospectral method and integrated in time [14].Next, we will show that (13) can be solved exactly.

Example with the Different Initial Condition.
The confining cigar trap potential is () = (1/2)( 2 + 2 +0.04 2 ).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.

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- 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.

Figure 4 :
Figure 4: The energy evolution according to .
is the scattering length of component  and  12 is that between components 1 and 2.Here ℎ is the Planck constant,   is the mass of the atom of component ,  = 1, 2, and   is the external trapping potential confining the gas.Generally, that is,  ( ⃗ ) = (  /2)( 2   2 +  2   2 +  2   2 )with   ( = , , ) representing the trap frequency in , , and  directions, respectively.The local mean-field   |  | 2 represents the s-wave interaction. dip is the longrange isotropic dipolar interaction potential between two dipoles and it is defined by