The numerical method for computing the ground state of the two-component dipolar Bose-Einstein condensate
© Li et al.; licensee Springer 2013
Received: 15 March 2013
Accepted: 25 June 2013
Published: 8 July 2013
A two-component Bose-Einstein condensate described by two coupled Gross-Pitaevskii (GP) equations in three dimensions is considered, where one equation has dipole-dipole interactions while the other one has only the usual s-wave contact interaction, in a harmonic trap. The singularity in the dipole-dipole interactions brings significant difficulties both in mathematical analysis and in numerical simulations. The backward Euler method in time and the sine spectral method in space are proposed to compute the ground states. Numerical results are given to show the efficiency of this method.
Since 1995, the Bose-Einstein condensation (BEC) of ultra-cold atomic and molecular gases has attracted much attention both theoretically and experimentally. Most of the properties of these trapped quantum gases are governed by the interactions between particles in the condensate . Over the past decade, 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 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 multi-component BEC [4–6]. In these papers, the studies of the binary condensates were limited to the case of s-wave interactions, while recently the dipolar BEC has drawn a great deal of attention.
In this work, a numerical method for computing the ground state of the two-component dipolar BEC is considered, where one equation has dipole-dipole interactions and the other has only the usual s-wave contact interaction. However, since the dipole-dipole interactions are long range, anisotropic and partially attractive, and the computational cost in three dimensions is high, the nontrivial task of achieving and controlling the dipolar BEC is thus particularly challenging.
where θ 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.
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 ground states
where , , .
Linear system (7) can be iteratively solved in a phase space very efficiently via discrete sine transform under the conditions and .
3 Numerical results
Here, , , . We solve this system on with and .
This work was supported by Natural Science Foundation of China (No. 11171032) and Beijing Municipal Education Commission (Nos. KM201110772017, 71D09111003).
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